REESE LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class 'he D. Van Nostrand Company intend this booh to be sold to the Public at the advertised price, and supply it to the Trade on terms which will not allow of reduction THE THEORY OF ELECTRIC CABLES AND NETWORKS THE THEORY OF ELECTRIC CABLES AND NETWORKS By ALEXANDER RUSSELL, M.A., D.Sc. Member of the Council of the Physical Society, Member of the Institution of Electrical Engineers. OF TH UNIVERS %/FOR U Y :' NEW YORK D. VAN NOSTRAND COMPANY 23 MURRAY AND 27 WARREN STREETS 1909 REESE BUTLER & TANNER, THE SELWOOD PRINTING WORKS, FROME, AND LONDON. PREFACE THERE is nothing more conducive to the satisfactory work- ing of an electric supply station than having a thoroughly trustworthy and economical network of cables connecting the dynamos with the lamps and motors of the consumer. It is necessary therefore that the engineer have a thorough knowledge of the phenomena connected with the flow of current along conductors and across dielectrics. He must also have a working knowledge of the dielectric strengths of insulating materials and the electric stresses to which they are subjected under working conditions. In addition, the thermal conductivity of the dielectric has to be con- sidered and its effect on the temperature of the conductor. The author gives some information on these points in this book. His experience in practical testing, and with the difficulties which sometimes arise in interpreting " specifica- tions " and " rules and regulations " has convinced him that the solutions of these problems are of practical use and ought to be more widely known. In fact many of the problems discussed were originally suggested by these diffi- culties. Questions in connexion with the electrostatic capacity and the inductance of cables have not been considered, as the author has discussed these points fully in his Treatise on the Theory of Alternating Currents. He has also omitted many elementary theoretical considerations as the V 196523 vj PREFACE reader is supposed to know the elements of the theory of electricity and electrical engineering. In Chapter I, the fundamental electrical principles are stated and a description is given of the various gauges in use for specifying wires. Conductivity is discussed in Chapter II, and special attention is devoted to the effect of the " lay " on the weight and conductivity of stranded cables. In Chapter III, the standard methods of measuring insulativity are described. The design of distributing networks is explained in Chapter IV, particular stress being laid on " feeding centres " and on the importance of calculating their positions. The theorems given in this Chapter can easily be expanded so as to enable satisfactory solutions to be obtained for the very complex problems which sometimes arise in practice. In Chapters V, VI, and VII methods of measuring the insulation resistance of house wiring and distributing net- works are given. The author only gives those methods which he has found useful in practice. The problem of the calculation of a suitable resistance to put in the earth con- nexion with the middle wire was suggested to him by Mr. A. P. Trotter. The dielectric strength of materials is discussed in Chapter VIII. Unfortunately very few accurate data are yet obtainable, but the author hopes that by applying the methods he suggests, engineers will be able to obtain satis- factory physical " constants " for dielectric strengths. An examination of many published results will show that the experimenters have neglected elementary theoretical con- siderations which must be taken into account if the results obtained are to be of any value. In Chapter IX, the grading, and in Chapter X, the heating, of cables is considered. It is only of recent years PREFACE vii that the former of these subjects has been recognized to be of practical importance. In Chapter XI, the very interest- ing subject of electrical safety valves is considered, but only a few types are discussed, as it is probable that the standard safety device has not yet been evolved. The author has added a Chapter on lightning conductors, in which he has made extensive use of the classical paper on the subject read to the Institution of Electrical Engin- eers by Sir Oliver Lodge in 1889. He has to thank several friends for the kind help they have given him by making suggestions or revising proofs. In particular he has to thank Dr. Chree, F.R.S., for much information about atmospheric electricity and Mr. G. F. C. Searle, F.R.S., for his helpful criticisms of Chapters I and II. He has also to thank Mr. J. N. Alty, A.I.E.E., for his able assistance in drawing the diagrams and reading proofs and his old pupil, the Hon. E. Fulke French, for checking most of the mathematical formulae given. A. R. 10, RICHMOND BRIDGE MANSIONS, TWICKENHAM. August, 1908. CONTENTS PAGE PREFACE .......... v CHAPTER I FUNDAMENTAL PRINCIPLES CHAPTER II CONDUCTIVITY . . . . . . . . .19 CHAPTER III INSULATIVITY . . . . . . . . .49 CHAPTER IV DISTRIBUTING NETWORKS ....... 67 CHAPTER V INSULATION RESISTANCE OF HOUSE WIRING ... 95 CHAPTER VI INSULATION RESISTANCE OF NETWORKS , , . . . Ill CHAPTER VII FAULTS IN NETWORKS . . , , , f 139 x CONTENTS CHAPTER VIII PAGE DIELECTRIC STRENGTH .... .163 CHAPTER IX THE GRADING OF CABLES . . . . . .187 CHAPTER X THE HEATING or CABLES . . . . . .211 CHAPTER XI ELECTRICAL SAFETY VALVES . . . 225 CHAPTER XII LIGHTNING CONDUCTORS. . .243 INDEX . . . 205 FUNDAMENTAL PRINCIPLES CHAPTER I Fundamental Principles Isotropic bodies Ohm's law Joule's law Example Resistances in series KirchhofP s first law The potential of the common junction Minimum heating Conductors in parallel Kirch- hoff' s second law Minimum heating of a loop of a network Volume resistivity Section variable Volume resistivities of metals Conductance and conductivity Circular mil Gauges Table of gauges References. IN this chapter we shall first give a resume of the elemen- tary electric principles which guide the electrician both in the design of a direct current network for the distribution of electric energy and in the measurements of the electric properties of conducting and insulating materials. We shall also give an account of the various wire gauges used in practice. Isotropic * n di scussm g the electric properties of con- bodies ductors it is customary to assume that the con- ductors are homogeneous in substance, and that the re- sistance they offer to the flow of current through them is the same in all directions. It has to be remembered, however, that violent mechanical forces like those used in hammering, rolling and wire drawing, produce permanent deformation of the substance of the metal, and alter by varying amounts its electrical properties in different direc- tions. In this chapter we shall assume that the conductors and insulators are isotropic, that is, that their substances 4 ELECTRIC CABLES AND NETWORKS have the same physical properties in all directions. Care- fully annealed copper may be considered to be practically isotropic as no tests we can apply can detect any difference in its physical properties in different directions. Carefully annealed glass also is practically isotropic. As all physi- cists, however, now accept the theory of the molecular structure of bodies, we must admit that if the portion examined were so small that it contained only a few molecules of the substance, the tests for isotropy would not be satisfied. All substances are in fact irregular when the dimensions of the portion examined are compar- able with the dimensions of a molecule of the substance. If R be the resistance of an electric circuit, Ohm's law E the electromotive force round it, and C the current flowing in it, Ohm's law states that 0=E/B (1), where the current is measured in amperes, the electro- motive force in volts and the resistance in ohms. In general, if r be the resistance of a part of a circuit containing sources producing a resultant electromotive force E, and if Fi and V 2 be the potentials of A and B the ends of this portion of the circuit, we have Cr=Vi V 2 + E (2). In this equation the current C is positive when it flows from A to B, and E is positive when it tends to produce a current in the same direction. Equation (2) shows that it is possible to have the poten- tials Fi and F 2 at two points on one member of a net- work of conductors the same, and yet have a current E/r flowing from one point to the other. In this case the local electromotive force E is entirely expended in maintaining the current C flowing through the resistance r. It is obvious that we may short-circuit the two FUNDAMENTAL PRINCIPLES 5 points without affecting in the least the working of any part of the network. We can also have a current flowing from a point of lower potential through a source of elec- tromotive force to a point at a higher potential. Joule's -^y *ke definition of electromotive force given law in treatises on electricity it follows that if a current of C amperes flow for t seconds through a wire having a potential difference of V volts between its terminals, the work done will be VCt joules. If all this work be expended in heating the wire, that is, if no mechanical work, such as causing an armature to rotate, and no chemical work, such as charging accumulators, is done, we have JH = VCt (3), where H is the number of water gramme Centigrade units of heat (calories) developed, and J is the number of joules in a calorie. The law expressed by this equation is called Joule's law, as he was the first to employ it to determine the mechanical equivalent of heat. The value of J is very approximately 4-18 ergs per water gramme degree Centi- grade, and hence, by Ohm's law, we can write JH=C*rt=(V*/r)t (4), or H = 0-239 <7 2 rZ= 0-239(7 2 /r)t . . .. (5), approximately. Since the work done in t seconds is VCt joules, when V and C are maintained constant, it follows that the rate at which work is being done is VC joules per second, that is VC watts. As an illustration of the application of (5), we Example , . shall find the rise of temperature per second in a coil formed of a copper wire 0*1 mm. in radius and 1,000 metres long when placed between the hundred volt mains, supposing that no heat is lost by radiation and neglecting the effect of the rise of resistance due to rise of temperature. 6 ELECTRIC CABLES AND NETWORKS At 16 C.,the resistance of this wire would be about 508 ohms. By (5), the heat which is generated per second is 0-239 (100) 2 /508, that is, 4-7 calories nearly. Assuming the specific gravity of copper to be 8-9, the mass of the copper wire will be 8-9 X7r(0-01) 2 x 100,000, that is, 280 grammes nearly. Hence taking the specific heat of copper to be 0-095 the rise of temperature per second will, on the given assump- tions, be 4-7/(280 x 0-095), that is, 0-18 C. nearly. We conclude, therefore, that, when the coil is connected with the mains, its temperature rises initially by about 0-18 C. per second. As it warms, the heat lost by radia- tion gradually increases and so the rate at which the tem- perature rises gradually diminishes until the temperature attains a steady value, when the rate at which heat is being lost by radiation equals the rate at which heat is being generated in the wire. If we have n coils of resistance r l5 r 2 , . / r n respec- Resist- ances in tively, connected in series, and if C be the series current flowing through them, we have, by (2), Cr i = V i ~V 2 ,Cr 2 = V 2 -V^ .. Cr n = V n -V n+1 .. (6), where V v and V v+l are the potentials at the ends of the resistance r p . It follows, by adding equations (6), that We see, therefore, that r^+r 2 + .. +r n is the resultant resistance R of the n coils, so that B = ri +r a + .. +r n ...... (7). Hence the resistance of n coils, in series equals the sum of the resistances. When we have n conductors connected with Kirch- hoff's first a point and the currents in them have attained law their steady values, the algebraical sum of all the currents in these conductors must be zero. For if not, FUNDAMENTAL PRINCIPLES 7 the quantity of electricity at would continually increase or continually diminish, which is obviously impossible. In algebraical symbols we may express Kirchhoff's first law as follows C7i+C7 a + .. +0 W =0 ...... (8), or simply, %C=Q, a current C p being positive when it is flowing towards the common junction. The poten- Let ^ ^ e tne potential of the common junc- "cornL? 6 tion of n arms of a network, and let F l5 F 2 , . . junction y^ be the p 0tentia i s of the other ends of the arms, then, by (2) and (8), we have (F 1 -F+^ 1 )/r 1 +(F 2 -F+# 2 )/r 2 + .. +(V n -V+E n )/r n =0 .......... (9), and thus, VS(l/r) =S( V v +E v )/r v . In analogy with the nomenclature of alternating current theory, conductors having a common junction will be said to be star connected. If we assume that the potentials at the ends Minimum of n branches, star connected, of a network are heating constant, and that their resistances and the electromotive forces in their circuits are also constant, we see almost at once, by the differential calculus, that (9) determines the value of F which makes a minimum. Hence the actual value of the potential of the common junction is the theoretical value which makes the heating as determined by Joule's law a mini- mum. When we have n resistances r 1} r 2 , . . r n , in^puaUei connec ted in parallel between two mains each of negligible resistance, and when the potential difference F between the mains is constant, then the cur- 8 ELECTRIC CABLES AND NETWORKS rents d, C 2 , . . C n , in the resistances are given by and therefore, 7=0^=00*= .. =C n r n .. .. (10). Now if C be the current in the main, we have, by KirchhofFs first law, C=0 1 +0,+ .. +C n , and hence, by (10), <7 = F (l/r 1 +l/r 2 + .. +!/>) .. (11). If R be the value of the single resistance which when placed between the mains would allow a current C to flow, we have and thus, by (11), l/U = l/r 1 +l/r a + ... +l/r n =Sl/r, .. (12). Hence the reciprocal of R equals the sum of the reciprocals of the resistances of the coils. We shall call R the equivalent resistance. Since 7= CR, we find by (10), C l =C(R/r,) 9 C a =C(R/r 2 ) .. .. (13). If in a network we take any of the conductors Kirchhoff's second which form a closed circuit, the algebraical sum of the currents multiplied by the resist- ances of these conductors equals the algebraical sum of the electromotive forces round the closed circuit. This theorem was enunciated by Kirchhoff and is known as his second law. It follows at once from (2), for SCr=S(V,-V, +1 +li:,)=SE .. .. (14), since in a closed circuit 2(F P V f+1 ) = Vi F 2 +F 2 F 3 If in a network we choose any system of con- f ductors which form a closed circuit and if the a network resu ltant electromotive force round this circuit be zero, then, for all values of the currents in FUNDAMENTAL PRINCIPLES 9 these conductors which are consistent with Kirchhoff's first law, the values which make the heating of the con- ductors a minimum satisfy Kirchhoff's second law. Let 7*1, r 2 , . . be the resistances of the various con- ductors and let C t , C 2 , . . be the currents in them. Since C 1 C 2 gives the resultant value of the currents flowing into or out of the circuit at the common junction of r and r 2 , and as this value is to be the same whatever hypothetical values we give to the currents, we see that these values must be C -{-x, C 2 -{- x, C 3 + x -> - where x may be positive or negative. The total heating W, therefore, is given by But by Kirchhoff 's second law 2(7r = ^E and is therefore zero. Hence and W, therefore, has its minimum value %C 2 r when x is zero, that is, when the values of the currents are in accord- ance with Kirchhoff 's second law. The volume resistivity p of a conducting V sSvity e " su bstance at a given temperature is the resist- ance offered at that temperature by a centi- metre cube of the substance to a flow of electricity from one face to the opposite face of the cube, the lines of flow being perpendicular to these faces. In practice, p is usually expressed in microhms (millionths of an ohm). Since, by Ohm's law, the fall of potential from one face to the other of the cube is uniform it follows that the resist- ance of a rectangular prism one square centimetre in cross section and the nth part of a centimetre long is p/n. We see that the resistance of a rectangular prism, one square cm. in cross section and I cms. long is the same 10 ELECTRIC CABLES AND NETWORKS as that of nl prisms of the same section and of length l/n arranged in series. It is therefore nl x (p/ri), that is, pi. If we now suppose this prism divided up into m parallel prisms, the areas of the ends of which will be the mth part of a square centimetre, the resistance of each of these elementary prisms will be mpL If we have a prism of length I and cross sectional area S, we may suppose it to consist of mS elementary prisms arranged in parallel. Its resistance would therefore be mpl/(mS), that is pl/S. If R, therefore, be the resistance of this prism in microhms we have R= P l/S (15). It has to be remembered that this is true also for cylindrical conductors of any section since a cylinder is a particular case of a prism. The only assumption made is that the current flow is parallel to the axis. Section When the section of a wire varies slightly, it is customary in calculating its resistance to measure the cross sectional areas at equidistant points along the wire, and to substitute the mean of the values thus found for S in formula (15). To see the nature of the error made by this assumption, let us consider the resistance R of a series of n cylinders each of length l/n and of cross sectional areas $1, 82, S n . We shall suppose that the cylinders are joined to one another by a material infinitely thin, having absolutely no resistivity, and spread uniformly over their ends. This will ensure that the flow of the current at every point in each of the cylinders is parallel to its axis and hence we can at once write down the value of all the resistances. The actual value of the resistance of the whole will be greater than this, for the stream lines of current will be curved. We have, by (15), FUNDAMENTAL PRINCIPLES The formula ordinarily used is h .. +S n )/n} 11 Now, by algebra, since Si+S 2 + . . +S n is greater than n(SiS 2 . . S n ) l/n and, for the same reason, l/Si+l/S 2 + . . + \/S n is greater than n(8i8 a . . S n )~ l/n , we have, therefore, (^+^2+ .. +S n )(l/8 i + l/S a + .. + 1/SJ greater than n 2 , and hence R is greater than R', pro- vided that Si, S 2 , . . are not all equal. Since the actual value of the resistance of the n rods in series is greater than R, it is a fortiori greater than R'. Therefore the value of the resistance of a wire calculated by means of (15) by making the customary assumptions is too small. Conversely the value of the volume resistivity, calculated from the value of the resistance found by a Wheatstone's bridge by aid of (15) is too great. If the wire be nearly uniform in cross section the error due to neglecting the curvature of the lines of flow is very small. In the following table the values of the vol- Volume re- sistivities of ume resistivities of pure metals at C., found metals by J. Dewar and J. A. Fleming (Phil. Mag. p. 299, Sept. 1893) are given. The metals were in all cases soft and annealed. Metal P u microhms Metal p \ microhms Aluminium . . . Cadmium .... Copper Gold '. .... 2-665 10-02 1-561 2-197 Nickel . . . .. . Palladium . Platinum Silver / '. 12-32 10-22 10-92 1-468 Iron 9-065 Thallium . 17-63 Lead. 20-38 Tin * * . . f 13-05 IVEasrnesium . . 4-355 Zinc . . * 5-751 12 ELECTRIC CABLES AND NETWORKS The conductance K of a conductor is mea- Conduct- ance and sured by the current flowing in the conductor conductivity . . . when unit potential difference is applied at its terminals. Hence, by Ohm's law, and so, K = l/E. The conductivity K of the substance of a conductor is measured by the current which flows, parallel to an edge, through a unit cube of the substance, when unit difference of potential is maintained between the two faces perpen- dicular to the edge. Hence it readily follows that the conductance K of a wire of conductivity K, length I, and cross section S t is given by K=*(8/l). As K and K are simply the reciprocals of R and p, it is unnecessary to tabulate their values as the values of the latter quantities for various wires and substances are given in tables. It is also unnecessary to discuss methods of measuring conductivity separately from methods of mea- suring resistivity, as any method which measures the one quantity will also give the other. Circular ^ e sna ^ now describe how the dimensions of mi1 the conductors used in practice are specified. On the Continent of Europe, thin wires are usually specified in terms of their diameters measured in millimetres. In England and America they are generally specified in terms of certain gauges or in terms of the diameters measured in mils, a mil being the thousandth part of an inch. Cable manufacturers call the area of a circle one mil in diameter a " circular mil." If, for instance, the diameter of a wire were d mils, its area would be d 2 circular mils or 0-7854 d 2 /(l,000) 2 square inches approximately, since the value of a circular mil is 0-7854/(l,000) 2 square inches. In practice, FUNDAMENTAL PRINCIPLES 13 it is convenient to use the expression circular mil as it is a perfectly definite unit and by its use we avoid the necessity of multiplying the square of the diameter by 7r/4, i.e. by 0-7854. Various gauges are used for the measurement Gauges of wires. The oldest of them is the Birmingham Wire Gauge (B.W.G.). In this gauge the thickest wire which is tabulated has a diameter of 500 mils and is denoted by No. 00000. The thinnest wire has a diameter of 4 mils and is called No. 36. In England this gauge has been replaced by the British Legal Standard or as it is generally called the Standard Wire Gauge (S.W.G.). As in the B.W.G., the thickest wire which is tabulated has a dia- meter of 500 mils, but it is denoted by No. 0000000 or 7/0. The thinnest wire is No. 50 and is 1 mil in diameter. From the tables given below it will be seen, that the diameters of the wires corresponding to the various numbers do not proceed by any regular law. The number of sizes is ample for all practical purposes. In electric lighting practice, conductors having a larger sectional area than that of a No. 14 S.W.G. wire are stranded. The trolley wires used in electric traction are generally No. 0, 3/0, or 4/0, S.W.G. and have diameters of 324, 372, and 400 mils respectively. In America, the Brown and Sharpe Gauge (B. & S.) is in general use. It is the only gauge that has been calculated on scientific principles. It is founded on the Birmingham Wire Gauge but the diameters, and consequently also the areas, of the cross sections of the wires corresponding to the various numbers are in geometrical progression. The largest wire is 4/0 and has a diameter of 460 mils. The smallest is No. 40 with a diameter of 3-14 mils. The diameter of every wire in this gauge is practically double 14 ELECTRIC CABLES AND NETWORKS that of the sixth consecutive wire succeeding it or half that of the sixth consecutive wire preceding it. It follows that the area of the cross section of every wire is practically half that of the area of the wire which is three above it, or double that of the wire which is three below it. For instance : B. & S. Gauge Diameter in mils Area in Circular mils Mass in Ibs. of 1,000 yds. Cu. wire No. ... No. 3 .... 325 229 105,600 52 630 958 477 No. 6 . . . . 162 26,250 238 It will also be seen that the weight of a yard of No. n wire will be half that of a yard of No. (n 3) wire and double that of No. (n-}-3) wire. To find the value of the ratio x of the diameters of consecutive wires in this gauge, let us calculate from the diameters of No. and No. 10 wire respectively. These are 324-95 and 101-89 mils. We have, therefore, 324-95 = 101-89 x w and hence, 10 log x = log 324-95 log 101-89=0-5036849 and thus, a; = 1-123 very nearly. The diameter of 4/0 wire would be given by 324-95 (1-123) 3 , that is, 460-2 mils nearly. The diameter of No. 40 wire would be 324-95 (1-123)- 40 which equals 3-14 mils nearly. The following table gives the diameters of the wires in the Standard Wire Gauge, the Birmingham Wire Gauge, and Brown and Sharpe's Gauge. The masses of a thousand yards of a pure copper wire of the various sizes are given in the second table for purposes of comparison. FUNDAMENTAL PRINCIPLES 15 THE ENGLISH AND AMERICAN GAUGES (IN MILS). No. S.W.G. B.W.G. B. & S. No. S.W.G. B.W.G. B. &S. 4/0 400 454 460-2 19 40 42 35-9 3/0 372 425 409-6 20 36 35 32-0 2/0 348 380 364-8 21 32 32 28-5 324 340 324-9 22 28 28 25-3 1 300 300 289-3 23 24 25 22-6 2 276 284 257-6 24 22 22 20-1 3 252 259 229-4 25 20 20 17-9 4 232 238 204-3 26 18 18 15-9 5 212 220 181-9 27 16-4 16 14-2 6 192 203 162-0 28 14-8 14 12-6 7 176 180 144-3 29 13-6 13 11-3 8 160 165 128-5 30 12-4 12 10-0 9 144 148 114-4 31 11-6 10 8-9 10 128 134 101-9 32 10-8 9 7-9 11 116 120 90-7 33 10-0 8 7-1 12 104 109 80-8 34 9-2 7 6-3 13 92 95 72-0 35 8-4 5 5-6 14 80 83 64-1 36 7-6 4 5-0 15 72 72 57-1 37 6-8 16 64 65 50-8 38 6-0 17 56 58 45-3 39 5-2 18 48 49 40-3 40 4-8 MASS OF 1,000 YARDS OF COPPER WIRE IN POUNDS, WHEN ITS SPECIFIC GRAVITY is 8*90. S.W.G. No. Mass Lbs. S.W.G. No. Mass Lbs. S.W.G. No. Mass Lbs. S.W.G. No. Mass Lbs. 4/0 1452 8 232-3 19 14-52 30 1-395 3/0 1256 9 188-2 20 11-76 31 1-221 2/0 1099 10 148-7 21 9-293 32 1-058 952-7 11 122-1 22 7-115 33 0-9076 1 816-8 12 98-16 23 5-228 34 0-7682 2 691-3 13 76-82 24 4-393 35 0-6404 3 576-3 ! 14 58-08 25 3-630 36 0-5242 4 488-5 15 47-05 26 2-940 37 0-4196 5 407-9 16 37-17 27 2-440 38 0-3267 6 334-6 17 28-46 28 1-987 39 0-2454 7 281-1 18 20-91 29 1-679 40 0-2091 16 ELECTRIC CABLES ASD NETWORKS REFERENCES. J. J. Thomson, Elements of the Mathematical Theory of Electricity and Magnetism* J. Dewar and J. A. Fleming. " The Electrical Resistivities of Metals and Alloys at Temperatures approaching the Absolute Zero.** Phil. Mag. [5] voL xxxvi, p. 271, 1893. F. C- Raphael, " The Electrician " Wireman's Pocket Book. CONDUCTIVITY CHAPTER II Conductivity The elastic constants of metal wires Hard drawn and annealed copper The density of copper The standard density Mass resistivity Resistance temperature formulae Resistivity tem- perature formulae Numerical example Tinning Measuring the rise of temperature Temperature coefficients of metals Stranded cables Effect of lay on the mass of the conductor Effect of lay on the resistance of the conductor Permissible current in cables Resistance of cables High frequency alternating currents Data for calculations References. THE elasticity of an isotropic metal depends constant^ ^f on two qualities of the metal, its resistance to change of volume and its resistance to change of shape. The former depends on the compressibility and the latter is called the rigidity. If a piece of metal recovers its original volume and shape exactly when the forces applied to it are removed, it is said to have been strained within the limits of perfect elasticity. Within these limits Hooke's law that the effects produced are pro- portional to the applied forces is true, and we may speak of the metal as being perfectly elastic. It is of importance that engineers should know the elastic constants of metals as these are an indication of their suitability for certain purposes. If when a body is subjected to any forces every cubical portion of it remains a cube, although its volume has altered, this strain is called a compression when the volume has diminished, and an expansion when the 19 20 ELECTRIC CABLES AND NETWORKS volume has increased. The bulk modulus k of the sub- stance is the ratio of the stress to the strain. If a stress of dp dynes per square centimetre uniformly applied to the surface of a body of volume F alter its volume to V dV, the compression is measured by dV/V, and thus, by Hooke's law k = stress/strain = dp/(dV/V)= V(dp/dV). In the preceding case we have considered change of volume but not change of shape. We shall now consider change of shape without change of volume. If two pairs of the bounding faces of every elementary cubical portion of a piece of metal remain squares while the other pair of faces become parallelograms having angles 90 -\-A and 90 A respectively, the piece of metal is said to be sheared. A simple way of producing a shear on a cube is to apply tangential stresses of p dynes per square centimetre to the four faces which remain squares, the stresses on opposite faces being oppositely directed. If 6 be the circular mea- sure of A, will equal 77^4/180, and the rigidity n is given by n stress/strain = p/0 dynes per square centimetre. As an example, let us suppose that a tangential stress of 10 8 dynes per square centimetre produces a shear of 7rxlO~ 4 radians in copper. In this case n=(l/7r)xW l2 =3-I8xlQ 11 dynes cm~ 2 approx. If a uniform stress of T dynes per square centimetre of cross section pull out a uniform rod from a length I to a length l-\-\ cm., we have ; Young's modulus =E = stress/strain =T/(\/l) dynes cm.~ 2 . These constants are not independent of one another. It is proved in treatises on elasticity that for an isotropic material and therefore E cannot be greater than 3n. CONDUCTIVITY 21 G. F. C. Searle (Phil. Mag. p. 199, Feb. 1900, or Experimental Elasticity, p. 113) gives the following values of E and n for various metals and alloys, obtained on the assumption that the wires experimented on were isotropic :- Metal E dynes cm. 2 n dynes cm. 2 " Silver " Steel 98 X 10 12 7-87 x 10 11 Brass (hard drawn) Phosphor Bronze Silver (hard drawn) . 02 20 78 3-72 4-36 2-82 Copper (hardened by stretching) Copper (annealed) 24 29 3-88 4-02 In the last two cases E is slightly greater than 3r&. This shows that the wires were not strictly isotropic. As a rule there is an alteration of temperature when the volume or shape of a body is altered. Hence, strictly speaking, the values of the elastic constants are indeter- minate unless the alteration, if any, of the temperature is specified. It is customary to consider two cases, namely, (1) when no heat is allowed to enter or leave the body during the application of the forces, and (2) when the temperature of the solid is maintained constant. The constants ob- tained in the first case are the adiabatic constants k^, E^ and n^, and in the second case the isothermal constants k t) E t and n t . It may be shown by aid of the principles of thermodynamics that in the case of copper at C., k^ is about 3 per cent, greater than k t , that E$ is about 0*3 per cent, greater than E t , and that n^ and n t have practi- cally the same value. For most purposes, therefore, we may disregard the differences between E t , n t and E^n^. Hard drawn * n ma king copper wire, a drawplate of hard nealed 1 " steel pierced with several holes of graduated copper sizes is mounted on the draw-bench. The wire 22 ELECTRIC CABLES AND NETWORKS is drawn in succession through smaller and smaller holes which are widest where the wire enters and taper slightly to where it leaves. During each operation it is wound on a reel on the draw-bench. After this process the wire is hard-drawn. It is usually circular in section having been drawn through conical holes. By properly designing, how- ever, the holes in the drawplate, the wire can be drawn so as to have a cross section of any desired shape. Copper is annealed by heating it to redness and cooling it suddenly. The result is that it is rendered soft and malleable. In electrical work it is customary to divide copper wires into " hard drawn " and " annealed." The Engineering Standards Committee (England) define a hard drawn copper wire as one which will not elongate by more than 1 per cent, with- out fracture. Such wire is generally used for overhead con- ductors where mechanical strength is desirable, as its break- ing stress is about double that of annealed copper, and its conductivity is only about 2 per cent, less than that of the soft copper wires used for insulated conductors in electric lighting. Benton's ^' -^' Benton nas made an investigation experiments (Ph ys i ca l Review, vol. xiii, p. 234, 1901) on the effect of " drawing " on the elasticity of copper wire. The copper wire was first annealed by heating it elec- trically to redness and cooling it suddenly. The effect of successive drawings on its elastic constants, determined on the assumption that the "drawn" wire was isotropic, was then investigated. The wire was finally annealed and its constants were again found. CONDUCTIVITY 23 Treatment Wire Diameter in cm. E dynes cm. 2 n dynes cm. 2 Annealed Drawn once , , twice A 0-1504 0-1391 4-017 x 10 11 3-946 ,, 3 times .... 