A LIST OF WORKS publisfoefc 1 / I II LIBRARY OF THE UNIVERSITY OF CALIFORNIA. No. Deceived ^Accession No. 7 3 ff /^\3 Class Experiments, Tesla, 4^. 6J. Fleming, 2 vols. 25^. Hospitaller, 3^. 6d. Kapp, 4J. 6.Y. Alternate Current Transformer De- sign, 2S. Analysis, Iron, Blair, i8s. Steel Works, Arnold, icu. 6d. Anatomy, Brodie,4 parts, or I vol. 2.1. 2s. Anderson's Heat, 6s. Andreoli's Ozone, 2s. 6a. Arithmetic, Electrical, is. Armature Windings, Parshall, 30^. Drum, Weymouth, "js. 6d. Armorial Families, 5/. 55. Arnold's Steel Works Anal., los. 6d. Artillery, Modern French, 50^. Astronomy, Chambers, 2s. 6d. Atkinson's Static Electricity, 6^. 6J. Atlantic Ferry, 7-r. 6d. & 2s. 6d. Ballooning, May, 2s. 6it. Back's Electrical Works. Bales' Mod. Shafting and Gearing, 25. 6:/. Biggs' Elec. Engineering, 2s. 6d. Black's First Prin. of Building, 3^. 6V r ork, is. 6d. Fitting, 5.5-. English Minstrelsie, 4/. Ewing's Induction, los. &/. Explosives, Guttmann, 2 Vols., 2/. 2s. Fairbairn's Book of Crests, 2 vols., 4/. 45. Findlay's English Railway, 7^. 6d. Fitting, Horner, 5_y. Electric Light, Allsop, $s. Fitzgerald's Nav. Tactics, is. [net. Flatau's Atlas of Human Brain, i6s. Fleming's Transformers. Vol. I., I2S. 6d. Vol. II., I2S. 6d. Electric Lamps, 7^. 6d. Electric Lab. Notes, I2s. 6ft. net. Fletcher's Steam-Jacket, 7^. 6V/. F'oden's Mechanical Tables, is. 6d. Forbes' Electric Currents, 2s. 6d. Forestry, Webster, 3^. 6d. Formulse for Electrical Engineers, 7s. 6d. Forth Bridge, 55-. Foster's Central Station Bookkeeping, los. 6d. Fox-Davies' Book of Crests, 4/. 41. Armorial Families, 5/. 5*. Gaseous Fuel, is. 6d. Gatehouse's Dynamo, is. Gearing, Helical, 7^. 6d. Geipel and Kilgour's Electrical Formulse, 7^. 6d. Geology, Jukes-Browne, 4^. German Technological Dictionary, 5-f. Gibbings' Dynamo Attendants, is. Godfrey's Water Supply. Gore's Electro-chemistry, 2s. Electro-deposition, i^. 6d. Metals, lew. 6d. Gray's Influence Machines, 4^. 6d. Griffiths' Manures, 7^. 6d. Guttmann's Explosives, 2 vols., 2/. is. Guy's Electric Light and Power, 5^. Hatch's Mineralogy, 2s. 6d. Haulbaum's Ventilation, is. Hawkins' and Wallis's Dynamo, I os. 6d. Heat Engines, Anderson, 6^. Heaviside's Electro-magnetic Theory, Vol. I., 12s. 6d. Helical Gears, 7^. 6d. ALTEENATING CUEEENTS OF ELECTEICITY AND THE THEOEY OF TEANSFOEMEES BOOKS FOR STUDENTS IN ELECTRICAL ENGINEERING. STILL'S ALTERNATING CURRENTS AND THE THEORY OF TRANSFORMERS. KAPP'S ELECTRIC TRANSMISSION OF ENERGY. 105. 6) that the magnetic resistance of iron, nickel, or cobalt does not depend solely on the nature of the material, but is also a function of the induction, whereas in the electrical analogy the resistance is in every case an attribute of the material itself, and has a physical meaning apart from its defi- nition as the ratio of E.M.F. to current. Having stated these objections, and thereby re- moved any possibility of misconception on the part of the reader, we shall proceed to consider the effect of replacing the non-magnetic core in the arrangement under discussion by an iron one. 5. Magnetic Induction in Iron. Assuming the magnetising force to be the same as before, the induction is now greater than it was when the interior of the helix was occupied by air, and it is MAGNETIC INDUCTION IN IRON 11 denoted by B. The ratio B/H gives us the permea- bility, or the m/uttiplying power of the iron. This, as is well known, is not constant, even for a given sample of iron, but depends upon many things, among others on the magnitude of the magnetising force and the previous magnetic condition of the iron. The relation of B to H , or of B to the exciting ampere-turns, must therefore always be determined experimentally ; it is usually expressed by the aid of a curve, the general characteristics of which are too well known to necessi- tate their being dwelt upon here. In the expression B=/JL\-\ it is customary to define H as the number of lines per square centimetre which the magnetising coil would produce in the space occupied by the iron, on the assumption that the iron core were removed, the resultant magnetising force remaining the same as before. The point which is not generally clearly explained is that there is no necessity whatever to consider the iron core removed, or even to imagine longitudinal holes drilled through the mass of the iron, in order to understand what is meant by H in the above relation. There is no such thing in nature as an insulator of magnetism, and the magnetic intensity represented by H is a function only of the resultant magnetising force, or difference of magnetic potential, and the geometrical dimensions of the magnetic circuit, or portion of magnetic circuit, considered ; it has just as real an existence in a mass of iron which has practically 12 MAGNETIC PRINCIPLES reached its saturation limit as in a vacuum, or in air ; and when we say that the number of lines represented by B are made up of the number of lines represented by H in addition to the number of lines due to the magnetic condition of the iron, this is not an arbi- trary or artificial division, but, on the contrary, a scien- tific analysis which shows what actually takes place. It is well known that, with very strong magnetising forces, the magnetic condition of iron may be brought very near to the saturation point ; but, on the other hand, the quantity represented by B does not approach a saturation value, but increases, to all appearances, without limit. This is because the component H of the resultant induction B increases always in proportion to the magnetising force. It is therefore usual to write : where the quantity 4nr\ represents the number of mag- netic lines per square centimetre added by the iron. | is called the intensity of the magnetisation of the iron, and is a measure of that physical condition of the iron to which the additional number of lines 4?r| are due. The factor 4-7T is simply a multiplier which depends upon our system of units, the meaning of which does not concern us at present, but is easily understood when dealing with the fields surrounding the ends of bar magnets. In practice, especially in connection with alternating current work, the inductions used are generally low, in MAGNETIC INDUCTION IN IRON 13 which case the component H of the induction B may be neglected, as it is very small in comparison with the term 4?rl ; it is, however, usual to read the values of the induction directly off a curve which gives the rela- tion between B and the resultant magnetising force for the particular kind of iron to which the calculations apply. 6. The Magnetic Circuit. Straight Bar. In the case, just considered, of a closed iron ring, all the magnetic lines pass through the iron, and, in order that N in equation (4) may still stand for the total flux of induction, we must now write >*< N=-j . . .(5), Txji where the quantity l\Ap is still called the magnetic resistance of the circuit. It is hardly necessary to point out that the above equation is merely a convenient way of stating the relation between the induction in the iron core and the magnetising ampere-turns, and that it can in no respect be compared with Ohm's law for the electric circuit, which is founded upon the quality of constancy of the electrical conductivity of substances irrespective of the value of the current, whereas the multiplier /* is a variable quantity which can only be exactly determined by experiment. Let us, now, in the place of the ring, consider a 14 MAGNETIC PRINCIPLES straight iron bar, also uniformly wound with Si ampere- turns per centimetre of its length. The path of the magnetic lines is now partly through the iron and partly through the surrounding air. If the bar is short as compared with its section, the average induction in it will be determined almost entirely by the resistance of the air path. If, on the other hand, the bar is made longer and longer without limit, the resistance of the air path rapidly diminishes, and, if we consider an iron wire of which the length is about 300 times the diameter, the resistance of the air return path may be neglected, and the induction in the centre portions of the wire will be practically the same as in the case of the closed iron ring. In order to get some idea of the distribution of a magnetic field in any particular case, it must be re- membered (a) that all the lines of induction due to the exciting coils are closed lines, and (6) that every line will always choose the path of least resistance ; or, in other words, that in any portion of a magnetic circuit the flux of induction will always be directly proportional to the magnetic difference of potential between the ends of the section, and inversely proportional to what has been called its resistance. It is only in the case of the very simplest arrange- ments of magnetic circuits that the distribution of the induction can be correctly predetermined ; but it should always be possible to get a general idea of the magnetic THE MAGNETIC CIRCUIT 15 field likely to result from any particular arrangement of the magnetising coils. If we place in the interior of a solenoid an iron core of comparatively small section, only a certain amount of the total flux of induction will pass through the iron ; in fact, we have here an almost perfect electrical analogy in the case of a glass tube filled with mercury, through which there is a uniform flow of current. If we place in such a tube a length of copper wire, the , E, s E* iS FIG. 1 lines of flow will crowd into the copper, on account of its greater conductivity, more or less in the manner indicated in fig. 1 . If we assume a constant difference of potential between the two equipotential surfaces E l and E 2 , we observe, after the introduction of the copper wire, that the current density in the mercury on the section s S is less than before, unless the copper rod a b is very long and extends beyond the sections E t and E 2 ; and it is only in the case of such a long copper rod that the current density in the mercury will be uniform throughout the section s, and will bear the same relation 16 MAGNETIC PEINCIPLES to the density in the copper as the conductivity of the mercury bears to that of the copper. The large fall of potential in the neighbourhood of the ends of the rod, where the lines of flow crowd together on entering and leaving the copper, is the reason why the difference of potential available for