tSSSSSSSSSSt "PHYSICS =n UNIVERSITY OF CALIFORNIA Received rU^HJ^t^lH^^^ ^ J ^ f 9 Accessions No./..Q./..Q... Book No. ../.*>. DEPARTMENT OF COMMERCE OF THE BUREAU OF STANDARDS S. W. STRATTON, DIRECTOR No. 74 RADIO INSTRUMENTS AND MEASUREMENTS ISSUED MARCH 23, 1918 PRICE, 60 CENTS Sold only by the Superintendent of Documents, Government Printing Office mgton, D. C. WASHINGTON GOVERNMENT PRINTING OFFICE 1918 \ TK57V/ CONTENTS PART I. THEORETICAL BASIS OF RADIO MEASUREMENTS 5 Introduction 5 The fundamentals of electromagnetism 6 1. Electric current i 6 2. Energy 9 3. Resistance n 4. Capacity 13 5. Inductance 14 The principles of alternating currents 19 6. Induced electromotive force 19 7. Sine wave 21 8. Circuit having resistance and inductance 23 9. Circuit having resistance, inductance and capacity 25 10. " Vector" diagrams 28 11. Resonance 31 12. Parallel resonance 39 Radio circuits 41 13. Simple circuits 41 14. Coupled circuits 45 15. Kinds of coupling 48 16. Direct coupling : 52 17. Inductive coupling 56 18. Capacitive coupling 60 19. Capacity of inductance coils. 62 20. The simple antenna 69 21. Antenna with uniformly distributed capacity and inductance. ... 71 22. Loaded antenna 75 23. Antenna constants 81 Damping 86 24. Free oscillations 86 25. Logarithmic decrement 90 26. Principles of decrement measurement 92 PART II. INSTRUMENTS AND METHODS OP RADIO MEASUREMENT 96 27. General principles 96 Wave meters 97 28. The fundamental radio instrument 97 29. Calibration of a standard wave meter 99 30. Standardization of a commercial wave meter 104 Condensers .' 108 31. General 108 32. Air condensers no 33. Power condensers 120 34. Power factor 122 35. Measurement of capacity 129 Coils 131 36. Characteristics of radio coils 131 37. Capacity of coils 132 38. Measurement of inductance and capacity of coils 136 Current measurement 139 39. Principles 139 3 I / . . 4 Contents PART II. INSTRUMENTS AND METHODS OF RADIO MEASUREMENT Continued. ' Current measurement Continued. p age 40. Ammeters for small and moderate currents 141 41. Thermal ammeters for large currents 144 42 . Current transformers 1 50 43. Measurement of very small currents 15 J' 44. Standardization of ammeters 170 Resistance measurement 175 45. High-frequency resistance standards 175 46. Methods of measurement 177 47. Calorimeter method 177 48. Substitution method 178 49. Resistance variation method 180 50. Reactance variation method 185 51. Resistance of a wave meter 187 52. Resistance of a condenser 190 53. Resistance of a coil 193 54. Decrement of a wave 195 55. The decremeter 196 Sources of high-frequency current 200 56. Electron tubes 200 57. Electron tube as detector and amplifier 204 58. Electron tube as generator 2 10 59. Poulsen arc 221 60. High-frequency alternators and frequency transformers 223 61. Buzzers 227 62. The spark 228 PART III. FORMULAS AND DATA 235 Calculation of capacity 235 63. Capacity of condensers 235 64. Capacity of wires and antennas 237 65. Tables for capacity calculations 241 Calculation of inductance 242 66. General 242 67. Self-inductance of wires and antennas 243 68. Self-inductance of coils 250 69. Mutual inductance 269 70. Tables for inductance calculations 282 Design of inductance coils 286 71. Design of single-layer coils 286 72. Design of multiple-layer coils 292 73. Design of flat spirals 296 High-frequency resistance 299 74. Resistance of simple conductors 299 75. Resistance of coils 304 76. Stranded wire 306 77. Tables for resistance calculations 309 Miscellaneous formulas and data 312 78. Wave length and frequency of resonance 312 79. Miscellaneous radio formulas 313 80. Properties of metals 317 APPENDIXES 319 Appendix i. Radio work of the Bureau of Standards 319 Appendix 2. Bibliography 324 Appendix 3. Symbols used in this circular 330 RADIO INSTRUMENTS AND MEASUREMENTS PART L THEORETICAL BASIS OF RADIO MEAS- UREMENTS INTRODUCTION In the rapid growth of radio communication, the appliances and methods used have undergone frequent and radical changes. In this growth, progress has been made largely by new inventions and by the use of greater power, and comparatively little attention paid to refinements of measurement. In consequence the methods and instruments of measurement peculiar to radio science have developed slowly and have not yet been carried to a point where they are as accurate or as well standardized as other electrical measurements. This circular presents information regarding the more important instruments and measurements actually used in radio work. It is hoped that the treatment will be of interest and value to Govern- ment officers, radio engineers, and others, notwithstanding the subject is not completely covered. Many of the matters dealt with are or have been under investigation in the laboratories of this Bureau and are not treated in previously existing publications. No attempt is made in this circular to deal with the operation of apparatus in sending and receiving. It is hoped to deal with such apparatus in a future circular. The present circular will be revised from time to time, in order to supplement the information given and to keep pace with progress. The Bureau will greatly appreciate suggestions from those who use the publication for improvements or changes which would make it more useful in military or other service. The methods, formulas, and data used in radio work can not be properly understood or effectively used without a knowledge of the principles on which they are based. The first part of this circular, therefore, attempts to give a summary of these principles 5 6 Circular of the Bureau of Standards in a form that' is as simple as is consistent with accuracy. A large proportion of this publication is devoted to the treatment of fundamental principles for the reasons, first, that however much the methods and technique of radio measurement may change the same principles continue to apply, and second, that this will make the present circular serve better as an introduction to other circulars on radio subjects which may be issued. A familiarity with elementary electrical theory and practice is assumed. Introductory treatment of electrostatics and mag- netic poles, electric and magnetic fields, the laws of direct currents, and descriptions of the more common electric instruments and experiments may be found in many books. A list of publications suitable as an introduction to the theory given in this circular may be found in the bibliography (p. 324). The common explana- tion of electric current as similar to the flow of water in a pipe, while adequate for most of the phenomena of direct current is not suitable for alternating currents and particularly for radio. The explanations here given attempt to give a better insight into the behavior of electric current. Most of the treatment of principles is a presentation of the theory of low-frequency alternating currents, arranged with its radio applications in mind. There is little in the way of special theory before section 24, which deals with damped waves, and yet the underlying principles of the chief radio phenomena are covered. Furthermore, damped waves are of less importance than formerly, since modern practice tends toward the exclusive use of continuous or undamped waves. The principles of radio measurements are thus nearly identical with those of any other alternating-current measurements. THE FUNDAMENTALS OF ELECTRO MAGNETISM 1. ELECTRIC CURRENT Electric current is the rate of flow of a quantity of electricity. The most familiar and most useful properties of an electric cur- rent are (i) the heating effect produced in a conductor in which it flows, and (2) the magnetic field surrounding it. The latter is by far the most important property in radio work. The study of the combined effects of electricity and the magnetism accom- panying an electric current constitutes the subject of electro- magnetism. When a current flows continuously in the same direction, as the current from a battery, it is called a direct current. When the Radio Instruments and Measurements 7 current periodically reverses in direction, it is an alternating cur- rent. The alternation of current is accompanied by a reversal of direction of the magnetic field around the current. On this account alternating currents behave very differently from direct currents. The uses of alternating currents may be divided, roughly, into three groups, separated according to the frequency of alternation of the currents used: Electric power applications, 20 to 100 per second. Telephony, 100 to 20 ooo per second. Radio, 20 ooo to 2 ooo ooo per second. Displacement Currents. Direct currents can flow continuously in conductors only, while alternating currents flow also in insula- tors. Suppose a circuit contains a condenser, consisting of two large metal plates separated by air or some other insulating me- dium. If a battery is connected into the circuit, a momentary flow of current takes place, accompanied by an electric strain in the insulating medium of the condenser. This strain is opposed by an electric stress, which soon stops the current flow. The action is much like the flow of gas under pressure into a gas tank, as described below in the section on capacity. The flow of gas stops as soon as the back pressure of the compressed gas in the tank is equal to the applied pressure. The flow of electric current stops as soon as the back electric pressure (called " potential difference ") of the electrically strained medium is equal to the electric pressure ("electromotive force".) of the battery. If there is a source of alternating current in a circuit containing a condenser, the electric strain in the insulating medium reverses in direction for every alternation of the current. The electric strain of which we have been speaking is called electric displace- ment and its variation gives rise to a so-called displacement cur- rent. The electric strain is of two kinds: First, there is the strain in the actual material dielectric. This part is a movement of electricity in the same sense as the transference of a definite quantity of electricity through a wire is a movement of elec- tricity, the only difference being that in the insulator there is a force (which we called electric stress) which acts against the electric displacement. Second, there is the strain which would exist in the ether if the material dielectric were absent. Some prefer to think of this as a sort of electric displacement in the ether, of a kind similar to that in the matter; but a knowledge of the physical nature of electric displacement is unnecessary for practical purposes ; all that is necessary is a statement of how the 8 Circular of the Bureau of Standards displacement current is to be denned as measured, and of the rela- tion of the quantity so measured to other electromagnetic quantities. The effect of electric displacement at any point in a medium is handed on to adjacent points and so spreads out through space. Under certain conditions a considerable quantity of this moving displacement and the magnetic field accompanying it become detached from the circuit. This process is what constitutes the radiation of electromagnetic waves, which makes radio commu- nication possible. Electrons. The flow of an electric current in a conductor is not opposed by electric stress as in an insulator. A current in a con- ductor is believed to consist of the motion of immense numbers of extremely small particles of electricity, called electrons. All electrons are, so far as known, strictly identical, are, for ease of calculation, assumed to be spherical in shape, and have the fol- lowing dimensions, etc.: Radius, i X io~ 13 centimeter; mass, 8.8X io~ 28 gram; electric charge, 1.59 x io~ 19 coulomb. An electron is thousands of times smaller than any atom. Views as to the transference of the electric current by the motion of electrons in a conductor have undergone considerable changes during the last few years. Some of the electrons in a conductor are bound to the molecules while others are free to move about. The latter are in constant motion between the molecules, in zigzag paths because of repeated collisions with the molecules. The motion of the free electrons is thus very similar to the heat mo- tions of the molecules; in fact, the average kinetic energy of an electron is equal to that of a molecule at the same temperature. The average velocity of the electrons is about 100 km per second at o C. This increases with temperature. Until recently it was supposed that conduction in a solid takes place almost entirely through the agency of the free electrons. The electric current was nothing more than a slow drift (a sort of electric wind) super- posed upon the random motions of the electrons by the electric field. This view, although suggestive and fruitful, is attended with many difficulties, and the present tendency is toward some form of theory in which the conduction is brought about by the spontaneous discharge of electrons from one molecule to another, the function of the field being to influence the orientation of the discharge, which would, in its absence, be perfectly random. The study of electrons has recently led to great improvements in Radio Instruments and Measurements 9 apparatus for producing and detecting currents of radio frequency. (See sec. 56.) When electricity is in equilibrium in a conductor, the electric charge is in a very thin layer upon its surface; thus the phenom- ena of electrostatics arise. An insulator is believed to contain no free electrons. The electrons are bound to the molecules in such a way that they can be slightly displaced by an electric force but return to their positions of equilibrium when it is removed. This motion of the bound electrons, together with the electric strain in the ether itself, constitutes the electric displacement in the insulator, and determines its dielectric constant. Another type of motion of which the electrons attached to molecules are capable is vibration about their positions of equilibrium. They thus give rise to waves that travel outward in the form of light and heat radiation. 2. ENERGY Most useful operations in physics involve the movement of something from one place to another; and in general, as for example when the body moved is held by a spring, or when its velocity changes during the motion, force has to be exerted to cause the motion. A useful quantity which figures in the discus- sion of such motions is the quantity called "work." The work done upon a body is defined as the product of the force which acts upon the body into the distance moved; or (when the force varies during the motion) as the sum of such products for each element of path described by the body. When a body or system of any kind possesses .the power to do work in virtue of its posi- tion, velocity, chemical constitution, or any other feature, it is said to possess energy; and the measure of the change of energy which it experiences in doing such work is the amount of work done. One of the fundamental principles of science is the "con- servation of energy." The amount of energy in existence is con- stant; energy can not be created nor destroyed; it can only be transformed from one form into another. It is often very helpful to the understanding of a process to consider what energy changes are taking place. The transformation from one form into another is always accompanied by a dissipation of some of the energy as heat or some other form in which it is no longer available for the use desired. Thus while none of the energy is lost during an energy change, more or less of it becomes no longer available. io Circular of the Bureau of Standards Kinds of Energy. The familiar kinds of energy are mechanical energy, heat, chemical energy and electrical energy. To these may be added radiant energy, but this is considered to be a form of electrical energy. Mechanical energy is of two kinds, kinetic energy and potential energy. When an object is in motion it is said to possess kinetic energy. If the motion is stopped the kinetic energy of the object changes into some other form. For example, if the moving object is stopped by suddenly striking an immovable obstacle its kinetic energy is converted into heat. If when stopping it starts another object moving, the second object then has kinetic energy. Any object of mass ra moving with velocity v has a kinetic energy = - m i; 2 . The energy which an object possesses in virtue of its position is called potential energy. A stone, lifted a certain distance above the earth, will fall if released. It then acquires kinetic energy in falling. It had a certain amount of potential energy when at the highest point, simply in virtue of its position above the earth. As it falls this potential energy is being changed into kinetic energy, and when it is just about to strike the ground the potential energy has all been converted into kinetic. This is a simple exam- ple of the principle of conservation of energy. When a change of energy from one form into another -occurs, work is done. When an object falls to the earth from a height there is a change of potential energy into kinetic and work is done upon the object by the force of gravity, the amount of which is equal to the product of the force by the distance through which the object falls. Again, when a body is moved against a force tending to oppose the motion, work must be done by the agency which moves the body. The product of these two factors, the force acting and the displacement of the object, is the amount of work done in an energy transformation. Electrical Energy. There are two kinds of electrical energy, similar to the two kinds of mechanical energy. Corresponding to potential energy there is electrostatic energy, which is the energy of position of electricity at rest; this is the form in which electric- ity is stored in a charged condenser. Corresponding to kinetic energy there is electrokinetic energy (also called magnetic energy) , which is the energy of electricity in motion. The latter is the energy of the electric current, and is associated with the mag- netic field accompanying the current. In accordance with the law of conservation of energy the sum of the electrostatic and Radio Instruments and Measurements 1 1 the magnetic energies in any electrical system is constant if the system as a whole does not receive or give out any energy, or if energy is being supplied at the same rate at which it is being dissipated. Electrical energy can readily be converted into other types of energy ; if this were not so it would not be the important factor in the life of man that it now is. As far as radio science is con- cerned, the two principal forms of energy into which electrical energy is transformed are heat and electromagnetic radiation. From any electrical system there is a continuous dissipation or loss of electrical energy going on, and in general the evolution of heat in the circuit plus the energy radiated as electromagnetic waves equals the diminution of the sum of the electrostatic and magnetic energies. The energy of electromagnetic waves is a form of electric energy, being a combination of electrostatic and magnetic energies. Inasmuch, however, as it travels through space entirely detached from the sending circuit, it represents a loss of energy from that circuit. 3. RESISTANCE The dissipation or loss of electrical energy is expressible in terms of resistance. The rate of evolution of energy at any instant in a conductor is the product of the electromotive force acting in the conductor by the current flowing. This energy usually mani- fests itself in the form of heat. The time rate of energy is called power. Thus, w E> -, & p = - = e i = R i" = -p t j\. Resistance is defined by R = -. Power (p) is generally expressed i in watts, energy (w) in joules, electromotive force (e) in volts, current (i) in amperes, and resistance (R) in ohms, unless other- wise stated. The resistance of a conductor depends on the mate- rial of which it is made, the size and shape of the conductor, and the frequency of the current. The characteristic property of the material is called its resistivity. Denoting by p the ordinary or volume resistivity, by / the length, by S the cross section of the conductor, and by R the resistance to direct current, ^ = P 5 The resistance of a system of conductors is readily found by the simple laws of series and parallel combination, for direct currents. 1 2 Circular of the Bureau of Standards With alternating currents, however, the calculation is more diffi- cult, and it is usually found convenient to utilize the relation *- < The resistance of a single conductor to alternating currents is found by the aid of the same relation. It may be shown that the distribution of direct current in a system of conductors or over the cross section of a single conductor is such as to make the production of heat a minimum ; and it results that in a single uniform conductor the current is uniformly distributed over the cross section. When alternating current flows in a conductor, it tends to flow more in the outer portions of the conductor than in the center. In consequence of this change of current distribu- tion, the power which is converted into heat increases. The higher the frequency the farther does the current distribution depart from the direct-current distribution, and the greater does the power p become. It follows, in accordance with equation (i), that the resistance increases as frequency increases. Radio-Frequency Resistance. With alternating currents the departure from uniform distribution of the current is spoken of as the skin effect. At high frequencies the current flows in a thin layer at the surfaces of conductors, and the skin effect is thus large in all except very thin wires; the resistances of ordinary con- ductors at radio frequencies may be many times their low-fre- quency resistances. The ratio of resistance at any frequency to the direct-current or low-frequency resistance can be calculated for certain simple forms of conductors. Formulas for this are given below in sections 74 to 76. In most practical cases, how- ever, the radio resistance can be obtained only by measurement. In addition to the resistances of conductors, resistance is intro- duced into radio circuits by three other causes, viz, sparks, dielec- trics, and radiation. Dielectric resistance is treated in section 34, below. The energy radiated from a circuit per unit time in elec- tromagnetic waves is proportional to the square of the current in the circuit. It is thus analogous to the energy dissipation as heat in a conductor, and, therefore, the radiation increases the effective or equivalent resistance by a certain amount. This added resist- ance is conveniently called radiation resistance. It can be cal- culated for a few simple types of circuit. It is, in general, large enough to be appreciable only when the circuit has the open or antenna form, or when a closed circuit is of large area and the frequency is high. Radio Instruments and Measurements 13 4o CAPACITY Electrostatic energy may be stored in an arrangement of con- ductors and insulator called an electrical condenser. The action of a condenser is somewhat similar to that of a gas tank used for the storage of gas. The amount of gas a tank will hold is not a constant, fixed amount; it depends on the pressure. If the pres- sure is doubled, twice the mass of gas is forced into the tank. The internal or back pressure of the gas opposes the applied pressure. If the applied pressure is released and an opening is left in the tank, the gas rushes forth. The amount of electric charge given to a condenser depends on the electric pressure, or potential difference; and in exact simi- larity to the gas, the charge is proportional to this potential dif- ference. The constant ratio of charge to potential difference is called the capacity of the condenser. In symbols, ^ = C, The capacity of a condenser depends on the size and distance apart of its plates, and on the kind of dielectric between the plates. (Vari- ous kinds of condensers are described in sec. 32, etc., and formu- las for calculating capacity are given in sees. 63 to 65.) The applied potential difference is opposed by a sort of elastic re- action of the electricity in the condenser, just as the internal pressure of the gas in a tank opposes the external applied pressure. If the plates of a charged condenser are connected by a conductor, with no applied electromotive force, the condenser discharges. The Dielectric. The insulating medium in a condenser is called a dielectric. The process of charging causes electric displacement in the dielectric. When a body is moved against a force tending to prevent the motion, work is done, and similarly, when a con- denser is charged against the quasi-elastic reaction of the dielectric, work is done upon the condenser. The energy of the charging source is stored up as electrostatic energy in the dielectric. The two factors upon which the energy depends are the charge and the potential difference, Since p. = C, we have also ~~2 14 Circular of the Bureau of Standards The pressure upon the gas in a tank can not be increased indefinitely, for the tank will ultimately yield and break. Simi- larly there is a limit to the potential difference which can be applied to a condenser, for the dielectric will be broken down, or punctured, if the limit is exceeded. The potential difference at which a spark will pass and the dielectric be punctured is called the " dielectric strength." Capacity is one of the two quantities of chief importance in radio circuits. The other is inductance, treated in the following section. 5, INDUCTANCE Magnetic Flux. The physical quantity called inductance is dependent upon the magnetic field which surrounds every electric current. The intensity of this magnetic field at any point is proportional to the current. The direction of the magnetic field around a straight wire car- rying a current is given by Am- pere's right-hand rule : Close the right hand with the thumb ex- FIG. i. Direction of magnetic field around tended; point the thumb in the a wire carrying a current direction of the Current flow ; the magnetic field is then in the direction in which the fingers point, in circles in planes perpendicular to the wire. The magnitude of the magnetic field intensity can be easily computed for some sim- ple forms of circuit from the principle that its line integral * in a path completely around the current is equal to : - times the cur- rent in amperes. As an example, suppose a current i flowing in a very long solenoid of N turns, radius r and length /. The magnetic field intensity H is parallel to the axis and of constant value in- side the solenoid; it may be shown to be zero in the space outside, and the effects of the ends may be neglected. The WT ^ c . ., , ., J TIG. 2. simple solenoid line integral of H along any path completely around the current is HI inside the solenoid 1 The line integral of a quantity along any line or path is the sum of the products of the length of each element of the path by the value of the quantity along that element. If the quantity has a constant value along the whole path, the line integral is simply the product of this value by the length of the path. Radio Instruments and Measurements and is zero for the rest of the path. i, N times. Hence The path incloses the current , 10 4 Ni 10 / (2) The magnetic field in the medium surrounding a conductor carrying a current produces a magnetized condition of the medium. This condition is a sort of magnetic strain in the medium and is analogous to displacement produced in a dielectric by electric potential difference. The amount of this magnetic strain through any area is called the magnetic flux. This quantity (for which the symbol is used) is equal to the product of the three factors, magnetic field intensity, area and the magnetic permeability. Permeability is a property of matter or of any medium which indicates, so to speak, i t s magnetizability. Its numerical value is equal to unity for empty space and for air and most sub- stances. Iron may have a permeability as high as 10 ooo or even more. Self - inductance. Inductance is a quan- tity introduced as a convenient means of dealing with magnetic fluxes associated with currents. The self-inductance of a circuit is simply the total mag- netic flux linked with the circuit due to a current in the circuit, per unit of current. In symbols, d> f . L-? (3) The magnitude of L depends on the shape and size of the circuit and is a constant for a given circuit, the surrounding medium being of constant permeability. If the circuit has N turns each traversed by the same magnetic flux , then when L is expressed in terms of the usual unit, called the " henry," N , \ L = (4) FlG " 3 lux around a solenoid in which a current is flowing. io 35601 c i6 Circular of the Bureau of Standards Analogy of Inductance to Inertia. The magnetic flux x asso- ciated with a current is analogous to the momentum associated with a moving body. Because of its inertia or mass (m) , a body in motion with velocity (v) opposes any change in its momentum (mv). Inertia is obviously a very different thing from friction, which always resists the motion and tends to decrease the velocity or momentum. Inertia only opposes a change in momentum, \J "V ano ^ hence does not affect a motion with constant velocity. Inductance may be spoken of as electrical inertia or mass. The inductance (L) of a cir- cuit in which a current (i) is FIG. 4. A body having a mass (m) and a velocity flowing Opposes any change in (v) opposes any change in its momentum (mv) ^h e flux (d>) . Electrical resist- ance and inductance are very different, for resistance behaves like mechanical friction, opposing even a constant current, while in- ductance only opposes a change in the current. Thus inductance has no effect on constant direct currents but is one of the deter- mining factors in the flow of alternating currents. The analogy of inductance to inertia, of current to velocity, and of flux to momentum, will be brought out further in the next section. Mutual Inductance. A part of the magnetic flux from a circuit may pass through or link with a second circuit. The amount of FIG. 5. Linking of magnetic flux of one circuit with another; the basis oj the conception of mutual inductance this flux linked with circuit 2, per unit of current in circuit i, is called the mutual inductance of the two circuits. If < 12 denotes the flux mentioned and % = current in circuit i , the mutual inductance = Radio Instruments and Measurements 1 7 It is also true that if < 21 =the flux from circuit 2 linked with circuit i , and i 2 = current in circuit 2, The magnitude of any mutual inductance depends on the shape and size of the two circuits, their positions and distance apart, and the permeability of the medium. If there are N^ turns in the first circuit and N 2 turns in the second and the same amount of flux from one passes through every turn of the other, then using the < 's to denote that part of the flux from one turn of either circuit which passes through each turn of the other circuit, and using the usual units, If, however, the < 's denote the total flux from either circuit pass- ing through each turn of the other circuit, this becomes M=^ (5) 10% For any of the definitions of 0, the ratio -^ = -^, and is a quan- i 2 i 1 tity depending only on geometrical configuration. Calculation of Inductance. It is frequently convenient to deal with the self-inductance of a particular coil or with the mutual inductance of limited portions of two circuits. Inductance is strictly defined only for complete circuits. The self-inductance of a part of a circuit is understood to be such that the inductance of the complete circuit is equal to the sum of the self -inductances of all the parts and the mutual inductance of every part with every other part. Inductances are computed by the aid of equation (3) , together with the principle given above that flux is field intensity times area times permeability. For example, to find the inductance of the long solenoid of Fig. 2, 1 8 Circular of the Bureau of Standards in which 5 = the area of the circular cross section of the' solenoid and jot = i , the permeability of air. Substituting the value of H from (2) , and putting 5" = irr 2 , _ N 4-n-Ni _47T 2 7VV 2 X j~ X irr X I -f o j o T 10 s l 10 / I0 9 / Change of Inductance with Frequency. When it is desired to calculate inductance with great accuracy, account must be taken of the magnetic flux within the conductor carrying the current, as well as the flux outside the conductor. The flux in a wire is greatest at the circumference and zero at the axis of the wire, because the flux is due only to the current which it surrounds. Any change in the distribution of current within the wire changes also the flux distribution and hence the inductance. As has been stated in the section on resistance, the current distribution is different for different frequencies. Consequently inductance varies with frequency. As the frequency is increased, less cur- rent flows near the axis of the wire and more flows in the surface portions. The flux in the central parts of the wire is thus dimin- ished, and the inductance decreases as frequency increases. This effect is small, because the whole flux within the wire is a small part of the total flux. (See formulas (131) to (138) in sec. 67.) There is a similar change of mutual inductance with frequency, but it is so small as to be wholly negligible. Series and Parallel Arrangement of Inductances. Inductances in series add like resistances. When the coils or conductors which are combined are so far apart that mutual inductances are negli- gible, inductances in parallel are combined like resistances in parallel. Taking account of mutual inductance, the total induc- tance of any number of inductances in series is Some or all of the mutual inductances may be negative. For two coils in parallel, the total inductance is L 1 +L 2 -2M The last term in the denominator changes sign if the coils are so connected that M is negative. This expression applies at radio frequencies, but at low frequencies the resistance of the coil may have to be taken into account. For more than two inductances in parallel, the expression for the total inductance is complicated. Radio Instruments and Measurements 19 THE PRINCIPLES OF ALTERNATING CURRENTS 6. INDUCED ELECTROMOTIVE FORCE When the magnetic flux through any circuit is changing, an electromotive force is produced around the circuit, which lasts while the change is going on. The change of flux may be caused in various ways; a magnet may be moved in the vicinity, the circuit or a part of it may be moved while near a magnet, the cur- rent in a second near-by circuit may be altered, or either circuit or a part thereof may be moved. The electromotive force thus caused is called an induced electromotive force, and the result- ing current in the circuit is an induced current. The direction of the induced emf and current is given by L/enz's law, viz, an induced current always flows in such a direction as to oppose the action which produces it. For example, if the current is induced by bringing a magnet near a circuit, the current in the circuit will be such as to repel the magnet. The energy of the induced current results from the work necessary to bring up the magnet against the repelling force. The magnitude of the induced emf at any instant is in every case equal to the rate of change of the magnetic flux through the circuit. This is expressed by the formula where e represents the instantaneous value of the induced emf in a circuit consisting of a simple loop or single turn, and -~ is an expression called the derivative 2 of flux with respect to time and which tells the instantaneous rate of change of the flux. This quantity -37 is the change of flux during a very small inter- val of time divided by the time, and its value may vary from instant to instant. If it remains constant for a certain length of time, then its value is the whole change of flux during that interval divided by the interval. If the changing flux through the circuit is the flux 21 from a second circuit, e = jr- This flux is expressible in terms of mu- 2 While derivatives are used in a few places in this circular, it is believed that the treatment can, never- theless, be understood by a person not familiar with calculus. To avoid the use of derivatives entirely would require circumlocution such that the treatment would doubtless be even less clear. 2O Circular of the Bureau of Standards tual inductance and the current in the second circuit, thus, 4> 2l = Mi 2 . Consequently, _d(Mi i ). dt If the circuits are fixed in position, M remains constant, so this becomes e = M-^ (7) Thus the electromotive force induced in a circuit by variation of current in another circuit is equal to the product of the mutual inductance by the rate of change of current in the second circuit. An emf may also be induced in one circuit by a variation of the current in the circuit itself. Since the flux associated with a current is tf>=Li, it follows that the self -induced emf is given by That is, the emf induced in the circuit is equal to the product of the self -inductance by the rate of change of the current. When the flux $ through a coil of N turns is changing, the total emf e. induced in the whole coil is N times that induced in one turn. The simple equation (6) becomes, in terms of the usual units, N d* , . e = ^~dt (9) Equations (7) and (8) are correct when emf is expressed in volts, inductance in henries, current in amperes, and time in seconds. They were obtained from the simpler equation (6) for induced emf, but they need no modification on this account, because equa- tions (4) and (5) for self and mutual inductance also contain the factors N and io 8 which cancel those in (9). Mechanical Analog. The fact that a change of magnetic flux gives rise to an electromotive force which opposes the change may be understood by recalling that a change of mechanical momentum of a body is opposed by the force of inertia. This force is equal to the rate of change of momentum. The elec- trical and mechanical cases are strictly analogous. Flux corre- sponds to momentum, electromotive force to mechanical force (F), current to velocity (u), and inductance to mass (m). If the mass is constant, we have ~ dv Radio Instruments and Measurements 21 as the expression that force equals rate of change of momentum, analogous to di 7. SINE WAVE The ordinary dynamo is the most familiar application of the principle of induced emf. The field magnets give rise to mag- netic flux, and coils of wire (constituting the armature) are caused to move across this flux by some outside source of power. The simplest type of dynamo generates an al- ternating current FIG. 6. Simple dynamo illustrating how the revolving con- ductor cuts magnetic lines R and is sketched in Fig. 6. The single turn of wire shown is in such a position that maximum magnetic flux passes through it. When it is rotated in either direction, the flux passing through it is changed, and hence an electromotive force is induced in it. If the turn of wire is continuously rotated at constant angular speed w, the rate of change of magnetic flux through it will be greater in some positions than in others, and the electromotive force at the slip- rings A A will vary in a certain manner, this variation being repeated each revo- lution. In Fig. 7 let POP represent the end view of the turn of wire. As the wire revolves to the successive positions PiPPi, P 2 OP 2 , etc., the emf is propor- tional to the sine of the angle formed by the revolving wire with POP. The mag- nitude of the emf at any instant may, therefore, be represented by the vertical lines P^M^ P 2 M 2 , etc., drawn from the horizontal axis to the end of a line revolving with the angular velocity co. A diagram may be drawn, taking the distance along a hori- zontal line to represent time and the vertical distance from \/ FIG. 7. Successive positions of revolving conductor; the emf generated is propor- tional to the sine of the angle formed by the revolving con- ductor and POP 22 Circular of the Bureau of Standards this line to represent induced electromotive force. This emf curve has the mathematical form of a sine wave. Many dyna- mos in actual use generate electromotive forces very nearly of this form, and on account of its mathematical simplicity the sine wave is assumed in most of alternating-current theory. It should not be forgotten, however, that sine-wave theory is in many prac- tical cases only an approximation because the emf is not rigor- ously of sine-wave form. Letting e = emf at any instant, E = maximum emf (that at the crest of the wave as shown in Fig. 8) , t = time, co = angular velocity of the turn of wire in Fig. 6, = E sin (10) This emf alternates in direction. Starting at a, Fig. 8, it passes through a set of positive values, then through a set of negative \ ,'P Time FIG. 8. Sine wave developed from circle diagram values, and at b begins to repeat the same "cycle. " In the time of one complete cycle, represented by the distance a b, the revolv- ing radius OP makes one complete revolution, or passes through the angle 2?r radians. The time required for one complete cycle being represented by T, it follows that 27T The time T is called the "period" of the alternation. It is the reciprocal of the "frequency," which is the number of times per second that the electromotive force passes through a complete cycle of values. It follows, denoting frequency by /, that (H) Radio Instruments and Measurements 23 In considering the effect of frequency in electrical phenomena the quantity w is found more convenient and appears oftener than /. Effective Values of Alternating Quantities. The instantaneous rate at which heat is produced in a circuit is proportional to the square of the instantaneous current. According to the equation p = Ri 2 , the average rate of heat production must be proportional to the average value of i 2 . The average heating effect deter- mines the deflection of such an instrument as a hot-wire ammeter, which thus indicates a current / fulfilling the condition, average power = RI 2 . The indicated current / must therefore be the square root of the average value of &. The square root of the mean square of an alternating current or emf is called the "effec- tive " or " root-mean-square " current or emf. All ordinary amme- ters and voltmeters used in alternating-current measurements give effective values. When an electromotive force has the sine-wave form, e=E sin cot, the mean square value is proportional to the average value of sin 2 wt during a half cycle, which is equal to 0.5. The effective value is proportional to the square root of this, so that the effective value is V^5 E< or E = 0.707 E Similarly in the case of current, 7 = 0.707 7 8. CIRCUIT HAVING RESISTANCE AND INDUCTANCE When an emf is suddenly impressed on a circuit containing inductance as well as resistance, say by closing a switch, a cur- rent begins to flow but does not rise to its full value instantly. The magnetic flux accompanying the current causes a self -induced emf which by Lenz's law opposes the increase of current. There are then acting in the circuit two emf's, the impressed emf e and di the emf of self-induction, which by equation (8) is L-rf The minus sign is used to indicate that the induced emf opposes the impressed emf. This is similar to the action of a mechanical force on a mate- rial object; the applied force is opposed by the force of inertia and some time is required before the body moves with the final velocity determined by the applied force and the friction. The opposing force of inertia in the mechanical case is given by the Circular of the Bureau of Standards product of mass by the time rate of change of velocity. The force of inertia corresponds to induced emf, the mass to inductance, and velocity to current. The total emf acting to produce current through the resistance is the sum of the impressed emf and the emf of self-induction, thus: e-L%-Ri at This equation gives the relation between current and applied emf at any instant. It is usually convenient to consider this equation in the form 4 at (12) which indicates that the applied emf is opposed by the resistance and the inductance. When R is relatively large or L relatively small the current comes very quickly to its final value. FIG. 9. Circuit with resistance and inductance in series Impedance. In alternating-current and radio work the most common and the simplest type of electromotive force is the sine wave. Such an emf is expressed by equation (10). Supposing a sine-wave emf to be impressed on a circuit, equation (12) becomes di -r. + Ri = E sin ut d3) The solution of this differential equation (neglecting a term which represents the transient phenomena when the current is started) is (R sin uit coL cos or +0) 2 L 2 sin (ut 8) (14) d5) Radio Instruments and Measurements 25 where 6 is defined by tan0=~ (16) The current which flows in a circuit containing constant resistance and inductance due to a sine emf is also a sine wave. The emf varies as sin (wf), the current as sin (at 6), hence the current lags behind the emf by the angle 6. This angle is called the phase angle. The instantaneous current becomes a maximum (7 ) when sin (at 6) i , E and, since the effective values / and E are equal to 0.707 7 and 0.707 Eo, respectively, This has the form of Ohm's law, the quantity ^ 2 +w 2 L 2 occurring in place of R. This quantity is for this circuit the value of the impedance, which is defined as the ratio of emf to current. Since w = 27r times the frequency it is clear that impedance is a function of frequency as well as of resistance and inductance. Power. The power expended in the circuit is at any instant the product of the instantaneous electromotive force and current; p = ei. The average power is the mean taken over a complete cycle of this instantaneous product. Performing the calculation, the average power is found to be P = E7cos0, where E and 7 are effective values and 6 is the phase angle, defined p by equation (16) above. The ratio . = cos 6 is called the power factor of the circuit. 9. CIRCUIT HAVING RESISTANCE, INDUCTANCE AND CAPACITY When a circuit contains a condenser in series with resistance and inductance the applied electromotive force is opposed by the potential difference of the condenser in addition to the opposition of the resistance and inductance. The potential difference of the condenser equals which may be written *j= so that ec l ua tion (12) becomes di fidt , . 26 Circular of the Bureau of Standards Taking e = E sin cot, and differentiating the equation, dH di i The solution (neglecting terms representing the transients, which die out very quickly after the current is started) is E sin (ut 8) The phase angle 6 is given by a tan = -Fr- R RaC R d9) FIG. 10. Circuit with resistance, inductance, and capacity in series; a typical radio circuit The maximum value of the current is E /0 = The impedance is V- The relation between effective current and emf is E (20) Radio Instruments and Measurements 27 It is to be noted that the terms in L and C have opposite signs in this equation. Thus one tends to neutralize the other, and comparing with equation (17) for a circuit with resistance and inductance only, the impedance of an inductive circuit can be reduced and the current increased by putting a condenser of suitable value in series with the inductance. This increase of the current has sometimes been called resonance or partial resonance, but the term " resonance" is usually reserved for the production of a maximum current, as treated in section 1 1 , below. Special Cases. It is of interest to consider the following special cases : 3 I. L = o and C = oo . II. C=co. III. L = o. Case I represents a circuit with resistance alone. The equations just above give for this case I = -~, 8 = 0. The impressed electro- motive force and the current are in phase, and their ratio is the resistance, just as with direct current. FIG. n.i Circuit with resistance and capacity in series Case II is that of a circuit with resistance and inductance, which has already been treated in section 8. Case III is that of a circuit with resistance and capacity in series. Equations (20) and (19) give E tan B = - RaC 8 Putting C=oo is mathematically equivalent to the statement that the condenser is short-circuited. As the distance between the plates of a condenser is decreased the capacity increases without limit. We may consider, then, that when the plates touch together and the condenser is short-circuited the capacity is infinite. 28 Circular of the Bureau of Standards The case is of special importance when the resistance term is entirely due to energy losses within the condenser. It is frequently convenient to deal with the "phase difference" ^, which = 90 6, rather than with the phase angle. The phase difference is given by tan \fr = If R is very small tan \f/ = ^, and the phase difference = RwC. 10. "VECTOR" DIAGRAMS Writing equation (17) in the form E = ^/R 2 P +w 2 L 2 P, it is evi- dent that E has such a value as would be given by the diagonal of a rectangle having sides equal to RI and coL/. It is therefore pos- sible to determine the value of E by the aid of a vector diagram RI cuLI cLI FIG. 12. Vector combination of electromotive forces FIG. 13. Vector diagram for resistance and in- ductance in series such as is used for calculating the resultant of mechanical forces. u>LI is represented as a vector perpendicular to RI, and their re- sultant is E. The current / is represented as a vector in the same direction as RI; since, from equations (10), (15), and (16), the current and electromotive force differ in phase by the angle whose tangent is -=-, and this is equal to the angle 6 in Fig. 13. If equation (20) is written in the form E = it is evident that E is calculable as the result of adding the three vectors RI, wLI, and ^, drawn as in Fig. 14. Radio Instruments and Measurements 29 These three quantities are the emf 's across the resistance, induc- tance, and capacity, respectively. The emf ^ is drawn downward from the origin, opposite in direction to uLI, corresponding to the minus sign in the equation. The phase angle between / and the resultant E is 6. From the Fig. 14, tan0 = I ^c (21) R in agreement with (19) above. When coL is greater than ^, E OIL, is above the horizontal line as shown, 6 is positive, and the cur- coLI J_ cuC e\ RI FIG. 14. Vector diagram for resistance, inductance, and capacity in series rent is said to lag behind the electromotive force. When ~ is coC greater than coL, 8 is negative, and the current is said to lead the electromotive force. The component of emf in phase with the current is RI. The component at 90 to the current is ( uL ^ j 7. The ratio of this component emf to the current is called the reactance. Its value here is (coL -^j. The ratio of the re- sultant emf to the current is called the impedance, which is here equal to J R 2 -fvL-^V If all the electromotive forces in Fig. 14 be divided by /, the component vectors then become resistance and reactances and 3O Circular of the Bureau of Standards these combine vectorially to give the impedance as a resultant/ It is sometimes convenient to speak of resistance and reactance as impedance components. Reactance (usually denoted by the symbol X) is expressible in ohms just as resistance is. The reactance in the case under consideration consists of two parts, the "inductive reactance" coL and the "capacitive reactance" ~. These may be denoted, respectively, by XL and Xc. From the expression (21) above it is seen that the tangent of the phase angle is a ratio of impedance components. X X^L Xc SS ~R = - R When the capacitive reactance is greater than the inductive reactance, the total reactance X and the phase angle have negative R HWiiH FIG. 15. Circuit with resistance and capacity in parallel values. In any case the ratio of reactance to resistance is the tangent of the phase angle. Phase Difference. In a circuit consisting of resistance and capa- city only, or resistance and inductance only, in series, it is more convenient to deal with the phase difference than with the phase angle. The tangent of the phase difference is the reciprocal of the tangent of the phase angle. When an angle is small it is equal to its tangent, and consequently the phase difference is equal to the ratio of resistance to reactance, in a series circuit in which the re- sistance is small compared with the reactance. This is in agree- ment with the case discussed above on page 27, where it was shown that the phase difference of a condenser with resistance in series Radio Instruments and Measurements 3 1 = RcoC. Similarly, the phase difference of an inductance with resistance in series = r* coL Vector Addition of Currents. For series circuits the emfs and the components of impedance combine vectorially just as, with direct current, the emfs and resistances combine algebraically. With direct current in parallel circuits, on the other hand, the currents and conductances add up algebraically. For parallel circuits with alternating current, the currents combine vectorially, and so do the components of the ad- mittance (reciprocal of impedance) . Suppose, for example, a condenser and resistance in parallel (Fig. 1 5) . Current / is the vector sum of the currents 7 R = ~ an( i 7 C = wCE. The impressed emf and the _ FiG. io. v ector diagram for re- Current = are in the Same direction, Distance and capacity in parallel K. while the current uCE leads the impressed emf by 90. The resultant current / leads E by the angle 6, where tan 6 = From Fig. 16, and the admittance (ratio of resultant current to emf) is \ -&- 2 + co 2 C 3 . 11. RESONANCE In a circuit consisting of inductance, capacity and resistance in series, the effective current has been shown to be L __LY ( 22 > When coL = ^ the impedance is a minimum and the effective coC current is a maximum. This condition for maximum current is called resonance. The ratio of the current at resonance to the current in the circuit with the condenser removed has been called the "resonance ratio"; this quantity is practically the same in radio circuits as the "sharpness of resonance" denned below. At a given "frequency resonance may be brought about by varying either the capacity or the inductance. On the other 35601 18 3 32 Circular of the Bureau of Standards hand, for a circuit of given L and C, there is some particular frequency at which resonance occurs. The condition is equivalent to i ^C LC i (23) (24) The relation (24) is of the greatest importance in high-frequency work. It is the fundamental equation of the wave meter, for instance. A number of other important ways of expressing the same relation are given in section 78. Simplified Current Equation at Resonance. At resonance the inductive reactance is equal to the capacitive reactance, the total reactance is zero, and the impedance equals simply the resistance. That is, at resonance, equation (22) reduces to - r ~R This means that the impressed emf is strictly equal to RI T . The potential difference across the condenser and that across the inductance may be greater than this, and in fact may be many times the impressed emf. Being equal and opposite, they neutralize each other and contribute nothing to the resultant emf opposing the applied emf. Mechanical Illustration. The phenomenon of resonance is well FIG. 17. Simple mechanical system which can exhibit the phenomenon oj resonance illustrated by the vibration of a spring with a mass attached. When a force F acts on the mass m, it is opposed by the stiffness of the spring, by the inertia of the mass, and by friction. The analogy to the electrical case is not perfect, since friction due to sliding is not proportional to the velocity. If the force is applied periodically, there will be a certain particular frequency for Radio Instruments and Measurements 33 which a more vigorous oscillation is produced than for any other. When the frequency of the applied force is just equal to the frequency of resonance, the applied force is all used in overcoming friction; the elasticity of the spring and the inertia of the mass constitute two equal forces opposing each other. These two opposite forces may be much greater than the applied force. For instance, the vibration may become so violent as to break the spring although the impressed force is far too small to do so. Magnification of Voltage. Similarly, in the case of the electrical circuit, there is danger of breaking down the condenser in a resonant circuit because the potential difference across the con- denser is much greater than the applied electromotive force. The ratio of the voltage across the condenser to the applied voltage is greater the smaller the resistance (including under resistance not only the ordinary "ohmic" resistance of the con- ductors but all sources of energy loss, such as dielectric loss in the condenser). In comparing the electrical circuit with the vibrating spring, one should remember that the mass is the analog of the inductance and the spring the analog of the condenser. It is unfortunate that the diagram generally used for an inductance is the same as that used above for a spring. Vector Diagram of Resonance. Resonance phenomena are shown in an interesting manner by means of vector diagrams. In Fig. 14, which illustrates the vector diagram of emfs for a circuit with resistance, inductance, and capacity in series, the inductive reactance coL is taken to be greater numerically than the capacitive reactance -^ so that the resultant vector E has a direction corresponding to a positive rotation from the direction of RI through an angle 6. Suppose that the frequency is de- creased; ooL decreases and ^ increases numerically. When they become equal, the total reactance ( coL -~ j and the angle 6 become zero, and E equals RI. The diagram then becomes Fig. i 8. Resonance Curves. As already stated, in a circuit of given L and C there is some frequency at which resonance occurs, given bycoL= -~- At all other frequencies the inductive reactance and the capacitive reactance are unequal, and their difference 34 Circular of the Bureau of Standards enters the expression for impedance. At frequencies less than the frequency of resonance the capacitive reactance is the larger, and consequently we may say that at low frequencies the capacity keeps down the current, while at high frequencies it is the induc- tance that keeps the value of the current down. For any depar- ture from the condition of resonance, whether by variation of fre- quency, of inductance, or of capacity, the current is diminished. The process of varying either the capacity or the inductance to obtain the setting at which the circuit is in resonance with the frequency of the applied electromotive force is called "tuning" the circuit or tuning the circuit to resonance. The reduction of the current on both sides of resonance is shown in Fig. 19, in which the square of current is plotted against capacity, the emf being constant. Such curves are called reso- nance curves. o>LI RI cuC FIG. 18. Vector diagram of series circuit in resonance The three curves shown are for an actual circuit, with its normal resistance of 4.4 ohms, with 5 ohms added, and with 10 ohms added. The inductance is 377 microhenries and the frequency 169 100 cycles per second. The curves show theoretical values as obtained from the formula E 2 The theoretical values were closely checked by actual observa- tions, using a pliotron as a source of alternating current and a thermocouple and galvanometer to measure the current. The Radio Instruments and Measurements 35 square of current is plotted instead of current, simply because the galvanometer deflections were proportional to the square of current. The ordinates are thus in terms of galvanometer deflec- tions. The value of the constant impressed emf is given by E = RI T , where 7 r = current at resonance. In the arbitrary units resulting from the expression of current-square in galvanometer deflections and R in ohms, for curve A the value of E = ^X 1/19 = 19.2. The emf across the inductance = coL7, so that at resonance its FIG. 19. Resonance curves for series circuit with different resistances value is 2ir x 169 100 x 377 X io~ 6 X V^9 J 75- The emf across the condenser at resonance = ~ , which equals the same value, 1750. Note that this is much greater than the applied emf, 19.2. In the case of curve B, the applied emf is the same, but the emf across the inductance and the equal emf across the condenser at resonance = 27r Xi 69 100X377 X io~ 6 X V4-i6 = 8i8. l n the case of curve C, the equal emfs across inductance and capacity at resonance each equal 534. The applied emf having the same value, 19.2, for each curve, this clearly illustrates the statement previously made that the ratio of condenser voltage to applied voltage is greater the smaller the resistance. 