6' .... ,, 9 ,, .... Re-annealed 0-1306 0-1122 0-0941 0-0932 1-387 xlO 12 1-390 1-420 1'190 3-919 3-876 3-863 4-322 Annealed B 0-1617 1-282 4-177 Drawn once . 0-1508 1-321 4-015 ,, 3 times .... 5 .... ? / 55 .... 9 .... Re -annealed . . . . . 0-1366 0-1230 0-1140 0-1001 0-0986 1-290 1-326 1-340 1-332 1-109 3-961 3-948 3-945 3-897 4-305 These results apparently indicate that the effect of suc- cessive drawings is to increase the value of Young's modulus and to diminish the value of the rigidity, but since E is in most cases greater than 3n the assumption of isotropy is not strictly permissible. On re-annealing the wire the value of Young's modulus is appreciably smaller than its initial value, and the value of the rigidity is appreciably greater. Instead of considering how the electrical The density of resistance of a copper wire varies with the length copper . and the area of the cross section, it is often convenient in practice to consider how it varies with the length and the mass of the wire. The experiments of Fitzpatrick (B. A. Report, 1894) prove that the densities (mass in grammes per cubic centimetre) of most kinds of commercial copper at ordinary temperatures lie between 8-90 and 8-95. Since the coefficient of linear expansion of copper for rise of temperature is 0-0000168, we have l t = l o { 1-|-0-0000168^}, approximately, 24 ELECTRIC CABLES AND NETWORKS where 1 , l t are the lengths of a copper wire at and t C. The volume V t of a lump of copper at t is, therefore, given by V t =V { 1 + 0-0000168^3 = V { 1 +3 x 0-0000168Z } approximately = V {1 +0-0000504Z}. Hence, the volume of a given mass of copper increases by about the half of 1 per cent, for a rise in temperature of 100 C. Thus the density of copper varies appreci- ably with the temperature, diminishing by about 0-005 per cent, per degree as the temperature rises. In order to simplify the arithmetical work The standard necessary in making calculations and to assist density the memory, it is customary in England to assume that copper weighs 555 Ib. per cubic foot at 60 F. (15-6 C.). Hence the volume of 555 Ib. of copper at C. is taken to be 1, 728/(l +0-0000504 x 15-6), that is, 1,726-6 cubic inches. Hence at C. 1 Ib. of copper has a volume of 3-111 cubic inches and 1 cubic inch has a mass of 0-3214 Ib. at the same temperature. Since there are 453-6 grammes in a pound and 16-39 cubic centimetres in a cubic inch, it will be seen that the standard density of copper at 15-6 C. is taken as 555 x 453-6/(l,728x 16-39), that is, 8-890. The standard density of copper at C. is, therefore, taken as 8-89 (1+0-0000504 x 15-6), that is, 8-897 approximately. Mass The resistance in ohms at 60 F. of a wire resistivity one me tre long and weighing one gramme is called the mass resistivity of the metal forming the wire. We shall denote mass resistivity by p '. For annealed high conductivity copper the standard value of p' assumed in England is 0-1508, and for hard-drawn high conductivity copper it is taken to be 0-1539. CONDUCTIVITY 25 If the mass of a wire be ra grammes and its length be L metres, then, if // be the mass resistivity and R the resistance of the wire, we have To prove this, notice that the resistance of a metre of the wire weighing m/L grammes would be p '/(m/L), i.e. p'L/m, and hence the resistance of L metres of this wire, having a total weight of m grammes would be p'L 2 /m ohms. We have already seen that if p be the volume resistivity of the copper (p. 10), where R is measured in ohms, p in microhms, L in metres, and S in square centimetres. Hence, if F be the volume of the copper and d its density (8-89) at 60 F., Therefore, 10 *p'(L 2 /m) = p (L*/m) 8-89 x 10 4 , and hence, p'=P* 0-0889, and p = //x 11-25, at 60 F. For annealed high conductivity copper, for instance, p = 0-1508x1 1-25 = 1-6965, at 60 F., and for hard drawn high conductivity copper, p = 0-1539x11-25 = 1-731, at 60 F. These values of p and p / are taken as the standard values in England, and are generally referred to as Mat- thiessen's Standards. It has to be remembered, however, that they are only consistent with one another when the specific gravity of the copper is 8-90. But the values of the specific gravities met with in practice may vary 26 ELECTRIC CABLES AND NETWORKS from 8-88 to 8-96. It will be seen, therefore, that the per- centage conductivity of a sample of copper wire in terms of Matthiessen's Standard as ordinarily determined may vary by as much as the half of 1 per cent, according as we take the mass resistivity or the volume resist- ivity standard. The conductivity of the best quality copper used in practice is often 2 or 3 per cent, greater than " Matthiessen's Standard." Resistance ^^ e res i s ^ ance of copper wire alters con- te ta P re ra ~ siderably with temperature. As the result of formulae an ex tensive series of experiments Matthiessen (Phil. Trans., Roy. Soc., 1862) gave the following formula for the connexion between the conductance K t of a wire at t C. and its conductance K at C. K t =K \l 0-0038701+0-000009009Z 2 ]. Hence, by means of the binomial theorem, it is easy to show that the formula connecting the corresponding resistances R t and R is, ^ =J R o [l +0-00387^+0-0000060^ -0000000 12 3 . . .]. As the error introduced by the assumption that the relation between R t and R is a linear one, at least up to 50 C., is not large, this assumption is generally made in practice. We assume, therefore, that B t =R (l+at) and give such a value to a that the errors due to this assumption are as small as possible. The following values of a, found experimentally, were quoted by Dr. Glazebrook in the Electrician, vol. 59, p. 65. NDUCT1V1TY 27 Observers Range in Deg. Cent. a Matthiessen in Phil. Trans. 1862 . . j 0-50 0-100 0-00412 0-00422 (These mean values of a are deduced from Matthiessen' s formula.) 0-18 0-0042 4 Dewar and Fleming, Phil. Mag. 1893. - 0-60 0-100 0-0042 7 0-0042 8 I 0-200 0-0042 6 Swan & Rhodin, Proc. Roy. Soc. 1894J 13-90 0-0040 8 0-0041 6 Fitzpatrick, B. A. Report, 1894. . 0-0040 5 For temperatures between and 50 C. the value cus- tomarily taken for a in England in 1908 is 0-00428. In America and Germany 0-0042 is taken as the standard value. It is probable that a varies appreciably with the kind of copper used and the method of treatment to which it has been subjected, but sufficient data on this point have not yet been obtained. We shall take 0-0042 as the standard value for the temperature coefficient for temperatures from to 50 C. The formula is therefore Resistivity temperature formulae Assuming that the formula RJM* = 1 + at gives the value of R t accurately, we shall find the temperature coefficients for the volume resistivity and for the mass resistivity. For the volume resistivity, we have Pt =E t (S t /l t ) and Po = R (S /1 ), and thus, Pt /p =(R t /R )(S t /S )(l /l t ) where 7 is the coefficient of the linear expansion of copper for rise of temperature. Hence 28 ELECTRIC CABLES AND NETWORKS Pt Po { l +( a +7)* } approximately. We also have P \=R t (m/L\) and p' =R (m/L* ), and thus, p' t = P ' (R t /R )/(L\/L\) =p' { 1 + ( a 2y)t } approximately. If a = 0-00420 and y= 0-000017, we may write ^=^(1+0-004220 and p' t = p' (l+W041M). It is customary to assume that the values of the temper- ature coefficients for R t , p t and p' t are the same. We see that the maximum error which arises from this neglect of the thermal expansion of the metal, when t is less than 50 C., is less than 0-2 per cent. Numerical ^- s an application of the formulae we shall example cons ici er the problem of finding the percentage conductivity in terms of " standard " copper of a copper rod one centimetre in diameter. We shall suppose that when a current is flowing through it, the potential difference between two knife edges at a distance of 200-0 cms. apart as read by an accurately calibrated millivoltmeter is 0-1948 volt. We shall also suppose that the temperature of the rod is 35-6 C. and that the current flowing through it and the voltmeter in parallel, as read by a Kelvin balance is 398-9 amperes. If the resistance of the millivoltmeter with its connecting leads be 7-30 ohms the current flowing through it will be 0-1948/7-3, or 0-027 ampere nearly. The current in the conductor may therefore be taken as 398-9 amperes, and hence, the resistance between the two equi- potential surfaces passing through the points of contact of the knife edges will be 0-1948/398-9, that is, 0-0004883 ohm. If we assume that the lines of flow of the current are parallel to the axis of the conductor so that the equipotential surfaces are planes perpendicular to this axis, then since 0-7854 is the area of the cross section of the rod, we have CONDUCTIVITY 29 =0-0004883 x 0-7854/200 = 1-917 microhms, and therefore, ^ 15 . 6 = 1.917(1 +0-0042 x 15-6)/(l +0-0042x35-6) = 1-917/1-079 = 1-777. Now at this temperature the standard volume resistivity is 1-731. The percentage conductivity of the copper forming the rod is, therefore, 100x1-731/1-777, that is, 97-4. As a 2 per cent, variation from the adopted standard is considered permissible by manufacturers and engineers, this conductor would legally satisfy a specifica- tion insisting on a 99 per cent, conductivity but not one insisting on a 100 per cent, conductivity. In practice many tests must be taken, and the conditions of the experiment or the method adopted must be varied in some of the tests, be- fore the experimenter can make certain that his maximum inaccuracy is less than the half of 1 per cent. It has to be re- membered that, since the resistance is found by dividing the reading of the millivoltmeter by the reading of the ammeter, the percentage error in the computed resistance is sometimes equal to the sum of the two instrumental percentage errors. Some of the substances, sulphur for instance, Tinning in the materials used to insulate copper wires, attack copper. When, therefore, these substances are used the wires are given a coating of pure tin. As the conduct- ivity of tin is less than that of copper, the conductivity of a tinned copper conductor will be slightly less than that of a pure copper conductor of the same diameter. For this reason the conductivity of all tinned copper conductors whose diameters lie between 0-104 and 0-028 inches (No. 12 and No. 22 S.W.G.) inclusive, is allowed to be 1 per cent, lower than that of pure copper. 30 ELECTRIC CABLES AND NETWORKS Lines of flow In measuring a resistance by the fall of potential method, as described in the last section, we assumed that the lines of flow of the current were straight lines. If the metal is not homo- geneous or if its diameter vary ap- preciably, this as- sumption is not per- missible and errors may arise from this cause. For example, in measuring the re- sistance of the cop- per bonds used for rails in electric trac- tion the result de- pends on the equi- potential surfaces chosen. This is illustrated in Fig. 1. Here ab is a short cylindrical copper conductor connect- ing two large copper cylinders A and B. The lines and arrow-heads indicate the direction of the flow of the current through the conduc- CONDUCTIVITY 31 tors. The curved lines AA, BB^ aa^ and bb 1? cutting the lines of flow at right angles are sections of equipotential surfaces by the plane of the paper. If we were to put the knife contacts at A and B, and proceed as in the last section but one, the resistance measured would be that between the surfaces AA, and BB^ and unless these surfaces were accurately known and also the temperatures at the various sections, it would be impossible to deduce the conductivity of the copper. Similarly if the knife edges were placed at a and b we would get another value of E, but as the equipotential surfaces are still appreciably curved an error would be introduced if we made calculations on the assumption that they were planes. Measuring ^ n ^ e re P or ^ ^ the Standardization Com- o^tempera- m ^^ ee f the American Institution of Electrical ture Engineers (Journal, 1907), it is recommended that the rise of temperature in all conductors should, when practicable, be determined by their increase of resistance. The resistance may be measured either by a Wheat- stone's bridge method or preferably by an ammeter and voltmeter. The temperature calculated in this way is usually higher than that obtained by placing a thermometer against the conductor. It is also recommended that, when a thermometer is placed against the surface of the object of which the temperature is being measured, the bulb should be covered by a pad of definite area. For instance, a con- venient pad may be made of cotton waste contained in a shallow circular box about 1J in. in diameter. The bulb of the thermometer is inserted through a hole in the side of the box. If the pad be too large it interferes with the natural radiation of heat from the metal surface and thus introduces complications into the test. 32 ELECTRIC CABLES AND NETWORKS The formula B=B (l+OW42t) ........ (1), which shows the connexion between the resistance E of copper at t C., and its resistance R at C., enables us to calculate the temperature rise when the values of the initial and final resistances, R and R', are known. If R' be the resistance when the temperature is t + x, we have by (1) Hence from (1) and (2), E'/E = { 1 +0*0042( +x) } / { 1 +0-0042* } =l+42/(lp,000-t-42*), and therefore, x= (238+t)(R'/Rl) ........ (3). As an example of the use of (3), let us suppose that R is 81-8 ohms at the initial temperature of 12 C., and that the resistance is finally 85-8 ohms. By (3), we have a=(238 + 12)(86-8/81-8 1) =1,000/81-8 = 12-2 C. As another example let us take the case of an armature winding. Before the test let us suppose that its resistance was 0*230 of an ohm and that the temperature was 25 C., and that after carrying a current for some time the re- sistance rises to 0*271 of an ohm. In this case x (238+25)(0*271/0*23 1) = 46-9 C. J. Dewar and J. A. Fleming (Phil. Mag., [5], coeffideiS 6 vol. 36 > P- 27 1, 1893) give the following values of the mean temperature coefficients of pure metals for temperatures from to 100 C. CONDUCTIVITY 33 Metal a Metal a Aluminium . Cadmium Copper .... Gold 0-00435 0-00419 0-00428 0-00377 Nickel .... Palladium . Platinum . . , Silver . 0-00622 0-00354 0-00367 0-00400 Iron Lead .... 0-00625 0-00411 Thallium . Tin . . . . . 0-00398 0-00440 Masrnesium 0-00381 Zinc 0-00406 Stranded cables It is interesting to notice that the temperature coefficient of platinum is practically the same as the temperature co- efficient of the pressure of a gas at constant volume. When the area of the cross section of the copper in a cable has to be greater than 6,400 circular mils, that is, than the area of the cross section of a solid cylindrical conductor 0-08 of an inch in diameter (No. 14 S.W.G.) it is customary to form the conductor of several strands of wire. In general there is one cen- tral wire and round this wire is a layer of six wires, and after this the number of wires in successive layers increases in arithmetical progression, the com- mon difference being 6. The number of strands, for instance, in the section of the cable shown in Fig. 2 is 1+6+12, that is, 19. When there are n layers, the total number N of strands is given by Thus FIG. 2. Nineteen strand cable. =(n+l) 2 n 2/3 34 ELECTRIC CABLES AND NETWORKS FIG. 3. Cross section of a cable containing 37 strands of wire. The middle wire is straight, and consecutive layers are spiralled in oppo- site directions. Hence (N/3) 1 ' 2 is greater than n but less than n+1. Con- sequently the number of layers in a cable of N strands is the integral part of (JV/3) 1/2 . For example if N were 331, the num- ber of layers would be 10, for (N/3)V* equals (110-3. .) 1/2 , and the integral part of this radi- cal is obviously 10. We should therefore have ten layers contain- ing 6, 12, 18 . . 60 wires in addition to the central wire. In ordinary cables the number of strands used are 1, 7, 19, 37, 61, 91 or 127. A cable consisting of N strands of No. M wire is called an N/M cable. In Fig. 3 the cross section of a cable consisting of 37 strands is shown. It will be noticed that after the first layer the sections of the strands do not necessarily touch the sections adjacent to them. In practice, consecutive layers of the strands are given a slight twist in oppo- site directions, the effect being that the centres of the sections of the strands in each layer lie on a circle concentric with the section of the central wire. Since the wires in the layers are helical, their sections by a plane perpendicular to the axis of the cable will not be exactly circular. If all the strands were parallel circular cylinders and if FIG. 4. Cross section of a stranded cable of 37 wires when the wires are all parallel. Notice that the difference between the numbers of wires in consecutive layers is six. CONDUCTIVITY 35 the cable had to be as compact as possible the section would be hexagonal in shape (Fig. 4), and every conductor inside the outside layer would touch the six adjacent conductors. FIG. 5. Stranded cable. The strands in successive layers are spiralled in opposite directions. The effect of giving a helical form to the layers is to make them bind together. The inner and outer boundaries of the layers (Fig. 5) touch concentric cylindrical surfaces. The radius of the inner cylindrical surface which every wire on the nth layer touches is (2n l)r. We shall now consider the number of strands it would be possible to get on the nth layer on the assumption that the sections of the strands are circles. In Fig. 6, let r be FIG. 6. the radius of each of the small circles which touch one another and the large circle. Let the radius of the large circle be (2n l)r. The angle c/> subtended at the centre of 36 ELECTRIC CABLES AND NETWORKS the large circle by the line joining the centres of the two small ones is given by sin( + . . .+0 B Z W ) =(2 P /8)SCl .............. (1). If I denote the distance from M of the centre of 70 ELECTRIC CABLES AND NETWORKS parallel forces equal to C^ C 2 , . . . acting at the points AI, A 2) . . . when the mains are straight, we have and thus, from (1), 8=(2p/pjlSC ...... (2). Hence, when p is given, (2) determines the cross section of the main. Copper -^ or c PP er mains, when the temperature is mains 1Q o Q ^ ^ = 1665 xlO- 9 ohms, and thus (2) becomes If / be measured in metres, and S in square mms., we get =(T/30p)5C ........ (3), approximately. This formula is often used by French electricians and is convenient in practice. When making calculations instead of showing both mains in the diagram it is sufficient to show one only (Fig. 13), since, in practice, we may regard the return main as identical with the outgoing main. M*. A A. c, c. 13. Distri- buting centre If G be the point in the main MA n , at which, if all the current 2C were taken, the voltage drop between M and A n would be the same as in the actual case, G is called the distributing centre of the load. If we suppose that the main is stretched straight DISTRIBUTING NETWORKS 71 and that weights equal to Ci, C 2 , . . . C n are placed at A 1} A 2 , . . . A n , respectively, G will be the centre of gravity of these weights. Hence we can use the ordinary statical formula lSC=CJi+CJ a +. . .+C n l n , to determine the length I of MG. Let us suppose that there are five distri- Example buting points each 25 metres apart, and that the distance of the first distributing point from the station is 50 metres. Let the currents required at A l9 A 2 ,. . . A 6 , be 5, 10, 30, 10 and 5 respectively. A 3 is obviously the centre of gravity and thus 1 = 100. Hence substitut- ing in (3), we find that 8 = 100 x60/30p=200/p sq. mms. The From (3) we have economy of high p=(l/ZOS)2C (4). pressure If we increase the pressure of supply n times the permissible value of p is generally increased n times also, as the Board of Trade rule fixes the percentage variation of the pressure, and not its absolute magnitude. We see, therefore, from (4) that with the same mains we can supply n times the current. But we have also increased the pressure n times, and hence the load we can supply with the same mains is increased n 2 times. If, for example, we increase the pressure from 100 to 250 volts, we can increase the maximum permissible load (250/100) 2 , that is, 6-25 times. For this reason it is economical to supply at the highest permissible pressure. Let us now suppose that the currents are Uniformly distributed taken from points A l5 A 2 , A z , , . . A n (Fig. 14) at equal distances apart and that the main is fed from M . Let us also suppose that the currents are all equal to c, that MA^=a, and that A i A 2 =A 2 A 3 = . . . 72 ELECTRIC CABLES AND NETWORKS =x. The distributing centre of the currents is at a dis- tance (n l)x/2 from A lt and thus^= MG=a-\-(n I)x/2, and 2c=nc=C. M, FIG. 14. Hence, by (2), p=(2 P /S){a+(n l)x/2}C =(2p/S)[a/2+ { a+(n-l)x } /2]C =(pa/S)C+EC where R and E are the resistances of the whole main M A n , and the part MA t respectively. It follows that if the load be uniformly distributed along the main from A to A we have n , If the load had been concentrated at A n , p would equal 2EC. Hence, except in the case when MA^ is negligibly small, the voltage drop with a uniformly distributed load is slightly more than half the value it has when the load is concentrated at the far end. If H be the power in watts expended in heating the mains, then in the case represented in Fig. 14, we have H/2=(pa/S)C*+(px/S){(n l) 2 +(n 2) 2 +- . .+ {2 l/n}C 2 /6 For a load uniformly distributed n is infinite, and thus H/2=(2/3)E 1 C 2 -\-(l/^)EC 2 . If the load had been concentrated at the far end of the DISTRIBUTING NETWORKS 73 line, the value of H/2 would have been RC 2 , and there- fore, if MA be small, the power expended in heating the mains when the load is uniformly distributed is very little more than one-third of its value with all the load at the far end. Let us now consider the case of a main ML from both (Fig. 15) uniformly loaded and supplied from ends both ends. If I be the length of the main and C be the total current required, (7/2 will be the current flowing in at each end, and the greatest permissible voltage drop p will be at the middle of the main. Hence p=(R/2) ((7/2) where R is the resistance of the whole main, M.. A 3 A., A,., A. -. L FIG. 15. and thus RC=4=p. If the main had been supplied from one end only, the greatest value C' of the current would be given by RC' =p. For the same maximum voltage drop, therefore, we could supply four times as much current when we feed from both ends of the main, but the losses in heating the mains would be sixteen times greater in the latter case. It sometimes happens that the cross section current of the main found by formula (2) makes the current density too high. In this case, the greatest permissible current density is chosen. In only a few cases would it be advisable to choose a current density as high as 2'5 amperes per sq. mm. (approxi- mately 1,600 amperes per sq. in.). Suppose, for 74 ELECTRIC CABLES AND NETWORKS instance, that p is 2, 1=20 metres and %C=30 (Fig. 16). Formula (2) gives =20x30/(30x2)=10 sq. mms. M Zo Ao FIG. 16. This would give a current density of 30/10, that is, 3 amperes per sq. mm. It would be better, therefore, to make the area of the cross section 15 sq. mms. so as to reduce the current density to 30/15, that is, 2 amperes per sq. mm. Main Let us suppose that the main Mab (Fig. 17) ^ as a branch ca joining it at a. Let us also suppose that Mab is the main circuit so that the section of Mab is uniform. Let Ci, c 2 , . . . be the branched circuit M c FIG. 17. currents tapped off between M and a, at distances di, d 2 , . . . from M . Let c/, c 2 ', . . . be the currents tapped DISTRIBUTING NETWORKS 75 off between a and b, at distances di t d 2 ', . . . from a, and let Ci", c 2 ", ... be the currents taken between a and c. We see, by (4), that the drop of voltage between a and b is (a<7 2 /30$)3V, where g 2 is the distributing centre of the currents c/, c/, etc. Similarly if g v and g 3 be the distributing centres of the currents Ci, c 2 , . . . and C/' c/, . . . respectively, the voltage drop p between M and b is given by p = {Mg l .C+Ma.(C'+C")+ag 2 .C'}/WS, where C, C' and C"' stand for Jc, 3c' and c" respectively. If we write d for Mg 1} d' for agr 2 , and / for Ma, this formula becomes p = { dC+l(C'+C") +d'C' } /30tf, and hence, S = {dC+l(C'+C")+d'C'}/30p .. .. (5). The voltage drop p^ from M to a is given by and the voltage drop p 2 from a to c by where d" equals ag 3 . Hence, if p^-\-p 2 p y we must have p 2 =p pi 9 and therefore, S"=d"C"/W(pp,)=S(d"C"/d'C') . . (6). Hence if d"C" be greater than d'C', S" will be greater than S. We have now to consider whether it would be more economical to make the section of ab or the section of ac the same as that of Ma. Let F 6 denote the volume of the copper required in the first case, and V c the volume required in the second. If the lengths of ab and ac are V and I" respectively, we have F 5 /2 =8(1+1') +S(d"C"/d'C')l" =[ { dC+l(C'+C") +d'C' } /30p] {l+l'+l"(d"C"/d'C') } . We also have 76 ELECTRIC CABLES AND NETWORKS V c /2 =[ { dC+l(C' + C"} +d"C" and hence, V b V c =2{ (d'C'd"C")/mp } [l(l'/d"C" If therefore (d'C' d"C") and [I (l f /d"C"+l" /d'C') (dC-\-l(C f -\-C ff )}'\ have the same sign, V b is greater than V c and thus ac should be made the principal branch. If they have not the same sign, ab should be made the principal branch. Sections -^ e ^ us now su PP ose that the sections of the mainsail three mains Ma, ab and ac (Fig. 17) are all different different but that there is the same voltage drop between M and b and between M and c. Let V be the volume of the copper used and x the voltage drop to a. Then using the same notation as in the last section we have V/2 = {dC+l(C'+C") }l/3Qx+d'C'l'/30(p x) +d"CT/M(px) =A/x+B(px), where A= {dC+l(C'+C")}l/W, and B = {d f C'l'+d"CT} /30. By the differential calculus the rate at which V varies as x increases equals -A/x*+B/(p x)* ...... (7). This vanishes when x=p/{ 1+(J5/M) 1/2 }. Since x must be less than p we take the positive sign, and it is easy to see that when x=p/{ l-\-(B/A) l/2 }, V attains its minimum value. This can be seen from first principles as follows. When x is very small the amount of copper used in Ma (Fig. 17) must be excessive. As x increases, (7) shows that the volume of copper required is rapidly diminishing. It attains its minimum value when (7) vanishes, and when x is nearly equal to p, the volume required is again very large as the voltage drops in ab and ac have to be very small. DISTRIBUTING NETWORKS 77 The rule therefore is to choose the cross sections so that }V*] . .(8). (9) Hence, by (3) we must make S = {dC+l(C'+C")}/30x, \ S'=d'C'/W(p-x), and S"=d"C"/W(p x), ) where the value of x is computed from (8). Numerical -Let the numerical data of the problem be example ag g j ven m pj g 18? so t ^ at we have ^ = 100, d'=300, and N be feeding ^ e Ceding centres which we suppose are all centres maintained at the same potential. Let x be the current in Ma, and let currents d, C 2 and C 3 be tapped from the loop at points a, b, and c, between M and L. Then, if Ma=li, ab=l 2 , bc=l 3 and cL=l i} we have DISTRIBUTING NETWORKS 83 )l z +(xC l C 2 )h+(xCiC. f C 3 )lt =0, and therefore, X = {C 1 (1 2 + 1 3 + 1,)+C 3 (1 3 + 1,)+C 3 1,}/(1 1 +1 2 + 1 3 ^1.). If the value of x found from this equation be less than M FIG. 22. <7i, a will be the point of minimum potential. If x be greater than d, but less than Ci-\-C 2 , & will be the point of minimum potential and if x be greater than Ci+C 2 , c will be the point of lowest potential between L and M . Let us first suppose that x is less thand. In this case, by (3), S= liX/SOp. If the value of x lies between C t and Oi-\-C 2) the section of the loop between M and L would be given by and when the value of x is greater than (7i+(7 2 , the equation for S is S =Z 4 (d +C 2 +C s x)/30p. Ring main We shall now consider the case of a ring feeding main and in order to simplify the formulae we shall suppose that it forms a circle (Fig. 23), with the power station S at its centre, and that the feed- 84 ELECTRIC CABLES AND NETWORKS ing centres are equally spaced round it. We shall also suppose that the load is evenly distributed so that the points of minimum potential are midway between the feeding centres. If there are n feeders, and C is the total A, FIG. 23. current output, C/n will be the current in each feeder and half (C/2n) of this current will flow in one direction round the circle and half in the other. Let pi be the drop of potential from 8 to any of the feeding points. Then, by (3), the section of each feeder is given in square millimetres by where a is the radius of the circle in metres. The section S' of the ring main, in square millimetres, is given by (see p. 72) S'=(C/2n) (27ra/2n)/QO(pp i ). Hence, if V be the volume of the copper required in cubic centimetres, we have V/2 =n(C/n)a^/30 =Ca 2 /30pi -f <7a By the differential calculus, V has its minimum value when p i =np v/2/(7r -\-n v/2). DISTRIBUTING NETWORKS 85 In this case, If n were infinite, the volume V of the copper required would equal 2<7a 2 /30p, and thus The following table shows how this ratio varies as n increases. n 1 2 3 4 5 6 1 8 9 10 100 v/v 10-4 4-46 3-03 2-42 2-09 1-88 1-73 1-63 1-56 1-49 1-04 The proper the^power anc ^ station o f It will be seen that a substantial saving in copper is effected by increasing the number of the feeders. When the positions of the feeding centres curren ^ s they require are fixed, the f ee d ers varies largely with the site of the generating station. We shall now prove that the most economical site is the " centre of gravity " of the various loads at the various feeding centres. By the centre of gravity of the load is meant the centroid of masses, proportional to the loads at the various feeding centres, placed at these centres. Let us suppose that A i} A 2 , . . . A n (Fig. 24) are the feeding centres, and that d, G 2 , . . . C n are the currents required for them. Then, if p be the maximum permissible voltage drop in the mains between the generating station S and the feeding centres, the section Si of the main joining S and A! is given by and the volume of this main by 2(7iZi 2 /30p. Hence, if V be the total volume of the copper required 86 ELECTRIC CABLES AND NETWORKS Now it is a well-known theorem in statics (see Thomson and Tait's Elements of Nat. Phil. 196) that SCI 2 has a mini- mum value when S coincides with the centroid G of masses Ci, C 2 , . . . C n placed at AI, A 2 , . . . A n re- spectively. If V m denote the minimum value of F, it also readily follows that (F F.)/2=(0 1 +0,+. . .