sending current through the centre portions of the rod is relatively less in the case of a short rod. It is hardly necessary to return from this electrical analogy to our starting point of the iron wire in a large solenoid, the similarity between the two cases being sufficiently evident. This is not, however, the usual way of explaining magnetic phenomena, but it is more consistent with what has gone before, and possibly less confusing than the theories of free magnetism and self demagnetising effects of short bars, although, as a little consideration will show, all the observed phenomena may be equally well explained by assuming the rod a b in our electrical analogy to produce a back E.M.F. depending upon the number of amperes which pass into and out of it at the ends. 7. Magnetic Leakage. Effect of Satura- tion. If, in the case of the uniformly wound iron ring already referred to, an air gap had been introduced in some part of the magnetic circuit, or even if the winding on the ring had not been uniform, but had been con- fined to a small portion of the ring, a certain amount of magnetism would have escaped laterally from the MAGNETIC LEAKAGE 17 surface of the iron, thus following an alternative path through the surrounding air. The magnetism which leaves the main path of the magnetic lines in this way, and which rarely serves a useful purpose but, on the contrary, may often, as in the case of transformers, be decidedly objectionable is generally known as leakage magnetism. In most cases arising in practice the amount of this leakage magnetism will be approximately propor- tional to the magnetising force, and indeed this is FIG. 2 probably true of any conceivable arrangement of the magnetic circuit, provided the magnetisation of the iron in the circuit is not carried above a value equal to about half its saturation value ; but as the iron becomes more and more nearly saturated, the amount of the leakage magnetism will, in certain cases, diminish, as will be readily understood by reference to fig. 2. Here we have the case of a uniformly wound helix or solenoid with a discontinuous iron core. The ends shown broken off in the figure must be considered either as extending in both directions to a considerable c OF THK UNIVERSITY 18 MAGNETIC PRINCIPLES distance, or as being bent round so as to join, in which case the air gap a b may be looked upon as a dividing slot in an otherwise complete iron ring. When a current of electricity is sent through the coil, it is evident that, so long as the permeability of the air gap is less than that of the iron, a certain amount of the total magnetic flux will escape laterally from the iron and pass outside the coil in the manner indicated by the dotted lines. This ' leakage ' mag- netism will depend upon the length of the air gap ; in fact, it may be experimentally shown that the ratio of the length I to the diameter d determines, within certain limits, the relation between the amount of magnetism which finds its way outside the winding and the amount which passes from iron to iron, through the air gap, within the coil. This relation is also found to remain practically unaltered notwithstanding variations in the magnetising current, unless the latter is increased to such an extent that the iron begins to show signs of saturation. When this stage is reached the leakage coefficient which may be defined as the ratio of the total flux in the iron to that portion of it which crosses the air gap within the coil, and whfch hitherto has remained practically constant begins to decrease, and continues to do so until, if it were possible to reach the point of saturation of the iron, there is every reason to believe that the leakage coefficient would become unity, whatever might be the dimensions of the gap ; in other MAGNETIC LEAKAGE 19 words, the magnetic lines would all pass within the coil between the two opposing faces of the iron core. The reason why this point has been dwelt upon at some length, is not so much because it may be of practical importance when dealing with alternating current phenomena indeed, questions of saturation of iron are not likely to arise in this connection but principally because it is usual to estimate the value and distribution of leakage magnetism without giving proper consideration to the question of saturation. For instance, it is only permissible to compare a system of magnetic conductors to a similar system of electric con- ductors immersed in water, if it is clearly understood that the magnetisation of the iron is not carried beyond about half the saturation value ; otherwise the analogy will, in all probability, be worthless, if not entirely mis- leading. Those acquainted with the theories of magnetism will have no difficulty in explaining the observed re- duction of the leakage magnetism with the higher magnetising forces in the special arrangement which we have just considered : it is a case for the application of Kirchhoff's law of saturation, in accordance with which we may say that, if we increase the magnetising ampere- turns, the direction of the magnetisation, I, will, at every point of the iron, agree more and more nearly with the direction of the field, H, due to the coil alone, as the iron approaches the condition of saturation. c 2 20 ALTERNATING CURRENTS 8. Graphical Representation. The varia- tions, both in strength and direction, of an alternating current may be accurately represented by means of a curve such as the one drawn in fig. 3. The lapse of time is measured horizontally from left to right, while the current strengths are measured vertically; hence the position of any point P on the curve indicates that, after an interval of time represented by the distance o a y the instantaneous value of the current is given by the length of the line a P ; and further, since this measurement is made above the horizontal datum line ox, we conclude that the current is flowing in a positive direction. If the measurement had been made below this datum line, it would have been an indication that the current was flowing in a negative direction. The distance o b is evidently equivalent to the time of one complete period or alternation, after which the current must be considered as rising and falling periodi- cally in identically the same manner. In practice, the time of one complete period is usually somewhere between the fiftieth and the hundredth part of a second, GEAPHICAL REPRESENTATION 21 the tendency in England at present being towards the employment of comparatively low frequencies, i.e. about fifty complete periods per second. If n stands for the frequency, or number of complete periods (^/) per second, then the number of reversals per second is 2n, and the time of one complete alterna- tion is - seconds, which is numerically equivalent to n the distance o b in fig. 3. FIG. 3 Let us suppose that the resistance of the circuit con- veying the current represented in fig. 3 is r ohms; then, if i is the current at any instant, the rate at which work is being done in heating the conductors will, at that particular instant, be expressed by the quantity i 2 r. Hence, if we wish to know the average rate at which work is being done by an alternating current, 22 ALTERNATING CURRENTS it will be necessary to calculate the mean value of the square of the current and multiply this quantity by the resistance r. In other words, if we imagine a very large number of ordinates to be drawn, and the average taken of the squares of all these ordi- nates, the product of this quantity by the ohmic resis- tance of the circuit will give us the watts lost in heating the conductors. It follows, therefore, that when we speak of an alternating current as being equal to a certain number of amperes, we invariably allude to that value of the current which, when squared and multiplied by the resistance of the circuit through which it is flowing, will give us the actual power in watts which is being spent in overcoming the resistance of the con- ductors. Thus it is the square-root-of-the-mean-square value of an alternating current which is of primary importance, and which, as far as power measurements are concerned, enables us directly to compare a periodically varying current with a continuous current of constant strength : it is the product of this value of the current and the corresponding value of the effective or resultant E.M.F. to which it owes its existence which, in all cases, is a measure of the power absorbed by the circuit. The readings of nearly all commercial measuring instruments for alternating currents depend upon the /V/mean square value of the current or E.M.F. ; and it is only in exceptional cases that we require to know either the maximum or the true mean value of an alternating GRAPHICAL REPRESENTATION 23 quantity, whether current or E.M.F., although it will be evident that questions of insulation must be discussed with reference to the maximum value of the E.M.F., which will be great or small according to the shape of the E.M.F. curve, even if the \/ mean square value remains unaltered. With regard to the mean value of a periodically varying quantity, it is hardly necessary to point out that this also depends upon the shape of the curve, and that it is by no means the same thing as the \/ mean square value. It may, in fact, differ, even widely, from the latter ; but we are very little concerned with it at pre- sent. From an inspection of fig. 3 it will be seen that we have merely to take the average of the ordinates of the current wave, or divide the area of the curve opd by the length of the line od, in order to obtain the true mean value of the current. It must, however, be clearly understood that it is the \/ mean square value of an alternating current or E.M.F. with which we are principally concerned ; and when mention is made of amperes or volts in connection with variable currents, it is always this value which is alluded to, unless special reference is made either to the inaxK mum, or the mean, or to an instantaneous value of the current or E.M.F. 9. Clock Diagram. The best and simplest way of dealing with alternate current problems is, undoubtedly, by the aid of vector, or ' clock,' diagrams, such as the one shown in fig. 4. These diagrams were first intro- 24 ALTERNATING CURRENTS duced by Thomson and Tait, and they are now exten- sively used for the graphical solution of alternate current