36 Circular of the Bureau of Standards Sharpness of Resonance. One of the principal applications of the phenomenon of resonance is the determination of frequency. Since the current is a maximum when w = , , the frequency VLC is determined when L and C are known. The precision with which frequency can be determined by this method depends upon the sensitiveness of the current indication to a given change in C or L at resonance. This sensitiveness is obviously greater the sharper the peak (Fig. 19). The precision of determination of frequency, therefore, depends upon what may be called the sharpness of resonance, a quantity which measures the fractional change in current for a given fractional change in either C or L at resonance. (This quantity has also been called selectivity; see also statement on p. 31 regarding the term resonance ratio.) The sharpness of resonance is an important characteristic of a circuit and is very simply related to the phase differences and other constants. It may be denned in mathematical terms by the following ratio (C,-Q C where the subscript r denotes value at resonance, and 7 t is some value of current corresponding to a capacity C which differs from the resonance value. The numerator of this expression is some- what arbitrarily taken to be the square root of the fractional change in the current-square instead of taking directly the fractional change of the first power of current. This is done because of the convenience in actual use of this expression (since the deflections of the usual detecting devices are proportional to the square of the current) and also because of its mathematical convenience. It is readily shown that the sharpness of resonance thus defined is equal to the ratio of the inductive reactance to the resistance. Since /.- ( 25 ) the relation r 2 _ . Radio Instruments and Measurements 37 becomes 772 ,o>C r This, together with gives (26) The right-hand member of the equation is the ratio of the capaci- tive reactance at resonance to the resistance. In virtue of equa- tion (25), it is also equal to the ratio of the inductive reactance to the resistance; thus sharpness of resonance = -~-' It is of interest to note the relation of the sharpness of reso- nance to phase difference. As shown above on page 31, the phase difference of a series combination either of resistance and induct- ance or of resistance and capacity is equal to the ratio of resist- ance to reactance. If in a circuit having an inductance coil and a condenser in series the only resistance is that of the inductance coil, it follows that the sharpness of resonance is equal to the reciprocal of the phase difference of the coil. If, on the other hand, the resistance of the circuit is all due to energy loss in the con- denser, the sharpness of resonance is equal to the reciprocal of the phase difference of the condenser. A measurement of the sharpness of resonance thus gives the phase difference directly. If the resistance is partly in the coil and partly in the condenser, each has a phase difference and the sharpness of resonance is equal to the reciprocal of the sum of the two phase differences. Circular of the Bureau of Standards It has been mentioned that there is danger of breaking down a condenser in a resonant circuit because the potential difference across the condenser may be many times higher than the applied electromotive force. This danger is directly in proportion to the sharpness of resonance, or inversely as the phase difference of the condenser. For if the applied emf is RI, the condenser voltage The ratio of the condenser voltage to the at resonance is coC r i applied emf is ^ ~ which is the sharpness of resonance or the reciprocal of the condenser phase difference. Application to Radio Resistance Measurement. Formula (26) above, which gives the relation between the sharpness of resonance FlG. 20. Simple circuit for measurements of resist- ance or wave length and the phase difference of the condenser, has been shown to have important applications to the precision of frequency measure- ment and to the rise of voltage on the condenser. Another appli- cation of great importance is the measurement of resistance. This is seen by writing the equation in the form (c,-o [-77- (27) Thus the resistance of a simple circuit as in Fig. 20 is measured by observing deflections of the indicating instrument A for two settings of the variable condenser, one setting at resonance C T and any other setting C. This is one of the principal methods of measuring high-frequency resistance, and may be called the ' ' reactance variation ' ' method. Other ways of using the prin- ciple of reactance variation are described in section 50 below. Radio Instruments and Measurements 39 The method is rigorous, involving no approximations, provided the applied emf is undamped. The resistance so measured is the effective resistance of the entire circuit, including that due to condenser losses and radiation. 12. PARALLEL RESONANCE When a coil and a condenser are in parallel in a circuit, the phenomena are strikingly dif- ferent from those of the series arrangement. The total cur- rent / is the vector sum of the currents in the two branches, 7 L and 7 C . The current through the coil depends on- its resistance and inductance, thus, E Also FIG. 21. Parallel circuit having capacity in parallel with inductance and resistance; un- der certain conditions the current in either of the branches exceeds that in the main line assuming the condenser loss to be negligible. Taking the sum of the two currents, with due regard to their ^ phase relation, the total current is /- When // coL V / R V V \ ~R 2 + tfU) \R* 4*VLV (28) the total current is in phase with the emf, and has the value, ER (29) This is the minimum current for varying values of C and is very nearly the minimum current for varying values of L or w. Equation (28) is the condition for what may be called inverse resonance or parallel resonance. At parallel resonance, the total current in the external circuit is less than the current in the coil. This is because the currents in condenser and coil are in FIG. 22. Vector diagram for parallel circuit 40 Circular of the Bureau of Standards opposite directions as regards the external circuit, and thus tend to neutralize each other in that circuit. Simple Case. When the resistance of the coil is very small compared with its reactance, as is usual at radio frequencies, equation (28), the condition for parallel resonance, becomes (using co to denote the value of o> at parallel resonance) , or i Wo == / _ _ VLC The total current at this frequency is ER (30) FIG. 23. Vector diagram illustrating resonance in simple parallel circuit o i ex /o w ia FIG. 24. Resonance curve showing the con- dition of parallel resonance The current in the condenser is very closely equal to that in the coil, the value being T ' The total current is the vector sum C0 i^ of the currents in coil and condenser, and is thus smaller than the r> current in either by the ratio j- As suggested by a compari- COo/-* son of Fig. 22 and Fig. 23 the resultant current may be yanish- ingly small. The combination of coil and condenser acts like a Radio Instruments and Measurements 41 very large impedance in the main circuit, the value of this im- pedance being ^ . For any variation of frequency, induc- tance, or capacity, from the condition of parallel resonance, the total current increases. At low frequencies the inductance carries the greater part of the current, and at high frequencies the condenser is the more im- portant. Comparison of Series and Parallel Resonance. It is interesting to compare the resonance phenomena in a series circuit with the phenomena of parallel resonance. In the former case the indi- vidual voltages across the coil and condenser exceed the result- ant voltage across both, whereas in the latter case the separate currents exceed the resultant current. The impedance intro- duced into the circuit by the series combination is yanishingly small, and the impedance due to the parallel combination is very 17 large. Comparing equation (29) with/= , . the ratio' of the total current at parallel resonance to the current in the circuit r> with the condenser removed is /r ., = Thus the current is reduced in parallel resonance in the same ratio that it is increased in series resonance. The two kinds of resonance are discussed further in the next section. RADIO CIRCUITS 13. SIMPLE CIRCUITS A typical radio circuit comprises an inductance coil, a con- denser, and a source of electromotive force, in series. The source E may be a small coil in which an alternating electro- motive force is induced by the current in a neighboring circuit. Some of the phenomena in such a circuit have already been treated under "The principles of alternating currents." What is there given applies to high as well as low frequencies. Some of the phenomena and their mathematical treatment are much simplified at high frequencies. Electromotive forces of sine-wave form are assumed in this discussion; the results obtained apply equally to slightly damped waves. It will be recalled that the reactance of an inductance is uL, and the reactance of a capacity is It is essential to remember coC 42 Circular of the Bureau of Standards that co is 27T times the frequency. In fact, the physical meaning of co in reactance expressions is the same thing as frequency; the 27T is a factor result ing from the way the units are defined. The expression coL tells us that the reactance of an inductance is pro- portional to frequency. Series Circuit. The simple circuit of Fig. 25 is, in fact, the principal circuit used in radio transmitting sets, receiving sets, and wave meters. Some of its chief properties are conveniently brought out by a graphical study of the variation of its reactance with frequency. Advantage is taken of the fact that resistance is a negligible part of the impedance (except at resonance), to obtain very simply an idea of the way the current varies with frequency. Small current corresponds to large reactance (either positive or negative), and vice versa. FIG. 25. Simple series circuit The reactance of the circuit is ( coL -?=,) The inductive react- ance, coL, is the predominating portion of this at high frequencies. It is represented by the line coL in Fig. 26. The capacitive react- ance predominates at low frequencies; it is represented by the line ~' The sum of these two is represented by the line marked total reactance, crosses the axis- At the point co' -i. e., where coL = coC where this reactance curve the current is a maximum. The resonance curve, showing variation of the current with fre- quency, would rise to infinity at the point co', where the total reactance is o, if it were not for the resistance in the circuit. While the current at resonance is determined by the resistance, the frequency of resonance is given accurately by the reactance curve which takes no account of resistance. The most important Radio Instruments and Measurements 43 aspect of resonance phenomena is thus shown by the simple react- ance curve, the plotting of the current curve being unnecessary. Use of Reactance Curves. Complex circuits can be studied and much useful information easily obtained by the use of reactance curves. The effect of any auxiliary circuit upon a wave meter or a transmitting apparatus can be determined, as will be shown later. In any such diagram the points where the reactance curve crosses the co axis give the frequencies at which the current is a maximum. 2000 600 500 2000 1500 FIG. 26. Reactance diagram for simple series circuit, showing the capacitive and induc- tive reactances and their resultant at different frequencies . The current in the circuit is a maximum at u f Parallel Circuit. An inductance and a capacity placed in paral- lel in a circuit behave very differently from the series arrangement of inductance and capacity already discussed. As shown in Fig. 27, the same electromotive force is impressed upon the terminals of both L and C by the source E, which may be a spark gap, another condenser, a coupling coil, etc. The total current / is the sum of the currents in L and C, or 7 = 4~-coCE coL / The ratio, ^ is equal to the reciprocal of the impedance, and when 44 Circular of the Bureau of Standards resistance is negligible this is a quantity called the susceptance. The total susceptance is here made up of two parts, the inductive susceptance -y which predominates at low frequencies, and the FIG. 27. Simple parallel circuit capacitive susceptance wC, which predominates at high frequencies. Each of these two susceptances is the reciprocal of the correspond- ing reactance, but this is true only when resistances are negligible. 0.008 FIG. 28. Reactance diagram for simple parallel circuit, showing the capaciti-ve and inductive susceptances at different frequencies, together -with their resultant and the resultant reactance The curve marked "Total susceptance" in Fig. 28 was obtained by addition of the two curves, 7- and coC. The curve "Reactance " coL was obtained by taking reciprocals of the points on the curve of total Radio Instruments and Measurements 45 susceptance. The reactance of the circuit is small at very low and very high frequencies, but at co , the point of parallel resonance, both branches of the reactance curve go to infinity. The current in the circuit is a minimum at co and would be strictly zero if there were actually no resistance in either branch of the circuit. Thus it is seen that while a series combination of inductance and capacity has zero reactance when wL = ^ a parallel arrange- ment has infinite reactance under the same condition. A series arrangement is therefore used when it is desired to make current of a given frequency a maximum, and a parallel arrangement is used when it is desired to suppress the current of that frequency. 14. COUPLED CIRCUITS Circuits which are more complex than those already discussed may be considered as combinations of simple circuits. The component simple circuits in general have certain parts in com- FIG. 29. Simple case of coupled circuits in which a parallel circuit u combined with a series circuit mon, and these parts are said to constitute the coupling between the circuits. Suppose, for example, the simple series circuit and the simple parallel circuit are combined as shown in Fig. 29. The coil L is the coupling between the circuit QL and the circuit C 2 L. Elimination of Interference. A great deal of information about coupled circuits may be obtained from their reactance diagrams. A curve of the variation of reactance with frequency tells in a very simple way at what frequencies the current is either large or small. The reactance of C 2 and L in parallel (Fig. 29) is as shown in Fig. 28 and designated by X" in Fig. 30. This com- bination is in series with -C : . Adding the curve of condenser 46 Circular of the Bureau of Standards reactance to the curve X", the curve X is obtained, giving coC t the reactance to current flowing through the ammeter. At the frequency corresponding to ', the reactance is zero and the current a maximum. At co the reactance is infinite and the current is a minimum. It is easily seen, therefore, that such a circuit is very useful where it is desired to have current of a certain frequency in a circuit but to exclude current of a certain other frequency. For example, if it is desired to receive radio messages of a certain wave length from a distant station, and a 1000 750 5OO 250 FIG. 30. Reactance diagram for combination circuit of Fig. 29; curve X is the resultant reactance of the system, near-by station operating on a different wave length emits waves so powerful as to interfere with the reception, the interfering signals can be greatly reduced by using this kind of circuit. The circuit C 2 L is first independently tuned to resonance with the waves which it is desired to suppress. The setting of condenser Ci is then varied until the main circuit is in resonance with the desired waves. If the resistances in the circuit are very small, interference is readily eliminated in this manner. The same thing is accomplished by other types of coupled circuits, as explained below. Suppression of Harmonics. Such a circuit is useful also in sending stations or in laboratory set-ups, where certain wave Radio Instruments and Measurements 47 lengths need to be eliminated. For example, the emf from an arc generator is not a pure sine wave but contains harmonics in addition to the fundamental frequency. Some harmonic may be especially strong and it may be desired to suppress it. This can be accomplished in some cases by connecting a condenser across a loading coil (which is not a coil used to introduce the emf into the circuit) either in the closed circuit or the antenna and tuning the combination of loading coil and condenser to the ob- jectionable frequency. Various modifications of this simple scheme can be used, which may be more convenient under certain circumstances. Thus, instead of a condenser only, a condenser and coil in series can be connected around the main inductance as in Fig. 31. The circuit L b MC 2 is independently tuned to FIG. 31. Coupled circuits involving two simple series cir- cuits; -various modifications of this circuit are used in trans- mitting sets in which it is desired to emit certain frequencies and suppress others the harmonic which is to be suppressed. The main circuit is then tuned to the frequency which is to be emitted. The reactance to the emitted frequency is thus made zero, while the reactance to the objectionable frequency is made very large, as shown in Fig. 32. The reactance of the parallel combination of M with L b and C 2 is found by the method used before to be the curve X." with two branches. The condenser reactance -- is added to this, giving the heavy curve X of react- ance to current flowing through the ammeter. One of the characteristic properties of coupled circuits is brought out by Fig. 32, viz, the reactance is zero at two frequen- cies. That is the current is a maximum for two different fre- 35601 18 4 48 Circular of the Bureau of Standards quencies. Between these two, indicated by a/ and co", is the frequency of infinite reactance or minimum current, indicated by co . Thus, it is possible to suppress a certain frequency and tune the circuit to a different frequency either larger or smaller than the one suppressed. There will be current maxima at both fre- quencies (corresponding to w' and ") in the circuit if the source of emf supplies these two frequencies simultaneously. The current will have a minimum at the intermediate frequency of infinite reactance only provided the resistances of the circuits are small. JOOO FlG. 32. Reactance diagram for the simple coupled circuits of Fig. 31; the curve X of resultant reactance is zero for two values of frequency IS. KINDS OF COUPLING Circuits may be connected or coupled together in a number of ways. When there are two circuits, the one containing the source of power is called the primary, the other circuit the secondary. These are generally coupled in one of the follow- ing ways: (a) By direct connection across an inductance coil; (6) by electro-magnetic induction; (c) by direct connection across a condenser. In the first kind, called "direct coupling," an inductance coil is common to the two circuits as illus- trated in Fig. 33 (a). In the second kind, "inductive coup- ling," shown in (6), the two circuits are connected only by mu- tual inductance. An example of the third kind, "capacitive Radio Instruments and Measurements 49 coupling," is shown in (c); a condenser is common to both cir- cuits in place of the coil M of Fig. 33 (a). It is characteristic of coupled circuits that the impedances in each circuit affect the current flowing in the other. This reaction of the circuits upon each other is the more marked when the common portion of the two circuits is a larger proportion of their impedances. When this is large the coupling is said to be "close" and when small the coupling is "loose." In the case of extremely loose coupling, the back action of the secondary on the primary circuit is negligible, and the considerations of coupled circuits do not apply ; the two circuits act practically as independent circuits, the primary merely applying an electromotive force to the secondary. (c) FIG. 33. Types of coupling; (a) direct coupling, (b) inductive coupling, (c) capacitive coupling Coupling Coefficient. The closeness of coupling is specified by a quantity called the coupling coefficient. This is defined as the , where X m is the mutual or common reactance \ ratio (either inductive or capacitive) and X^ is the total inductive or capacitive reactance in the primary circuit and X 2 the total similar reactance in the secondary. Thus, in the case of direct coupling, Fig. 33 (a), the coupling coefficient is coM V(L a +M)co(L b +M) Denote the total inductance of primary and secondary by L t and L 2 , respectively, and the coupling coefficient by k; then k M 50 Circular of the Bureau of Standards This also gives the coupling coefficient for inductively coupled circuits, as illustrated in Fig. 33 (6) , L l and L 2 being the respective total inductances of primary and secondary, each measured with the other circuit removed. As suggested by the identity of expression for coupling coefficient, inductively coupled circuits may be considered as equivalent to direct-coupled circuits having the same M, C lt and C 2 , and in which L SL = L 1 M and Lb =L, M. The coupling coefficient in Fig. 33 (c) is: coC r l -+- L -} Cb wC m / CoCb Denote by C\ the total capacity of the primary circuit, and by C 3 the total capacity in the secondary. ii i A i i i r = T=T + =r~ and. r - r ~~ whence, c L, Cb L FIG. 34. Special case of capacilive coupling From these expressions the coupling coefficient may be obtained for particular cases. Thus, for Fig. 31, which is a special case of the kind of coupling shown in Fig. 33 (a) , M L b +M Similarly, Fig. 34 shows a special case of capacitive coupling. Radio Instruments and Measurements The coupling coefficient is readily found to be V* +c m Use of Coupled Circuits to Select Frequencies. Any of the systems of coupled circuits which have been mentioned may be used for the purpose of suppressing current of one frequency while responding to current of another frequency or wave length. This was discussed above in connection with the simple case of direct coupling in Fig. 29. It may be shown that each of the more general circuits in Fig. 33 will accomplish the same thing. This FIG. 35. Reactance diagram for case of capacitive coupling shown in Fig. 34 is also true of the simple case of capacitive coupling in Fig. 34, as may be seen from its reactance diagram Fig. 35. The curve X" gives the reactance of the parallel combination of C m with L 2 and Ct>. Adding the reactance of L t to this, the heavy curve X is obtained, showing the total reactance to current in the primary circuit. As before, the reactance is zero at two fre- quencies and is infinite at one intermediate frequency. Thus any of these arrangements of coupled circuits may be used to remove an objectionable frequency while tuning to some other frequency either higher or lower than the one suppressed, provided the resistances of the circuits are not large. In every 52 Circular of the Bureau of Standards case, co corresponding to the frequency suppressed in the primary circuit is given by i where L 2 = total inductance of the secondary circuit and C 2 = total capacity. 16. DIRECT COUPLING The above discussion shows how a qualitative comprehension of the action of coupled circuits may readily be obtained. The exact frequencies to which a coupled system responds may be obtained by calculation in the manner here shown for direct coupling, upon the assumption that resistances can be neglected. The emf E in the primary (Fig. 36) is opposed by the impedance ^ G . 3 6.-Cucuitsin W Mngdirectcoupling of La^fC,, and of the parallel com- bination of M with L b and C 2 . Denoting by 1^ and / 2 the currents in primary and secondary, re- spectively, the current in M is 1^ - I 2 . >-/,) (32) The emf across M is the same as that across L b and C 2 in series, hence Therefore, E i T =coL a ^ (33) The last term is X" ', the reactance of the parallel combination of M with L b and C 2 , which may be shown, as before, to vary with Radio Instruments and Measurements 53 frequency according to the curve marked X" in Fig. 32. Adding to this the curve of L a 77-. the total reactance curve X is coCi obtained (Fig. 37). This curve is the graph of equation (33). E . The value of co at which j- is oo , or the current in the primary a M minimum, is obtained when the denominator of the last term is zero, co(L b +M) 7=r=o. Thus co 2 = . - The symbol co, is used to indicate that this is the value for resonance in the sec- ondary circuit C 2 LbM when the primary circuit is open. The values of co at which the primary current is a maximum are given by equating (33) to o and solving for co. A similar ex- pression involving the secondary current may be treated in the same way, and it is found that the secondary current has maxima at the same values of co as the primary current. Expressed in terms of the inductances and capacities, the solution is rather complicated. It is more convenient to express it in terms of k, the coupling coefficient, and ^ and co 2 , the respective values of co for resonance in the primary circuit C t L a M alone and in the secondary circuit C 2 L b M alone. Using the relations, , M i i (L b +M)' ( 7ZC+H53T- ~V(L b +M)C 2 ' the following two values are found for which the currents have maxima, / V = A V co, 2 + co 2 2 - W - co 2 2 ) f , 2(1 -k 2 ) w - " 2 2 (36) K ) Example. The theory was experimentally verified in the fol- lowing case. Two circuits were direct-coupled as in Fig. 36. The following capacities and inductances were used : C l =0.0023 x io~ 6 farad C 2 =0.00093 X I0 ~ 6 farad Lf& = 56 X iO" 6 henry L b = 209 X iQ- 6 henry M = 241 X io~ 6 henry 54 Circular of the Bureau of Standards The coupling coefficient and the values of w for resonance in the primary circuit alone and in the secondary circuit alone are found by (34) to be =0.659 IO 6 The frequencies for maximum current in the coupled system are found by (35) and (36) to be o/ = i.o37X io 6 eo" = 2.392 X io 6 &OOOCL 1OOO 1O00 20OO FlG. 37. Reactances and current in cas of direct coupling It is thus evident that the effect of coupling is to spread out or separate farther the two independent frequencies. Values of primary reactance were calculated b)^ equation (33) for a number of values of w, giving the curve X of Fig. 37. It will be noted that it crosses the co axis at the values just given for a' and co" ', and that it goes to + and infinity at 1.545 X io 6 , the value of co 2 . Current was produced in the primary circuit by induction from a buzzer, the buzzer circuit being varied to supply different fre- quencies. The coil L a was inductively coupled to the buzzer circuit, the coupling with that circuit being so loose that the emf could be considered as applied at one point of the circuit L Radio Instruments and Measurements 55 Current was measured by a galvanometer and a crystal detector attached to a circuit inductively coupled to the secondary circuit L b MC 2 . The galvanometer deflections were approximately pro- portional to the square of the current. As shown, the curve of observed galvanometer deflections for varying frequency has two maxima corresponding closely to co' and co". The slight discrep- ancies are probably 'due to inaccuracies in the values used for Lj and L 2 ; the inductances of these coils were later found to vary slightly with frequency, whereas a constant value was assumed for each in computing the reactance curve. Special Cases. In the special case when co 1 =co 2 , equations (35) and (36) become ">'=^r (37) (38) When k is very .small; that is, when M is very small compared with L a or L b , co'=co"=co t (39) In this case, where the coupling is very loose, the system responds to only one frequency instead of two, and this is the frequency of resonance of either circuit by itself. When, on the other hand, L a and L b are very 'small compared with M, the coupling is said to be very close, and k approaches the value unity. As k increases to this value, the two frequencies become more widely separated and in the limit '=-:=! (40) -V/2 co" = oo (41) Practically this means that when oj 1 =w 2 and L a and L b are negligible in comparison with M, there is only one frequency and this is given by co' = \~wiir' ^ e reactance curve of such a sys- " tern is of the type showTi in Fig. 30. The curve X crosses the co axis at co', a value less than ^ (called co in the figure), and touches the co axis again at infinity. A particularly interesting special case of direct coupling is that in which co 1 = co z and L l =L 2 . This is obtained when L a =L b and 6 Circular of the Bureau of Standards \ = C 2 . The values of co for maximum current in the primary are (42) co = In this case one of the frequencies is constant, not varying when Lg. is kept constant and M is varied. In the reactance diagram, Fig. 37, the point co" remains fixed, and co' moves farther to the left as M is increased. When M is extremely small, the two fre- quencies are equal, and the equations (42) and (43) reduce to (39). When M is very large in comparison with L a , the equations reduce to (40) and (41). 17. INDUCTIVE COUPLING The applied emf E in the primary of two inductively coupled circuits must satisfy : E = , - coM/ 2 (44) The primary current and reactance can be found by writing down a similar equation for the secondary circuit and solving. This is not necessary, however, as the solution already obtained for direct coupling applies to this case also. Consider L t to be made up of two inductances in series, M and L a , the latter being given by La^^i M. Similarly consider L 2 to consist of two parts in series, M and L b =L 2 -M. and equation (44) becomes E = Then Fig 38 is replaced by Fig. 36, (45) cod or, -M FIG. 38. Circuits involving inductive coupling This is the same as equation (32) for direct-coupled circuits. The two cases are, therefore, equivalent. Radio Instruments and Measurements 57 Equivalent Direct Coupling. Thus, an inductively coupled system may be considered to be replaced by the direct-coupled system of Fig. 36, in which L b =L 2 -M The reactance curves are the same, and the frequencies of maxi- mum current, given by ' and w", (35) and (36), are the same. Equations (34) are more convenient in the following form: M i i = w, : The example given in Fig. 37 was actually for direct-coupled circuits, but corresponds also to a case of inductively coupled circuits in which L! = 297 X io~ 6 henry, L 2 = 450 x io- 6 henry, and C lt C 2 , and M are the same as before. The special cases treated above, in which co 1 = co 2 may be con- sidered as special cases of inductive coupling as well as of direct coupling, except that the last case, where 1^ = 1^, is of no par- ticular interest when the coupling is inductive, because when M is varied L a is not usually kept constant. With inductive coup- ling, M is usually varied by moving the coils with reference to one another, L x and L 2 remaining constant. Example. A test of this theory of inductively coupled circuits was made by a set of measurements upon two circuits arranged as in Fig. 38. The coupling was varied by changing the distance apart of the two coils L t and L 2 . The effect of varying coupling is shown in Fig. 39. As the coils are brought closer together, increasing the coupling, the resonance points co' and co" be- come more widely separated. The constants were as follows: C v = 0.000244 microfarad, C 2 = 0.000098 microfarad, = 103.5 microhenries ! . , L 2 == 246.9 microhenries, M = o.6, 2.0, 5.1, 25.0 microhenries, successively. 58 Circular of the Bureau of Standards The reactance curves were calculated from these data and the preceding formulas. The curves of current squared were plotted from observations of deflections of a galvanometer connected 20 FIG. 39. Effect of "varying the coupling upon reactance and resonance curves for induc- tively coupled circuits to a thermocouple loosely coupled to the secondary circuit, as a function of the frequency of the current which was induced in the primary by coupling loosely to a pliotron circuit. Each Radio Instruments and Measurements 59 mutual inductance was measured by two measurements of the self -inductance of the two coils connected in series, the connections of one coil being reversed for the second measurement. While there are slight discrepancies in the agreement between the points of zero reactance and maximum current, due to slight changes of the inductances with frequency, the agreement is considered very good. Effect of Coupling on Currents. To calculate the current in the coupled circuits requires that account be taken of the resistances. A specially important case is that in which the primary and secondary circuits are both tuned, so as to be separately in resonance with the applied electromotive force, 03 = ^cTvfe (46) Letting ./?! = resistance of primary circuit and R 2 resistance of secondary, it may be shown 4 that FIG. 40. Variation of current 7 2 -with coupling in tuned circuits inductively coupled M in these formulas is supposed to be in henries. For varying values of M the current in the secondary is a maximum when This also holds for maximum current for a variation of co, pro- vided the relations (46) are maintained by variations of the capacities. Other cases of this sort are solved in the reference cited below. 4 4 See reference Nos. 15 and 24, Appendix a. 6o Circular of the Bureau of Standards 18. CAPACITIVE COUPLING The phenomena in a pair of coupled circuits joined by capaci- tive coupling may be shown in a manner similar to the above discussion of difect coupling. Denoting by I and 7 2 the currents in the primary and secondary, respectively, the current in C m is 7 X 7 2 , and coL, cod Therefore, E / I i " =0)^1 k When the coupling is very loose, k approaches o, and The system responds simply to the frequency of resonance of either circuit by itself. When, on the other hand, C m is small in comparison with C a and d,, the coupling is close and in the limit (48) and (49) reduce to co=o 62 Circular of the Bureau of Standards Practically this means that the system responds to only one fre- quency, given by <*>' =\ T r . It should be noted that l =u 2 and L t =L 2 (of course also C a = C b ), As in the similar case of direct coupling, one of the frequencies is constant, not varying when C a is kept constant and C m is varied. More General Cases. The kind of capacitive coupling treated in the foregoing is a simple case of the more general type of capaci- tive coupling shown in Fig. 43. The expressions for coupling 1 3> "^ I> [[ * :> ii 0> I> ^ ^>L. cj r> D - ha z> :> o> r FIG. 43. Generalized case of capacitiue coupling FIG. 44. Special type of case shown in Fiff- 43 coefficient, etc., which are complicated in the general case, are treated by E. Bellini (La Lumiere Electrique, 32, p. 241; 1916). Another simple case which has been found useful is that shown in Fig. 44. For this kind of capacitive coupling, r V(c'+c 3 ) (C"+c 3 ) Here again, in the special case of Wj =co 2 and C' =*C", one of the frequencies is constant, not varying when C' is kept constant and C 3 is varied. For further information on coupled circuits, calculation of the currents, transformation ratios, etc., the reader is referred to Fleming's The Principles of Electric Wave Telegraphy and Te- lephony, Chapter III. 19. CAPACITY OF INDUCTANCE COILS The small capacities between the turns of a coil are of such im- portance in radio design and measurements that a coil can seldom be regarded as a pure inductance. The effect of this distributed Radio Instruments and Measurements capacity is ordinarily negligible at low frequencies, but it modi- fies greatly the behavior of a coil at radio frequencies. For most purposes a coil can be considered as an inductance with a small capacity in parallel as shown in Fig. 45. This fictitious equiva- c< FIG. 45. Circuit which is equivalent to a coil having distributed capacity FIG. 46. Coil having capacity, with emf in series; a case of parallel resonance lent capacity is called the capacity of the coil. Investigations have shown that in ordinary coils its magnitude does not vary with frequency. Thus a coil may in itself constitute a complete oscillating circuit even when the ends of the coil are open. fOOOO 8000 2.000 O S ~~ZO 15 FIG. 47. Reactance diagram for coil having capacity with emf in series Emf in Series with the Coil. If such a coil is placed in a circuit with an electromotive force in series, the case is one of parallel resonance. The reactance curve will be as shown in Fig. 47, which has the same shape as the left branch of the resultant in 35601 18 5 64 Circular of the Bureau of Standards Fig. 28. The right branch is of no interest and is not shown here, because for higher frequencies than co (at which the react- ance becomes infinite) the coil no longer functions as an induct- ance. If the resistance is negligible, the current due to the elec- tromotive force E is - co 2 C<>L T- coL 5oo 400 900 200 JOV i Meters JOO OO 30O 400 FIG. 48. Variation of apparent inductance of a coil with wave length The apparent inductance of the coil, which would be obtained by measurement of the coil as an inductance, is L a in L Ju Comparing with the above expression, L a = T +^c L When 2 C L is small compared with i , this becomes L a = L(i+co 2 C L), (50) Radio Instruments and Measurements 65 at frequencies remote from co , and for C in farads and L in henries. It is usually convenient to calculate the apparent inductance in terms of wave length. (See sec. 78.) Equation (51) becomes where C is in micromicrofarads, L in microhenries, and X in meters. This holds except for wave lengths near that corresponding to BO ieo fro iso Smq of Wavemeter Condenser FIG. 53. Calibration curve of a commercial wave meter with discontinuity caused by distributed capacity of the condenser d is varied, the frequencies to which the circuit responds are changed. For settings in the neighborhood of the natural frequency of the circuit ll'C strong currents are obtained at both the resonant frequencies. For frequencies considerably more remote, the current of only one frequency is appreciable. This behavior is sometimes experimentally found in wave meters. As the setting of the condenser is varied, the frequencies or wave lengths to which the current responds vary, and in the neighborhood of a certain wave length there are two wave lengths at which resonance occurs for every condenser setting. This is shown in Fig. 53, which is an experimentally obtained calibration curve of a commercial wave meter. Radio Instruments and Measurements 20. THE SIMPLE ANTENNA 6 9 Distributed Capacity and Inductance, The current flowing into a condenser is given by I = EwC, and the voltage across an induct- ance is given by E = IuL. Thus the current into a condenser (voltage constant) and the voltage across an inductance (current constant) increase as the frequency increases. Both of these facts tend to make the small capacities between different portions of a circuit more important, the higher the frequency. At low frequencies in general the current at different points in a circuit is the same, and displacement currents are present only where relatively large condensers have been intentionally inserted in the circuit. The inductance and capacity are definitely localized or ' ' lumped. ' ' At very high frequencies, however, or when the dimensions of the circuit are comparable to the wave length, the 'j i> ii i ' ' *\ A / ( M ,( '{ / V if *f >r Vl i ( X A ^ J 1 ' 5) _ : b-- , i . i f&\ ^ fa FlG. 54. Circuit representing distributed capacity and inductance capacities between different parts of the circuit become impor- tant and the current may vary appreciably from point to point in the circuit. Some of the current leaks away from or onto the conductor through the capacity to other parts of the circuit, the current through inductances in different parts of the same circuit will be different, and hence their inductive effect will be different. In such a case the equivalent capacity and inductance of the circuit will depend upon the frequency and the separate condensers and inductances must be considered with regard to their position in the circuit i. e., one has to deal with ' ' distributed ' ' inductance and capacity. Fig. 54 represents a circuit or line of two long parallel wires supplied with current from a generator and closed at the far end. The inductance of the wires and the capacity between them are represented by condensers and inductances drawn in dotted lines. A number of ammeters are, for conven- ience, supposed to be inserted at points in the circuit. At a low yo Circular of the Bureau of Standards frequency very little current will flow into the condensers and all of the ammeters will read the same. If the frequency is in- creased, more current will flow through the condensers and the ammeter readings will decrease successively from the generator to the far end. As a result of the changed distribution of current in the line, the equivalent inductance, capacity, and resistance of the line will vary with the frequency. Simple Antenna. The simplest form of antenna is a single ver- tical wire, the lower end of which is connected to ground. This forms an oscillatory circuit, the inductance is due to the wire and the capacity is that between the wire and the ground. Thus Fig. 55 shows diagrammatically the capacity and inductance and the flow of current at an instant of time. Some of the current from the wire is continually flowing off by the capacity paths to ground, FlG. 55. Distributed capacity and indue- tance in a vertical wire, a simple form of antenna ViS^^^^/^^^W^ FlG. 56. Distribution of current and voltage in the simple antenna when oscillating at its fundamental frequency so that the maximum current is flowing at the base of the antenna, while at the extreme top there is no current flowing in the wire. The amplitude of the voltage alternations is zero at the ground and is a maximum at the top. The distribution of current and volt- age in approximately sinusoidal and is shown in Fig. 56. This represents the fundamental oscillation of a simple antenna. The length of the wire is equal to the distance from node to loop or is one-fourth of the wave length. It is possible for the simple an- tenna to oscillate with other distributions of current and voltage, in which,however, the top must always be a node for current and the bottom a node for voltage. Thus in Fig. 57 is shown the next possible oscillation. Here the length of the wire is three-fourths of a wave length. Hence the wave length is one-third of the fundamental or the frequency is three times as great. Other pos- sible oscillations have frequencies of five, seven, nine, etc., times the fundamental. Antenna with Large End Capacity. Suppose that a number of long horizontal wires are attached to the top of the vertical wire Radio Instruments and Measurements of the simple antenna, thus forming an inverted " L " antenna as in Fig. 58. In this case only a small proportion of the current in the vertical portion flows off to ground through capacity paths, the main capacity flow taking place from the horizontal portion. Thus the current throughout the vertical portion will be very nearly constant. The total capacity will be much larger than that of the simple antenna and the inductance likewise somewhat larger, hence the wave length will be considerably increased. There are a number of other forms of antennas which also have large capacity areas at the top of the vertical lead, such as the "T," "umbrella/' etc. FIG. 57. The distribution of current and -voltage in the simple antenna when oscillating in the first harmonic FIG. 58. Form of antenna of large capacity 21. ANTENNA WITH UNIFORMLY DISTRIBUTED CAPACITY AND INDUCTANCE The mathematical treatment of currents in circuits having dis- tributed capacity and inductance are generally concerned with the case where these quantities are uniformly distributed. Because of end effects, this condition can not be strictly realized except with circuits of infinite length, such as two parallel wires, a single wire .with a concentric cylindrical return, or a single wire (or num- ber of parallel wires), and ground return. Such theory applies approximately, however, to the simple vertical-wire antenna or to the horizontal portion of an inverted " I/' antenna. ,The fol- lowing notation is used in this discussion : Z=Iength of antenna, CD. (Fig. 59), L t =inductance per unit length. C\=capacity per unit length. L =^L 1 inductance for uniform current. C =ZC!=capacity for uniform voltage. L ft =low-frequency inductance of antenna=i/3 L . C a =lovv-frequency capacity of antenna=C . X=reactance of antenna. X L = reactance of loading coil. Xc=reactance of series condenser. L=loading coil. C=series condenser. 7 2 Circular of the Bureau of Standards In Fig. 59 an inverted "I/' antenna is drawn to represent a circuit with uniformly distributed capacity and inductance, the distributed quantities being represented by dotted lines. The resistance is assumed to be negligible. CD is the horizontal por- tion, BE the ground, and BC the lead-in which is supposed to be free from inductance or capacity, excepting when a coil or con- denser or both are inserted at A. Low-Frequency Capacity and Inductance of Antenna. It is desirable to explain the significance of the quantities L t and C lf and the quantities C = /C 1 and L = /L 1 . If the portion CD were uni- formly charged to unit positive potential, then the charge on each unit of length of CD would be numerically equal to C l and the total charge on CD would be C = IC 1 . The antenna would be charged in this way if a constant or slowly alternating emf were introduced at A, and hence the quantity C = lC t is sometimes ii! i'li'.iiliiiii 'i 1 1; vVVVvvvvV i i i 1 i ! i i r i 1 \l V VCoL or ~v~ for different values of y- from which then co, /, or X may be determined. -i^o Table i gives only the lowest value of co-v/C L or TT-, corresponding A to the fundamental oscillation of the loaded antenna. In any actual antenna the wave length would be greater than that given by this calculation because of the inductance and capacity of the vertical portion; this discussion deals only with the horizontal portion. As an example of the method let us assume the quantities used in Figs. 6 1 and 62. Let the length of the antenna be / = 60 meters and the static capacity C = IC V = 0.0008 microfarad. Then since I 60 L = 50 microhenries. 8 c 3 X io 8 In the case of the unloaded antenna, the natural wave lengths would be 4/, 4/3/, etc. that is, 240, 80, etc., meters; the natural (c == "^ x io^\ /= x meters/ WOuld be 1 - 2 5>< lo6 3-75 X io 8 , etc., cycles per second; the periodicities (w==2ir/) 7.85 X io 6 , 23.6 X io a , etc., radians per second agreeing with the values in Fig. 62. If now an inductance L = 100 microhenries is introduced in the lead-in, we have 7- = 2. From Table i we find that L>o _ 27T/ co VCoLo = -y = 0.653, hence 0.653 co= , "^ Vo.oooS X io- 6 X 50 X io- 6 and x X = ^ = 577 meters. 0-653 78 Circular of the Bureau of Standards This corresponds to the lowest frequency of oscillation as shown in Fig. 62. Introducing the inductance has increased the wave length of this oscillation from 240 to 577 meters. The harmonic oscillations are of importance in some cases. If the emf which is applied to the antenna has the fundamental frequency of the loaded antenna, the oscillations will be of that frequency alone. If, however, the antenna is first charged and then set into oscillation by the breaking down of a spark gap in the antenna as in the original Marconi antenna, frequencies corre- sponding to all of the possible modes of oscillation will be emitted FIG. 63. Reactance curve for antenna -with series condenser (Z on curves corresponds to X in text) by the antenna. Or in the case of the arc, which in itself generates fundamental and harmonic frequencies, if a harmonic of the arc coincides somewhat closely with one of the harmonic modes of vibration of the antenna, this oscillation will be strongly rein- forced, a large amount of energy will be wasted, and interference will be caused. Antenna with Series Condenser. Though not as important practically as the case just considered, there are occasions when a condenser is inserted in the lead-in of an antenna to shorten the wave length. If its capacity is C, its reactance will be Xc = ~ In Fig. 63 the cotangent curves representing the aerial reactance X. are again shown as is also the parabola representing Xc. The Radio Instruments and Measurements 79 sum Xc + X is drawn in heavy solid lines and crosses the axis at points corresponding to the natural frequencies of oscillation of the loaded antenna. It will be noted that the frequencies of oscil- lation are increased i. e., the wave length shortened by the insertion of the condenser and that the harmonic oscillations are not integral multiples of the fundamental. Fig. 63 shows that o) is increased from 7.85X10* to 1 0.96X10* or the wave length decreased from 240 to 172 meters by the insertion of a 0.0005 microfarad condenser. Simple Calculation of the Wave Length of a Loaded Antenna. The ordinary formula for the frequency of oscillation of circuits with lumped inductance and capacity may be applied to the antenna with distributed constants in the case of an inductance coil in the lead-in, and the error in computing the frequency or wave length will be small. The inductance and capacity of the aerial at all frequencies is supposed to be the same as the low- frequency values; i. e., -- and C . If the loading coil has an O inductance L, the total inductance will be L -f Hence o ' I the frequency is and the wave length where in the last equation X m means wave length in meters, inductance is expressed in microhenries, and the capacity in microfarads. Applying this to the numerical example worked out above by the exact theory, in which *-*o \ s~* / \ (55) ( L + - ' J = 1 1 6.67 microhenries C =0.0008 microfarads 35601 18 - 6 8o Circular of the Bureau of Standards we have A = 575 meters, which differs only one-third of i per cent from the value A = 577 obtained before. The magnitude of the errors in using the simple formula -1C, is also shown in Table i. In the second column, as pointed out before, are the values of L or -* f r different values of y- A Z-*o as computed from the exact cotangent formula. The simple formula gives > 3 and in the third column are given for comparison the values of w-iJCoLo computed on this basis. These values are too high; i. e., result in too high a value for the frequency or too low a value for the wave length. The per cent error is given in the last column. The maximum error is 10 per cent for L=o i. e., at the funda- mental of the antenna but the error rapidly decreases as L increases and is less than i per cent for L equal to or greater than L . It has been stated in several publications that very large errors would result from applying the ordinary theory of circuits with lumped constants to the case of an antenna which has distributed quantities. This misconception has arisen because the quantity Lo which occurs in the formula for the distributed case was used for the inductance of the aerial in applying the formula for the case of lumped constants. We have pointed out that L could not be the inductance of the aerial at any frequency. When , instead of L , is used the agreement is very close. In fact, o since this error is usually less than i per cent, it is practically never worth while to use formulas based on the precise theory to calculate wave length, because of the uncertainty introduced by the vertical portion of the antenna. Equation (55) is therefore sufficient for all ordinary calculations. Radio Instruments and Measurements TABLE 1. Data for Loaded Antenna Calculations 81 1 Differ- 1 Differ- L Lo uVCoLo VcH ence, per cent L Lo "VCoLo l L + ~\Lo3 ence, per cent 0.