+CJQ8*/30p, and this is the volume of the copper saved by moving the generating station from S to G. It is to be noticed that we have chosen the volume of the copper in the two cases so that the power expended in the mains namely p2C is the same in the two cases. Let us suppose that the feeding centres Example A i, A 2 , .. . A n were equally spaced round a circle of radius a, and that a current C was required at each. Then, if the generating station were at the centre of this circle, the volume of the copper required would be given by DISTRIBUTING NETWORKS 87 If S were at a distance lea from G, the volume F of the copper required would be found from It is therefore very important in practice that Ic should be small. The feeding ^et ML (Fig. 25) be a straight main which Straight we suppose to be uniformly loaded and let A be the position of the generating station. It is required to find the position of a point F in M L, so that A L-x. FIG. 25. when ML is of uniform section, the copper required for the feeder AF and the main ML may be a minimum, subject to the condition that the voltage drop from A to the farthest point of ML must not be less than p. From A (Fig. 25) draw AN at right angles to LM or LM pro- duced. If we take any point F at a distance x from M as the feeding point and if we suppose that x is less than 1/2, L will be the point of minimum potential. We shall now find the sections of the feeder AF and the main ML so that the copper used in them is a minimum, the voltage drop from A to L being p. Let $1 be the section of the feeder AF and y its length. Then, by (3), where pi is the voltage drop between A and F. If we 88 ELECTRIC CABLES AND NETWORKS suppose the main ML to be uniformly loaded, the section $ 2 will be given by where I is the length of ML. Hence the volume F of the copper in cubic centimetres is given by V/2=Cy*/30 Pl +C(l xy/W(p p,), The volume of the copper required, therefore, varies as PI, and has its extreme values when and hence, \/2y/pi = (l x)/(p p the positive sign being taken as this gives the only ad- missible value, and in this case V has its minimum value F'. Hence the minimum possible volume V of copper when the feeding centre is F, is given by x+ { 2d We have now to find out what position of F makes this the absolute minimum. By the differential calculus it follows, almost at once, that when x equals d a, V is the absolute minimum F and hence miTC , It is to be noticed that in Fig. 27 we have taken a positive when N is to the left of M . If, therefore, N lies between M and C (Fig. 25) at a distance a from M, MF d-\-a when the volume has its minimum value (C/3Qp) (I a-\-d) z . Now MF cannot be greater than 1/2 or our assumption that the minimum potential is at L is no longer true. We see, therefore, that if d a, when N is to the left of M, or d-\-a, when N lies between M and C, be not greater than 1/2 the most economical solution is to make x equal to d a or d-\-a according as N is to the left or right of M. DISTRIBUTING NETWO&KS 89 If the given quantities be greater than 1/2, then, the most economical solution is to make the middle point C of ML the feeding centre. Lastly when N is to the left of M and d a is negative, M is the proper feeding centre. Practical From the symmetry of the arrangement rule when M lies to the right of C similar solutions apply in the various cases. The analytical results lead to the following practical rule for finding the feeding centre for a straight main ML, when the distributing centre is at any point A (Fig. 25). Draw AN at right angles to LM or LM produced. Make the angle NAF equal to 45, where F lies on LM or LM produced. Then if F lie between N and M, M is the feeding centre, but if it lie between M and C, or on C, F is the feeding centre. Finally, if it lie to the right of C, C is the feeding centre. The following is a graphical illustration of the rule. Let ML (Fig. 26) be the main which we suppose to be uniformly loaded. Make the angle M'Ma equal to 45 and draw Cb parallel to Ma, where C is the middle point of ML. Similarly make the angle L'La' equal to 45 and draw Cb' parallel to La'. Let us suppose that the generating station A is above the line ML. If A lie within the angle M'Ma, M is the feeding centre. If it lie between the parallel lines Ma and Cb, then F is the feeding centre, 90 ELECTETC CABLES AND NETWORKS where AF is parallel to Ma. If it lie within the right angle bCb', C is the feeding centre. If it lie between Cb' and La' we draw AF parallel to Cb', and finally if it lie within the angle a'LL ', L is the feeding centre. An ex- actly similar solution applies when A is below the line ML. When the foot of the perpendicular from A on ML falls between M and C' where C" is the middle point of M C, we have Now at all points on M T, d equals a, and thus if the generating station A be situated on MT, V mint equals (C/3Qp)l 2 and is therefore constant. If with centre C and radius CT we describe the quadrant TT' of a circle, then, if A be situated at any point on this quadrant, F miw> will have the same value. Consequently, if A be situated at any point inside MTT'LCM, V min , will be less than if it were situated at M 9 and if A be situated above MTT'L, Vmin. wu l be greater than if A were at M . It is now easy to see that the locus of A for which F min> is constant is a quadrant of a circle between Cb and Cb', a straight line parallel to M T between Cb and Ma, a quad- rant of a circle between Ma and TM produced, etc. In practice, in order to reduce the initial cost of the copper required when designing a distributing network, it is customary in certain cases to put a " boosting " dynamo or " booster " in series with a feeder, so as to maintain the potential of the distributing centre constant however the load may vary. A booster (Fig. 27) has two directly coupled rotating armatures. One of these is the armature of a shunt wound motor driven from the mains, the other the armature of a series dynamo connected in series with the feeder. When no current is passing through the dynamo, the DISTRIBUTING NETWORKS 91 field is practically unexcited and the E.M.F. generated by the rotating armature is negligibly small. When, however, there is a current in the feeder the field magnets FIG. 27. Direct current booster. are excited and an E.M.F. e is generated. If R be the resistance of the dynamo windings and of the outgoing and return feeder, C the current, and E the initial potential of the feeding point, then the new potential will be E-\-e CE. If the first part of the characteristic of the dynamo be a straight line, it is possible to arrange that e CR is practically zero for all the values of the current during normal working. The power expended in the feeding circuit is C 2 R and we have now to consider whether it is more economical to use a booster or to increase the weight of the feeder. Let us suppose that the booster is so designed The economy of that at full load the drop p of the potential at the distributing centre is the same as if a single feeder of resistance R were used. Let us suppose also that e=np. Then, at full load, we have np CBi = p. or CE =(n+l)p 9 and thus, Ei=(n+l)E. Hence, when a booster is used, the copper required is only the (n-\-l)ih part of that required for a feeder main by itself. It has to be remembered that the losses will be (n-j-1) times greater, but they are only heavy at full load. Hence, for a small distributing centre at a considerable 92 ELECTRIC CABLES AND NETWORKS distance from the station, the use of a booster often effects considerable economies. If the interest saved on the initial cost, by using the booster and the lighter main, be greater than the annual increase of the generating charges together with the cost of the maintenance of the booster, it will be more economical to use a booster. When the distributing centre is large a special dynamo must be used. REFERENCES. J. Herzog and L. Stark, " Ueber die Stromvertheilung in Leitungs- netzen." Elektrotechnische Zeitschrift, vol. ii. pp. 221 and 445, 1890. O. Li Gotti, " Sur une Methode pour le Calcul des Reseaux de Dis- tribution." Eclairage Blectrique, vol. 44, pp. 281-286, 1905. Marcel Leboucq. " Methode Pratique de Calcul de Reseau Blec- trique, d'Eclairage et de Transport de Force." Societe Beige d'Electriciens, Bulletin, 23, pp. 109-137, 1906. INSULATION RESISTANCE OF HOUSE WIRING CHAPTER V Insulation Resistance of House Wiring Institution Rules Ohmmeter and Generator Megger Electro- static voltmeter method Earth lamps References. WHEN a building has been wired for the electric light it is necessary to make certain electrical tests to find out whether the mains are properly insulated from one another and from earth. In the wiring rules (1907) issued by the Institution of Electrical Engineers the methods of testing, etc., are described as follows Institution "97. The insulation resistance to earth of rules the whole or any part of the wiring must, when tested previously to the erection of fittings and electroliers, be measured with a pressure not less than twice the intended working pressure, and must not be less in megohms than 30 divided by the number of points under test. For this purpose the points are to be counted as the number of pairs of terminal wires from which it is proposed to take the current, either directly, or by flexibles, to lamps or other appliances. "98. Current must not be switched on until the follow- ing test has been applied to the finished work : " The whole of the lamps having been connected to the conductors and all switches and fuses being on, a pressure equal to twice the working pressure must be applied and 96 ELECTRIC CABLES AND NETWORKS the insulation resistance of the whole or any part of the installation must not be less in megohms than 25 divided by the number of lamps. When all lamps and appliances have been removed from the circuit, the insulation resist- ance between conductors must not be less than 25 megohms divided by the number of lamps. The insulation of any individual sub-circuit must not fall below 1 megohm. Any motor, heater, arc lamp or other appliance may be connected to the supply of electrical energy provided that the insulation of the parts carrying the current measured as above, is greater than 1 megohm from the frame or case. " 99. The value of systematically inspecting and test- ing apparatus and circuits cannot be too strongly urged. Records should be kept of all tests, so that any gradual deterioration of the system may be detected. Cleanliness of all parts of the apparatus and fittings is essential. " 100. Before making any repairs or alterations, the circuits which are being attended to must be entirely dis- connected from the supply." It is advisable to make two insulation tests between the mains. For the first test all the switches should be turned off and all lamps and appliances should be in position. The result of this test will show whether any switch is faulty or not. A second test should be made with the switches turned on and all the lamps and appli- ances removed. This will show whether the insulation resistance between the " flexibles " connecting the ceiling roses with the lamp holders, etc., is satisfactory. The results of insulation tests give only a partial indi- cation of the way in which the wiring of a building has been done and the quality of the materials used. If the house be damp the insulation will probably come out INSULATION RESISTANCE OF HOUSE WIRING 97 low no matter how carefully the wiring has been done. If the house be dry the insulation resistances will probably come out very high even although the insulating materials used be of poor quality and the joints be made in the most careless manner. The forty-first of the Board of Trade Regulations for securing the public from a " bad and inefficient supply of the electric light " is as follows : " The undertakers shall not connect the wiring and fittings on a consumer's premises with their mains unless they are reasonably satisfied that the connexion would not cause a leakage from those wires and fittings exceeding one ten thousandth part of the maximum supply current to the premises ; and where the undertakers decline to make such connexion they shall serve upon the consumer a notice stating their reason for so declining." This is usually taken to mean that, if V be the declared pressure at the consumer's terminals and F be the insulation resistance to earth of the house wiring, V/F must be less than the ten thousandth part of the maximum supply current. V/F, however, is a purely imaginary current. To make this clear we shall consider the case of a house the wiring of which is connected with two of the mains of a direct current 3- wire system of supply. On open circuit, the potentials to earth of the house mains are the same as the potentials of the supply mains to which they are attached. As the potential difference drop enclosed circuit is at the most 2 per cent., we see that no great error is made by the assumption that the potential to earth of a main is constant at all points of its length whatever may be the load. In order to simplify the theory we shall make this assumption. As the insulation resist- ance of the coverings of the mains is not infinite, leakage H 98 ELECTRIC CABLES AND NETWORKS currents will always be flowing either from the copper to the earth or vice versa. It is convenient to divide the paths of the leakage current into three groups. In a path of the first group, the current flows between the copper of one main and the earth. In a path of the second group it flows between the copper of the other main and the earth, and in a path of the third group the current flows from one main to the other without passing through the " earth." The path, for example, may be from one main to the other across the surface of a porcelain switch which may be excellently in- P N sulated from the earth. AA/VVVV P The point of this path, therefore, which is at zero potential must not be considered as belonging to the "earth." Let P and N (Fig. 28) denote the cross sec- tions of the conductors of the house mains. Let x denote the resultant resistance of the first group of leakage paths which we suppose connects P with the earth E. Similarly let y denote the resistance of the second group connecting N with E, and a the resistance of the third group of leakage paths which are all insulated from the earth. Strictly speaking the values of x, y, and a vary with the number of switches closed and with the number of lamps which are taken from their sockets. To fix our ideas we shall suppose that the readings are taken when all the switches are on and all the lamps are removed from their sockets, E FIG. 28. INSULATION RESISTANCE OF HOUSE WIRING 99 the mains being put in metallic connexion during the insulation test to earth. In this case the insulation re- sistance F to earth is given by F=xy/(x+y), and the insulation resistance R between the mains by The values of x, y, and a, therefore, cannot be determined from a knowledge of F and R only. If Fi and F 2 be the potentials of the mains P and N, the leakage current from the main P to earth will be V/x and the leakage current from N to earth will be Vs/y. We shall also have a leakage current (Fi F 2 )/a beween the mains. It is not clear, however, whether the greatest of these currents taken singly or the sum of the numerical values of the three is the " leakage current " specified by the Board of Trade rules. From the point of view of the public, the considerations which limit the magnitudes of the leakage currents are the risk of fire and the damage done by electrolysis. The fire risk is the more important. From this point of view the heating effects which are measured by Vf/x, V 2 2 /y and (Fi F 2 ) 2 /a respectively govern the danger. If the rules are to be equitable, the maximum permissible heating effects should be the same in all cases. If we double the voltage, therefore, the insulation resistance should be quadrupled. It is to be noticed that for given values of x, y and a the danger will be less the more distributed are the leakage paths, and the danger will be greatest when the leakage paths are concentrated at one spot. If, how- ever, the values of x, y, and a are sufficiently high, the leakage power will be so small that there is no danger of fire even if there is only one fault. It is important, therefore, to know their values. 100 ELECTRIC CABLES AND NETWORKS Measure- ^he best instrument to use for the measure- Se^auft men t of the insulation resistance of the wiring resistances Q a biding j s a portable high voltage gener- ator and an ohmmeter. These are combined in an instru- ment called the " megger " described below. The method of procedure is as follows : 1. Measure the resistance X between P and E, when N is connected with E by a piece of wire. A water-pipe makes an excellent earth connexion. In practice it is customary to make this measurement at the main fuse block. We take out the fuses and connect one terminal of the ohmmeter to the end of the house main P where it joins the fuse block. N is connected with the water- pipe and so also is the other terminal of the ohmmeter. On turning the handle of the generator, the pointer of the ohmmeter gives the value of X directly. 2. Measure the resistance Y of N to earth when P is earthed. 3. Measure the insulation resistance F between P and N in parallel and the earth. Our equations are, ...... (1), ...... (2), and l/x+l/y = l/F, ...... (3). Hence, by addition, we find that and therefore, by (2), we have by (1), l/y=(l/X+l/Y+I/F)/2, and by (3), I/a=(l/X+l/Yl/F)/2. The reciprocals of x, y, and a are thus found in terms of measured quantities and so x, y, and a can be found. It INSULATION* RESISTANCE OF HOUSE WIRING 10 1 is to be noticed that when the sum of the reciprocals of two of the quantities X, Y, and F is nearly equal to the reciprocal of the third quantity, a small percentage error in the determination of any of them will make a large percentage error in the computed value of one of the quantities x, y, or a. As an example let us suppose that X , Y, and F are found to be 198, 2-38 and 4'09 megohms respectively. In this case l/a?=(l/l'98 l/2-38+l/4-09)/2 =(0-5051 0-4202+0-2445)/2 =0-1647, and therefore, #=6O7 megohms. The above calculation is best made with the help of a table of the reciprocals of numbers. Similarly we find that y = 12-5, and a =2-94 megohms. The fault resistance, therefore, of the main N is practically double that of the main P. Hence, unless there is any special reason to the contrary, it would be advisable to connect N with the supply main which is at the higher potential. In practice, the resistance to earth xy/(x+y), which equals 4O9 megohms, and the insulation resistance a(x-\-y)/(x -\-y-\-a), which equals 2*54 megohms, are the quantities which are measured. But as a knowledge of these two quantities only is not sufficient to enable us to find out the values of x, y, and a, we cannot determine the leakage power or the leakage currents. We know that both x and y are separately greater than the insulation resistance to earth, and that a is greater than the insulation resistance between the mains. Hence we see that 4-09 is the minimum possible value of either x or y, and that a is not less than 2-54. Since, however, the actual values of x, y, and a can be found by the above method in a few minutes it is always advisable to find them as they give important information 102 ELECTK1C CABLES AND NETWORKS about the relative values of the insulation resistance of the two mains. In making the above test we have supposed that the readings are taken when all the switches are closed, and all the lamps and other appliances are removed from their sockets or disconnected. In this case the main part of the leakage is generally taking place across the flexible wires used in the fittings, and the value of a found by the test corresponds to the value of a when all the lights, etc., in the building are switched on. If now all the lamp switches are turned off, and all the lamps are in position, a new test can be made to see if there is any important alteration in the values of x, y, or a. The values found in this case enable us to find the leakage currents when all the switches of the consuming devices are turned off. There is still a possible source of leakage that we have not yet considered, namely, the direct leakage between the terminals of the glow lamp itself. The terminals usually consist of pieces of brass separated from one another and from the collar of the lamp by plaster of Paris. If they are not well made, there may be considerable leakage taking place between the terminals or, if the socket for the lamp be in connexion with the earth, between the ter- minals and the collar. Leakage to earth through the collar of a lamp lowers the apparent fault resistance of a main. If, however, we make a test with the lamps in position, and another with the lamps removed, we can easily find out if the lamps are at fault between the leading in wires and the collar. To measure the insulation re- sistance of the plaster between the contact pieces is diffi- cult as they are directly connected by the filament. It is advisable, therefore, to break the filament of a sample lamp in order to test this resistance. In good lamps it INSULATION RESISTANCE OF HOUSE WIRING 103 ought to be exceedingly high, but the standard of 1,000 megohms suggested by the Engineering Standards Com- mittee (1907) is generally considered to be excessive. X CQ O i- Ohmmeter and generator For testing the insulation resistance of the electric wiring in a building an ohmmeter and generator is usually employed. The generator 104 ELECTRIC CABLES AND NETWORKS consists of a small hand dynamo D (Fig. 29) enclosed in a portable box. Instruments are made giving pressures of 100, 200, 500, or 1,000 volts. Another little box contains the ohmmeter. Two coils of wire A and B (Fig. 29) are placed with their axes making a fixed angle with one another, and a small soft iron needle ns is placed between the two. The connexions for testing the insulation re sistance of the mains to earth are shown in the figure. When the handle of the generator is turned a current passes through the coil B and the resistance in series with it. If the insulation resistance of the cables to earth be infinite no current will pass through A . The needle, there- fore, will set itself in the direction of the resultant mag- netic force which will be parallel to the axis of the coil B. In this position the pointer will be opposite infinity on the scale of the instrument. Similarly when the insulation resistance to earth is zero, practically all the current will pass through A, and the needle will be parallel to the axis of this coil, the reading now being zero. For other values of the insulation resistance, an appreciable current passes through both coils, and the needle takes up an intermediate position. The instrument may be calibrated by putting known high resistances between its terminals and turning the handle of the generator. By means of a two-way switch, the resistance in series with B can be altered so as to increase the range of the instrument. In practical work, the read- ings can be trusted to within 2 or 3 per cent. In another instrument made by Messrs. Evershed and Vignoles, called the megger, the ohmmeter and generator are combined so that they form a single instrument. The manner in which the ohmmeter principle is applied in this case is shown in Fig. 30. The ohmmeter and the generator have the same INSULATION RESISTANCE OP HOUSE WIRING 105 magnetic circuit. The ohmmeter has two coils called the pressure and current coils. They are mounted on a moving axle with their axes inclined to one another. The field in the annular gap in which the current coil FIG. 30. The Evershed Megger. moves is uniform, but the pressure coil, starting from a position midway between the poles, is dragged into a field of gradually increasing strength. When there is no current in the current coil, the pressure coil is at rest with its plane midway between the poles, and the pointer reads infinity. If the resistance be zero, a large current will pass through the current coil, and the moving system will be dragged round by the forces acting on this coil into a new position of equilibrium where the pointer will read zero. For other values of the resist- ance and therefore of the current in the current coil, the position of equilibrium will be intermediate between thes e two positions and the pointer will give definite readings, and so the scale can be graduated. By suitably designing the shape of the poles so that the resistance offered by the magnetic forces acting on the pressure coil to the motion increases at a certain rate, instruments with open and evenly divided scales can be produced. The generators 106 ELECTRIC CABLES AND NETWORKS are usually wound for voltages of 100, 200, 500, or 1,000. The low range instruments read from to 100 megohms, and the high range instruments from 10 to 2,000 megohms. In order to eliminate possible errors due to external fields, a differential system of winding is adopted for the pressure coil. The only thing that has to be guarded against is the demagnetisation of the magnetic circuit. A centrifugal friction clutch is sometimes used with the generator so that, when it runs above the slipping speed, its velocity, and consequently, the E.M.F. generated is very approximately constant. When the capacity between the circuits, the insulation resistance of which is being measured, is greater than one microfarad, an appreciable condenser current will flow through the current coil if the E.M.F. make rapid periodic variations, and this current will affect the reading of the instrument. For this reason it is advisable to use a " constant pressure megger " in these cases. Both types of instrument are practically dead beat. As the total weight of the instrument is only about 18 Ibs., it is extremely convenient for those tests which have to be made outside the testing room. Electro- Another method of measuring insulation voltmeter resistance is by means of an electrostatic volt- method meter and a known resistance. Let us suppose, for example, that the insulation resistance of the wiring of a building has to be measured, and let the two mains and earth be denoted by P, N, and E, respectively (Fig. 28). The procedure is as follows : 1. Disconnect the supply main connected with N from the fuse box. Let the volt- meter reading between P and N and between N and E be Fi and F 2 respectively. Then, by Ohm's law, we have y /a = V 2 /V 1 (1). 2. Disconnect the supply main connected with P and INSULATION RESISTANCE OF HOUSE WIRING 107 connect the other main again to N. Let the voltmeter readings between N and P and between P and E be F/ and F/ respectively, then 3. Finally, without altering the connexions place the resistance r between P and N, and read the voltage between the same points again. If the readings be now F/' and F 2 ", we have x/{ar/(a+r)}=V 2 "/Vi" .. .. (3). Hence, from (2) and (3), a= r(Vi'/V*') (VS/VS JY/JY). The value of a is thus found and the values of x and y follow readily from (1) and (2). In connexion with this method three small carbon resistances 1, 04, and 001 of a megohm, will be found useful. L__ , TT M X ~ E FIG. 31. Fault indicator. The following method of automatically indi- lamps" eating when a fault occurs on either of the method mains of a 2-wire distributing system is known as the " earth lamps " method. Two 8-candle power lamps are connected in series between the mains at the distributing board. The wire joining them is con- nected with a water pipe by means of a switch 8 2 (Fig. 31). If this switch is open and $1 is closed both lamps will 108 ELECTRIC CABLES AND NETWORKS burn dimly as the pressure between their terminals will only be half that of the supply mains. Let us now suppose that the switch S 2 is closed. If the fault resist- ance of each main be the same, no change in the relative brightness of the lamps will ensue, but if the fault resistance of one of them be appreciably lower than that of the other, the lamp next the faulty main will be duller than the other. On a 100 volt installation, having an insulation resistance greater than 0-1 of a megohm, the effect of earthing either of the mains through a 5,000 ohm resistance can easily be detected by the earth lamps. It has to be carefully noticed, however, that the mere fact that opening and closing the switch S 2 has no appre- ciable effect on the relative brightness of the lamps is not a certain indication that there are no faults on the mains. It may only indicate that the faults are equally balanced between the two mains. If the lamp connected with M glow brightly when $1 is open and S 2 closed this will show that the fault resistance of L is small compared with the resistance of the lamp (about 300 ohms). REFERENCES. A. Russell, " Insulation Resistance and Leakage Currents." The Electrician, vol. xli., p. 206, 1898. " The Megger," Electrical Engineering, vol. i., p. 205, vol. ii.,'p. 677, 1907. INSULATION RESISTANCE OF NETWORKS CHAPTER VI Insulation Resistance of Networks Insulation resistance Measuring fault and insulation resistance in a 2-wire system 3-wire system Graphical construction for potentials General theorem Measurement of insulation resistance Example Regulating the potential of the mains Leak in the positive outer Leak in the middle main Numerical example Energy expended in earth currents Leakage currents Numerical examples The values of / lf / 2 , and / 3 References. Insulation THE practically universal adoption of pressures of supply greater than 200 volts has brought into prominence the importance of knowing the insulation resistance of the various portions into which a network of wires, for supplying electric power, can usually be divided. The insulation resistance of a network to earth is defined to be the resistance between all the conductors of the network connected in parallel and the earth. In this chapter we shall describe methods of measuring this resistance and we shall also show how a knowledge of its value gives us important information as to the leakage currents and consequent power losses in the network. When a regular record is made of the insulation resistance not only of the whole network but also of the various portions of it, timely notice is often given, by a gradual fall in the value of the resistance, of the development of a fault. This fault can in most cases be readily located by the methods described in the next 111 112 ELECTRIC CABLES AND NETWORKS chapter, and rectified. We shall first describe how the fault resistance of each of the mains of a 2- wire network can be found by means of a voltmeter, and an ammeter or a resist- ance of known value. In some cases the resistance of the voltmeter itself can be utilized as the known resistance. Fault ket P anc ^ ^ (-^8- 32) denote the cross sections resistance Q f tne two mamSj the pressure V between which FIG. 32. is kept constant by means of a dynamo or battery. By the fault resistance /i of the positive main P, we denote the com- bined resistance of all those stray paths from it to earth along which leakage currents flow, and similarly / 2 denotes the resultant resistance of the paths, in the insulating materials used, through which the current flows from the earth to the negative main N. We do not consider that a conductor at zero potential belongs to " earth " unless it is in good elec- trical connexion with earth. A metallic portion of a switch, for instance, mounted on a porcelain base may be at zero potential, and yet, the resistance of any stray paths from it to either main are not included in /i or / 2 . Similarly any part of the path between P and N at zero potential does not belong to " earth " unless the resistance between it and " earth " water pipe for example is comparable in magnitude INSULATION RESISTANCE OF NETWORKS 113 with the joint resistance of the direct paths to earth from P orN. In practice we may consider that, to a first approximation, /! and / 2 are independent of the load between P and N. When we switch on a lamp between P and N, a portion of the connecting wire leading to the lamp is added on to the posi- tive main. This portion previously to switching on would be at the same potential as the negative main and would be virtually part of it. If there was a leak in the portion of it beyond the lamp, we see that when the switch is open this leak is credited to the negative main but after it is closed to the positive main. In this case /i and / 2 will vary with the load. If FI denote the potential of the positive main we shall assume that FI//I gives the value of the leakage current from this main. On open circuit, this assumption is admissible. On a heavy load, it is admissible as a rough approximation. If the voltage drop be not more than 5 per cent, and if the service circuits are well insulated, the inaccuracy intro- duced by our assumption will not, in the great majority of cases, be greater than 5 per cent. The insulation resistance F of the network is given by l/F = l/A + l// 2 (1). Hence, if closing a switch transfer a leaky path from / 2 to / 13 the value of F is unaltered. In a 2-wire system when a voltmeter is Measuring fault and available, the ratio of the fault resistances of insulation resistance the two mains can be determined immediately. in a 2-wire We shall first suppose that the voltmeter is not system electrostatic and that its resistance is R. When it is connected between P (Fig. 32) and a water pipe or other good earth let the reading be F/. Similarly when con- nected between N and earth let it read F 2 '. In this case Fs' will be a negative quantity. In the first case, since by 114 ELECTEIC CABLES AND NETWOKKS KirchhofFs law, the sum of the currents through E and /! must equal the current flowing from the earth through / 2 , we have V 1 '/f i + V l '/K=(V FO//2 .. (2). In the second case, we have -F 2 7/2-F 2 7^=(F+F 2 ')//i .. (3). Hence it readily follows that A// 2 - - - Fi'/F,' (4), /, = - - BUVVS+Vt'W} .. (5), and / 2 = R{(V Fi'+FO/Fi'} .. (6). From (1), (5), and (6), we see also that F=E{V/(Vi' F 2 ') 1} .. .. (7). For example, suppose that the resistance R of the volt- meter is 1,000 ohms, that F= 220, Fi' = 160 and F 2 ' = 20 volts, respectively, then by (5), /! = 1,000 {(220 160 20)/20 } = 2,000 ohms, by (6), / a = 1,000 {(220 160 20)/160} = 250 ohms, and by (7), F = 1,000(220/180 1 } = 222 ohms nearly. Let us now suppose that an electrostatic voltmeter is used, and let Fi and F 2 be the potentials of the two mains to earth respectively. In this case, the reading of the volt- meter when connected between the positive main and earth will give Fi directly, and similarly the reading between the negative main and earth will give F 2 . Equations (2) and (3) may now be written. Fi/A = (F F 1 )//,= F/(/ 1 +/,) - .. (8), and -F 2 // 2 -(F+F 2 )// 1 =F/(/ 1 +/ 2 ) .. .. (9). These equations show us that f /f 2 = Fi/( F FI), and hence, when F is known, a single reading FI of the voltmeter gives us the ratio of the fault resistances. In order to find their absolute values, however, further measurements must be made. For example, we may connect between the positive main P and earth a resistance and a milli-ammeter in series. INSULATION EESISTANCE OF NETWOEKS 115 If C be the reading of the ammeter and Ft' the new read- ing of the electrostatic voltmeter, we have, by Kirchhoff's law and by (8), Vi/h=(VVi)/f* 9 and thus, subtracting, (Fi FI')//I C= (V, F/)// 2 . Hence, ^= (V.V^/C ... . (10). From (8) and (9), we also have, f i= (V/V 2 )F ...... (11), and fs=(V/V,)F ......... (12). As an example, let us suppose that V= 200, Fi = 150, and F/ = 50 volts, and that (7 equals 0-0010 of an ampere. We find, by (10), that ^=(150 50)/0-001 =100,000 ohms, and by (11) and (12), that /!