problems. Let the line OB, in fig. 4, be made equal in length to the maximum value of the alternating current or E.M.F. (CB or C I E I in fig. 3) ; and let us suppose this M line to revolve round the point as a centre in the direction indicated by the arrow. If, now, we consider the projection of this revolving line upon any fixed straight line such as the vertical diameter MN of the dotted circle it will be seen that the speed of OB can be so regulated that the length of this projection will, at any moment, be a measure of the instantaneous CLOCK DIAGRAM 25 value of the variable current or E.M.F. It will also be evident that since the current must pass twice through its maximum value, and twice through zero value, in the time of one complete period the line OB must, in all cases, perform one complete revolution in - seconds ; where n is the frequency, or number of 71 complete alternations per second. Also, in order that this diagram may give us all the information needed, it will be convenient to assume that all measurements, such as od, which are made above the centre o, corre- spond with the positive values of the variable quantity, whereas all measurements made below will apply to the negative values. When the line OB (fig. 4) is vertical, its projection od is equal to it ; we therefore conclude that the alter- nating quantity is at that moment passing through its maximum positive value. As OB continues to move round in a clockwise direction, od will diminish, until the point B has moved to Q, when od will be zero ; after which it will again increase in length, but this time since it is now below the line PQ -the flow of current, or the direction of the E.M.F. , is reversed. At N the maximum negative value will be reached, only to fall again to zero at P ; after which it rises once more to the positive maximum at M. If the line OB revolves round o at a uniform rate, the point d will move to and from the centre o with a 26 ALTERNATING CURRENTS simple periodic, or simple harmonic motion. It follows that, if the length of the projection o^ represents the variations of an alternating E.M.F., this E.M.F. must be understood to be rising and falling in a simple periodic manner ; and since the length od will now be propor- tional to the sine of the time angle 6, the shape of the wave (fig. 3) will be that of a curve of sines, the characteristic feature of which is that every ordinate, such as a P, will be proportional to the sine of its hori- zontal distance from o : this distance being now ex- pressed, not in time, but in angular measure, it being, of course, understood that 360 degrees correspond to the time of one complete period. It is on account of the almost general use of ' clock ' diagrams that intervals of time are frequently expressed in angular degrees; but, once the reason of this is understood, no confusion is likely to arise from the use of the expression. 10. Addition of alternating E.M.F.s. In fig. 5 the vectors e l and e 2 must be thought of as repre- senting the maximum values of two distinct alternating E.M.F.s of the same frequency of alternation, but with a. phase difference proportional to the angle 6. These two E.M.F.s may be considered as being produced by a couple of alternators, having equal numbers of poles, and which are rigidly coupled together so as to be driven at the same constant speed. The armature windings of these machines being joined in series, the ADDITION OF ALTEKNATING E.M.F.S 27 question arises as to what will be the resultant E.M.F. of the combined machines If the alternators are so coupled that the magnet poles of both of 'them are opposite the centres of the armature coils at exactly the same instant, the two E.M.F.s will be said to be in phase, and the lines oe^ and oe 2 will coincide. As it is, we have supposed that the E.M.F. wave produced by one alternator reaches its maximum value at a time when the E.M.F. due to the other alternator has already passed its maximum, and is falling towards its zero value ; the exact fraction of a period which corresponds to this lag of the one E.M.F. 28 ALTERNATING CURRENTS behind the other being measured by the ratio which the angle 6 bears to the complete circle. The resultant E.M.F. generated will, at any instant, be equal to the sum of the projections oa and ob of the vectors e l and e. 2 upon the line MN ; and if we construct the parallelogram of forces in the usual way, and project the resultant OE upon MN, it will be seen that since bc = oa the projection oc of the vector E will, at any instant, be a measure of the total E.M.F. generated ; it being understood that OE revolves round the centre o, at the same uniform rate as the two com- ponent vectors e l and e y 11. Practical Application of Vector Dia- grams. By means of such a diagram as the one just described, it is, of course, possible to add together any number of alternating E.M.F.s of the same perio- dicity ; and in this way we are enabled to predeter- mine not only the maximum value of the resultant E.M.F., but also its instantaneous value at any particu- lar moment. In order to do this, it is, however, neces- sary to assume that the component E.M.F.s are sine functions of the time ; in other words, that the rise and fall of the E.M.F.s (including the resultant) is in accordance with the simple harmonic law of variation. In actual practice this condition is not always fulfilled, and the E.M.F. waves produced by different alternators may be of various shapes ; some having more or less pronounced peaks, corresponding to a APPLICATION OF VECTOR DIAGRAMS 29 comparatively large maximum value ; others being flatter, and more rectangular in shape than the simple sine curve. It is not often that we require to determine instan- taneous values of an alternating current or E.M.F. It is generally sufficient as already mentioned if we know the V mean square values, such as can be read off any alternate current ammeter or voltmeter. Consider again two alternators, A and B, joined in series. They must still be supposed to have the same number of poles, and to be driven at the same speed ; but the E.M.F. waves produced by the two machines, instead of being sine curves, may now be of any other shape, and the form of wave due to A may differ entirely from the wave due to B. Let us suppose three voltmeters to be connected as shown in fig. 6. These voltmeters must be such as may be used indifferently on alternating or direct current circuits ; that is to say, they must measure the \/ mean square values of the alternating volts. They may, for instance, be either electrostatic, or hot wire instruments ; but they should not depend upon the electro-magnetic actions of coils having iron cores, FIG. 6 \ \ \ 30 ALTERNATING CURRENTS because the induction in the iron cores will depend upon the mean value of the applied E.M.F., and the read- ings of such instruments will, therefore, be more or less dependent upon the wave form of the impressed volts. The voltmeters e^ and e 2 will give us the volts due, respectively, to the alternators A and B ; whereas E will measure the resultant volts at terminals. If the volts measured by E are equal to the arithmetic sum of the volts e } and e. D the two machines would be said to be in phase^ though it is quite possible that the majximum values of the component E.M.F.s might not agree, on account of ', the waves produced by 'e g the two alternators being of different shapes. As a rule, the reading on E will be less than the arithmetic sum of e l and e 2 . We will suppose these three values to be known. From the centre o (fig. 7) describe a circle of radius OE, the length of which is a measure of the volts E. Now draw oe l in any direction, to represent the volts e p From e l as a centre describe an arc of radius gjE, the length of which is proportional to the volts e. 2 ; it will cut the arc already drawn at the point E. Join OE, and complete the parallelogram. Then the angle of lag 0, between the vectors e { and e 2 , is a measure of APPLICATION OF VECTOR DIAGRAMS 31 what we must now understand as the phase difference between two alternating quantities, which are of the same frequency, but which do not necessarily follow the sine law of variation. 12. Current Flow in Circuit without Self-induction. Let us consider an electric circuit which is practically without self-induction, or electro- static capacity. It may consist of a wire doubled back upon itself (in the manner adopted in winding resis- tance coils for testing purposes), or of glow lamps, or of a water resistance. If an alternating E.M.F. is applied to the terminals of such a circuit, the current at any instant will be equal to the quotient of the instantaneous value of the E.M.F., divided by the total resistance of the circuit ; or, L= e -> r from which we see that the current wave will be of the same shape as the E.M.F. wave, and in phase with it ; a state of things which is the evident result of the fulfilment of Ohm's law : for there is no reason for supposing that Ohm's law is not equally applicable to variable as to steady currents ; it is only necessary to bear in mind that, in the case of variable currents, the applied E.M.F. and the effective or resultant E.M.F. in the circuit (to which the current is due) are not necessarily one and the same thing. In the case under considera- 32 ALTERNATING CURRENTS tion, of a circuit supposed to be without self-induction or capacity, there is only one E.M.F. tending to produce a flow of current, i.e. the E.M.F. supplied at the terminals of the generator : the current will therefore rise and fall in exact synchronism with the applied E.M.F. The yower at any instant will be equal to the pro- duct of the E.M.F. and corresponding current, or By drawing the curves of current and E.M.F., and plotting the product W L for a number of ordinates, we can readily obtain the power curve, the mean ordinate of which will give us the average rate at which work is being done by the current in the circuit. The power at any instant being equal to e- L x i iy this may also be written and ifr. Hence, if e and i stand for the ^/ mean square values of E.M.F. and current, it e follows that w= ^.i 2 r=ei. where w=the mean ordi- r nate of the power curve, which gives us the average rate at which energy is being supplied to the circuit. We therefore see that, in the case of a circuit which may be considered non-inductive and without appre- ciable capacity, we have merely to take simultaneous readings of the volts and amperes and multiply these together in order to obtain the true watts going into the circuit. 