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.571 1.429 1.314 1.220 1.142 1.077 1.021 0.973 .931 .894 .860 .831 1.732 1.519 1.369 1.257 1.168 1.095 1.035 0.984 .939 .900 % .866 '.835 10.3 6.3 4.2 3.0 2.3 1.7 1.4 1.1 0.9 .7 .7 .5 5 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.5 5.0 5 5 0.539 .532 .524 .517 .510 .504 .4977 .4916 .4859 .4801 .4548 .4330 4141 0.540 .532 .525 .518 .511 .504 .4979 .4919 .4860 .4804 .4549 .4330 4141 0.1 .1 .1 .1 .1 .0 .0 .0 .0 .0 .0 .0 1 3 779 782 4 6 3974 .3974 1 4 757 760 4 6 5 3826 .3826 1 e 736 739 4 7 3693 .3693 1 6 717 719 3 7 5 3574 .3574 ' 1 7 699 701 3 8 3465 .3465 1 8 683 685 3 8 5 3366 .3366 1 9 668 689 3 9 o 3275 .3275 2 Q 653 655 3 9 5 3189 3189 2 1 640 641 2 10 3111 .3111 2 2 627 628 2 11 2972 2972 2 3 615 616 2 12 2850 .2850 2 4 604 605 2 13 2741 .2741 2 5 593 594 2 14 2644 .2644 2 6 583 584 2 15 2556 2556 2 7 574 574 2 16 2476 2476 2 g 564 565 1 17 2402 2402 2 9 556 556 1 18 2338 .2338 3 547 548 1 19 2277 2277 20 2219 2219 23. ANTENNA CONSTANTS Antenna Resistance. The power supplied to maintain oscilla- tions in an antenna is dissipated in three ways: (i) Radiation; (2) heat, due to conductor resistance; (3) heat, due to dielectric absorption. (At high voltages there is a further power loss due to brush discharge; this will not be considered in the following.) The first of these represents the only useful dissipation of power since it is the power which travels out from the antenna in the form of the electromagnetic waves which transmit the radio signals. The amount of power radiated depends upon the form of the antenna, is proportional to the square of the current flowing at the current antinode of the antenna, and inversely proportional to the square of the wave length of the oscillation. Since the 82 Circular of the Bureau of Standards dissipation of power is proportional to the square of the current, it may be considered to be caused by an equivalent or effective resistance, which is called the radiation resistance of an antenna. Thus the radiation resistance of an antenna is that resistance which, if inserted at the antinode of current in the antenna would dissipate the same power as that radiated by the antenna. The radiation resistance varies with the wave length in the same way as the radiated power; i. e., inversely as the square of the wave length. Curve i of Fig. 64 represents the variation of this com- ponent of the resistance of an antenna. FlG. 64. Variation of antenna resistance with wave length The second source of dissipation of power, that due to ohmic resistance, includes the losses in the resistance of the wires, ground, etc., of the antenna. Due to eddy currents and skin effect in both the wires and ground, this resistance will vary somewhat with the wave length, being greater at shorter wave lengths. But in an actual antenna these changes are so small compared to other variations that we may regard this component of the total antenna resistance to be almost constant, as it is represented by the straight line 2 of Fig. 64. The third source of power dis- sipation i. e., that due to dielectric absorption is a result of the fact that the antenna capacity is an imperfect condenser. The magnitude of this power loss will depend upon the nature and position of imperfect dielectrics in the field of the antenna. Thus it has been found that a tree under an antenna may increase the resistance of the antenna enormously; buildings, wooden masts, and the antenna insulators also affect the absorption of the antenna capacity. It is pointed out in section 34 on con- Radio Instruments and Measurements 83 densers that the effective resistance of an absorbing conden- ser is proportional to the wave length. In the antenna, there- fore, the loss of power due to dielectric absorption may be rep- resented as taking place in a resistance which increases in propor- tion to the wave length. This component of the antenna resist- ance is represented in Fig. 64 by the straight line 3. The curve of the total antenna resistance is obtained by com- bining these three resistance components as in curve 4 of the same figure. This is the typical resistance curve of an antenna. (See also Fig. 91 , p. 1 26.) In the case of some antennas the resist- ance curve shows one or more humps at certain wave lengths. This indicates the presence of circuits with natural periods of oscil- lation in the vicinity of the antenna, possibly the stays of the an- tenna, another antenna, or the metal structure of a building; and the humps indicate that, at the particular wave length at which they occur, these extraneous circuits are in tune with, and are ab- sorbing power from, the antenna. The resistance curve of an an- tenna may be determined by several of the methods of resistance measurement given in sections 47 to 50 below. Measurement of Capacity and Inductance of an Antenna. An antenna is ordinarily used with a series loading coil. In the case of uniform distribution of capacity and inductance, a formula and table have been given (pp. 80 and 8 1 ) which permit the wave length of resonance to be calculated for a given loading coil L when the quantities C and L are known. C is the low-frequency capacity and - ! the low-frequency inductance of the antenna. It has furthermore been shown that the resonance wave length can be calculated with sufficient accuracy from the simple formula appli- cable to a circuit with lumped capacity and inductance. The capacity in the equivalent circuit is taken to be the low-frequency capacity C of the antenna, and the inductance is the sum of the inductance of the loading coil L and the low-frequency induc- tance of the antenna Thus the low-frequency values of an- tenna capacity and inductance are sufficient for wave-length calculations by either formula. These low-frequency values will be called simply the capacity C a and inductance L of the antenna. In terms of the previous notation C =C 1C *^8 *^O '*-' 1 , _L _lL l J-^a, 3 3 84 Circular of the Bureau of Standards and the simple formula for the wave length becomes "c; (56) where inductance is in microhenries and capacity in microfarads. Measurement by the Use of Two Loading Coils. In order to de- termine C a and L a experimentally, two loading coils of different and known values are successively inserted in the antenna and the wave lengths determined for which the antenna is in resonance. This may be done as in Fig. 65, which shows the inserted inductance, S a source of oscillations, and W a stand- FlG. 65. Circuits for determining the ca- pacity and inductance of an antenna ardized wave meter. The wave length of the source is varied until the antenna is in resonance as indicated by the ammeter or other indicating device on the antenna circuit. Then the antenna is detuned and the wave length of the source determined by the wave meter. Two coils L and L ' are inserted and the corresponding wave lengths X and X' are determined. Using the simple equation (56) for lumped capacity and inductance C a ' = i88 4A /(L'+L a ) C a (57) Eliminating Co between these two equations and solving for L a , we obtain r L>\*-L\ /,) ^ X /2 -X 2 V5 ' Radio Instruments and Measurements 85 From the known values of L, L r , X, and X' we obtain, therefore, the value of L a and substituting this value in one of the original equations (preferably the one corresponding to the larger loading coil) , we obtain the value of C a . Example, An illustrative check upon this method will now be given. Let us suppose that the antenna has uniformly distrib- uted capacity and inductance of certain values so that we can compute the wave lengths which would be observed by experi- ment when the loading coils L and L' are inserted. Then, from these wave lengths and the values of L and L' we will compute by the above formula (58) the values of L a and C a , and see how closely they agree with the original values -- and C . We will and capacily in series same way, the current in the circuit under consideration is reduced very rapidly if the resistance is large. A high-frequency current of continuously decreasing amplitude is called an "oscillatory" current, although the term "oscil- latory" is sometimes applied to any current of very high frequency. The decrease of amplitude is called "damping," and the current or wave is called a "damped" current or wave. (In contradistinction to a damped wave, a wave in which the amplitudes do not continuously decrease is called a persistent or sustained wave or oscillation.) The frequency of oscillation in a freely oscillating circuit depends only on the inductance and capac- FIG. 67. Wave train with a logarithmic ity of the circuit, if the resistance decrement of 0.2 j s not very large ^he damping is determined by the resistance together with the inductance and capacity. The resistance is thus important in determining the character of the phenomena. Damped currents are in this respect distinct from those of the sine-wave form. The oscillations which occur in a simple circuit upon which no external alternating emf is applied are called the "free" oscil- lations of the circuit. "Forced" oscillations, on the other hand, 88 Circular of the Bureau of Standards are those impressed on the circuit by an alternating emf from a source outside the circuit. When free oscillations are produced by the sudden discharge of a condenser, all of the energy which was stored in the condenser before discharge is lost from the circuit during the oscillations. The potential difference of the condenser, therefore, becomes lower and lower at every alternation of the current. Since there is no emf applied from outside the circuit, the potential differences of condenser, resistance, and inductance must balance, and their algebraic sum be zero. T di fidt / x L dt +Rl+ c =0 (59) This is the same as equation (18) given above in simple alternating- current theory, except that e, the applied emf, is here equal to o. If the circuit contains a spark gap, the resistance R is not a con- , stant, and the solution given immediately below does not apply. Free oscillations may be produced by another method than the sudden discharge of the condenser in the circuit. If current is produced in the circuit by induction and the inducing emf is suddenly cut off, free oscillations are produced. The quenched gap is the means utilized in practice for suddenly cutting off the inducing emf, the gap being in a circuit closely coupled to the oscillating circuit. The solution of equation (59) for any circuit in which the resistance is not extremely great, is i = I e~ at sin u>t where / is the initial current amplitude, a is the damping factor, and (a is 2ir times frequency of oscillation. The values of these constants are, V being the initial potential difference across the condenser, /o = R , which is not rigorously true when the decrement is very large. Radio Instruments and Measurements 91 equal to TT times the ratio between the resistance and capacitive reactance, 7 for co = p= or coL = and therefore VLC coC Thus the logarithmic decrement is TT times the reciprocal of the sharpness of resonance (discussed above on p. 36). The logarithmic decrement expressed in terms of the three quantities resistance, inductance, and capacity is 8 Interpretations of Logarithmic Decrement. The decrement has been given in terms of (i) a ratio of current amplitudes, (2) a ratio of impedance components, and (3) the reciprocal of the quantity called sharpness of resonance. A fourth interesting interpretation is in terms of an energy ratio. It is readily shown that 8 = ^2 the ratio of the average energy dissipated per cycle to the average magnetic energy at the current maxima, as follows: RP -^-JL- JL where / is the effective current as measured by an ammeter. -j- = average energy dissipated per cycle, since RP = average energy dissipated per second. LP = average (L/ k 2 ) (where 7 k 2 = average current square during the k 'th cycle) , since P = average / k 2 , (I 2 \ L - J, (where / max = maximum current during the fc'th cycle), since 7 k 2 = /4/ max 2 , just as in the case of undamped currents. / I 2 \ But ( L max j is the magnetic energy at the current maximum during the k 'th cycle. Therefore LP = average magnetic energy at the current maxima. Hence the decrement is one-naif the ratio of the average energy dissipated per cycle to the average magnetic energy at the current maxima. This is true when the energy is lost from the circuit by radiation as well as when lost by heating. 7 Reactance is ordinarily calculated from the simple expression for w, which is not rigorously true when the decrement is very large. 8 This expression for the decrement does not hold when the decrement is extremely large. 92 Circular of the Bureau of Standards Number of Oscillations in a Wave Train. There are theoret- ically an infinite number of oscillations in a wave train. In prac- tice, however, the wave train may be considered ended when the oscillations are reduced to a negligible amplitude. The fraction of the initial amplitude that is considered negligible depends on the use to which the oscillatory current is put. For a given ratio of initial amplitude to final amplitude, -p, the number of complete *n oscillations n is given by /o log* - and the number of maxima or of semioscillations /o 2 lo 5 The oscillations after the amplitude is reduced to o.oi of its initial value can usually be considered negligible. For -2 = 100, the * n number of oscillations = 4. For example, the number of oscil- 5 lations in which a current having 5 = 0.2 falls off to i per cent of its initial value is equal to 23. 26. PRINCIPLES OF DECREMENT MEASUREMENT A number of so-called methods of measuring decrement are in reality measurements of resistance. From the resistance the D [7* logarithmic decrement of a circuit is calculated by 5== j=r- or one of the related formulas. Any method for measuring the re- sistance of a circuit thus enables one to calculate the decrement. The value so obtained is the decrement of the current that would flow in the circuit if free oscillations were suddenly started, but not in general the decrement of the current used in making the measurement. The methods available for such resistance meas- urements are summarized below in sections 47 to 50. Only those methods in which damped oscillations are used can be considered actual measurements of decrement. .There are two classes of genuine decrement measurement, one in which free oscillations are used and one in which a damped electromotive force is impressed on the circuit so that both free and forced oscillations contribute to the current. Free oscilla- Radio Instruments and Measurements 93 tions are obtained in the case of pure impulse excitation. It is very difficult in practice to obtain such excitation. Assuming, however, that free oscillations are produced by the sudden dis- charge of the condenser at a constant potential difference N times per second, the effective current is given by equation (60), Since 4" co5 8> p This may be written, 7 2 5 = constant. Let / be the current for a certain resistance in the circuit, and I v be the current when a resistance R^ is added so as to increase the decrement by an amount 8 t . 7 2 5=7 1 2 (5 + 5 1 ) 5 = 5 iFZ71 (61) Bjerknes Methods. A method of measuring the high-frequency resistance of a circuit using undamped currents has been described on page 38 above and others are described in sections 49 and 50 R FIG. 69. Inductively coupled circuits for decrement measurements below. These methods have been extended to the measurement of resistance and decrement by the use of damped waves. When a damped emf is impressed on the circuit both free and forced oscillations exist, and the measurement is an actual measurement of decrement. For a circuit 77 (Fig. 69) very loosely coupled to a circuit 7 in which oscillations are generated, Bjerknes 9 showed that when the two circuits are in resonance, JVEo 2 i6L 2 a'a(a' 9 See reference No. 42, Appendix 2. 94 Circular of the Bureau of Standards where E is the maximum value of the impressed electromotive force, N the number of trains of waves per second, a.' the damping factor of the emf due to circuit / impressed on circuit II, and a the damping factor of the second circuit. The circuit / may be a great distance from II, and may even be a distant radiating antenna. The equation holds only when a' and a are small in comparison with iL, ^ L =L * The mean based on several values should be obtained. In the above comparisons the leads of the coils have been con- sidered as a part of the coil, contributing both to the inductance and capacity of the whole. These leads should be fixed and definite and should be of sufficient length and otherwise designed so that the coil constants will not be appreciably altered on account of eddy currents in the condenser case or capacity to it, when the coil is connected to the condenser to form a wave meter. 30. STANDARDIZATION OF A COMMERCIAL WAVE METER A commercial wave meter is generally equipped with one or more indicating devices, of which the hot-wire ammeter and crystal detector with phones are especially important. In addition, it is customary to provide a buzzer circuit, so that oscillations of known wave length may be generated by the wave meter. The indications of the wave meter will be some- what different, depending upon the way it is operated that is, whether one or the other of the indicating devices is used or whether it is used as a source of oscillations. Hence, in each of these cases it will usually be necessary to have a separate cali- bration. Calibration of wave meter used as a source is treated on page 1 08 below. Wave Meter with Ammeter. When the hot-wire ammeter is in use, it is either inserted in the circuit directly, when, on account of its high resistance, it is generally shunted by a small inductance, or it may be tapped across a number of turns of the inductance coil. The calibration is effected by comparing the wave meter with the standard wave meter in a manner similar to that outlined above for intercomparing the standard wave-meter circuits. The condenser of the commercial instrument is set at a given reading Radio Instruments and Measurements 105 and the wave length of the source adjusted until the ammeter of the wave meter indicates maximum current. The wave meter is then detuned, and without changing the source the standard wave meter is adjusted until resonance is obtained. The wave length as indicated by the standard corresponds to the chosen setting of the commercial wave meter, and, repeating the obser- vations for other settings, a curve may be obtained giving the wave length as a function of the setting. Or if it is desired to engrave the scale of the commercial instrument so as to read wave lengths directly, the standard circuit may be set at a chosen integral wave length, the source adjusted to this wave length, and the corresponding setting of the commercial instrument found and PELATIVE. AUDIBILITY. .85 55 FIG. 72. Various wave-meter circuits, using detector and phones marked. In these comparisons a source of either damped or undamped oscillations may be used. The latter, however, per- mits a higher precision in the measurement on account of the sharper tuning. The ammeter of the commercial wave meter can only be used when it is possible to draw a considerable amount of power from the source. It is practically indispensable when the wave meter is to be used for measurements of resistance or loga- rithmic decrement. Use of Crystal Detector. When the source supplies only a small amount of power, it is necessary to use a sensitive indicator, such as a crystal detector and phones. When such a detecting circuit is connected or coupled to the wave-meter circuit, the wave io6 Circular of the Bureau of Standards length calibration and the resistance of the wave meter will be changed somewhat, depending upon the type of detecting circuit. The changes will also depend to some extent upon the adjust- FlG. 73. Increase of wave length for different condenser settings due to the addition of detector circuit ment of the crystal contact, so that it is important in the design of a wave meter to choose a detecting circuit which will least 0.150 FlG. 74. Increase in decrement of the wave-meter circuit due to the detector circuit affect the wave-meter constants. The wave meter should then be calibrated with the detecting circuit connected as for use. Circuit o in Fig. 72 represents the wave meter without detector, Radio Instruments and Measurements 107 while the circuits numbered 1-6 show the detecting circuits frequently used. Typical examples of the effects of these circuits are shown in Figs. 73 and 74. Fig. 73 illustrates the increased wave length for different condenser settings caused by the addition of the detector circuit. The increase is extremely small in the case of circuits 4 and 6 i. e., when the detecting circuit is connected to the wave-meter circuit at one point or when it is loosely coupled FlG. 75. Arrangement of circuits for compar- ing -wave meters by impact excitation to the wave-meter circuit. In Fig. 74 the effect of the detector circuit in increasing the decrement (or resistance), and hence in impairing the sharpness of tuning of the wave meter is shown, and here again the circuits 4 and 6 appear to produce the least effect. As shown by the relative audibility values in Fig. 72, these circuits are not as sensitive as those which withdraw more power from the main circuit. FlG. 76. Arrangement of circuits for deter- mining resonance by coupling to an aperi- odic detector circuit Use of Buzzer. In order that an audible note may be heard in the phone, the calibration must be carried out using either damped oscillations with a wave-train frequency that is audible or un- damped oscillations that are interrupted or "chopped" at an audible frequency. A simple and accurate method of comparison is that shown in Fig. 75. Here circuit / is the buzzer circuit described on page 227, which excites the standard wave-meter cir- io8 Circular of the Bureau of Standards cuit // by impact excitation. The commercial wave meter /// is loosely coupled to the standard, and resonance is indicated by the setting for maximum response in the phones. Sharpness in setting is facilitated by reducing the coupling between // and /// until the phone responds only when /// is very nearly in resonance with //. The standard circuit has no detector or buzzer attached ; hence, its calibration is unaffected if it is sufficiently loosely coupled to / and ///. The buzzer circuit is generally not strictly aperiodic and will show a very broad tuning at its natural fre- quency. It is necessary to use it at frequencies differing con- siderably from its natural frequency or errors will be introduced. FIG. 77. Arrangement of circuits for com- paring a wave meter with a standard buzzer circuit, the resonance point being indicated by the aperiodic detector circuit Another method makes use of a tuned buzzer source, and each circuit is separately tuned to the source in the same manner as described above for the comparison when ammeters are used. Resonance in the standard circuit is indicated by an aperiodic detector circuit loosely coupled to that circuit as indicated in Fig. 76. When the commercial wave meter is used as a source, the buzzer is usually connected as shown in Fig. 77, circuit /. The leads to the buzzer will add capacity to the circuit and the lengths of the waves emitted will be increased, in particular at low condenser settings. The calibration is simply carried out by the circuits of Fig. 77. Circuit // is the standard wave meter and /// an aperiodic detector circuit loosely coupled to the standard cir- cuit. The coil of circuit /// may be so oriented as not to be directly affected by circuit /. Then either circuit // may be set Radio Instruments and Measurements 109 at integral wave lengths and the settings found at which these wave lengths are emitted by /, or / is set and the wave length cor- responding to the setting is found by tuning //. CONDENSERS 31. GENERAL A condenser is an apparatus so designed that electrostatic capacity is its important property. It consists of a pair of con- ductors with their surfaces relatively close together, separated by an insulating medium called the dielectric. When the two con- ducting plates are parallel, close together, and of large area, the capacity of a condenser is given by C =0.0885 Xio where C is in microfarads, S = area of one side of one conducting plate in cm 2 , r = thickness of dielectric between the plates in centimeters, and K, the dielectric constant, = i for air and is be- tween i and 10 for most ordinary substances. Formulas for capacities of various combinations of conductors, antennas, etc., are given on pages 237 to 242. These formulas assume that the charge is uniformly distributed over the surfaces of the conductors, no corrections being made for edges or end effects. It is seldom worth while, however, to apply a correction on this account, because the capacity to the condenser case or other conductors is ordinarily not calculable, so that the actual capacity of a con- denser can be calculated only approximately. The actual value is likely to be in excess of that calculated. When very accurate values are required they must be obtained by measurement. The usual methods for measuring capacities at radio frequencies are discussed on pages 129 to 131. Series and Parallel Connection. When two or more condensers are connected in series, the resultant capacity is given by -- C CC 2 Cj The resultant capacity of a number of condensers connected in series is always less than the smallest capacity in the series. The series connection is used when it is necessary to use a voltage higher than a single condenser would stand without breakdown. no Circular of the Bureau of Standards When condensers are connected in parallel, their capacities are simply added, thus : The laws of series and parallel combination of condensers are thus the inverse of the laws for resistances. In combining condensers, care must be exercised that there are no appreciable mutual capacities between the parts combined. Stray Capacities. It is very difficult to concentrate the total capacity in a radio circuit at a particular point in the circuit. Every part of the apparatus has capacities to other parts, and these small stray capacities may have to be taken into account as well as the capacity of the condenser which is intentionally inserted in the circuit. The stray capacities are particularly objectionable because they vary when parts of the circuit or con- ductors near by are moved. Thus, they make it difficult to keep the capacity of the circuit constant. The disturbing effects may be minimized in practice, as follows: (i) Keeping the condenser a considerable distance away from conducting or dielectric masses; (2) shielding the condenser, i. e., surrounding the whole condenser by a metal covering connected to one plate; (3) using a condenser of sufficiently large capacity so that the stray capacities are negligible in comparison. The first of these methods reduces only the stray capacities of the condenser itself to other parts of the circuit. This is also true of the second method, which is none the less a desirable precaution. One of the chief causes of variation in the stray capacities is the presence of the hand or body of the operator near some point of the circuit. Shielding the condenser reduces the capacity variation from this cause. The third method is, in general, the best for reducing or eliminating these errors. On account of the stray capacities of its various parts, the whole circuit is in effect a part of the condenser, and their effect is best rendered negligible by making the condenser capacity relatively great. Imperfection of Condensers. In an ideal condenser, the con- ductors or plates would have zero resistance and the dielectric infinite resistivity in all its parts. In case an alternating emf is applied to such a condenser, current will flow into the condenser as the voltage is increasing and flow out as the voltage is decreasing. At the moment when the emf is a maximum no current will be flowing, and when the emf is zero the current will be a maxi- mum. Hence, in a perfect condenser the current and voltage are 90 out of phase. In actual condensers the conditions as to Radio Instruments and Measurements in resistance in the plates and dielectric are not fulfilled, and in con- sequence an alternating current flowing in a condenser is not exactly 90 out of phase with the impressed voltage. The differ- ence between 90 and -the actual phase angle is called the "phase difference. " In an ideal condenser there would be no consumption of power; the existence of a phase difference means a power loss, which appears as a production of heat in the condenser. The amount of the power loss is given, as for any part of a circuit, by P = EI cos 6, where is the phase angle between current and voltage and cos 6 is the power factor. This is equivalent to where ^ is the phase difference and sin $ is the power factor of the condenser. In all except extremely poor condensers, \l/ is small, sin \{/ = \}/, and thus the phase difference and power factor are synonymous. The power loss is given by P = coC 2 sin^ (68) This shows that, for constant voltage, the power loss is propor- tional to the frequency, to the capacity, and to the power factor. Information on the power factors of condensers is given in section (34) below. Change of Capacity with Frequency. Another effect of the im- perfection of dielectrics is a change of capacity with frequency. The quantity of electricity which flows into a condenser during any finite charging period is greater than would flow in during an infinitely short charging period. In consequence the measured or apparent capacity with alternating current of any finite fre- quency is greater than the capacity on infinite frequency. The latter is called the geometric capacity (being the capacity that would be calculated from the geometric dimensions of the con- denser on the assumption of perfect dielectric) . The capacity of a condenser decreases as the frequency is increased, approaching the geometric capacity at extremely high frequencies. For this reason, when dielectric constants are measured at high frequencies of charge and discharge, smaller values are obtained than with low frequencies. When the phase difference of the condenser is due to ordinary leakage or conduction through the dielectric or along its surface, the apparent capacity at any frequency is readily shown to be 35601 18 H2 Circular of the Bureau of Standards where C is the geometric capacity in microfarads, $ the phase difference, and R the leakage resistance in ohms. It is evident that the apparent capacity decreases very rapidly as frequency increases. For example, suppose a condenser whose geometric capacity is o.ooi microfarad to have a leakage resistance of 10 megohms, the dielectric being otherwise perfect. Its capacity at 60 cycles will be 0.001070 microfarad, at 300 cycles will be 0.001003 microfarad, and at all radio frequencies will be equal to the geometric capacity. When the phase difference of a condenser is due to dielectric absorption (a phenomenon discussed below, p. 124), the capacity decreases as the frequency increases, as before, approaching the geometric capacity at infinite frequency, but the amount of the change can not be predicted from a knowledge of the phase dif- ference. The change with frequency is large in condensers that have large phase difference. In certain cases the change of capac- ity with frequency has been found to be roughly proportional to the reciprocal of the square root of frequency. A series resistance in the plates or leads of a condenser causes a phase difference but does not give rise to a change of capacity with frequency. When the leads inside the case of a condenser are long enough to have appreciable inductance the capacity measured at the terminals appears to be greater than it actually is. The magnitude of the effect is given by where C a is the apparent or measured capacity, and in the paren- thesis C is in microfarads and L in microhenries. Thus, the inductance of the interior leads makes the apparent capacity of a condenser increase as frequency increases, while the imperfection of the dielectric makes the capacity decrease with increase of frequency. 32. AIR CONDENSERS Electrical condensers are classified according to their dielectrics. The plates are relatively unimportant, their only requirement being low resistance. This requirement is met in the materials used for condenser plates, viz, aluminum, copper, brass. When the plates are thin the material must not have too high a resistivity. Various dielectrics are used; the one most frequently used in radio measurements is air. Radio Instruments and Measurements 113 A condenser which is to be used as a standard of capacity in measurements at radio frequencies is itself standardized at low frequencies, and its construction must be such that either the capac- ity does not change with frequency or the change can be calcu- lated. The capacity of a condenser with a solid dielectric changes with the frequency in an indeterminate manner, and hence it is practically impossible to calculate the capacity at high frequencies from that measured at low frequency. Air is very nearly a perfect dielectric, hence a condenser with only air as a dielectric should show no change in capacity with the frequency, and thus the capacity at radio frequencies should be the same as that for low frequency. It is on this account that air condensers are quite generally used as standards of capacity in radio measurements. Phase Difference of Air Condensers. Air condensers are valuable in radio measurements for another reason. Their perfection, from a dielectric standpoint, involves freedom from power loss. The phase difference of an ideal air condenser is zero; there is no component of current in phase with the electromotive force, and thus the condenser acts as a pure capacity and introduces no resistance into the circuit. It is consequently advantageous to use them in circuits in which it is desirable to keep the resistance very low. In resistance measurements at high frequencies it is often necessary to assume that the resistance of the condenser in the circuit is negligible. This requires the use of a properly con- structed air condenser. Only the most careful design, however, can produce an air condenser which is close to perfection. In order to support the two conductors or sets of plates and insulate them from each other it is necessary to introduce some solid dielectric. There is necessarily some capacity through this dielectric, and since all solid insulators are imperfect dielectrics this introduces a phase difference. The magnitude of the phase difference is determined by the quality of the solid dielectric and the relative capacities through this dielectric and through the air. The effect is magni- fied by the concentration of the lines of electric field intensity in solid dielectric due to its high dielectric constant. Some air con- densers tested by this Bureau, in which the pieces of dielectric used as insulators were large and poorly located, had phase differ- ences or power factors many times greater than those of commercial paper condensers. Although purporting to be air condensers they were actually poorer than ordinary solid-dielectric condensers ii4 Circular of the Bureau of Standards because the insulating pieces used to separate the plates were very poor dielectrics. In a variable air condenser the phase difference varies with the setting and is approximately inversely proportional to the capacity at any setting. The equivalent resistance (defined on p. 125) is inversely proportional to the square of capacity at any setting. The questions of materials and construction of air condensers are further dealt with below in connection with design. It is seldom safe to assume that an air condenser has zero phase difference. Antennas are subject to this same imperfection. An antenna is essentially an air condenser and is similarly subject to power loss from poor dielectrics u in its field. Simple Variable Condenser. In the most generally used types of air condensers the capacity is continuously variable. Variable condensers are extensively used, because most radio measure- ments involve a variation of either inductance or capacity, and it is relatively difficult to secure sufficient variation of an induct- ance without variation of the resistance and capacity in the cir- cuit. The most familiar type of variable air condenser has two sets of semicircular plates (see Fig. 78 , facing p. 1 1 8) , one set of which can be revolved, bringing the plates in or out from between the plates of the fixed set. The position of the movable plates is indicated by a pointer moving over a scale which is marked off in arbitrary divisions. These may be degrees, o corresponding to the posi- tion of the plates when they are completely outside of the fixed set and the capacity is a minimum, and 180 when the capacity is a maximum. It is preferable to divide the range into 100 divisions rather than into degrees. The capacity of a condenser is proportional to the area of the plates. In a variable condenser of the semicircular type the effective area of the plates is changed by rotating the movable plates and, neglecting the edge effects, it is changed in proportion to the angle of rotation. As a result the capacity is approximately proportional to the setting through- out a wide range, provided that the condenser is well constructed and the distance between the two sets of plates is not affected by rotation of the movable set. Fig. 79 shows a typical capacity curve for such a condenser. Throughout the range in which the capacity curve is a straight line the capacity is given by the formula 11 See reference No. 197, Appendix 2. Radio Instruments and Measurements where a and b are constants and 6 is the setting in scale divisions. The coefficient a represents the change in capacity for one division and may be computed by taking the difference of the capacities at, say, 20 and 80 and dividing by 60. The constant b represents the capacity which the condenser would have at o if its linear character were maintained down to this setting. It may be positive, negative, or zero, depending upon the setting of the pointer relative to the movable plates. Its value is determined by subtracting the value of aff at, say, 30 from the actual value of the capacity at that setting. When a condenser has a capacity curve that is closely linear, it may be found easier to compute 2800 2600 240O 2200 2000 (800 1600 MOO I2OO toco / / A / / / / / < / < / 5 a / 2 / / COO / / / / o y Sc< -r r VISIO *s 10 20 JO 4O iO 60 70 80 90 ICO FIG. 79. Typical capacity curve for con- denser with semicircular plates the capacity for given settings by means of the formula rather than to read off the values from a curve. Uniform Wave-Length Type. Some condensers have been spe- cially designed to give a capacity curve different from the cus- tomary linear curve. In a wave meter in which a semicircular plate condenser is used, for any one coil the wave length varies as the square root of the capacity and hence as the square root of the setting of the condenser. If it is attempted to make the wave meter direct reading i. e., to substitute a wave-length scale in place of the scale of setting in degrees this wave-length scale will be nonuniform and either crowded together too closely at the low settings or too open at the high settings. In order to obtain a uniform scale of wave lengths, it is necessary to have n6 Circular of the Bureau of Standards the capacity vary as the square of the displacement or rotation. Tissot 12 proposed a condenser which had two sets of square plates (Fig. 80) which moved relative to each other along their diagonals. The same result can be obtained in a rotary con- denser if one set of plates is given the proper shape. 13 It is required that the capacity and, hence, the effect- ive area between the plates shall vary as the square of the angle of rotation. Thus, FIG. 80. Form of condenser plates for which the ca- pacity varies as the square of the dis- placement But in polar coordinates the area is equal to -!/ . 2 Differentiating these two values of A, dA i r = In the condenser as actually made the fixed plates may be semicircular and the moving plates given the required shape to FIG. 81. Form of rotary condenser plates for which the capacity varies as the square of angular displacement make the effective area vary as the square of the angle of rotation. This effective area is the projection of the moving plates on the fixed. To provide clearance for the shaft of the moving-plate system a circular area of radius r 2 must be cut from the fixed 12 See reference No. 81, Appendix 2. 13 See reference No. 82, Appendix 2. Radio Instruments and Measurements 117 plates, and, taking this into account, the equation of the boundary curve of the moving plates becomes In Fig. 8 1 the form of the plates is shown and the effective area denoted by shading. Decremeter Type. Another special shape of plates is utilized in the direct-reading decremeter 14 developed at this Bureau. As shown in section 55, logarithmic decrement may be measured by the per cent change in capacity required to reduce by a certain amount the indication of an instrument in the circuit at resonance. In order that equal angular rotations may correspond to the same decrement at any setting of the condenser, it is necessary that the per cent change in capacity for a given rotation shall be the same at all parts of the scale. Thus we have the requirement j/"* a = constant = per cent change of capacity per scale division. By integration, log C = ad + b where b = a constant, or C = FIG. 82 . Form of rotary condenser plates in which the per cent change of capacity is the same throughout the entire range of the condenser C = e 6 = capacity when 6 = O. Since the area must vary as the capacity A=- 2 = ~ = r <* dd~ 2 ~ L ae r = J'2Crfrt* (69) M See reference No. 196, Appendix 2, and description of decremeter on p. 199- 1 1 8 Circular of the Bureau of Standards This latter is then the polar equation of the bounding curve required to give a uniform decrement scale. The shape of the condenser, as actually made, is shown in Fig. 82 and Fig. 218, facing page 320. A small semicircular area is omitted from the fixed plates, to provide clearance for the metal washers which must hold the moving plates together. Taking account of this omitted area, of radius r 2 , the equation of the boundary curve becomes r = The effective area is denoted by shading in Fig. 82. A wave-length scale placed on such a condenser is somewhat crowded at high settings just the opposite of the effect with a semicircular plate condenser. It is much more nearly uniform than in the case of a semicircular-plate condenser, as might be expected from the similarity of shape of Figs. 81 and 82. Important Points in Design. In a standard condenser it is required that the capacity remain constant and be definite. The former condition requires rigidity of construction, which is diffi- cult to secure in a variable condenser. The pointer and movable plates must be securely fastened to the shaft so that no relative motion is possible. A simple set screw is not sufficient to hold the pointer in place. It is preferable to have no stops against which the pointer may hit. Particular care must be exercised in insulating the fixed and moving plates from each other. The sus- pension of heavy sets of plates from a material such as hard rubber, which may warp, is objectionable. In some cases the high temperature coefficient of expansion of insulators may pro- duce relative motions of the two sets of plates, resulting in a high temperature coefficient of capacity. In order to make the capac- ity definite and also minimize power loss, it is desirable to sur- round the condenser with a metal case which is connected to one set of plates and grounded when the condenser is calibrated and used. The inductance of the leads inside of the condenser should be a minimum, for the apparent capacity of a condenser at high frequencies will increase with the frequency due to induc- tance in the leads in a similar manner to the variation of apparent inductance of a coil with distributed capacity. The connections from the binding posts to the plates should therefore be short and thick; this minimizes both inductance and resistance. Bureau of Standards Circular No. 74 FlG. 78. Commercial types of variable air condensers FIG. 84. Quartz-pillar standard condenser of FIG. 85. Type of -variable con- fixed value denser suitable for high voltages, with double set of semicircular moving plates Bureau of Standards Circular No. 74 FIG. 86. Leyden jar type of high voltage condenser FIG. 87. Mica condensers suitable for high voltages Radio Instruments and Measurements 119 The resistance of leads and plates and contact resistances between the individual plates and separating washers should be kept as low as possible, in order to minimize the phase angle. For the same reason the dielectric used to support one set of plates and insulate the two sets from each other should be as nearly perfect as possible. The insulation resistance must be high, because otherwise it would introduce an error in the meas- ured capacity at low frequencies (see p. in). Power loss in the ELEVATION SHOWING CASE BROKEN AWAV FIG. 83. Quartz-pillar variable condenser used as primary standard of capacity at high frequencies dielectric should be kept small by using suitable insulating ma- terial, locating it where the electric field intensity is not great, and using pieces of such size and shape that the capacity through it is small. Built-up mica, bakelite, formica, and similar materials have been found to have bad dielectric losses. Hard rubber and porcelain are better. For further information on dielectrics, see section 34 below, on Power Factor. In special cases quartz may be used, as in the special condensers next described. 1 20 Circular of the Bureau of Standards Bureau of Standards Type. In the customary design of variable air condensers, the movable plates are insulated from the fixed by rings of insulating material around the bearings at top and bottom. This introduces some difficulties in the design of the bearing; the capacity through the dielectric is likely to be large and the choice of insulator is limited. In the condensers used as standards at this Bureau, instead of the moving plates being insulated from the case of the condenser, they are in electrical connection with the case and the bearings are metal. The fixed plates are insulated from the moving plates and the case by pieces of quartz rod inserted in the uprights which support the fixed plates. The capacity through the insulator is small. The use of quartz is desirable on account of very high insulation and low temperature coefficient of expansion. The condensers of this type which have been tested have shown per- manence, no measurable power loss, and a very low temperature coefficient of capacity. On account of the brittleness of the quartz and the heavy weight supported by it, these condensers are not very portable and must be dismounted in shipping. Views of the inside and the case of these condensers are shown in Fig. 214, facing page 318. Two sizes, of 0.007 and 0.0035 microfarad capacity, are shown. Data on the construction are given in Fig. 83. Fixed Condensers. Air condensers in which the two sets of plates are fixed relative to each other are particularly valuable when well constructed on account of their permanence. They serve to standardize variable air condensers, which are more likely to change, and are of value in standardizing solid-dielectric con- densers. In Fig. 84, facing page 1 18, is shown a type designed at this Bureau, of o.oi microfarad capacity, which has two sets of square plates and which is insulated by quartz supports in a man- ner similar to the variable condensers just described. Great care is necessary in the construction and maintenance of these con- densers. The brass plates must be thoroughly annealed; the air must be kept quite dry. 33. POWER CONDENSERS The condensers used in radio transmitting circuits carry large amounts of power and are called power condensers. They are operated at high voltages, and this requires special construction. The voltage required in transmitter use may be found as follows : Radio Instruments and Measurements 121 The power input is measured by the energy stored in the capacity at each charge multiplied by the number of charges or discharges per second. Thus P = y^CE^N where N is the number of discharges per second, or the spark frequency. 2 p Hence E Q 2 = - In a typical case N = 1000, C = o.oi6 X io~ 6 farad and P = 2ooo watts; then / 2 (2i '~ V 1000(0. = 1 6000 volts. 016)10' Air and Oil Condensers. The foregoing indicates the order of the voltage requirement for power condensers. In the case of air condensers of ordinary construction this high voltage would neces- sitate very large spacing between the plates, and this, in turn, would necessitate a very big volume for a moderate capacity. If, however, the air condensers have a strong air-tight case, so that the air inside the case can be compressed up to 15 or 20 atmospheres, the breakdown voltage becomes very high, even for a small distance between the plates. Thus voltages as high as 35 ooo may be used with 3 -mm spacing between the plates. At the same time the brush discharge losses are reduced so as to be negligible. Such condensers have the advantage of low power loss, but the disadvantage of being quite bulky. A size com- monly used has a capacity of about 0.005 microfarad. Another way of utilizing the air condenser type for high voltages is to fill the condenser with oil. Using ordinary petroleum oils, this about doubles the capacity. Condensers with oil as a dielectric are very well suited for power condensers. The breakdown voltage is very high, dielectric and brush losses low, and on account of the high dielectric constant of some oils, it is easy to get a large capacity in a moderate volume. A variable condenser suitable for oil immersion is shown in Fig. 85, facing page 118. Only condensers with liquid or gas dielectrics can have continuously variable capacity. This is a great advantage, and such a condenser as this is a most valuable supplement to the fixed condensers of solid dielectric used in a high-voltage circuit, since it makes possible easy variation of the wave length. Glass Condensers. Condensers with glass for a dielectric, and especially the cylindrical Leyden jars with copper coatings, are very extensively used for power condensers. Special types have 122 Circular of the Bureau of Standards been devised to increase breakdown voltage and reduce brush discharge. This is accomplished with the ordinary types by immersion in oil, which also improves the insulation. Glass con- densers are relatively cheap. They must, of course, be carefully handled. A stock size of Ley den jar has a capacity of 0.002 microfarad. The power factor of glass condensers is rather large (see next section). When used in the primary circuit of a quenched gap transmitter, however, moderate power loss in the condensers is probably not important in reducing the efficiency of the set since the power losses in the rest of the circuit are high and the circuit is operative, if proper quenching is secured, for but a small fraction of the time. (See Fig. 86, facing p. 119.) Mica Condensers. These are coming into use to a considerable extent in radio work, both as power condensers and as standards. They have the advantages of low power loss, small volume, and are not fragile. Stock sizes include 0.002 and 0.004 microfarad. (See Fig. 87, facing p. 119.) In order to withstand high voltages, the mica sheets are comparatively thick and several sections are joined in series. When they are properly made and have very small phase difference, they may be used as standard condensers ; and very conveniently supplement the air condensers ordinarily used as standards, being obtainable in larger capacities. They are valuable as standards both on account of their permanence and the large capacity obtainable in a small volume, but must be standard- ized both in respect to capacity and power factor for the range of frequencies at which they are to be used. While the capacity and power factor may be considerably different at high frequencies from the values at low frequency, it is fortunate that throughout the range of radio frequencies both of these quantities are practically con- stant. An exception must be made in the case of the power factor if any considerable portion of the power loss is due to ohmic resistance in the leads or plates; for under these circum- stances, the power factor will increase with increasing frequency, as pointed out below. In properly constructed condensers, how- ever, power loss from this source is negligible. 34. POWER FACTOR The power loss in a condenser may be due either to imperfection of the dielectric or to resistance in the metal plates or leads. The dielectric may cause a power loss either by current leakage, by brush discharge, or more commonly by the phenomenon described later under the head of " Dielectric absorption." Radio Instruments and Measurements 123 Leakage. The leakage of electricity by ordinary conduction through the dielectric or along its surface contributes to the phase difference at low frequencies but is generally negligible at high frequencies. The effect of leakage on the power factor may be seen as follows: A condenser having leakage may be represented by a pure capacity with a resistance in parallel. The current divides between the two branches, the current /R through the resistance being in phase with the applied E, and the current Io through the capacity leading E by 90. The resultant / leads E by an angle which is less than 90 by the phase difference \f/. From Fig. 88, T tan $ = RwC The effect of R may be shown by an example. A condenser of o.oi microfarad capacity with an insulation resistance as low R AAAAMM FIG. 88. Equivalent circuit and -vector diagram for condenser having leakage as 10 megohms has, at a frequency of 60 cycles per second, a power factor = - r = 0.027 = 2.7 per cent. This is a (io) 7 377 (10)-" very appreciable quantity; 2.7 per cent of the current flows by conduction instead of by dielectric displacement. This effect, however, decreases as the frequency increases, for the dielectric current increases in proportion to the frequency while the leak- age current does not, For instance, at io ooo cycles, the power factor = 0.0001 6. Thus at radio frequencies the power factor 124 Circular of the Bureau of Standards due to conduction through any but an extremely poor condenser is wholly negligible. Series Resistance. A resistance within a condenser in series with the capacity affects the power factor very differently from a resistance in parallel. The series resistance includes the resist- ance of plates, joints, or contacts, and the leads from binding posts to plates. The E t across the resistance is in phase with the current /, and the emf Eo across the capacity is 90 behind / in phase. The power f actor = sin $, and since ^ is usually small, it may be taken as = tan \l/, which from Fig. 89 is rcoC. If r = i ohm and C=o.oi microfarad the power factor at 60 cycles = 3. 8 (io)~ 6 . This is utterly negligible. However, at a frequency of i ooo ooo cycles, the power factor = 0.063 = 6. 3 per cent, which is so large as to be serious. Thus, it is important to minimize series resistance in condens- 1[ /ty\/yUAAAA__ ers for radio work, while condenser leakage on the other hand has its chief _______^_ importance at low frequencies. In the foregoing example, r was taken as i ohm. In actual condensers it is sometimes greater than this. A plate resistance of several ohms is common in the ordinary paper condenser. In FIG. 89. Equivalent circuit and most other condensers, a high series re- vector diagram for condenser s i sta nce indicates a defect. having dielectric losses or plate ^ . , . ~, resistance ^ n account of skin effect, the series resistance in a condenser increases to some extent with frequency. In consequence, the power factor increases in proportion to a power of the frequency slightly greater than unity. The magnitude of this effect in typical condensers is not known. Dielectric Absorption. When a condenser is connected to a source of emf such as a battery, the instantaneous charge is fol- lowed by the flow of a small and steadily decreasing current into the condenser. The additional charge seems to be absorbed by the dielectric. Similarly, the instantaneous discharge of a con- denser is followed by a continuously decreasing current. It follows that the maximum charge in a condenser cyclically charged and discharged varies with the frequency of charge. The phe- nomenon is similar to viscosity in a liquid, and is sometimes called " dielectric viscosity. " Radio Instruments and Measurements 125 Dielectric absorption is always accompanied by a power loss, which appears as a production of heat in the condenser. The existence of a power loss signifies that there is a component of emf in phase with the current. The effect of absorption is thus equivalent to that of a resistance either in series or in parallel with the condenser. It is found most convenient to represent absorption in terms of a series resistance, which is spoken of as the "equivalent resistance" of the condenser. An absorbing condenser is, therefore, considered from the standpoint of Fig. 89, and the power factor = rcoC. The equivalent resistance r z q u u I V) u tt CO JIV AL- E:N r f ?E \s- ~f\t ICE / OF GU AS sc ON DE *s CR > J CAI A :IT Y' = 0. 501 KI if / / / / / / / / ' - / / / / / f j / / v 1000 2000 3000 WAVE LENGTH .METERS FIG. 90. Variation of dielectric loss in glass with wave length is constant for a given frequency but is different for different frequencies. The variation of the power factor and the equivalent resistance with frequency is a complicated matter, the laws of which are not accurately known. To a first approximation, however, the power factor of an absorbing condenser is constant. Since rcoC is approximately constant, r is inversely proportional to fre- quency (and therefore directly proportional to wave-length). This is well shown by the nearly straight line in Fig. 90; the 126 Circular of the Bureau of Standards crosses show observed resistances for a glass condenser. For most condensers used in radio circuits, the power factor is con- stant over the range of radio frequencies and has nearly the same value at low frequencies. This fact, viz, that power factor or phase difference is approximately independent of frequency, is very convenient and easily remembered. The same law holds for antennas at frequencies less than those for which the radiation resistance is large; the equivalent resistance is inversely proportional to frequency. Otherwise expressed, the equivalent resistance is proportional to wave length, for wave lengths greater than those at which radiation is appreciable. This is shown by Fig. 91, and is discussed on page 81. It is believed -5 -j 600 1000 1 500 ZCOO 2600 3OOO WAVE LENGTH, METERS FIG. 91. Variation of antenna resistance with wave length to be due to the presence of imperfect dielectrics (such as build- ings, insulators, and trees) in the field of the antenna; thus an an- tenna is like an air condenser in respect to the effect of poor dielec- trics in its field, as well as in other ways. Values of power factor and equivalent resistance are given for typical radio condensers in Table 2. The power factor does not vary with the size of the condenser; it is a function of the dielectric and not of the particular condenser. The equivalent resistance, on the other hand, is inversely proportional to the capacity of the condenser, since power factor = ruC. The data in the table for Ley den jars in oil are really typical of the glass dielectric, while the Radio Instruments and Measurements 127 relatively large power factor given for Ley den jars in air is due to brush discharge. The brush discharge is a function of voltage (see Brush Loss below), and is only appreciable at voltages above 10 ooo. For all voltages below this, the power factor of a Ley den jar in air is about the same as the value given for a Ley den jar in oil, about 0.003. TABLE 2. Power Factors of Radio Condensers (at 14 500 volts) Kind of condenser Power factor Capacity Equivalent resistance at 1000 m Compressed air'* 0.001 0.0058 0.14 .003 .0060 28 Molded (Murdock), new .004 .0054 .41 Glass plates in oil .005 .0042 .58 Glass (Moscicki type) .006 .0055 57 Glass (Leyden jar) in air .016 .0061 1 4 Molded micanite .023 0041 2 9 Paper .024 .0058 2.2 o Calculated from determinations by Austin, Bull. B. S., 9, p. 77; 1912. & The power factor observed for the compressed-air condenser is attributable not to the air but to the resistance in its leads and plates, to eddy current loss in the metal case, and to the insulating material used to separate the two sets of plates. Except for the compressed-air condenser and the Leyden jar in air, the power factors do not vary with voltage nor to a great extent with frequency. The power factors of solid dielectrics are principally due to dielectric absorption. If an appreciable portion of the power factor were due to a series resistance, such as a high plate resistance, it would be manifested by an increase of power factor with increasing frequency. At frequencies higher than 300000 (corresponding to a wave length of 1000 meters), this is to be expected in the case of paper condensers, which have long tin-foil plates; the power factor will increase, and the equiva- lent resistance can not decrease to a value lower than the plate resistance. When the power factor is due purely to dielectric absorption the equivalent resistance decreases in proportion to an increase of frequency (or increases in proportion to an increase of wave length) . The dependence of power loss, power factor, and equivalent resistance upon the frequency, in the case of absorbing condensers, condensers with leakage, etc., is summarized in the following table. P is the power loss, r the equivalent series resistance, \f/ the phase difference (same as power factor) , o> the usual 2ir times frequency, 35601 18 9 128 Circular of the Bureau of Standards and X wave length. Much of the information in the table is con- tained in the equations The table shows the powers of frequency and wave length to which P, \f/, and r are proportional. An arbitrary notation is used in the exponents of w and X; the symbols > and < for "greater than" and "less than" are used to indicate the mag- nitudes of the powers. TABLE 3. Variation of Phase Difference, etc., with Frequency Kind of system 1 -> I' 1 Condenser with leakage w x W- 1 X 1 w- 2 X 2 Absorbing condenser at very low frequencies w< ! X>- J o>< X> oK- 1 X>* Absorbing condenser at radio frequencies w 1 X- 1 w X w- 1 X 1 Constant series resistance Skin effect in conductors CO 2 w> 2 X- 2 X<- 2 w 1 w> ! X- 1 x<- 1 o> w>0 x x< Brush Loss. When a condenser is operated in air at high voltage, more or less ionization occurs at the edges of the plates. When the edges of the plates are entirely exposed to the air, as in a Ley den jar, this effect is large and a considerable power loss occurs at high voltages. At very high voltage the discharge is so great as to be evident in the form of a visible brush from the edges of the conducting plates, and is especially strong at corners and points. At such voltages the power factor due to the brush discharge is large compared with that caused by absorption and other causes. Austin 15 has found that the power factor and, hence, the equivalent resistance of a Ley den jar in air do not vary with voltage up to about 10 ooo volts; between 10 ooo and 22 ooo volts they increase approximately as the square of the voltage; and at voltages greater than 22 ooo increase faster than the square of the voltage. Since the power loss is proportional to the square of voltage and to the power factor, a constant power factor means power loss proportional to the square of the voltage. Thus, the 16 See reference No. air, Appendix 2. Radio Instruments and Measurements 1 29 power loss in Ley den jars below 10 ooo volts is proportional to the square of the voltage. Between 10 ooo and 22 ooo volts, where brushing occurs and the power factor is proportional to the square of voltage, it follows that the power loss varies as the fourth power of the voltage. Above 22 ooo volts the power loss increases faster than the fourth power of voltage. 