= (200/50) 100000 =400,000 ohms nearly, and f 2 = (200/150)100000= 133,000 ohms nearly. By (8) and (9), we see that the power expended in the leakage currents to earth F 1 2 //i + F 3 2 // 2 , equals F 2 /(/i+/ 2 ). Hence any diminution in the value of /i+/2 always increases the power loss due to leakage currents. Again, since = (l/F){ F! (*//,) F}*+F.V(/i+/.), we see that, if we regard FI as the only variable quantity, the expression for the power lost has its minimum value, when Vi=(F/f 2 )V, that is, when Ohm's law is obeyed. We con- clude therefore that if the potential difference between the mains be maintained constant, then as the fault resistances vary, the potentials of the mains vary always in such a way that the energy expended in leakage currents is a minimum (see Chapter I). 116 ELECTKIC CABLES AND NETWORKS 3-wire system We shall now consider how the potentials of the three mains in a 3-wire system of distribution vary with the fault resistances of the three mains. As practically all direct current networks are supplied on the 3-wire system, this problem is one of considerable practical importance. Let P, M, and N (Fig. 33) be the sections of the positive, E FIG. 33. middle, and negative mains of the system and let / l5 / 2 , and / 3 , be the fault resistances of these mains respectively. By the fault resistance fi we denote the resultant resistance of all the leakage paths from the main P to earth which do not pass through the main M. If the potential of the main M be positive and there are lamps switched on between P and M , it is obvious that there will be leakage paths to earth through these lamps and then through the insulation of the main M . Even when there is no load between P and M we may have current flowing along leakage paths from P and M, and then to earth. It has to be remembered that these leakage paths directly connecting the mains and insulated from earth are not included in the fault resistances /i, / 2 , and / 3 . These values merely give the resultant resistances of the direct leakage paths to earth from each main. INSULATION RESISTANCE OF NETWORKS 117 The insulation resistance F of the network is defined by the equation l/F = 1//1 + 1//2 + 1//3. It is therefore the insulation resistance between the three mains in parallel and the earth. F generally remains ap- proximately constant at all loads, for when a switch is turned on between the positive main and the middle main, for instance, some of the leakage paths may be taken from one main and given to the other, but usually l//!+l// 2 remains very approximately constant. When, however, a double pole switch is used for a leaky subcircuit, F is diminished when the switch is turned on. In Fig. 33 let FI, F 2 , and F 3 , be the potentials of the three mains P, M, and N. Since there can be no accumulation of electricity in the earth, we have, by Kirchhoff's law, Fi//i + F a // a + F 3 // 3 =0 .. .. (13). We may either have F 2 and F 3 negative, or F 3 alone may be negative. At the supply station the potential differences between the mains P and M , and between M and N, are each maintained constant and equal to F (suppose). Hence Substituting for Fi and F 3 from (14) in (13) we get a simple equation from which F 2 is easily found in terms of F, /i, / 2 , and / 3 . Hence also, from (14), we find FI and F 3 in terms of these quantities. The following graphical construction is quite as simple as this method and is easier to apply in practice. Draw a line PN (Fig. 34) and make PM cSSon MN = V - Place Particles of mass I//,, l// 2 , potentials and l /f 3 ' at P > M > and N > res pectively and let be their centre of gravity. We shall consider that lines measured in the direction GP are positive and in 118 ELECTRIC CABLES AND NETWORKS the direction ON negative. Taking moments about G, we have ...... (15). H I V / v t *- a V K* n G A/ /: I r _> i ' S FIG. 34. Statical diagram illustrating the connexions between the potentials and the fault resistances of a three-wire distributing system. (16). We also have OP=OM+V\ and GN=GM V) " " Comparing (15) and (16) with (13) and (14), we see at once that GP = Vi, GM = V 2 and GN = V 3 . To find the potentials of the mains, therefore, when the fault resistances /i, / 2 , and / 3 , are known, we proceed as follows : Choosing a suitable scale draw a straight line NP (Fig. 34) to represent 2V, where V is the voltage of supply. Bisect this line in M, and find the centre of gravity G of masses 1// 1? l// 2 , and l// 3 , placed at P, M, and N respectively. Then the potentials of the three mains are GP, GM, and GN respectively. General ^ n g enera ^' ^ we nave n mains whose fault theorem resistances are /i, / 2 , / 3 , . . and if the potential differences between them are V, V, V", . . the potentials Fi, F 2 , Fa, . . of the mains are given by the following construction. Draw a straight line PiP n the length of which represents F+F'-f-F"-}- .. . Mark the points P 2 , P 3 , . . on it, where P i P 2 = V, P 2 P 3 = V, etc. Place particles of mass l// l5 l// 2 , . . at P l9 P 2 , . . respec- INSULATION RESISTANCE OP NETWORKS 119 lively, and let G be their centre of gravity. Then it is easy to see that V^P.G, V 2 =P 2 G, .. . Measure- -Let us su PP ose that the middle main is con- futation nected with earth at the generating station resistance through a small resistance and an ammeter, the reading of which is C. Let us also suppose that the volt- meter connected between the middle main and earth reads F 2 . If we now break the current in the earth circuit so that the ammeter reads zero, the voltmeter will assume a new value F 2 ', which will be numerically greater than F 2 . If the voltmeter be electrostatic, the insulation resistance F of the network is given by F=(V 2 'V 2 )/C (17). If the voltmeter have a resistance R, we obviously have FK/(F+E) = (V 2 ' V 2 )/C=F' (suppose), and thus, F=F'R/(RF') (18). In either case the insulation resistance is found almost at once. We may prove formula (17) as follows. Let x denote the resistance of the earth connexion with the middle main, and let the voltmeter be electrostatic. Then, by Kirchhoff's law, we have, Fi// 1 +F a // J + F,/*+F 8 //3 = .. .. (19), and F 1 7/i + F,7/,+Fa7/3 = .. .. (20). We also have Ft F, - F 2 -F 3 = F, and F/ F 2 ' = F 2 ' F 3 ' = F, and therefore, F/ V, = V 2 'V 2 = F 3 ' F 3 .. (21). Hence, by subtracting (19) from (20), we get ( F/-F 1 )//i+( F 2 '-F 2 )// 2 +( Fa' Fa)//. = F 2 /* = C, and therefore, by (21), i// 1 + i// a + i// 3 = C7/(F a '_F 2 ), and thus, F=(V 2 'V 2 )/C. 120 ELECTRIC CABLES AND NETWORKS When the voltmeter has a resistance R, we may consider that it forms one of the leakage paths to earth on the middle main, and hence, as we have shown above, the formula can be suitably modified without difficulty, Let us suppose that initially the potential of the middle main was 8 volts, and that the reading on the ammeter was 3-5 amperes. Let us also suppose that when the earth connexion was broken the voltmeter read 112. Then, if the voltmeter is electrostatic, we get by (17), F=(ll2 8)/3-5= 29-7 ohms nearly. If the voltmeter had a resistance of 400 ohms, we find, by (18), that ^=29-7 x 400/(40029-7) = 32-1 ohms. In practice, it is sometimes more convenient to connect the positive outer through a resistance and an ammeter to earth. The earth connexion on the middle main being opened, let Fi be the potential of the positive outer. When the switch on the artificial leak on the positive outer is closed, let C be the reading of the ammeter and F/ the new reading of the voltmeter. Then, proceeding as before, it is easy to show that The maximum pressure of supply, between Regulating the " any pair of terminals," to the ordinary con- potentials J J of the sumer is fixed by the Board of Trade at 250 volts. mains The object of this regulation is to prevent shocks, at pressures greater than 250 volts, being accidentally re- ceived. If, however, the absolute value of the potential to earth of any terminal be greater than 250 volts, it is obvious that possible shocks can be obtained between this terminal and a gas or water pipe or a damp wall or floor. To carry out the object of the regulation, therefore, it is necessary to INSULATION RESISTANCE OF NETWORKS 121 prevent the potential of any terminal from being perma- nently greater than 250 volts. We have seen above that the values of the potentials of the mains depend only on the pressure maintained between them, and on the fault resistances. The graphical construction for these potentials (Fig. 34) also shows us that by making a large artificial leak on the middle main, so that l// 2 is large compared with either l// t or l// 3 , we can anchor the potential of the middle main so that it never differs much from zero, and so, also, that the potentials of the positive and negative outers never differ much from + V and V respectively, where V is the pressure of the supply. It is found, in practice, that the insulation resistance of the negative outer of a 3-wire network is generally much smaller than the insulation resistances of the other mains. The flow of leakage current from the earth seems to force moisture, by a phenomenon similar to endosmosis, into the insulating covering of the main, and thus lowers its resistance. In practice, the negative outer of an insulated 3-wire network is generally at a small negative potential. For example, in a large 3-wire system in London, the po- tentials of the mains were generally about 190, 85, and 20 volts from earth respectively for many years. If the voltage of supply had been doubled the potential of the positive main would have been 380 and it would be clearly undesir- able to have parts of lampholders and switches in damp cellars, etc., at this potential. It would therefore have been necessary to prevent the potential of the positive outer from exceeding 250 volts, and this could be done by making an artificial leak on either the positive outer or the middle main. We shall now calculate the values of the resistances of the leaks which would be necessary in order to reduce the potential of the positive outer to a given value. 122 ELECTRIC CABLES AND NETWORKS Leak Let us suppose that F l5 F 2 , and F 3 , are the positive potentials of the three mains, and that F is the insulation resistance of the network to earth. Let x be the resistance which has to be connected between P and earth in order to reduce its potential Fi to the required value F/. If C be the current in the leak we have, by (17), but G is also equal to Vi'/x, and thus x = V,'F/(V,V, / ) ...... (22). Leak ^ we eartn the middle main through a resist- Jm'ddfe ance V> we have main F 2 '/ . In the particular case, when the middle main is dead earthed so that both x and F 2 ' are zero, the formulae become .',.. .. .. (a*), (6"), (c"), and P" A = VV*/F ........ (d"). K 130 ELECTRIC CABLES AND NETWORKS The above formulae show that whether we earth the middle wire or not, (F+F 2 )/2^, and a fortiori V/F, is a superior limit to the value of the earth current to or from any main. They also show that V 2 /F is a superior limit to the power expended in earth currents. For instance, if V = 220 volts, and F = 25 ohms, the power expended in leakage currents cannot be greater than 220 2 /25, that is, 1-936 kilowatts, and no earth current can be greater than V/F, that is, 8-8 amperes. If the values of F 2 and x be known, we can in general reduce these values considerably. The following numerical examples illustrate how readily the above formulae, which are due to the author, can be applied in practice. Numerical -^et us suppose that the maximum output examples of a 3 _ wire dj rec t current station with 400 volts between the outers is 3,000 kws. We shall calculate the lowest insulation resistance which will ensure that no earth leakage current is greater than the thousandth part of the maximum supply current, the potential of the middle main being 40 volts. In this case F=200, F 2 =40, the maximum current of supply is (3,000,000/400), that is 7,500 amperes, and therefore the maximum leakage current must not exceed 7*5 amperes. Substituting these values in (a), we get 7-5 ={ (200 40) /F } { (200+40)/400 } , and therefore -F = 12'8 ohms. Hence if the insulation resist- ance of the network be greater than 12-8 ohms the maximum value of the leakage current from any part of the three mains will be less than the thousandth part of the maximum supply current. The maximum possible value of the power expended in the currents to earth in this case is, by (c), 40 2 )/12'8=3 INSULATION RESISTANCE OF NETWORKS 131 If the middle wire had to be dead earthed we see, by (a"), that F=IQ ohms, and by (c"), P maXf = 200 2 /12-8= 3-125 kws. Let us now suppose that the pressure between the outer mains was reduced to 200 volts, so that F 100, and V 2 =20. If the output of the station remained the same the maxi- mum permissible leakage current would be 15 amperes. In this case for the insulated network, by (a), F must not be less than {(100 20)/15} {(100+ 20)/200}, that is, 3 '2 ohms, and for the earthed network, F must not be less than 4 ohms. The values of the leakage power ex- pended in the leakage paths would be as before 3 and 3-125 kws. In practice, it is not permissible to have a voltage drop in the mains greater than 4 per cent., and hence (Chapter V) the maximum load on a low pressure network varies as the square of the voltage. The maximum permissible load, therefore, when the pressure is halved is only one-quarter of its original value, and thus the maximum permissible leakage current is only 3-75 amperes, and the value of F, therefore, is 16 ohms, the same value as before. As a further example, let us suppose that the potentials of the mains of a 3-wire direct current system are 300, 100, and 100 volts, respectively. Let us also suppose that the insulation resistance F of the system is found to be 10 ohms. We shall find the limits between which the greatest of the earth currents must lie and also the limits between which the leakage pow r er must lie, both when the network is insulated and when the middle wire is connected with earth through a resistance of 2 ohms. By formulae (a), (6), (c), and (d), we at once find that 132 ELECTRIC CABLES AND NETWORKS C m(KC =(200 100)(200+100)/400 x 10 =7-5 amperes, <7 miri =(200 100)100/200 xlO =5 amperes, P^ =400 x7-5 =3 kws., and P min . =200x5 =1 kw. When the resistance x of the earth connexion of the middle main is 2 ohms, we have, by (26), V 2 '={x/(F+x)}V 2 = 100/6= 16-7 volts approx., and therefore, F/ = 216-7 and V 3 '= 183-3 volts. We easily find by (a 7 ), (&'), (O> and (d'), that C' max . = (20016-7)300/400 x 10 = 13-75 amperes, 0'mfci. = (20016-7)100/200 x 10 = 9-17 amperes, P' waa , = 200 2 /10 (1/6)100 2 /10 =3-83 kws., and P' min< = 200 x 100/10 (1/6)100 2 /10 = 1-83 kws. In this case, the current in the earth connexion is 16-7/2, that is 8-35 amperes. If finally we suppose that the middle main is dead earthed so that x is zero, we have by (a"), (&"), (c"), and (d"), C" m(KK .= (200+100)/2 x 10= 15 amperes, C"' m . n 100/10 = 10 amperes, ^"ma*. = 200 2 /10 =4 kws., and, P" min . = 200 x 100/10 = 2 kws. We could have predicted at once that the maximum leakage current would in any case have been less than V/F, that is, 20 amperes and that the maximum leakage power could not have been greater than V 2 /F, that is, 4 kilowatts. The more complicated formulae, however, give us valuable additional information. As a final example, we shall take the values obtained by measurements made in 1900 on a large supply network in London. In this case F t =190, F 2 =85, F 3 = 20, and P=2-5. We shall find the limits between which the maximum value of the earth current to any of the mains must lie, INSULATION RESISTANCE OF NETWORKS 133 and also the limits for the leakage power. By formulae (a) and (b), we have, O maB .=(105 85)(105+85)/210 x2-5=7-24 amperes, and <7 min =85 X20/105 x2*5 =6-48 amperes. Whatever may have been the actual values of the fault resistances of the mains, the value of the leakage current to the negative main cannot have been less than 6-48 am- peres or greater than 7-24 amperes. Similarly, by (c) and (d), we find that the value of the leakage power cannot have been less than 0*68 kw. or greater than 1-52 kws. If the middle main of this network had been earthed the current to the negative main would have had some value between 34 and 38 amperes, and the power expended in leakage currents would have had a value between 3-57 and 4-41 kws. The current in the earth connexion also would have been 34 amperes. In this case, the only advantage gained by earthing the middle wire would have been the reduction of the potential of the positive main from 190 to 105 volts. On the other hand, the leakage current to the negative would now have been doing five times the amount of electrolytic damage, and in order to maintain the balance of the potentials about 3 kws. would have to be expended in the leakage paths all the year round. These examples illustrate that a knowledge of V 2 and F gives us most useful information about the leakage currents and the leakage power in a 3- wire network. j There is no good practical method of determin- of A, /2, ing the values of /i, / 2 ,and / 3 . When it is per- missible to arrange that, during the brief time required to take the necessary readings, the voltage between the positive and the middle may be 10 or preferably 20 per cent, different from the voltage between the negative and 134 ELECTRIC CABLES AND NETWORKS the middle, the following method will give approximate values of the three fault resistances. Measure in the ordinary way, first of all, F, F 1? F 2 and F 3 . This gives us the two equations, 1//1 + 1//2 + 1//3 =l/F ..... (1), and F 1 // 1 + F 2 // 2 + 7 3 / 3 = ...... (2). Next upset the balance of the pressures so that the voltages between the two sides of the 3-wire supply are appreciably different, and measure the new values Fi', F/, and F 3 ', of the potentials of the mains. This gives us the further equation - F t 7A+F,7/>+F 8 7/3 = o ...... (3). Hence we have three equations to determine three unknown quantities l// l5 l// 2 , and l// 3 , and thus solving by deter- minants, or otherwise, we get and l/A-tFiF/ F 2 where A= F 2 F 3 7 F 3 F 2 / +F 3 F 1 / F 1 F 3 / +F 1 F 2 / F 2 F/. The solution shows that a small percentage error in the value of a voltmeter reading may sometimes introduce a large error in the value of the fault resistances deduced by the formulae. Suppose, for instance, that FiF 2 ' F 2 Fi' is 12 and that FiF 2 ' is 2,000, then a 1 per cent, error in the reading of either Fi or F 2 ' could give to / 3 an impossible negative value. Hence this method has to be used with caution. It is to be noticed that it is necessary to upset the balance of the voltages in order to get equation (3). If we merely make an artificial leak in one of the mains we get where C is the current in the leak. But since Fi Fi" ' = F 3 F 2 " = F 3 F 3 ", and F= (V. V^/C, equation (4) can INSULATION RESISTANCE OF NETWORKS 135 be deduced from (1) and (2) and is therefore not an inde- pendent equation. REFERENCES. A. Russell, " The Regulation of the Potentials to Earth of Direct Current Mains." Journ. Inst. EL Eng., vol. 30, p. 326, 1900. A. M. Taylor. " Network Tests and Station Earthing." Journ. Inst. EL Eng., vol. 32, p. 872, 1903. W. E. Groves. " Localization of Faults on Low- tension Networks," Journ. Inst. EL Eng., vol. 33, p. 1029, 1904. FAULTS IN NETWORKS CHAPTER VII Faults in Networks Faults in networks House wiring Earths Short circuits Breaks Distributing networks The localization of faults Detecting faulty mains by flashing General methods The Hopkinson 3-ammeter method The final methods of localization The fall of potential method Loop test 2-ammeter method Induction method Blavier's test Example References. Faults in ^HE f au ^ s that most commonly arise in practice networks are ( j ue f. Q causes which can be roughly classi- fied under three headings, short-circuits, earths, and breaks. A short circuit, or as it is generally called in America, a " cross," occurs when two conductors of opposite polarity get connected by a path of very small resistance. The consequent dangers, of fire and of the dynamos being over- loaded, arising from this type of fault, are obviated in practice by means of fuses or automatic cut-outs. An earth, or a " ground " as it is sometimes called, occurs when any conductor of the network makes contact through a path of small resistance with the earth. Water pipes, for instance, make effective contact with the earth, and if a metal conductor touch a water pipe in such a way that the contact resistance is very small it makes what is called a " dead earth." Sometimes, however, the resistance of the fault is appreciable and we get what is called a partial earth. If systematic tests of the in- sulation resistance to earth of the wiring be not made 139 140 ELECTRIC CABLES AND NETWORKS periodically, this kind of fault may remain undetected for a long time, and in the event of a fault developing on a main of opposite polarity there will be a risk of fire. The third kind of fault occurs when there is a partial or a total break in a conductor. It may arise owing to a terminal screw working loose, and the end of the conductor ceasing to make contact, or it may be due to an actual break in the conductor itself. The localization of faults in a distributing network is an operation demanding not only considerable skill, but also a tJiorough knowledge of how the cables and feeders are arranged in the network. A detailed plan of the wiring is therefore almost essential, and ought always to be readily available. Having access to this plan, it is, as a rule, not difficult to devise a method of procedure which must ulti- mately locate the fault. It is always best to make the search in a methodical and thorough manner. The youth- ful engineer, for instance, sometimes neglects to test part of a section simply because it is easier to disconnect and make rough tests at the sectional pillars than at the under- ground manholes. Hence a partial earth, which could easily be detected at a manhole, may be left undetected for months. If the search be made methodically the fault or faults cannot fail to be discovered. Sometimes the first test indicates the position of the only fault, and sometimes the faulty section is only found after having isolated and tested all the others. House ^ s an introduction to the more difficult wiring case o a distributing network let us consider the method of testing for faults in a house wiring circuit. To simplify the problem, we shall consider the case of a house installed on the 2-wire system. FAULTS IN NETWORKS 141 Earths Let us first suppose that the insulation re- sistance to earth of the wiring shown in the diagram (Fig. 35) has been found to be below the standard. We have therefore to locate the section on which the partial or dead earth is situated. In Fig. 35, CMF re- presents the company's main fuse, M the meter, MS the main switch, M F the main fuse, and MDB the main dis- FIG. 35. Diagram of House Wiring. tributing board. The distributing boards for the various floors are marked DB. We shall suppose that the insulation test (see p. 100) to earth has been made at the company's main fuse CMF, all the lamps being in their sockets and all the switches being closed. We first open the double pole switch MS, and repeat the test. If the testing instrument, ohmmeter let 142 ELECTRIC CABLES AND NETWORKS us suppose, now read infinity we see that the fault is not in connexion with the meter or the main switch. If, however, it reads practically the same as when the switch was closed, the fault is in the meter or the base of the switch. In the former case, a bare wire is probably making contact with the meter cover, and in the latter, the base of a switch may be a conductor. The slate, for instance, of which it is made may have metallic veins. This can generally be remedied by bushing the fixing screws with ebonite. We have next to test the mains on the house side of the main switch MS. Turn off the switches on the main distributing board, one by one, and take the reading of the ohmmeter between each operation. Let us suppose, for example, that when the fourth switch is turned off the reading changes very appreciably. In this case, the fourth switch obviously controls a faulty section. By discon- necting the leads from the distributing board on No. 4 circuit, we can readily test at the main switchboard whether the fault be in the mains connecting the two boards. If these mains be free from faults we next proceed to No. 4 distributing board, and test it in the same way as the main switchboard. We thus finally locate the faulty lamp circuit. The fault may be due to a defective switch having been placed on damp plaster or an abrasion of the cover- ing of one of the wires or flexibles may provide a path of small resistance to a neighbouring gaspipe or a steel girder. Faults are often found also in ceiling roses or lamp brackets. Suppose that when all the switches are turned off on the main switchboard the insulation resistance measured from MS still reads very low. "In this case we first re- move the fuses in MF, and test the insulation resistance of the section between MS and MF, and of the base of FAULTS IN* NETWORKS 143 MF. If these resistances were satisfactory the fault or faults must be on the main switchboard. To test the main switchboard we disconnect all the mains from it. We then join all the metallic parts on the face of the board with binding wire, and measure the in- sulation resistance between this wire and earth. By dis- connecting the binding wire from each of the metallic portions in turn, and reading the ohmmeter between each operation, we test each portion, and thus readily locate the fault or faults. Slate of inferior quality, badly insu- lated from the fixing screws by defective bushing, may easily develop bad earth faults. Short - e seen that to locate an earth fault circuits rapidly, more especially when its resistance is high, an ohmmeter or some other suitable testing instru- ment is necessary. The location of a short circuit, how- ever, requires no instrument and is usually exceedingly simple. The blowing of the fuse generally locates the faulty circuit. We have then to examine the lamp, holder, ceiling rose, and flexible cord to find out where contact between conductors of opposite polarity is taking place. In ordinary installations, short circuits can occur in the lamp holders, and in the flexible wires used to support the lamps. In these cases the fuse protecting the circuit generally blows at once, and thus they are not dangerous. When, however, a flexible wiring system is used, or when a switch is connected with flexible wires, a more dangerous partial short circuit can occur. Let us suppose, for ex- ample, that the switch S (Fig. 36) controls the lamps L. If a short circuit occurs at A or (7, the fuse will immedi- ately blow, but if the short circuit occurs at B, between the wires connected with the switch, the lamps will still be in circuit. Hence, although an arc will start at J5, the fuses 144 ELECTRIC CABLES AND NETWORKS will not blow. In this case the arc will probably move slowly up the flexible until the mains are involved, when the fuse will almost certainly blow. The risk of fire will MAINS - : i HH ! LI 1 L D.PF. B_ - \ FIG. 36. A Short Circuit at B is dangerous in flexible Wiring Systems. therefore be much greater when the short occurs at B, than when it occurs at either A or C. To obviate this risk a safety device, due to Coninx, is sometimes employed. A third wire (Fig. 37) connected with the opposite main is A C MAINS HH DRF i L B "") FIG. 37. The Safety Wire. twisted with the flexible required for the switch connexion S. Hence in the event of an arc occurring on a switch flexible, a dead short will be sure to occur very quickly FAULTS IN NETWORKS 145 between the mains. This will blow the fuses before the arc has time to set fire to neighbouring objects, and so the fire risk will be minimized. A break in the continuity of the conductors Breaks J is generally easily located when a portable volt- meter is available. If the switch be turned on we can find whether two parts of a conductor are of opposite polarity by noticing the reading of the voltmeter when its terminals are connected by means of suitable flexible conductors across the two parts. If no pressure be indicated, they are both on the same side of the break, but if the full pres- sure be indicated they are on opposite sides of the break. By pushing needles through the insulation, contact can be made with the conductor and the exact position of the break can often in this way be rapidly located. Methods of rapidly finding the position of Distribut- i ing faults on a distributing network are of con- networks .IT,. i siderable importance to the station engineer. In ordinary low-tension networks, the location is only diffi- cult, when the network is closely netted by numerous feeders. Attention is often directed to the fault at once by the complaints of consumers, and blown fuses in the manhole section boxes or the section pillars indicate the faulty section. Let us consider the case of a 3- wire low-tension network with its neutral earthed through a resistance of 2 ohms. In order that faults may be detected as soon as possible it is essential that daily tests of the insulation resistance of the whole system to earth be made. The