33 SELF-INDUCTION 13. Coefficient of Self-induction. The self- induction of a circuit for any given value of the current i passing through it is the total magnetic flux of induc- tion through the circuit, which is due to the cuwent i. If this current is a variable one, the self-induction will vary also. If the induction is produced by a coil of wire with an air core, it will vary in the same manner as the current ; but if there is iron in the magnetic circuit, the law of variation of the self-induction is a less simple one. The coefficient of self-induction of a circuit generally denoted by L is a quantity which it is occasionally useful to know ; it is made much use of in the analytical treatment of alternate current problems, and leads to some simplifications when it is permissible to assume a complete absence of iron in the magnetic circuit ; in which case L is constant, and may be defined as the amount of self-enclosing of magnetic lines by the circuit when the current has unit value. For the coefficient of self-induction takes into account the number of times that the total induction N is threaded through the circuit : for instance, if a circuit takes one or two turns D 34 SELF-INDUCTION upon itself, it will be the same thing as regards the magnitude of the induced E.M.F. as if it formed only one loop, but enclosed two or three times the amount of magnetism which we have called the self-induction. Thus, if we consider a coil of wire of 8 turns, the self- induction (if there is no iron) will be proportional to the ampere-turns Si ; but the coefficient of self-induc- tion, L, will be proportional to S 2 . If the coil of wire had consisted of only a single turn, the coefficient of self-induction might have been denned as the number of magnetic lines which would be threaded through the circuit if one (absolute) unit of current were flowing. Let us denote this by N ; then, if, instead of having only one turn, the coil be supposed to consist of $ turns, the self-induction will be S times N , and the amount of self-enclosing of magnetic lines will be N $ 2 . This may be written N$, where N stands for the actual number of lines threaded through a coil of 8 turns when unit current is passing through the coil. When there is iron in the path of the magnetic lines, the coefficient of self-induction will no longer be constant, but will depend to a certain extent upon the value of the current. We can, however, still consider L as a coefficient which, when multiplied by the maximum value of the current, will give us the total amount of self-enclosing of magnetic lines, if we put .... (6) COEFFICIENT OF SELF-INDUCTION 35 where / stands for the maximum value of the alternating current actually flowing in the circuit ; and this defini- tion of L applies equally well to the case of a circuit in which there is no iron, only, as N is then proportional to I, L will have a constant value which does not depend upon the current. 14. Electro-motive Force of Self-induc- tion. Under the heading MAGNETIC PRINCIPLES, the relation between the magnetic flux and the induced E.M.F. has already been discussed. It was there shown that the E.M.F. generated in a coil is pro- portional to the rate of ' cutting,' or of enclosing or withdrawing magnetic lines. This relation between the changes of magnetism and the induced E.M.F. is, of course, in no wise altered if the magnetism producing this E.M.F. is due to the current flowing in the coil itself. Let / be the maximum value of an alternating current passing through the coil, and N the total amount of magnetic flux produced by this current ; then, since in one complete period the magnetic lines denoted by N are twice created and twice withdrawn, it follows that the mean value of the induced E.M.F., or E.M.F. of self-induction, will be ... (7) where n stands for the frequency of alternation, in com- plete periods per second ; and this equation is true, D 2 36 SELF-INDUCTION whatever may be the shape of the current wave. Putting in the place of N its value LI/ 8, which may be deduced from (6), and assuming the current wave to be a sine curve, in which case the mean value of 9 the current, I m , is equal to - times its maximum value, 7T J, we obtain the expression E m =2imLxI m . . (8) which is well known in connection with the analytical treatment of the subject. Since, in the above equation, L is measured in absolute C.G.S. units, both E m and I m will, of course, be expressed in the same units. If the E.M.F. is to be given in volts and the current in amperes, we should have to write e m = 2 r=\/2, which makes p=2irn^ as already stated. INDUCTANCE 39 16. Inductance. In the expression e=pLi, the quantity pL is sometimes called the inductance * of the circuit ; and it follows that, if we multiply the induc- tance by the current flowing, we obtain the E.M.F. of self-induction, e, or that component of the applied E.M.F. (exactly equal and opposite to e) which is required to overcome the inductance of the circuit and thus allow the current i to flow through it. This way of looking at the question of the self- induction of a circuit conveying an alternating current has found favour on account of the resemblance between the expressions =r = i and Ohm's law, ^=i', but it pL R must be remembered that since, as already pointed out, the coefficient of self-induction, L, will sometimes depend upon the strength of the current the inductance, pL, of a given circuit will not necessarily be constant, even with constant frequency. Also, in the case of Ohm's law, the component e of the impressed E.M.F. which produces the flow of current i through the non-inductive resistance R is exactly in phase with the current, and the product e x i always represents expenditure of energy; whereas the E.M.F. of self-induction being exactly one quarter period behind the current producing it, the power represented by the product of e and i in the expression e (pL)i is always equal to zero. This will be made clearer by what follows. * This quantity (pL) is also called the reactance-, the name in- ductance being then given to the coefficient of self-induction (L). 4U SELF-INDUCTION 17. Current Flow in Circuit having Appreciable Self-induction. In order to get a better understanding of the whole question of self- induction in connection with alternating currents, let us consider an alternating current flowing in a circuit which has both ohmic resistance and inductance, as, for instance, a coil of wire of many turns, which, for the \ FIG. 8 present, we will assume, has no iron core. Such a current is shown graphically by the curve c in fig. 8, where intervals of time are measured, as usual, hori- zontally from left to right. The magnetism due to the current c will vary in amount and direction in accordance with the variations of the current. It may CIRCUIT HAVING SELF-INDUCTION 41 be calculated in the usual way for any given value of c, provided we know the length and cross-section or the magnetic resistance of the various parts of the magnetic circuit, and the number of turns of wire in the coil. Let the curve m represent the rise and fall of this magnetism. Since the induced or back E.M.F. due to these variations in the magnetic induction will be proportional to the rate of change in the total number of magnetic lines threaded through the circuit, we shall have no difficulty in drawing the curve E 15 which represents the E.M.F. of self-induction. It is only necessary to remember that one hundred million C.G.S. lines enclosed or withdrawn per second will generate a mean E.M.F. of one volt per turn of wire in the coil. Regarding the direction of the resulting induced E.M.F., a very simple rule will enable us to determine this with absolute certainty in every possible case. We have merely to bear in mind that the direction of this E.M.F. is always such as will tend to produce a flow of current opposing the changes in the magnetic induction. Thus, during the time that the magnetism is rising from its zero value to its maximum positive value, the induced E.M.F. will be negative ; and all the while that the magnetism is falling from its maximum positive value to its maximum negative value, the induced E.M.F. will be in a positive direction; thus tending to produce a current which would prevent the fall, or check the rate 42 SELF-INDUCTION of decrease, of the magnetic flux. It follows that the induced E.M.F. must always pass through zero value at the time when the magnetism threaded through the circuit is at its maximum. A graphical method of obtaining the curve of induced E.M.F. from the curve of magnetisation will be considered in due course. With regard to the relation between the direction of the magnetising current c and that of the magnetic flux m, this is too well known to necessitate its being dwelt upon here. The analogy between the forward motion of a corkscrew and the positive direction (i.e. from S to N) of the magnetic flux, on the one hand, and the right-handed rotation of the corkscrew and the clockwise circulation of the current which produces the forward flow of magnetism, on the other hand, is a very useful one, and is of more general application than many others. Having drawn the curve E l (fig. 8), which, it will be seen, lags, as already stated, exactly one quarter period behind the current wave, we are now in a position to determine the potential difference which must exist at the terminals of the circuit in question in order that the current C will flow through it. Draw the curve E a to represent the E.M.F. required to overcome the ohmic resistance. It will be in phase with the current, because its value at any point is simply C * -R, where R stands for the resistance of the circuit. Now add the ordinates of E 2 to those of an imaginary CIRCUIT HAVING SELF-INDUCTION 43 curve exactly similar but opposite to E n and the resulting curve E will evidently be that of the impressed potential difference which, if maintained at the ends of the circuit under consideration, will cause the current c to flow in it. Thus we see how the relation between the impressed E.M.F. and the resulting current may be graphically worked out for any given case. From a study of the curves in fig. 8 it is evident that the effect of self-induction is to make the current lag behind the impressed E.M.F. If the E.M.F. re- quired to force the current against the ohmic resistance is small in comparison with the induced E.M.F., the lag will be very considerable; it cannot, however, exceed one quarter of a complete period, which limit is only reached when the E.M.F. of self-induction is so large, and the ohmic resistance of the circuit so small, as to render the E.M.F. required to overcome this resistance of no account. In order to briefly sum up the principles governing the flow of an alternating current in a circuit having self-induction, we may say that the varying current produces changes of magnetism, which again produce a varying E.M.F., called the E.M.F. of self-induction. This, together with the E.M.F. already existing (and without which no current would flow), produces the effective or resultant E.M.F. By dividing the value of this resultant E.M.F. at any instant by the total ohmic resistance of the circuit, the corresponding current 44 SELF-INDUCTIOX intensity is obtained. This condition must always be fulfilled, otherwise Ohm's law will not be satisfied. 