35. MEASUREMENT OF CAPACITY A capacity measurement at radio frequencies is usually a com- parison of the unknown capacity with a standard variable condenser. The most precise measurements are made by means of a direct substitution. A primary standard for such work is a condenser which has been calibrated at low frequency and is so designed as to have extremely low absorption and high insulation resistance, so that its capacity is the same at radio and at low frequencies. For ordinary measurements it is possible to use a condenser with moderate absorption and insulation resistance which has been compared at radio frequencies with a primary standard. A condenser with high absorption or low insulation resistance may show changes in capacity in the range of radio frequencies, will increase the resistance of the circuit into which it is introduced, and is not suitable for a standard. The inductance of the leads within a standard condenser should be small in comparison with that of the coil, or the apparent capacity will vary with the frequency. The unknown condenser may be fixed or variable. It is inserted in series with an inductance coil in a circuit which is provided with a device to indicate resonance; that is, a wave- meter circuit. The source of oscillations is loosely coupled to this circuit and the frequency of the source adjusted until resonance is obtained. The unknown condenser is then removed and the variable standard substituted in its place and the setting of the standard found for which resonance is again obtained. The ca- pacity of the standard at this setting is equal to that of the unknown. In order to attain a high accuracy in the comparison, certain precautions must be observed. The capacity of the rest of the circuit to each of the two condensers must be either very small or the same. On this account it is desirable that the leads to the condenser be fairly long and fixed. Proximity of the metal shields, etc., of the condensers to the coil will reduce its inductance 1 30 Circular of the Bureau of Standards on account of eddy currents, and since this effect may vary with the two condensers, it is desirable to eliminate it by the use of long leads. In order that the capacities should be definite, both condensers should be shielded, the shield connected to one con- denser terminal and connected to ground, the same lead from the inductance coil being connected to the earthed terminal in both cases. When the unknown condenser is also a variable and is to be cali- brated at a number of points, it is convenient to use a throw-over switch to change from one condenser to the other. In this case it is necessary that the inductance and capacity of the two pairs of leads running from the switch to the condensers and mutual capacities to the rest of the circuit should be very nearly the same in the two positions. Any error from this cause may be checked by using the same condenser first on one set of leads and then on the other, and comparing the settings for resonance. Calibration of Large Variable Condenser. A large variable may be calibrated against a much smaller standard variable in the fol- lowing manner. The large unknown is first set at a low setting and directly compared with the standard as outlined above. This gives the capacity of the large condenser at the low setting with a high accuracy. Then the two condensers are connected in par- allel, the terminals of the two condensers which are connected to the shields being connected together. The large condenser is set at the known point and the small condenser at its maximum ca- pacity and the source is adjusted to resonance. Then keeping the frequency of the source unchanged, the setting of the large con- denser is increased by the desired steps, while the setting of the standard is reduced to compensate. From the reduction in capacity of the standard, the increase in capacity of the large condenser from that at the known point can be computed. When the stand- ard can be no longer reduced, the large condenser is set at the highest determined value, the standard is again set at its maximum, and the wave length of the source increased until resonance is again obtained. The process of increasing the capacity of the unknown and decreasing that of the standard is then repeated. Effect of Internal Lead Inductance. If the unknown condenser has internal leads which have appreciable inductance, its apparent capacity will increase as the wave length is decreased. To deter- mine this inductance the condenser is measured at a very long wave length with a large coil where the effect of the leads will be negli- gible and then with a small coil of inductance L. Calling the ca- Radio Instruments and Measurements 131 pacity at long wave lengths C\ and the apparent capacity at short wave lengths C 3 and the inductance of the leads / we may write or C C \- 2 ^1 This method requires the use of a standard condenser with leads of negligible inductance. COILS 36. CHARACTERISTICS OF RADIO COILS To introduce a certain amount of inductance into a circuit, a coil of copper strip or specially stranded wire is ordinarily used. This fa) FIG. 92. Typical forms of inductance coils is usually mounted or wound on a form made of such insulating materials as wood or bakelite or other composition. The form is usually hollow, and in some cases the conductor is supported only by strips of insulator at intervals. The turns of wire or strip are usually circular, but sometimes are polygonal. Iron is not used as a core in inductance coils for radio use, because such coils have high effective resistance, the power losses increasing with frequency. Three types of coil winding are widely used in radio work ; the sin- gle layer, the flat spiral or pancake, and the multiple layer. Cross sections of these windings are shown in (a), (6), and (c) of Fig. 92. 132 Circular of the Bureau of Standards The types (a) and (6) are universally used for coils of low or mod- erate inductance, the type (6) is especially convenient for portable instruments on account of its compactness. The multiple-layer coil is generally used where it is desired to obtain a large inductance in a compact form. The important electrical characteristics of a coil are its induc- tance, resistance, and capacity. In Part III are given formulas, with tables and examples, which cover the calculation of the in- ductance of practically all types of coils which are used in radio circuits. On account of the change of current distribution within a con- ductor, the self -inductance tends to decrease as the frequency in- creases. There is no appreciable change of mutual inductance with frequency. The change of self-inductance is very small, and can be calculated only in a few simple cases. The inductance of a coil decreases somewhat more than the inductance of the same length of wire laid out straight. The effect of distributed capac- ity, however, is to increase the inductance. At low frequencies the design of coil form is usually determined by the requirement of minimum resistance with a given inductance. This is treated on page 287. At radio frequencies, however, the capacity of coils is of very great importance and the choice of coil form is largely determined by the requirement of small coil capacity. 37. CAPACITY OF COILS In section 19 it has been pointed out how the capacity of coils may lead to circuits which resonate to two wave lengths and how overhanging or dead ends of coils may, on account of the coil capacity, seriously affect the reactance and resistance of a circuit. In this section other cases will be treated which show how the capacity of coils may considerably increase the resistance of radio circuits. Effect of Capacity on Inductance and Resistance. In the first case it is assumed that an emf (see Fig. 93(0)) is introduced into a circuit by means of a coupling coil L' of few turns of negligible inductance and not by coupling with the main coil L. Under these conditions the capacity of the coil L will affect both its resistance and apparent inductance and is not merely added to that of the condenser as would be the case if the coupling coil were removed and the emf introduced into the circuit by induction in the coil L itself. Instead, the coil L and its capacity form a parallel circuit as shown in Fig. 93(6). Radio Instruments and Measurements 133 If we write Z = ^R^+uPL^, where R & and L a are the apparent resistance and inductance of the parallel circuit, it may be shown that \7) > 2 Co 2 # 2 + (i-a> 2 LC ) 3 L(i-co 2 LC )-Cy? 2 The terms u*-C Q *R 2 and C R Z are very small and negligible, except- ing when very close to the frequency for which (i co 2 LC ) equals zero. This is the frequency with which the coil L would oscillate by itself that is, closed only by its own capacity. At this fre- - vvw - vvvvvvr- mm "c c. e _ '. FIG. 93. Equivalent circuit of a coil having distributed capacity placed in series with a condenser quency these terms determine the resistance and inductance of the coil. For other frequencies we may write R (i-co 2 LC ) a L : (i-o> 2 LC ) (72) (73) From these equations we see that the resistance and inductance start at the values R and L at low frequencies (co = o) and increase as the frequency increases, the resistance increasing about twice 134 Circular of the Bureau of Standards as rapidly in per cent as the inductance. When approaching the frequency co= , - the resistance becomes very great and \LC beyond this frequency falls off again, finally becoming zero. The inductance also becomes very great, but just before the frequency and R & = , " ^ = i . 1 1 R, an increase of 1 1 per cent. (o.95) 2 = 1.05 L, an increase of 5 per cent. & - ( - r The wave length corresponding to this frequency is A = 1885 meters, and the capacity C of the condenser for resonance is 950 micromicrofarads. If (o = 1.5 X io 6 corresponding to a wave length of 1250 meters, the resistance increase is 26 per cent and the inductance increase 12 per cent. The capacity C would then be 400 micromicrofarads. These are examples of conditions that may readily occur in practice. It is very desirable in designing radio circuits to arrange whenever possible that the emf be introduced in the main coil L itself. Effect of Dielectric Absorption in Coil Capacity. Even in the case of a simple oscillatory circuit of coil and condenser with the emf applied by induction in the coil (in which case the coil capacity may be considered as in parallel with that of the condenser) the resistance of the circuit may be increased. This is not due merely to the capacity of the coil, but because the coil capacity is, in Radio Instruments and Measurements 135 general, a highly absorbing condenser. The dielectric is the insulation of the wire and the material of the coil form, and the resulting phase difference of the coil capacity may be several degrees. The resistance added to the circuit on this account will be larger the greater the absorption of the coil capacity, and the larger this capacity relative to the condenser capacity, and also the longer the wave length. To illustrate the importance of this effect, let us take the same data as used in the preceding examples, with the further assump- tion that the coil capacity C has a phase difference of 2. In the first example co = io 6 , C = 5O micromicrofarads, and C = 95 2 \ =M T _L 2 1 ~M 2 V I+ co 2 L 2 2 > / [ L . M\ 2C0 2 L 2 : (75) (R 2 being small in comparison with wL 2 ) . The current ratio is that given by equation (74) , with a small correction term added. These calculations assume sine-wave currents, but apply to slightly damped currents as well. If the FIG. 106. Inductance shunted ammeter in- ductively coupled. Principle of the current transformer logarithmic decrement of the current is greater than a few hun- dredths, an additional correction term is needed. In using equa- tion (75), the actual high-frequency value of L 2 must be used. Transformer Without Iron. The current transformers used in the measurement of radio currents are of two types, with and without an iron core. The iron core has advantages in certain circumstances; this is discussed below. The simple transformer without a magnetic core is, however, satisfactory if used care- fully. A form which has been found 18 successful has a second - 18 See reference No. 103, Appendix 2. Radio Instruments and Measurements 153 ary winding consisting of a single layer of stranded wire on an insulating cylinder and a primary of one or more turns of thicker stranded wire near the middle of the secondary winding. To avoid induction from the leads and other parts of the circuit two such coils are used, connected so as to give astaticism, and the ends of the primary winding are brought out to a considerable distance. Equation (74) for the current ratio has been found to apply to such a transformer, within the accuracy of observation, for ratios as high as 100 to i. The ratio may vary a few per cent with the frequency because of the resistance of the instrument connected to the secondary. At frequencies below the radio range the correction becomes very large. Iron-Core Transformer. The iron-core radio current trans- former 180 consists of a laminated iron ring with a close winding of one or a few layers of fine wire upon it and a small number of primary turns of heavy stranded wire linking with it, as shown in Figs. 107 and 109, facing page 157. Very thin silicon-iron sheet is a satisfactory core material. The ring may be very small, of the order of 5 cm diameter. Little care need be taken to avoid induction from other parts of the primary circuit, as in the use of the transformers without iron. The iron core greatly increases L 2 and insures close coupling between the primary and secondary turns. The current ratio is readily found in terms of the ratio of pri- mary and secondary turns. The self-inductance of the secondary winding is given by I0d where n 2 = number of secondary turns, A = area of cross section of iron, ju a = apparent permeability of iron at the actual frequency used, and d = mean diameter of the iron ring. The mutual in- ductance is io 9 d t being number of primary turns. It follows that L 2== n2 M~n t iso See reference No. 103, Appendix 2. 1 54 Circular of the Bureau of Standards Since the ratio of primary to secondary current was found above to be approximately rl it follows that in the iron-core trans- former the current ratio is approximately the ratio of turns. The equation previously derived for current ratio does not apply exactly to the iron-core transformer because of the assumption of no energy loss. There is an energy loss in the iron due to eddy currents and hysteresis. This requires an energy current in the primary, which disturbs the current ratio. Taking account of this, the ratio may be shown to be ( ^,. (76) in which a is a quantity depending on the energy loss in the iron, having a value which is usually slightly less than unity. This assumes that all the magnetic flux from the secondary circuit links with the primary turns, a condition which is not fulfilled unless the secondary is uniformly and closely wound and the inductance of the instrument connected to the secondary is negligible. Because of the iron core the secondary inductance L 2 is so large that the correction term in equation (76) is ordinarily negligible at radio frequencies. Thus an advantage of the iron- core transformer is that the current ratio does not vary appre- ciably with frequency and does not depend upon the instrument connected to it. Careful design is necessary to secure this con- stancy of ratio, and even then it holds only for radio frequencies. At low frequencies these current transformers have large errors. The reason is easily seen, since the correction term in equa- tion (76) increases as w decreases. The increase of the correction is not proportional to the decrease of co because L 2 is smaller for high frequencies than for low. The value of L 2 is proportional to the apparent permeability of the iron, which decreases with increase of frequency because the skin effect reduces the effective cross section of the iron. Thus in a certain transformer 19 the iron had a permeability of 1000 at 50 cycles and an apparent permeability of 30 at 200 ooo cycles. The correction term in the current ratio equation was 14 per cent at the lower frequency and 0.2 per cent at the higher. 19 See reference No. 104, Appendix 2. Radio Instruments and Measurements 155 The apparent permeability depends on the thickness of the iron laminations. Making them thinner improves the accuracy of the transformer in two ways it increases the apparent permeability and thus increases L 2 , and it also makes a larger proportion of the magnetic flux from the secondary link with the primary turns. It is interesting to note that, at a given frequency, the apparent permeability of the iron does not vary with the current, because the fluxes in the iron are so very small that the permeability is" practically constant. Ad-vantages of Transformer. The ring form of current trans- former has also been used without the iron core. Under certain conditions, as when an ammeter of extremely small resistance is connected to the secondary, this may be a very good form of instrument. Neither this nor the iron-core transformer has been exhaustively studied. Both forms, however, are definitely known to have the following advantages as devices for measuring large currents: (i) They conform to the requirement of simplicity of circuit, for the primary turns have very little inductance and capacity; (2) they utilize the magnetic effect of the current and are thus, in themselves, free from the inherent limitations of thermal ammeters such as thermal lag and dependence on sur- rounding conditions; (3) the measuring circuit is electrically insulated from the main circuit and there is thus no conducting path to the indicating instrument and so the capacity of the latter can not cause so great a loss of current from the main circuit. Volt-ammeter Employing Current Transformer. The current transformer is used in a portable measuring instrument designed at the Bureau of Standards for the use of the radio inspectors of the Bureau of Navigation, Department of Commerce. Two views of the instrument are shown in Figs. 108 and 109, facing page 157. This instrument, which is called a volt-ammeter, is a combination of a current-square meter, two high-frequency current transform- ers, and series resistances which may be thrown into the circuit when it is desired to use the instrument as a voltmeter. The current-square meter is of a standard commercial type, requiring approximately o.i ampere for full scale deflection. Three scales are provided, running from o to 100, o to 5, and o to 25, respectively. The o to 100 scale is of use only in wave length or decrement measurement where relative values of current square are desired. The other two scales indicate amperes, and depend for their calibration on the accurate adjustment of the number of 156 Circular of the Bureau of Standards turns on the secondaries of the two current transformers contained in each instrument. Views of the transformers are shown in Fig. 109. The cores of both transformers are composed of a number of ring-shaped laminations of very thin silicon steel. These are bound tightly together and served with a layer of empire cloth tape. Over this is uniformly wound a single layer of fine wire comprising the secondary. The number of secondary turns is determined from the relation n^n/f- (77) where n 2 = number of secondary turns in series, n^ = number of primary turns in series, 7 2 = secondary current (about o. i ampere) , I 1 = primary current, either 5 or 25 amperes. In these transformers the number of primary turns (nj adopted was 2, so that expression (77) becomes ^ W2 ~ 2 / 2 This relation holds very closely at high frequencies if, as in this case of a toroid winding, the magnetic leakage is small or negligible. Small magnetic leakage requires that the secondary be wound in a single layer as uniformly as possible, covering the entire core length. The primary is wound in the manner indicated in Fig. 107, facing page , the wire being supported at a distance from the secondary and core. The terminals of the primaries of the transformers are brought out to four large binding posts at the bottom of the instrument. The secondary terminals are connected to opposite sides of a double-pole double-throw switch which is arranged to place the meter in series with either of the two windings. In operation, when it is desired to measure the antenna current of a transmitting set, the meter is connected in series with the antenna and ground at the large binding posts marked 5 or 25 amperes, depending upon the magnitude of the current to be meas- ured. The double-throw switch is then thrown to place the proper transformer in circuit and the readings obtained. The instrument will, of course, work equally well in a closed circuit. Bureau of Standards Circular No. 74 FIG. 96. Hot-wire type current-square meter FIG. 102. Hot-strip ammeter -with cylindrical arrangement of heating elements FIG. in. Mounted thermocouple -with protecting cap removed Bureau of Standards Circular No. 74 FIG. 107. Iron-core current transformer FIG. 108. Volt-ammeter employing the current transformer FIG. 109. Rear -view of volt-ammeter, showing the current transformers Radio Instruments and Measurements 157 As previously mentioned, the accuracy of the current scales depends upon the proper adjustment of the number of secondary turns. It is also affected by the frequency, the error growing larger as the frequency is decreased. These transformers are so constructed and calibrated that working over a range of fre- quencies corresponding to wave lengths between 150 and 1000 meters the current scales are accurate to better than 2 per cent. The rubber-covered binding posts at the top of the meter, in conjunction with the push button marked "Voltage", are used for voltage measurement. Three ranges are provided, 2.7, 40, and 1 50 volts. These were adopted so as to enable the inspector to make proper voltage test on the storage batteries of the auxiliary power supply. Voltage calibration curves are supplied with the instru- ment. The two sockets at the left marked decremeter connect directly to the terminals of the current-square meter. They are used when the meter is employed as a current indicator in wave length or decrement measurement. This volt-ammeter was designed with the primary object of reducing the weight and number of instruments which the radio inspector must carry with him. In the wavemeter previously used by the inspectors the current-square meter was contained within the wavemeter case. In the latest type the current- square meter has been omitted, thus providing a much smaller and lighter wavemeter. This, together with the volt-ammeter here described, provides an equipment for making all the required measurements. 43. MEASUREMENT OF VERY SMALL CURRENTS A number of methods are used for measuring currents of a few milliamperes or less. In addition to the thermal ammeter and the current transformer, used for larger currents, use is made of the electrostatic and the magnetic effects of the current, and rectification into unidirectional current. Instruments operating on these various principles are described below. The measure- ment of very small currents is free from some of the difficulties of measuring larger currents, principally because conductors of very small cross section may be used which do not change in resistance with frequency. Crossed-Wire Thermoelement. Sensitive thermoelements are easily made and are extensively used to measure small high-fre- quency currents. They consist essentially of two wires of different 1 58 Circular of the Bureau of Standards metals in contact, one or both of them being of very small diameter. The heat due to the R I 2 in the fine wire raises the temperature of the junction, thus giving rise to a thermoelectromotive force which is indicated by a direct-current galvanometer. A simple type is the crossed-wire thermoelement. Two fine wires are used, crossed in either of the ways shown in Fig. no. (Practically the same type was shown in Fig. 98) . The high-frequency current is led in through the heavy copper wires A and B, and the galvanometer is connected to a and b. The sensitivity of the thermoelement depends on the diameter, thermoelectric properties, and resistivity of the wires, on the length of the wires if very short, on the intimacy of contact of the two wires, and on the air pressure. Permanent contact of the two wires may be insured by soldering the junction. The use of a minute particle of solder does not appreciably reduce b B FIG. no. Crossed-wire thermoelements the sensitivity. If the junction is not soldered, some thermoele- ments are occasionally found to have an abnormally high sensitiv ity, probably owing to a particularly poor contact ; the high resist- ance of the contact causes a production of heat just at the junction which is large relatively to the heat produced in the wires. In some thermoelements this poor contact remains sufficiently constant so that the thermoelement can be relied upon, and in some it does not. The materials ordinarily used for these thermoelements are constantan and steel, and constantan and manganin. Using wires of the order of 0.02 mm diameter and 4 mm long, a con- stantan-steel thermoelement has a resistance of about i ohm and gives about 40 microvolts for 1 5 milliamperes high-frequency current. Using a galvanometer with a sensitivity of 2.5 mm per microvolt, it follows that a deflection of 100 mm is produced by a high-frequency current of 15 milliamperes. The electromo- tive force is very closely proportional to the square of the high- frequency current. Thus, the typical thermoelement just men- tioned gives a deflection of only i mm for 1.5 milliamperes. Radio Instruments and Measurements 1 59 These thermoelements have practically no thermal lag because of being made of such fine wire; it therefore pays to use a very quick-acting galvanometer with them. A thermoelement of this type made at the Bureau of Standards is shown in Fig. 1 1 1 , facing page 156. Smaller currents can be measured by the use of thermoelements made of wires of still smaller diameter. The resistance, the RP, and therefore the temperature, would be higher for a given cur- rent. High resistance is, however, objectionable in most radio circuits, so that the more sensitive thermoelements of higher resistance have little application. Another way to increase sensitivity at the expense of increasing the resistance of the circuit is to connect the thermoelement, not directly into the circuit, but to the low-voltage side of a current transformer in the circuit. This has the advantage that the galvanometer is not metallically connected to the main circuit and thus its capacity is less likely to cause leakage of high-fre- quency current from the main circuit. The exact limitations of this method are not known; it seems likely that the calibration will change with frequency and with the resistance and inductance connected to the transformer. The above figures on sensitivity are for thermoelements in air at ordinary atmospheric pressure. No variation of sensitivity with the ordinary barometric pressures has been observed, but the sensitivity may be greatly increased by placing the thermo- element in a vacuum. It has been found that an air pressure of about o.oi mm of mercury or lower is necessary to gain much in sensitivity, but that in such low vacua the sensitivity of thermo- elements of polished metal wires may be increased as much as 25 times. The removal of the air eliminates the cooling of the thermoelement by convection; a given current therefore raises it to a higher temperature. The temperature of a hot body in a vacuum is limited only by radiation of heat from its surface; thus the temperature of a polished metal surface, which is a poor radiator, rises higher than that of a dull metal surface, which is a good radiator. Self-Heated Thermoelement. In the type of thermoelement described in the preceding section the high-frequency current does not pass through the wires of the thermocouple itself. The thermocouple wires (Oa and Ob, Fig. no) touch the wires (OA and OB), which carry the high-frequency current, at one 35601 18 11 1 60 Circular of the Bureau of Standards point only. There is no reason why the thermocouple wires themselves may not carry the high-frequency current and heat the junction. A simple thermocouple so used may be called a self-heated thermoelement. By using this type, Austin 20 has found it possible to utilize tellurium as one of the metals of a thermoelement and obtain very high sensitivity. A couple consisting of tellurium and platinum gives about 25 times as great an emf as constantan and platinum at the same temperature. These thermoelements are made by the following process. Two copper wires are placed side by side about 3 mm apart and embedded in insulating material with their ends protruding (Fig. 112). To the end of one of these is soldered about 5 mm of 0.02 mm platinum or constantan wire. To the other is soldered a short bit" of 0.8 mm platinum wire, to the end of which a bead of tellurium had previously been attached when the platinum was white hot. (White-hot platinum wire when inserted in tellurium forms practically a resistance-free contact.) The end of the fine wire is next allowed to rest against the tellurium and the two are welded together electrically by means of a small induction coil with a high resistance in series with the secondary. The contact will FIG. ^.-Platinum- be less fra ile if the weldin g is done in an oxy- tellurium thermocou- gen-free atmosphere. The resistance of a ther- pie of the self-heated moelement prepared in this way may be any- where from 5 to 50 ohms, according to the condi- tions of welding and the resistance of the fine wire, the lower resistance being somewhat more difficult to obtain. The thermo- element is next inclosed in a test tube, and, if likely to be handled roughly, the whole may be inclosed in a larger test tube with cotton or felt between the two. These thermoelements remain constant over considerable periods, but some have been found to lose their sensitiveness after a year or two. A 3 2 -ohm thermoele- ment was found to give a deflection on a very sensitive galva- nometer of i mm for 120 microamperes. Such thermoelements are very satisfactory when used in connection with portable microammeters of the pointer type. Thermoelements of the self-heated type, can, of course, not be used on direct current, since the galvanometer forms a shunt on the heating wire. At high frequencies the current is kept out of the galvanometer because its impedance is so much greater than 20 See reference No. 207, Appendix 2. Radio Instruments and Measurements 161 that of the short lengths of thermocouple wires which it shunts. It should be noted that the heating current passes through both junctions of the thermocouple but that the temperature of only one rises appreciably because a fine wire of high resistance is used at one junction only. Large cross section at the other junction prevents a large heat production and temperature rise. Thermogahanometer. Currents of several hundred microam- peres may be conveniently measured by the Duddell thermo- galvanometer. This is a compact combination of hot wire, thermocouple, and galvanometer. The galvanometer coil is a single slender turn of wire, with the bismuth-antimony thermo- couple attached to its lower end. The junction between these Bi VSb Hot wire FlG. 113. Duddell thermogahanometer two metals is directly over, but not in contact with, the heater which is either a hot wire or thin gold leaf or film of platinum on glass. Currents as low as 10 microamperes may be measured when a heater of several thousand ohms resistance is used, but this great sensitivity is not available for radio measurements, which generally require heaters of less than 50 ohms resistance. The thermogalvanometer differs from the thermoelements de- scribed above in that the thermocouple is not in metallic contact with the heater, and thus the capacity of the galvanometer is less likely to cause leakage of high-frequency current from the main circuit. Bolometer. An instrument in which heat is measured by the change of resistance which it produces in a conductor is called a bolometer. Precise measurements of radiant heat, for instance, l62 Circular of the Bureau of Standards are made with a bolometer which consists of a blackened metal strip together with a Wheatstone bridge for measuring its resistance. The bolometer used in radio measurements consists essentially of a fine wire, through which the high-frequency cur- rent is passed, connected to a Wheatstone bridge. The resistance of the wire increases as it is heated by the current, and this change FlG. 114. Various methods of using the bolometer for measuring small radio currents of resistance either causes a deflection of the bridge galvanometer, or the bridge is balanced by a change in one of the bridge arms so as to keep the galvanometer deflection unchanged. In the latter method the resistance in the changed bridge arm is a measure of the current. The direct current used in the Wheatstone bridge produces more or less heat in the bolometer wire, and in order to avoid Radio Instruments and Measurements 1 63 error it is necessary to: (i) Keep it very small, or (2) keep it constant for all measurements, or (3) have an auxiliary wire similar to the bolometer wire in an adjacent arm, which will keep the bridge balanced as far as heating by the direct current in the bridge is concerned. The various forms shown in Fig. 114 have been used. Figures (6) and (d) are similar to (a) and (c), re- spectively, except for the auxiliary wires in a bridge arm adjacent to the one containing the bolometer wire. It is necessary to keep the high-frequency current out of the parts of the bridge other than the bolometer wire as the latter would not then carry the whole current to be measured. This is done in either of the two ways shown. Choke coils may be used on either side of the hot wire, as in (a) and (6); or, the bolometer wire may have the rhombus form as in (c) and (d). The heating current divides between the two halves of the rhombus, and the bridge connections are made at two points of equal potential so that there is no tendency for the heating current to flow into the bridge. For the bolometer wire, use has been made of iron, gold, plati- num, tungsten, and carbon. The smaller the wire the smaller the currents measurable. A gold wire 0.002 mm diameter has been found to give 10 scale divisions on a pointer- type galvanometer for 500 microamperes and a 0.0005 mra platinum wire 10 scale divi- sions on the same galvanometer for 34 microamperes. These fig- ures 21 are for the bolometer wire in air. The sensitivity may be increased by placing in a vacuum. A current of 5 microamperes has been measured by the use of a carbon filament in a vacuum. The bolometer has also been used as a means of measuring large currents up to 10 amperes. (Seep. 172 below.) Wire of com- paratively large diameter is used but not so large as to change in resistance with frequency. It is immersed in oil to keep down the the temperature rise. Electrometer. It is possible to measure fairly small currents by the aid of an electrometer shunted across a condenser. The deflections of an electrometer are proportional to the square of the effective voltage when the vane is connected to one plate. The current through the electrometer is proportional to its capa- city, to the frequency, and to the applied voltage. The form shown in Fig. 116 has been used. Connection is made to the fixed plates PP. The light metal vane V is suspended by a delicate fiber between them. The suspending fiber also 21 Data from B. Gati (Electrician, 58, p. 983, 1907, and 78, p. 354, 1916), who uses an arrangement for measuring resistance somewhat different from a Wheatstone bridge, and who calls his device a ' ' barretter." 164 Circular of the Bureau of Standards carries a damping-vane and a mirror for reading from a distance. The voltage impressed on the plates PP charges them with electric- ity of opposite signs. One of them being connected to the vane V, they thus exert a torque on it. FlG. 115 . Method of using electrometer (vane should be connected to one terminal) These instruments have the advantage of introducing no appre- ciable resistance into the circuit. They have a small capacity and therefore should not be used in parallel with any except a large condenser. They require very careful manipulation, as they are delicate and are very sensi- ^/"~V-\ * lve * s ^ ra y electrostatic IV ~^~~\A char S es - The method has ^ ' not been used in the Bureau FIG. 116. Schematic ar- of Standards laboratory. rangement of plates Electrodynamometer. The and -vane of the electro- instruments of this type used meter J \ at low frequency consist of a fixed and moving coil connected either in series or in parallel. Capacity between the two coils renders them unsuitable for measuring high- frequency currents. A somewhat different type has, however, been successfully used. The in- strument consists of a small coil of fine wire which carries the current to be measured, and p IG II7 .Typeofelc- a flat ring or disk of silver or copper suspended by a delicate fiber concentric with the coil and with its plane at 45 to the plane of the coil. Current in the coil induces an opposing current in the ring, which is then repelled. The torque acting on the ring is proportional to LNtfP trodynamometer suit- able for radio current measurement Radio Instruments and Measurements 165 where R and L are the resistance and inductance of the metal ring and N the number of turns in the coil. The deflection will therefore depend on the frequency, which is a serious limitation on the usefulness of the instrument. For frequencies so high, how- ever, that R is small compared with coL, the deflections are inde- pendent of frequency. Care should be taken to keep the instru- ment away from other parts of the circuit, so that no large mag- netic fields may act upon it. Crystal Detector, Currents too small to be measured by any of the preceding instruments, as for example the received currents in antennas, can be measured by the aid of a crystal detector. 1 c 1 1 1 1 I U . FIG. 118. Crystal detector circuits used in measuring small currents The exact action of these detectors is a complicated matter, but for practical purposes it is sufficient to regard them as unilateral conductors ; that is, they have a greater resistance to current flow- ing through them in one direction than to current flowing in the opposite direction. Thus, when an alternating emf is impressed on a crystal detector, more current flows in one direction than in the other, and a direct-current instrument in the circuit will be operated. The resistance of the ordinary crystal detectors in the low-resistance direction is of the order of 1000 to 10 ooo ohms, and the resistance in the opposite direction about 10 times as great. 1 66 Circular of the Bureau of Standards In combination with a sensitive galvanometer, a crystal detector may be used to measure currents of a few microamperes. The galvanometer and crystal may be connected to the circuit LC, the current in which is to be measured, in any of the ways shown in Fig. 1 1 8. Using the first mode of connection, Austin 22 , has ob- tained a deflection of 100 mm for 91 microamperes. A perikon detector (chalcopyrite-zincite) was used, with a galvanometer of 2000 ohms resistance giving i mm for 1.3 x io~ 9 ampere direct cur- rent. The deflections have been found to be proportional to the square of the high-frequency current (for this and some other crys- tals) . The sensitiveness of the crystal detector is hundreds of times that of the thermoelement, but it is not constant. It is always calibrated just before or after use (or both) by comparison with a thermoelement in the LC circuit, using current from a buzzer or other source. (See p. 1 74 below.) A telephone may be used in place of the galvanometer in any of the arrangements shown in Fig. 118, when periodically interrupted current is to be measured. Telephone measurements can not be made of uninterrupted undamped currents. Quantitative measurements may be made with the telephone in two ways. In both, the current through the telephone is reduced until the sound is just barely audible. (The limit of audibility is sometimes taken to be that at which dots and dashes can just barely be dis- guished.) In the first method a resistance is placed in parallel with the telephone and reduced until the limit of audibility is reached; this is the "shunted telephone" method. The second method employs variable coupling between the detector circuit and the main circuit. A measure of current in the shunted telephone method is obtained as follows: If t is the impedance of the telephone for the frequency and wave form of the current impulses through it, s the impedance of the shunt, 7 t the least current in the telephone which gives an audible sound, and / the total current flowing in the com- bination of telephone and shunt, I _s + t /t s s + t This ratio, - , is called the audibility. It is approximately pro- o portional to the square of the high-frequency current. It can be expressed in units of current if calibrated at one or more values of current by some deflection device in the high-frequency circuit. 22 See reference No. 206, Appendix 2. Radio Instruments and Measurements 1 67 The observed settings depend on the frequency, on the wave form of the pulses passing 'through the telephone, on the constants of the circuit, and on the frequency of interruption of the current used, and involve the assumption that the crystal and the sensi- tiveness of the operator's ear remain constant. It is desirable to minimize the variations of resistance which have to be made in the detector circuit by using a fixed resistance J?! in series with the detector and shunt the telephone across a variable portion R 2 of this. The apparatus should be calibrated under the exact condi- tions of use, both before and after each set of measurements. An accuracy of 10 per cent is difficult to obtain; the method is nevertheless very useful in measurements of radio currents in re- ceiving sets. For a frequency of interruption of the radio current of looo per second, currents of the same order of magnitude can Ri M FlG. 119. Circuit for measuring audi- bility ratios be measured with a crystal detector by the use of a telephone as by the use of a galvanometer. About 10 microamperes is the least current which can be detected by the ordinary crystal and telephone. In the variable coupling method the telephone is not shunted, but the coupling between the detector circuit and the main circuit is varied until the sound in the telephone is just barely audible. The greater the high-frequency current to be measured the looser is the coupling. The arrangement is calibrated by making at least one simultaneous observation of the coupling for barely audible sound and current as measured by some other device, such as a previously calibrated crystal and galvanometer connected to the main circuit, together with the plotting of a curve between coupling and current in the main circuit. The coupling may be measured in any arbitrary way, as by the distance apart of the coupling coils. Audion. The audion (described below in section 56) may be used for measurements of current, just as the crystal detector is, 1 68 Circular of the Bureau of Standards in conjunction with either a telephone or a galvanometer. With the ordinary audion connections, as in Fig. 120, the sensitivity is about the same as that of the best crystal detectors. The actions of the audion and other electron tubes as detectors, amplifiers, etc., pAMMMA FIG. 120. Use of the audion for measuring small currents in terms of audibility ratios are discussed on pages 204 to 2 10. The connections shown here are for the shunted telephone method. The variable coupling method can also be used. In the figure, L is the coil used for coupling to the circuit in which the current is to be measured, C l is a small FIG. 121. Oscillating ullraudion circuit used with crystal detector and galvanometer for measuring small currents fixed condenser, and T the telephone shunted by the variable R. The audibility is approximately proportional to the square of the high-frequency current, as in the case of the crystal detector. A galvanometer can be used with the audion, but it must not be put directly in place of the shunted telephone, because a con- Radio Instruments and Measurements 169 tinuous current would flow through it from the B battery. One arrangement is to place the primary of a transformer in series with the telephone and connect to its secondary a crystal detector in series with a sensitive galvanometer. The changes in current which affect the telephone give rise to alternating currents in the secondary which are rectified by the crystal detector and thus cause a deflection of the direct-current galvanometer. This arrangement is particularly advantageous when the oscillating ultraudion connections are used. (For description of the ultraudion see section 58 below.) The connections are given schematically in Fig. 121. This is suitable for the measurement of undamped currents. The note in the telephone T is produced FIG. 122. Oscillating audion circuits for quantitative measurements on undamped waves from distant radio stations by the beats between the impressed and the local currents. The condenser C 4 must be adjusted for maximum deflection of the galvanometer. Austin 23 has found that the deflections are pro- portional to the square of the high-frequency current, which means that the current in the telephone is proportional to the first power of the high-frequency current. (This law holds only for the oscillating condition. When the audion is not oscillating, the deflections are approximately proportional to the fourth power of the high-frequency current.) This constitutes a method of remarkable sensitiveness for measuring small high-frequency currents. Austin found that for signals of minimum audibility on the simple audion, the oscillating ultraudion gave audibilities 23 See first article in reference No. 108, Appendix 2. 1 70 Circular of the Bureau of Standards from 300 to 1000 times as great; that is, it would measure currents hundreds of times as small. For convenience in measuring received radio currents from distant stations the shunted telephone is used in connection with the oscillating ultraudion. The arrangement shown in Fig. 122 has been used by Austin. 24 The shunt s is used on the telephone T 2 . The audibility is approximately proportional to the current in the antenna. The sensitivity is always measured at the time of use by comparison with a silicon detector and galvanometer, which combination is in turn calibrated by comparison with a thermoelement. This arrangement has been used to make quan- titative measurements on undamped waves from radio stations 4000 miles away, the least high-frequency current detectable in the receiving antenna being 4 x io~ 9 ampere. 44. STANDARDIZATION OF AMMETERS The instruments for high-frequency current measurement may be grouped as follows, from the standpoint of standardization: (i) Those whose deflections are the same at all frequencies, such as suitably designed instruments of the hot-wire type; (2) those whose deflections are accurately calculable at all frequencies, such as electrometer ammeters; (3) those which are constant at all radio frequencies but not at lower frequencies, such as properly designed current transformers; and (4) those which have to be calibrated at the particular frequency used, such as the electro- dynamometer, crystal detector, and audion. Only the first and second of these groups are suitable to serve as standards for the calibration of high-frequency ammeters. The second group, electrometers, is not actually used for this purpose, so the ultimate standards used in practice are instruments of the hot- wire type. Small and Moderate Currents. The instruments described in subsection a above under this head are all simple hot-wire types. If properly constructed, if the hot wire is fine enough, and the design is otherwise correct, they are themselves standards and need no calibration at high frequency. Such instruments are calibrated at low frequency (50 to 3000 cycles per second). In no case should they be used without calibration. While thermo- couples and some detectors give deflections approximately propor- tional to the square of the current, they do not follow this law 24 See article in "{Electrician," reference No. 108, Appendix 2. Radio Instruments and Measurements 171 exactly. In some cases the calibration can be made with direct current. In other cases this is not desirable; in the thermoelec- tric ammeter of the crossed-wire type, for instance, the Peltier effect at the junction causes a current through the galvanometer, and in addition some of the direct current passes through the galvanometer inasmuch as the junction has some resistance and thus the galvanometer and the junction constitute two parallel paths for the current. Both of these effects may be eliminated by reversing the direct current and taking the mean deflection of the galvanometer (not the mean reading, in case the scale is calibrated in terms of the heating current) . Reversing the direct current is equivalent to using alternating current for calibration. The use of direct current involves another possible error, leakage to the galvanometer, which may or may not be reversed when the current is reversed. It is on the whole good practice to use alter- nating current rather than direct for standardizing high-frequency ammeters, using as comparison instruments any reliable low-frequency ammeters. If there is any doubt as to a high- frequency ammeter's independence of frequency, it should be compared with o o a reliable standard at radio frequencies | LOW FREQ. by the methods given below. Large Currents. The design of most ^ ^--Method of testing am- meters for the effect of change of ammeters for large currents of high fre- frequency quency is such that it is not safe to assume them independent of frequency. They should be standard- ized by comparison with instruments known to be reliable at several radio frequencies. The comparison is made as indicated in Fig. 123. The instrument to be tested, X, is in series with a standard instrument, N, and their deflections are simultaneously observed when supplied alternately with high-frequency and low- frequency current. The high-frequency circuit LC is coupled to a source of current such as a spark or arc set or a pliotron, and the low-frequency current is obtained from an alternator through a step-down transformer and a rheostat. The two ammeters could, of course, be compared using the high-frequency current only, but this would give no information as to the change of reading from low to high frequency. Also, the variation of the readings at different radio frequencies would 172 Circular of the Bureau of Standards not be as accurately obtained. The particular advantage of using an auxiliary low-frequency comparison current is that it enables one to determine accurately the difference between the high and low-frequency readings independently of zero shift, temperature variation, and other accidental errors. The experi- mental procedure is to pass high-frequency current through the two ammeters for a certain length of time, say, one minute, recording the deflections, then quickly throw the switch (5 in Fig. 123) and allow an approximately equal low-frequency cur- rent to flow the same length of time, recording the deflections; then high frequency again, then low frequency, and finally high frequency again. Thus, three high-frequency observations are obtained with two low-frequency observations sandwichedbetween them, and errors from thermal or other drifts are eliminated. The lack of constancy of a spark or arc source limits the precision FlG. 124. Standard ammeter using the bolometer principle of an observation to a few tenths per cent. This can be excelled with a pliotron source. Several variations of the hot-wire principle are available as standard ammeters. The simplest is a single wire as used in various instruments previously described, and capable of measuring up to about 2 amperes. An ammeter of about 10 amperes range can be calibrated by comparison with such a standard at the lower end of its range, 2 amperes or less. Such a calibration is, of course, not so satisfactory as one covering the whole range. As a standard ammeter for measurements up to 10 amperes an application of the bolometer principle may be used. The instrument consists essentially of a fine copper wire soldered to four upright posts, two of which carry the high-frequency current and the other two connect to a Wheatstone bridge as in Fig. 124. The wire rhombus is placed in oil. The high- frequency current has two paths in the instrument and hence Radio Instruments and Measurements 1 73 great care is necessary to insure that the resistances and induct- ances of the two paths are equal. With the most careful con- struction the current distribution between the two paths doubtless varies somewhat with frequency, but it is to be noted that the resistance depends upon the heat production in the whole instru- ment and not on that in one branch only. Small changes of current distribution do not appreciably affect the resistance of the instrument, and it is consequently a perfectly reliable high- frequency standard if carefully constructed. A curve i's plotted between the measured current and the resistance in the rheostat arm of the Wheatstone bridge. In the arrangement shown diagrammatically in Fig. 124, K is a tapping key in the battery circuit. A closed galvanometer circuit is used, thus eliminating errors of false zero. G is a sen- sitive moving-coil galvanometer. The current through the fine wire from the bridge battery should be o.oi of the heating current or less. It is not convenient to calibrate this standard on direct current, although it is theoretically possible to do so. For a heating current entering at L and M, X and Y need to be so adjusted as to be equipotential points; then no portion of the heating current flows in the bridge used to measure the resistance. However, it is difficult to make this adjustment exactly, and it is moreover unnecessary, as a calibration by low-frequency alter- nating current is just as good as a direct-current calibration. Consequently, the points X and Y are simply made approximately equipotential points, but not adjustable. For currents greater than 10 amperes, standard ammeters of the cylindrical type, described on page 146 above, are used. As there stated, the design of these instruments by no means insures their accuracy. If properly constructed of high-resistance metal, instruments of this type can be used as standards for moderate ranges of current and frequency. Standard instruments for the largest currents used in radio work are now under investigation at the Bureau of Standards. Very Small Currents. Thermoelements are usually used as standards in calibrating the instruments used to measure very small currents. As already explained, it is in general better to standardize them with low-frequency alternating current than with direct current. In particular, thermoelements of the self- heated type can not be used on direct current. It is possible to use a bolometer as a standard, and it likewise should be calibrated with low-frequency' alternating current, since if direct current 174 Circular of the Bureau of Standards were used some of it would be very likely to get into the bridge galvanometer. The thermogalvanometer and the electrometer, on the other hand, may theoretically be calibrated with direct current but the experimental errors are likely to be larger than with low-frequency alternating current. In the use of the crystal detector and the audion to measure current, with either a galvanometer or telephone, it is necessary to calibrate the arrangement at the time of use. These very sensitive devices are variable with time and with the conditions of the circuits. The calibration of a crystal detector is made by placing a thermoelement directly in the circuit to which the crystal is coupled or attached, as in Fig. 125, and currents used FIG. 125. Method of calibrating a detector in terms of a thermocouple such as to give small deflections on the thermoelement galvano- meter. These observations fix the value of current for one or more points corresponding to large deflections of the galvano- meter attached to the crystal. The law of variation of deflection of the latter galvanometer with respect to current in the main circuit is determined by a separate experiment in which the detector is coupled more loosely to the main circuit so that the deflections of the two galvanometers are more nearly equal. In Fig. 125, L is an inductance used to couple the main circuit to the source of current, Th is the thermoelement, and G t its galvanometer. L 2 is used to couple the detector circuit to L!, C 2 is a fixed condenser, and G 2 is the detector galvanometer. Ordinarily the deflections of the two galvanometers are approxi- mately proportional, and the ratio between the small thermoele- ment deflection and the large crystal deflection obtained in the Radio Instruments and Measurements 175 calibration is used as a multiplier to obtain the equivalent thermo- element deflection from any smaller crystal deflection subse- quently observed. The currents so measured thus depend, in a certain sense, on an extrapolation. Great accuracy is not expected nor required. When an oscillating ultraudion is used it is calibrated by comparison with a detector in the same way as the detector is calibrated in terms of the thermoelement. The exceedingly small currents measured with the oscillating ultra- udion thus depend upon two extrapolations. RESISTANCE MEASUREMENT 45. HIGH-FREQUENCY RESISTANCE STANDARDS Standards of resistance are required for some of the methods of resistance measurement described below. The resistance of the standard must be accurately known at all frequencies, and it is very desirable to have it remain constant over all the frequencies at which it may be used. This requirement practically limits the form of such a standard to a very fine wire. Very thin tubes would be satisfactory from the theoretical standpoint, but it is extremely difficult to obtain very thin metal tubes of sufficient uniformity that the current distribution, and hence the resistance, does not change with frequency. The accuracy of ordinary measurements requires that the re- sistance of the standard be constant to i per cent. The maximum size of wire of various materials that may be used can be found from Table 1 8, page 310, for the highest frequency that is to be used. The diameter required for any given accuracy and a given limiting frequency may be calculated from (207) , page 300 . These fine wires which must be used as high-frequency standards will not carry much current without serious heating. They must therefore be used in measurements with caution. When a stand- ard is required to carry large currents it can not, in general, be obtained by putting several fine wires in parallel. The design of such a resistance standard is, in fact, identically the same problem as the design of high-frequency ammeters for large currents. (Sec. 41, p. 144.) The ideal resistance standard would not change any of the con- stants of a circuit except resistance when it is inserted in the circuits. A wire, however, has some inductance, and since the inductances used in radio circuits are small the inductance of a wire standard of resistance can not be neglected. The induc- 35601 18 12 1 76 Circular of the Bureau of Standards tance must be made extremely small by using very short wires, or its effect must be minimized, by substituting for the resistance wire a copper wire of the same length whenever the resistance wire [is removed from the circuit. The first of these alterna- tives is followed in the use of a short slide wire, which gives a continuous variation of resistance; the contact with such a slide wire must, of course, be of small and constant resistance. The second alternative is followed in sets of resistance standards. The inductance of the copper wire link will be practically the same as the inductance of the resistance standard substituted for it, and the slight difference between them will not ordinarily affect the total inductance of the circuit provided the resistance wire and the copper link are short. A set of resistance standards for high frequency may, therefore, be a set of short wires of approximately equal lengths, a portion of the length consisting of a very fine wire. The wire must be of very small diameter to obtain fairly large resistances in the short length allowed. This fine wire is of resistance material, the length is adjusted to give the required resistance, and the remainder of the length is of relatively thick copper wire. In a set used by this Bureau, the resistance material is manganin, used because the resistance does not change appreciably with temperature. In order to protect the delicate wires from breakage each is mounted in a glass tube. The copper ends of each standard are amalga- mated for insertion in small mercury cups; the amalgamation must be renewed at frequent intervals. They are 7 cm long; the resistances have values from 0.2 to 40 ohms; the resistance wires have lengths from 0.5 to 6 cm, and diameters from 0.03 to 0.12 mm. On account of the wires being of such small diameter, care is neces- sary to avoid using currents that would overheat them. The inductances vary from 0.15 microhenry for the 4O-ohm standard to 0.08 microhenry for the o-ohm copper link. The difference, 0.07 microhenry, is negligible except in rare cases. The resistances have been found to remain satisfactorily constant for several years. Decade resistance boxes are also useful as standards when properly made and used with caution. The resistance units must be made very short and of sufficiently fine wire. When such a set is used in a circuit of low inductance, variation of the resistance setting may vary the inductance of the circuit. In some methods of resistance measurement this merely requires retuning to resonance. Radio Instruments and Measurements 177 46. METHODS OF MEASUREMENT The methods of measuring high-frequency resistance may be roughly classed as: (i) Calorimeter method, (2) substitution method, (3) resistance-variation method, and (4) reactance- variation method. The fourth has frequently been called the "decrement method," but it is primarily a method of measuring resistance rather than decrement, exactly as the resistance-variation method is. Thus, the measurement of decrement is the same problem as the measure- ment of resistance. When applied to determine the decrement of trains of waves, radio-resistance measurement accomplishes something similar at high frequencies to what is done at low fre- quencies by wave analysis. Either may be used to measure the decrement of a wave under certain conditions and, in fact, the results of resistance measurement by any method may be ex- pressed in terms of decrement. All four methods may be used with either damped or undamped waves, though in some of them the calculations are different in the two cases. They are all deflection methods, in the sense of de- pending upon the deflections of some form of high-frequency ammeter. In the first and second, however, it is only necessary to adjust two deflections to approximate equality, while in the third and fourth the deflections may have any magnitude. 