chart of the recording ammeter in the earth connexion should also be closely studied to see if there is any periodic variation. If there is, it is probably due to a fault in a private instal- lation periodically brought in and cut out of the network L 146 ELECTRIC CABLES AND NETWORKS by the consumer's switch. A continuous rapid oscillation of the ammeter pointer indicates that there is a defective motor armature in the circuit. If the ammeter in the earth connexion be polarized so that it indicates the direction as well as the magnitude of the current, we can tell at once which of the outers has the greater fault resistance. If the current be flowing from the middle conductor to earth the negative outer has the lower fault resistance, but if it flow in the other direction the positive outer has the lower fault resistance. When the reading of the recording ammeter in the earth connexion is very small, it has to be particularly noticed that the insulation resistance is not necessarily high. If the fault resistance of the middle main be very low or if the fault resistances of the two outers be nearly balanced, no matter how low they may be, the reading of the recording earth ammeter may be very small. A small reading of this instrument gives no certain indication of the magni- tude of the earth faults on the system, but a large reading indicates that the insulation resistance of at least one of the mains is very low. The daily test of the insulation resistance of the network (see Chapter VI) gives much more information about the magnitude of the faults than the readings of the earth ammeter. A sudden fall in the value of this quantity indicates that at least one fault has suddenly developed in the network. An inspection of the chart of the record- ing ammeter may indicate the exact time at which this fault developed, and the direction of the flow of current in the earth connexion in the case of serious faults usually indicates the main in which the fault has taken place. If the earth ammeter does not indicate the direction of the current it can be readily found by Ampere's rule with the FAULTS IN NETWORKS 147 help of a small compass. It is advisable to leave this small compass permanently in position so that the direction can always be ascertained by a glance. The polarity of the middle main with reference to the earth can also be easily found by pole testing paper. Having found the outer on which the fault exists (the negative suppose), we increase the resistance in the earth connexion, and momentarily close an artificial leak of small resistance in the sound outer. If the fault exist on a / ' s s RA r E ^ L s> \ ^1 ~\ r ~\ FIG. 38. Earth on the Middle Wire. consumer's circuit, the fuse in series with it will blow, and his complaints will determine the position of the fault. If closing momentarily the artificial leak, that is, if flashing does not clear the fault, it must lie on a part of the main protected by a large fuse, or there must also be a large fault on the middle main. us su PP ose that a central zero ammeter 8) or a current direction indicating am- meter can, by opening the switch L or by taking Detecting flashing UNIVERSITY O . Or 148 ELECTRIC CABLES AND NETWORKS out a plug, be readily put in series with the middle main. Choose some time of the day when the load is very small, and put the ammeter in circuit by opening L. Increase also the resistance of the earth connexion S 2 E, or preferably, if permissible, open the switch S 2 of this connexion. Now flash the positive main P, by closing the switch S\, and notice whether this operation produces a throw of the ammeter pointer. If it does there must be a fault on the middle main. In large networks there may always be a slight deflection of the ammeter pointer when either P or N is flashed, owing to the great length, and consequent small fault resistance of the middle main. The engineer- in-charge knows approximately the magnitudes of the throws to be expected and so an increase in the value of the throw indicates that a fault has developed. The absence of a throw also might indicate that a fault had been cleared. When the above operation indicates that there is a fault on the middle main, we have to determine the portion of the main or feeder on which the fault is situated. If we put an ammeter in circuit with each of the neutral feeders in turn, and notice the effect in each case on the ammeter pointer of flashing the outer, a faulty feeder will be indicated by the large throw obtained when the testing ammeter is in circuit with it. If all the feeder mains are sound, the fault is on the middle distributing main, and by putting an ammeter in circuit at various points of the middle main in turn, the first point at which no abnormal throw is observed on flashing the outer is the end of the section of the middle main on which the fault is situated. By noticing the readings of the ammeters in circuit with the feeder mains of the negative outer, when the positive outer is flashed, a faulty negative feeder can often be de- FAULTS IN NETWORKS 149 tected. Similarly a fault on a positive feeder is indicated by a throw on its ammeter when the negative is flashed. Let us now suppose that the existence of a fault on one of the negative feeders has been discovered. The various distributors branching from this feeder should then be transferred, one at a time, to another feeder. This may be easily done at the section pillars or at the manholes. The flashing test is repeated after each transfer, and thus, the faulty distributor is found when the transfer stops the t ' s a RA E ^-J- ^ *v r\ r ~\ r E FIG. 39. Earth on the Negative Outer. throw on the feeder ammeter due to flashing. If the fault be small, it may be necessary to put a large resistance in the earth connexion of the middle main during the test. This test is sometimes laborious owing to the frequent journeys to the station and back between each discon- nexion. It is to be noticed that none of the tests described hitherto interfere with the supply to the consumers. When it is permissible to open the earth connexion of 150 ELECTRIC CABLES AND NETWORKS the middle main, and to disconnect various sections of the network in turn, we may proceed as follows. Let us sup- pose, for instance, that there is an earth (Fig. 39) on the negative main. In this case there will be an appreciable reading on the recording ammeter HA. When we open the switch S 2) if the fault be a bad one, the potential of the positive outer from earth will be practically double of the pressure of the supply, and the potential of the middle main will be practically the same as the supply pressure. In this case a lamp will burn brightly when connected be- tween the middle main and earth. In general, if there is an appreciable fault on the negative main, a lamp con- nected between the middle main and earth will glow more or less brightly. If then we connect, at the nearest net- work box, a lamp between the middle main and the earth, the appearance of the filament will indicate the value of the voltage V 2 - The various service lines are disconnected in turn. If the faulty service line is connected with this box, there will be a sudden change in the brightness of the filament when it is disconnected. If not, we have to pro- ceed to the various network boxes in turn and repeat the test, until the faulty service line is discovered. When a suitable portable voltmeter is available it is better to use it instead of a lamp. When the fault is in the middle main there will be practi- cally no current indicated by the ammeter RA in the earth connexion. When disconnexion of the service lines is permissible we may proceed as follows. Open the switch S 2 (Fig. 39) and make a small artificial leak in the negative outer. We then disconnect the service lines as in the last section, and proceed as before, the only difference being that the lamp will now glow when the faulty section is disconnected. FAULTS IN NETWORKS 151 It will be seen that these methods are very simple and easy to apply. A drawback to their use is the necessity of breaking for a few seconds the continuity of the supply to individual consumers in the sections under test. The following methods of locating faults in distributing networks, described by F. Fernie, will be found trustworthy and expeditious. In modern networks, the different feeder sections are linked by fuse General methods FIG. 40. Arrangement of Switchboards during the test. switches in pillars above ground which are opened by a key. By sending a man round on a bicycle, therefore, it is a simple matter to divide the network (Fig. 40) into two distinct sections A and B, by removing the requisite posi- tive, negative and neutral fuses in the various pillar boxes. The neutral fuse is usually a stout piece of copper wire. If there is only one neutral bus bar at the station, a second must be extemporized. The A section is fed by one set 152 ELECTRIC CABLES AND NETWORKS of dynamos and the B section by another set. The balancing of the A section may be done by the storage battery, and the balancing of the B section by the balancer. The neutral bus bars belonging to the A and B sections respectively are earthed through separate ammeters. If there is a fault on one of the outers the earth ammeter of the section on which the fault is situated will give a large reading. Groups of feeders on this section together with the machine to which they are connected are now transferred method- ically to the other. The earth ammeters are inspected between each operation. When the faulty group is trans- ferred, the reading on one of the ammeters will suddenly drop and the reading on the other suddenly rise. The fault is thus localized to this group. The feeders of the group on which the fault is situated are now transferred back to the other section, one by one, until the faulty feeder is discovered. This faulty feeder may then be subdivided further, and the fault be localized to a few streets. By disconnecting at the service boxes, it can then be determined whether it is in a consumer's installation or in the mains themselves. In the latter case, the fault has sometimes been detected by noticing a dry patch on a wet pavement. If the fault be on a neutral feeder, it can be readily de- tected by putting a few secondary cells or a booster in series with the earth ammeters in the A and B sections in turn. If there be a fault in the A section, it will be possible to get a large constant current when the cells are in series with the earth ammeter of the A section. Now remove the feeders from the A section to the B section in turn. The faulty feeder will be detected by its trans- ference reducing very considerably the reading of the A ammeter. FAULTS IN NETWORKS 153 In this method a section of the network is Hopkinson isolated and the readings of the ammeters on three . ammeter the positive, neutral, and negative feeders of this section are taken. If the sum of two of the readings be not equal to the third there must obviously be a leak on the given section of the network. It has to be noticed, however, that even when the sum of two of them is approximately equal to the third there may be a fault on the neutral if its potential is small. Hence We must test for a neutral leak by putting a few cells in cir- cuit with the neutral and an earth connexion in the manner described above. This method can only be successfully used when the load is very steady. If there is a motor load on the section, it is exceedingly difficult to get con- sistent simultaneous readings of the three ammeters. If the system be a " drawn in " system and methods of the fault has been localized to a particular section, then, if the cables are lead covered and unbraided we can often by feeling the lead at the service boxes detect by the slight shock generally experi- enced, the portion of the cable in which the fault is situated. Sometimes, also, more especially with rubber and vulcanized bitumen cables, the fault can be localized by the smell of the overheated insulating material. If, however, the cables are " solid-laid," or are armoured and laid direct in the ground, one or other of the following methods,which have been found useful in practical work, can be used. Let us suppose that the fault lies in the parti- The fall of potential cular loop LFM (Fig. 41). Earth one pole of method the battery and connect the other through a resistance R, and an ammeter A , with the end L of the main LM . As ERLFE forms a closed circuit, a current will flow which can be read on the ammeter. An electros tatiQ 154 ELECTRIC CABLES AND NETWORKS voltmeter placed between L and M will read the voltage V between L and F, for F and M are at the same potential since there is no current in FM . If x be the resistance of LF, we have x = V/A t where A is the ammeter reading, E FIG. 41. Fall of Potential Method. and thus x is found. Knowing the resistance of a yard of cable the distance to the fault can now be readily calcu- lated with considerable accuracy. It is to be noticed that we have made the assumption that there is only one fault in the main. If there were more than one fault, this calculated length would be greater than the distance to the first fault. It is therefore always advisable to repeat the test, connecting the end M of the cable with the battery. If the two results obtained on the assumption that there is a single fault agree in locating this fault at the same point of the cable, it is highly pro- bable that a fault will be found at this point. When a spare drum of the same cable as the faulty main is available at the station, the following modification of the above test will be found convenient. Replace the ammeter A and the resistance E (Fig. 41) by the drum of cable, every- thing else remaining as before. Take the reading V of the voltmeter when placed across the terminals of the drum and the reading V between L and M . Then if I' be the FAULTS IN NETWORKS 155 length, of the cable on the drum and I the distance LF to the fault we have l=(V/V')l'. The principle of the loop test can be readily understood from Fig. 42. A wire bridge pq is connected across the terminals L and M of the loop of cable in which the fault lies, and a galvanometer G is also placed between the terminals. One pole of a battery is connected with the jockey of the bridge, the other pole being earthed. The jockey is then moved about until the deflection of the galvanometer is zero. In this case we 1 ^ E FIG. 42. Bridge Method of Testing. have x/y=p/q, and thus y = {q/(p-\-q)}(x-\-y). Hence, if I be the length of the whole loop, the length of LF is In this test care has to be taken that the resistance of the connexions at L and M are negligibly small. It is immaterial whether the fault at F is polarizable or not as it is in series with the battery. This is one of the most generally useful and accurate of the methods of localizing faults. In certain cases the following test can be easily applied. Place ammeters A l and A 2 (Fig. 43), of negligible resistance, in series with each branch of the loop. Now connect L with one pole Two ammeter method 156 ELECTRIC CABLES AND NETWORKS of a battery, the other pole of which is earthed. If A^ and A 2 be the readings of the ammeters, then, by Ohm's law, the potential difference between L and F will equal xAi, and it will also equal yA 2 . Hence y(A l -\-A 2 ) =(x-{-y)Ai, and thus, if I be the total length of the loop, the distance LF to the fault will equal { FIG. 43. Two Ammeter Method. In the above tests, it has been assumed that the section of the cable is uniform throughout. If it is not uniform, they can still be applied, provided the lengths and the resistances of the various portions of the cables are known. Let us suppose that the lengths LL , L L 2 , . . . are of different sections and that their resistances are R lt B 2 , . . . respectively. Let us also suppose that the resistance y of LF lies in value between R^-\-E 2 and B 1 -\-R 2 -\-R 3 . The fault will obviously be on the section L 2 L 3 , and the resist- ance of the conductor between L 2 and the fault will be y (Ri+Rz). Hence, knowing the resistance per yard of the section L 2 L 3 of the cable, we can easily find the position of the fault. induction The following " induction method " is ex- method tensively used in practice. The great advan- tage of this method is that no knowledge is required of FAULTS IN NETWORKS 157 the resistance of the main under test, and so uncertainties due to partial breaks or bad joints do not affect it. It can also be applied when there are several earth faults on the cable. Let us suppose that there is an earth fault at F in the cable LM (Fig. 44). Insulate one end M of the cable and FIG. 44. Induction Method. connect the other end with the terminal of a generator of alternating or pulsating E.M.F. If the other end of this generator be connected with the earth, an interrupted current will flow in the cable to the fault and return to the generator by the earth. C is a wooden triangular framework wound with several turns of insulated wire the ends of which are connected with a telephone T. When the plane of the triangle is vertical and its base is parallel to the cable carrying the interrupted current the continual fluctuation of the lines of magnetic induction, linked with, the triangular coil, will induce a fluctuating E.M.F. in it, and so a buzzing sound will be heard in the telephone re- ceiver. If the framework, held in this manner, be carried or wheeled directly over and along the route of the cable, a cessation or change of the note generally indicates the position of the fault. If the conductor carrying the current 158 ELECTRIC CABLES AND NETWORKS be enclosed in a lead sheath or in metallic casing, the earthed terminal of the generator should be connected with the sheath or the casing. Biavier's When the resistance of the line is high, the following method is sometimes useful. Let us suppose (Fig. 45) that there is a fault of resistance z at F. Let the resistances also of LF and FM be x and y respectively. In practice, the resistance R of the whole line LM is known. WVWV T M E FIG. 45. Biavier's Method. We first measure at L the insulation resistance R of the line, when the end M is insulated. We next measure its insulation resistance E 2 , when the end M is connected directly with the earth. We thus obtain the three following equations to find x, y and z. x+y=R (1), x+z=E, (2), and x+yz/(y+z)=E 2 (3). Eliminating y and z from (3), by means of (1) and (2), we have x+(Ex)(E i x)/(E+E i 2x) =R 2 , and thus, by solving this quadratic equation we get x =E 2 { (EE 2 )(E,E 2 ) } V 2 , the negative sign being prefixed to the radical, since by the conditions of the problem yz/(y+z) can only be posi- tive, and hence, by (3), x must be less than E 2 . FAULTS IN NETWORKS 159 There are four assumptions made in this test that have to be remembered in practical work. In the first place we assume that there is only one fault, secondly that the fault is non polarizable, thirdly that the temperature of the mains is uniform, and fourthly that the resistances are so high that the errors due to imperfect contacts are negligibly small. In electric lighting networks the fourth assumptipn is the most serious and in many cases it is not permissible. In a power transmission line, R was 44 ohms, Example and the values of E t and E 2 found by measure- ment were 25-9 and 24-3 respectively. In this case x =24-3 { (25-9 24-3)( 4424-3) } J / 2 = 18-7. By (2), 18-7+z=25-9, and therefore, 2 = 7*2. Similarly, by (1), y=25-3. When tests can be made at both ends of the line it is advisable to make them. If the results differ largely it is probable that there is more than one fault. REFERENCES. F. Charles Raphael. The Localization of Faults on Electric Light Mains. J. A. Fleming. Handbook for the Electrical Laboratory and Testing Room. G. L. Black. " The Maintenance of Underground Mains." Electri- cian, vol. 56, p. 507, 1906. F. Fernie. " Localization of Faults on a Three-Wire Network." Electrical Review, vol. 60, p. 451, 1907. A. Schwartz. "' Flexibles,' with Notes on the Testing of Rubber." J.I.E.E., vol. 39, p. 31, 1907. DIELECTRIC STRENGTH M CHAPTER VIII Dielectric Strength Dielectric strength Disruptive discharge Spherical condenser Single core main Effect of shape of conductor Equipotential surfaces round a pointed conductor Spherical electrodes Composite dielectrics The maximum electric stress between equal spherical electrodes American rules Failing cases in practice Measuring the dielectric strength of gases Dielectric strength of liquids Dielectric strength of isotropic solids Dielectric strength of eolotropic solids References. Dielectric ^ N power transmission, whether by direct or strength alternating current, the saving of copper effected by using very high pressures has directed the attention of manufacturers to the construction of cables which will withstand pressures of 100 kilovolts and upwards. To design cables which will successfully withstand these pres- sures, a knowledge of the electric stresses to which the various insulating materials round the core will be sub- jected under working conditions is essential, as well as an accurate knowledge of the dielectric coefficient (specific inductive capacity), dielectric strength, and resistivity of each of the insulating wrappings. In this chapter we shall discuss the dielectric strength of insulating materials and the methods of measuring it. By the dielectric strength of an isotropic insulating material in a given physical condition is meant the maximum value of the electric stress which it can withstand without break- 163 L64 ELECTRIC CABLES AND NETWORKS ing down. The substance of a homogeneous solid is called isotropic (see p. 3) when a spherical portion of it tested by any physical agency exhibits no difference in quality how- ever it is turned. Substances which are not isotropic are called eolo tropic. From an electrical point of view we can regard gases and pure liquids as isotropic. In his Experimental Researches in Electricity, vol. i, p. 436, Faraday states that " discharge probably occurs not when all the particles have attained to a certain degree of tension, but when that particle which has been most affected has been exalted to the subverting or turning point." He therefore considers that there is a definite " subverting or turning point " for each particle of the material ; that is, that it has a definite dielectric strength. In order to test Faraday's conclusion it is necessary to be able to calculate the electric stress at the point be- tween two electrodes where it has its maxiumm value. We shall first consider how electric stress is measured. From symmetry, it is obvious that the equipotential surfaces round a charged spherical conductor surrounded by a homogeneous dielectric, and at a considerable distance from all other conductors, are spherical in shape. 'If q denote the charge on the conductor, and v the potential at a point at a distance r from the centre of the sphere, we have v=q/r, and therefore the potentials of the equi- potential surfaces surrounding the sphere vary inversely as their radii. Let us suppose, for example, that the spherical conductor shown in Fig. 46 is at a potential of 10,000 volts, then the potentials of the various circles drawn in the figure are 9,000, 8,000, . . . 1,000, volts respectively. The equipotential surface of zero potential would be at infinity. It is to be noticed that close up to the sphere the surfaces are crowded together. The DIELECTRIC STRENGTH 165 spherical stratum of the dielectric included between the conductor and the first equipotential surface is obviously subjected to the greatest stress. The average stress on any of the spherical layers shown in Fig. 46 is inversely proportional to its thickness. By increasing the number of equipotential surfaces indefinitely until the concentric FIG. 40. Section of the equipotential surfaces round a charged sphere, the potential difference between consecutive surfaces being constant. layers of the dielectric become indefinitely thin, we see that the electric stress at a point is measured by {v (v+dv)} /dr, that is, by dv/dr. This quantity is called by electricians the potential gradient at the point. It follows at once from the definition of potential that the resultant force at a point in a dielectric is equal to the rate at which the potential diminishes as we move 166 ELECTRIC CABLES AND NETWORKS along the line of force through the point. Hence the potential gradient is the resultant force, and is the force with which a unit positive charge placed at the point would be urged if it could be placed there without disturbing the distribution elsewhere. This force measures the electric stress on the dielectric. A good way of picturing what happens in a dielectric is by means of Faraday's tubes of force. We picture one end of one of these tubes anchored to a unit positive charge on the positive electrode, and the other end anchored to a unit negative charge on the other electrode. In the case of the spherical conductor considered above, q tubes will start from the surface of the sphere, and thus the number of tubes passing through a square centimetre of the equipotential surface which is at a distance r from the centre of the sphere is q/^jrr 2 . But the potential gradient at a distance r from the centre of the sphere is q/r 2 , and hence 4?r times the number of Faraday tubes which pass through unit area of the equipotential surface is the numerical value of the potential gradient at all points on that surface. Another name for the electric stress at a point is the electric " intensity " at the point. It has to be remembered when reading the literature of the subject that, " the resultant electric force," " the potential gradient " and " the electric intensity " are all used to denote the re- sultant electric stress R at a point. In symbols, E= dv/dr= 7rN. If R were constant over the equipotential surface passing through the point under consideration, N would be the num- ber of Faraday tubes per square centimetre of this surface. Disruptive Greatly to the disappointment of the earlier discharge physicists, it Was found that the disruptive voltage between metal electrodes when close together DIELECTRIC STRENGTH 167 was apparently not governed by the maximum value of the potential gradient between them. As early as 1860, however, Lord Kelvin was led to infer, from his experi- ments with pressures of between 5 and 6 kilovolts obtained from a battery of 5510 Daniell cells in series, that at high voltages the disruptive discharge between large metal electrodes will take place the moment the electric stress attains a definite value. Recent experiments at high voltages have amply confirmed Lord Kelvin's conclusion, and it forms the basis of the practical methods of measuring dielectric strengths. It has to be remembered that part of an insulating material can break down without a disruptive discharge necessarily ensuing. When brush discharges, for instance, occur from an electrode in air part of the air surrounding the electrode has become a true gaseous electrolyte, and its insulativity therefore has broken down. The air at the boundary of this electrolyte has not broken down, because the electric stress to which it has been subjected has not reached the " subverting " value, which measures the dielectric strength of the medium. In Fig. 46, when the stress close up to the sphere equals the dielectric strength of air, which is about 38 kilovolts per centimetre at ordinary temperatures and pressures, the spherical layer round it breaks down and becomes luminous. If we raise the potential still higher the sphere is surrounded by a luminous spherical envelope called the corona, the radius of which is proportional to the potential to which we raise the conductor. Spherical * n ^ e case ^ a spherical condenser we have condenser a me t a lli c sphere concentric with a metallic spherical envelope. If the radius of the inner sphere be a, and the radius of the outer sphere be b, we have v=q/r, 168 ELECTRIC CABLES AND NETWORKS and q = Vab/(b a), where F is the potential difference between the two conductors. The equipotential surfaces are thus the same as in Fig. 46, and since R =dv/dr = Vab/r*(ba), we see that R has its maximum value R m , when r=a. Hence K m = Vb/a(ba). If we suppose that the sphere is surrounded by air, then, when R m attains the value of the dielectric strength of air R max ^ the air surrounding the sphere breaks down and becomes a conductor, but a disruptive discharge does not necessarily ensue. If the breaking down of the first stratum of air makes the new value of R m equal to or greater than jR max< , a disruptive discharge will ensue but if it makes it less than R maaif there will be no disruptive discharge, and a corona will be formed. If i be the radius of the corona formed, we have and thus dV/da^ = (b2a 1 )R maXf /b, assuming that R max . and b are constant. We see that V increases with i until a^ gets equal to 6/2, it then diminishes as i increases. Hence a corona can exist if i be less than 6/2, for the value of R m in the stratum immediately surrounding the corona is less than R maXm It cannot, how- ever, exist if ! be greater than 6/2, for the value of R m in the stratum surrounding it would be greater than RUM,.. We see, therefore, that the size of the inner sphere has no practical effect on the disruptive voltage provided that its radius be less than 6/2. We see also that a spherical condenser can be used to measure the dielectric strengths of gases or liquids provided that the radius of the inner conductor be not less than 6/2. In this case DIELECTRIC STRENGTH 169 where V is the voltage between the conductors at the instant of the discharge. Single core * n ^%- ^ tne equipotential surfaces for a mam single core cable with a homogeneous dielectric are shown. Let us suppose that a is the radius of the FIG. 47. Section of the equipotential surfaces in a single core cable- The dotted circle is the outer radius of the broken-down dielectric at the instant of the disruptive discharge. cylindrical core, and that b is the inner radius of the coaxial cylindrical lead sheath. The potential gradient R at a point P in the dielectric which is distant r from the axis of the core, is given by E =( 1 /X)dv/dr = F/rlog e (6/a), where V is the potential difference between the core and 170 ELECTRIC CABLES AND NETWORKS the sheath, and \ is the dielectric coefficient of the insu- lating material. R has its maximum value R m when r= a, and thus R m = V/alog e (b/a). Let us first suppose that the insulating material is a gas of dielectric strength E maXm . The conditions that a cylindrical corona of radius a be formed are that and that dv/da is a positive quantity when a =a . The second condition is true when log e (6/tfi) 1 is positive, that is, when i is less than 6/e, where e =2*718 ... is the base of Neperian logarithms. When the insulating material is a homogeneous solid sub- stance of dielectric strength R^^ the same formulae apply, at least to a first approximation. If the radius of the core be less than b/e, then, when F is greater than a log e (b/a)R max , and less than (b/e)R max ., some of the dielectric surrounding the inner core, which we suppose to be a smooth cylinder, has broken down and become a conductor. When the voltage V exceeds (b/e)R ma , K a disruptive discharge will ensue. A comparison of Figs. 46 and 47 will show that the electric stresses close to a spherical conductor are greater than close to a cylindrical conductor of the same radius and at the same potential. It will be noticed that the equipotential surfaces are more crowded together round the sphere than round the cylinder. Since we have supposed the cylinder to be infinitely long, the change of potential as we recede from it will obviously not be so rapid as in the case of the finite sphere. In Fig. 48 the effect of the shape of a con- Effect of shape of ductor on the electric stresses in the medium conductor surrounding it is illustrated. In the ngure the potential difference between any adjacent pair of DIELECTRIC STRENGTH 171 FIG. 48. Section of the equipotential surfaces when the core is a strip of copper with rounded ends. \ \ 5,000 2.600 FIG. 49. Section of the equipotential surfaces round a tapered copper conductor maintained at 30 kilovolts. 