18. Graphical Method of Deriving the Curve of Induced E.M.F. from the Curve of Magnetisation. Let the curve m in fig. 9 represent (as in fig. 8) the variations in the magnetic flux through the circuit which we have been considering. FIG. 9 Since the vertical distances above or below the datum line are a measure of the total number of magnetic lines passing at any moment through the circuit (the lapse of time being measured horizontally), it follows that the * slope ' or ' steepness ' of the curve m will give us, for any point on the curve and, therefore, at any particu- lar instant the rate at which the magnetic flux is CURVE OF INDUCED E.M.F. 45 changing. Thus, by drawing at any point P the tangent OP to the curve m, and then dividing the amount of the magnetic flux Pp by the lapse of time op, we obtain a number which is proportional to the induced volts pe^ and which enables us to plot the point e^ of the curve E p which will be that of the induced E.M.F. For instance, if the ordinate ?p represents 100,000 C.G.S. lines, and the distance op the two-hundredth part of a second, then, if the circuit makes one hun- dred turns upon itself, the instantaneous value (pe^ of the induced volts will be 10 5 x 200 x 100 x 10~ 8 = 20 volts, which must be plotted below the datum line, because this E.M.F. will be such as will tend to produce a current in a negative direction, i.e. such a current as would oppose the variation of the mag- netic flux, which is increasing in amount. At the point A, when the magnetism has reached its maximum positive value, the tangent to the curve is horizontal : the rate of change in the magnetism is there- fore zero, and the point a will be on the datum line. The maximum value of Ej will correspond to that point of the (falling) magnetisation curve which is steepest ; but this is not necessarily the point 5, where the curve m crosses the datum line. It is, however, interesting to note that the line bE always divides the curve aEt into two equal areas, abE and bEt y whatever may be the shape of the magnetisation curve. If the latter is 46 SELF-INDUCTION a curve of sines, the curve E l will also be a curve of sines, of which the maximum ordinates will correspond with the zero values of the curve m. To those unacquainted with the differential calculus it may not be quite clear why the ' slope ' of the tangent OP is a measure of the rate of growth or of decrease FIG. 10 of the number of magnetic lines threaded through the circuit. Let us therefore consider (see fig. 10) a magnified portion of the curve m of fig. 9, at the point P. Let Q be another point on the curve m, situated at a small distance from P. Join PQ : then, since the amount of the magnetic flux through the coil has y v ^=*^ .M.F. 47 CURVE OF INDUCED E increased by an amount dN (represented by the distance Qr) in the time dt, it follows that the average rate of increase of the magnetisation, while changing from its value at P to its value at Q, will be given by the fraction SI or , which will therefore be a measure of the pr dt mean value of the induced volts during the time taken by the magnetism in growing from P to Q. But the fraction is the trigonometrical tangent of the angle QPr which the line PQ makes with the horizontal line Pr. If, therefore, we imagine the point Q to move nearer and nearer to the point P, the limiting position of the line PQ will be the tangent to the curve at the point P ; and the ordinate pe l of the curve of induced E.M.F. (fig. 9) will therefore be proportional to the trigonometrical tangent of the angle (. = tVB*,, . .(18) where =the thickness of plates in inches, n =the frequency in periods per second, B,,=the maximum induction in C.G.S. lines per square inch. That this formula is not theoretically correct will be evident from the fact that the loss of power is assumed to be proportional to the square of the maximum induction B, which must depend, to a certain extent, upon the shape of the applied potential difference wave. Strictly speaking, the eddy current losses should be expressed in terms of the E.M.F. of self-induction generated in the coil itself, for they will be proportional to the square of this quantity. Although the power wasted by eddy currents is actually spent in heating the iron which lies in the path EDDY CURRENTS 59 of the magnetic lines, it will, of course, have to be put into the magnetising circuit in the form of electrical energy ; that is to say, there will be a certain com- ponent of the total current required solely to balance the eddy currents, and which will therefore be unavail- able for magnetising the core. Thus, in the case of a choking coil through which the current is kept constant whatever may be the eddy current loss, the E.M.F. of self-induction will be less when eddy currents are present than it would be if these could be entirely eliminated. If, on the other hand, the volts across the terminals of the choking coil be kept constant, the current in the coil will increase in such a manner as to neutralise the demagnetising effect of the eddy currents, and thus leave the induction (and consequently the induced E.M.F.) practically the same as it would be if there were no eddy currents in the iron core. In fact, the effect of the eddy currents upon the mag- netising coil will be almost exactly the same as if current were taken out of a secondary coil, as in the case of transformers (which will be discussed in due course), and there will be a current component added which will be in phase with that component of the applied potential difference which balances the in- duced E.M.F. It follows that if we multiply these two quantities together we shall obtain the measure (in watts) of the power wasted by eddy currents in the iron core. 60 SELF-INDUCTION 24. Hysteresis. Although no energy is lost in producing changes of magnetism when the path of the magnetic lines is through air only, as soon as iron is introduced to convey and increase the amount of this magnetism, power is spent in magnetising and demagnetising the iron core, owing to what Professor Ewing has called Hysteresis. This second source of loss, it should be clearly understood, is quite independent of the eddy current loss, and in no wise depends upon whether the iron is laminated or not. It is well known that all iron, even the softest and purest, retains some magnetism after the magnetising force has been removed. By applying a magnetising force in the opposite direction this residual magnetism is destroyed, and the magnitude of this force or, in other words, the amount of work which has to be done to withdraw this magnetism depends upon the quality of the iron. Soft annealed wrought iron retains most magnetism ; but, on the other hand, it parts with it more easily than the harder qualities of iron and steel, and for this reason requires the least expenditure of energy to carry it through a given cycle of magnetisa- tion. In fig. 16 is clearly shown the effect of hysteresis, which is to make the changes in the magnetism lag behind the changes in the exciting current. This curve is the ordinary magnetisation curve of a sample of HYSTERESIS 61 transformer iron, and it gives the exciting force and corresponding induction for a complete cycle. The magnetising current is measured horizontally, on each side of the centre line, and may be considered positive when to the right, and negative when to the left of the latter. The length of the ordinates above or below the horizontal centre line is a measure of the induction in the iron for any particular value of the exciting cur- rent; the direction of the magnetism will be positive if the ordinates are measured above this datum line, and negative if measured below. Starting at a, and follow- ing the curve in the direc- tion of the arrow, we see that it has required a certain definite positive current indi- cated by the distance oa to entirely withdraw all the negative magnetism. From a to b both current and induction rise to their maximum values ; but from b to c, although the current falls from its greatest value to zero, the induction changes very little; in fact, the residual magnetism at c is not much below the maxi- mum amount. However, as soon as a negative exciting force is applied, the magnetism decreases rapidly until FIG. 16 62 . SELF-INDUCTION when d is reached, and all the positive magnetism has been removed, it will be seen that the current has reached a negative value od, exactly equal to its positive value at a. If the sample experimented upon were a closed ring which had been magnetised to what may be considered the saturation point, the residual magnetisa- tion (represented by the distance oc) might be expressed as a percentage of the saturation value, and it would then depend only upon the magnetic properties of the iron used in the test. Again, on the assumption that the point b represents the practical limit of magnetisation, the distance oa which is a measure of the magnetising force required to withdraw the whole of the residual magnetisation will also have a definite value depending upon the quality of the iron under test, and it will not even depend upon the shape of the sample, but only upon the limiting value of the magnetisation. For these reasons we