47. CALORIMETER METHOD This method may be used to measure the resistance either of a part or the whole of a circuit. The circuit or coil or other appa- ratus whose resistance is desired, is placed in some form of calo- rimeter, which may be a simple air chamber, an oil bath, etc. The current is measured by an accurate high-frequency ammeter, and the resistance R* is calculated from the observed current / FIG. 126. Method of measuring resistance by calorimeter method and the power, or rate of heat production, P. P = RJ> (77) 178 Circular of the Bureau of Standards While P might be measured calorimetrically, in practice it is always measured electrically by an auxiliary observation in terms of low-frequency or direct current. Thus, it is only necessary to observe the temperature of the calorimeter in any arbitrary units when the high-frequency current flows after the temperature has reached a final steady state, and then cause low-frequency current to flow in the circuit, adjusting its value until the final temperature becomes the same as before. Denoting by the subscript o the low-frequency values P = R tion (66) and the relations 8 = TTJ and 5, = TTJ-^, Lu Lea KI 2 (87) where * = ' + yVa (88) This is, of course, not an explicit solution for R, since K involves 5 and therefore R, but gives a ready means of finding R or 5 when the sum of the two decrements 8' + 8 is known from some other measurement, such as the reactance-variation method described below. Thus, a combination of the two methods gives both 5' and 5, or 5' and R. There are two interesting special cases in which the measure- ment is simplified. When the decrement 8' of the supplied emf is very small and is negligible compared with 5, equation (87) reduces to #= (89) identical with (80) above, the equation for the use of undamped emf. This is to be expected, since undamped emf is the limiting case of small decrement. When, on the other hand, 5 and 8 t are both very small compared with 8', K becomes unity and equation (87) reduces to R=RI - (90) This happens to be the same as equation (84) above, the equa- tion for the use of impulse excitation. The proof given here Radio Instruments and Measurements 185 can not, however, be regarded as a deduction of equation (84) for impulse excitation, as it has been by some writers, since Bjerknes' equation (p. 187) is involved, which assumes that 8' and 5 are both small. 50. REACTANCE VARIATION METHOD This has been called the decrement method, a name which is no more applicable to this than to the other methods of resistance measurement since all measure decrement in the same sense that this does. That the method primarily measures resistance rather than decrement is seen from the fact that in its simple and most accurate form it utilizes undamped current, which has no decre- ment. FlG. 128. Circuit for measuring resistance by the reactance variation method The method is analogous to the resistance-variation method, two observations being taken. The current I T in the ammeter is measured at resonance, the reactance is then varied and the new current 7 t is observed. The total resistance of the circuit R is calculated from these two observations. The reactance may be varied by changing either the capacity, the inductance, or the frequency, the emf being maintained constant. The reactance is zero at resonance and it is changed to some value X t for the other observation. With undamped emf E, the currents are given by E 3 From these it follows that r> -y t\. -^-i (91) This has a similarity to R=R t l , the equation (80) for * ~A the resistance-variation method. It is also interesting that when 1 86 Circular of the Bureau of Standards the reactance is varied by such an amount as to make the quantity under the radical sign equal to unity, the equation reduces to -*, (92) This is similar to R=R 19 which is the equation for the quarter- deflection and half-deflection resistance variation methods. Special Cases of Method. When the reactance is varied by changing the setting of a variable condenser, the equation (91) becomes (27) given on page 38. For variation of the inductance, (91) becomes - 2 (93) For variation of the frequency, (91) becomes r -Y/T^ (94) It must be noted that variation of the frequency requires some alteration in the source of emf , and the greatest care is necessary to insure that the condition of constant emf is fulfilled. A convenient method which differs slightly from those just described is to observe two values of the reactance both corre- sponding to the same current 7j on the two sides of the resonant value 7 r . For observation in this manner of two capacity values Ci and C 3 , _ i C.-Q /-jrT- - Decrement Calculation. It is often convenient to calculate directly the decrement of the circuit instead of the resistance. Formulas for decrement exactly corresponding to the resistance formulas already given are obtained by application of the simple relations between resistance and decrement and are the same as formulas (96) to (100) below with 5' omitted. The formulas thus obtained are rigorous, as are the foregoing resistance formu- las, for undamped emf, and hold with sufficient accuracy for damped emf when the damping is negligibly small. When the damping of the supplied emf is appreciable, the same procedure is followed in making the measurement, and the equations are only slightly different. When the emf is supplied by coupling to a primary circuit in which current is flowing with a decrement 8', Bjerknes' classical proof shows that the sum of Radio Instruments and Measurements 187 the primary and secondary decrements is given by the same expression as that which gives the decrement 5 of the secondary when the emf is undamped. Thus (27), (93), (94), and (95) correspond to rtjj (98) ; _.i (99) Formula (98) is also equivalent to ^ / JM l r ( I0 ) These formulas are correct only when: (i) The coupling between the two circuits is so loose that the secondary does not appre- ciably affect the primary, (2) 5' and 5 are both small compared (C C) with 2Tr, and (3) the ratio -^-^ - and the corresponding ratios are small compared with unity. In any of these methods the calculation is obviously simplified if the reactance is varied by such an amount as to make If = ^2/ r 2 . This is done very easily when the current measuring instrument is graduated in terms of current squared. The quantity under the square-root sign in all the preceding equations becomes unity, greatly simplifying the formulas. A still further simplification by which all calculation is eliminated is utilized in special decre- meters as described below, section 55. 51. RESISTANCE OF A WAVE METER The accurate measurement of resistance or decrement of a wave- meter circuit is of first importance because the wave meter is fre- quently used to measure other resistances and the decrements of waves. It is the calibration of a resistance-measuring standard. Several forms of the resistance-variation and the reactance- variation methods may be used. 1 88 Circular of the Bureau of Standards The resistance of a wave meter is, of course, not a single, con- stant value. It varies with frequency and with the detecting or other apparatus connected to the wave-meter circuit. Usually both the resistance and the decrement of the circuit vary with the condenser setting. It is usually desirable to express either re- sistance or decrement in the form of curves for the several wave- meter coils, each for a particular detecting apparatus or other condition. An example of such a curve is given on page 190. Resistance Variation. Any of the forms of the resistance- variation method may be used. The apparatus and procedure are the same in all cases. The wave-meter coil is loosely coupled to the source. The current is read on the indicating device shown schematically as A in Fig. 129. A resistance standard of the type already described is then inserted at R! and the current read FIG. 129. Measurement of wave meter resistance again. The calculation of resistance depends on the damping of the source and the kind of current-measuring device. When a pliotron, arc, or other source of undamped waves is used, formula (80) above is used. When the current-measuring device is a current-square meter, thermocouple or crystal detector with galvanometer, or other apparatus which is so calibrated that deflections are accurately proportional to the square of the current, and when in addition a continuously variable resistance standard is used, the quarter-deflection method may be employed elimi- nating all calculation. As explained on page 182, the resistance of the circuit is equal to the inserted resistance required to reduce the deflection to one-quarter. When a buzzer or other damped source is used, some auxiliary measurement or special condition is needed, in order to evaluate or eliminate the decrement of the source. If this decrement is known, the decrement or resistance of the wave-meter circuit may be obtained. The solution is, however, complicated and, as a matter of fact, this method in not used. Instead of 5', the decrement of the source, being known explicitly, the more usual case is that 8' +6, the sum of the decrements of source and Radio Instruments and Measurements 189 wave meter, is known from a measurement by the reactance-vari- ation method. The wave-meter resistance is then calculated by and resistance and decrement are related o +o r> by 8 = r and the similar formulas given on page 316. The calcu- lation is considerably simplified in the two special cases of d f very small or 5 very small; formulas (98) and (99), respectively, apply. The latter is identical with the equation for impulse excitation, but with that exception these methods may be used only when 5' and 5 are both small. For impulse excitation from a buzzer or other source, equation (84) is used to calculate the resistance. When the current indi- cator is calibrated in terms of the square of current and the resistance standard is continuously variable, the measurement is conveniently made by the half-deflection method. In this case the resistance of the circuit equals the inserted resistance. Reactance Variation. This method may be used with either a damped or undamped source. When the emf is undamped or of extremely small damping, formulas (27) and (93) to (95) apply. It is customary to reduce the labor of computation by varying the / 72 reactance by such an amount that .*/ * =i, in which 1 r ./i (Cr-Q case r> taC f C R=a(L-L r ) (102) g_ L (co 2 ay 2 ) (103) i C 2 Cj (104) .TV = 7; TT~ 2CO C 2 L! When damped emf is used, formulas (96) to (100) apply. They also are simplified in practice by making */ : \ IT *j They require either that 5', the decrement of the source, be known, or that another relation be obtained between 5' and 5 by an independent measurement. It is not often that a source 190 Circular of the Bureau of Standards of fixed, known decrement is maintained in a laboratory, as the decrement varies with frequency and every other condition of use. Hence, the usual procedure is the combination of this measurement with a resistance-variation measurement as described above. An example of measurement of wave-meter resistance expressed in terms of decrement is given in Fig. 130. This shows the results of two independent measurements, one by the resistance-variation 004 0.03 OiOZ I =* RESISTANCE VARIATION (IMPULSE EXCITATION) X REACTANCE VARIATION CJMOAMPED EXCITATION) 20 <** 6O SO 100 ;20 J4-O t6O ISO FIG. 130. Variation of the decrement of a wave meter with condenser setting method, using impulse excitation and equation (84), and another by the reactance-variation method employing equation (27). 52. RESISTANCE OF A CONDENSER The methods ordinarily used for measurement of resistance of a condenser or of an inductance coil require a variable condenser whose effective resistance must be either negligibly small or accu- rately known. This condenser is used to retune the circuit to resonance after the unknown is taken out of the circuit, The standard condensers of negligible resistance used at the Bureau of Standards are described on page 119. These measurements may be made with an ordinary wave meter, provided the resist- ance of the circuit is accurately known for different condenser settings. Simple Methods. The simplest method is that of substitution. The condenser to be tested is connected in series with a coil and an ammeter of some sort, and loosely coupled to a source. The condenser is then replaced by the standard condenser and a series resistance. The resistance required to make the deflection at resonance the same as before is taken as the resistance of the condenser. This method is not very accurate, because the change of condensers changes the emf's electrostatically induced in the circuit. Radio Instruments and Measurements 191 Another method utilizes the principle of reactance variation. The frequency supplied to the condenser circuit is varied by changing the setting of a condenser in the supply circuit. Under certain conditions, equation (27) or (95) applies. The method can be made to give phase differences directly by use of a suit- able scale of phase differences on the condenser in the supply circuit. (See sec. 55.) Precision Method. Accurate measurements may be made by the resistance- variation method. The circuit is tuned to resonance by varying the frequency supplied, and the total resistance of the circuit is measured, with the unknown condenser in circuit. It is replaced by the standard condenser, the setting of which is varied until resonance is obtained, and then the resistance of the circuit is measured again. The difference of the two measured values is QJ v^ Source O R O -WWWWSM FIG. 131. Circuit used for measurement of high-frequency resistance of a condenser the resistance of the condenser under test. As previously noted, precautions are necessary to avoid changing the stray electro- static emf's in the circuit when the resistance R l is introduced. In respect to this it has been found desirable to insert R t between the condenser and the current-measuring device, and, if a ground wire is used, to connect it to the shielded side of the condenser or to the ammeter case. Also, the coupling must be loose enough so that too much power is not withdrawn from the source. The manipulation is made more convenient by using a double- throw switch to place the two condensers in circuit. The base of the switch must be a material which has very small phase differ- ence; paraffin has been found suitable. The resistance standards, when in the shape of short links, may be used as part of this switch, as shown at R! in Fig. 132. Another refinement of the measurement is to place a small variable condenser C\ of negligible resistance in parallel with the inductance coil. This gives a fine adjustment to resonance. Example. An example of a measurement at one frequency with a pliotron as a source of undamped emf is given below. The 35601 18 13 Circular of the Bureau of Standards condenser is a fixed condenser with molded dielectric. The column headed "d" gives deflections of galvanometer attached to FIG. 132. Circuit for precision measurement of con- denser resistance -with switching device and small tuning condenser a thermocouple; deflections are proportional to the square of the current. TABLE 4. Observations on Resistance of a Mica Condenser [0-0.00406 M'. L 40 nh, X 760 m, /?x ^?N=o.o9 ohm= resistance of condenser X; phase difference 0.09(4,060) 6-s ITT -3 -J c Ri Galvanometer Vd . R R ' Zero Deflection d Mean d ar 1 / d -l Vdi X N 0.503 .810 1.042 .810 .503 .503 .810 1.042 .810 .503 13.95 13.95 13.92 13.90 13.95 13.90 13.92 13.98 13.92 13.92 13.88 13.88 13.90 13.95 45.70 31.68 27.30 24.82 27.18 31.75 45.95 48.58 32.45 27.72 25.15 27.70 32.60 48.50 31.75 17.73 13.38 10.92 13.23 17.83 32.03 34.60 18.53 13.80 11.27 13.82 18.70 34.55 31.89 17.78 13.30 10.92 1. 480] 1.479ll.477 1.472 1.390 1.391 1.38s 1.387 0.340 .548 .708 34.58 18.62 13.81 11.27 .362 .582 .752 Radio Instruments and Measurements 193 The resistance of a condenser is generally measured at several frequencies. If the resistance is mainly due to dielectric absorp- tion, the resistance is generally inversely proportional to fre- quency. Variable condensers are usually measured at several settings. For a variable air condenser with semicircular plates having a small resistance mainly due to dielectric absorption in the separating insulators, the resistance is inversely proportional to the square of the setting. 53. RESISTANCE OF A COIL The resistance of a coil depends upon its position in the circuit, i. e., whether the emf acting upon the circuit is impressed in the coil itself or at some other point in the circuit. This difference arises from the effects due to the capacity of the coil. When the emf is induced in the coil itself, the capacity of the coil is to be considered in series with the inductance of the coil but in parallel vwwwwv-0 ' FIG. 133. Circuit for measuring high-fre- quency resistance of a coil by the substi- tution method with the rest of the circuit. When, however, the emf is impressed at some other point in the circuit, the coil capacity and inductance are in parallel with each other. When the resistance of a coil is measured by any of the methods given in this section, the value of the resistance obtained is valid for the coil only when used in the same position relative to the driving emf. The simplest method is that of substitution. In a wave meter circuit coupled to a source of emf by an independent coupling coil, the deflection is first observed with the coil whose resistance is desired in circuit. If the wave meter condenser resistance is known for various settings, the coil is then replaced by a stand- ard coil whose resistance is known, the condenser retuned to resonance, and resistance inserted until the deflection is the 1 94 Circular of the Bureau of Standards same as before. The change of coils may be made by a double- throw switch. If a variable inductor of known resistance is available, the procedure is still simpler, as the condenser setting need not be changed; the coil under test is replaced by the variable inductor, which is used to obtain resonance, and then resistance is inserted to equalize deflections. As in previous cases, the sub- stitution method is not an accurate one, but is valuable for speedy determination of approximate values. More accurate measurements may be made by either the resist- ance-variation or the reactance-variation method. Either of these methods may be used to determine the resistance in each of the three following procedures. Known Circuit. The circuit consists of the unknown coil, a condenser, the current-indicating instrument, and connecting leads. The emf is introduced into the circuit by coupling to the unknown coil. The resistance of the total circuit is obtained by the resist- ance-variation or the reactance- variation method and the coil resistance determined by sub- tracting the resistance of the rest FIG. 134. Coil resistance measurement in ^ terms of known circuit or standard coil of the Circuit. ^ The Condenser should be practically perfect or of known resistance, and the leads should be of negligible or calculable resistance. If the indicating instrument is a thermo- element or current-square meter with fine wire heating element, its resistance may be determined with direct current. In doubtful cases it can be determined at the frequency of the measurement by a separate experiment as outlined on page 179. Thermo- elements of low resistance, as described on page 157, are especially suited for this measurement, since the precision is higher if the resistance of the indicating device is a small part of the total resistance of the circuit. Known Coil. Two resistance measurements at high frequency are required when the unknown coil is compared with a standard coil of known resistance. The inductance of the standard coil should be of the same order of magnitude as that of the unknown, but need not be equal to it. The standard is substituted for the unknown and the condenser varied for resonance and the resistance of the circuit obtained in each case. Auxiliary Coil. The third procedure is to measure the resistance of a simple radio circuit consisting of a condenser, ammeter, Radio Instruments and Measurements 195 and any coil, all in series, and then insert the coil to be tested, re tune to resonance, and measure the resistance again. The coil must be inserted at such a place in the circuit that it has no mutual inductance with the auxiliary coil, and so that there is no emf induced in it by the source. 54. DECREMENT OF A WAVE Any measurement of the resistance of a circuit by the methods already given is in a sense a measurement of decrement, since it enables calculations of the decrement which the circuit would have when oscillating freely. In particular, the use of impulse excitation with the resistance-variation method measures the decrement of the oscillations actually existing in the measuring circuit during the measurement, and therefore the decrement of the wave emitted by the measuring circuit. There is a class of decrement measurements entirely apart from the measurement of decrement or resistance of a circuit. This is the determination of decrement of an emf, due either to a nearby antenna or other circuit or to a wave traveling through space. The fundamental principles of decrement measurement have been given in section 26 above. A simple wave-meter circuit is placed so as to receive the wave and a decrement measurement is made by either the resistance-variation or the reactance-variation method. If the resistance of the wave-meter circuit is known for the frequency and other conditions of the measurement, the decrement of the wave is calculated. For the resistance-variation method 8', the decrement of the wave, is given by formula (63), page 63, or one of the simplified formulas (64) or (65) . For the reactance-variation method the decrement is given by (96) to (100), page 187, or, in case the simplified procedure is followed, making I? = yZIr 2 , y+g-x (c ^" Q (105) g' + s =7r (j ;- Lr) ( I0 6) L. T '-) (I07 ) (108) 196 Circular of the Bureau of Standards The measurement may also be made by a direct-reading decre- meter (see next section), and (5 ' +8) read directly from the scale of the instrument. When the resistance or decrement of the wave-meter circuit is not known, measurements are made by both methods: and the combination of the two yields the value of d f , making use of equa- tion (63). 55. THE DECREMETER A decremeter is a wave-meter conveniently arranged for meas- urements of resistance or decrement. The forms usually employed make use of the reactance-variation method. Ways have been devised for manipulating the instrument in such a way that the decrement may be read directly from a scale. While, of course, resistance can be calculated from a measured value of decrement, the principle application of the decremeter is in the measurement of the decrement of a wave. Another important use is in the measurement of phase differ- ence of a condenser. Since the decrement due to a condenser is TT times its phase difference, a measurement of decrement gives directly the phase difference; if desired, the scale may be cali- brated in terms of phase difference instead of decrement. Determination of the Scale of a Decremeter. Any wave meter whose circuit includes some form of ammeter may be fitted with a special scale from which decrements may be read directly. The procedure for a wave meter having any sort of variable con- denser is given here. In the usual use of the reactance variation method of deter- mining decrement, the current I T is observed when the condenser is adjusted to the value C r to produce resonance, and the con- denser is then changed to another value C and the current / t read. When the second condenser setting is such that the If = 7 r 2 , the decrement is calculated by <-+-, -Q A certain value of decrement therefore corresponds to that dis- placement of the condenser's moving plates which varies the capacity by the amount (C r -C). The displacement for a given decrement will in general be different for different values of C, the total capacity in the circuit. At each point of the con- denser scale, therefore, any displacement of the moving plates which changes the square of current from 7 r 2 to - 7 r 2 means a certain value of (8 ' Radio Instruments and Measurements 197 A special scale may therefore be attached to any condenser with graduations upon it and so marked that the difference be- tween the two settings, when the square of current is 7 r 2 and - 7 r 2 , is equal to the decrement. The spacing of the graduations at different parts of the scale depends upon the relation beween capacity and displacement of the moving plates. When this rela- tion is known, the decrement scale can be predetermined. A scale may therefore be fitted to any condenser, from which de- crement may be read directly, provided the capacity of the circuit is known for all settings of the condenser. The decrement scale may be attached either to the moving-plate system or to the fixed condenser top. It is usually convenient to attach it to the unused half of the dial opposite the capacity scale. The value of decre- ment determined by this method is (d f + 6) , where 6 is the decre- ment of the instrument itself. This must be known from the calibration of the instrument (e. g., as in sec. 51), the value of 5', the decrement of the wave under measurement, being then ob- tained by subtraction. Simple Direct-Reading Decremeter. It is particularly easy to make a decremeter out of a circuit having a condenser with semi- circular plates. Such condensers follow closely the linear law, where 6 is the angle of rotation of the moving plates and a and C are constants. It can be shown that the decrement scale ap- plicable to such a condenser is one in which the graduations vary as the logarithm of the angle of rotation. Furthermore, the same decrement scale applies to all condensers of this type. This scale has been calculated and is given in Fig. 135. It is calculated to fit equation (108), for observations on both sides of resonance, and not for equation (105). This scale may be used on any con- denser with semicircular plates. The scale may be cut out and trimmed at such a radius as to fit the dial and then affixed to the condenser with its O point in coincidence with the graduation which corresponds to maximum capacity. This usually puts it on the unused half of the dial opposite the capacity scale. If the figures are trimmed off, they can be added over the lines in red ink. If it is desired not to mutilate the page, the scale may be copied. This scale gives accurate results if the capacity scale is so set that its indications are proportional to the capacity in the circuit. 198 Circular of the Bureau of Standards A measurement of decrement is made by first observing the current-square at resonance, then reading the decrement scale at a setting on each side of resonance for which the current-square FlG. 135. Direct-reading decrement scale for any semicircular- plate condenser is one-half its value at resonance. The difference between the two readings on the decrement scale is the value of 8 ' + 8. The scale permits accurate measurement of fairly large decre- ments, but offers no precision in +he measurement of very small decrements, particularly at the low-capacity end of the scale. Radio Instruments and Measurements 199 A similar scale is readily made to read phase differences di- rectly. The readings of Fig. 135 are all multiplied by 18.24, t give phase differences in degrees. The instrument is then specially valuable in measuring the phase differences of condensers. Decremeter with Uniform Decrement Scale. Just as it is pos- sible to determine a decrement scale to fit a condenser having any sort of law of capacity variation, it is equally possible to design a condenser with capacity varying in such a way as to fit any specified decrement scale. A uniform decrement scale i. e., one in which the graduations are equally spaced is particularly con- venient, and is the kind used in the Kolster decremeter. A uni- form decrement scale requires that the condenser plates be so shaped that for any small variation of setting the ratio of the change in the capacity to the total capacity is constant. The condenser required to give this uniform scale has its moving plates so shaped that the logarithm of the capacity is proportional to the angle of rotation of the plates. This condenser is dis- cussed above on page 116. The decremeter is fully described in Bulletin of the Bureau of Standards, 11, page 421 ; 1914 (Scientific Paper No. 235). A view of the inside arrangement is shown in Fig. 218, facing page 320, of this circular. The spiral shape of the condenser moving plates is shown. Figures 219 and 220, facing page 32, show different forms in which the instrument is made. The decrement scale is not attached directly to the moving plates, but is on a separate shaft geared to the moving plates at a 6 to i ratio. The decrement scale is thus opened out, so that very precise measurenemts may be made. To measure decrement the condenser setting is first varied to obtain maximum deflection of the current-square meter, and then varied until half this deflection is obtained. The movable decre- ment scale is then set at zero and clamped to its shaft, and the condenser setting is varied until the same deflection is obtained on the other side of the maximum. The reading on the decrement scale is the value of decrement sought. This decremeter is used in the inspection service of the Bureau of Navigation, Department of Commerce, and by radio engineers in the Army and Navy and elsewhere. 2OO Circular of the Bureau of Standards SOURCES OF HIGH-FREQUENCY CURRENT 56. ELECTRON TUBES For the purposes of measurement as well as in the transmission of radiograms, sources which furnish undamped currents are coming into more general use than those which supply damped oscillations. Thus the various forms of spark, used almost exclusively in the past, are giving way to the electron-tube gen- erator, the arc, and the high-frequency alternator. These sources are used both in radio communication and in laboratory work, but for the latter the electron tube is preeminently the best. When the source furnishes undamped current many methods of measurement are simplified, and since the sharpness of tuning is increased, a higher precision is obtainable in methods which depend upon tuning to resonance. Since most of the measurements at radio frequencies are based upon deflection methods, the primary requirement of a source for such measurements is that the intensity and frequency of the generated current be constant. It is only within the past few years that an almost ideal source has become available, viz, elec- tron-tube generators, such as the pliotron, audion, oscillion, etc. These electron tubes consist of an evacuated bulb containing the following three elements: (a) A heated filiament which acts as a source of electrons. (&) A metal ' ' plate ' ' placed near the electron source. (Across the plate and filament, outside of the bulb, is connected a bat- tery, so that an electron current flows from filament to plate.) (c) A "grid" consisting of fine wire or of a perforated metal sheet placed between plate and filament so that the electrons have to pass through the grid to get from filament to plate. Using these sources, undamped high-frequency current can be obtained which is as steady as the current from a storage battery and strictly constant in frequency. Furthermore, these genera- tors are extremely flexible as to the frequencies which can be obtained, the same tubes having been used to generate currents ranging in frequency from i cycle in two seconds to 50000000 cycles per second. 25 A high-frequency output of 500 watts or more has been attained with a single tube. For ordinary meas- uring purposes with sensitive indicating devices, such as a thermo- element and galvanometer, about 5 watts of high-frequency 25 See reference No. 138, Appendix 2. Radio Instruments and Measurements 201 output are sufficient. When it is desired to use low-range hot- wire ammeters, 10 to 20 watts may be required. On account of the extreme importance of the three-electrode tubes, both as generators and as detectors, and since the full realization of their utility and a satisfactory explanation of their functioning are of recent date, it is worth while to outline rather fully the phenomena upon which their operation is explained. Thermionic Emission. The modern conception of current flow in metals assumes that the conduction of electricity consists in the motion of electrons (see p. 8) under the action of an applied electromotive force. When not acted upon by an external emf these small negatively charged particles move about in the metal in zigzag paths in all directions, colliding with the atoms of the metal. Their mean velocity of motion depends upon the tem- perature, increasing with the temperature. At the surface of the metal, according to the theory of Richardson, 26 the electrons are restrained from leaving the metal by electric forces entirely simi- lar to the molecular forces which cause the surface tension of a liquid. Further, just as in the evaporation of a liquid, a certain number of electrons will in each second attain a high enough velocity to escape from the metal, and since the mean velocity increases with the temperature the number of electrons escaping per second will increase with the temperature. The heated fila- ment in the audion or pliotron is in this manner a source of electrons. The withdrawal of the negative electrons from the heated filament leaves it positively charged, thus tending to draw them back again and a state of equilibrium may be attained in which the same number are drawn back per second as are being emitted. When, however, a body maintained at a positive potential relative to the filament is brought into the field a certain proportion of the electrons will be attracted to the positive body and constitute a current of electricity between the filament and the positively charged body. In the electron tube this body is the "plate" and its potential is maintained positive with respect to the filament by a battery commonly called the B battery. If in the case of a tube with an extremely high vacuum the voltage of the B battery is increased, the flow of electrons or "plate current" will increase up to a point where practically all of the electrons emitted by the filament are being drawn over to the plate. If, on the other hand, the plate voltage is kept constant and the filament temperature increased, thus increasing the 28 See reference No. 131, Appendix 2. 202 Circular of the Bureau of Standards number of electrons emitted per second, the plate current will also increase up to a certain temperature, but beyond this tempera- ture will remain practically constant even though more electrons are being given off. The explanation of this behavior 27 is that the stream of negative electrons flowing through the tube acts as a space charge of negative electricity which neutralizes the field due to the positive plate. In consequence only a limited number of electrons can flow to the plate per second with a given plate voltage, and the remainder are compelled to return to the filament again. Grid Control. If in any way this space charge is neutralized, there will be an increase in the plate current; on the other hand, Grid Voltage FlG. 136. Variation of plate current (usu- ally in milliamperes) and grid current (us- ually in microamperes} with grid -voltage anything that will aid the space charge will result in a decrease in the plate current. In the audion or pliotron these effects are brought about by the grid of wires between the plate and filament. If this grid is charged positively with respect to the filament, the effect of the space charge will be neutralized to an extent depend- ing upon the charge on the grid, and the electron current through the tube will increase until the field due to the grid charge is also neutralized by the space charge. 28 Some few electrons will strike the 27 See reference No. 133, Appendix 2. 28 In tubes which are not evacuated to a high degree, the residual gas may become ionized and markedly affect the behavior of the tube. The ionization of the gas tends to neutralize the space charge, thus per- mitting larger currents to pass through the tube. To a certain extent such ionization is of value in the use of the tube as a detector, though when the ionization becomes intense and the tube shows a blue glow, so large a current passes through the tube that it is unaffected by variations of the grid voltage and its detect- ing qualities are lost. Radio Instruments and Measurements 203 grid wires and there will result a flow of current in the grid circuit, but this will, in general, be small relative to the plate current. If the grid is charged negatively with respect to the filament, the charge on the grid will then aid the space charge in driving the electrons back to the filament, resulting in a lowering of the plate current. In this latter case the number of electrons striking the grid will be very small and in consequence practically no current will flow in the grid circuit. The control of the plate current by the grid voltage and also the dependence of the current in the grid circuit upon the grid voltage are shown in curves of Fig. 136. Curve A shows the current in the plate circuit when the B battery is kept constant, but different voltages are applied between the grid and that terminal of the filament to which the negative of the filament battery is connected. Curve B represents on a magnified FIG. 137. Scheme of connections for determining characteristic curves scale the current in the grid circuit for different voltages of the grid with respect to the negative terminal of the filament. The ordinates and shape of these so-called characteristic curves depend upon a number of factors, such as B battery voltage, fineness and spacing of the grid wires, location of the grid relative to the other elements, etc. Fig. 137 shows the scheme of connections which may be used in determining such curves. The ammeter A l measures the current in the grid-filament circuit and A 2 measures the plate current. By means of the sliding contact on the shunt resistance to the battery C, the voltage between the filament and grid may be varied and made positive or negative, the voltage being read by the voltmeter V. The B battery voltage is held constant while the curve is taken. 204 Circular of the Bureau of Standards 57. ELECTRON TUBE AS DETECTOR AND AMPLIFIER As Detector of Damped Oscillations. A single tube may per- form separately or simultaneously the functions of a detector, amplifier, and generator. It will first be considered as a simple detector of damped oscillations. The circuits shown in Fig. 138 indicate one possible way of using the tube as a detector. The circuit LC\ is tuned to the oscillations in the antenna A. The C battery with variable resistance permits the adjustment of the grid potential with respect to the filament, so that the tube may be worked at any point on the characteristic curve of the plate cur- rent. Suppose that this voltage is adjusted to correspond to the FIG. 138. Possible circuits for using the electron tube as a detector of damped oscillations point X (Fig. 1 39) where the change in slope of the curve is large. If now a train of oscillations is set up in the antenna and hence in the secondary circuit, the alternating voltage across the condenser terminals will be superimposed upon the steady voltage of the C battery. It will be seen from the characteristic curve that an increase in voltage, say, from a to b, produces a large increase in the plate current (i. e., from x to y) , while a decrease in voltage of the same amount from a to c produces a much smaller change (from x to z) in the current. Thus, as the result of a wave train such as (i) in Fig. 140, the plate current will be changed about its normal value in some such way as (2) which is equivalent to a resultant increase in plate current. This increase of plate current Radio Instruments and Measurements 205 during a train of waves gives rise to a pulse of current in the tele- phone as shown in (3). This pulse will act upon the telephone diaphragm, and if the wave trains and hence the pulses in the telephone current are arriving at the rate of 1000 per second (cor- responding to the spark frequency at the transmitting station), a looo-cycle note will be heard in the phone. This use of the tube as a detector is entirely similar to the use of a crystal detector. Condenser in Grid Lead. If a condenser is inserted instead of the C battery in the lead to the grid, as C 2 in Fig. 141, the behavior cab O Grid Voltaqe, FIG. 139. Plate characteristic, showing region of curve where rectifying action is large of the tube as a detector of damped 29 oscillations is altered and depends to a great extent upon the characteristic curve of the grid current. The grid is insulated from the filament by the con- denser C 2 , excepting for such leakage as may take place through this condenser or in or about the tube. Suppose first that the tube is put into operation with the grid and filament at the same potential and with no incoming oscillations. It will be seen from 29 Although damped oscillations are referred to here and in the usual treatments of the subject, the same considerations apply to undamped oscillations which are periodically interrupted either in the trans- mitting or receiving circuits so that the tube receives groups or trains of waves. 206 Circular of the Bureau of Standards the filament-grid curve that there will be a flow of electrons to the grid and the grid will become negative with respect to the filament, FlG. 140. Action of the electron tube as a detector: (7) Incoming oscil- lations, (2) variations in plate current, (j) effective telephone pulses thereby reducing the flow to itself and to the plate until the leakage away from the grid is equal to the flow to it. In some tubes the FIG. 141. The electron tube as a detector of damped oscillations, using a con- denser in the grid circuit grid may be so highly insulated that it accumulates a negative charge sufficiently high to reduce the plate current practically to Radio Instruments and Measurements 207 zero. In such cases it is necessary to provide an artificial leak through a high resistance across C 2 . Suppose now that the grid has attained its equilibrium potential and the plate current its corresponding value and a series of wave trains impinges upon the antenna as in (i) of Fig. 142. The oscillations in the circuit LC\ will cause the grid potential to oscillate about its normal value. When the grid becomes positive Incom'mq Oscillations J Grid Potential 4- Plate Current Pulses m phone FIG. 142. Action of the electron tube as a detector connected as in Fig. 141 relative to its normal value there will be a considerable increase in the flow of electrons to it, overbalancing the reduction in the flow when on the negative half of the wave. Thus, during a wave train the grid will accumulate a negative charge and its mean potential will be lowered, as in (2) of Fig. 142. In conse- quence the mean plate current will be reduced. However, between wave trains the excess charge on the grid will leak off, restoring the plate current to its normal value. This is shown 35601 18 14 208 Circular of the Bureau of Standards in (3) of Fig. 142. Each wave train will produce a reduction in the current through the phones as in (4) of the same figure and a note corresponding to the wave train frequency will be heard. FiG. 143. Use of electron tube as an amplifier Amplification. If, as in Fig. 143, a source of alternating emf were interposed between the filament and grid of aji audion or pliotron, the potential of the grid with respect to the filament would alternate in accordance with the alternations of the FIG. 144. Variations of plate current with grid -voltage generator. These variations of the grid potential produce changes in the plate current corresponding to the plate characteristic. If the mean potential of the grid and the amplitude of its alter- nations are such that the plate current is always in that portion of Radio Instruments and Measurements 209 its characteristic where it is a straight line, then the alternations of the grid potential will be exactly duplicated in the variations of the plate current and the latter will be in phase with the former, at least in a high vacuum tube. Thus, if (a) of Fig. 144 represents the alternating potential of the grid, then (6) would represent the fluctuations of the plate current. For a given amplitude in (a), the amplitude of the alternating component in (b) will depend upon the steepness of the plate characteristic, increasing with increasing slope. The alternator in the grid lead supplies only the very small grid-filament current, thus the power drawn from it is extremely small. The power represented by the alternating component of the plate current is, however, considerable; thus there is a very large power amplification. This larger source of 5 P FIG. 145. Use of electron tube as a regenerative amplifier power might be utilized by inserting the primary P of a trans- former in the plate circuit, as in Fig. 143, in which case the alter- nating component alone would be present in the secondary 5. This illustrates the principle of a vacuum tube as a relay. The voltage in 5 might again be inserted in the grid lead of a second vacuum tube and with proper design a further amplification ob- tained in the plate circuit of the second tube. This may be carried through further stages and illustrates the principle of multiple amplification. Regenerative Amplification. It has been shown by E. H. Armstrong 80 that amplification similar to that obtained with * See reference No. 134, Appendix a. 2IO Circular of the Bureau of Standards several stages may be secured with a single tube. Instead of feeding the voltage of the secondary coil 5 into the grid circuit of a second tube it is fed back into the grid circuit of the same tube so as to increase the voltage operating upon the grid. This results in an increased amplitude of the plate-current alternations which likewise being fed back into the grid circuit increases the voltage operating upon the grid, etc. One form of the so-called feed-back circuit for rectifying and amplifying damped oscillations is shown in Fig. 145. The oper- ation of the circuit, used as a receiving device, is the same as that described above for the case of a condenser in the grid lead. The condenser C 2 is merely to provide a path of low impedance across the phones for the high-frequency oscillations. The coils P and 5 constitute the feed back by means of which the oscillations in FIG. 146. Use of electron tube as a generator the tuned circuit are reinforced. The mutual inductance between S and P must be of the proper sign so that the emf fed back aids the oscillations instead of opposing them. 58. ELECTRON TUBE AS GENERATOR Generation of Oscillations. If the coupling between the coils P and 5 in Fig. 145 is continuously increased and the values of L, S, and C and the resistance of this circuit are suitable within certain limits, the emf fed back by the coil P into the oscillatory circuit at any instant will become greater than that required to just sustain the oscillations in the circuit. In this case any oscil- lation, however small in the circuit L, S, C, will be continuously built up in amplitude until a limit determined by the character- istics of tube and circuits is reached. In other words, the tube self-generates alternating current of a frequency determined by the natural frequency of the oscillatory circuit. Radio Instruments and Measurements 211 Numerous circuits have been devised to produce oscillations. Fig. 146 shows a method of connection which is suitable for pro- ducing large currents. The oscillatory circuit is in the filament- plate circuit and a coil between filament and grid. The operation of this circuit is somewhat different from that outlined above. Instead of transferring all of the energy necessary to sustain the oscillations from the plate to the grid circuit as in the preceding case, only an emf which serves as a control is here transferred. Thus, the grid circuit plays a similar part to that of the slide valve in a reciprocating engine. The path of the current flow within FIG. 147. Generating circuits in which the oscillatory circuit is inductively coupled to both the grid and plate circuits the tube from plate to filament may be regarded as a variable resistance, the value of which depends upon the potential of the grid. If the potential of the grid is alternating, the resistance will increase and decrease in accordance, thus throwing an alter- nating emf upon the oscillatory circuit in series with this resistance. The oscillatory circuit which determines the frequency may be a separate circuit, as in Fig. 147. Here the coupling M 2 supplies the emf to reinforce the oscillations and M x furnishes the emf to the grid. The condenser Q is a large fixed condenser which serves as a path of low impedance across the battery for the high- frequency alternations in the plate circuit. 212 Circular of the Bureau of Standards In addition to the above types of circuit in which electromag- netic coupling between the plate and grid circuits is used to trans- fer emfs from one to the other, there are also circuits in which electrostatic coupling is utilized. This is illustrated in Fig. 148, in which the condenser C 2 serves as the coupling. The induc- tances L! and L 2 should be variable and approximately equal. Cf is a fixed condenser which serves as a path of small impedance for the high frequency around the battery. The frequency is primarily determined by the inductances L x and L 2 and the con- denser C 2 . The parallel connection of C\ and L t serves as an "absorbing" circuit that is, as C l is increased from a very low Ca FIG. 148. Generating circuits in which the plate and grid circuits are electrostatically couphd value the current circulating around this circuit will increase up to a certain point and may considerably exceed the current in the other portions of the circuit. Reception of Undamped Oscillations. If two sources, which separately furnish undamped oscillations of, say, 100 ooo and 101 ooo frequency, as shown in (a) and (6) of Fig. 149, act together upon the same circuit, the resultant oscillations in the circuit, obtained by adding the components, will be of the form shown in (c). The mode of adding the components is illustrated in Fig. 150. The amplitude of the combined oscillation will rise and fall, Radio Instruments and Measurements 213 becoming a maximum when the component oscillations are in phase and a minimum when they are 180 out of phase. The beats or periodic rise and fall in amplitude occur at a rate equal to the difference in frequencies of the two oscillations. Thus, the FIG. 149. Principle of heterodyne reception; (a) incoming oscillations, (6) oscillations produced by the tube, (c) result- ant current beat frequency in the case assumed above would be 101 ooo loo 000 = 1000 per second. If rectified, these beats will produce a note in a telephone of like frequency. In the reception of undamped signals by this method, called the heterodyne method, FIG. 150. Mode of addiitg component oscillations the incoming signals represent one component oscillation. The other oscillation is generated in the receiving apparatus and both act in the same circuit. The rectified resultant furnishes a musical note in the phones, the pitch of which can readily be 2I 4 Circular of the Bureau of Standards altered by varying the frequency of the local source of oscilla- tions. The electron tube may serve as a convenient source of local oscillations and at the same time as an amplifier and detector of the received signals. This is called the autodyne method. Numerous circuits may be utilized to produce these results, of which that shown in Fig. 145, page 209, may serve as an illustra- tion. Incoming signals set up oscillations in the antenna. By means of the coupling between the antenna and coil L oscillations FlG. 151. Variations of mean grid voltage and mean plate current as beat oscillations are being produced of the same frequency are set up in the circuit L C, and as ex- plained above are amplified on account of the feed back between S and P. Further, the coupling between 5 and P is such that the tube oscillates, the frequency of these oscillations depending largely upon the constants of the circuit L C. If this latter fre- quency is adjusted to be slightly different from that of the incom- ing oscillations, beats will result and the potential of the grid will follow the beat oscillations. Just as explained before in the case Radio Instruments and Measurements 215 of reception with a grid condenser, there will be an increased flow of negative electricity from the filament to the grid when this latter is positive and its mean potential will be lowered. Thus, as the oscillations in the beat are increasing the potential of the grid will become lower. The plate current will follow the varia- tions in potential of the grid, reproducing the beat oscillations and decreasing in mean value as the mean potential of the grid is lowered. The curve (a) of Fig. 151 represents the beat oscillations in the circuit L C. In (6) is shown the oscillations of the grid potential, the mean potential being indicated by a dotted line. FlG. 152. " Ultraudion" circuit for receiving undamped oscillations In (c) is shown the plate current, the mean value of which is also shown by a dotted line. The telephone current will likewise cor- respond to this mean value and hence the note will correspond to the beat frequency. In Fig. 152 is shown the connections for the circuit used by L. De Forest, the inventor of the audion, for the reception of undamped oscillations and called the " ultraudion. " The oscillatory circuit is connected between the grid and plate with a condenser in the grid lead. The variable condenser C, shunted across the plate battery and phones is important in the production of oscillations; in general, its value can not be increased beyond a certain point without stopping the oscillations. 