172 ELECTRIC CABLES AND NETWORKS equipotential surfaces is the same. The section of the core is elliptical in shape, and the maximum value of the electric stress at the rounded ends of the core is ten times the minimum stress at the middle part of the core. The equipotential surfaces show, however, that the electric stress is practically constant in the layer of the insulating material next the lead sheath. In Fig. 49, the equipotential surfaces round Equipoten- tial surfaces a tapered copper conductor, ellipsoidal in pointed shape, are shown. The electric stress on the insulating material in contact with the rounded point is very great. When electrodes of this shape are used for testing it is extremely difficult to calculate FIG. 50. Section of the equipotential surfaces round two spheres having equal and opposite charges. The potential gradient in the dielectric is obviously greatest at the points of the electrodes which are closest together. the value of the maximum potential gradient, and hence, only rough comparative tests can be made with them. Spherical * n ^- 50 ' ^e equipotential surfaces round electrodes ^ wo spherical electrodes maintained at equal and opposite potentials are shown. The potential difference DIELECTRIC STRENGTH 173 between any two adjacent surfaces is the same. The potential gradient is obviously a maximum at the points of the electrodes which are closest together. The value of the potential gradient at these points can be easily cal- culated by means of the tables given below. It can be shown that if the spheres be not farther apart than twice their diameter, a disruptive discharge will take place the moment the portions of the insulating material which are FIG. 51. Flux lines between cylindrical electrodes. The potential gradient is a maximum at the corners of the electrodes. subjected to the maximum stress break down. Hence, the disruptive voltage enables us to find the dielectric strength of the medium by which they are surrounded. For practical testing, spherical electrodes are generally the best. Composite ^ke effect of introducing a piece of insulating dielectrics material between two metal electrodes main- tained at constant potentials is illustrated in Figs. 51 and 174 ELECTRIC CABLES AND NETWORKS 52. The insulating material is supposed to have a high dielectric coefficient. The capacity and consequently the number of Faraday tubes between the electrodes is con- siderably increased. The stress on the air which is measured by 4?r times the number of Faraday tubes per unit area is increased also. Hence the introduction of a piece of glass near the electrodes sometimes causes a disruptive discharge FIG. 52. Flux lines when a glass sphere is introduced between the elec- trodes, the potential difference being maintained the same as in Fig. 51. Notice the increase in the total flux, and consequently the increase in the stress on the dielectric. between them. Since the dielectric coefficient of a metal is infinite, introducing a metal conductor between the electrodes increases the stress more than a piece of insu- lating material of the same size would. The calculation of the maximum potential gradient when the insulating materials have different dielectric coefficients is in general very difficult. DIELECTRIC STRENGTH 175 The easiest way of finding the dielectric maximum strength of insulating materials is by finding 6 stres? the disruptive voltage between two equal spherical electrodes embedded in the material. The author has shown (Phil. Mag. (6), vol. ii, p. 258, 1906), that, if the spheres be at a less distance apart than twice the diameter of either, a disruptive discharge will ensue the moment the maximum electric stress between the spheres equals the dielectric strength of the material. In order to calculate the maximum electric stress at the instant of discharge we must know the potential and size of each sphere and the distance between them. Let a be the radius of each sphere, and let x be the mini- mum distance between them. Let us first suppose that one sphere is at the potential FI and that the other is at zero potential. In this case the maximum electric stress, It max., between them is given by JWp.=(Fi/*)/i. where the values of /i can be found from Table II. A proof of this formula is given in the author's paper (quoted above). In the important practical case when Fi = F 2 = F/2, where F 2 is the potential of the second sphere, we have where / can be found from Table I. In general, if Fi and F 2 be the potentials of the two spherical electrodes, and Fi be numerically greater than F 2 , we have where / and /i are functions of x/a, the values of which can be found from Tables I and II. Hence by this formula we can calculate the dielectric 176 ELECTRIC CABLES AND NETWORKS strength of the material from the potentials of the electrodes at the instant of the disruptive discharge. TABLE I. VALUES OF /. x/a. /. x/a. /. o-o 000 2 1-770 O'l 034 3 2-214 0-2 068 4 2-677 0-3 102 5 3-151 0-4 137 6 3-632 0-5 173 7 4-117 0'6 208 8 4-604 0-7 245 9 5-095 0-8 283 10 5-586 0-9 321 100 50-51 1-0 359 1,000 500-5 1-5 559 10,000 5,000-5 TABLE II. VALUES OF x/a. A- x/a. /I. o-o 000 2 2-339 o-i 034 3 3-252 0-2 068 4 4-201 0-3 106 5 5-167 0-4 150 6 6-143 0-5 1-199 7 7-125 0-6 1-253 8 8-111 0-7 1-313 9 9-100 0-8 1-378 10 10-091 0'9 1-446 100 100-0 1-0 1-517 1,000 1,000 1-5 1-909 10,000 10,000 DIELECTRIC STRENGTH 177 The strengthof air In the following table the values of x and V are taken from Dr - Zenneck's work, Elek- tromagnetische ScJiwingungen und Drahtlose Telegraphic, 1905, p. 1011. They are due to J. Algermissen and are deduced from the average of the values obtained on different days under varying conditions. It has been assumed that the potentials of the electrodes are +F/2 and F/2 respectively at the instant of discharge. As the results in the last column are very approximately con- stant the assumption is justified : TABLE III. J. Algermissen. 5-cm. spheres (a =2-5). x is measured in centimetres, and V in kilo volts. X. x/a. /(calc.). V(obs.). Rmax (calc.). 2-0 0-80 1-283 58-2 37-3 2-2 0-88 312 62-8 37-5 2-4 0-96 342 67-0 37-5 2-6 1-04 374 70-8 37-4 2-8 1-12 406 74-4 37-4 3-0 1-20 437 78-0 37'4 3-2 1-28 469 81-3 37-3 3-4 1-36 500 84-7 37-4 3-6 1-44 533 88-0 37'5 3-8 1-52 566 91-2 37-6 4-0 1-60 599 94-2 37-7 4-2 1-68 632 97-2 37-5 has been calculated by means In the above table of the formula From the above results, and from many other experi- mental results obtained with both alternating and direct pressures, the author concludes that the dielectric strength N 178 ELECTKTC CABLES AND NETWORKS of air under normal conditions is about 3-8 kilo volts per millimetre. American ^ke practical constancy of the dielectric strength rules o f a j r un( j er ordinary atmospheric conditions is recognized in the Standardization Rules (1907) of the A.I.E.E. For instance, in 243, when discussing the value of the spark-gap safety-valve, it is stated that " a given setting of the spark-gap is a measure of one definite voltage, and, as its operation depends upon the maximum value of the voltage wave, it is independent of wave form, and is a limit on the maximum stress to which the insulation is subjected. The spark-gap is not conveniently adapted for comparatively low voltages." The reason for the limitation given in the last sentence of the above quotation will be discussed below. In Appendix D of the American Rules, the following table of the sparking distances in air between " opposed sharp needle-points " for sine-shaped voltage waves is given. The numbers are calculated from the experimental results given in a paper on the " Dielectric Strength of Air," by Professor Steinmetz, published in the Transactions of the American Institute of Electrical Engineers, vol. xv, p. 281. Under normal atmospheric conditions, with new sewing needles supported axially at the end of linear con- ductors which are at least twice the length of the gap, the maximum difference between the observed disruptive voltages and the values given in the table will probably be well within 5 per cent, of either voltage. Care must be taken that the potentials of the needles are always equal and opposite and that no foreign body is near the spark- gap. In practical work it is also important that a non- inductive resistance of about one-half of an ohm per volt should be inserted in series with each terminal of the gap DIELECTRIC STRENGTH 179 so as to keep the discharge current between the limits of one-quarter ampere and two amperes. The object of limit- ing this current is to prevent the surges of the voltage and current which might otherwise occur at the instant of breakdown. TABLE IV. SPARKING DISTANCES BETWEEN NEEDLE POINTS. Effective Kilovolts. Inches. Cms. Effective Kilovolts. Inches. Cms. 5 0-225 0-57 140 13-95 35-4 10 0-47 1-19 150 15-0 38-1 15 0-725 1-84 160 16-05 40-7 20 1-00 2-54 170 17'10 43-4 25 1-30 3-3 180 18-15 46-1 30 1-625 4-1 190 19-20 48-8 35 40 2-00 2-45 5-1 6-2 200 210 20-25 21-30 51-4 54-1 45 2-95 7-5 220 22-35 56-8 50 3-55 9-0 230 23-40 59-4 60 4-65 11-8 240 24-45 62-1 70 5-85 14-9 250 25-50 64-7 80 7-10 18-0 260 26-50 67-3 90 8-35 21-2 270 27-50 69-8 100 9-60 24-4 280 28-50 72-4 110 10-75 27-3 290 29-50 74-9 120 11-85 30-1 300 30-50 77-4 130 12-90 32-8 Failing cases in practice When spark-gaps between needle points are used to measure very low voltages very un- satisfactory results are obtained. Even when the electrodes are spherical it is very difficult to obtain consistent results when the distance between them is less than a millimetre. When the electrodes are at microscopic distances apart the above formulae cannot be applied in practice. G. M. Hobbs (Phil. Mag. [6], 10, p. 617) has shown that when the minimum distance x between the 180 ELECTRIC CABLES AND NETWORKS spheres is less than 3^, where ^ = 10~ 6 metre, the spark- ing potentials are practically independent of the nature of the gas between the electrodes. They depend, however, on the metal of which the electrodes are made. When the electrodes are very close together, it has to be remem- bered that our assumption of an isotropic medium bounded by smooth rigid equipotential surfaces is no longer per- missible. If the surfaces were magnified sufficiently they would be seen to be rough, and the dielectric surrounding the microscopic projections would probably be ionized. In these circumstances, therefore, accurate calculations would be difficult. Hence, in determining dielectric strengths, it is necessary to have the electrodes at appreciable distances apart, and therefore high voltages, must be used. It is not safe to calculate dielectric strengths from the observed disruptive voltages when the electrodes are less than a millimetre apart. When a maximum inaccuracy of more than 1 per cent, is not permissible, they should be at least half a centimetre apart. It has also to be remembered that the formulae for the maximum value of the electric stress on the medium be- tween spherical electrodes have been obtained on the assumption that the Faraday tubes are in statical equili- brium. In the case of impulsive rushes of electricity (see Chapter XII), or with alternating pressures at exceedingly high frequencies, the disruptive voltages seem to be inde- pendent of the shape of the electrodes. The dielectric strength of a gas may be the deduced from experiments on the sparking strength of voltages between spherical electrodes. The containing vessel for the gas should be large with the spherical electrodes near the centre. The dia- DIELECTRIC STRENGTH 181 meter of the supporting rods should be small compared with the diameter of the electrodes, and care should be taken that no conducting materials or insulating materials having dielectric coefficients different from the gas are in the immediate vicinity of the electrodes, otherwise the distribution of the Faraday tubes between the electrodes will be altered and our formulae will not apply. It is usually best to earth the middle point of the secondary coil of the transformer, or the middle point of the batteries used, so as to make the potentials of the electrodes equal and opposite at the instant of discharge. If E/2 and E/2 be the potentials of the electrodes, at the instant of discharge, when direct voltages are used, we have where R max . is the dielectric strength of the gas, x the mini- mum distance between the electrodes, and / a number which can be obtained from Table I. The nearest points on the electrodes should not be closer than about half a centimetre, and their diameter should be about 5 cm. With air at atmospheric pressure a voltage slightly less than 20 kilovolts would be required when x was 0*5 cm. When alternating pressures are used it is absolutely necessary to know the ratio of the maximum voltage E to the effective voltage V. Let this ratio, which is some- times called the amplitude factor, be denoted by k, then our formula is Steinmetz's method of putting the electrodes into nitrate of mercury, and rubbing them with a clean cloth, before and during the experiments is to be commended. This is especially necessary when the electrodes are only a small distance apart. 182 ELECTRIC CABLES AND NETWORKS The pressure, temperature, and humidity of the gas must be given. J. N. Collie and W. Ramsay (Proc. Roy. Soc., vol. lix., p. 257, 1896) give interesting comparative values of the sparking potentials for various gases contained in glass tubes. The electrodes were of platinum with slightly rounded ends. Owing to the dielectric coefficient of the glass tube being different from that of the gas, and owing to the great electric stress at the electrodes causing ex- cessive ionization, absolute values cannot be found from their results, but the following table shows that the di- electric strengths of the gases differ considerably : Sparking Gas. Distances in mms. Oxygen 23 Air 33 Hydrogen 39 Argon ...... 45'5 Helium .... greater than 250. The dielectric strength of helium, therefore, is extra- ordinarily low compared with that of other gases. The liquid to be tested is generally placed Dielectric strength of in a vertical glass cylinder about 2 in. in dia- liquids meter. Spherical electrodes about half an inch in diameter are immersed in the liquid, and the distance between them is varied by means of a micrometer screw. The formulae for deducing the dielectric strength from the disruptive voltage are the same as for a gas. The electrodes should not be less than 0-3 of a centi- metre apart, and at this distance 40 or 50 kilovolts will be required to break down good insulating oils. In some cases when water is present much smaller voltages suffice. In order to find the true dielectric strength of an oil, it DIELECTRIC STRENGTH 183 is necessary to thoroughly dry it before the test. This can be done by letting hot air bubble up through it. It is inadvisable, however, to heat the oil above 100 C. as con- siderable discolouration often results and its physical state alters. When oils are dried in this way perfectly con- sistent results can be obtained. As a numerical example, let us suppose that the dis- ruptive voltage for an oil between 1 cm. spherical electrodes, 0*3 of a centimetre apart, is 28 kilovolts, V\ being equal to F 2 , and the amplitude factor being 1-5. By Table I, we get B wafl .=(l-5x28/0-3)xl-21 = 168 kilovolts per centimetre. If the spherical electrodes can be entirely Btrength^of em ^edded in the insulating material then we iS g*JJJj can proceed as for liquids and gases, the same formulae being employed. The method frequently adopted of putting thin sheets of the insulating material between metal electrodes in air is of doubtful value. As the voltage is increased the air surrounding the electrodes is broken down long before the disruptive voltage is reached. The insulating material heats excessively, and the maximum electric stress to which it is subjected cannot be calculated as the temperature is rarely uniform throughout, and the insulativity of the medium and the dielectric coefficient vary with the tem- perature. Results obtained by neglecting the variations of the electric stress due to temperature are useful only when all the conditions of the experiment are mentioned. Dielectric When the insulating material is eolotropic S eo e io?ropic f ^ e GS ^ CU ^^ OU f tne electric stresses is very solids difficult. They vary with the dielectric co- efficients and the insulativities of the various constituents, 184 ELECTRIC CABLES AND NETWORKS and, as we have just mentioned, these quantities vary rapidly with the temperature. Accurate measurements of the mean dielectric strength are therefore in many cases almost impossible. REFERENCES, H. W. Turner and H. M. Hobart, The Insulation of Electric Ma- chines. A. Schuster, " The Disruptive Discharge of Electricity through Gases." Phil Mag, vol. xxix, p. 192, 1890. C. P. Steinmetz, " Note on the Disruptive Strength of Insulating Materials." Trans. Amer. Inst. Elect. Eng., vol. x, p. 85, 1893. T. Gray, " The Dielectric Strength of Insulating Materals." Phys. Rev., vol. vii, p. 199, 1898. C. P. Steinmetz, "Dielectric Strength of Air." Trans. Amer. Inst. Elect. Eng., vol. xv, p. 281, 1898. C. E. Skinner, " Energy Loss in Commercial Insulating Materials when Subjected to High Potential Stress." Trans. Amer. Inst. Elect. Eng., vol. xx, p. 1047, 1902. E. Jona, " Dielectric Strength of Air, Oil and Various Liquids." Atti deW Associazione Elet. Italiana, vol. vi, p. 3. H. J. Ryan, " The Conductivity of the Atmosphere at High Voltages." Trans. Amer. Inst. Elect. Eng., vol. xxiii, p. 101, 1903. E. Jona, " Limits of High Tension Tests on Polyphase Apparatus." Elettricista, Rome, vol. 13, p. 113, 1904. A. Russell, " The Dielectric Strength of Air." Phil. Mag. [6], vol. ii, p. 258, 1906. A. Russell, " The Dielectric Strength of Insulating Materials." Journ. of the Inst. of Elect. Eng., vol. 40, p. 6, 1907. R. P. Jackson, " Potential Stresses as affected by Overhead Grounded Conductors." Trans. Amer. Inst. EL Eng., vol. 26, p. 435, 1907. H. W. Fisher, " Spark Distances." High Tension Power Trans- mission. St. Louis El. Cong., vol. 2, p. 91, 1904. H. J. Ryan, " Some Elements in the Design of High Pressure Insu- lation." High Tension Power Transmission. St. Louis EL Cong., vol. 2, p. 278, 1904. THE GRADING OF CABLES CHAPTER IX The Grading of Cables The grading of cables Concentric main Suitable dimensions for a concentric main Grading single core cables for alternating pressures Grading single core cables for direct pressures Jona's graded cables The effects of leakage currents on the grading of concentric mains Numerical example The British standard radial thicknesses for jute and paper dielectrics The effects of stranding on the electric stresses Conclusions References. The grading ^ Y ^ e grading of cables is meant the arranging of cables o f ^he or( j er anc [ thickness of properly chosen insulating wrappings so that each bears its due share of the total electric stress to which the insulating material is sub- jected. In addition, the electric stress on every wrapping must be as uniform as possible throughout its substance. We shall see that it is only possible to secure absolute uniformity of stress in a wrapping by making the dielectric coefficient of the insulating material diminish in a regular manner as we proceed outwards from the axis of the cable. In the special case of a single core main with a homo- geneous dielectric, the maximum electric stress -fi^. occurs next the core, and the minimum R min , next the lead sheath. If a be the radius of the core, the surface of which we sup- pose to be smooth, and 6 the inner radius of the lead sheath, we have fl M . = F/alog e (6/a), and jR min . = F/6 log e (6/a), where F is the potential difference between the core and the lead sheath. We see that in this case E max> /R min =b/a, "mm. 187 188 ELECTRIC CABLES AND NETWORKS and if b/a be large, the material next the core will have to withstand a stress much greater than the average stress, which equals V/(b a). The layer of insulating material, therefore, has to be made very much thicker than if it had merely to insulate from one another two infinite plane surfaces at the given potentials. If it were possible to make the electric stress on the dielectric of a single core main uniform throughout so that its value was V/(b a), the thickness of the layer would be considerably reduced, and a considerable economy could be effected by the smaller amount of sheathing and armouring required. We shall show how this can be done by using insulating materials having different dielectric coefficients and arranged in concentric wrappings of suitable thicknesses, and in a given order. Before doing this, however, we shall discuss the formula R = V / { xlog e (b/a) } for a concentric main, where R is the electric stress at points the distance of which from the axis of the cable is x. A proof of this formula is given in the author's treatise on Alternating Currents, vol. i, p. 95. Concentric ^he formula shows us that the value of R at a point in the dielectric is independent of the absolute values of the potentials of the mains, and depends merely on the difference of the potentials and the distance of the point from the axis. We obviously have If V and b remain constant V da - { lo g ,(&/)}* {1 - 10g ' (6/0)K Hence, if a be less than b/e where e is the base of Neperian logarithms, jR^ will diminish as a increases. In this case, we see that the breaking down of the dielectric round the inner core actually diminishes the maximum stress to which the dielectric is subjected. It is only when the radius of THE GKADING OF CABLES 189 the charred dielectric gets greater than b/e that a disruptive discharge ensues. Jona (Trans. Int. Cong., St. Louis, vol. ii, p. 550) describes an experiment on the disruptive voltages of two single-core cables of very different diameters, but each wound with the same thickness (1-4 cm.) of paper insulation. The core of one consisted of a thin wire O'l cm. in diameter, while the other was a copper cylinder 2-9 cm. in diameter. The former broke down at 40 kilo volts, and the latter at from 75-80 kilo volts. The former also got exceedingly hot after being subjected to 30 kilo volts for an hour, whilst the latter was still cold after 50 kilovolts had been applied for the same time. If we calculate the maximum electric stress on the dielectric surrounding the thin wire, on the assumption that no part of it is broken down before the disruptive discharge ensues, we get ""**" 0'05log e (145/0-05) = 238 kilovolts per centimetre. Similarly, the experimental results with the thick cable make -R^oa.. lie between 76-5 and 81*6 kilovolts per centi- metre. This experiment is quoted by Jona to show that the ordinary formula cannot be applied when b/a is large. If, however, we assume that the disruptive discharge does not occur until the outer radius of the charred dielectric becomes equal to b/e, the experiment on the thin wire gives us _40 x 2-718 **.- 145 =75 kilovolts per centimetre, nearly, which, being in substantial agreement with the results given by the test on the thick cable, is a striking confirma- tion of the theory outlined above. 190 ELECTRIC CABLES AND NETWORKS Let us suppose that the maximum working dimensions voltage F, the density of the current in the concentric inner conductor, and the maximum permissible stress to which the dielectric may be subjected, are fixed. Let us first suppose that the inner cylindrical conductor is solid and that its radius is a. If, then, V/l be the maximum permissible stress, we have F F alog e (b/a) I ' and thus, b = ae l/a . Hence also, da \ a If, therefore, a be greater than I, d b/d a is positive, and therefore b increases as a increases, but if a be less than Z, b diminishes as a increases. In the latter case it Would obviously be advantageous to make the inner conductor hollow, its section remaining constant, so as to increase the value of a and diminish the value of b. The quantity of armouring and insulating material used would be diminished by this procedure. We conclude, therefore, that if a solid inner conductor of the required cross-section would have a radius less than I, the inner conductor should be made hollow and its outer radius should not be less than I. In some cases it would be advantageous to make the inner conductor of aluminium. Although the inner radius of the outer conductor begins to increase when a gets greater than I, the following reason- ing shows that the quantity of the dielectric required diminishes until a gets greater than 1-25 L Using the same notation, the area of the cross-section of the dielectric of the cable is TT (b 2 a 2 ), and we have to THE GKADING OF CABLES 191 find the value of a that makes a 2 (e 2l/a 1) a minimum. Differentiating with respect to a and equating to zero, we get e zlfa =a/(al). Let a = nl, then 2/n = log e nlog e (nl). By trial, we find that ft, = 1*2550 . . . satisfies this equation, and hence, when n has this value the quantity of insulating material required is a minimum. In this case a = 1-2551, 6=2'784Z, and b=2'2lSa. As the saving of insulating material effected by increasing a from Z to 1*25 I is only about 3 per cent, it is of little importance compared with the increased cost of the armouring. We conclude, therefore, that high-pressure concentric cables, having isotropic dielectrics, for use at a maximum voltage V, should be constructed so that b =ae l/a , where V/l is the maximum permissible working stress to which the dielectric may be subjected, and a should never be made less than I. We shall first make the supposition that all single core the insulating wrappings used have the same alternating dielectric strength, and that the maximum and minimum stresses to which they are subjected, when working, are to be the same for them all. We shall also suppose that the leakage current across the dielectric can be neglected in comparison with the capacity current. Let us suppose that there are n insulating wrappings the inner radii of which are a, r 2 , r 3 , ... r n , respectively, where a is the outer radius of the lead tube encasing the inner core, and let b equal the inner radius of the lead sheath. Since the ratio of the maximum to the minimum electric stress is to be the same in all the wrappings, we must have r 2 r 3 b 192 ELECTRIC CABLES AND NETWORKS We see, therefore, that the radii should be in geometrical progression, the common ratio being (b/a) l/n . The thick- nesses of the layers also form a geometrical progression having the same ratio (b/a) yn . Let Fi, F 2 , . . . F w+1 , be the potentials of points at dis- tances a, r 2 , . . . b from the axis of the cable. Then, since the layers form n condensers in series, the potential difference across a layer will be inversely proportional to the capacity of the layer, and thus we have Fi F, F 2 -F 3 F n -F n+1 (lAi)log.(r a /a) (l/X a )log e (r 3 /r a ) UA n )log e (&/rj' Hence, since the P.D.s are all in phase, each of these ratios equals V{S(l/*>J log e (r n+1 /rj} 9 where F is the voltage applied between the core and the sheath. If R m denote the maximum electric stress on the rath layer, we have -p _ ' F - F ' m v m + 1 =( V/\ m r m )/S(l/\ m )log e (r m+l /r n ). Now, since the maximum stress on every layer is to be the same, we must arrange so that \ 1 a Therefore a Hence \i, X 2 , ... \, are the terms of a geometrical progression whose common ratio is (b/a) 1/n -. If Rmax, denotes the maximum electric stress in the graded cable, we have -^1- ~ a li b/ct 1 T /7 / \ 1/n _ 1} log (6/ft) = R f max b/al THE GKADING OF CABLES 193 where R' max . stands for V/a log (b /a) the maximum stress in a cable of the given dimensions with an isotropic dielectric. If R min , denote the minimum electric stress in the dielectric of the graded cable, We have In the ideal cable n would be infinite, and thus the stress would be the same at all points, and would equal V/(b-a). The capacity per unit length of a single-core cable, with isotropic dielectric equals X/ { 2log e (b/a) } . The capacity of the graded cable equals \ 1 n{(b/a) ln 1 }/{(b/a l)21og e (6/a)}, When n is infinite this equals X 1 a/{2(6 a)}. If X be the dielectric coefficient of the cable with the isotropic dielectric, and X max be the dielectric coefficient of the inner coating of a graded cable having n layers, the capacities of the cables will be equal if >W =X(6/*-l)/ { (&/)' "-1 } . If the value of X,^ be less than this, the capacity of the graded cable will be the smaller. For example, if there are 4 layers and b/a equals 3, we find that the capacities are equal when X maa . 1-58 X. In this case the minimum value of X in the graded cable is 0-69 X. To illustrate how the value of the maximum electric stress diminishes as the number of wrappings is increased we shall work out a few numerical examples. (i) Two wrappings (n=2) b/a 2 3 4 5 R mas /R min graded dielectric . 1-414 1-732 ; 2 2-236 X\i m x ./R/ min . isotropic dielectric . 2 3 4. 5 J^max yil'max 0-828 0-732 ! 0-667 0-618 Per cent, increase of the per- missible voltage due to grading 21 37 50 62 194 ELECTRIC CABLES AND NETWORKS (ii) Three wrappings (n=3) b/a 2 3 4 5 Rmax./ R mh,. graded dielectric . T260 1-442 1-587 1-710 R'max /R'min isotropic dielectric . 2 3 4 5 R /R' 0-780 0-663 0-587 0-532 Per cent, increase of the per- missible voltage due to grading 28 51 70 88 (iii) Four wrappings (n = 4) b/a 2 3 4 5 Rmax./R m in. graded dielectric . 1-189 1-316 1-414 1-495 R'max./R'min. isotropic dielectric . 2 3 4 5 R max /R'jnk 0-756 0-632 0-552 0-495 Per cent, increase of the per- missible voltage due to grading 32 58 81 102 (iv) Ideal uniformly graded cable (r&=infinity) b/a 2 3 4 5 R max./R'max. 0-693 0-549 0-462 0-402 Per cent, increase of the per- missible voltage due to grading 44 82 116 149 We have assumed above that the dielectric strengths of all the insulating wrappings are the same. If, however, the dielectric strengths are known accurately and are not all the same, another solution may be preferable. If R m be the safe working stress for the mth layer, we have Since it is advisable to make the ratio R m(K c./Rmin. the same for all layers, the ratio r m+l /r m will be constant, and as before r 2 , r 3 , ... r n will be the n 1 geometrical means between a and b. The above equation shows that we must have THE GKADING OF CABLES 195 Since a, r 2t ... r n , are in an ascending order of magnitude, X t EI, A 2 E 2 , ... \E n , must be in a descending order. We see, therefore, that it is necessary to put the wrappings whose constants are (X, E) over the wrapping whose con- stants are (X', R'), if X E be greater than X' R', even although X' may be less than X. When, however, the main object we have in view is to make, at all costs, the factor of safety of the cable as high as possible, it is, in general, advisable to put the insulating material having the greatest dielectric strength in contact with the core, and, if possible, grade the dielectric by using outer layers having smaller dielectric coefficients. At the moment of switching on the direct Grading single core pressure, the distribution of the electric stresses cables for direct depends on the dielectric coefficients of the pressures wrappings, but after a few seconds, when the leakage current attains its steady value, the values of the stresses depend on the insulativities of the various wrappings. If the pressure be always applied gradually at the start we may neglect the dielectric coefficients and grade the cable for the steady pressure taking the insulativities only into account. Let o-j, o- 2 , ... be the insulativities of the various wrappings and C the leakage current, then (p. 51) the resistance of the mth cylinder to the flow of the current C across it is Hence we have, c=, F - F = (o- 1 /2?r)log (r 2 /a) (o- 2 /27r)log e (r 3 /r 2 ) and therefore, y n Y n+l 196 ELECTRIC CABLES AND NETWORKS Also for the mth layer d v For reasons stated above, we choose the radii of the bound- aries between the wrappings so that they are the n 1 geometric means between a and b. Hence, if the factor of safety is to be the same for all the layers, we must have r m R m /i and < 2 denote respectively the phase- difference between Fi and F 2 and the effective value of *. Then if / be the frequency, and a>=2 TT/, we have tan i = uKiRi =/\i, it is easy to see that FI increases with w when X 2 o- 2 is greater than \!o-i. In this case, the electric stresses in the medium next the inner conductor increase as the frequency increases, and the stresses in the outer medium diminish. We see from (B) that, when X 2 <7 2 is greater than X^i, Fi/F has its minimum value when o> is zero, that is, with steady pressures, and its maximum value when a>is infinite, that is, with an alternating voltage of very high frequency. Similarly when X 2 o- 2 is less than X^i, Fi/Fhasits maxi- mum value E i /(E i -\-E 2 ) with steady voltages, and its mini- mum value K 2 /(K L -\-K 2 ) with alternating voltages of very high frequency. 