are justified in giving to the demag- netising force, defined as above, a name of its own ; and Dr. Hopkinson has called it the co&rcive force. Instead of being expressed in terms of the current passing through the exciting coil, the coercive force may also be defined as the demagnetising field, H (see 5, under MAGNETIC PRINCIPLES) required to neutra- lise the residual magnetism, and this is the usual definition of the term. HYSTERESIS 63 In fig. 17 the curve of magnetisation, m, has been drawn in the same manner as in fig. 8 (p. 40), its maximum ordinate being scaled off fig. 16. The cur- rent curve c has then been plotted from measurements taken on the curve, fig. 16, which gives the current inten- sity corresponding to any particular value of the induction. FIG. 17 In order to show quite clearly the manner in which the current curve c is derived from fig. 16, a few corresponding points on the two curves have been marked and numbered. Thus, in fig. 16, the distance of the point 5 from the vertical centre line gives us the magnetising current for a certain definite value of the induction. This magnetising current is repre- sented in fig. 17 by the distance of the point 5 from UNIVERSITY TAl 64 SELF-INDUCTION the horizontal datum line, and the ordinate of the magnetisation curve which passes through this point will be found to be exactly equal to the ordinate of the corresponding point in fig. 16. By comparing fig. 17 with fig. 8, it will be seen that the introduction of an iron core in the place of an air core (even if eddy currents are assumed to be absent) has considerably distorted the current wave, which now no longer rises and falls in synchronism with the magnetisation wave, but, in general, pre- cedes it ; although it should be noted that except when eddy currents are present the maximum values of the induction and FlG> is current still occur at the same instant. 25. 'Wattless' and 'Hysteresis' Com- ponents of the Magnetising Current. The dotted centre line hob, in fig. 18, may be taken as representing the relation between exciting current and magnetism on the assumption that hysteresis is absent. Hence it follows that, at any point in the real cycle, the portion of the current which may be considered as doing COMPONENTS OF MAGNETISING CURRENT 65 the work against hysteresis is indicated by the length of the current line comprised between the dotted line and the outside curve. This current will be positive when measured to the right of this new centre line, and negative when to the left. The short horizontal lines in the figure represent the successive positive values of this hysteresis current during the change of FIG. 19 the magnetism from its negative to its positive maximum. In. fig. 19 the curve of magnetisation is marked m as before. C E is the imaginary exciting current for the assumed condition of no hysteresis in the iron, it is plotted from measurements taken on hob in fig. 18, while C H is what we may call the hysteresis component of the current, and is also F 66 . SELF-INDUCTION drawn from measurements made on fig. 18. The resultant or true magnetising current is, at every point, equal to the algebraic sum of these two. 26. Power lost owing to Hysteresis. The impressed potential difference required in order that the current c may flow in the circuit will consist as already fully explained of two components : one equal to c x R, which is required to overcome the ohmic resistance ; and the other exactly equal and opposite to the E.M.F. induced in the circuit owing to the rise and fall of the magnetism, m. It is evidently the pro- duct of this last component of the impressed potential difference and the current c which will give us the power supplied to the circuit on account of the hysteresis losses in the core : the latter, it should be added, being due to a kind of internal or molecular friction which has the result, as in the case of the eddy current losses, of heating the iron core. Now, the phase of the induced E.M.F. being always, as previously explained, exactly one quarter of a period behind that of the magnetism, it follows that the current component C E is wattless; but if we take the ordinates of C H and multiply them by the ordinates of the induced E.M.F. curve (not shown in fig. 19), we obtain the rate (in watts) at which work is being absorbed through hysteresis. Note also that if the frequency remains constant, the induction B (fig. 18) is a measure of the induced E.M.F., and that the lost POWER LOST OWING TO HYSTERESIS 67 watts are, therefore, directly proportional to the product of C H and B, or to the area of the curve fig. 18. There is another way of proving that the area of the hysteresis curve is a measure of the energy spent in carrying unit volume of the iron through one complete cycle of magnetisation. Let us suppose that in fig. 20 we have plotted the magnetising force in ampere-turns (Si) per unit length of the iron core, and the corresponding induction B in C.G.S. lines per unit cross-section of the core, for one complete cycle of magnetisation. Consider the right-hand portion of the curve, from n to b. The magnetising ampere-turns, Si, have risen from zero value at n to their maximum positive value gb ; while the magnetic induction B has changed from its negative residual amount on, to its positive maximum amount og. Let us suppose this change to have taken place in the time t seconds. Then, if A is the cross-section of F 2 FIG. 20 68 SELF-INDUCTION the core and Z its length, the mean induced back E.M.F. due to the change in the magnetism will evidently be: Ax (nq) x 81 and since the quantity of electricity which has been moved against this E.M.F. is equal to the mean value, i m , of the current between its value at n and at b, multiplied by the time t, it follows that the work done is equal to : Al jQi x Al or to x the area nabg. Thus we see that a distinctly appreciable amount of energy has been spent in raising the magnetic induction from its value on to its maximum value +og, if, now, we destroy this magnetism, the whole or a part of the energy required for its production will be restored to the circuit, owing to the fact that the E.M.F. induced by the withdrawal of the magnetism will be in the opposite direction to what it was before, i.e. it will now help the exciting current instead of opposing it. If hysteresis were absent, the magnetism would fall in the same way as it rose, thus restoring the whole of the energy to the circuit. But in fig. 20 the magnetism falls from b to c along the curve fee, during the with- drawal of the whole of the magnetising force, thus only POWER LOST OWING TO HYSTERESIS 69 restoring to the circuit the amount of energy represented by the area gbc. It follows that the amount of energy lost during one half period that is to say, while the current rises from zero to its maximum value, and falls again Al to zero will be equal to x the shaded area nabc. Exactly the same arguments apply to the left-hand por- tion of the curve (chri) , which is a repetition of the curve nbc ; and, since the product Al is equivalent to the volume of the iron core, we may write : work expended per cycle volume of iron in core , = - 8 - x area of hysteresis curve ; and since we have taken the back E.M.F. in volts and the current in amperes, the above amount of work will be expressed in joules, or practical units. But, since H = r } Si, instead of plotting B and Si, we may plot B and H, and, in order to express the energy lost in absolute C.G.S. units, we must convert joules into ergs (1 joule = 10 7 ergs) and take all measurements in centimetres ; then ergs per cycle per cubic centimetre = area of hysteresis curve 4-7T The work absorbed by hysteresis depends very con- siderably upon the quality of the iron. In any one sample it is approximately proportional to the l'6th power of the limiting induction, if the whole range of magnetisation is considered. For the straight part of 70 SELF-INDUCTION the magnetisation curve which corresponds to the low inductions, as used in nearly all alternate current apparatus, it has been pointed out by Professor Ewing that the losses do not increase so rapidly, but more nearly as the l'5th power of the induction; and it is for this reason that the law B 155 is adopted in the formula given below. The number of alternations per second does not appreciably influence the hysteresis loss per cycle, excepting when the latter is very slowly performed ; hence the watts lost per pound of iron for any par- ticular maximum value of the induction will be propor- tional to the frequency. The following formula applies to a good quality of transformer iron and gives the watts lost per Ib. ,,'. - .(20) where B,, is the maximum value of the induction in C.G.S. lines per square inch, and n is the frequency, or number of complete periods per second. If B stands for the induction in lines per square centimetre, the above formula becomes : To obtain the watts lost per cubic inch, multiply by 28. The curve, fig. 21, gives the relation between the POWER LOST OWING TO HYSTERESIS 71 limiting values of the induction and the watts lost per lb., at a frequency of 100, for the particular sample of transformer iron to which the above formulae apply. Curve giving hysteresis loss in watts per lb. of good quality transformer iron at a frequency of 100 9,000 a 5 000 ^ ^ 7 000 / ^ w 6000 S 5,000' / / 6 4,000 / 3 5 5 3,000 / / ( s 2,000 / 1 1,000 / H / 3 -2 -4 -6 -8 1 1-2 1-4 1-6 1-8 Power lost in watts per lb. at frequency n= 100 FIG. 21 An induction of 2,500 lines per square centimetre (approximately 16,000 lines per square inch) and a frequency of 100 are often taken as standard conditions for purposes of comparison of hysteresis losses in 72 SELF-INDUCTION various samples of iron. Only a few years ago, a loss per Ib. of 0-38 watts, under the above conditions, was not considered excessive for an average quality of com- mercial transformer iron : but great improvements have recently been made in the production of iron suitable for use in alternate current apparatus. The curve fig. 21 applies to a sample of iron giving a loss of 0*25 watts at B = 2,500, and an exceptionally good specimen tested by Professor Ewing in 1895 had a loss as low as 0-16 watts. TABLE GIVING TOTAL Loss IN WATTS PER POUND OF GOOD QUALITY, WELL INSULATED TRANSFORMER IRON STAMPINGS, '014 INCH THICK C.G.S. ifnes per sq. in. Frequency =50 