2l6 Circular of the Bureau of Standards By this beat method high sensitiveness and selectivity are attained in receiving. Interference is minimized because even slight differences in frequency of the waves from other sources result in notes either of different pitch or completely inaudible. Tone Modulation of Radio Currents from Electron Tube. In undamped wave radio transmitters, the radio-frequency currents may be modulated by the use of what may be called "tone cir- cuits." It is then possible to take advantage of the very selective tuning obtainable with undamped w r aves without employing beat methods of reception. Since undamped high-frequency currents can be produced from electron tubes, it is particularly convenient to apply to these tubes devices for impressing an audible tone on the currents generated. This means >' \ that a periodic variation of the ampli- tude of the radio-frequency current is produced, this periodic variation being of audible frequency. Radio currents modulated in this way may be pro- duced from electron tubes in the three ways described below. Modulation of Electron Stream. The electron current through the tube may be modified by placing the tube in a strong magnetic field which varies in strength with an audible period. The circuits shown in Fig. 146 or in Fig. 1 60 may be used. A coil wound around the electron tube is supplied with cur- rent from a 5oo-cycle generator, as in Fig. 153, or a direct current through the coil may be interrupted by means of a buzzer. Modulation of Grid Potential. Instead of modulating the current by external means, advantage may be taken of the characteristics of the tube itself. The potential of the grid with respect to the filament may be varied with a relatively slow period by means of the arrangement shown in Fig. 154, where an audio-frequency circuit is inserted in the lead to the grid of the electron tube. The circuit L 3 C 3 is tuned to resonance with the L 2 circuit which is coupled loosely to it. Any of the various methods of generating alternating currents may be used for this purpose if a circuit such as L 3 C 3 is inserted in the grid lead and an alternating current of an audible frequency induced in it. For FIG. 153. Modulation of gener- ated current by action of low- frequency magnetic field on electron stream Radio Instruments and Measurements 217 example, the L 2 circuit may be supplied from a 5oocycle genera- tor, with L 3 equal to 50 millihenries and C 3 equal to 2 microfarads. The audio-frequency current may, like the radio-frequency current, be generated by an electron tube. In this case the L, FIG. 154. Modulation of generated current by means of periodic changes of grid voltage circuit referred to above is replaced by the oscillatory circuit of the audio-frequency generator as in Fig. 155. It will be noted that the same type of circuit is used for generating audio as for generating radio currents, it being necessary merely to provide FlG. 155. Method of using an electron tube for producing periodic changes of grid voltage in another tube generating radio-frequency current suitable values of L 4 , L 3 and C 2 . There must be mutual induc- tance between L 4 and L 2 of Fig. 155 just as between the coils in the grid and plate circuits of Fig. 146. The audio and the radio frequency generators may be operated from the same batteries. 218 Circular of the Bureau of Standards Self -Modulating Tube. In the methods previously described, means for modulating are provided outside of the radio-frequency tube. This, however, is not necessary, for it is possible to generate FlG. 1 56. A rrangementfor producing modulated radio-frequency current by use of a single tube both audio and radio frequency currents simultaneously from the same tube. Two arrangements of circuits whereby this may be done are shown in Figs. 156 and 157. In these two diagrams the FIG. 157. Method of coupling tone-circuit generator to an antenna radio-frequency circuit is Lf^ The circuits L 2 C 2 and L 3 C 9 are of audio-frequency. There is mutual inductance between L 8 and L, as well as between L and Lj. Radio Instruments and Measurements 219 Any of the arrangements described above may be used to produce modulated radio-frequency currents in an antenna by coupling the antenna to the radio-frequency coil L l as in Fig. 157. For making signals a key may be inserted in the lead to the grid or in the connection between the filament and the B battery, or in the audio circuit L 3 C 3 as shown in Fig. 1 57. The pitch of the note given by these tone transmitters may be varied at will by changing the constants of the audio circuits L 3 C 3 and L 3 C 2 . Thus, several transmitters may operate using the same wave length but having different modulating tones. The receiver, if provided with means for tuning to the audio as well as to the radio frequency, will be free from interference even by other stations using the same wave length. This method of transmitting offers the considerable advantage that the tone is a pure musical note and does not change in pitch with slight changes in the tuning of the receiving station as in beat meth- ods of reception. Electron Tube as Generator for Measurement Purposes. It is desirable in generating oscillations for measurement purposes that the amplitude and frequency of the generated current shall be constant and that the set-up shall be simple and flexible. By the latter term is meant that a wide range of wave lengths may be obtained with the same apparatus. Constancy of amplitude and frequency are easily obtained. The main requirement being steadiness in the batteries supplying the filament heating current and the electron current between plate and filament. High-frequency current, constant both in magnitude and in frequency to better than one-tenth of i per cent over long intervals of time, is readily obtained. When two or more tubes are operated in parallel on the same B battery changes occur in the intensity of the current furnished by one tube at the instant the second tube is put into operation or when the operation of the second tube is changed. Independent fila- ment batteries should always be used. The circuit shown in Fig. 146 is simple and flexible. The frequency generated is ordi- narily varied by changing the capacity C\. With given coils, as the capacity is increased, there comes a point where the oscilla- tory current falls off and finally "breaks". It is then necessary to use coils of greater inductance in order to obtain longer wave lengths. Another circuit similar to the above and which has shown itself to be convenient is shown in Fig. 160. Here the coils L x 22O Circular of the Bureau of Standards and L 2 may be wound in a single layer adjacent to each other on the same form. Taps may be brought out on each coil so as to use the number of turns desired. The condensers C 2 and C s are large fixed-value condensers which should be of low resistance. Q is the tuning condenser. A tungsten lamp is introduced in series with the B battery to protect the filament of the tube in case of an accident. The measuring circuit may be coupled directly to the coils L lt L 2 , or to a special coil of a few turns inserted in series with either of these coils, preferably on the side connected to the B battery since this point is held at constant * potential by the large capacity of battery to ground. The B battery may be inserted directly in the lead from the plate instead of adjacent to the filament as shown above. With Plate, FIG. 160. Scheme of. connections for pliotron generator such connection, however, care must be taken that there is very little capacity between the two batteries or their leads; if the batteries or their leads are not well separated and insulated from each other, the high-frequency current is much reduced. An advantage of locating the B battery adjacent to the plate is that a single continuous coil may provide all the inductances required in the circuits. Thus, as shown in Fig. 161 below, connections may be made to the coil LL from filament, grid, plate, condenser, and high-frequency ammeter by movable contacts. Great lati- tude of adjustment of the several inductances is thus allowed, and the connections are very simply shifted from one type of circuit to another, so that the proper connections to give maxi- mum current for any wave length are made by simply sliding Radio Instruments and Measurements 221 these contacts. An advantage of the mode of drawing the cir- cuits shown in Fig. 161 is that it brings out that the several types of connection are equivalent. 59. POULSEN ARC Another valuable source of undamped oscillations for measure- ments with moderate or high power is the Poulsen arc. If, as in Fig. 162, an ordinary direct-current carbon arc in air is shunted by a circuit containing capacity and inductance in series, oscilla- tions may be obtained in the shunt circuit. Since the oscillations obtained with this simple arc are, in general, of audible frequency, the arrangement is called the singing arc. Numerous attempts FIG. 161. Pliotron generator using a single coil with sliding contacts have been made to utilize the arc in air as a generator of high- frequency currents, but it was found that the power of the oscillations rapidly decreased with increasing frequency so that it was impossible to attain frequencies higher than about 10 ooo. V. Poulsen, however, found that by modifying the arc in the following respects, high powers could be obtained at least up to moderately high radio frequencies: 1 . The arc is surrounded by a hydrocarbon atmosphere such as coal gas or alcohol vapor. 2. Copper instead of carbon is substituted for the positive electrode. Further, it is desirable to cool the copper electrode by water circulation, to rotate the carbon electrode, and (particularly for high powers) to provide a transverse magnetic field across the arc 222 Circular of the Bureau of Standards to blow it out. The source supplies several hundred volts. The action of the arc in generating oscillations is roughly the following : When no current is flowing through the gap a high voltage is re- quired to start the arc. Immediately, however, upon starting the arc the path of the discharge is ionized and the resistance of the arc is greatly reduced ; in fact, the greater the current through the arc the greater the ionization and the lower its resistance. In series with the direct-current source of supply are choke coils and regulating resistances which tend to keep the supply current con- stant. Suppose that the arc is suddenly extinguished and deionized. On account of the magnetic energy in the supply cir- cuit the voltage across the arc will rise very rapidly, at the same time charging the condenser in the oscillatory circuit until a suffi- cient voltage is reached to strike the arc and again ionize the path of the discharge. The resistance of the arc immediately falls, hence the condenser discharges through it, and on account of the inertia of the discharge, becomes charged again in the opposite sense. It then starts to dis- charge through the arc again, but in the opposite direction to the flow of cur- rent from the source. Thus, the result- FIG. 162. Production of high- ant current through the arc is reduced, frequency currents by means of and when the discharge current of the tne Poulsen arc condenser increases up to that of the supply, the resultant becomes zero. At this point the arc is ex- tinguished, and, as a result of the features introduced by Poulsen, is rapidly deionized. The supply current completes the con- denser discharge and again charges up the condenser to the point where the arc will strike again, and the cycle is repeated. It can readily be seen that while the discharge of the condenser is de- pendent upon the natural period of the oscillatory circuit, the charging depends upon such factors as arc length, constants of the supply circuit, etc., so tliat the period of the oscillation like- wise depends upon these latter factors. Further, the voltage to which the condenser is charged, and hence the amplitude of the oscillation, depends upon the length of the arc, rapidity of deioniza- tion, etc., so that the one factor of arc length affects both the fre- quency and intensity of the high-frequency oscillations. In order to obtain constancy in the oscillations such as is neces- sary for measuring purposes it is necessary that the arc length remain constant. When the arc is burning it tends to eat into the Radio Instruments and Measurements 223 electrodes, and thereby increase its length, and then move to another spot with a shorter gap. This results in unsteadiness in both the frequency and amplitude of the oscillations. In some constructions the arc is caused to revolve slowly around a cylin- drical electrode by means of a radial magnetic field, in others one of the electrodes is slowly revolved. In either case care must be taken to insure that the distance between the electrodes shall be constant ; otherwise slow changes in frequency and intensity will result. In all cases a transverse field used to gain high power will increase the irregularities. In general, the fluctuations are min- imized as the capacity in the oscillatory circuit is decreased, the wave length increased, and the supply current increased. It is practically impossible to attain reasonable steadiness in the opera- tion at wave lengths much shorter than 1000 meters, though satis- factory operation is attainable at longer waves. 60. HIGH-FREQUENCY ALTERNATORS AND FREQUENCY TRANSFORMERS The direct generation of high-frequency currents by means of alternators is a difficult problem; in general, very high speeds of rotation are required, and the losses in the machine from eddy currents, hysteresis and dielectric absorption are likely to be very great. However, two types of generators have been successfully evolved and have been developed to very high powers for radio transmission. Inductor Alternator. The first of these is of the inductor type, which has been developed by E. F. W. Alexanderson, of the Gen- eral Electric Co. This machine has stationary field and arma- ture windings and a solid steel rotor provided with slots cut at equal intervals near the circumference and filled with a nonmag- netic material. As the rotor revolves the magnetic circuit of the field is closed alternately through the nonmagnetic material filling the slots and the steel between the slots ; thus the magnetic flux due to the field is alternately decreased and increased. This flux threads the armature coils, setting up an alternating emf in these. By providing the rotor with 300 slots around the circumference and driving it at a speed of 20 ooo revolutions per minute a fre- quency of 100 ooo cycles per second is attained. In a later design the frequency has been increased to 200 ooo cycles per second. It is stated that there is no difficulty in attaining a constant speed with this machine, and hence it should be of great value for measuring purposes within the range of frequencies covered. A 35601 18 15 224 Circular of the Bureau of Standards further extremely valuable feature is that the frequency can be determined absolutely from the speed. Goldschmidt Alternator. A second type of alternator is the so- called reflection type due to R. Goldschmidt. The rotor and stator are each laminated and provided with windings. The principle upon which the operation of this generator is based is as follows: If an alternator is excited with alternating current of frequency N lt it will generate current of two frequencies N l + N 2 and N! N 2 where N 2 is the frequency which would be generated with direct-current excitation. If N 1 = N 2 = N then the frequencies would be 2N and o. If the current of frequency 2N is used to excite the field of another similar generator running at the same speed, generated frequencies of 2N + N and 2N N that is, 3-/V and TV would result. Thus, a series of generators running at a moderately high speed could be used for generating high-frequency currents. In the Goldschmidt generator this frequency multipli- cation is attained in one machine. The stator is excited with direct current and current of frequency N is generated in the rotor. Since the induction of currents depends only upon the relative motion of rotor and stator we may consider that the rotor field is excited with current of frequency N and that the stator is rotating in this field. Consequently, currents of frequency 2N and o will be gen- erated in the stator. The fields of these currents in turn react upon the rotor, producing in it currents of frequency $N and N, and in this manner the frequency is successively stepped up, the frequencies in the rotor being odd multiples of N and those in the stator even multiples. In order that the flow of current of these frequencies may not be prevented by the reactance of the circuits, the principle of resonance is utilized and tuned circuits are pro- vided for each frequency up to that which is to be used. The flow of current corresponding to the lower frequencies is sup- pressed to a great extent. For as we have seen, starting with the fundamental frequency N, after two ' ' reflections ' ' we again have an induced frequency N in company with $N. It may be shown that these two currents of frequency N will be opposite in phase and hence tend to neutralize each other. This is likewise true of the magnetic fields so that the losses due to hysteresis and eddy cur- rents will be caused mainly by the field of the utilized frequency alone. While these machines have been developed very satisfac- torily for radio transmission purposes, it is doubtful whether they could be readily utilized for measuring purposes in the laboratory Radio Instruments and Measurements 225 since the multiplicity of tuned circuits would render frequency changes difficult. "Static" Frequency Transformers. Several methods of fre- quency multiplication have been devised which are based upon the distortion of the wave of magnetic induction in iron from that of the impressed magnetizing force. Since these frequency multi- pliers have no moving parts they are called static frequency trans- formers. The principle is well illustrated in the method of tripling the frequency, due to Joly. Fig. 1 63 is a typical curve showing the variation of induction in iron with the magnetizing force. As the magnetizing force is increased from zero the resultant flux of induc- tion in the iron at first increases rather slowly, then very rapidly, and then less rapidly, again becoming almost constant at a value called the saturation value. If the magnetizing force is alternating and sinusoidal and of such an amplitude that the maximum value comes on the steep part of the induction curve as at A , Fig. 1 63, the resulting alter- nating wave of magnetic induction will be peaked, as in b, Fig. 164. If, however, the maximum magnetizing force has a | value sufficiently high to bring up the in- J duction to the flat part of the curve where it is changing very slowly, as at B, Fig. Mag^t*^ r mt , 163, then the resulting alternating wave FlG - 163- Variation of mag- of induction will be flat topped, as in c, tic induciio f n in iron with magnetizing force Fig. 1 64. The wave form b indicates that there is a strong harmonic oscillation of three times the fundamental frequency impressed upon the fundamental oscillation and differ- ing in phase from it by 180. The wave form c likewise indicates the presence of a strong harmonic of three times the fundamental frequency but which is in phase with the fundamental. If, there- fore, the two waves b and c can be combined in such a manner that the fundamental frequencies are 180 out of phase and hence neutralize each other, the harmonics of triple frequency will be in phase and will exist alone. This is illustrated in curve d which is obtained by subtracting the ordinates of the curve c from those of curve b. This method is applied by means of transformers as illustrated in Fig. 165. The alternator supplies current of the fundamental frequency / to the primaries P l and P 2 . P l has few turns and P 2 many turns, so that the iron is magnetized more in- tensely in 2 than in i. The two secondaries Sj and 5 2 are so 226 Circular of the Bureau of Standards wound and connected that the emf 's of the fundamental frequency neutralize each other, but the triple harmonics cause a current flow in the tuned circuit of frequency 3/. Thus with an initial frequency of 10 ooo cycles per second, a frequency of 30 ooo can FIG. 164. Method of combining alternating waves of magnetic induction so as to triple the frequency be obtained with one transformation. This corresponds to a wave length of 10 ooo meters and is suitable for long-distance transmission. Large powers may be generated. For measuring purposes it would be possible to step up the frequency through several stages obtaining 3, 9, 27, etc., times the FIG. 165. Use of two transformers for producing fre- quency transformations fundamental frequency. This might furnish a valuable method of determining high frequencies in terms of lower frequencies, which latter can be determined absolutely from the speed and number of poles of the alternator. Radio Instruments and Measurements 61. BUZZERS 227 The buzzer is a very convenient source of damped oscillations for measurement purposes. Since, in general, it furnishes only very small power, it is used in conjunction with very sensitive detecting instruments. A number of different modes of connec- tion may be used in generating oscillations with a buzzer. That shown in Fig. 166 has been found to be very satisfactory. The current from the battery B flows through the adjustable resistance R, the coils F, armature contact A, and coil L t . When through the action of the buzzer the con- tact is opened, the energy due to the current in the coil L t is transferred to the condenser, C lt giving it a charge. The con- FiG. 166. Use of buzzer as a source of current of definite frequency denser then discharges, causing a train of oscillations in the circuit C t L!, the frequency of which depends upon the constants of this circuit with a small correction for the capacity added by the leads, etc., of the buzzer circuit. Thus, each break of the buzzer sets up a train of oscillations in the circuit C\ L v The circuit L 2 C 2 is a measuring circuit coupled to the driving circuit L t C\. The current in the measuring circuit may be indicated by a gal- vanometer and thermoelement (T) inserted directly in the circuit or any other sensitive device. Constancy of the high-frequency cur- rent depends upon the steadiness of the buzzer action. This is obtained by using a good buzzer giving a note of high pitch, such as the Ericsson, by adjust- ment of the buzzer contacts and resist- ance R until the buzzer emits a clear and steady musical tone, by employing a constant battery, preferably a low- voltage storage battery to insure steady direct current, and by preventing sparking at the contact. This latter requirement is attained by sending only a moderate current through the buzzer and by using a fairly large fixed condenser C 3 across the buzzer field coils to absorb the magnetic energy stored therein which otherwise would produce a high voltage and sparking at the con- tact on break. FIG. 167. Buzzer circuit capable of producing currents by shock excitation 228 Circular of the Bureau of Standards Another form of buzzer circuit which is frequently used and is capable of furnishing somewhat larger currents is shown in Fig. 1 67. In this case the condenser C\ is charged to the voltage of the battery when the buzzer contact is open and discharges through LI when the contact is closed. A possible objection to this circuit is the presence of the buzzer contact in the oscillatory circuit. If Ci in the above is a fixed condenser of several microfarads capacity and L l a small inductance of only one or two turns, then the oscillations in the circuit L t C\ will be very highly damped and will last for a very short time, possibly only one or two oscillations. Under these conditions an oscillatory circuit coupled to L! will be shocked into oscillations by what is called impact excitation, the frequency and damping of the oscillations will FIG. 168. Typical spark circuit for producing high-frequency oscillations be those natural to this circuit and independent of the circuit L l C\. On this account this method of impact excitation is very useful in many measurements. In place of the very convenient buzzer many other forms of circuit interrupters may be used, such as the vibrating wire, tuning fork, rotating and mercury interrupter. 62. THE SPARK Certain forms of spark gap are simple and inexpensive sources of damped currents and so are often used as sources in high-fre- quency measurements. In some kinds of measurement it is neces- sary or advantageous for the oscillations to have a decrement. Simple Spark Gap. In Fig. 168 is shown a typical circuit for the generation of high-frequency oscillations by means of a spark discharge. The alternator supplies the low-voltage wind- ing P of a step-up transformer. The high-voltage side S leads Radio Instruments and Measurements 229 to the terminals of the condenser C, across which is an inductance L and spark gap G in series. The coil L is loosely coupled to the measuring circuit L m C m (or to the antenna in transmitting). During an alternation, as the voltage across 5 increases, the con- denser C becomes charged up to the point where the voltage is sufficient to jump the spark gap. The condenser then discharges through the inductance L and the gap G. The discharge consists of a train of oscillations of a frequency approximately corre- sponding to the inductance and capacity of the circuit. It is possible to adjust the voltage of the transformer and the length of the gap so that the discharge takes place when the voltage is at a maximum, either positive or negative. In this case one spark and one train of oscillations is obtained per alternation of the supply, thus with a 6o-cycle generator the spark frequency FIG. 169. Groups of oscillations for case of two spark discharges per cycle will be 1 20. By shortening the gap or raising the voltage several discharges per alternation may be obtained. These are called partial discharges and occur somewhat irregularly. The first case is illustrated in Fig. 169. In (a) is shown the transformer secondary voltage as it would be if the spark gap were absent and in (6) the current oscillation in the condenser discharge. In Fig. 170 is shown the effect of the spark gap upon the damping of the oscillations in the high-frequency train. In a circuit with constant resistance the amplitude would decrease exponentially as indicated by the dash curve i, in the case of a circuit with a spark gap the decrease of amplitude tends to become linear, as shown by the dash line 2. This is due to the increase in resistance of the spark as the amplitude of the current decreases, the effect depending upon the material of the electrodes, etc. 230 Circular of the Bureau of Standards Use of Resonance Transformer. A serious difficulty in the operation of the spark circuit is caused by the short-circuiting of the transformer secondary by the spark. As a result there is a heavy flow of current through the gap causing the formation of an arc which reduces the amplitude of the oscillations and destroys the electrodes. In order to eliminate this difficulty the resonance transformer is used. The alternator, transformer and secondary condenser are adjusted to make a system which is in resonance for the alternator frequency. When the condenser is short-circuited by the spark the condition of resonance is destroyed, and in effect this is equivalent to the sudden insertion of a reactance FlG. 170. Linear damping produced by the increase of spark resistance as the amplitude of current decreases in the transformer primary. As a result, there is no heavy flow of current through the gap. The theory of the adjustment of the system of alternator, transformer, and secondary condenser to resonance is as follows. If we have a simple circuit of inductance and capacity in series across the terminals of the alternator, as in Fig. 171, the condi- tion for resonance for a frequency / is where L p is the total inductance of the circuit including the alternator. The combination of transformer and secondary condenser can be reduced to this simple case. Assuming that all the induction linked with the primary winding of the transformer also passes through the secondary turns that is, that there is no magnetic leakage and that the ratio of the number of secondary turns to primary turns is n, it may be shown that a capacity C B in the secondary is equivalent to a capacity C 9 = n 2 C s in the primary. The effect of inductance in the secondary is decreased in the ratio of i : n 2 when transferred back to the primary, hence inductance is inserted in the primary to tune to resonance. The total primary inductance FIG .171 . Simp le circuit equiva- lent to Fig. 168 Radio Instruments and Measurements 231 consists of that inserted plus the inductance of the alternator and that due to transformer leakage. This latter is small in the case of a closed-core transformer. Experimentally a fairly close adjust- ment to resonance may readily be obtained by lowering the gen- erator voltage until no spark passes the gap and then varying either the primary inductance or secondary condenser until the primary current or secondary voltage is a maximum. The primary inductance may conveniently consist of a solenoidal winding with an iron core that can be moved in or out to vary the inductance value. When adjusted to resonance the voltage across the secondary condenser may rise to a value much higher than that correspond- ing to the voltage of the alternator and the transformer ratio. In Fig. 172 is shown the way the voltage rises with each alternation until it is sufficient to jump the spark gap discharging the con- denser. The voltage then begins to rise again until the next spark takes place. The alternator voltage or spark length can be Spark Spark Spark FIG. 172 Condenser voltage when the transformer system is adjusted to resonance with the generator adjusted to obtain either one spark per alternation or one spark in several alternations, as shown in the Fig. 172. In order to obtain constant high-frequency current with a simple spark gap it is desirable to use a low spark frequency in order to prevent heating of the gap which would lead to arcing. Magnesium electrodes have been found to give the best results and to furnish oscillations most closely logarithmic in damping. Zinc is also a good material. The gap, the voltage and reso- nance conditions should be adjusted to give a spark of moderate and uniform frequency. The alternator must run at constant speed, otherwise the voltage and resonance conditions will vary. Under these conditions it is possible to attain high-frequency oscillations of a constancy which is probably not excelled by any other source of damped oscillations. 232 Circular of the Bureau of Standards If the resonance transformer is not utilized, the arcing across the gap may be reduced by inserting resistance or inductance coils in the primary of the transformer and by employing an air blast to blow out the arc. Or, in place of the simple gap, a rotary gap, as shown in Fig. 173, may be utilized. Its character- istics are intermediate between those of the simple and the quenched spark gap. Quenched Gap. It was found by M. Wien that if a series of short spark gaps be substituted for a single long gap and a dis- charge passed through them, the discharge path returns much more quickly after discharge to its initial condition of high resistance. This is a result of the more rapid deionization of the gap and is called the quenching action. The quenching action FIG. ijiRotat- . . j . r ,, - r ,, . ., ing spark gap 1S lncrease d the surfaces of the gaps are of silver or copper and the gap is kept cool and air-tight. In Fig. 1 74 is shown a cross section of a single gap showing the insulating gasket between the plates which renders the gap air- tight, the silver sparking surfaces and the flanges to provide a large cooling surface. The insulating gasket may be of paper, mica or rubber, and is about 0.2 mm thick. Its thickness is exaggerated in the figure. A number of such gaps are stacked in series and clamped together, and either the leads to the gap are provided with clips so that the number of gaps used may be varied or means are provided for short-circuiting as many of the gaps as desired. A plate of an improved quenched gap designed at the Bureau of Standards is shown in Fig. 175, facing page 323. The con- struction is such as to permit air circulation on both sides of each gap. This is accomplished by inverting alternate plates. The assembled FlG ; , T-V tion of auctioned gap quenched gap is shown in Fig. 221, page 322. late While close coupling with the secondary circuit in the case of ordinary spark gaps is to be avoided, since it causes the generation of two frequencies (the so-called coup- ling waves, see p. 48) of which only one can be utilized, good working of the quenched gap, on the other hand, requires a fairly close coup- ling between the primary and secondary circuits. This secures high efficiency and still permits a single wave to be obtained. The ex- planation is as follows : Assume first that the primary circuit contains an ordinary spark gap, the secondary (which may be an antenna) is fairly closely coupled to the primary, and that the two circuits Radio Instruments and Measurements 233 when separated have the same natural frequency. Due to the coupling, oscillations of two frequencies, one lower and one higher than that common to the uncoupled circuits, will result in both circuits after the discharge takes place in the primary. The com- bination of the two frequencies will result in beats in both cir- cuits, the amplitude of the resultant oscillation will rise to a maximum and fall to a minimum in each circuit, being a maxi- mum in the primary when a minimum in the secondary, and vice versa. As a result, the total energy of the oscillations (excepting that dissipated) is transferred back and forth between the two circuits. Although the current in the primary circuit may pass FIG. 176. Current in (a) primary and (b) secondary when using an ordinary gap through a zero value, the rapidity of deionization of the ordinary spark gap is not sufficient to render it nonconducting in the short interval of time available and the spark reignites. The phenomena are shown in Fig. 176 where (a) represents the voltage oscillations in the primary and (6) the oscillations in the secondary. If, on the other hand, a quenched gap is used and the coupling between the primary and secondary is favorable, it will become deionized when the primary oscillations are a minimum and thus prevent reignition. At this time all of the energy has been transferred to the secondary and, since the primary has become inoperative, this energy will be dissipated in a train of oscilla- tions of which the frequency and damping are determined 234 Circular of the Bureau of Standards entirely by the constants of the secondary circuit. The oscilla- tions of primary and secondary are shown in (a) and (6) of Fig. 177. In ideal operation, the time during which the primary circuit is operative will be extremely short, there will be only the one frequency, and, since the major loss of power takes place in the high-resistance primary circuit, the efficiency will be high. With poorer operation the primary circuit may remain in opera- tion until the second or third minimum. In this case three FIG. 177. Current in (a) primary and (6) secondary -when using aquenched gap waves may be observed, the two coupling waves and the inter- mediate wave corresponding to the oscillations of the secondary by itself. The connections for the quenched gap are similar to those for a plain spark, using a resonance transformer. Best operation is generally obtained when the inductance in the primary circuit is somewhat greater than that required for resonance. On account of the rapid quenching of the gap, the supply alternator may have a frequency of 500 cycles and adjustments made so as to obtain one spark per alternation. PART EL FORMULAS AND DATA <& CALCULATION OF CAPACITY 63. CAPACITY OF CONDENSERS Units. The capacities given by the following formulas are in micromicrofarads. This unit is io" 12 of the farad, the farad being defined as the capacity of a condenser charged to a potential of i volt by i coulomb of electricity. The micromicrofarad and the microfarad (one-millionth of a farad) are the units commonly used in radio work. Radio writers have occasionally used the cgs electrostatic unit, sometimes called the "centimeter." This unit is 1.1124 micromicrofarads. In the formulas here given all lengths are expressed in centi- meters and all areas in square centimeters. The constants given are correct 31 to o.i per cent. PARALLEL PLATE CONDENSER Let S = surface area of one plate r = thickness of the dielectric K = dielectric constant (K = i for air, and for most ordinary substances lies between i and io). <^ C = 0.0885^ micromicrofarads. (no) r If, instead of a single pair of metal plates, there are N similar plates with dielectric between, alternate plates being connected in parallel, ~ l)5 (in) In these formulas no allowance is made for the curving of the lines of force at the edges of the plates; the effect is negligible when T is very small compared with 5. 81 The constants given in the formulas are correct for absolute units. To reduce to international units the values in absolute units should be multiplied by 1.00052. This difference need not be considered when calculations correct to i part in 1000 only are required. 235 236 Circular of the Bureau of Standards VARIABLE CONDENSER WITH SEMICIRCULAR PLATES Let N = total number of parallel plates r^ = outside radius of the plates r 2 = inner radius of plates T = thickness of dielectric K = dielectric constant Then, for the position of maximum capacity (movable plates between the fixed plates) , - 'W^ (,,2) This formula does not take into account the effect of the edges of the plates, but as the capacity is also affected by the contain- ing case it will not generally be worth while to take the edge effect into account. Formula (112) gives the maximum capacity between the plates with this form of condenser. As the movable plates are rotated the capacity decreases, and ordinarily the decrease in capacity is proportional to the angle through which the plates are rotated. ISOLATED DISK OF NEGLIGIBLE THICKNESS Let d = diameter of the disk then C = o.354io and may be interpolated from Table 6, page 242. These formulas assume a uniform distribution of charge from point to point of the wire. VERTICAL WERE Formula (119), omitting the k 2 in the denominator, is sometimes used to calculate the capacity of a vertical wire. It applies accurately only when h is large compared with Z, and gives very rough values for a vertical single-wire antenna, the lower end of which is connected to apparatus at least several meters above the ground. 238 Circular of the Bureau of Standards CAPACITY BETWEEN TWO HORIZONTAL PARALLEL WIRES AT THE SAME HEIGHT Let d = the diameter of cross section of the wires / = length of each wire & = the height of the wires above the earth D = distance between centers of the wires. The capacity is denned as the quotient of the charge on one wire, divided by the difference in potential of the two wires, when the potential of one wire is as much positive as the other is negative. 0.1208 1 (120) In most cases d/l and D/l may be neglected in comparison with unity, and we may write 0.1208 / , TWO PARALLEL WIRES, ONE ABOVE THE OTHER For the case of one wire placed vertically above the other, the formula (121) may usually be used, taking for the value of h the mean height of the w r ires, - The potential of one wire is assumed to be as much positive as the other is negative. CAPACITY OF TWO PARALLEL WIRES JOINED TOGETHER Let / = the length of each wire D = distance between centers h = their height above the earth d = diameter of cross section. The wires are supposed to be parallel to each other and to lie in a horizontal plane. They are joined together so that they are at the same potential. The capacity is denned as the quotient of the sum of their charges by the potential above the earth. 0-4831 1 , N (122) 4* ,. 10 - " ' 10 J /?v which, in those cases where d*/P and I -r I may be neglected in comparison with unity, may be written in the following forms: Radio Instruments and Measurements 239 T? = For u = 4.6o5 ] p l2 = 4.605 or, 4^ 'T~ J 2 h rp -ij = 4.605! log 10 ^-M (I2 5 ) (126) 35601 1 -16 240 Circular of the Bureau of Standards the approximate capacity of the n wires in parallel will be (127) the quantities k, k^ and k 2 being obtained from Tables 6 and 7, page 242. Example. To find the capacity of an antenna of 10 wires 0.16 inch in diameter, in parallel, each wire 1 10 feet long, the spacing between the wires being 2 feet and their height above the ground 80 feet. For this case 4&// = or 7/4/1=0.344 and Table 6 gives k 1 10 =0.146. 2XI2XIIO 2/ 2l/d=~ 2 = 16500, Iogi -r =4.2175 o. i o a "~ = 1.7404 .-. p n = 4.605 [4.218 -0.146] = 18.75 p n = 4.605 [1.740 -0.146]= 7.340 and from formula (127) and Table 7 the capacity is, reducing the length of the wires to cm = 584 MM/ =0.000584 M/- Example. A second antenna of 10 wires, 3/32 inch diameter, 155 feet long, spaced 2.5 feet apart, and stretched at a distance of 64 feet from the earth. For this case l/^h = =0.606, k 2 =0.249 2l/d = 39680, Iog 10 ^- = 4- l/D =62, log 10 //D = 1.7924 P ' -2.05=6.35 Radio Instruments and Measurements 241 If the length of the antenna had been 500 feet, with the height 4h 2 56 dh unchanged, then V = -;- =0.512, ^-0.026, log, ^-=4.5154, i 5^^ logic ^-1.7093; by (125) p n = 20.67, i2 = 7-75, = 2.05, 1.112X500X30.5 C = - = 0.002426 fjif. 6.99 65. TABLES FOR CAPACITY CALCULATIONS TABLE 5. For Converting Common Logarithms Into Natural Logarithms Common Natural Common Natural Common Natural Common Natural 0.0000 25.0 57. 565 50.0 115. 129 75.0 172. 694 1.0 2. 3026 26.0 59. 867 51.0 117. 432 76.0 174. 996 2.0 4. 6052 27.0 62. 170 52.0 119. 734 77.0 177. 299 3.0 6.9078 28.0 64. 472 53.0 122. 037 78.0 179. 602 4.0 9. 2103 29.0 66. 775 54.0 124. 340 79.0 181. 904 5.0 11.513 30.0 69. 078 55.0 126. 642 80.0 184. 207 6.0 13. 816 31.0 71. 380 56.0 128. 945 81.0 186. 509 7.0 16. 118 32.0 73.683 57.0 131. 247 82.0 188. 812 8.0 18. 421 33.0 75. 985 58.0 133. 550 83.0 191. 115 ' 9.0 20. 723 34.0 78. 288 59.0 135. 853 84.0 193. 417 10.0 23. 026 35.0 80.590 60.0 138. 155 85.0 195. 720 11.0 25. 328 36.0 82. 893 61.0 140. 458 86.0 198. 022 12.0 27. 631 37.0 85. 196 62.0 142. 760 87.0 200. 325 13.0 29. 934 38.0 87. 498 63.0 145. 063 88.0 202. 627 14.0 32. 236 39.0 89.801 64.0 147. 365 89.0 204. 930 15.0 34.539 40.0 92. 103 65.0 149.668 90.0 207. 233 16.0 36.841 41.0 94. 406 66.0 151. 971 91.0 209. 535 17.0 39. 144 42.0 96. 709 67.0 154. 273 92.0 211. 838 18.0 41. 447 43.0 99. Oil 68.0 156. 576 93.0 214. 140 19.0 43. 749 44.0 101. 314 69.0 158. 878 94.0 216. 443 20.0 46. 052 45.0 103. 616 70.0 161. 181 95.0 218. 746 21.0 48. 354 46.0 105. 919 71.0 163. 484 96.0 221. 048 22.0 50. 657 47.0 108. 221 72.0 165. 786 97.0 223. 351 23.0 52. 959 48.0 110. 524 73.0 168. 089 98.0 225. 653 24.0 55. 262 49.0 112. 827 74.0 170. 391 99.0 227. 956 100.0 230. 259 The table is carried out to a higher precision than the formulas, e. g., 2.3026 is abbre- viated to 2.303 in the formulas. Examples. To illustrate the use of such a table, suppose we wish to find the nat- ural logarithm of 37.48. The common logarithm of 37.48 is 1.57380. If we denote the number 2.3026 by M, then from the table i. 5 M= 3 . 4539 . 073 M= . 1681 . 00080 M . 0018 3. 6238=^37.48 To find the natural logarithm of 0.00748: The common logarithm is 3-87390, which may be written 0.873903. Entering the table we find 0.87 M=2. 00325 3M= 6. 9078 . 0039 M= . 00898 sum 2. 0122 6. 9078 4. 8956 =natural log of 0.00748 242 Circular of the Bureau of Standards TABLE 6. For Use in Connection with Formulas (118), (119), (123), (124), (125), and (126) 4h/l ki l/4h k 2 4h/l ki l/4h k 2 0.6 0.035 0.6 0.247 0.1 0.001 0.1 0.043 .7 .045 .7 .283 .2 .004 .2 .086 .8 .057 .8 .318 .3 .009 .3 .128 .9 .069 .9 .351 .4 .016 .4 .169 1.0 .082 1.0 .383 .5 .025 .5 .209 TABLE 7. Values of k in Formulas (127) and (146) k n k n k n k 2 6 1.18 11 2.22 16 2.85 3 0.308 7 1.43 12 2.37 17 2.95 4 .621 8 1.66 13 2.51 18 3.04 5 .906 9 1.86 14 2.63 19 3.14 10 2.05 15 2.74 20 3.24 CALCULATION OF INDUCTANCE 66. GENERAL In this section are given formulas for the calculation of self and mutual inductance in the more common circuits met with in prac- tice. The attempt is here made, not to present all the formulas available for this purpose, but rather the minimum number re- quired, and to attain an accuracy of about one part in a thousand. So far as has seemed practicable, tables have been prepared to facilitate numerical calculations. In some cases, to render inter- polation more certain, the values in the tables are carried out to one more significant figure than is necessary. In such instances, after having obtained the required quantity by interpolation from a table, the superfluous figure may be dropped. In all the tables the intervals for which the desired quantities are tabulated are taken small enough to render the consideration of second differ- ences in interpolation unnecessary. Most of the formulas given are for low frequencies, this fact being indicated by the subscript zero, thus L , M . The high-frequency formulas are given where such are known. Fortunately it is possible by proper design to render unimportant the change of inductance with frequency, except in cases where extremely high precision is required. The usual unit of inductance used in radio work is the micro- henry, which is one millionth of the international henry. 32 The w The constants in the formulas for inductance given here refer to absolute units. To reduce to inter- national units multiply by 0.99948. Since, however, an accuracy of the order of only one part in a thousand is sought here, it will not be necessary to take this difference into account. Radio Instruments and Measurements 243 henry is defined as the inductance "in a circuit when the electro- motive force induced in this circuit is one international volt, while the inducing current varies at the rate of one ampere per second." i henry = 1000 millihenries = io 8 microhenries = io 9 cgs electro- magnetic units. ^n the following formulas lengths and other dimensions are expressed in centimeters, unless otherwise stipulated, and the inductance calculated will be in microhenries. Logarithms are given, either to the natural base or to the base io, as indicated. The labor involved in the multiplication of common logarithms by the factor 2.303 to reduce to the corre- sponding natural logarithms will be very materially reduced by the employment of the multiplication table, Table 5, page 124, which is an abridgement of the table for this purpose usually given in collections of logarithms. All of these formulas assume that there is no iron in the vicinity of the conductor or circuit of which the inductance is to be calcu- lated. Thus, the formulas here given can not be used to calculate the inductance of electromagnets. A much more complete collection of inductance formulas with numerical examples is given in the Bulletin of the Bureau of Standards, 8, pages 1-237; 1 9 12 > also known as Scientific Paper No. 169. 67. SELF-INDUCTANCE OF WIRES AND ANTENNAS STRAIGHT, ROUND WIRE If / = length of wire d = diameter of cross section ju = permeability of the material of the wire L = o.oo2/ log e ~r i-f microhenries (128) = o.oo2/ 2.303 Iog 10 - - i +- microhenries (129) [_ a 4 J For all except iron wires this becomes L = o.oo2/ 2.303 Iog 10 ^ -0.75 (130) For wires whose length is less than about 1000 times the diameter of the cross section ( -7 < 1000 ), the term, should be added inside \a / 2/ the brackets. These formulas give merely the self-inductance 244 Circular of the Bureau of Standards of one conductor. If the return conductor is not far away, the mutual inductances have to be taken into account (see formulas (134) and (136)). As the frequency of the current increases, the inductance diminishes, and approaches the limiting value. 2.303 Iog 10 ^ - i (131) which holds for infinite frequency. The general formula for the inductance at any frequency is L= 0.0021 2.303 Iog 10 ^ - i +M5 (132) where 5 is a quantity given in Table 8, page 282, as a function of x where /3 VP (133) / = frequency. 1 p = volume resistivity of wire in microhm-centimeters p c = same for copper /x = i for all except iron wires. For copper at 20 C, # c = 0.1071 d -JJ. The value a c of x for a copper wire o. i cm in diameter at different frequencies may be obtained from Table 19, page 311. For a copper wire d cm in diameter x c = 10 d a c and for a wire of some other material x = 10 d a c - I ^ V P The total change in inductance when the frequency of the current is raised from zero to infinity is a function of the ratio of the length of the wire to the diameter of the cross section. Thus, the decrease in inductance of a wire whose length is 25 times the diameter is 6 per cent at infinite frequency; and for a wire 100 ooo times as long as its diameter, 2 per cent. Example. For a copper wire of length 200 cm and diameter 0.25 cm at a wave length of 600 meters, that is / = 500 ooo, the value of x is 18.93, and from Table 8, 5 =0.037. A* = i, = 3200, Iog 10 3200 = 3.51851 Radio Instruments and Measurements 245 (From Table 5) log e 3200 = 8.0590 414 12 8.IOI6 For zero frequency L = o.4 [8.102 i +0.25] = 2. 941 microhenry For / = 500 ooo L = o.4 [8. 102 -i +0.037] = 2.856 microhenry a difference of 2.9 per cent out of a possible 3.4 per cent. For an iron wire of the same length and diameter, assuming a resistivity 7 times as great as that of copper, and a permeability of loo, the value of x is .*/- - times as great as for the copper wire, or 71.5, and for this value of x, 5=o.oio (TableS) L = o.4 [32.10] = 12. 84 ph L =0.4 [8.102] =^.24 nh at 500 ooo cycles. The limiting value is Loo =2.84 ph. TWO PARALLEL, ROUND WIRES-RETURN CIRCUIT In this case the current is supposed to flow in opposite direc- tions in two parallel wires each of length / and diameter d. Denot- ing by D the distance from the center of one wire to the center of the other, 1 2.303 log 10 -^--j+M5j (i34) L = 0.004 / The permeability of the wires being /x, and 5 being obtained from (133) and Table 8, page 282. For low frequency 6=0.25. This formula neglects the inductance of the connecting wires between the two main wires. If these are not of negligible length, their inductances may be calculated by (132) and added to the result obtained by (134), or else the whole circuit may be treated by the formula (138) for the rectangle below. 246 Circular of the Bureau of Standards STRAIGHT RECTANGULAR BAR Let / = length of bar. b, c = sides of the rectangular section. L =0.002 / 2.303 Iog 10 ^ +0. 5 +0.2235 ( -~^ (I 35 ) The last term may be neglected for values of / greater than about 50 times (b + c) . The permeability of the wire is here assumed as unity. RETURN CIRCUIT OF RECTANGULAR WIRES If the wires are supposed to be of the same cross section, 6 by c, and length /, and of permeability unity, and the distance be- tween their centers is D, L = 0.004 ^2.303 lo Sio b^~ c + l~l + a22 35 ^y^ I (136) FIG. 178. The two conductors of a return circuit of rectan- gular wires For wires of different sizes, the inductance is given by L =L t + L 2 2M in which the inductances L t and L 2 of the individual wires are to be calculated by (135), and their mutual inductance M by (174) below. SQUARE OF ROUND WIRE If a is the length of one side of the square and the wire is of circular cross section of diameter d, the permeability of the wire being M, L=o.oo8 a I 2.303 log 10 -^ + -0.774+M5J (137) in which 8 may be obtained from Table 8 as a function of the argument x given in formula (133). The value of d for low fre- quency is 0.25, and for infinite frequency is o. Radio Instruments and Measurements 247 RECTANGLE OF ROUND WIRE Let the sides of the rectangle be a and a lf the diagonal g = Vo 2 + a? and d = diameter of the cross section of the wire. Then the inductance at any frequency is L= 0.0092 1 (a + ajlogj L + 0.004 [M^ (a + a^+2 (g + d/2)2 (a + aj] (138) The quantity 6 is obtained by use of (133) and Table 8. Its value for zero frequency is 0.25, and is o for infinite frequency. RECTANGLE OF RECTANGULAR-SECTION WIRE FIG. 179. Rectangle of rectan- gular -wire Assuming the dimensions of the section of the wire to be b and c, and the sides of the rectangle a and a lt then for nonmag- netic material the inductance at low frequency is = 0.0092 1 where ^- I Iog 1 + 0.004 [,- -a Iog 10 (a + g) -a l Iog 1 - + 0.447 (b+c) (139) INDUCTANCE OF GROUNDED HORIZONTAL WIRE If we have a wire placed horizontally with the earth, which acts as the return for the current, the self-inductance of the wire is given by the following formula, in which I = length of the wire h = height above ground d = diameter of the wire ju = permeability of the wire 5 = constant given in Table 8, to take account of the effect of frequency (see p. 282). [ 4k d~\ - / x 248 Circular of the Bureau of Standards which, neglecting j , as may be done in all practical cases, may be written in the following forms convenient for calculation: For y^i, A.H L = o.oo2 l\ 2.3026 log 10 -T- P + /IO (141) and for -7^1, 2h L = 0.002 / 2.3026 Iog 10 ~-Q+tJ,8 (142) the values of P and Q being obtained by interpolation from Table 9. Mutual Inductance of Two Parallel Grounded Wires. The two wires are assumed to be stretched horizontally, with both ends grounded, the earth forming the return circuit. Let I = length of each wire d = diameter of wire D = distance between centers of the wires h = height above the earth Then _r ^h*+D 2 = o.oo 4 6o 5 / [lo glo * L_ _ + l og + 0.002 / [~y// 2 + D 2 + 4/fc 2 V^ 2 + D 2 +D -\/D 2 + 4/ 2 ] (143) which, if we neglect -^ and ( -r ) 2 may be expressed in the follow- ing forms : For y -j> etc. Then / a T .TLi + Cn-i) M t ,"] L = l\ - - - - -o.ooikl (146) in which n is the number of wires in parallel and k is a function of n tabulated in Table 7, page 242. Example. An antenna of 10 wires in parallel, each wire 155 feet long and -fa inch in diameter, spaced 2.5 feet apart, and sus- pended at a height of 64 feet above the earth. Find the inductance at 100 ooo cycles per second. We have here -r-=; = 0.826, and using this as argument in ^J \j Table 9, P= 0.6671. From (133) x = 8.07, and thence from Table 8, 6=0.087. i2X =32768, Iog 10 = 4. 128 logi 0.2 , L =0.01257 a {2.303 Iog 10 -j 2+/x5[ (147) I J in which 8 will be obtained from (133) and Table 8, page 282. Its value for zero frequency is 0.25. TUBE BENT INTO A CIRCLE Let the inner and outer diameters of the annular cross section of the tube be d l and d 2 , respectively, and the mean radius of the circle a, then neglecting -^ and -~ i6a d, 2 L = 0.01257 a I 2.303 Iog 10 "^~~ I -75~ 2 M2_ f j2) For infinite frequency this becomes LOO =0.012570! 2.303 log 10 ^-2J (i49) 68. SELF-INDUCTANCE OF COILS CIRCULAR COIL OF CIRCULAR CROSS SECTION For a coil of n fine wires wound with the mean radius of the turns equal to a, the area of cross section of the winding being a circle of diameter d, i6a an 2 2.303 Iog 10 r -i-75 I a ) Radio Instruments and Measurements 251 This neglects the space occupied by the insulation between the wires. TORUS WITH SINGLE-LAYER WINDING A torus is a ring of circular cross section (doughnut shape) . Let R distance from axis to center of cross section of the winding o = radius of the turns of the winding n = number of turns of the winding (151) FIG. 180. Torus of single layer winding TOROIDAL COIL OF RECTANGULAR CROSS SECTION WITH SINGLE-LAYER WINDING A coil of this shape might also be called a circular solenoid of rectangular section. Let r x = inner radius of toroid (distance from the axis to inside of winding) r 2 = outer radius of toroid (distance from axis to outside of winding) h = axial depth of toroid. Then L = 0.004606 n 2 h log t (152) FIG. 181. Toroidal coil of rec- tangular section with single layer winding The value so computed is strictly correct only for an infinitely thin winding. For a winding of actual wires a correction may be calculated as shown in Bulletin, Bureau of Standards, 8, page 125; 1912. The correction is, however, very small. 252 Circular of the Bureau of Standards SINGLE-LAYER COIL OR SOLENOID An approximate value is given by L ^_ 0.03948 on' g ( I53 ) where n = number of turns of the winding, a = radius of the coil, measured from the axis to the center of any wire, b = length of coil = n times the distance between centers of turns, and K is a 2a function of -r- and is given in Table 10, page 283, which was calcu- lated by Nagaoka. (See Bulletin, Bureau of Standards, 8, p. 224, 1912.) For a coil very long in comparison with its diameter, JC-x, Formula (153) takes no account of the shape or size of the cross section of the wire. Formulas are given below for more accurate calculation of the low-frequency inductance. The inductance at high frequency can not generally be calculated with great accuracy. Formulas which take account of the skin effect, or change of current distribution with frequency, have been devel- oped. The change is very small when the coil is wound with suitably stranded wire. The inductance at high frequencies depends, however, also on the capacity of the coil, which is gen- erally not calculable. If the capacity is known, from measure- ments or otherwise, its effect upon the inductance can be cal- culated by L a = L[i+co 2 CL(io)-'] (154) where L a is the apparent or observed value of the inductance, C is in micromicrofarads, and L in microhenries. The inductance of a coil is decreased by skin effect, and is increased by capacity. The changes due to these two effects sometimes neutralize each other, and in general, formula (153) gives about as good a value of the high-frequency inductance as can be obtained. Round Wire. The low-frequency inductance of a coil wound with round wire can be calculated to much higher precision than that of formula (153) by the use of correction terms. Formula (153) gives strictly, the inductance of the equivalent current sheet, which is a winding in which the wire is replaced by an ex- tremely thin tape, the center of each turn of tape being situated at the center of a turn of wire, the edges of adjacent tapes being separated by an infinitely thin insulation. The inductance of the actual coil is obtained from the current-sheet inductance as follows : Radio Instruments and Measurements 253 Putting L 8 = inductance of equivalent cylindrical current sheet, obtained from (153) Lo = inductance of the coil at low frequencies n = number of turns a = radius of coil measured out to the center of the wire D = pitch of winding = distance from center of one wire to the center of the next measured along the axis b = length of equivalent current sheet = nD d = diameter of the bare wire Then L = L 8 0.01257 na (A + B) microhenry (155) in which A is constant, which takes into account the difference in self-inductance of a turn of the wire from that of a turn of the current sheet, and B depends on the difference in mutual induc- tance of the turns of the coil from that of the turns of the current sheet. The quantities A and B may be interpolated from Tables ii and 12, page 284, which are taken from Tables 7 and 8 of Bul- letin, Bureau of Standards, 8, pages 197-199; 1912. Example. A coil of 400 turns of round wire of bare diameter 0.05 cm, wound with a pitch of 10 turns per cm, on a form of such a diameter that the mean radius out to the center of the wire is 10 cm. ^o, ^ = 400, D = o.i, =0.5 2/r The value of K corresponding to -r- = o. 5 is 0.8 1 8 1 (Table 10) . IOO L s =0.03948 (4Oo) 2 - - 0.8181 =0.03948X400 000x0.8181 4 = 12 919 microhenries =0.012919 henry log 0.03948 = 2.59638 log 400 ooo = 5.60206 log 0.8181=1.91281 4.11125 Entering Tables 1 1 and 1 2 with -^ = 0.5, n = 400, we find A = 0.136 B= 0.335 A+B= 0.199 The correction in (155) is, accordingly 0.01257 (400) 10 (0.199) = 9-99 microhenries. 