202 ELECTRIC CABLES AND NETWORKS Numerical Let us assume that the radius of the inner example con d uc t O r is 1 cm. (a = l) the radius of the boundary 1-5 cm. (r 1-5), and the inner radius of the outer conductor 2*25 cm. (6=2-25). Let us also assume that for the outer jute wrapping, cr 2 = 10 12 , X 2 =2, and that for the vulcanized rubber inner wrapping, o-^lO 16 , Xi=4. If the direct voltage applied to the conductors be 30,000, then, putting a>=0 in (1), we find that F! =30,000, and F 2 =0, very approximately. Thus practically all the electric stress comes on the rubber. Let us now suppose that an alternating pressure of very high frequency is applied between the conductors. In this case, putting o> equal to infinity in (1), we get Vi__ X 2 log e (fr/V) J_ V Xilog t (r/a)+Xilog.(^A) 3' and thus, Fi is 10,000 volts and F 2 is 20,000. Hence, as the frequency increases from to infinity, FI diminishes from 30,000 to 10,000 volts, and F 2 increases from to 20,000. From (A), we see that tan/>!=2 x!0 5 //9, and tan < 2 = 10//9. Hence, at ordinary frequencies, the error made by assuming that $j and 2 are both 90 is small. If / is greater than 9, Fi/F 2 is nearly equal to 1/2. In practice, therefore, we see that in the case considered the maximum pressure across the outer layer with alter- nating pressures may be very much larger than when a direct pressure is applied between the conductors, the value of which equals the maximum value of the alternating pressure between the conductors. On the other hand, the electric stresses on the inner dielectric may be much less with the alternating pressures. THE GBADING OF CABLES 203 The nominal area of the cross-sections of the conductors and the radial thicknesses (b a) of the dielectric for concentric cables given in the following table are taken from a report issued by the Engineering Standards Committee (E.S.C.) in August, 1904 (p. 8) The British standard radial thicknesses for jute and paper dielectrics 660 Volts. 11, 000 Volts. 8. a. b - a. R b -a. Rm sq. in. 0-025 in. 0-089 in. 0-08 K.V. per mm. 0-64 in. 0-35 K.V. per mm. 4-3 0-050 0-126 0-08 0-59 0-35 3-7 0-075 0-155 0-08 0-57 0-35 3-3 o-ioo 0-178 0-09 0-50 0-36 3-1 0-125 0-199 0-09 0-49 0-36 3-0 0-150 0-219 0-09 0-49 0-36 2-9 0-200 0-253 0-09 0-48 0-36 2-7 0-250 0-282 o-io 0-43 0-37 2-6 In the above table, 8 represents the cross-sectional area, a the radius of the cylindrical conductor whose cross- sectional area is S, b a the thickness of the dielectric given by the E.S.C., and E m the maximum working electric stress when the amplitude factor of the applied alternating pres- sure is \/2. It will be seen that the electric stresses on the dielectric are very different in the high-pressure cable from what they are in the low-pressure cable, and the dielectrics in cables of different sizes are subjected to appreciably different stresses. In the first five of the high-pressure cables, the dielectric surrounding the high-pressure conductor will begin to be broken down before the disruptive discharge takes place, because in these cables the ratio of b /a is greater than e 204 ELECTRIC CABLES AND NETWORKS (2-718). The specified thicknesses, therefore, are not economical. Take, for instance, the main in which the nominal cross-sectional area of the conductor is 0-025 sq. in. With a solid cylindrical conductor a equals 0-089 in., and b is, therefore, equal to 0-35+0-089 =0-439 in. Thus &/a=: 4-92. If we make the inner conductor hollow and a =0-1 42 in., b =0-3865 in., we get the same maximum stress on the dielectric, but its thickness has been reduced by 33 per cent, and the outer radius by 12 per cent. As the armouring, etc., would also be substantially reduced, the cable would be less costly. If we merely kept b =0-439 in., but increased a to 0-1616, so that b/a=e nearly, then the carrying capacity of the cable would be nearly quadrupled, the thickness of the dielectric diminished 20 per cent., and the maximum electric stress would have been reduced to 3-8 kilo volts per millimetre. The fact that the dielectrics of cables are not quite isotropic is sometimes advanced as a reason for making the radius of the inner conductor smaller than the value indi- cated by theory. This practice, however, is founded on a misapprehension, as the effect of diminishing the radius is to increase the electric stress, and there is no reason why dielectrics of heterogeneous substance should be subjected to greater stresses than those of homogeneous substance. The want of isotropy may possibly be a reason for increasing the diameter of the inner conductor, the thickness of the insulating covering remaining the same. In designing cables, it has to be remembered that the insulation resistance, the capacity, the electric stresses, and, as we shall see in the next chapter, the thermal con- ductance have to be considered. For low-tension cables, the insulation resistance must be comparatively high, and hence, it is not possible to use too small a value of b/a. THE GRADING OF CABLES 205 Similarly, for high-tension cables, although a small value of b/a makes the electric stresses small and increases the thermal conductance yet it makes the capacity large, and the consequent large capacity current may be a serious drawback in practical work. The effects In order to simplify the formulae for the of strand- , , . , ing on the electric stress, we have assumed that the inner stresses conductor is a smooth cylinder. In practice, the inner conductor is nearly always stranded, and it is necessary therefore to consider the effect of the stranding. Owing to the greater curvature of the surface of the strands, we can see, from first principles, that the effect will be to increase the maximum stress. Jona found experimentally that the brush discharges from solid wires and stranded or braided wires having the same external size begin at prac- tically the same voltages. Hence we may infer that the stranding of the conductor does not much affect the dielectric strength of the cable. It is important, however, to be able to calculate the stress exactly, and this can be done by means of a formula due to Professor Levi-Civita (vide Jona I.e. ante). The formula is given in terms of Gauss's hypergeometric series, but Jona has computed these series for useful values of the variables, so that approximate solutions can be readily obtained. The results show that the effect of the stranding is generally to increase the maximum stress on the inner dielectric by about 20 per cent. It is worth while, therefore, to prevent this increase in the stress on the inner wrapping by making the sur- face of the conductor smooth. This can be done by covering, as Jona does, the inner conductor with a thin lead tube. For extra high-pressure cables the gain in the strength is well worth the slight increase in the cost of the cable. 206 ELECTRIC CABLES AND NETWORKS We may sum up the results arrived at in Conclusions this and the preceding chapter as follows. (1) When part of the dielectric under stress breaks down, a disruptive discharge ensues only when the effect of this partial breakdown is to increase the electric stress on the remaining portion. (2) The dielectric strength of air under given conditions can be found accurately by finding the disruptive voltages between spherical electrodes at distances greater than 0-5 of a centimetre apart. Under normal conditions it is about 3-8 kilovolts per millimetre. (3) The dielectric strength of other gases can be found in a similar way experimentally by the -help of the tables given on p. 176. The dielectric strengths of the mon- atomic gases helium and neon are small. (4) The dielectric strength of oils can be found by noticing the disruptive voltages between spherical electrodes im- mersed in them, provided that the distance apart is greater than 0-3 of a centimetre. An excellent way of drying oils is by letting heated air bubble through them. (5) In finding the dielectric strength of solids it is ad- visable, when possible, to embed the spherical electrodes in the material under test. (6) High-pressure concentric cables having an isotropic dielectric, for a maximum working pressure F should be constructed so that b=at l/a , where V/l is the maximum permissible working stress to which the dielectric may be subjected, b is the inner radius of the outer conductor, and a is the outer radius of the inner conductor. The smallest permissible value of a is I. When the core is stranded it should be encased in a thin lead tube. THE GRADING OF CABLES 207 (7) With a composite dielectric subjected to alternating pressure, the P.D.s across the layers are usually out of phase with one another. It is only in a limited number of cases, however, that the increase of the stress due to this cause has to be considered, as the leakage currents are usually negligibly small in comparison with the capacity currents. (8) The effects of alternating and direct pressures in producing stresses in the dielectric are sometimes quite different. (9) High-pressure cables for alternating or direct current circuits should be graded so as to make the maximum electric stress on the dielectric as small as possible, and stranded conductors should be encased in thin lead tubes. REFERENCES. Mervyn O'Gorman, "Insulation on Cables." Journ. Inst. EL Eng., vol. xxx, p. 608, 1901. E. Jona, " Insulating Materials in High Tension Cables." Trans. Int. Congress. St. Louis, vol. ii, p. 550, 1904. A. Russell, " The Dielectric Strength of Insulating Materials and the Grading of Cables." Journ. Inst. El. Engin., vol. 40, p. 6, 1907. THE HEATING OF CABLES CHAPTER X The Heating of Cables The heating of cables Temperature gradient in a concentric main Numerical example Effects of heat on the dielectric Effect of the temperature gradient on the electric stress The thermal conductance of a single core main Numerical example The temperature in the substance of conductors carrying uniform currents The thermal conductance of a concentric main The thermal conductance of polycore cables The heating of bare conductors References. The heating ALTHOUGH the rise of temperature in under- of cables ground cables is a problem of considerable practical importance, yet very little information on this subject is available. It is usual to consider that a cable of given dimensions is carrying its full load when the current density in the core has a given value. As a rule the effect of the thermal conductivity of the insulating wrappings is not taken into account. The value of this physical constant, however, determines the difference of temperature between the core and the sheath, and hence the temperature, and consequently the electrical conductivity of the copper must be considerably affected by the value of the thermal con- ductivity of the dielectric. For high-pressure cables also the thermal gradient in the dielectric affects, in some cases very seriously, the dielectric strength. We shall therefore briefly consider the laws governing the flow of heat across the dielectric of cables. 211 212 ELECTRIC CABLES AND NETWORKS Temperature When a concentric main is carrying a current, *f*^fji! I n the temperature of the dielectric is not uniform i concen- trie main ow i n g to the heat generated in the inner con- ductor. If the dielectric is isotropic, the temperature at any point after the flow of heat has become steady can be readily written down, if we assume that the thermal con- ductivity k of the dielectric remains approximately constant over the range of working temperatures. i If 6 be the temperature at all points at a distance r from the axis of the main, we have, since the heat entering per second an elementary cylinder of the dielectric, coaxial with the main, must equal the heat leaving it, , d r \ dr J neglecting the flow of heat near the ends parallel to the length. Hence d 0__ __ A ~dV~ ~T y where A is a constant. We have, therefore, where 2 is the temperature of the outer conductor, the inner radius of which is b. Let us suppose that the inner conductor, supposed of copper, is solid and of radius a, and that i is the current density in it. Then, if p be the volume resistivity of the copper in ohms, we have 4-2 x ZTra j dr and thus, A = a 2 i 2 p/(8-k). Hence 6 = 2 +( a 2 i 2 p/S-4: k) log (6/r), and 61 0. 2 = (a*i V/8-4 k) log e (6/a), where is the temperature of the surface of the inner conductor. THE HEATING OF CABLES 213 We have assumed above that the thermal conductivity k of the dielectric does not vary appreciably with the tem- peratures likely to occur in practice. C. H. Lees (Trans. Roy. Soc., p. 433, vol. 204A, 1905) has proved that this assumption is permissible for paraffin wax, glycerine, and various other insulating materials. There appears to be a slight tendency, however, towards lower conductivity as the temperature increases. G. F. C. Searle (Proc. Camb. Phil. Soc., xiv, 2, p. 189, 1907) has devised an exceedingly simple method of determining the thermal conductivity of rubber, the value of which he finds to equal 0*0004 nearly. Numerical To illustrate the values of #1 2 likely to occur in practice, let us suppose that 6 = 1*649 cm., and a = l cm. Let us also suppose that the current density i is 150 amperes per sq. cm., that p=l'S xlO~ 6 and that k =0-0006. The author has no trustworthy data with reference to the conductivities of the dielectrics used in actual cables, and so he takes the value of Tc for paraffin wax, which has been found accurately by Lees (I.e. ante). Substituting in the formula, we get e _ (150)2 x 1-8 xlO- 6 1 8.4x0-0006 =4 C. nearly. It is easy to see from the formula for 2 , that for a given value of b and for a given current density, the difference of temperature between the inner and outer conductors is a maximum when a = b/ v7 = 6/1-649 = 0-6065 6, which is the case we considered. We see, therefore, that the difference of temperature between the inner and outer conductors is probably not greater than 10 deg. in the most unfavourable circumstances. 214 ELECTRIC CABLES AND NETWORKS It is known that the dielectric coefficient and Effects of heat on the the electric insulativity of an insulating material vary rapidly with the temperature. Jona (p. 207) mentions a case where a rise of temperature of 20 C. made the insulation resistance of a paper insulated cable fall to one- thirtieth of its original value, and even more striking instances could be given. The table, also, quoted in the last chapter (p. 196) for oven-dried cellulose shows that the dielectric coefficient varies from 6-7 to 7 -5 as the temperature rises from 20 to 70 degrees Centigrade. In most cables it is noticed that the capacity current increases and the insulation resistance diminishes as the temperature rises. The rise of the capacity current is probably generally due to the increase in the value of the dielectric coefficient. We see therefore that in practice, since the temperature of the dielectric is not uniform, both p and X vary, even when the insulating covering is made of homogeneous material. We shall now consider what effect this has on the electric stresses in the insulating material. Let us suppose that a steady pressure E is the applied across the inner and outer conductors temperature f . . . , . . ... , gradient of a concentric mam having an isotropic dielec- eiectric trie. The momentary stresses set up initially are the same as if the resistivity were infinite. Now imagine that the dielectric is split up into an infinite number of concentric cylindrical tubes, the material of each tube being at the same temperature. Since these tubes form condensers in series between the conductors, the quantity of electricity per unit length in each condenser will be the same, and thus X - - dv = constant, where X is the value of the dielectric coefficient at a distance THE HEATING OF CABLES 215 r from the axis, and v is the potential at the same distance. Hence _ dv _ A ~^dr~ \r 9 where A is a constant. Now X diminishes as the tempera- ture diminishes, it therefore diminishes as r increases. We see, therefore, that the effect of A, varying with the tem- perature is to make the electric stress on the dielectric more uniform. If we assume that X varies with temperature according to the linear law, we may write = X {l+log e (6/r)}, where B = a a 2 i 2 p /&-4Jc. It readily follows that the electric stress is a minimum where r=be l ~ B)/B . In practice, B is very small compared with unity, and hence the electric stress diminishes as we pass from the inner to the outer conductor. Let us now suppose that the direct pressure E has been applied sufficiently long to make the electric stresses and the leakage currents assume their steady values. In this case, by Ohm's law, duff log e (&/a) where B is the temperature of the dielectric at points distant r from the axis of the cable, B and 6 2 the temperatures of the core and sheath respectively, and Q is the thermal flow per unit length in calories per second. Hence the thermal conductance per unit length for a single-core cable is 27rk/log e (b/a)-. It is therefore equal to 47T&/A, times the corresponding electrostatic capacity. If ri be the electric resistance of the conducting core per unit length and C the current flowing in it, then, when the thermal flow attains its steady value, Q=C7 2 ri/418, and thus 6 Q 2 can be readily found if k be known. 218 ELECTRIC CABLES AND NETWORKS Numerical -^et us assur &e that the diameter of the con- examples d uc tor of a single-core main is 1 centimetre, and that a current of 100 amperes is flowing in it. Let us also assume that 6/a=2'718, and that k is 0-0004. In this case, n will be 0-2xlO~ 5 ohms approximately, and thus Q= 2 r t /4-18 =0-02/4-18 =0-004785. Hence since we readily find that 0! 2 = l-9 C. nearly. If we suppose that k is greater than 0-0004 the difference of the temperatures will be less than this. The thermal We shall now consider a concentric main. conduc- tance of a Let the conductivity of the isotropic dielectrics concentric main between the two conductors and between the outer conductor and the lead sheath be & t and k 2 respectively. Then if 0^ ,0 2 , and 6 3 be the temperatures of the two con- ductors and the sheath, we have Q/2=27rk i (e i a )/log e (6/a), and where Q/2 is the heat generated per unit length in each conductor per second, and c and d are the radii of the outer dielectric. Hence Q 27T If we can write b = c, and ki = k 2 , without appreciable error, we get 2rf 1 /log e (d/v/i&) for the thermal conductance. The problem of the thermal conductance of a The thermal conduc- polycore cable is much more difficult than that poiycore of the single-core cable. When the dielectric is isotropic, formulae for the electrostatic capacity between the cores in parallel and the sheath are given in THE HEATING OP CABLES 219 Russell's Alternating Cur- rents, vol. i, chap. v. The corresponding ther- mal conductances can be at once deduced from these formulae by multi- plying the capacities by 47T&/X. In a three-core cable, for instance, of a certain " clover leaf " pattern (Fig. 55), if Q be the heat generated in the three cores, per unit length, we have Q . FIG. 55. The section of a Three-core Main of which the thermal conduc- tance can be accurately calculated. c 3 )]' exactly, where R is the maximum inner radius of the lead sheath and b and c are the maximum and minimum dis- tances of points on the cores from the axis of the cable. By using Kelvin's method of images, it may be shown that if the centres of the n cores are symmetri- cally situated on a circle of radius a (Fig. 56), and if the cross-sections of the COr6S ar6 Sma11 Cir " FIG. 56. Section of another ideal Three core Main, of which the thermal con- cles of ductance can be calculated approxi- mately. have, radlUS T, WG 220 ELECTRIC CABLES AND NETWORKS Q approximately, where Ris the inner radius of the lead sheath. When n = 3 this formula agrees with that given above, provided that r/a and (a/E) 3 can be neglected compared with unity. It is to be noticed that all the logarithms given in this chapter and the preceding two chapters are Neperian. Their values are best found directly from Bottomley's Tables. They may also be found with an accuracy sufficient for practical purposes by multiplying the corresponding ordinary logarithms by 2-3. In many cases they may be computed easily, without tables, by means of the formula where ={(&/<) 1} {(&/)+!}. For instance log. 1-5= 0-4+0-016/3+ ... 0-405. We shall conclude this chapter by discussing The heating of bare the variation of the temperature of a bare cylindrical conductor, through which electrical currents are passing, when suspended horizontally. We have seen that no appreciable error is made by the assump- tion that the temperature of the conductor is uniform over its cross-section. We shall make the further assumption that the rate at which the wire radiates heat is proportional to its surface and to the difference of temperature between the wire and its surroundings. Hence for rises of tem- perature greater than about 50 C. our results are only rough approximations. The temperatures of the conductor also will not be the same during a gale or when it is raining as they would on a calm dry day. The formulae, however, THE HEATING OF CABLES 221 indicate how the temperatures vary with the dimensions of the conductor. Let us first suppose that when the switch is closed a current C flows in the conductor, and that this current is maintained constant. Let p be the volume resistivity, D the density, h the emissivity for heat of the surface, c the specific heat, / the length, and r the radius of the con- ductor. If 6 be the temperature of the wire, at the time t after the switch has been closed, we have since the electric power, expressed in calories per second, being expended in the wire equals the rate at which heat is being radiated from the surface together with the rate at which it is being stored in the substance of the conductor. Assuming for the present that p is constant, we see that the solution of the above equation is 0=^(16-^), where = -239(7 2 /?/(2/br 2 r 3 ) and m=2h/Drc. The final temperature of the wire is 6 which varies directly as C 2 p and inversely as hr 3 . Let us next suppose that the voltage V at the terminals of the wire is maintained constant. The equation now becomes and thus, where, Hence the final temperature varies directly as (V 2 /p) and r, and inversely as hi 2 . The greater the value of m the more rapidly does the temperature rise to its steady value. The greater the value of the emissivity, therefore, and the 222 ELECTRIC CABLES AND NETWORKS smaller the value of the density, the radius, and the specific heat, the more rapid will be the rise of temperature. Let us now suppose that p varies with the temperature. In this case we may assume that p = p (I+a6) where a is a constant. When the current is constant the temperature equation now becomes and hence, 0=041 IT**), where t = 0-239CV /(2for a r 8 0-239(7 z Po a), and ra = (2hr-Q&39C*p a/ir*r*)/Dre. Again when the applied voltage is constant, noticing that l/p=l/p aO/p approximately, we have 0-239 V*(7rr*/ Po l) = { 27rr^+0-239F 2 a(7rr 2 //> } +D7rr 2 lc(d0/dt), and thus 6' = 0-239 V 2 r/(2hl 2 p +0-239 F 2 r), and m' = {2^-f 0-239F 2 a(r/^ 2 )} /Drc, and the temperature at any instant is given by REFERENCES. G. Mie. " tiber die Warmeleitung ineinem verseilten Kabel." Elek- trotechnische Zeitschrift, vol. 26, p. 137, 1905. R. V. Picou. " Capacite et BchaufTement des Cables Souterrains." V Industrie Electrique, vol. 15, pp. 245 and 281, 1906. A. Russell. " The Dielectric Strength of Insulating Materials." Journ. Inst. EL Engin., vol. 40, p. 6, 1907. A. E. Kennelly. " The Heating of Copper Wires by Electric Cur- rents." Amer. Inst. Elect. Engin. Proc., vol. 26, p. 39, 1907. G. F. C. Searle. " A Method of Determining the Thermal Conduc- tivity of 'India Rubber." Camb. Phil. Soc. Proc., vol. xiv., Part II, 1907. ELECTRICAL SAFETY VALVES CHAPTER XI Electrical Safety Valves Electrical safety valves Intermittent safety valves Siemens and Halske horn arrester The Seibt safety valve Multiple gap lightning arresters Pressure safety valves on a 3-phase line Continuous arresters Electrolytic arresters References. IN practical working, the machines, apparatus, Electrical safety and cables, used for high pressure networks are often subjected to abnormal stresses owing to a sudden rise of pressure. This rise of pressure may be due to atmospheric electricity, to resonance due to a har- monic of the applied E.M.F. wave having the same period as a free period of vibration of the system, or to electro- mechanical resonance between the prime mover and the oscillating electrical energy. It may also be due to resonance of the high frequency oscillations often set up when an arc occurs at a short circuit. Hence excessive rises of pressure can occur on a direct current network when an arc giving rise to Duddell currents exists on any part of the circuit. The impulsive rush of electricity also, at these high frequencies, often causes momentary disruptive dis- charges, making pinhole marks, through the dielectric at places where the rush meets with inductive resistance. Between the first two turns, for instance, of a coil in the circuit or at a sudden bend of the conductor. The most frequent cause of breakdown is due to the oscillating arc, 225 Q 226 ELECTRIC CABLES AND NETWORKS but the most disastrous occur when the period of one of the free oscillations of a network has the same value as a harmonic of low order of the impressed oscillations. In this case the amplitude of the pressure between two points may reach enormous values, and considerable power may be expended where a breakdown occurs. To prevent the breakdown of the cables from these causes, or of the insulation of the armatures of the dynamos and other appliances in the network, electrical " safety valves " must be provided. These devices may be classed under two main heads, (1) intermittent safety valves, that is, devices which only act when the pressure exceeds a certain critical value, and (2) continuous safety valves which are always in operation. The first type acts by providing a safety path for the oscillating charge when its value gets excessive. The second type acts by conduction. It pre- vents the accumulation of an excessive charge by allowing it to leak away by a path of small resistance which is always in circuit. In the usual types of intermittent safety Inter- valves, the two electrodes are separated by a mittent safety suitable dielectric which is usually either air valves or oil. One electrode is connected with one main, and the other with another main of different polarity. When the difference of the pressure between the two ex- ceeds a certain value the dielectric breaks down, and the ensuing arc having a very small resistance, the pressure between the mains to which the device is joined cannot attain a high value. The ensuing arc is broken in several ways, some of which are described below. If one of the electrodes is connected with a main, and the other with an earth plate, the device is generally called a lightning arrester. A device used to limit the pressure ELECTRICAL SAFETY VALVES 227 Siemens and hora Ske arrester M can also be used as a lightning arrester. The breadth of the gap between the electrodes, and the design of certain auxiliary apparatus, however, is generally different in the two cases. The Siemens and Halske horn arrester 57) is perhaps the one most extensively used on power circuits. When the pressure between the main M and the earth E exceeds a certain value, an arc ensues between the narrowest portion of the gap between the two horns. It then rapidly travels upwards until its length gets too great for the voltage at its terminals when it automatically rup- tures. Although the rupture of the arc is accelerated by the convection currents of air yet it is partly also an electromag- netic phenomenon. If we invert the ar- rester, for instance, the arc travels downwards if it is started below the nar- rowest point (see A. Moore, Elect. Engineer, vol. 34., p. 520, 1904). The Oerlikon Company use a horn arrester with a non- inductive resistance in series with it. This is formed of nickeline wire immersed in oil and bent so as to be practic- ally non-inductive. The use of this resistance is to obviate the dangers arising from possible high frequency oscillations being set up at the arc, and also from oscillations being set FIG. 57. Horn Lightning Arrester. 228 ELECTRIC CABLES AND NETWORKS up by a sudden rush of current through a path of small resistance. The Allgemeine Elektricitats Gesellschaft have employed for the series resistances 150 watt, 150 volt glow lamps connected in series, a sufficient number being em- ployed to prevent any of them burning out and breaking the circuit. To prevent oscillations, and to provide a path of small resistance for the discharge, it is important to have the inductance of these circuits as low as possible. In practice, it is difficult to make the inductance of a wire circuit small enough, and hence carbon cylinders, water resistances, and even wet sand are sometimes employed. The function of these resistances is to get rid of most of the destructive energy contained in violent atmospheric discharges. W. H. Patchell (Journ. Inst. EL Engin., vol. 36, p. 97) has found that a spark-gap, with one electrode of copper and the other of carbon, in a glass enclosure is very effective as a safety-valve. The travelling of the spark upwards in the gap between the horns is accelerated by the chimney action of the enclosure, and many tests have proved that this type can be calibrated more accurately, and adjusted within narrower limits, than the ordinary open horn type with copper electrodes. A liquid resistance is used con- sisting of a solution of glycerine and water contained in earthenware vessels, but it has not proved altogether satis- factory as the values of the resistances are liable to change. The width of spark-gap employed is 4-5 mms. for 10,000^3" that is, about 5,800 volt working. A spark will jump the gap and start the arc at a pressure of 12,000 effective volts when the horns are clean and the atmosphere is normal. When these safety-valves were first installed in the City of London Works of the Charing Cross Company, the irregular times at which they acted attracted attention. ELECTRICAL SAFETY VALVES 229 To discover the cause a detector was extemporized for ex- perimental use. The primary of a small transformer was inserted in the earth wire of the spark-gap resistance. The secondary acted a relay which rang a bell and thus attracted the attention of the engineer. It was found that irregu- larities in starting a machine, although not sufficient to prevent easy synchronizing and switching in, were a frequent cause of spark discharges. An interruption in the supply due to a faulty insulator nearly always caused a discharge. It is probable, therefore, that the rises of pressure in the FIG. 58. Seibt Lightning Arrester. network were due to the superposition of the free oscilla- tions set up by the disturbance on the normal oscillations. The indicating device has proved so useful that it has been adopted permanently. A time recorder could also easily be actuated by the transformer, the inductance of the primary of which can be made very small. A drawback to the use of the horn arrester The Seibt safety is the large factor of safety that has to be valve allowed in order to avoid unnecessary sparking. 