Frequency =100 II Hysteresis Eddy Currents Total Hysteresis Eddy Currents Total 8,000 036 005 041 072 -019 091 9,000 041 006 047 082 -024 106 10,000 047 007 054 094 -030 124 11,000 054 009 063 108 036 144 12,000 061 Oil 072 122 043 165 13,000 069 013 082 138 050 188 14,000 077 015 092 154 058 212 15,000 086 017 103 172 067 239 16,000 -095 019 114 190 076 266 17,000 102 021 123 204 086 290 18,000 111 024 135 222 096 318 19,000 122 027 149 244 107 351 20,000 135 030 165 270 118 388 22,000 150 036 186 300 143 443 24,000 175 043 218 350 170 520 26,000 200 050 250 400 200 600 28,000 222 058 280 444 232 676 30,000 248 067 315 496 266 762 35,000 312 091 403 624 363 987 40,000 382 118 500 764 473 1-237 45,000 460 150 610 920 600 1-520 POWER LOST OWING TO HYSTERESIS 73 In the table on p. 72 the hysteresis losses have been taken from the curve fig. 21, and the eddy current losses have been calculated by means of formula (18). It is interesting to note that the eddy current losses are of more relative importance, both for the higher frequencies and the higher inductions. With low frequency and induction the eddy current losses are almost negligible in well-laminated iron. Thus, for B,, = 15,000 and %=50 the eddy current loss in plates 0'014 inch thick is only equal to 16 per cent, of the total loss. On the other hand, if an exceptionally good quality of iron is used, thus permitting the use of higher inductions, the eddy currents at a frequency of 100 may form a considerable part of the total losses. For this reason tests made on transformers have sometimes shown the eddy current loss under working conditions to be fully equal to half the hysteresis loss. 27. Vector Diagram for Inductive Circuit containing Iron. In 23 (p. 57) it was shown that the power absorbed by eddy currents in the iron was supplied to the circuit in the form of a component of the total current practically opposite in phase to the induced volts, i.e. in phase with that component of the impressed volts which balances the induced volts ; let us call this current component Cs. In 25 it was shown that the remaining portion of 74 - SELF-INDUCTION the total current, i.e. that portion of it which is available for producing the magnetic induction, may be divided into two components the c wattless ' component C E , in phase with the magnetisation wave, and the { hysteresis ' FIG. 22 component C H , exactly one quarter period in advance of C E , and therefore in phase with the eddy current com- ponent C s . In fig. 22 let oe l represent the induced E.M.F. Then C E , the wattless component of the exciting VECTOR DIAGRAM FOR INDUCTIVE CIRCUIT 75 current, will have to be drawn 90 in advance of oe r Draw c s and C H (both opposite to e^ and add them together, to form c w , the total ' work ' component of the current. Now add the vectors C E and c w in the usual way, and draw oc, which will represent the total current. If J?=the ohmic resistance of the circuit, e^ the effective volts (in phase with c), will be equal to c x R. Add e. 2 and the vector e\ (exactly equal and opposite to e^, and draw oe, which will represent the necessary impressed potential difference in order that the current C may flow through the circuit. It is interesting to compare fig. 22 with fig. 12 (p. 53). The current in fig. 12 (which applies to a circuit or choking coil without an iron or metal core) lags exactly 90 behind e\ ; whereas in fig. 22 the current is no more c wattless '" with respect to this component of the total E.M.F. Also, an inspection of fig. 22 will show that the true watts supplied to the circuit are still equal to the product c x e cos 0, and that the cosine of the angle may therefore still be defined as the ratio of the true watts to the appa- rent watts, or the ' power factor ' of the circuit. In order to make this quite clear, the vector oe has been projected upon oc. Then ocxoe cos 0= ocx om =oc xoe 2 + oc x e 2 m = C 2 jR losses + oc x e 76 SELF-INDUCTION But oc x e 2 m=oc x the projection of oe\ on oc = oe\ x the projection of oc on oe\ = 06^ X OC x OC _ ( watts lost by) (watts lost by ~ tedd currents] ( hsteresis y currents] ( hyste The product c x e cos 6 is therefore equal to the total watts lost, both in the copper of the circuit and the iron of the core. It follows also that the three volt- meter method of measuring the power supplied to an inductive circuit, which was described in 22 (p. 55), may still be used when a portion, or even the greater part, of the power supplied is transformed into heat in the iron core. 28. Design of Choking Coils. Although we have gone at some length into the question of the losses which occur when iron is introduced into an alternating current circuit, it must not be supposed that, in order to design a simple choking coil such as might be used, for instance, on an arc lamp circuit it is necessary to predetermine, with any degree of accuracy, the losses which will occur in the iron core. We have only to be careful that the induction in the iron is reasonably low in order that the temperature rise may not be excessive ; the actual losses will then be practically negligible. The above remarks do not, however, apply to trans- formers, in which the iron losses are of the greatest importance; they must be kept very small in order (OF THB CTNIVERS DESIGN OF CHOKING COILS CTNIVERSITY CALIEORH\* that the efficiency of the transformers at light loads may be as high as possible. Let us suppose it is required to design a choking coil for use in series with a 10 ampere arc lamp on a 100 volt alternating current supply; the object of the choking coil being to reduce the voltage at the lamp terminals to 40. Since the E.M.F. of self-induction is very nearly 90 out of phase with the current, and therefore with the effective E.M.F. (for we are assuming the iron losses to be negligible), the back E.M.F. to be produced by the choking coil must not be (100 40) = 60 volts, but more nearly >y/T00 2 -40 2 =92 volts. This, therefore, is the voltage which would be measured across the terminals of the choking coil, on the assumption that the ohmic resistance of the winding is also negligible. In order to produce this back E.M.F., it is necessary that the fundamental formula e m = (see p. 37) should be satisfied. But this gives the relation between the induction and the mean value of the induced volts. We must therefore multiply e m by the ' wave constant ' in order to get the \/ m ean square value of the induced 4N $Vi volts. Hence, e=m - , and, on the assumption that the induced E.M.F. follows the sine law which in nearly all cases will be sufficiently near to the truth 78 SELF-INDUCTION for our purpose ??i=l'll, which enables us to write : _ Q9 _4-44N?i , 10* Knowing the frequency ?i, we can therefore calculate the product N$. It will be evident that we can vary the two factors, N and S (the number of turns in the coil) in whatever way we like, provided their product remains constant ; in other words, 8 must vary inversely as N . Again, for any given arrangement of the magnetic circuit there is only one definite value of S which will give the required result, since the current passing through the coil is of constant strength (in this case 10 amperes). It is therefore necessary to know what value of the induction, or of the total flux N, corre- sponds to any given value of the magnetising ampere- turns. This relation may be determined either by experiment, or, if no great accuracy is required, by calculation in the usual way from measurements of the (proposed) magnetic circuit, and with the aid of the usual magnetisation or permeability curves. It will almost certainly be found that a closed magnetic circuit is not suitable in the case we have taken as an example, because the number of turns, 8, would have to be very small in order to keep the induc- tion in the iron within reasonable limits, thus making it necessary for N and therefore the cross-section of the core to be very great. However, by introducing an DESIGN OF CHOKING COILS 79 air gap in the magnetic circuit, the resistance of the latter may be readily increased so as to allow of a proper number of turns being wound on the core. In calculating the ampere-turns, it will generally be sufficiently accurate to assume that if we multiply the current by 1*4 we shall obtain its maximum value (in this case 14 amperes), which, as far as magnetic effects are concerned, is what we require to know. The following empirical formula may also be of use : it gives the ampere-turns required per inch length of the iron circuit for a good quality of transformer iron : p Ampere turns per inch = -^-- + 2 . . (22) where B,,=the induction in the iron in C.G.S. lines per square inch of cross-section. The above rule applies only to cases where B,, lies somewhere between 12,000 and 40,000, and it is based on the assumption that the curve connecting exciting force and induction is a straight line between these limits. For air, the formula : Ampere-turns per inch = 3 1 3 B , , . (23) is true for all values of B,,. 