254 Circular of the Bureau of Standards The total inductance is 12 919 10 = 12 909 microhenries. Example. A coil of 79 turns of wire of about 0.8 mm bare diameter. The mean diameter is about 22.3 cm and, for deter- mining the pitch, it was found that the distance from the first to the 79th wire was 9.0 cm. We have, then, b =nD = 79X0.115 =9.12 2a__ d _ 0.08 = M45 ' = The value of K is given by Table 10 as 0.4772, so that (n is) 2 L g =0.03948 (79) 2 - '^- 0.4772 = 1602.8 microhenries log 0.03948 = 2. 59638 ^ d ^ , For n = 79, 7^=0.7, Tables n 2 log 79=3.79526 D 2 log 11.15=2.09454 and 12 give log 0.4772=1.67870 A =o. 200 5=0.326 4. 16488 log 9.12=0.95999 (A +B) =0.526 3- 20489 The correction is 0.01257 X 79 X 11.15 X 0.526 = 5. 8 microhen- ries, and the total is 1597.0 microhenries. The measured in- ductance of this coil is 1595.5. COIL WOUND WITH WIRE OR STRIP OF RECTANGULAR CROSS SECTION Approximate values may be obtained for a coil wound with rectangular-section wire or strip by using the simple formula (153), as already explained. More precise values for the low- frequency inductance could be calculated in the same manner as for round wire above, using different values for A and B. It is simpler, however, to use formula (156) below, which applies to the single-layer coil, if the symbols are given the following meaning: a = radius measured from the axis out to the center of the cross section of the wire; 6= the pitch of the winding D, multiplied by the number of turns n; c=w = the radial dimen- sion of the wire; Z=the axial thickness of the wire. The cor- rection for the cross section of the wire is obtained by using iv t formulas (161) and (162), using v = ^> T=r\' Radio Instruments and Measurements 255 Example. A solenoid of 30 turns is wound with ribbon J4 by iV inch thick, with a winding pitch of X inch to form a sole- noid of mean diameter 10 inches. Here = 5X2.54 = 12.70 cm, w=c = - (2.54) =0.635 cm 4 6 = 30X^(2. 54) =19.05 cm, c/b = ^,D= 0.635 for the equivalent coil. Solving this by Rosa's formula (156), using ^ = -># =0.6230 (Table 10), - = 30, ^,=0.693, #8=0.3218, 03 ^ we find L u = 182.55 fj.h. The value obtained by Stefan's formula (157) is very slightly in error, being 182.5. w i To obtain the correction, we have v = j=: = i, r = -> and therefore u 4 = 0.470 1.25 '.-D 2Q 28 27 26 B.= -2 1 0.060 H 0.018 +- 0.008 H 0.005 1 30 30 30 30 21 1 + . . .+ o.ooi = -0.188 30 so that the correction is (0.01257) 30 (12.70) (0.285) =1.36 /*& and the total inductance is 183.9. INDUCTANCE OF POLYGONAL COILS Such coils, instead of being wound on a cylindrical form, are wrapped around a frame such that each turn of wire incloses an area bounded by a polygon. No formula has been developed to fit this case, but it is found that the inductance of such a coil (when the number of sides of the polygon is fairly large) may be calculated, within i per cent, by assuming that the coil is equivalent to a helix, whose mean radius is equal to the mean of the radii of the circumscribed and inscribed circles of the polygon. That is, if r = the radius of the circumscribed circle, Fig. 182 (which can be measured without difficulty for a polygon for which the number of sides N is an even number), then the modified radius a r cos 2 -^ is to be used for a in the formulas (153) and (155) of the preceding section. 35601 18 17 256 Circular of the Bureau of Standards Examples. The following table gives the results obtained by this method for some 1 2-sided polygonal coils, the measured inductance being given for comparison. For N = i2, a = Coil r Oo n D b L calculated M* Lo measured P* A 6.35 6.24 23 0.32 7.3 63.0 61.7 B a 25 8.10 28 .32 9.0 124.7 126.3 C 11.43 11.22 52 .212 11.0 638.0 630.5 D 11.43 11.22 34 .388 10.8 274.9 274.6 E 13.97 13.73 64 .211 13.1 1119. 5 1115.5 F 19.05 18.71 117 .158 18.5 5389 5387 MULTIPLE-LAYER COILS Different formulas are used for long than for short coils. . Far long coils of few layers, sometimes called multiple-layer solenoids, the inductance is given, approximately, by , . (0.693+5.) (156) FIG. 182. Polygonal coil where L 8 = inductance, calculated by (153), letting n = number of turns of the winding a = radius of coil measured from the axis to the center of cross section of the winding b = length of coil = distance between centers of turns, times number of turns in one layer c = radial depth of winding = distance between centers of two adjacent layers times number of layers B 6 correction given in Table 13, page 284, in terms of the ratio - c Radio Instruments and Measurements 257 Values obtained by this formula are less accurate as the ratio c/a is greater, and may be a few parts in 1000 in error for values of this ratio as great as 0.25, and - as great as 5. For accurate results a correction needs to be applied to L u (see (159) below). The solution of the problem for short coils is based on that for the ideal case of a circular coil of rectangular cross section. Such a coil would be realized by a winding of wire of rectangular cross 'Axis FIG. 183. Multiple-layer coil with winding of rectangular cross section section, arranged in several layers, with an insulating space of negligible thickness between adjacent wires. Let a = the mean radius of the winding, measured from the axis to the center of the cross section 6= the axial dimension of the cross section c = the radial dimension of the cross section d = V& 2 + c 2 = the diagonal of the cross section n = number of turns of rectangular wire. Then, if the dimensions b and c are small in comparison with a, the inductance is very accurately given by Stefan's formula, which, for b > c, takes the form L u = 0.01257 an 2 b 2 where y l and y 2 are constants given in Table 14, page 285. \ 258 Circular of the Bureau of Standards For disk or pancake coils, b 'log. - fc\ r~T o c "7^ t ; 5 )(22 5 y 2 p I + 3(o. 3 ) 2 + (o. 3 ) 2 u ^U.Ul Z$ 1 ) \ 96 -0.8483+- ^o.8i6J log 8 =0.90309 2.76310 1.00375 lo \ log 0.18=7.62764 17269 104 8a ~d = 2> 9478 -y,= .8483 104 8a 8a 0.09 , -j =1.27545 2. 93683= log. -^- -^0.8 1 6 2. 104 Iog 10 2.i04 =o-3 2 35 log 10 225 =4.70436 Iog 10 o.oi 257 = 2.09934 L u = 6694 microhenries. Iog 10 5 =0.69897 3.82572 The correction for insulation is found from (159), as follows: f-IS'-f' log.of =.0 9 6 9 I, log.|-0.22 3 0.138 E = o. 017 0.378 correction = (0.0125 7) (5) (225) 0.378 =3.34^ 260 Circular of the Bureau of Standards The total inductance is 6697 microhenries = 6. 69 7 millihenries. The correction could, in this case, have been safely neglected. Example. A coil of 10 layers of 100 turns per layer, mean radius = 10 cm, the wires being spaced o.i cm apart. For this case n = 1000, a = 10, b = 10, c = i . 2a Using formula (156) with -r- = 2, K = 0.5255, b(c = io L 9 = (0.03948) l ~ * IO 0.5255 =207 400 microhenries. 10 For the correction, Table 13 gives for - = io C 0.693 B a =0.279 0-973 so that the correction = (0.01257) io 6 0.973 = 12 200 and the inductance is L u = 207 400 1 2 200 = 195 200 microhenries = 195.2 millihenries. The formula (157) gives a value about one part in 900 higher than this. INDUCTANCE OF A FLAT SPIRAL Such a spiral may be wound of metal ribbon, or of thicker rectangular wire, or of round wire. In each case, the inductance calculated for the equivalent coil, whose dimensions are measured by the method about to be treated, will generally be as close as i per cent to the truth, the value thus computed being too small. If n wires, Fig. 184, of rectangular cross section are used, whose width in the direction of the axis is w, whose thickness is t, and whose pitch, measured from the center of cross section of one turn to the corresponding point of the next wire is D, then the dimen- sions of the cross section of the equivalent coil are to be taken as b=w, c = nD, and as before d = -y6 2 + c 2 . The mean radius of the equivalent coil is to be taken as a = Oi + /4(ni)D, the distance a^ being one-half of the distance AB (see Fig. 185) measured from the innermost end of the spiral across the center of the spiral to the opposite point of the inner- most turn. The inductance L u of the equivalent coil is to be calculated using the above dimensions in (158), assuming for n the same number of turns as that of the spiral. Radio Instruments and Measurements 261 If round wire is employed, the same method is used for obtain- ing the mean radius a and the dimension c, but it is more con- venient to take b as zero, and use for the calculation of the induc- tance of the equivalent coil the special form of (158) which follows when 6 is placed equal to zero. Sa 80 f Sa i 0.01257 n*a 2.303 Iog 10 - FlG. 184. Sectional -view of flat spiral wound with metal ribbon FIG. 185. Side -view of flat spiral The correction for cross section may, in each case, be made by subtracting 0.01257 na (A l +5 1 ) from the value of inductance for the equivalent coil. For round wires the quantities A x and B l may be taken as equal to A and B in the Tables 1 1 and 12, page 284, just as in the case of single-layer coils of round wire. In the case of wire or strip of rectangular cross section the matter is more complicated on account of the two dimensions of the cross section. TV t If we let jj v and JJ = T, then the quantities involved in the calculation of A t and B l may be made to depend on these two 262 Circular of the Bureau of Standards parameters alone. The equations are then with sufficient accu- racy: 1 V + l V+l I H \ A! = log. ;^.- 2.303 tag^j (161) |~n I. n 2 s n 3 s , J s "1 #!=-2 5 12 H S 13 + -flu + . . +-S n (162) L n n n n J in which 6 12 , 5 13 , etc., are to be taken from Table 15, page 285. Example. For a spiral of 38 turns, wound with copper ribbon whose cross sectional dimensions are 3/8 by 1/32 inch, the inner diameter was found to be 2a t = 10.3 cm and the measured pitch was found to be 0.40 cm. The dimensions of the equivalent coil of rectangular cross sec- tion-are, accordingly, 6 = 3/8 inch = 0.953 cm, a = ^ + -^-37 (o-4) =12.55, = 38 X 0.40 = 15. 2. For this coil b/c = 0.0627 which gives (Table 14) ^=0.5604, d 2 Sa ^3 = 0.599,^= 1.472, log. -j-i.886. Hence from (158), L u = (0.01257) (12.55) (38) 2 [1.015 (1.886) -0.5604 + 0.055] = 323.3 microhenries. For this spiral ^ = 2.38, r = 0.198 7 jfi A, = 2.303 Iog 10 ^fg =0.270 = ~ 2 (0>028) + (0\\C X 2 J ) i \ / /__\ logio r = ~ Iog 10 * + -( i - 7i ) log! 2 \ c / 2.303 When the distance x between the, planes of the coils is chosen equal to the pitch D of their windings, the calculation of their inductance, when joined in series, may be obtained in a simpler manner. Putting b = 2D and n i = 2n, the number of turns of the two windings in series, a b + c L =0.008 n^a] 2.303 Iog 10 7 (-0.2235 1-0.726 L b+c a J +0.008 n^a I 2.303 Iog 10 j + 0.153 J (172) for a square coil, and V = 0.0092 1 o tt! 2 (a + aj Iog 10 r - 1 - a Iog 10 (a + g) (/ * I C I tt ~{~ di J L 2 J + 0.004 ^i ( + Oi) I 2-303 Iog 10 ^ + o. 1 53 J ( 1 73) for a rectangular coil <7 = -\ja 2 + a t 2 , J = diameter of bare wire. Example. As an example of the use of these formulas, take the case of an actual coil of two sections, each being a flat, square coil of 5 turns of 0.12 cm wire, wound with a pitch of .0 = 1.27 cm, the distance of the planes of the coils being # = 1.27 cm. The length of a side of the outside turn was 101 cm. Putting n = 5, a = 101 -4X1. 27 =95.9, = 5X1.27 = 6.35, and d/D=o.i, formula (169) gives ^ = 66.28 + 6.14 = 72.42^, for a single section. Radio Instruments and Measurements 269 To obtain the mutual inductance, we find by (170) for S_1.27 "6.35 = 0.2 3 3, 2.303 Iog 10 k=2. 303X0.04 (-0.699) +0.2 if -- ^(0.04) -^(o.ooi 6) = 0.06444-0.6283 1.50.06 0.0001 = -0.9962 Iog 10 fe= -0.4326 = 1.5674 k =0.3693 and r = 0.3693 X 6.35 = 2.344 Putting this value of r in place of D in (184) with = 95.9 M =0.008 X 5 X 5 [2.303 X 95-9 Iog 1 191.86 + 2.34 =56. For the two coils in series, then + 135-62 .82 L' = 2(72.42 +56.82) =258.5 ph and for the parallel arrangement The inductance of the coils in series may also be found by putting = 95-9* b = 6.i,5, ^ = 2.54, ^ = 10 in (163) and (159) and we find L = 239.8 + 18.8 = 258. 6 ph in agreement with the other method. 69. MUTUAL INDUCTANCE The following formulas for mutual inductance hold strictly only for low frequencies. In gen- eral, however, the values will be the same at high frequencies. A TWO PARALLEL WIRES OR BARS SIDE BY SIDE Let / = length of each wire or bar. D = distance between centers of the wires. The following expression is exact when the FIG. igo.Two wires have no appreciable cross section, but is kl wires side b v side sufficiently exact even when the cross section is large if / is 270 Circular of the Bureau of Standards great compared with D. Within these limits the shape is im- material. = 0.002/ 2.303 I 2. D '] (174) = o.oo2/ 2.303 Iog 10 ^ i +-T nearly. ( J 75) TWO WIRES END TO END WITH THEIR AXES IN LINE Let the lengths of the two wires be / and m, their radii being supposed to be small. Then, l + m M = 0.002 303 /Iog 10 -j l + rnl -^- f ,\ (176) TT\ FIG. 191. Two wires end to end in same straight line FIG. 192. Two wires in same straight line but separated TWO WIRES WITH THEIR AXES IN THE SAME STRAIGHT LINE BUT SEPARATED Let their lengths be / and m and the distance between the nearer ends be Z. M = 0.002303 [(/ + m + Z) log xo (/ + m + Z) + Z Iog 10 Z - (/ + Z) Iog 10 (/ + Z) - (m +Z) lo glo (m + Z)] d77) Radio Instruments and Measurements 271 TWO WIRES WITH AXES IN PARALLEL LINES If AD, AD', AC, AC', etc., represent the distances shown in the figure, the general formula is Dl ^ C' &' i m B FIG. 193. Two wires with axis in parallel lines r ;1 { AD - AD >X A c + AC'\ AD + AD' BD-BD' iAD+AD' AC-AC' BD-BD' BC+BC'\~] 1 8">\AD-AD' X AC+AC' X BD+BD' X BC-BC'\] -o.ooi (AD-AC-BD+BC) the distances being AD' = l + m + Z, AD = -\lx 2 + (l + m + Z) 2 , etc. This formula holds for Z=o, but not when one wire overlaps on the other. When they overlap, as in Fig. 194, M = M 1(34 + M 23 + M 24 (179) in which M 1)34 is to be calculated by the general formula, using Z =o and putting the segment PV for / and ST for m, while for M, 4 the length VR is put for / and WT for m with Z=o. The 35601 18 IS 272 Circular of the Bureau of Standards mutual inductance M 23 of the overlapping portions is obtained by (174)- T R V w S FIG. 194. Two wires with axis in parallel lines; a particular case of Fig. 193 Special Cases. For the case shown in Fig. 195 M = 0.001 -D D- (180) FIG. 195. Two wires with axes in parallel FIG. 196. Two wires with axes in parallel lines, lines; another particular case of Fig. 193 -with one end of each on the same perpendicular and for the wires of Fig. 196 M=o.oo, [4.605, logl llblj Radio Instruments and Measurements 273 MUTUAL INDUCTANCE OF TWO PARALLEL SYMMETRICALLY PLACED WIRES 2* t 2JI, FIG. 197 . Two parallel symmetrically placed wires Putting for the lengths of the two wires 2/ and 2/j (2/ the shorter) and for their distance apart D M = 0.002 I 2.303(2/)log 10 | l V D l (182) TWO EQUAL PARALLEL RECTANGLES Let a and a t be the sides of the rectangles and D the distance between their planes, the centers of the rectangles being in the same line, perpendicular to these planes ) a ,- M = 0.009210 a log 8 . 2 L [a + V + a i ^ J - -Ja 2 +D* - Jaf+D* + D] (183) 274 Circular of the Bureau of Standards TWO EQUAL PARALLEL SQUARES If a is the side of each square and D is the distance between their planes, then the preceding formula becomes :is 4 ) + 0.008 + D 2 - 2 a 2 +D 2 +D] MUTUAL INDUCTANCE OF TWO RECTANGLES IN THE SAME PLANE WITH THEIR SIDES PARALLEL M = 2(M W +M 27 ) - 2(M 18 +M 25 +M 33 +M 47 ) (185) ,.6 '8 FIG. 198. Two rectangles in the same plane with their sides parallel the separate mutual inductances being calculated by formula (182), if the sides are symmetrically placed, and by (182) and (178) if that is not the case. If the rectangles have a common center M 18 = M 38 , M 45 = M, 7 , M 18 = M 3e , M 25 =M 47 and for the case of concentric squares, we have = 4 (M 18 -M 18 ) (186) TWO PARALLEL COAXIAL CIRCLES This is an important case because of its applicability in calcu- lating the mutual inductances of coils (see below) . Let a = the smaller radius (Fig. 199). A =the larger radius. D = the distance between the planes of the circles. Then - Radio Instruments and Measurements 275 (i8 7 ) where F may be obtained by interpolation in Table 16 for the calculated value of ~ -'"? f x = the longest distance between the circumferences. r 2 = the shortest distance between the circumferences. TWO COAXIAL CIRCULAR COILS OF RECTANGULAR CROSS SECTION If the coil windings are of square, or nearly square, cross section, a first ap- proximation to the mutual inductance is (188) where n^ and n 2 are the number of turns on the two coils and M is the mutual inductance of two coaxial circles, one located at the center of the cross section of one of the coils and the other at the center of the cross section of the other. Thus, if ,Vv / 1 r \ \ \ \ A \ \ a \ \ r V-D ^ \ \ \ \ \ \ \ ' \ \ 1 \ 1 1 i , V 1 i \ 1 \ FIG. 199. Cross sections of two parallel coaxial circles FIG. 200. Two paral- lel coaxial coils with windings of rectangu- lar cross sections a = mean radius of one coil, measured from the axis to the center of cross section, A = mean radius, similarly measured, of the other coil, D = distance between the planes passed through the centers of cross section of the coils, perpendicular to their com- mon axis (Fig. 200). the value M will be computed by formula (187) and Table 16, using the values of a, A, and D, just defined. If the cross sections of the windings are square, this value will not be more than a few parts in a thousand in error, even with relatively large cross sectional dimensions, except when the coils are close together. 276 Circular of the Bureau of Standards A more accurate value for coils of square cross section may be obtained by supposing the two parallel circles to remain at the distance D, but to have radii where 6j and b 2 are the dimensions of the square cross sections corresponding to the coils of mean radius a and A, respectively. When the correction factors in (189) are only a few parts in 1000, the values of.rjr lt and hence F, are very little affected, and the fractional correction to the mutual inductance, to allow for the cross sections, is approximately equal to the geometric mean of the fractional corrections to a and A , so that an estimate of the magnitude of the correction to the mutual inductance may be gained with little labor. With rectangular cross sections the error from the assumption that the coils may be replaced by equivalent filaments at the center of the cross section is more important than in the case of coils of square cross section and rapidly increases as the axial dimension of one or both of the cross sections is increased, in rela- tion to the distance D between the median planes. The error may, easily, be as great as i per cent or more in practical cases. An estimate of the magnitude of the error, in any case, may be made by dividing the coils up into two or more sections of, as nearly as possible, square cross section, and assuming that each portion of the coil may be replaced by a circular filament at the center of its cross section. Suppose that coil A is divided into two equal parts, and replaced by two filaments i, 2, while coil B is likewise replaced by two filaments 3, 4, then, assuming that each filament is associated with a number of turns which is the same fraction of the whole number of turns in the coil as the area of the section is to the whole cross sectional area (one-half in this case) we have M ^ 2s M, 3 +"' B * M u +** Af, + ?-!=! M a 4 4 4 (190) in which M 13 is the mutual inductance of the two circular filaments i and 3, etc. Radio Instruments and Measurements 277 For a discussion of more accurate methods for correcting for the cross section of coils, the reader is referred to Bulletin, Bureau of Standards, 8, pages 33-43; 1912. If the coils are of the nature of solenoids of few layers, it is best to use the formulas for the mutual inductance of coaxial solenoids given in the next section. Example. Suppose two coils of square cross section 2 cm on a side, the radii being, a = 20, A = 25, and the distance between their median planes being D = io cm (Fig. 201). Further, suppose that one coil has 100 turns and the other 500. Then '0.24253 From Table 16 we find, corresponding to this value of > F = o.oi 1 1 3. Therefore, from (187) M =0.01 1 13-^25X20 = 0.2489^ and M = n^Mo = 100 X 500 X 0.2489 = 12 445 microhenries = 0.012445 henry. If we take account of the cross sections we have from (189) =2 (1-00042) FIG. 201. Exam- ple of two paral- lel coaxial coils with windings of rectangular cross section so that the correction factor to the mutual inductance will be of the order of about 1/1.00042 X 1.00027, or the mutual inductance should be increased by about 3.5 parts in 10 ooo only. Example. Fig. 202 shows two coils of rectangular cross section. For coil P, a = 20, 6 1 = 2, ^ = 3, 7^ = 600. For coil Q, A =25, 6 2 = 4 c 2 = i, n 2 = 400 and D = io. If, first, we replace each coil by a 278 Circular of the Bureau of Standards circular filament at the center of its cross section, we have the 1 same value of M as in the previous example, and M = 600 X 400 X 0.2489 microhenries. More precise formulas, involving a good deal of computation, show that the true value is M = 600 x 400 x o. 249844, \ so that the approximate value is about 3.8 parts in looo too small. Each coil is then subdivided into two sections and filaments p, q,r, s, imagined to pass through Another ^ cen ter of cross section of each of these subdivi- example of rig. 200 sions. The data for these filaments are as follows : Radius Filaments a A D rz/ri F p 19. 25 pr 19.25 25 9 0. 2365 0. 01140 q 20. 75 ps 19.25 25 11 .2722 . 009872 r 25 V 20.75 25 9 .2135 . 01255 s 25 qs 20.75 25 11 .2506 . 01077 We find then [0.2501+0.2166+0.2858+0.2452! M =600X400 4 =600X400X0.24942 a result which is 1.7 in 1000 too small. The increase in accuracy is hardly commensurate with the increased labor. MUTUAL INDUCTANCE OF COAXIAL SOLENOIDS NOT CONCENTRIC Gray's formula, given for this case, supposes that each coil approximates the condition of a continuous thin winding, that is, a current sheet. 1x_ ^ ! *""^-. n ~n. a, ~*^ f K-- _ _ y _ ^j P- *1 1 FIG. 203. Coaxial solenoids not concentric Radio Instruments and Measurements 279 Let a = the smaller radius, measured from the axis of the coil to the center of the wire A =the larger radius, measured in the same way 2/ = length of the coil of smaller radius = number of turns times the pitch of winding 2% = length of the coil of larger radius, measured in the same way n x and n 2 = total number of turns on the two coils D = axial distance between centers of the coils x-i = D x t\ = Then a 2 A 2 * M = 0.0098 70 & + K s k 3 + K 6 k s in which (191) A*T x^ x,-\ z a / x 2 2 = - TL^( 3 - fa) - f? (3 - ^ (z ^- 2 a~ This formula is most accurate for short coils with relatively great distance between them. In the case of long coils it is some- times necessary to subdivide the coil into two or more parts. The mutual inductance of each of these parts on the other coil having been found, the total mutual inductance is obtained by adding these values. Example. 20.53- 7.i&- 4 ' 6.44 t >-*-7.2-* 4.455 * FIG. 204. Example of coaxial solenoids not concentric 2X = 20.55 .4=6.44 ^i = I5 2/ =27.38 a =4.435 2 = 75 Distance between the adjacent ends of the two solenoids = 7.2 cm. 280 Circular of the Bureau of Standards Then *! = 20.89 k 1 K l =0.04294 2 = 4! -44 k 3 K 3 = .01827 = .00519 0.06640 /a 2 A 2 n.n\ and M = 0.009870! M 10.06640 = i .069 microhenries log 0.009870 = 3.99432 2 log a =1.29378 log 2^ = 1.31281 2 log A =1.61778 log 2/ =1.43743 lognjw, =3.05115 log 0.06640 =2.82217 2.75024 2.77920 2.75024 0.02896 = log M Dividing the longer coil into two sections C and D of 37 and 38 turns, respectively, and repeating the calculation for the mutual inductance of these sections on the other coil R (Fig. 204) , For MRC For M RD k l K l = 0.04889 k 1 K 1 = 0.01155 k 3 K 3 = .00652 k 3 K 3 = .00061 k 5 K 5 = .00005 0.05546 0.01216 and M = MHO + MRD = 0.891 7 4- o.i 956 = 1.087 M- Further subdivision showed that this last value is not in error by more than 5 parts in 10 ooo. The criterion as to the necessity of subdivision is the rapidity with which the terms kJK-i, k 3 K 3 , etc., fall off in value. In the first case k 7 K 7 and k^K 9 are not negligible. The expressions for these quantities are not here given because they are laborious to calculate, and it is easier to obtain the value of the mutual induc- tance by the subdivision method. COAXIAL, CONCENTRIC SOLENOIDS (OUTER COIL THE LONGER) The formula here given holds, strictly, only for current sheets. The lengths of the coils should be taken as equal to the number of turns times the pitch of the winding in each case. Then the Radio Instruments and Measurements 281 mutual inductance of the current sheets is not appreciably differ- ent from that of the coils. Let a = smaller radius ., / A = larger radius 2# = equivalent length of outer coil 2/ = equi valent length of inner coil g = ix*+A 2 = diag- onal. FIG. 205. Coaxial concentric solenoids, outer coil begin longer This formula is more accurate, the shorter the coils and the greater the difference of their radii, but in most practical cases the accu- racy is ample. In many cases the second term in (192) is negli- gible, and it is a good plan to make a preliminary rough calcula- tion of this term to see whether it will need to be considered. In the case of long coils, and of coils of nearly equal radii, the terms neglected in this formula may be as great as i per cent. A crite- o?A 2 rion of rapid convergence is, in general, the smallness of j-> but (1 2 \ 3-43 j and the corresponding coefficients of terms neglected in (192) may in some cases modify this condition for rapid convergence materially. Example. = 300 n 2 = 200 y 0.01974 !-?= 1 198.5 M = 1 198.5 (i + .001 15) = 1 199.9 microhenries. For the case, however, where 2% = 30 a = 2 n l = 300 2/ = 24 A = 5 n 2 = 960 282 Circular of the Bureau of Standards a 2 A 2 i / 7 2 \ although the value of - r =- - only, the coefficient (3-4 -= 1 9 5 \ a 2 / = 141, (the length of the coil is great compared with its radius) so a 2 A 2 is 0.0282, and investigation of the corn- that the term in plete formula shows that the succeeding terms are 0.0127 and 0.0048, so that their neglect will give an error of over 1.5 per cent. (For precision calculations see Bull., Bureau of Standards, 8, pp. 61-64, !9i2, for the complete formula.) CONCENTRIC COAXIAL SOLENOIDS (OUTER COIL THE SHORTER) I % / o t -< r^ ' / y ) - 6 "Z - i FIG. 206. Coaxial concentric solenoids, outer coil bing shorter In this case we have to put g = ^ and the formula is ('93) which is rapidly convergent in most cases. 70. TABLES FOR INDUCTANCE CALCULATIONS TABLE 8. Values of d in Formulas (132), (134), (137), (138), (140), (141), (142), and (147), for Calculating Inductance of Straight Wires at Any Frequency I z i 0.250 12.0 0.059 0.5 .250 14.0 .050 1.0 .249 16.0 .044 1.5 .247 18.0 .039 2.0 .240 20.0 .035 2.5 0.228 25.0 0.028 3.0 .211 30. . 024 3.5 .191 40. . 0175 4.0 .1715 50.0 .014 4.5 .154 60.0 .012 5.0 0.139 70.0 0.010 6.0 .116 80.0 .009 7.0 .100 90.0 .008 8.0 .088 100.0 .007 9.0 .078 00 .000 10.0 .070 Radio Instruments and Measurements 283 TABLE 9. Constants P and Q in Formulas (141), (142), (144), and (145) 2A I P I 2A Q 2A ( P I 2A Q .0000 0.6 0.5136 0.6 1.2918 0.1 0.0975 0.1 .0499 .7 .5840 .7 1. 3373 .2 .1900 .2 .0997 .8 .6507 .8 1. 3819 .3 .2778 .3 .1489 .9 .7139 .9 1. 4251 .4 .3608 .4 .1975 1.0 .7740 1.0 1. 4672 .5 .4393 .5 .2452 TABLE 10. Values of K for Use in Formula (153) Diameter Diameter Diameter E< Length K Difference Length Difference Length jK DiSerenca 0.00 1.0000 -0. 0209 2.00 0. 5255 -0.0118 7.00 0. 2584 -0.0047 .05 .9791 203 2.10 .5137 112 7.20 .2537 45 .10 .9588 197 2.20 .5025 107 7.40 .2491 43 .15 .9391 190 2.30 .4918 102 7.60 .2448 42 .20 .9201 185 2.40 .4516 97 7.80 .2406 40 0.25 0. 9016 -0.0178 2.50 0. 4719 -0.0093 8.00 0.2366 -0.0094 .30 .8838 173 2.60 .4626 89 8.50 .2272 86 .35 .8665 167 2.70 .4537 85 9.00 .2185 79 .40 .8499 162 2.80 .4452 82 9.50 .2106 73 .45 .8337 156 2.90 . 4370 78 10.00 .2033 0.50 0. 8181 -0.0150 3.00 0. 4292 -0.0075 10.0 0.2033 -0.0133 .55 .8031 146 3.10 .4217 72 11.0 .1*03 113 .60 .7885 140 3.20 .4145 70 12.0 .1790 98 .65 .7745 136 3.30 .4075. 67 13.0 .1692 87 .70 .7609 131 3.40 .4000 G4 14.0 .1605 78 0.75 0. 7478 -0.0127 3.50 0. 3944 -0.0062 15.0 0. 1527 -0.0070 .80 -7351 123 3.60 .3882 60 16.0 .1457 63 .85 .7228 118 3.70 .3822 58 17.0 .1394 58 .90 .7110 115 3.80 .3764 56 18.0 .1336 52 .95 .6995 111 3.90 . 3708 54 19.0 .1284 43 1.00 0. 6884 -0.0107 4.00 0. 3654 -0. 0052 20.0 0. 1236 -0.0085 1.05 .6777 104 4.10 .3602 51 22.0 .1151 73 1.10 .6673 100 4.20 .3551 49 24.0 .1078 63 1.15 .6573 98 4.30 .3502 47 26.0 .1015 56 1.20 .6475 94 4.40 .3455 46 28.0 .0959 49 1.25 0. 6381 -0.0091 4.50 0.3409 -0.0045 30.0 0.0910 -0.0102 1.30 .6290 89 4.60 .3364 43 35.0 .0808 80 1.35 .6201 86 4.70 .3321 42 40.0 .0728 64 1.40 .6115 84 4.80 .3279 41 45.0 .0664 53 1.45 .6031 81 4.90 .3238 40 50.0 .0611 43 1.50 0. 5950 -0. 0079 5.00 0. 3198 -0.0076 60.0 0.0528 -0.0061 1.55 .5871 76 5.20 .3122 72 70.0 .0467 48 1.60 .5795 74 5.40 .3050 69 80.0 .0419 38 1.65 .5721 72 5.60 .2981 65 90.0 .0381 31 1.70 .5649 70 5.80 .2916 62 100.0 .0350 1.75 0. 5579 -0.0068 6.00 0. 2854 -0. 0059 1.80 .5511' 67 6.20 .2795 56 1.85 .5444 65 6.40 .2739 54 1.90 .5379 63 6.60 .2685 52 1.95 .5316 61 6.80 2633 49 284 Circular of the Bureau of Standards TABLE 11. Values of Correction Term A in Formulas (155), (165), (168), and (169) d ~D A Difference d D A Difference d D A Difference 1.00 0.557 -0. 051 0.40 -0.359 -0.052 0. 15 -1. 340 -0.069 0.95 .506 54 .38 .411 54 .14 1.409 74 .90 .452 57 .36 .465 57 .13 1.483 80 .85 .394 61 .34 .522 61 .12 1.563 87 .80 .334 65 .32 .583 64 .11 1.650 96 0.75 0.269 -0.069 0.30 -0. 647 -0.069 - 0.10 -1.746 -0. 105 .70 .200 74 .28 .716 74 .09 1.851 .118 .65 .126 80 .26 .790 80 .08 1.969 .133 .60 .046 87 .24 .870 87 .07 2.102 .154 .55 - .041 95 .22 .957 96 .06 2.256 .173 0.50 -0.136 -0.041 0.20 -1.053 -0. 051 0.05 -2. 439 -0.223 .48 .177 43 .19 1.104 54 .04 2.662 .288 .46 .220 44 .18 1.158 57 .03 2.950 .405 .44 .264 47 .17 1.215 61 .02 3.355 .693 .42 .311 48 .16 1.276 64 .01 4.048 TABLE 12. Values of Correction B in Formulas (155), (165), and (169) Number of turns, 7i B Number of turns, n B 1 0.000 40 0.315 2 .114 45 .317 3 .166 50 .319 4 ..197 60 .322 5 .218 70 .324 6 0.233 80 0.326 7 .244 90 .327 8 .253 100 .328 9 .260 150 .331 10 .266 200 .333 15 0.286 300 0.334 20 .297 400 .335 25 .304 500 .336 30 .308 700 .336 35 .312 1000 .336 TABLE 13. Values of B e for Use in Formula (156) b c B. b C B, 1 0.0000 16 0. 3017 2 .1202 17 .3041 3 .1753 18 .3062 4 .2076 19 .3082 5 .2292 20 .3099 6 0. 2446 21 0. 3116 7 .2563 22 .3131 8 .2656 23 .3145 9 .2730 24 .3157 10 .2792 25 .3169 11 0.2844 26 0.3180 12 .2888 27 .3190 13 .2927 28 .3200 14 .2961 29 .3209 15 .2991 30 .3218 Radio Instruments and Measurements TABLE 14. Constants Used in Formulas (157) and (1S8) b/c or c/6 n Difference C/6 yi. Difference 6/c V3 Differ- ence 025 0.5000 .5253 0.0253 237 0.125 0.002 0.597 0.002 '.05 .5490 434 0.05 .127 5 0.05 .599 3 .10 .5924 386 .10 .132 10 .10 .602 6 0.15 0.6310 0.0342 0.15 0.142 0.013 0.15 0.608 0.007 .20 .6652 301 .20 .155 16 .20 .615 9 .25 .6953 266 .25 .171 20 .25 .624 9 .30 .7217 230 .30 .192 23 .30 .633 10 0.35 0. 7447 0.0198 0.35 0.215 0.027 0.35 0.643 0.011 .40 .7645 171 .40 .242 31 .40 .654 11 .45 .7816 144 .45 .273 34 .45 .665 12 .50 .7960 121 .50 .307 37 .50 .677 13 0.55 0.8081 0.0101 0.55 0.344 0.040 0.55 0.690 0.012 .60 .8182 83 .60 .384 43 .60 .702 13 .65 .8265 66 .65 .427 47 .65 .715 14 .70 .8331 52 .70 .474 49 .70 .729 13 0.75 0. 8383 0.0039 0.75 0.523 0.053 0.75 0.742 0.014 .80 .8422 29 .80 .576 56 .80 .756 15 .85 .8451 19 .85 .632 59 .85 .771 15 .90 .8470 10 .90 .690 62 .90 .786 15 0.95 0.8480 0.0003 0.95 0.752 0.064 0.95 0.801 0.015 1.00 .8483 1.00 .816 1.00 .816 TABLE 15. Values of Constants in Formula (162) t> Values of 12 V Values of 13 T=0 0.1 0.3 0.5 0.7 0.9 T=0 0.3 0.6 0.9 0.114 0.113 0.106 0.092 0.068 0.030 0.022 0.020 0.014 0.004 0.5 .090 .089 .083 .070 .049 .020 0.5 021 .018 .014 .004 1.0 .064 .064 .059 .050 .034 .013 1.0 019 .018 .013 .004 1.5 .047 .046 .043 .036 .025 .009 2.0 015 .015 .010 .003 2.0 .035 .035 .032 .027 .018 .007 4.0 008 .008 .005 .002 3.0 .022 .022 .020 .017 .011 .004 6.0 005 .005 .004 .001 4.0 .015 .015 .014 .012 .008 .003 10.0 003 .003 .002 .005 6.0 .008 .008 .008 .006 .004 .002 8.0 .006 .006 .005 .004 .003 .001 10.0 .004 .004 .004 .003 .002 .001 V Values of in V Values of 5is r=0 0.3 0.6 0.9 T=0 0.1 0.5 0.9 0. 009 0. 009 O.*006 0.002 0.005 0.005 0.004 0.001 1 .009 .008 .006 .002 5 .003 .003 .002 .001 3 .007 .006 .004 .001 10 .002 .002 .001 .000 5 . 004 . 004 .003 .001 10 .002 .002 .001 .000 V Values of ie V Values of 617 V Values of &w T=0 and 0.1 0.5 0.9 T=0 and 0. 0.5 0.9 T=Q and 0.1 0.5 0.9 0.003 0.003 0.001 0.002 0.002 0.001 0.002 0.001 0.000 5 .002 .002 .000 5 .002 .001 .000 5 .001 .001 .000 10 .001 .001 .000 10 .001 .001 .000 10 . 001 . 001 .000 NOTE. The maximum values of all further values of the i's are o.ooi or less. 286 Circular of the Bureau of Standards TABLE 16. Values of F in Formula (187) for the Calculation of the Mutual Inductance of Coaxial Circles rj/ri F Difference r 2 /ri F Difference r/ri F Difference 00 0.010 0. 05016 -0. 00120 0.30 0. 008844 0.000341 0.80 0. 0007345 -0. 0000504 -.011 4897 109 .31 8503 328 .81 6741 579 .012 4787 100 .32 8175 314 .82 6162 555 .33 7861 302 .83 5607 531 0.013 4687 -0. 00093 .34 7559 290 .84 5076 ' 507 .014 4594 87 .015 4507 81 0.35 0. 007269 -0.000280 0.85 0.0004569 -0.0000484 .016 4426 148 .36 6989 270 .86 4085 460 .018 4278 132 .37 6720 260 .87 3625 437 .38 6460 249 .88 3188 413 , 0.020 0. 04146 -0.00119 .39 6211 241 .89 2775 389 .022 4027 109 .024 3918 100 0.40 0. 005970 -0. 000232 0.90 0. 0002386 -0. 0000363 .026 3818 93 .41 5738 225 .91 2021 341 .028 3725 86 .42 5514 217 .92 1680 316 .43 5297 210 .93 1364 290 0.030 3639 -0. 00081 .44 5087 202 .94 1074 263 .032 .034 .036 .038 0.040 3558 3482 3411 3343 0. 03279 76 71 68 64 -0.00061 0.45 .46 .47 .48 .49 0. 004885 4690 4501 4318 4140 -0. 000195 189 183 178 171 0.95 .96 .97 .98 .99 0. 00008107 5756 3710 2004 703 -0.00002351 2046 1706 1301 703 .042 3218 58 Olfrt 0(v"i7QiO A fWUfifi 1.00 .044 3160 55 . j(j .51 . uujyoy 3803 ~ u. lA/vlOO 160 .046 3105 53 .52 3643 156 0.950 O."00008107 -0.00000494 .048 3052 51 .53 3487 150 .952 7613 482 .54 3337 146 .954 7131 4.70 0.050 0. 03001 -0. 00226 .956 6661 458 .060 2775 191 0.55 0. 003191 -0. 000141 .958 6202 446 .070 2584 164 .56 3050 137 .080 2420 144 .57 2913 133 0.960 0. 00005756 -0. 00000436 .090 2276 128 .58 2780 128 .962 5320 421 .59 2652 125 .964 4899 409 o-lOO .11 0. 02148 2032 -0.00116 104 0.60 0. 002527 -0. 000120 .966 .968 4490 4093 397 383 .12 1928 96 .61 2407 117 .13 1832 89 .62 2290 113 0.970 0. 00003710 -0. 00000370 .14 1743 82 .63 2177 109 .972 3340 356 .64 2068 106 .974 2984 341 0.15 .16 0. 01661 1586 -0. 00075 71 0.65 .66 0. 001962 IS59 -0. 000103 99 .976 .978 2643 2316 327 312 .17 .18 .19 1515 1449 1387 66 62 59 .67 .68 .69 1760 1664 1571 96 93 ' 90 0.980 .982 .984 0. 00002004 1708 1430 -0. 00000296 278 262 0-20 0. 01328 -0.00055 0.70 0. 001481 -0. 000087 .986 1168 242 .21 1273 52 .71 1394 84 .988 926 223 .22 1221 50 .72 1310 81 .23 1171 47 .73 1228 78 0.990 0. 00000703 -0. 00000201 .24 1124 45 .74 1150 76 .992 502 177 .994 326 148 0.25 0. 010792 -0. 000425 0.75 0. 0010740 -0.0000731 .996 177 115 .26 10366 408 .76 . -1001 704 .998 062 62 .27 0. 009958 388 .77 0930 680 .28 9570 371 .78 862 653 .29 9199 355 .79 797 628 DESIGN OF INDUCTANCE COILS 71. DESIGN OF SINGLE-LAYER COILS The problems of design of single-layt* < ^ils m1r be broadly classified as of two kinds. (i) Where it is required to design a coil which shall have a certain desired inductance with a given length of wire, the choice of dimensions of the winding and kind of wire to be used being unrestricted within rather broad limits. This class of problems of design includes a consideration of the question as to what Radio Instruments and Measurements 287 shape of coil will give the required inductance with the minimum resistance. (2) Given a certain winding form or frame, what pitch of winding and number of turns will be necessary, if a certain inductance is to be obtained. In the following treatment of the problem the inductance of the coil will be assumed as equal to that of the equivalent cylin- drical current sheet. This is allowable, since, in general, the correction for the cross section of the wire will not amount to more than i per cent of the total inductance, an amount which may be safely neglected in making the design. The formulas to be given may, of course, be used for making a calculation of the inductance of a given coil. Nevertheless, since their practical use is made to depend upon the interpolation of numerical values from a graph, for accurate calculations formulas (153) and (155) should be used. The inductances of coils of different size, but of identical shape, and the same number of turns, are proportional to the ratio of their linear dimensions. Every formula for the inductance should, accordingly, be capable of expression in terms of some single chosen linear dimension, all the other dimensions occurring in the formula in pairs in the form of ratios. Two formulas are here developed, the first applicable to the solution of problems of the first class, giving the inductance in terms of the total length of wire /, the second for problems presupposing a winding frame of given dimensions. Both show the dependence of the inductance on the shape of the coil. Coil of Minimum Resistance. The fundamental relations of the constants of a coil are a 2 L 8 = 47T 2 w 2 -r-K cgs units 2d the constant K being a function of the shape factor -r- , diameter *- lengtiigifable ic^p. 283). The expression for the inductance may be written as __2Traln L.--J-K and n may be eliminated by substituting for it the expression 35601 18 - 19 288 Circular of the Bureau of Standards obtained by multiplying together the two expressions involving n above. There results, then, , / 2O / jrr ., L e = u TT - r ~-K cgs units or s Z K I Z r. = T=T . / ,_ ? == -7= F microhenries yD loooy 6 v^ (194) FIG. 207. (j) Variation of F with different ratios of coil diameter to length; (2) "variations of v with ratios of diameter to length To aid in the use of this formula the curve of Fig. 207 has been I - - /TT Y 6 prepared, which enables the value of F IOOO to be obtained 2/TT for any desired value of -r The formula (194) and the curve enable one to obtain with very little labor the approximate value of the inductance which may be obtained in a coil of given shape with given / and D. On the same figure is also plotted the factor as a function of - (see example below) . 7T20 b Radio Instruments and Measurements 289 Coil Wound on Given Form. To obtain the second formula, we substitute for n its value yy and b 2 a 2 2aV b or L, IT IOOO20 .K" microhenries and, finally, (195) l :::: 20, FIG. 208. Variation of f and Iog 10 [fwith j- To aid in making calculations the curves of Fig. 208 have been rT OOO 2 CL I ^.^ r , W6 . w ^ w , VCMUM ~* , .^ ^&io/-^&io ~^ IT LTT/V bj 2 fl for different values of -j-- The value of Iog 10 / is plotted, rather than that of /, for large values polated with greater accuracy. than that of /, for large values of -r to enable values to be inter- 290 Circular of the Bureau of Standards From formula (194) and Fig. 207 it is at once evident that with a given length of wire, wound with a given pitch, that coil has the greatest inductance, which has such a shape that the diameter ratio -; . =2.46 approximately. Or, to obtain a coil of a certain desired inductance, with a minimum resistance, this relation should be realized. However, although the inductance diminishes rather rapidly for longer coils than this, changes in the direction of making the coil shorter relative to the diameter are not important over rather wide limits. Naturally, other considerations may modify the design appreciably. These other considerations include the distributed capacity of the coil and the variation of resistance with frequency. Example. Given the pitch of winding, the shape of the coil ( T~ )' and the inductance, to determine the length of wire necessary, the dimensions of the coil and the number of turns. Assuming D = 0.2 cm, -r- = 2.6, L 8 = 1000 microhenries, By formula (194) , l\ = - i (the value of F = 0.001322 being 0.001322 log 1000 = 3. taken from the curve of Fig. 207) or Klog 0.2 =1.65052 2 = 4850 cm. The number of turns may 2 6^052 ke obtained immediately from the relation log F = 3.12123 n = fi . / A. = /J and the graph of v. \D\ 2ira \ D 3 / 2 log/ =5.52929 =1.84310 log I =3.68619 Here n = ^r~ $ (0.350) =54.5 turns, and b=nD = io.<) cm, while Y 0.2 20 = 2.6 X 10.9 = 28.3 cm. If the pitch of the winding had been assumed greater, or a coil of much larger inductance were required, the design of the coil would call for larger dimensions, and cases may arise where the design may prove unsatisfactory, because the coil would be too large. The effect of changing the length and pitch, the shape being taken /I constant, may be seen from (194), which shows that Loc -T=, so that a given fractional increase in the length of the wire is more Radio Instruments and Measurements 291 effective in increasing the inductance than the same fractional decrease in the pitch. The number of turns depends on - / the shape of the coil being kept the same. Example. Formula (194) will also enable the question to be answered as to what pitch must be used if a given length of wire is to be wound with a certain shape of coil to give a desired inductance. If the pitch comes out smaller than the diameter of the proposed wire, the assumed length of wire must be increased. Suppose that an inductance of 10 ooo microhenries is desired 2a with 50 meters of wire, the value of -j- being taken as 2.6, as before. Then l\ _ (5000)* 0.001322 >T-F - - or /} =0.00218 cm, L 8 10 ooo which is manifestly impracticably small. The maximum inductance attainable with the given length of wire could be found by solving (194) for L with the smallest practicable pitch substituted for D, that value being used for F, which corresponds to the assumed ratio of diameter to length. Example. Suppose we have a winding form of given diameter 2a = io cm, how many turns of wire will have to be used for an inductance of looonh if the winding pitch is taken as 0.2, and what will be the axial length of the winding ? From (196) 1000 / = - =25 or Iog 10 / = i.398 1000X0.04 From Fig. 208 this corresponds to a value of -r= 0.225, or 6 must be 45 cm, and the number of turns n = jj = = 225. Such a coil would be too long to be convenient. A smaller pitch should be used. Example. Suppose we have given the same winding form, and we wish to find what pitch is necessary for an inductance of loooph, in order that the length of the coil shall not be greater than the diameter. For 2b (198) i6a : }V and for b > c L-+ -. 2 ( r 99) FIG. 209. Values of (G)for given -values of c_an(lb_ a c Both of these equations may be written in the form /! L = =G microhenries in which G is a factor whose value for given values of - and - may CL C be taken from the curves of Fig. 209. 294 Circular of the Bureau of Standards When / is known c D* / / M (c/a) 2 ( 201 ) From these curves one can see that, for a square cross section, b/c = i, the inductance of a given length of wire is a maximum for C 2 a value of - equal to about Investigation shows that this O , point is, more exactly, c/a = 0.662; that is, for a mean diameter of coil = 3.02 times the side of the cross section. Further, for a given resistance and shape of coil, the square cross section gives a greater inductance than any other form. tit 0)6 FIG. sic. Values of (g)for given -values of and a c The second design formula supposes that the dimensions a, c, and - of the winding form are given, together with the pitch of the C winding. The expressions (157) and (158) for the inductance may then be written L =0.01257 a =0.01257 an 9 microhenries (202) (203) Radio Instruments and Measurements 295 The curves of Fig. 210, which give g for different values of - and - allow of interpolation of the proper value in any given CL C case. Example. Suppose we have a wire of such a size that it may be wound 20 turns to the centimeter, and we wish to design a coil to have an inductance of 10 millihenries, to have a square cross section and such a mean radius as to obtain the desired inductance with the smallest resistance (smallest length of the wire). The latter condition requires that -=0.662. The given quan- CL tities are .0=0.05 cm, b/c i. From Fig. 209 we find that G = 0.000606, so that (200) becomes 10000 = 7 rj 0.000606, from which / = 6458 cm or 64.58 meters of wire. 2/3 log D = 1.13265 From the fundamental equation (201) io 7 5/3 log / = 6.35018 1/3 log/ =^27004 "' 2 log / = 7. 62O22 log/ = 3.8101 1 =1.80 and thence 6=c = o.662 X 1.80 = 1.19, and n = j^ -^ = 570. D 2 0.0025 This coil is rather too small to allow of its dimensions being accurately measured. If wire of double the pitch is used, the design works out with the following results / = 85. 22 meters c = 6 = 2.o8 ^ = 432 = 3.18 which is more suitable. Example. We have a form whose dimensions are 2 D 2ira which, on multiplication, give .1 a 'D (204) Radio Instruments and Measurements 297 and this introduced into the expression 47rcm 2 = 2/n gives finally g < 8 ~ logf (205) = pL// microhenries. FIG. 211. Value of (H) for given values of and The factor H, which may be determined from the curves of Fig. 211 is a function of c/a and b/c. The latter quantity may be expressed in terms of the known quantities by the equation -VI (206) Accordingly, the curves are plotted with H as ordinates, c/a as , , /27r abscissas, and 6 -%//n as parameter. An important deduction which may be made from the curves is that for the maximum inductance with a given length of tape the ratio c/a should be about ^, which means that the opening of the spiral should have a radius nearly as great as the dimension across 298 Circular of the Bureau of Standards the turns of the spiral. This point in design is in agreement with the practical observation that turns in the center of the spiral add a disproportionate amount to the high-frequency resistance of the spiral. Example. Find the length of tape 0.6 cm wide, wound with a pitch of 0.6 cm, to give an inductance of 200 ph, assuming such proportions that cja = i . Work out the design. Since / is not known, the parameter 6 -%T is not known. Assume a value of o.i for the latter. Then for the value c/a= i the curve (Fig. 211) gives H = 0.00123. Thence / T = - - or = 3287 cm. With this value of /, the parameter is 0.6 -t/- - or 0.0339, to which the value H = 0.00128 corresponds (with - == i). Repeating the calculation of / with this CL value of H, we find = 3370 cm as a second approximation. The next approximation gives a parameter of 0.0335 and the values of H and / are sensibly unchanged. Using this parameter in (206) , - =0.0335 orc = - =17. 9 and " \)OO the value of a = 1 7 .9 likewise. The number of turns will be n = ' 0.6 = about 30. Example. We have 17.50 meters of tape i cm wide, which we wind with a pitch of 0.5 cm, to such a shape that c/a=o.8. Here D = 0.5, / = 1 750 cm, b = i . The parameter is */:j- = 0.0847, V 75 to which, for c/a = o.8, H = 0.001248 corresponds. ( L = ,^ o.ooi 248 = 1 29.2 Vo-5 b 0.0847 - = - = 0.0947, by equation (206) -\O.o T = 10.56 cm. 0.0947 10.56 = ^8- := ' 3 - 2 and the number of turns, n = 21 nearly. 0.5 * Radio Instruments and Measurements 299 Example. The problem may arise as to how closely the tape in the preceding case would have to be wound, still keeping - =0.8, a to obtain an inductance of 200 ph. Changing the pitch D will change the parameter of the curves, and hence H. The changes in the latter will not be important, for small changes in D, so that to a first approximation the induc- tance will change inversely as Therefore [D 129.2 */ =- t or D= 0.2086 cm. Yo.5 200 Calculating the parameter with this value we find 0.1312, and thence H= 0.00121 6, so that the second approximation is / \ 3 ^JD = (0.001216), and D= 0.1981, and another approxima- 200 tion is 0.197, the parameter being 0.1346. The dimensions are found from 6 0.1346 i - = ^^ = 0.1505 c = 6.64 c Vo.8 0.1505 c 6.649 a = ^ = 8.30 n = - - = 34 nearly. 0.8 0.197 HIGH-FREQUENCY RESISTANCE 74. RESISTANCE OF SIMPLE CONDUCTORS Two principal causes act to increase the resistance of a cir- cuit carrying a current of high frequency, above the value of its resistance with direct current, viz, the so-called skin effect and the capacity between the conductors. This section deals ex- clusively with the skin effect or change of resistance caused by change of current distribution within the conductor. (See sec. 3.) Unfortunately, formulas for the skin effect are available only for the most simple circuits; and for other very common cases in practice only qualitative indications of the magnitude of the increase in resistance can be given. In what follows R =the resistance at frequency / J R = the resistance with direct current or very low frequency alternating current. 300 Circular of the Bureau of Standards The quantity of greatest practical interest is not R, but the D resistance ratio ==- Given this ratio for the desired frequency and the easily measured direct-current resistance, the high-frequency resistance follows at once. The skin effect in a conductor always depends, in addition to the thickness of the conductor, on the parameter -v in which H = permeability of the material, / = frequency of the current, p = the volume resistivity in microhm-cms, so that as far as skin effect is concerned, a thick wire at low frequencies may show as great a skin effect as a thin one at much higher frequency. The skin effect is greater in good conductors than in wires of high resistivity, and conductors of magnetic material show an exaggerated increase of resistance with frequency. Cylindrical Straight Wires. For this case accurate values of the resistance ratio are given by the formula and tables here given. If d is the diameter of the cross section of the wire in cm, the quantity must be calculated (or, in the case of copper, obtained for the desired frequency from Table 1 9 , p . 3 1 1 and formula ( 209) ) . Know- r> ing the value of x, the value of ^ r may be taken at once from Table K r> 17, page 309, which gives the value of _- directly for a wide range /Co of values of x. Table 1 9 gives values of a c =o.io7i-y/r (208) for a copper wire at 20 C, o.i cm in diameter, and at various frequencies. The value of x for a copper wire of diameter d in cm is x c = ioda c (209) For a material of resistivity p and permeability ju, the parameter x may also be simply obtained from the value which holds for a copper wire of the same diameter, by multiplying the latter value Radio Instruments and Measurements 301 The range of Table 19 may be considerably extended by remem- bering that a is proportional to V/ or \Ar' wnere X is the wave length. Table 18, page 310, will be found useful, when it is desired to determine what is the largest diameter of wire of a given mate- rial, which has a resistance ratio of not more than i per cent greater than unity. These values are, of course, based on certain assumed values of resistivity; temperature changes and differ- ences of chemical composition will slightly alter the values. In the case of iron wires /* is the effective permeability over the cycle. This will, in general, be impossible to estimate closely. The values given show plainly how important is the skin effect in iron wires. For a resistance ratio only one-tenth per cent greater than unity the values in Table 18 should be multiplied by 0.55, and for a 10 per cent increase of the high-frequency resistance the diameters given in the table must be multiplied by 1.78. The formulas above given apply only to wires which are too far away from others to be affected by the latter. For wires near together, as, for example, in the case of parallel wires forming a return circuit, the mutual effect of one wire on the other always TO increases the ratio -p- No formula for calculating this effect is -fVo available, but it is only for wires nearly in contact that it is impor- tant. At distances of 10 to 20 cm the mutual effect is entirely negligible. Tubular Conductors. The resistance ratio of tubular conduc- tors in which the thickness of the walls of the tube is small in com- parison with the mean diameter of the tube, may be calculated by the theoretical formula for an infinite plane of twice the thick- ness of the walls of the tube. The value of the resistance ratio for this case may be obtained directly from Table 20, page 311, in terms of the quantity where T = the thickness of the walls of the tube in cm * x = the parameter defined in formula (207) . For copper tubes the parameter $ c may be obtained very simply from the values of a c in Table 19, page 31 1 , and the relation 302 Circular of the Bureau of Standards For values of /3 greater than 4 no table is necessary, since we have simply, with an accuracy always greater than one-tenth of i per cent, Sufficient experimental evidence is not available to indicate an accurate method of procedure in the case of tubing where the ratio of diameter to wall thickness is not large. Measurements with tubing in which this ratio is as small as two or three indicate TJ that approximate values of -p- for this case may be calculated by KO using for T, in the calculation of the parameter /3, a value equal to two-thirds of the actual thickness of the walls of the tube. Tubing which is very thin in comparison with its radius has, for the same cross section, a smaller high-frequency resistance than any other single conductor. For this reason galvanized- iron pipe is a good form of conductor for some radio work, the current all flowing in the thin layer of zinc. A conductor of smaller resistance than a tube of a certain cross section is obtained by the use of very fine strands separated widely from one another; there are practical difficulties, however, in making the separation great enough. In a return circuit of tubular conductors the distance between the conductors should be kept as great as i o or 20 cm. For tubular conductors nearly in contact the resistance ratio may be double that for a spacing of a few centimeters. I x. "i * + -4- T- T FIG. 212. Cross section of strip conductors forming a return circuit with narrow surfaces in the same plane Strip Conductors. If two strips form together a return circuit and they are so placed that there is only a small thickness of dielectric between the wider face of one and the same face of the other (Fig. 212), the resistance ratio may be calculated by formula (210) , using for r the actual thickness of the strip. Radio Instruments and Measurements 303 As the thickness of the insulating space between the plates is increased, the accuracy of the formula decreases, but the error does not amount to more than a few per cent for values of this thickness as great as several centimeters. W -w- * 7- Thus the resistance ratio -5- is greater in a wide strip than in a ZV narrow one of the same thickness, and in every case the resistance ratio is greater than for the two juxtaposed strips of Fig. 212. R ^ For -p- between i and 1.5, the increase over formula (210) is E> usually not greater than 10 per cent. Strips of square, or nearly square, cross section have values of not very different from those which hold for round conductors of the same area of cross section, the values being greater for the square strip than for the round conductor whose diameter is equal to the side of the square. Simple Circuits of Round or Rectangular Wire. The ratio of the resistance at high frequencies to that with direct current may be accurately obtained from Table 1 7, page 309, for circles or rectangles of round wire and in fact for any circuit of which the length is 35601 18 20 304 Circular of the Bureau of Standards great compared with the thickness of the wire, provided no con- siderable portions of the circuit are placed close together. In the latter case, the resistance ratio is somewhat increased beyond the value calculated by the previous method and by an amount which can not be calculated. The resistance ratio for a circuit of wire of rectangular section may be treated by the same method as for a single strip. If por- tions of the circuit are in close proximity, the precautions men- tioned for two strips near together (p. 303) should be borne in mind. 75. RESISTANCE OF COILS Single-Layer Coil; Wire of Rectangular Cross Section. The only case for which an exact formula is available is that of a single-layer winding of wire of rectangular cross section with an insulation of negligible thickness between the turns, the length of the winding being assumed to be very great compared with the mean radius, and the latter being assumed very great compared with the thickness of the wire. If R = the resistance at high frequency R = the resistance to direct current T = the radial thickness of the wire b = the axial thickness of the wire p = the volume resistivity of the wire in microhm-cm p c = the volume resistivity of copper H = the permeability of the wire D =the pitch of the winding, E> then i^- may be obtained directly from Table 20, page 311, having KO r- /M/ calculated first the quantity /3 = lory 2 a, in which a = 0.1985 -/ - Values of a c for copper are given in Table 19, page 311, and the value of a for any other material is obtained from a c by the relation a = a c A p. For values of /3 greater than are included in Table V P 20 we have simply jr~ = $. KO In practice the ideal conditions presupposed above will not be realized. To reduce the value calculated for the idealized wind- ing corrections need to be applied: (i) For the spacing of the wire, (2) for the round cross stection of the wire, (3) for the curv- ature of the wire, (4) for the finite length of the coil. Radio Instruments and Measurements 305 Correction for Pitch of the Winding. To take into account the fact that the pitch of the winding is not in general equal to the axial breadth of the wire an approximation is obtained if for j8 the argument is substituted. D For values of D greater than about 36 the values of TT thus *Vo obtained are too small. Correction for the Round Cross Section of the Wire. For coils of round wire only empirical expressions are known, and more experimental work is desirable. To obtain an accuracy of perhaps 10 per cent in the resistance ratio the following procedure may be used: Calculate first by (210) and Table 20, page 311, the resistance r>/ ratio p-/' supposing the coil to be wound with wire of square -IVO cross section of the same thickness as the actual diameter, taking into account the correction for the pitch of the winding. Then r> the resistance ratio -^- for a winding of round wire will be found ft-o by the relation -R'-R<,'- / x (2,2) Effect of Thickness of the Wire. Although formula (210) holds only for a coil whose diameter is very great in comparison with the thickness of the wire, the error resulting from non-fulfillment of this condition will, in practical cases, be small compared with the other corrections and may be neglected. Correction for Finite Length of the Coil. For short coils the resistance ratio is greater than for long coils of the same wire, pitch, and radius, due to the appreciable strength of the magnetic field close to the wires on the outside of the coil. No formulas are available for calculating this effect, but experiment seems to show that for short coils of thick wire at radio frequencies the resistance ratio may be expressed by R A B , v in which the first term represents the value as calculated by the formulas of the preceding section for long coils, while the con- 306 Circular of the Bureau of Standards stant of the second term has to be obtained by experiment. At long wave lengths the first term will predominate, but at very short wave lengths the second term may be equal or even larger than the first. For round copper wires we may obtain the constant A by the relation A = 1 5 500 dR . Multiple-Layer Coils. For this case no accurate formulas have been derived. Experiment shows that the resistance ratio is much greater for a multiple-layer coil than for a single-layer coil of the same wire. Furthermore, the capacity of such a coil has, as already pointed out, a large effect on the resistance of the coil. Consequently, it is usually impossible to calculate even an approximate value for the change of resistance with frequency. At very high frequencies losses in the dielectric between the wires may cause an appreciable increase in the effective resistance of the coil. This effect is proportional to / 3 . 