230 ELECTRIC CABLES AND NETWORKS Hence it might easily happen that the excessive electric stress did serious harm before the valve acted. This difficulty is neatly surmounted in the Seibt safety valve (Fig. 58), by utilizing the well-known effect of ultra-violet radiation in lowering the value of the disruptive voltage required by an air gap. The primary P of a small trans- former is put in series with the line L, and the secondary 8 consists of many turns of fine wire. A vacuum tube T placed between the secondary terminals glows when high frequency oscillations are set up in the main, and the di- electric strength of the air in the spark-gap safety valve being lowered by the radiations from the vacuum tube, a disruptive charge takes place to the earth E, and thus the pressure is prevented from becoming excessive. The multiple gap lightning arrester (Fig. 59) Multiple . gap lightning is a type of arrester frequently used in power arresters . transmission circuits in America. Between the line M and the earth E there is a series of insulated con- Line M Series Air Gctps P Shunted Air G5. Model illustrating the laws governing the A Flash. On turning the handle of the Wimshurst machine a dif- ference of potential is gradually established between C and E. As soon as the potential gradient at a point of any of the conductors exceeds the dielectric strength of the air between the plates there will either be a disruptive spark between the conductor and <7, or there will be a brush dis- charge from the conductor. If there be only two model con- ductors on E, and if the centre of the small knob of one be closer to G than the centre of the large knob of the other, it will in general protect it, that is, the disruptive discharge will take place between the smaller knob and (7, even when the minimum distance between the small knob and G is very appreciably greater than that between the large knob and G. It is found that the resistance of the supporting pieces connecting the knobs with the lower plate has very little LIGHTNING CONDUCTORS effect on their liability to be struck. For instance, when the stands are of metal and a damp cloth is placed between the stand of the small knob and the lower plate, the small knob is still struck even although the resistance of its connexion with the lower plate is hundreds or thousands of times greater than that of the larger knob. With the arrangement shown in Fig. 65, model pointed conductors are so effective in dissipating the charge that it is almost impossible to obtain a spark at all when they are used. If a lighted gas burner be placed on the lower tray FIG. 66. Type of B Flash. The A Flash between the clouds causes the B Flash to the earth. it will as a rule protect the knobs, the spark readily passing to the flame through the heated products of combustion. If the top tray be replaced by a sieve into which water is poured, it is impossible to obtain a spark at all. The B flash which is caused by an impulsive rush of electricity occurs when the difference of potential between the cloud and the earth is established almost instantaneously. There are several varieties of this flash. In Fig. 66, for example, a discharge between two clouds alters by electrostatic induction the potential differ- The B flash 252 ELECTRIC CABLES AND NETWORKS ence between another cloud and the earth, and this voltage being greater than the air can withstand, we have a B flash between the cloud and the earth. An experimental illustration of this flash is shown in Fig. 67. As in Fig. 65, C and E are two sheets of metal represent- FIG. 67. Model illustrating the action of the B Flash. ing a cloud and the earth, L and M are two Ley den jars which we suppose to be placed on a badly insulating wooden table. Their inner coatings are connected with the discharge knobs D of a Wimshurst machine W, and the outer coatings are in metallic connexion with C and E. On turning the handle of the Wimshurst machine, the inner coatings are brought to a high difference of potential, and there will be large electrostatic charges induced on the out- side coatings of the jars at these high potentials. When a spark occurs at D the potentials of the inner coatings will be equalized, probably by an oscillating discharge, and the potential difference between the outside coatings of L and M , owing to the large charges on them of equal and opposite sign, will attain a high value. Hence also the potential difference between C and E (Fig. 67) which are in metallic LIGHTNING CONDUCTORS 253 connexion without the outside coatings will be high, and if the height of G above the model lightning conductors be not too great there will be a spark discharge. Before the spark occurs at D, the potential difference between the plates is very small, as each is practically at earth potential, for the inductive effects produced by the equal and opposite charges on the inner coatings of the Ley den jars practically neutral- izes the effects produced by the outer coatings. But when the spark occurs at D the potential difference between G and E is altered practically instantaneously, the spark between them is therefore of the B type. In this case the ac- tion of the model light- ning conductors is quite different to their action in the preceding case (Fig. 65). In Fig. 67, for example, where the cone and the tops of the knobs are all of the same height the B spark takes place between C and one or other of these conductors. If one be placed slightly nearer to the upper plate than the others it will protect them. Other varieties of the B flash are illustrated in figs. 68 and 69. An experimental illustration of these cases is shown in Fig. 70. When a spark takes place between the discharge knobs D, a B flash will pass between the highest lightning conductor and C provided that the distance between them be not too great. The shape of the end of the lightning conductor is quite immaterial in this case, the protective FIG. 68. A second type of B Flash. 254 ELECTRIC CABLES AND NETWORKS action being quite different to that which occurs with A flashes. There is no time for the ionization of the air which takes place at points to prepare a path of small resistance for the discharge. The rush of electricity apparently always takes place across the shortest path. Lodge compares the paths in the steady stress and in the impulsive rush cases to the paths taken down a hill side by a gentle stream of water and by an avalanche respectively. FIG. 69. A third type of B Flash. The A Flash from a cloud to the chimney stack causes the B Flash from a neighbouring cloud. As in the case of the A flash it is found that the absolute values of the resistances of the lightning conductors them- selves have little effect on their protective qualities. If we replace the top sheet of metal in Figs. 67 and 70 by a sieve into which water is poured, sparks still ensue. The flashes, like those which occur during thunderstorms, are sometimes very long and very irregular. They seem to make use of the rain drops as stepping stones. As thunderstorms in this country are nearly always ac- companied by rain, it is highly probable that most of the flashes which occur belong to the impulsive rush case. In the majority of cases also it is probable that the discharge is oscillatory, for we know both by theory and experiment LIGHTNING CONDUCTORS 255 that the spark discharge of a condenser is oscillatory pro- vided that the resistance of the path of the discharge be not above a certain value. The main function of a lightning rod is to dissipate the energy stored in the lightning flash harmlessly, and so prevent it from doing damage to neigh- bouring objects. Hence the conductor must not be too small in diameter or it will be deflagrated by the discharge. Lightning rods FiG. 70. Model illustrating second type of B Flash. A subsidiary function is to equalize the potential between the thunder cloud and earth by the " silent discharge " taking place from all points on the conductor. This action is probably less energetic than the discharging action of certain kinds of trees for instance, fir trees. Statistics prove that the cutting down of extensive fir forests in certain parts of Europe has led to a considerable increase in the number of destructive lightning flashes experienced in those districts. It is probable, therefore, that in towns where numerous lightning conductors with multiple points on them are employed, they will have the effect of diminishing the average number of the lightning flashes that occur, 256 ELECTRIC CABLES AND NETWORKS The When the discharge is oscillatory it must have C in r the an exceedingly high frequency, and when it is non-oscillatory the discharge is over in a very small fraction of a second. In either case we know from theory that the current in the lightning conductor flows so that the magnetization in the metal of the conductor is a minimum (c/. Chapter II, p. 43). It is therefore prac- tically confined to a thin layer of the metal on the outer surface of the conductor. In calculating the resistance of the path, therefore, we must not, as in electric light wiring, proceed on the assumption that the current density is uniform over the cross section. Lord Rayleigh has shown that when the frequency is very high the resistance R and the self -inductance L of a cylindri- cal rod for a symmetrical flow of current, obeying the sine law, are given by and L In these equations I denotes the length of the conductor, a its radius, p the resistivity in absolute units, ^ the permea- bility of the metal, / the frequency, and A a constant depend- ing on the dimensions, etc., of the return circuit. All the quantities in the equations are in C.G.S. units. Numerical ^et us su PP ose that the lightning conductor example j g a C yli n ^ r i ca l copper rod 100 metres long and 1 centimetre in diameter. In this case 1= 10,000, a = 0-5, p = 1,600 approximately, and /j,= 1. We shall suppose also that the frequency is 1,000,000, so that /=10 6 . Sub- stituting these values in the formula for R we find that 12 = (10 4 /0-5)y 1,600 x!0 6 =8 xlO 8 absolute units = 0-8 of an ohm. The resistance to a flow of direct current would be 0-02 of an ohm. Hence the resistance of the lightning conductor to an impulsive rush of electricity is forty times as great as that which it would offer to direct currents or LIGHTNING CONDUCTORS 257 alternating currents of the frequencies used for electric lighting. If the conductor had been made of iron, all the dimensions remaining the same, and if we take the resistivity of iron as nine times that of copper and assume that its average per- meability under the given conditions is 100, then, R will equal 24 ohms, and the resistance with direct current will be 0-18 of an ohm which is less than the hundredth part of the apparent resistance to the alternating impulsive rushes. It follows from Rayleigh's formula that the inductance of the conductor is lA-\-R/2jrf . Hence the reactance is 27rflA-{-R. In most cases R will be very small compared with 2-7T/L4, and hence the inductance and reactance of lightning conductors is practically independent of the material of which they are made. Rayleigh's formulae show that the greater the radius of a cylindrical rod the smaller will be its resistance and induct- ance. As the radius of the rod increases, however, the greater will be the ratio of the apparent resistance R a with alternating currents to the resistance R d with direct currents, for R a / R d = {(l I a)Vf}/( P l/7ra*)=7ratfp. Hence the ratio R a /R d varies directly as the radius of the rod, but the absolute value of R a diminishes as a increases. It is to be remembered that the longer the conductor, or the greater its resistance, the lower will be the frequency of the oscillations set up by the lightning flash. We have also to remember that in calculating the values of R no account has been taken of the energy lost by radiation into space, which at these high frequencies is probably appreciable. If a piece of wire is placed sufficiently close Side flash to a lightning rod, part of the charge will leave the rod and travel along the piece of wire as the reactance of the divided path is less than the path in the conductor s 258 ELECTRIC CABLES AND NETWORKS alone. This explains the phenomenon of side flash which is often observed when an object is struck by lightning. The potential differences existing between various parts of a lightning conductor when it is struck are obviously very high and hence the electrostatic field round it is very intense. The electric stresses ionize the air round the conductor and so a spark readily ensues to any neighbouring conductor. In setting up lightning conductors this tendency to side flash has to be remembered, as sparks due to this cause can ignite escaping gas and thus set fire to buildings. It has often been noticed that when a lightning rod is struck a peculiar noise is heard not unlike the pouring of water on a fire, and electric sparks are emitted from bodies in the neighbourhood. These phenomena are probably caused by brush discharges due to the breaking down of the air by the electrostatic stresses set up during the discharge. The Lightning Research Committee, appointed Research by the Royal Institution of British Architects and the Surveyors' Institution in 1901, have in their report made the following practical suggestions. 1. Two main lightning rods, one on each side should be provided, extending from the top of each tower, spire or high chimney-stack by the most direct course to earth. The diagrams shown in Fig. 71 illustrate this sugges- tion. In Y, which is the usual method, the conductor follows the outline of the building. In this case there is a tendency for the discharge to leave the conductors at the bends, as it always tends to make a path or paths for itself in addition to that provided by the lightning rod, so that the reactance of all of them in parallel may be a minimum. It thus some- times breaks away the brickwork, and in some cases the mechanical forces called into play break the conductor LIGHTNING CONDUCTORS 259 itself. The Research Committee recommend the method illustrated in X (Fig. 71), where the conductor is kept away from the building by suitable holdfasts, which may be made of iron. It seems to the author that the method X recommended is excellent for getting round sharp corners, or in cases where there is danger from side flash owing to the presence of neighbouring conductors. In general, however, when there FIG. 71. X is the method of fixing lightning conductors recommended by the Lightning Research Committee. is a straight run for the conductor there is no need to keep it away from the surface of the wall. 2. Horizontal conductors should connect all the vertical rods (a) along the ridge, (b) at or near the ground line. This recommendation of the Committee was probably suggested to obviate the risks of side flash from one con- ductor to the other and as a partial protection also for the space between the two. 260 ELECTRIC CABLES AND NETWORKS 3. The upper horizontal conductor should be fitted with aigrettes or points at intervals of 20 or 30 feet. 4. Short vertical rods also should be erected along minor pinnacles, and connected with the upper horizontal con- ductor. 5. All roof metals such as finials, ridging, rain water and ventilating pipes, metal cowls, lead flushing, gutters, etc., should be connected with the horizontal conductors. 6. All large masses of metal in the building should be con- nected with the earth, either directly or by means of the lower horizontal conductor. 7. Where roofs are partially or wholly metal-lined they should be connected with the earth by means of vertical rods at several points. 8. Gas pipes should be kept as far away as possible from the positions occupied by lightning conductors, and as an additional protection the service mains of the gas meter should be metallically connected with house services leading from the meter. Many useful suggestions will also be found in the Report issued by the Lightning Rod Conference held in 1882. Some of the rules given in this report, however, have to be amended as they proceed on the erroneous assumption that a lightning flash will follow the path of minimum resistance in exactly the same way that a steady direct current would. The end of the lightning conductor is usually connected with a copper plate embedded in moist earth in the neighbourhood of the building. If none of the earth in the immediate neighbourhood of the con- ductor is moist, it is advisable to dig a pit about 6 feet deep in which the sheet of copper about a square yard in area and one-eighth of an inch thick should be placed and then surrounded with charcoal or pulverized carbon. The ends LIGHTNING CONDUCTORS 261 of the carbons used in arc lamps do excellently for this purpose. Coke is sometimes employed, but its use is objectionable owing to the chemical and electrolytic effects produced in the copper. The pit should not be quite filled up with earth, so that there may be a sufficient depression on the surface over the pit to catch the rain during a thunderstorm and thus keep the earth in the neighbour- hood of the plate moist. The resistance of the " earth " is measured by finding, by a Wheatstone's bridge or otherwise, the resistance between the conductor and any neighbouring water pipe. On a dry day if this resistance be not greater than 100 ohms the " earth " may be considered satisfactory. Accurate mea- surements of this resistance are neither possible nor necessary. In the neighbourhood of towns supplied with electric light or tramways a permanent deflection is often obtained on the galvanometer owing to a leakage current from some of the supply networks. Instead of using a special " earth " it is sometimes convenient to connect the end of the lightning conductor with the water mains. Tubular Mr. Killingworth Hedges' " tubular earth " earth can often be advantageously used. It consists of a hollow perforated steel spike filled with granulated carbon and driven into moist earth. The lightning conductor is taken to the bottom of the tube. The earth in the neigh- bourhood of this device can easily be kept moist by con- necting it with the nearest rain-water pipe. In this case the earth resistance is negligibly small. The lightning conductors used in this The metal of the country are generally made of copper. Either conductor copper tape or copper wire rope is employed. In the former case the section is usually } inch broad by J- inch thick, this size being found ample in practice. In 262 ELECTRIC CABLES AND NETWORKS the latter case the rope is usually J inch in diameter. If smaller sized conductors are used there is a risk of them being deflagrated by a severe lightning flash. In those climates where there is little risk of the con- ductor being corroded either by the moisture or by chemical fumes, galvanized soft stranded iron rope is the most suitable lightning conductor. The higher specific heat of an iron conductor compensates for its smaller density, and so its higher melting point enables it to get rid of a larger amount of the electrical energy of the flash than a copper conductor of the same dimensions. The elevation rod or top of a lightning con- elevation ductor ought to be the highest point of the building, but there is no necessity to have it more than about a foot taller than the summit of a pinnacle or the brickwork of a chimney. Four or five well gilded or platinized " points " should be attached to the elevation rod. Town There is not much danger of town houses houses being struck by lightning, as the numerous gutters, ventilating and rain-water pipes afford them con- siderable protection. Occasionally metallic bonds are used to connect the various sections of rain- water pipes, and thus ensure their metallic continuity and so guard against damage by lightning. As most fire insurance policies issued in England cover damage done by lightning, these precautions are seldom taken for ordinary town houses. Important buildings are usually elaborately protected by lightning conductors. Even with very elaborate systems, however, possible dangers arise from side flash from the conductors to neighbouring gas pipes or stove pipes. The Hotel de Ville at Brussels, which is protected by a very complete network of wires, had a narrow escape from LIGHTNING CONDUCTORS 263 being burned down during a thunderstorm, as a spark from one of the lightning conductors to a neighbouring piece of metal set fire to gas which had escaped from a leak in a gas pipe. Lightning Heavy damp soils such as loam are particu- fataiities j ar jy u ' a bl e to be struck by lightning flashes. The most frequent fatalities in this country from lightning happen to people standing under trees which are struck, the lightning " side flashing " from the tree to the person whose body, or clothes if wet, forms a good conductor. Trees whose roots are near the water are particularly liable to be struck. Again a person in the centre of a field or crossing the brow of a hill might possibly be struck, as he would be the highest object in the neighbourhood. Horses, cattle, and sheep, especially when steam is rising from them owing to their being overheated, are sometimes struck. Deer in public parks are frequently killed owing to their habit of congregating under trees during thunderstorms. In America wire ropes are of ten used to hang clothes on to dry after being washed. Several fatalities occur every year to people taking the clothes off these ropes at the beginning of a thunderstorm. REFERENCES. R. Anderson, Lightning Conductors. Sir Oliver Lodge, " On Lightning, Lightning Conductors and Lightning Protectors." Journ. of the Inst. of EL Engin., vol. viii, p. 386, 1889. Sir Oliver Lodge, Lightning Conductors and Lightning Guards. Killingworth Hedges, Modern Lightning Conductors. Melsens, Des Paratonnerres d pointes, d conducteurs et d raccorde- ments terrestres multiples ; description detaillee des paratonnerres etablis sur r hotel de ville de Bruxelles. 1877. F. Linke, " Luftelektrische Messungen bei 12 Ballonfahrten." (Abh. der Koniglichen Gesell. der Wiss. zu Gottingen, Math- Phys. Klasse, Neue Folge, Band iii, 1904.) 264 ELECTEIC GABLES AND NETWORKS C. Chree, " A Discussion of Atmospheric Electric Potential Results at Kew, from selected days during the seven years 1898-1904." Phil. Trans., Series A., vol. 206, p. 299, 1906. Lord Rayleigh, " On the Self -Induction and Resistance of Straight Conductors." Phil. Mag. [5] vol. 21, p. 381, 1886, or Scientific Papers, vol. ii., p. 493. Report of the Lightning Research Committee. Journal of the Royal Institute of British Architects [111] vol. 12, p. 405, 1904. INDEX " A " flash, 249 Air, dielectric strength of, 177, 182 sparking distances in, 179 Algermissen, J., 177 Allgemeine Elektricitats Gesell- schaft, 228 Alternating currents, high frequency, 43, 256 American rules, 178 American Standardization Com- mittee, 31 Anderson, R., 263 Annealed copper, 21 Appleyard, R., 60, 64 Argon, dielectric strength of, 182 Arresters, continuous, 232 electrolytic, 235 horn, 227 intermittent, 226 lightning, 225 et seq. multiple gap, 230 water jet, 233, 235 Atmospheric electricity, 243 et seq. Atmospheric potential gradient, 224 " B " flash, 249, 251 B. and S. Gauge, 13 B.W.G., 13 Baur, C., 64 Benton, J. R, 22 Birmingham Wire Gauge, 13, 15 Black, G. L., 159 Blavier's test, 158 Board of Trade Regulations, 68, 97, 125 Booster, 90 Bose, M. von, 45 Breaks, locating, 145 Bridge method of testing, 155 British Legal Standard, 13 British Standard Radial Thick- nesses, 203 Brown and Sharpe Gauge, 13, 15 Bulk modulus, 20 Cables, grading, 187 et seq. insulation resistance of, 58 mass of, 15, 37, 40 resistance of, 39, 42 stranded, 33 Cadmium cell, 55 Calorie, 5 Campbell, A., 52, 196 Centre of gravity of load, 85 Centres, distributing, 70 Charing Cross Company, 228 Chree, C., 245, 246, 264 Circular mil, 12 Clark, W. S., 64 Collie, J. N., 182 Composite dielectrics, 173, 207 Concentric main, 43 electric stresses in, 188 grading of, 191 et seq. suitable dimensions for, 190 temperature gradient in, 212. Condenser, spherical, 167 Conductance, 12 Conductivity, 19 et seq. Conductors, in parallel, 7 in series, 6 lightning, 243 et seq. Copper, annealed, 21 density, 23 elastic constants, 21 hard drawn, 21 temperature coefficient of, 27 useful data for, 45 Current density, 40 265 266 INDEX Current, excessive, 73 high frequency, 43, 256 in lightning conductors, 256 permissible, 40 Data for calculations, 45 Density of copper, 23 standard, 24 Dewar, J., 11, 16, 27, 32, 46 Dielectrics, composite, 173 Dielectric strength, 163 et seq. coefficients for, 176 of air, 177 of eolotropic solids, 183 of gases, 180, 182 of iso tropic solids, 183 Disruptive discharge, 166 Distributing centres, 70 Duddell currents, 225 Dusaugey, 239 Earth currents, energy expended in, 123 Earth faults, 112, 116, 140 Earth, in house wiring, 140 in middle main, 147 in negative outer, 149 Earth lamps detector, 107 Earthing lightning conductors, 260 Elastic constants, 19 Electric intensity, 166 stress, 164 Electricity, atmospheric, 243 Electrolytic arresters, 235, 238 Electrostatic voltmeter method, 106 Elevation rod, 262 Endosmosis, 121 Energy expended in earth cur- rents, 123 Engineering Standards Com- mittee, 103, 203 Eolotropic solids, 183 Evershed and Vignoles, 104 Fall of potential method, 153 Faraday, M., 164 Fault resistance, 112 measurement of, 100, 113, 133 Faults in networks, 159 locating, 145 et seq. locating, by flashing, 147 final methods, 153 general methods, 151 Feeding centres, 73, 78 et seq. for straight main, 87 Fernie, F., 151, 159 Fisher, H. W., 184 Fitzpatrick, T. C., 23, 46 Flashing, 147 Fleming, J. A., 11, 16, 27, 32, 46, 159 Garrard, C. C., 227, 229 Gases, dielectric strength of, 180 Gauges, 13 tables of, 15 Gauss's Hypergeometric Series, 205 Glazebrook, R. T., 26 Gotti, O. Li, 92 Grade of insulation, 56 Gradient, potential, 165 Grading of cables, 187 et seq. Graphical construction for po- tentials, 117 Gray, T., 184 Groves, W. E., 135 Guard wire, Price's, 56 Gutta, 64 Hard drawn copper, 21 Heat, effects on dielectric of, 214 Heating of bare conductors, 220 of cables, 211 et seq. of wire, 5 Hedges, Killingworth, 261, 263 Helium, dielectric strength, 182 Herzog, J., 92 High frequency alternating cur- ents, 43, 256 INDEX 267 High pressure, economy of, 71 Hobart, H. M., 184 Hobbs, G. M., 179 Hooke's law, 19 Hopkinson, J., 153 Horn lightning arrester, 227 House wiring, faults in, 140 Hydrogen, dielectric strength of, 182 Induction method, 156 Institution Rules, 57, 95 Insulation resistance, 49 formula for, 51 measurement of, 53 of house wiring, 95 et seq. of three wire network, 119 of two wire network, 111 Institution rules for, 57 tables of, 58, 61 Insulation, specific, 50 Insulativity, 49 et seq. Intensity, electric, 166 Iso tropic bodies, 3 dielectric strength of, 183 Jackson, R. P., 184, 239 Jaeger and Kahle's formula, 55 Jona, E., 184, 189, 197, 205, 207, 214 Jona's graded cables, 197 Joule's law, 5 Kelvin, Lord, 246 skin effect, 44 Thomson and Tait, 86 Kelvin's law, 67 Kennelly, A. E., 222 Kew observations, 245 Kirchhoff's first law, 6 second law, 6 Langan, J., 64 Lay, definition of, 37 effect on mass of cable of, 37 effect on resistance of cable of, 39 Leak in middle main, 122 in positive outer, 122 Leakage currents, 125 effect on grading of, 199 Leboucq, M., 92 Lees, C. H., 213 Levi-Civita, 205 Lightning arresters, 228, 231, 233 Lightning conductors, 243 et seq. currents in, 256 metal of, 261 method of fixing, 259 Lightning fatalities, 263 Lightning Research Committee, 258, 264 Lightning Rod Conference, 260 Lines of flow, 30 Linke, F., 244, 263 Load, uniformly distributed, 71 centre of gravity of, 85 Lodge, Sir Oliver, 243, 249, 254, 263 Main, concentric, 43 feeding centre for straight, 87 grading, 188 et seq. single core, 169 thermal conductance of, 217 Mass of conductor, effect of " lay " on, 37 tables, 15 Mass resistivity, 24 Matthiessen, A., 45 Matthiessen's Standards, 25 Megger, the Evershed, 105 Melsens, 263 Microscopic spark lengths, 179 Mie, G., 222 Mil, 12 circular, 12 Minimum heating of networks, 7, 9, 124 Minimum insulation resistance, 58 268 INDEX Minimum radial thicknesses, 58, 203 Modulus, bulk, 20 Young's 20 Monatomic gases, 206 Moore, A., 227 Multiple gap arrester, 230 Neon, dielectric strength of, 206 Networks, distributing, 68 et seq. Nodon valve, 236 Norden, K., 236 Numerical data, 45 Oerlikon Company, 227 O'Gorman, M., 199, 207 Ohmmeter, 103 Ohm's law, 4 Oxygen, dielectric strength of, 182 Patchell, W. H., 228 Peck, J. S., 239 Permissible current, 40 Perrine, F. A. C., 64 Picou, R. V., 222 Pointed conductor, 172 Pollak rectifier, 236 Potential gradient, 165 Power station, site of, 85 Preece, Sir William, 50, 64 Pressure at consumer's ter- minals, 68 Price, W. A., 56 Ramsay, Sir W., 182 Raphael, F. C., 16, 159 Rayleigh, Lord, 256, 264 Rayner, E. H., 64 Rectifier, Pollak, 236 Nodon valve, 236 Regulation of potentials, 120 Resistance of stranded cables, 39 Resistances, in parallel, 8 in series, 6 Resistivity, mass, 24 volume, 9 Resultant electric force, 166 Rhodin, J., 27, 46 Ring mains, 81, 83, 84 Rubber, 63 Russell, S. A., 64 Russell, A., 108, 130, 135, 184, 201, 207, 219, 222 Ryan, H. J., 184 Safety valves, 225 et seq. continuous, 232 intermittent, 226 Schuster, A., 184 Schwartz, A., 64, 159 Searle, G. F. C., 21, 46, 213, 222 Sections, economical, of mains, 74 et seq. Seibt, G., 229, 239 Shape of electrodes, effect of, 171 Shear, 20 Short circuits, locating, 143 Side flash, 257, 263 Siemens Bros., 52 Siemens and Halske, 227 Single core main, grading of, 191 heating of, 217 Skin effect, 44 Skinner, C. E., 184 Societe d' Applications Indus- trielles, 234 Societe de Grenoble et Voiron, 231 Societe de Vizille, 233 Specific insulation, 50 Spherical condenser, 167 Spherical electrodes, 172 Standard wire gauge, 13, 15 Star connexion, 7 potential, 7 Stark, L., 92 Steimnetz, C. P., 184 Straight main, feeding centre for, 87 INDEX 269 Stranded cables, 33 et seq. Stranding, effect on electric stresses of, 205 Swan, J. W., 27, 46 Tables, gauges, 15 mass of wires, 15 maximum stress, 176 minimum radial thickness, 58 minimum insulation resistance, 58 RD. drop, 40 permissible currents, 40 resistance of cables, 42 section of cables, 40 sparking distances, 177, 179 Taylor, A. 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Electrical Testing for Telegraph Engineers. Illustrated. 8vo., cloth, 264 pp Net, $4.00 A 112=page Catalog of Books on Electricity, classified by subjects, will be furnished gratis, postage prepaid, on application. THE NEW FOSTER Fifth Edition, Completely Revised and Enlarged, with Four-fifths of Old Matter Replaced by New, Up-to-date Material. Pocket size, flexible leather, elaborately illustrated, with an extensive index, 1636 pp., Thumb Index, etc. Price, $5.00. ELECTRICAL ENGINEER'S POCKETBOOK The Most Complete Book of Its Kind Ever Published, Treating of the Latest and Best Practice in Electrical Engineering By HORATIO A. FOSTER Member Am. Inst. E. E., Member Am. Soc. M. E. With the Collaboration of Eminent Specialists Symbols, Units, Instruments Measurements Magnetic Properties of Iron Electro-Magnets Properties of Conductors Relations and Dimensions of Conductors Jnderground Conduit Con- struction Standard Symbols Zable Testing 3ynamos and Motors Tests of Dynamos and Motors CONTENTS The Static Transformer Standardization Rules Illuminating Engineering Electric Lighting (Arc) (Incandescent) Electric Street Railways Electrolysis Transmission of Power Storage Batteries Switchboards Lightning Arresters Electricity Meters Wireless Telegraphy Telegraphy Telephony Electricity in the U. S. Army Electricity in the U. S. Navy Resonance Electric Automobiles Electro-chemistry and Electro- metallurgy X-Rays Electric Heating, Cooking and Welding Lightning Conductors Mechanical Section Index D. VAN NO5TRAND COHPANY, Publishers and Booksellers, 23 MURRAY AND 27 WARPEN STREETS. 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