76. STRANDED WIRE The use of conductors consisting of a number of fine wires to reduce the skin effect is common. The resistance ratio for a stranded conductor is, however, always considerably larger than the value calculated by Table 19, page 311, and Table 17, page 309, for a single one of the strands. Only when the strands are at impracticably large distances from one another is this condition even approximately realized. Formulas have been proposed for calculating the resistance ratio of stranded conductors, 35 but although they enable quali- tatively correct conclusions to be drawn as to the effect of chang- ing the frequency and some of the other variables, they do not give numerical values which agree at all closely with experiment. The cause for this lies, probably, to a large extent in the impor- tance of small changes in the arrangement of the strands. The following general statements will serve as a rough guide as to what may be expected for the order of magnitude of the resist- ance ratio as an aid in design, but when a precise knowledge of the resistance ratio is required in any given case it should be measured. (See methods given in sections 46 to 50.) Bare Strands in Contact. The resistance ratio of n strands of bare wire placed parallel and making contact with one another is found by experiment to be the same as for a round solid wire '"See references 112 to 123 of the Bibliography. Radio Instruments and Measurements 307 which has the same area of cross section as the sum of the cross- sectional areas of the strands; that is, n times the cross section of a single strand. This will be essentially the case in conductors that are in contact and are poorly insulated, except that at high frequencies the additional loss of energy due to heating of the imperfect contacts by the passage of the current from one strand to another may raise the resistance still higher. Insulated Strands. As the distance between the strands is increased, the resistance ratio falls, rapidly at first, and then more slowly toward the limit which holds for a single isolated strand. A very moderate thickness of insulation between the strands will quite materially reduce the resistance ratio, provided conduction in the dielectric is negligible. Spiraling or twisting the strands has the effect of increasing the resistance ratio slightly, the distance between the strands being unchanged. Transposition of the strands so that each takes up successively all possible positions in the cross section as for example, by thorough braiding reduces the resistance ratio but not as low as the value for a single strand. Twisting together conductors, each of which is made up of a number of strands twisted together, the resulting composite con- ductor being twisted together with other similar composite con- ductors, etc., is a common method for transposing the strands in the cross section. Such conductors do not have a resistance ratio very much different from a simple bundle of well-insulated strands. The most efficient method of transposition is to combine the strands in a hollow tube of basket weave. Such a conductor is naturally more costly than other forms of stranded conductor. Effect of Number of Strands. With respect to the choice of the number of strands, experiment shows that the absolute rise of the resistance in ohms depends on the diameter of a single strand, but is independent of the number of strands. Since, however, the direct-current resistance of the conductor is smaller the greater the number of the strands, the resistance ratio is greater the greater the number of strands. Reducing the diameter of the strands reduces the resistance ratio, the number of strands remain- ing unchanged, but to obtain a given current-carrying capacity, or a small enough total resistance, the total cross section must not be lowered below a certain limit, so that, in general, reducing 308 Circular of the Bureau of Standards the diameter of the strands means an increase in the number of strands. With enameled strands of about 0.07 mm bare diameter twisted together to form a composite conductor the order of magnitude of the resistance ratio may be estimated by the following procedure. Calculate by Table 19, page 311, and Table 17, page 309, the resist- ance ratio for a single strand at the desired frequency (this value of R/R will lie very close to unity) , and carry out the same calcu- lation for the equivalent solid wire, whose diameter will of course be d^fn, where n = the number of strands and d = the diameter of a single strand. Then the resistance ratio for the stranded conductor will, for moderate frequencies, lie about one-quarter to one-third of the way between these two values, being closer to the lower limit. This holds for straight wires up to higher fre- quencies than for solenoids. (See critical frequency mentioned in second paragraph below.) Not all so-called litzendraht is as good as this by any means. For a woven tube the resistance ratio may be as low as one- tenth of the way from the lower to the upper limits mentioned. Coils of Stranded Wire. In the case of solenoids wound w r ith stranded conductor, the resistance ratio is always larger than for the straight conductor, and at high frequencies may be two to three times as great. It is appreciably greater for a very short coil than for a long solenoid. For moderate frequencies the resistance ratio is less than for a similar coil of solid wire of the same cross section as just stated, but for every stranded-conductor coil there is a critical frequency above which the stranded conductor has the larger resistance ratio. This critical frequency lies higher the finer the strands and the smaller their number. For 100 strands of say 0.07 mm diameter this limit lies above the more usual radio frequencies. This supposes that losses in the dielectric are not important, which is the case for single-layer coils with strands well insulated. In multiple-layer coils of stranded wire ; dielectric losses are not negligible at high frequencies. Radio Instruments and Measurements 309 77. TABLES FOR RESISTANCE CALCULATIONS TABLE 17. Ratio of High-Frequency Resistance to the Direct-Current Resistance [See formulas (207), (208), and (109)] z R R; Difference z R Ro Difference z R Ro Difference 1.0000 0.0003 5.2 2.114 0.070 14.0 5.209 0.177 0.5 1.0003 .0004 5,4 2.184 .070 14.5 5.386 .176 .6 1.0007 .0005 5.6 2.254 .070 15.0 5.562 .353 .7 1.0012 .0009 5.8 2.324 .070 .8 1. 0021 .0013 6.0 2.394 .069 16.0 5.915 0.353 .9 1.0034 .0018 6.2 2.463 .070 17.0 6.268 .353 18.0 6.621 .353 1.0 1.005 0.003 6.4 2.533 0.070 19.0 6.974 .354 1.1 1.008 .003 6.6 2.603 .070 20.0 7.328 .353 1.2 1.011 .004 6.8 2.673 .070 1.3 1.015 .005 7.0 2.743 .070 21.0 7.681 0.353 1.4 1.020 .006 7.2 2.813 .071 22.0 8.034 .353 1.5 1.026 .007 7.4 2.884 .070 23.0 8 387 .354 24.0 8.741 .353 1.6 1.033 0.003 7.6 2.954 0.070 25.0 9.094 .353 1.7 1.042 .010 7.8 3.024 .070 1.8 1.052 .012 8.0 3.094 .071 26.0 9.447 0.70 1.9 1.064 .014 8.2 3.165 .070 28.0 10.15 .71 2.0 1.078 .033 8.4 3.235 .071 30.0 10.86 .71 32.0 11.57 .70 2.2 1.111 0.041 8.6 3.306 0.071 34.0 12.27 .71 2.4 2.6 2.8 3.0 1.152 1.201 1.256 1.318 .049 .056 .062 .067 8.8 9.0 9.2 9.4 3.376 3.446 3.517 3.587 .070 .071 .070 .071 36.0 38.0 40.0 42.0 12.98 13.69 14.40 15.10 0.71 .71 .70 .71 3.2 1.385 0.071 9.6 3.658 0.070 44.0 15.81 .71 3.4 1.456 .073 9.8 3.728 .071 46.0 16.52 0.70 3.6 1.529 .074 10.0 3.799 .176 48.0 17.22 .71 3.8 4.0 1.603 1.678 .075 .074 10.5 11.0 3.975 4.151 .176 .176 50.0 60.0 17.93 21.47 3.54 3.53 4.2 1.752 0.074 11.5 4.327 0.177 70.0 25.00 3.54 4.4 1.826 .073 12.0 4.504 .176 80.0 28.54 3.53 4.6 1.899 .072 12.5 4.680 .176 90.0 32.07 3.54 4.8 1.971 .072 13.0 4.856 .177 100.0 35.61 5.0 2.043 .071 13.5 5.033 .176 oo 00 310 Circular of the Bureau of Standards o o i V I w I "8 I v>fot'-i''if y }wjv>^ 1 Hpjpr^-ooof^'tf-ino^ ill o o rl 2 rf^"*"?5\\r3c?-tCO goocQinfO'j^tr^u^ OOOOOOO^-tCO C M- \O o " " " CS oo t~ jj rj ov **; ro o vo oo Sx^J^XXX 001 ^* e> ' O 1 SSRSSSSSNve SSSSSSSiSSS i i i 1 CO o Oro*vjJ-<'rM IM 1 o c5 ' ' 8 .>ooBr>t^o\^>'r VHvHv-tf-Ot^t^C^v-lin OOOOMOOOCOVO S S SSI c5 ' * ' c> ' < o S r- ro O o oa vo O R-RS8*vofr)C^-* rMMfM&ooMpSwjrrs 111 C5 o I \p j to^rsa^-^ooo^^ic coco-'H>or~ooo>O OOO^^p^'^'^'-'t^^o 111 C> -< ci ' ' ::::::::::( Frequency *- 10 Wave length, meters i ^{lli B s J 1 1 11 1 f 1 I a B 1 ES 1 - 2 g-^-saiuSgSgg uKoSSSoooo I I I M Radio Instruments and Measurements TABLE 19. Values of the Argument for Copper Wire 0.1 cm Diameter and Resistivity 1.724 Microhm-cms f cycles per second at Difference X meters f cycles per second a. Difference X meters 100 1071 0.0443 50 000 2.395 0.229 6000 200 1514 .0341 60 000 2 624 .210 5000 300 1855 .0287 70 000 2.834 .195 4286 400 2142 0253 80 000 3 029 .184 3750 MB 2395 .0229 90 000 3.213 .174 3333 600 0. 2624 0.0210 100 000 3.387 0.761 3000 2834 .0195 150 000 4 148 .642 2000 800 .3029 .0184 200 000 4.790 .565 1500 900 .3213 .0174 250 000 5.355 .511 1200 3387 .1403 300 000 5 866 .318 1000 2000 0.4790 0. 1076 333 333 6.184 0.380 900 3000 5866 .0908 375 000 6.564 .452 800 4000 6774 .0799 428 570 7.012 .561 700 MAO 7573 .0723 500 000 7.573 .723 600 6000 82% 0664 600 000 8.296 .664 500 7000 .8960 .0619 700 000 8.960 0.315 429 8000 9579 0581 750 000 9.275 .304 400 9000 1.0160 .055 800 000 9.579 .581 375 10 000 15000 20 000 30 000 1.071 1.312 1.514 1 855 0.241 .202 .341 287 30 000 20 000 15 000 10 000 900 000 1 000 000 1 500 000 3 000 000 10.16 10.71 13.12 18.55 .55 2.41 5.43 333 300 200 100 40 000 2.142 .253 7500 TABLE 20. Values of for Use with Formula (210) ft R R Difference ft R c Difference ft R R. Difference o 1.000 1.0 .086 0.037 2.5 2.477 0.111 0.1 1.000 1.1 .123 .047 2.6 2.588 .109 .2 1.000 1.2 .170 .059 2.7 2.697 .106 .3 1.001 1.3 ' .229 .069 2.8 2.803 .104 .4 1.002 1.4 .298 .080 2.9 2.907 .103 .5 1.006 0.002 1.5 .378 .090 3.0 3.010 .101 0.55 1.008 .004 1.6 1.468 0.098 3.1 3.111 0.101 .60 1.012 .004 1.7 1.566 .166 3.2 3.212 099 .65 1.016 .005 1.8 1.672 .111 3.3 3.311 .099 .70 1.021 .007 1.9 1.783 .115 3.4 3.410 .099 .75 1.028 .008 2.0 1.898 .117 3.5 3.509 .099 0.80 1.036 0.009 2.1 2.015 0.117 3.6 3.608 0.098 .85 1.045 .011 2.2 2.132 .117 3.7 3.706 .098 .90 1.057 .013 2.3 2.248 .115 3.8 3.804 .098 .95 1.070 .016 2.4 2.364 .113 3.9 3.902 .098 1.00 1.086 2.5 2.477 .111 4.0 4.000 3 12 Circular of the Bureau of Standards MISCELLANEOUS FORMULAS AND DATA 78. WAVE LENGTH AND FREQUENCY OF RESONANCE X cm = i .8838 X io u -Y/LC (cgs electromagnetic units) (214) = 6.283 VL cgs electromagnetic C cgs electrostatic (2 1 5) Am = 0.0595 7 -\/L cgs electromagnetic C micromicrofarad (216) = 1-884 VL microhenry C micromicrofarad (217) = 1884 VL microhenry C microfarad ( 2 i8) = 5957 VL millihenry C microfarad (219) = i 884000 VL henry C microfarad (220) L henry C microfarad 5033 millihenry C microfarad 159 200 VL microhenry C microfarad i OOP _ _ V-L henry C microfarad _ 31620 millihenry C microfarad = _ 1 OOP OOP ___ VL microhenry C microfarad T I _ 27r ~J~~^ ( 22 7) , _ 2.998 X IP 8 Am J (228) _ 1. 884X10* ' ~^~ (229) Radio Instruments and Measurements 313 79. MISCELLANEOUS RADIO FORMULAS When units are not specified, international electric units are to be understood. These are the ordinary units, based on the international ohm and ampere, the centimeter and the second. Full information is given on electric units in reference No. 152, Appendix 2. Current in Simple Series Circuit. jl E Phase Angle. =^ = XL o Xo ( 2 3i) wL -- p 7 - in simple series circuit. (232) Sharpness of Resonance. C Current at Parallel Resonance. ER / r 2 -/ t A 3 _J ^ ( 2 ^\ E, .T" 1 t> \*3O/ V- KcoCr K (See p. 3 7.) r> 2 . e ,r t (234) (See p. 39.) Coefficient of Coupling. (235) M .- for direct and inductive coupling (236) = ^ ' for capacitive coupling. ( 2 37) Cm (See p. 49.) Power Input in Condenser P = 0.5 X io-WCE * watts (238) for C in microfarads, E, in volts, and N = number of charges per second. 3*4 Circular of the Bureau of Standards Power Loss in Condenser Condenser Phase Difference (240) for $ in radians, r in ohms, C in farads. rC 1^=0.1079 -^-degrees ( 24I ) for r in ohms, C in micromicrofarads, X in meters. rC t = 389- y seconds (242) for r in ohms, C in micromicrofarads, X in meters. . o.ooi X "*> X I^o XO ' I 54 ohms (243) for ^ in minutes, C in microfarads, X in meters. Energy Associated with Inductance W = \ LI ' (=44) Inductance of Coil Having Capacity: j L *~i-u*CL ( 2 45) for C in farads, L in the denominator in henries. L. =L (i +3.553 -jj-j-J approximately (246) for X in meters, C in micromicrofarads, L in the parentheses in microhenries. This formula is accurate when the last term is small compared with unity. Current Transformer A n/ I+ oA (247) J, n\ uLj (Seep. 154.) Audibility (248) (See p. 1 66.) Radio Instruments and Measurements 315 Natural Oscillations of Horizontal Antenna. = i, 3, 5, (249) m for X in meters, C = capacity in microfarads for uniform voltage, L = inductance in microhenries for uniform current. Approximate Wave Length of Resonance for Loaded Antenna. 37 (250) where L = inductance of loading coil in microhenries and other quantities are as in preceding formula. Radiation Resistance of an Antenna. (h V T- ) ohms (2 si) */ where h height from ground to center of capacity, and h and X are in the same units, and X is considerably greater than the fun- damental wave length. Electron Flow From Hot Filament. 7 8 = AT*6-| (252) where 7 B = electron current in milliamperes per centimeter 2 of fila- ment surface, T = absolute temperature, and A and b depend on metal of filament; for tungsten A = 2.5 x io 10 , b = 52500. Electron Current in ^-Electrode Tube. T \> ( F* ~\~ k 1) )^ ( / 2.Xt'l\ B \ B 1 I/ \ OO/ where EB = plate voltage, z 1 = grid voltage, & t = amplification con- stant. Resistance Measurement by Resistance Variation Method Using Undamped Emf. R = R ^j^rr i ( 2 54) Resistance Measurement by Resistance Variation Method Using Impulse Excitation. r> r> i /^^i-\ K Kfj ^- (255) L i Resistance Measurement by Reactance-Variation Method Using Undamped emf. R=X '\IT^T' (256) V ^r -*i where X 1 = change of reactance between the two observations of current. Various particular cases of this formula are given in section 50. 316 Circular of the Bureau of Standards Natural Frequency of Simple Series Circuit. (257) CO = : (2 5 8) Number of Oscillations 1o Reduce Current to i Per Cent of Initial Value in Wave Train. 4.6 n = -y (259) Logarithmic Decrement. 5 = log e y = (260) A 7 R = 7T =- = sharpness of resonance = TT X phase difference of condenser or coil, the resistance being in one or the other average energy dissipated per cycle _ 2 X average magnetic energy at the current maxima 5=0.00167 -j- (261) JLrf for R in ohms, X in meters, L in microhenries. 5 = 5918 -y- (262) for R in ohms, X in meters, C in microfarads. (263) for R in ohms, C in microfarads, L in microhenries. Current at resonance Produced by Slightly Damped emf Induced in a Circuit. /v F - 72 "* "^ O / s- \ = i6/ 3 L 2 5'5(6' + 5) (264) Decrement Measurement by Reactance Variation Method. (See p. 187 for variations of this formula.) Radio Instruments and Measurements 317 80. PROPERTIES OF METALS TABLE 21 Metal Microhm- centimeters at 20 C Temperature coefficient at 20 C Specific gravity Tensile strength, Ibs./in.v Melting point, "C Advance. See Constantan. Aluminum ..... . 2.828 41.7 120 7 7.6 87 49 1. 7241 1.771 92 33 2.44 10 22 4.6 44 95.783 5.7 42 100 7.8 11 7.8 10 1.59 10.4 11.9 18 70 15.5 47 11.5 5.6 5.8 0.0039 .0036 .004 .002 .0038 .0007 .00001 .00393 .00382 .00016 .0004 .00342 .0050 .0039 .004 .00001 .00089 .604 .0020 .0004 .006 .0033 .0018 .003 .0038 .005 .004 .003 .001 .0031 .00001 .0042 .0045 .0037 2.70 6.6 9.8 8.6 8.6 8.1 8.9 8.89 8.89 8.9 8.4 19.3 7.8 11.4 1.74 8.4 13. 546 9.0 8,9 8.2 8.9 12.2 8.9 21.4 10.5 7.7 7.7 7.7 7.5 16.6 8.2 7.3 19 7.1 30 000 659 630 271 900 321 1250 1190 1083 Bismuth Brass 70 000 Cadmium Calido. See Nicbrome. Climax 150 000 120 000 30 000 60 000 95 000 150 000 20000 Constantan Copper, annealed Eureka. See Constantan. Excello 1500 1100 1063 1530 327 651 910 -sag 2500 1300 1500 1452 1550 750 1755 960 1510 1510 1510 1260 2850 O^rman silver, 1R pe* ^eft ......... German silver, 30per cent. See Constantan. Gold la la. See Constantan. Ideal. See Constantan. Iron, 99.98 per cent pure ... Iron. See Steel. Lead 3 000 33 000 150 000 Magnesium Manganin - Mercury Molybdenum, drawn Monel rnetnl 160 000 150 000 120 000 39 000 25 000 50 000 42 000 53 000 58 000 100 000 230 000 Nichrome Nickel Palladium , , Phosphor bronze Platinum Silver Ftffl, F P P ^fe Steel, B.B w* Steel, Siemens-Martin *.T*;i"' Pte^l, T"angn t x >Kp . Superior. See Climax. Tantalum Therlo Tin 4000 500 000 10 000 232 3000 419 Tungsten, drawn Zinc The resistivities given in Table 21 are values of p in the equa- tion R = p-, where / = length in centimeters and s = cross section in s square centimeters . This formula gives the low-frequency or direct- current resistance of a conductor. For the calculation of resist- ances at high frequencies, see Tables 17 to 20, pages 309-311. 318 Circular of the Bureau of Standards The values given for resistivity and temperature coefficient of copper are the international standard values for commercial copper. Any departure from this resistivity is accompanied by an inverse variation in the temperature coefficient. This is true in a general way for other metal elements. In tjie case of copper the resistivity and temperature coefficient are inversely propor- tional, to a high degree of accuracy. The "temperature coefficient at 2oC" is a 20 in the equation Rt = R 2Q (i +a 20 [^ 20]). In some cases the temperature variation does not follow a straight-line law; in such cases a 20 applies only to a small range of temperature close to 20. Steel is an example, the resistance rise at high temperatures being faster than propor- tional to temperature. Constantan and the other wires (Advance, etc.) having substan- tially the same properties, are alloys of approximately 60 per cent copper and 40 per cent nickel. They are used in rheostats and measuring instruments. German silver is an alloy of copper, nickel, and zinc. The per cent stated indicates the percentage of nickel. Manganin contains about 84 per cent copper, 12 per cent man- ganese, and 4 per cent nickel. It is the usual material in resist- ance coils. Its very small thermal electromotive force against copper is one of its main advantages. The similar alloy, therlo, is used for the same purposes. Monel metal is an alloy containing approximately 71 per cent nickel, 27 per cent copper, and 2 per cent iron. Bureau of Standards Circular No. 74 FIG. 214. Variable condensers used as standards of capacity FIG. 215. Single-layer coils used as standards of inductance Bureau of Standards Circular No. 74 FIG. 216. Multiple-layer standard coil FIG. 217. Standard wave length circuit APPENDIXES APPENDIX 1. RADIO WORK OF THE BUREAU OF STANDARDS The functions of the radio laboratory of this Bureau include the maintenance of standards for radio measurements, the testing of instruments and apparatus, technical assistance in radio matters to various branches of the Government, and researches in the theory and practice of radio communication. The activities of the Bureau in some of these lines have been to a considerable extent covered in the foregoing sections. A more comprehensive account is given here of the facilities, accomplishments, and aims of this laboratory. This account does not include a description of the work of the United States naval radiotelegraphic laboratory or of the Signal Corps laboratory, both of which are located at the Bureau of Standards. They were in existence before the Bureau's own radio laboratory was established, and the publications of the Naval Laboratory are printed in the Bulletin of the Bureau of Standards. A list of these publications is given in the "Bibliography," page 329. 1. DEVELOPMENT AND MAINTENANCE OF STANDARDS Capacity. The qualities desirable in a condenser to be used as a standard at radio frequencies are: Constancy of capacity with varying frequency and temperature and other conditions, small resistance or phase difference, careful shielding, and con- venience of design. The quartz-pillar air condensers described above (p. 120) have these qualities and are satisfactory fundamental standards for radio measurements. They are the result of many years' experience at this Bureau in the measurement and design of condensers. The laboratory has a set of variable condensers of this type, with maximum capacities ranging from o.oooi to 0.0075 microfarad. Having con- tinuously variable capacity, they are very convenient to use in a standard circuit. Fixed-value condensers of the same general type with greater capacity are also used. Good mica condensers, well made and properly shielded, may also be used as standards at radio frequencies. The best mica condensers have lower phase differences than many air condensers of ordinary design, because of the solid dielectric used to insulate the plates in the air condensers. Fixed-value condensers for radio use should pref- erably be independent and not parts of a permanently connected set of condensers, on account of mutual capacities between the parts of such a set. The capacities of air condensers used as radio standards are determined by low- frequency measurements, either by the absolute Maxwell bridge method 38 at a fre- quency of too per second, or by alternating-current comparison 37 with standard con- densers at frequencies from 100 to 3000. The plate and lead resistances and inductances of these condensers are negligibly small, the insulation resistance is extremely high, and the phase difference due to absorption is very small. A few of the condensers have a phase difference which can just be detected at low settings. It is so small as to cause no change of capacity, as shown by agreement of the capacities at low and high settings at different frequencies. These condensers have practically zero temperature coefficient and have remained constant in capacity. 84 See reference No. 174, Appendix i. r See reference No. 176, Appendix a. 3*9 35601 18 - 21 320 Circular of the Bureau of Standards Inductance. The problem of developing standards of inductance for use at radio frequencies is mainly that of minimizing resistance and distributed capacity. These requirements are both met by the single-layer coil for inductances of moderate value. The shape of coil having the minimum length of wire (and hence minimum resistance if the cross section of the wire is specified) for a required inductance is given on pages 286 to 292. The capacity of a coil is roughly proportional to the radius of the coil and independent of the number of turns and length. For a single-layer coil having a close winding the value in micromicrofarads is approximately equal to the numerical value of the diameter in centimeters. For a single-layer coil with spaced winding on an open form, like those described below, the capacity in micro- microfarads is approximately equal to the numerical value of the radius in centi- meters. Hence a coil should be made longer and of smaller diameter than the theo- retical shape indicated for minimum resistance in order to reduce the capacity. The single-layer coils used as standards in this laboratory conform to these prin- ciples. Theset shown in Fig. 215, facing page3i8, ranges in diameter from 13 1x>38cm and in inductance from 60 to 5350 microhenries. The capacities of the coils with their leads range from 9 micromicrofarads for the smallest to 16 for the largest. The capacities are kept small by eliminating as much dielectric as possible from the neigh- borhood of the wire. The coils are wound with silk-covered "litzendraht", with the turns spaced. The open form gives the coils the shape of a i2-sided polygon instead of a circle. Multiple-layer coils are used as standards for inductances larger than any of these single-layer standards. These are satisfactory if the wires are spaced well apart and the amount of dielectric between the turns and layers is kept small. On this account it is not desirable to impregnate such coils with insulating compound. Such a coil is shown in Fig. 216, facing page 319. The inductances of the standard coils are determined by intercomparison in circuits using the standard air condensers referred to above. The basis of these intercompari- sons is a rectangular inductance consisting of a single turn of copper tubing. The inductance of this, obtained by calculation, is 9 microhenries. For discussion of these determinations see page 247. Wave Length. The wave-length standards consist essentially of standard circuits made up of the standard condensers and inductance coils just described. As shown in Fig. 217, facing page 313, the circuit includes a pair of leads; the inductance coil is considered to include these when its value is determined. A wire leading to ground is connected to the shielded side of the condenser. A current indication is obtained in a thermal ammeter in a separate circuit near the standard circuit. This separate circuit is placed by trial at such a distance that it does not affect appreciably the capacity or inductance of the standard circuit. An alternative method of observing the current in the standard circuit is the use of a thermoelement in series, the circuit being stand- ardized with the thermoelement in. The range of wave lengths obtained with the coils and condensers described above is from 100 to 13 ooo meters. The work of the Bureau in connection with wave-length standardization includes also the development of the decremeter described on pages 196-199. This instru- ment is a wave meter which has a more nearly uniform scale of wave lengths than wave meters employing the ordinary condenser with semicircular plates. It is built for wave lengths from 75 meters up. Current. High-frequency current standardization is at present based upon thermal ammeters. For small currents the standard instruments are thermoelements. These are made of such fine wire that the resistance does not change with frequency. When made with a resistance of i ohm or less, a thermoelement may be inserted directly in a radio circuit without reducing the current materially. Bureau of Standards Circular No. 74 FIG. 219. Small-typj decremeter IG. 220. Navy-type decremeter Radio Instruments and Measurements 321 For currents of intermediate value, a hot-wire ammeter with a single wire is taken as a standard. The principal precaution necessary is that the heated wire be fine enough to remain constant in resistance at all frequencies used. For measurements of large currents, instruments with multiple wires or strips are taken as standards, in which careful investigation has shown that the current distri- bution among the wires or strips does not change with frequency. Ammeters of the cylindrical type with a thermocouple on each wire or strip have been developed for this purpose. The questions of errors and design are treated above, section 41. The standard instruments thus far developed are suitable for measuring currents up to 50 amperes at frequencies up to i ooo ooo. Resistance and decrement. The standards used in the determination of resistance and decrement are of two classes. In the first class, in which resistance is measured by the substitution or the deflection method, resistance standards of manganin wire are used. These are short, straight lengths of fine wire, the substitution of which in a circuit does not appreciably change its inductance. (See p. 178-180 for further details.) In the second class of measurement, the determination of resistance depends ulti- mately on the reactance of a circuit and the deflections of an ammeter. From the variation of reactance required to produce a certain change of current in the circuit the resistance is obtained. In the decremeter described on page 196, which was devel- oped at this Bureau, the decrement is obtained directly by manipulation of the instrument without the necessity of any calculation. A dial is graduated in terms of decrements from o to 0.3 readable to o.ooi. Photographs of several types of the instrument are given in Figs. 218 to 220. 2. TESTING OF INSTRUMENTS AND MATERIALS Most of the radio apparatus which the Bureau is called on to test is standardized by direct comparison with the standards described in the preceding section. The fees which have been established for testing radio and other electrical apparatus, and instructions to applicants for tests, are given in this Bureau's Circular No. 6, Fees for Electric, Magnetic, and Photometric Testing. For tests not listed there, a special fee of nominal amount is charged. Unless otherwise specified, apparatus is tested with undamped current using a pliotron as the source. Such current is very steady and gives the maximum accuracy of measurement. In general, the Bureau does not certify an accuracy better than i per cent on any radio apparatus. Wave meters. A wave meter is tested by direct comparison with a standard circuit, both being coupled to the same source of high-frequency current. For the procedure see section 30. If a ground connection is to be used on the wave meter, it is tested with a ground on. It is usually most convenient to use an ordinary commercial wave meter without a ground connection. Grounding makes very little difference in the indications of the instrument, except in wave meters where the condenser has unusu- ally small capacity at the low settings. Wave meters which are to be used as instru- ments of precision are tested at the points specified by the applicant for test. Coils. An inductance coil is tested by a substitution method, other coils of nearly the same value being substituted in a circuit which is tuned to resonance by a variable condenser. By varying the capacity of the condenser, or if necessary inserting addi- tional inductance, the wave length is varied, and a curve of inductance against wave length may be plotted. Such a curve is shown on page 64. Condensers. Condensers are also tested by substitution, either by placing the test condenser and the standard successively in the same position or by using a double- throw switch. The use of variable standard condensers makes this measurement very simple. Ammeters. High-frequency ammeters are standardized by comparison with a Standard ammeter in series in the same circuit. Test is usually made at more than 322 Circular of the Bureau of Standards one wave length. No regular fees have been established for this as yet, each test being subject to a special fee depending on the time consumed in the measurement. Resistance Measurements. The resistances of high-frequency resistance standards, of wires or other conductors, and of coils, condensers, or circuits, are measured in any of the ways mentioned in sections 47 to 50 above. These include substitution or deflection methods in terms of standards, and the reactance-variation methods. In the case of stranded wire submitted for high-frequency resistance measurement, it is desirable to measiire also the low-frequency or direct-current resistance of each sepa- rate strand and the insulation resistance between strands. Insulating Materials. Tests are made of the dielectric loss or phase difference of insulating materials if submitted in large thin sheets. Measurements of dielectric strength are made with low-frequency voltage up to 100 ooo volts. An equipment has also been developed for voltage tests at radio frequencies up to 20 ooo volts. Operating Apparatus. Complete transmitting and receiving sets, accessories, and parts of sets are tested when the circumstances render the test of such importance as to justify the Bureau in undertaking the work. Tests of the performance of complete sets have not yet been standardized, as each set submitted presents a distinct prob- lem. Such a test may include: Output of transmitter, wave forms of current and voltage of power supply circuits, purity and decrement of generated wave, wave lengths of transmitter and receiver, selectivity and sensitivity of receiver. 3. RADIO ENGINEERING FOR THE GOVERNMENT The testing and research work is of direct value to the Army, Navy, and various other branches of the Government, but in addition to this the laboratory performs special services for Government Bureaus, in particular those of the Department of Commerce. Some of the special lines of work thus pursued are described below. Technical information is also furnished upon request. The subjects upon which information has thus been furnished include: The installing of transmitting and receiving equipment, the efficiency of radio apparatus, the adjustment of equipment to comply with the law, the design of measuring instruments, formulas, and data. Assistance is rendered the Government in the preparation of legislation on radio matters. Design of Instruments. Portable testing equipments have been developed for the radio inspectors of the Bureau of Navigation of the Department of Commerce. The decremeter and the voltammeter for this purpose (described in preceding sections) were designed, construction supervised, and calibrated here. Technical problems in connection with instruments, which have arisen in the radio inspections, have been referred to this laboratory for solution. Design of Radio Sets. Complete radio transmitting and receiving sets have been designed and furnished to three of the Bureaus of the Department of Commerce. These are in use on the ships of the Lighthouse Service, Bureau of Navigation, and Coast and Geodetic Survey. The transmitters are built in compact panel form and are supplied with i kw of power in a motor generator delivering 5oo-cycle current. This current flows in a closed-core transformer, adjusted for maximum efficiency, to the secondary of which are connected a quenched gap of special design, mica con- densers, and a flat spiral coil. A simple switch sets the wave length on 600, 750, and 1000 meters. The sets handle relatively little traffic, and have a range of about 260 km. Two views of the transmitter are shown in Figs. 221 and 222, facing page 322. The transformer and the inductance spirals, which were given special attention in the development work, are shown in Fig. 223, facing page 323. The receiver designed for these sets consists of two circuits, the antenna circuit and a closed detecting and measuring circuit inductively coupled to it. The closed circuit and the antenna loading coils and variable condenser are all contained in a Bureau of Standards Circular No. 74 FIG. 221. Transmitting set designed by Bureau FIG. 222. Transmitting set designed by Bureau of Standards (front -view) of Standards (rear view) Bureau of Standards Circular No. 74 FIG. 175. Quenched gap plate showing the circular silver sparking surface FIG. 223. Inductance spirals and transformer used in the transmitting set shown in Figs. 221 and 222 FIG. 224. Receiving set designed by Bureau of Standards Radio Instruments and Measurements 323 compact cabinet. The closed circuit includes a variable condenser of the decremeter type, and serves as a wave meter and decremeter as well as acting as a receiver by virtue of the crystal detector connected across the condenser. The receiver may be tuned to wave lengths from about 500 to 2500 meters. Two views are shown in Fig. 224, facing page 323. Fog Signaling Apparatus. The Bureau of Standards has been active in its efforts to promote safety at sea by means of radio apparatus. An equipment was designed and constructed for use at a lighthouse, which should efficiently supplement the light of a lighthouse during fog and prove of great assistance to navigation. An auto- matic transmitting device is arranged to send out a characteristic signal once every minute on a short wave length, so that it will be readily received by all ships within a few miles of the lighthouse. A direction finder was developed for use on ships receiving the signal, so that they can get their bearings by radio. Field Work. Inspection and other trips are made at the request of other Govern- ment bureaus. Assistance has thus been rendered to the Bureau of Navigation of the Department of Commerce in order to solve technical problems that have arisen in the radio inspections. Such problems have included the equipment of emer- gency radio sets on shipboard, cases of interference, use of instruments and testing equipment, etc. 4. RESEARCH WORK. Military Researches. The testing of instruments and materials is of direct or indi- rect benefit to the military departments. Additional service of military value is being rendered by the laboratory through the results of most of the investigations which are in progress. These investigations are of both a scientific and engineering nature. It is obviously impossible to publish any description of this work. Radio Instruments and Methods of Measurement. A number of problems in radio measurements are being studied in the laboratories of the Bureau of Standards. Some of these have been brought to the point where a publication has been issued or a testing routine established, but all of them remain fruitful fields for investigation. Among the more important problems is that of establishing wave-length standards. The standard circuits which have been developed are described above. The pro- duction and measurement of large currents and high voltages is another branch of the work. In this connection one publication has been issued, Scientific Paper No. 206, "High-Frequency Ammeters," and a special type of volt-ammeter has been designed. These investigations have shown that simplicity of circuit is a great desideratum for many radio measurements. The measurement of resistance and decrement has received considerable attention. A number of methods have been used, and their limitations studied. An apparatus for quick measurements has been developed; it is described in Scientific Paper No. 235, "A Direct-Reading Decremeter for Measuring the Logarithmic Decrement and Wave Length of Electromagnetic Waves." Properties of Conductors and Insulators. Data are obtained on the ratio of high- frequency to low-frequency resistance of stranded wire of various kinds. This work may be extended to strips, tubes, and other special forms of conductors. Insulating materials are studied for dielectric loss, dielectric constant and its variation with frequency, surface flashover voltage, etc. There is great need for systematic study both of the methods of measurement and of the properties of these materials. Inductance Coils. The capacity and the resistance of radio coils and their effect upon the inductance furnish an interesting problem. The effects of varying shape, size, pitch and size and kind of conductor, insulation of conductor, and material and kind of mounting, all require investigation, as well as the modes of connection to radio coils and the effects produced by combinations of coils. Electron Tubes. The characteristics and applications of three-electrode thermionic tubes constitute a most important field of investigation. These tubes have been 324 Circular of the Bureau of Standards found to be excellent sources of current for laboratory measurements. A number of applications to military uses are under development. The characteristic curves of tubes are studied, and different types of tubes compared as amplifiers, generators, and detectors. Special attention is given to the production of maximum current in generating circuits for particular purposes, and to the modulation of the radio- frequency current. Antennas. Some of the great variety of problems presented by the antenna are under study. The properties, functioning, and merits of antennas of various forms for particular purposes are investigated. The means of supplying current to the an- tenna are studied. The investigation includes the consideration of the behavior and transmission of the electromagnetic waves emitted from an antenna. Measurements of antenna resistance, inductance, and capacity are made. One publication has been issued, Scientific Paper No. 269, " Effect of Imperfect Dielectrics in the Field of a Radiotelegraphic Antenna." APPENDIX 2. BIBLIOGRAPHY This bibliography is by no means comprehensive. A few of the more important references are given for each of the subjects treated in the text. In many of the publi- cations listed here references are given to previous publications. Bibliographies of the current literature have been given bimonthly in the "Jahrbuch der drahtlosen Telegraphic" since 1907. Articles on radio measurements as well as other phases of radio communication appear in the bimonthly ' ' Proceedings of the Institute of Radio Engineers." ELEMENTARY ELECTRICITY. 1. Elementsof Electricity and Magnetism, J. J. Thomson;4th ed.. 1909 (Cambridge). 2. Modern Views of Electricity, O. J. Lodge; 1889 (MacMillan). 3. The Elements of Physics, Vol. II, Electricity and Magnetism, Nichols and Franklin; 1905 (MacMillan). 4. Electricity and Magnetism, R. T. Glazebrook; 1910 (Cambridge). 5. Elements of Electricity for Technical Students, W. H. Timbie; 1911 (John Wiley & Sons). 6. Magnetism and Electricity for Students, H. E. Hadley; 1910 (MacMillan). 7. The Elements of Electricity and Magnetism, Franklin and MacNut; 1914 (Mac- Millan). 8. Elementary Lessons in Electricity and Magnetism, S. P. Thompson; 7th ed., 1915 (MacMillan). 9. A Treatise on Electricity, F. B. Pidduck; 1916 (Cambridge). ga. Electricity and Magnetism, S. G. Starling; 1912 (Longmans, Green & Co.). ATLERNATING CURRENTS. > 11. Alternating Currents, Bedell and Crehore; 4th ed., 1901 (McGraw-Hill). 12. Alternating Currents and Alternating Current Machinery, D. C. and J. P. Jack- son; 1896 (MacMillan). 13. The Theory of Alternating Currents (2 vols.), A. Russell: 2d ed., 1914 (Cam- bridge). 14. Kapazitat tmd Induktivitat, E. Orlich; 1909. 15. Calculation of Alternating Current Problems, L. Cohen; 1913 (McGraw-Hill). 16. The Foundations of Alternating Current Theory, C. V. Drysdale; 1910 (E. Ar- nold). 17. Transient Electric Phenomena and Oscillations, C. P. Steinmetz; 1909 (McGraw- Hill). Radio Instruments and Measurements 325 COUPLED CIRCUITS. 21. Currents in Coupled Circuits; A. Oberbeck; Annalen der Physik, 291, p. 623; 1895. 22. Use of Coupled Circuits; F. Braun; Physikalische Zs., 3, p. 148; 1901. 23. Coupling phenomena; M. Wien; Annalen der Physik, 61, p. 151, 1897; 25, p. i, 1908. 24. Maximum Current in the Secondary of a Transformer; J. S. Stone; Physical Review, 32, p. 399; 1911. 25. Cisoidal Oscillations; G. A. Campbell; Trans. A. I. E. E., 30, p. 873; 1911. 26. The Impedances, Angular Velocities, and Frequencies of Oscillating-Current Circuits; A. E. Kennelly; Proc. I. R. E. 4, p. 47; 1916. 27. Alternating and Transient Currents in Coupled Electrical Circuits; F. E. Pernot; University of California, publications in Engineering, 1, p. 161; 1916. 28. Oscillograph Demonstrations of Coupled Circuits; G. W. O. Howe; Proc. Physical Society London, 23, p. 237; 1911. J.A.Fleming; Proc. Physical Society Lon- don, 25, p. 217; 1913. 29. Mechanical Models; T. R. Lyle; Phil. Mag., 25, p. 567; 1913. W. Deutsch; Physikalische Zs., 16, p. 138; 1915. ANTENNA CALCULATIONS. 31. Theory of Horizontal Antennas; J. S. Stone; Trans. Int. Elec. Congress, St. Louis, 3, p. 555; 1904. 32. Theory of Loaded Antenna; A. Guyau; La Lumiere Electrique, 15, p. 13; 1911. 33. Capacity of Radiotelegraphic Antennas; G. W. O. Howe; Electrician, 73, pp. 829, 859, 906, 1914; 75, p. 870; 1915. 34. The Electrical Constants of Antennas; L. Cohen; Elec. World, 65, p. 286; 1915. DAMPING. 41. Theory of Free Oscillations; Alternating Current Phenomena, C. P. Steinmetz; Appendix II, p. 709; 4th ed., 1908. 42. Decrements in Coupled Circuits; V. Bjerkhes; Annalen der Physik, 44, pp. 74, 92, 1891; 291, p. 121, 1895. M. Wien, Annalen der Physik, 25, p. 625, 1908; 29, p. 679, 1909. 43. Linear Decrement: J. S. Stone; Electrician, 73, p. 926; 1914. Proc. I. R. E-, 2, p. 307, 1914; 4, p. 463, 1916. ELECTROMAGNETIC WAVES. 51. A Treatise on Electricity and Magnetism; J. C. Maxwell; 1873. 52. Recent Researches in Electricity and Magnetism; J. J. Thomson; 1893. 53. Electromagnetic Theory (3 vols.); O. Heaviside; 1893. 54. Signaling Through Space Without Wires; O. J. Lodge; 1894. 55. Derivation of Equations of a Plane Electromagnetic Wave; E. B. Rosa; Phys. Rev., 8, p. 282; 1899. 56. Electric Waves; H. Hertz (translated into English by D. E. Jones); 1900. 57. Maxwell's Theory and Wireless Telegraphy; H. Poincare (translated into Eng- lish by F. K. Vreeland) ; 1904. 58. Researches in Radiotelegraphy; J. A. Fleming; Smithsonian Report for 1909, P- 157- RADIO MEASUREMENTS AND MISCELLANEOUS. 61. The Principles of Electric Wave Telegraphy and Telephony; J. A. Fleming; 3d ed., 1916. 62. Les Oscillations Electriques; C. Tissot; 1910. 63. Radiotelegraphisches Praktikum; H. Rein; 1912. 326 Circular of the Bureau of Standards 64. Wireless Telegraphy; J. Zenneck (translated into English by A. E. Seelig); 65. Wireless Telegraphy and Telephony, A Handbook; W. H. Eccles; 1916. 66. Radio Communication; J. Mills; 1917. 67. Standardization Rules, Institute of Radio Engineers; 1915. WAVE LENGTH. 71. Die Frequenzmesser und Dampfungsmesser der drahtlosen Telegraphic; E. Nesper; 1907. 72. Standard Wave Length Circuits; A. Campbell; Phil. Mag., 18, p. 794; 1909. Electrician, 64, p. 612; 1910. 73. Calibration of Wavemeters; G. W. O. Howe; Electrician, 69, p. 490; 1912. 74. Wavemeter Standardization; Diesselhorst; Elektrotechnische Zs., 29, p. 703; 1908. 75. Pointer-Type Wavemeter; Feme and Carpentier; Jahrb. d. drahtl. Tel., 5, p. 106; 1911. 76. Practical Uses of the Wavemeter in Wireless Telegraphy; J. O. Mauborgne; 1914. 77. Oval Diagram for Wave Length Calculations; W. H. Eccles; Electrician, 76, p. 388; 1915. CAPACITY. 81. Square- Plate Condenser for Uniform Scale of Wave Lengths; C. Tissot; Journal de Physique, 2, p. 719; 1912. 82. Rotary Condenser for Uniform Scale of Wave Length; W. Duddell; Jour. I. E. E., 52, p. 275; 1914. 83. A.-c. Resistance of Condensers; Fleming and Dyke; Electrician, 68, pp. 1017, 1060, 1912; 69, p. 10, 1912. G. E. Bairsto; Electrician, 76, p. 53, 1915. 84. Calculation of Capacity Using Method of Images; "Alternating Currents";. A. Russell; Vol. I, chaps. 5 and 6; 1914. INDUCTANCE. 91. The Effects of Distributed Capacity of Coils Used in Radiotelegraphic Circuits; F. A. Kolster; Proc. I. R. E., 1, p. 19; 1913. 92. Distributed Capacity of Single-Layer Solenoids; J. C. Hubbard; Phys, Review, 9, p. 529; 1917. 93. Development of Inductance Formulas; "Alternating Currents"; A. Russell; Vol. I, chaps. 2 and 3; 1914. "Absolute Measurements in Electricity and Magnetism"; A. Gray; Vol. II, part i, chap. 6. CURRENT MEASUREMENT. 101. Thermoelements for High-Frequency Measurements; Dowse; Electrician, 65, p. 765; 1910. 102. Hot-Strip Ammeters for Large High-Frequency Currents; R. Hartmann-Kempf; Elektrotechnische Zs., 32, p. 1134; 1911. G. Eichhorn; Jahrbuch d. drahtl. Tel., 5, p. 517; 1912. 103. High-Frequency Current Transformer; Campbell and Dye; Proc. Royal Soc., 90, p. 621; 1914. 104. Use of Iron in High-Frequency Current Transformer; McLachlan; Electrician, 78, p. 382; 1916. 105. Use of Galvanometer in Audion Plate Circuit; L. E. Whittemore; Phys. Review, 9, p. 434; 1917. 106. Measurement of Signal Intensity with Crystal Detector; J. L. Hogan; (Marconi) Year-Book of Wireless Telegraphy, p. 662; 1916. Radio Instruments and Measurements 327 107. Measurements With Crystal and Telephone; J. Zenneck; Proc. I. R. E., 4, p. 363; 1916. 108. Current Measurement With the Audion; L. W. Austin; Jour. Wash. Acad. Sci- ences, 6, p. 81; 1916. Proc. I. R. E., 4, p. 251; 1916. Electrician, 78, p. 465; 1917. Proc. I. R. E., 6, p. 239; 1917. HIGH-FREQUENCY RESISTANCE. in. Skin Effect in Round Wires; Lord Rayleigh; Phil. Mag., pp. 382, 469; 1886; Sci. Papers, Vol. II, pp. 486, 495. Skin Effect in Round Wires; Lord Kelvin; Math, and Phys. Papers, Vol. Ill, p. 491; 1889. 112. Skin Effect in Stranded Conductors to Oscillatory Currents; F. Dolezalek; Ann. der Phys., (4), 12, p. 1142; 1903. 113. Passage of High-Frequency Current Through Coils; M. Wien; Ann. der Phys., (4), 14, p. i; 1904. 114. Long Solenoids at High Frequencies, Mathematical Theory; A. Sommerfeld; Ann. der Phys., (4), 15, p. 673, 1904; (4), 24, p. 609, 1907. 115. Calorimetric Measurements of High-Frequency Resistance of Solenoids; T. Black; Ann. der Phys., 19, p. 157; 1906. 116. Measurements on Stranded Conductors; R. Lindemann; Verh. deutsch. Phys. Gesel., 11, p. 682; 1909. 117. Theory for Stranded-Conductor Solenoids; M6ller; Ann. der Phys., 36, p. 738, 1911; and Jahr. draht. Tel., 9, p. 32, 1914. 118. Measurements on Single and Multiple Layer Coils; Esau; Ann. der Phys., 84, p. 57; 1911. 119. Skin Effect in Flat Coils and Short Cylindrical Coils; Lindemann and Hiiter; Verh. deutsch. Phys. Ges., 15, p. 219; 1913. 120. The Alternating-Current Resistance of Long Coils of Stranded Wire, Theory; Rogowski; Arch. f. Elect., 3, p. 264; 1915. 121. Bibliography, and Measurements on Wires and Strips; Kennelly, Laws, and Pierce; Proc. A. I. E. E., 34, p. 1749; 1915. 122. Bibliography, and Measurements on Solid and Stranded Conductors; Kennelly and Affel; Proc. I. R. E-, 4, p. 523; 1916. 123. High-Frequency Resistance of Multiply-Stranded Insulated Wire; G. W. O. Howe; Proc. Royal Society London, 93, p. 468; 1917. 124. The Accuracy of High-Frequency Resistance Measurements; S. Loewe, Jahr- buch d. Drahtlosen Telegraphic, 7, p. 365; 1913. ELECTRON TUBES. 131. Theory of Thermionic Emission; O. W. Richardson; Phil. Trans., 202, p. 516; 1903. 132. Audion Detector and Amplifier; L. De Forest; Electrician, 73, p. 842; 1914. Elec. World, 65, p. 465; 1914. 133. Theory of Electron Tubes; I. Langmuir; Phys. Review, 2, p. 450; 1913. Proc. I. R. E., 3, p. 261; 1915. 134. Operating Features of the Audion, Amplification, etc.; E. H. Armstrong; Elec. World, 64, p. 1149; 1914. Proc. I. R. E., 8, p. 215, 1915; 5, p. 145, 1917. 135. Characteristic Curves, and Use as Source of High Frequency Current; J. Beth- enod; La Lumiere Electrique, 35, pp. 25, 225; 1916. 136. Generalized Equations for Audions; M. Latour; La Lumiere Electrique, Dec. 30, 1916. Electrician, 78, p. 280; 1916. 137. Characteristics of Audion Tubes Used in Radiotelegraphy ; G. Vallauri; L'Elet- trotecnica, 4, Nos. 3, 4, 18, and 19; 1917. 138. Use of Pliotron to Produce Extreme Frequencies, Currents, and Voltages; W. C. White; General Electric Review, 19, p. 771, 1916; 20, p. 635, 1917. 328 Circular of the Bureau of Standards MISCELLANEOUS SOURCES OF HIGH-FREQUENCY CURRENT. 141. Disturbing Short Waves in Buzzer Circuits; S. Loewe; Jahrb. d. drahtl. Tel., 6, p. 325; 1912. 142. Production of Undamped Oscillations; M. Wien; Jahrb. d. drahtl. Tel., 1, p. 474; 1908. Physikaltsche Zs., 11, p. 76; 1910. 143. Impulse Excitation Transmitter; E. W. Stone; Proc. I. R. E., 4, p. 233, 1916; & P- 133. I 9 I 7- 144. Frequency Multipliers; A. N. Goldsmith; Proc. I. R. E., 3, p. 55;i9i5- W. H. Eccles; Electrician, 72, p. 944; 1914. 145. High-Frequency Alternator of Induction Type; General Electric Review, 16, p. 16; 1913. 146. High-Frequency Alternator Employing Rotating Magnetic Fields; R. Gold- schmidt; Electrician, 66, p. 744; 1911. T. R. Lyle; Electrician, 71, p. 1004; 147. Duddell Arc; W. Duddell; Jour. Rontgen Soc., 4, p. i; 1907. 148. Arc generator for laboratory purposes; F. Kock, Phys. Zeitschr., 12, p. 124; 1911. 149. Impact excitation of undamped waves; E. L. Chaffee; Jahrb. d. drahtl. Tel. 7, p. 483; 1913. Proc. Amer. Ac. Arts & Sci., 47, No. 9; p. 267; 1911. UNITS AND INSTRUMENTS 151. Units of Weight and Measure; Circular No. 47; 1914. 152. Electric Units and Standards; Circular No. 60; 1916. International System of Electric and Magnetic Units; J. H. Bellinger; Bull., 13, p. 599; 1916 (S. P. 292). 153. Electrical Measuring Instruments; Circular No. 20; 2d ed., 1915. 154. Fees for Electric, Magnetic, and Photometric Testing; Circular No. 6; 7th ed., 1916. ELECTRICAL PROPERTIES OF MATERIALS 161. Copper Wire Tables; Circular No. 31; 3d ed., 1914. 162. Electric Wire and Cable Terminology; Circular No. 37; 2d ed., 1915. 163. Insulating Properties of Solid Dielectrics; H. L. Curtis; Bull., 11, p. 359; 1914 (S. P. 234). CAPACITY AND INDUCTANCE 171. The Testing and Properties of Electric Condensers; Circular No. 36; 1912. 172. Formulas and Tables for the Calculation of Mutual and Self Inductance; Rosa and Grover; Bull., 8, p. i; 1911 (S. P. 169). 173. Various papers on inductance calculations; see Circular No. 24, "Publications of the Bureau of Standards." 174. The Absolute Measurement of Capacity; Rosa and Grover; Bull., 1, p. 153; 1904 (S. P. 10). 175. Measurement of Inductance by Anderson's Method, Using Alternating Currents and a Vibration Galvanometer; Rosa and Grover; Bull., 1, p. 291; 1905 (S. P. 14). 176. The Simultaneous Measurement of the Capacity and Power Factor of Con- densers; F. W. Grover, Bull., 3, p. 371; 1907 (S. P. 64). 177. Mica Condenser as Standards of Capacity; H. L. Curtis, Bull., 6, p. 431; 1910 (S. P. 137). 178. The Capacity and Phase Difference of Paraffined Paper Condensers as Func- tions of Temperature and Frequency; F. W. Grover; Bull., 7, p. 495; 1911 (S. P. 166). Radio Instruments and Measurements 329 179. The Measurement of the Inductances of Resistance Coils; Grover and Curtis; Bull., 8, p. 455! 19" (S. P. 175)- 180. Resistance Coils for Alternating Current Work; Curtis and Grover; Bull., 8, p. 495; 1911 (S. P. 177). 181. A Variable Self and Mutual Inductor; Brooks and Weaver; Bull., 13, p. 569; 1916 (S. P. 290). RADIO SUBJECTS 191. The Influence of Frequency Upon the Self-Inductance of Coils; J. G. Coffin; Bull., 2, p. 275; 1906 (S. P. 37). 192. The Influence of Frequency on the Resistance and Inductance of Solenoidal Coils; L. Cohen; Bull., 4, p. 161; 1907 (S. P. 76). 193. The Theory of Coupled Circuits; L. Cohen; Bull., 5, p. 511; 1909 (S. P. 112). 194. Coupled Circuits in which the Secondary has Distributed Inductance and Capacity; L. Cohen; Bull., 6, p. 247; 1909 (S. P. 126). 195. High-Frequency Ammeters; J. H. Bellinger; Bull., 10, p. 91; 1913 (S. P. 206). 196. Direct-Reading Instrument for Measuring Logarithmic Decrement and Wave Length of Electromagnetic Waves; F. A. Kolster; Bull., 11, p. 421; 1914 (S. P. 235). 197. Effect of Imperfect Dielectrics in Field of Radiotelegraphic Antennas; J. M. Miller; Bull., 13, p. 129; 1916 (S. P. 269). PUBLICATIONS OF THE UNITED STATES NAVAL RADIOTELEGRAPHIC LABARATORY IN THE BULLETIN OF THE BUREAU OF STANDARDS. 201. Detector for Small Alternating Currents and Electrical Waves; L. W. Austin; Bull., 1, p. 435: 1905 (S. P. 22). 202. The Production of High- Frequency Oscillations from the Electric Arc; L. W. Austin; Bull., 3, p. 325; 1907 (S. P. 60). 203. Some Contact Rectifiers of Electric Currents; L. W. Austin; Bull., 5, p. 133; 1908 (S. P. 94). 204. A Method of Producing Feebly Damped High-Frequency Electrical Oscillations for Laboratory Measurements; L. W. Austin; Bull., 5, p. 149; 1908 (S. P. 95). 205. The Comparative Sensitiveness of Some Common Detectors of Electrical Oscilla- tions; L. W. Austin; Bull., 6, p. 527; 1910 (S. P. 140). 206. The Measurement of Electric Oscillations in the Receiving Antenna; L. W. Austin; Bull., 7, p. 295; 1911 (S. P. 157). 207. Some Experiments with Coupled High-Frequency Circuits; L. W. Austin; Bull., 7, p. 301; 1911 (S. P. 158). 208. On the Advantages of a High Spark Frequency in Radiotelegraphy; L. W. Austin; Bull., 5, p. 153; 1908 (S. P. 96). 209. Some Quantitative Experiments in Long Distance Radiotelegraphy; L. W. Austin; Bull. 7, p. 315; 1911 (S. P. 159). 210. Antenna Resistance ; L. W. Austin; Bull., 9, p. 65; 1912 (S. P. 189). 211. The Energy Losses in Some Condensers Used in High-Frequency Circuits; L. W. Austin; Bull., 9, p. 73 (S. P. 190). 212. Quantitative Experiments in Radiotelegraphic Transmisssion, L. W. Austin; Bull., 11, p. 69; 1914 (S. P. 226). 213. Note on Resistance of Radiotelegraphic Antennas; L. W. Austin; Bull. 12, p. 465; 1915 (S. P. 257). 330 Circular of the Bureau of Standards APPENDIX 3. SYMBOLS USED IN THIS CIRCULAR l?=magnetic induction. c= velocity of light=2.9982Xio 10 cm per second. C=electrostatic capacity. d=diameter. e=instantaneous electromotive force. JE=effective electromotive force. E =maximum electromotive force. 3==electric field intensity. /=frequency. F=force. ,j/^= magnetomotive force. 7/=magnetic field intensity. t = instantaneous current. 7=effective current. 7 =maximum current. p =coupling coefficient. /C=dielectric constant. 2=length. L=self-inductance. w=mass. M= mutual inductance. p= instantaneous power. P=average power. ^=quantity of electricity. Special symbols are denned where used in Part III and elsewhere. r =distance from a point. /?=resistance. R=reluctance. j=length along a path. S=area. /=time. T=period of a complete oscillation. 7>=velocity. F=potential difference of a condenser. w=instantaneous energy. W= average energy. AT=reactance. Z=impedance. 5=logarithmic decrement. t=base of napierian logarithms == 2.71828. 0=phase angle. X=wave length. ^t=permeability. volume resistivity. magnetic flux. phase difference. 27rXfreqiiency. microfarad. micromicrofarad. microhenry. or to the NORTHERN REGIONAL UBRARV FACILITY Bidg. 400, Richmond Field Static be recharged by bringin recharges may be made 4 day! prior to due date. DUE AS STAMPED BELOW 12,00001/95) RB 17-60m-12,'57 (703slO)4188 General Library University of California Berkeley r /J -C UN1VERSITY OF CALIFORNIA LIBRARY