80 CAPACITY 29. Definition of Quantity and Capacity. The unit quantity of electricity is the coulomb. A coulomb is defined as the quantity of electricity conveyed in one second when the current is one ampere. The capacity of a condenser is the number of coulombs required to be given to one set of plates in order to produce a difference of potential of one volt between the two sets of plates ; or, since the capacity of a condenser is constant per volt difference of potential at the terminals, whatever this difference of potential may be, it may be defined as the ratio of the charge, in coulombs, to the potential difference, in volts, between the coatings. Hence, if K = capacity of condenser in farads, F= volts applied at terminals, the charge in coulombs =Kx V. The microfarad is the one-millionth of a farad. To calculate capacity ; let ET, n =the capacity in micro- farads ; ^i = the area of each set of plates in square CAPACITY 81 inches; = the distance between the plates; then, if the plates are separated by air, Jf " i= 4-452xl0 6 x* * If the plates are not separated by air, the capacity will be found by multiplying the above value of K by the specific inductive capacity of the dielectric. This multiplier is about 3 for vulcanized rubber, 5 for mica, and 10 for very dense flint glass. Capacity of Cylindrical Condenser. If Z=the length in centimetres, D and d the diameters of the outer and inner coatings respectively, then K = l ~ 2 log e D where K a is the capacity in absolute electrostatic units, on the assumption that the specific inductive capacity is unity. Since the microfarad is approximately 900,000 times greater than the absolute electrostatic unit, the above expression must be divided by this number in order to convert it into practical units, and if we put \ for the length in feet, and convert the Neperian logs, into common logs., we may write : K - 7 ' 353 x *' " 10" X lo glo D. (25) d 82 CAPACITY The following measurements of the capacity per mile of high-tension electric light cables have been kindly furnished by the makers ; they all refer to 19/18 concentric cables, the capacity given being that between the inner and outer conductors : British Insulated Wire Co. (Paper) . -31 microfarads W. T. Glover & Co. (Vulcanized Eubber) -615 (Diatrine) . . -315 Condensers Connected in Parallel. The joint capacity of a set of condensers connected in parallel is evidently equal to the sum of the several capacities. Condensers Connected in Series. The capacity of several condensers connected in series will be less than that of any single one of the condensers, because the arrangement is evidently equivalent to increasing the distance between the plates of any one condenser, and the reciprocal of the total capacity will be equal to the sum of the reciprocals of the several capacities, or K= * A charged condenser must be considered as con- taining a store of electric energy which may be used for doing useful work by joining the terminals of the condenser through an electric circuit. In this respect it may with advantage be compared with a deflected spring, which, so long as it is kept in a state of strain, has a certain capacity for doing work. CIRCUIT HAVING CAPACITY 83 30. Current Flow in Circuit having appre- ciable Capacity. Consider an alternator (fig. 23) connected through the circuit R to the condenser K. If we neglect the resistance of the circuit, the amount of current passing will evidently be determined by the E.M.F. e, and the capacity K of the condenser. As long as the current is flowing in a positive direction, the condenser is being charged ; when the current reverses (second half of complete period), the condenser is discharging. It follows that the FIG. 23 maximum charge and therefore the maximum value of the condenser E.M.F. will always occur at the instant when the current is changing its direction. This is clearly shown in fig. 24. Here the curve c represents the current flowing through the condenser. At the instant b, when, from having been flowing in a positive direction, it is passing through zero value before flowing out of the condenser again, the charge in coulombs (represented by the curve q) has evidently reached its maximum positive value d, and will now fall, through zero, to its maximum negative value, which G 2 84 CAPACITY will occur at the moment when the current is changing from its negative to its positive direction that is to say, at the moment when the positive current has ceased to flow into the opposite set of con- denser plates. When a condenser is charged by means of a current flowing in a positive direction, it is evident that the condenser E.M.F. will be negative in other words, it will oppose the applied E.M.F. required to force the current into the condenser, and will therefore tend to expel the charge. Hence we may draw the curve e n exactly opposite to g, and such that its ordinates are at every point equal to ^, where K is the capacity K. of the condenser. Since we are neglecting the resistance of the alternator and leads, the curve e l which is that of the potential difference at the con- denser terminals, will be exactly equal and opposite to the alternator E.M.F. (fig. 23). Thus we see that the condenser E.M.F. is exactly one quarter period in advance of the current ; whereas, when dealing with the question of self-induction, we found that the back E.M.F. of self-induction lagged exactly one quarter period behind the current. 31. Determination of Condenser Current. Let K be the capacity, in farads, of a condenser con- nected in series with an alternating current circuit ; and let V stand for the maximum value of the volts DETERMINATION OF CONDENSER CURRENT 85 at condenser terminals ; then the total charge in coulombs at the end of each half-period will be equal to K x V. But quantity = current x time, hence maximum charge in coulombs = L x , where i /)t = the mean value i FIG. 24 of the current between zero and its maximum (i.e. from a to ?>, fig. 24) and = the time taken by the current to change from its maximum to its zero value, n being the frequency in complete periods per second. 86 CAPACITY It follows that = KV or .... (26) The quantity we generally require to know being the \/ mean square value of the current, let us write (see ?"' j-ssm, where i stands for the *J mean square "m value of the current flowing through the condenser ; and -=r 5 where v is the \/ mean square value of the condenser E.M.F. in volts. It follows that equation (26) may be written in the form : or, if the capacity is expressed in microfarads, i=pK m vxlQr 6 . . (27) If the current is a sine curve, the multiplier p is equal to 2Trn, as already shown ( 15), and . (28) 32. Vector Diagram for Current Flow in Circuit having appreciable Capacity. In fig. 25, oc is the current; oe 2 , the effective component of the impressed potential difference (equal to C x R, where R is the resistance of the circuit connected to the alternator terminals, see fig. 23); oe,, the condenser CURRENT IN CIRCUIT WITH CAPACITY 87 E.M.F., drawn at right angles to oc in the forward direction ; and oe is then the necessary impressed potential difference, obtained by adding the force oe 2 to the force oe\ (exactly equal and opposite to e^ in the usual way. The angle 6 is the phase difference between the impressed potential difference and the resulting current, and it is a measure of the amount by which FIG. 25 the latter is in advance of the former. This diagram should be compared with fig. 12, p. 53. Condenser in Parallel. A problem of more practical utility than the above is that of a condenser or condensers connected in parallel with the main circuit ; because this arrangement is almost exactly equivalent to the case of a concentric feeder, as used in nearly all systems of 88 CAPACITY alternate current supply in this country. In fact, for all calculations of capacity current, &c. in long con- centric mains, the concentric conductor rnay be considered as consisting of two ordinary conductors bridged across by a number of condensers. In fig. 26 the alternator, which generates e volts at its terminals, is supplying a circuit of resistance r + R, and which, for simplicity, we will suppose to be without appreciable self-induction. This circuit is bridged by FIG. 26 the condenser K, the connections to which may be assumed to have no appreciable resistance. Let c 2 be the current in the remote section, N ; it is evidently determined by the voltage e } at the con- denser terminals, and, since we are assuming no self- induction in the circuit, it will be in phase with e l and p equal to l . Also, the condenser current Cj is a quarter R period in advance of e l (for we are considering e l as the volts supplied to the condenser terminals) ; hence the total current, c, in section M, which'is equal to the CURRENT IN CIRCUIT WITH CAPACITY 89 sum of these two currents, will be represented by the expression */c 2 l + c 2 2 . In addition to the above, we have the relation e f =cxr 9 where e. f stands for that component of the applied E.M.F. which sends the total current c through the resistance r. We are now in a position to draw the vector diagram for this particular circuit. Let us suppose the E.M.F. at condenser terminals (e } ) to be known. Draw FIG. 27 oe l (fig. 27) to represent this force. Divide e l by R to get the current c 2 in the remote section, and lay down oc 2 to scale. Now calculate the condenser current c l =pKe l and draw OC L at right angles to oe l in the forward (clockwise) direction. The resultant oc of c 2 and Cj is evidently a measure of the total current c in the section M. By multiplying c by the resistance r we obtain the effective E.M.F. e r , required to overcome the resistance of the section M. It is now merely necessary to compound the two forces e l and e r in order to obtain 90 CAPACITY e, the potential difference required at the alternator terminals in order that the voltage at the condenser terminals shall be e r The angle <, which represents the phase difference between the impressed volts on the whole system and the volts at condenser terminals, can be shown to be such that its trigonometrical tangent is equal to $- -. 33. Capacity Current in Concentric Cables. In very long concentric feeders, such as are used in many alternating current electric lighting systems, the current flowing into the cable when there is no connection at the distant end is sometimes quite appreciable, and, especially when the cables are insulated with vulcanized rubber and the frequency is high, it may amount to as much as 10 amperes, or even more, on a 2,000-volt circuit. This is the capacity current ; and it follows from what has been said that if the load at the far end of the feeder be non-inductive, the in-going current at the station end will necessarily be greater than the out-going, or work^ current. If c=the in-going current, c 2 =the out-going current =60 amperes, and Cj = the condenser current = 10 amperes, then c=/v/c 2 2 + c 2 j = 60*82 amperes. With regard to the statement that the current due to capacity is in advance of the impressed E.M.F., this CAPACITY CURRENT IN CONCENTRIC CABLES 91 is sometimes objected to on the ground that the effect cannot precede the cause, and it has even been suggested that we should speak of the current, not as leading by one quarter period, but as lagging by an amount equal to three quarters of a complete period. There is certainly no great objection to doing so, seeing that we are dealing with two periodically varying quantities, in connection with which there is no beginning and no end to be considered ; but it is to be feared that such a way of stating the case would lead to some confusion regarding the manner in which an alternating current flows into arid out of a condenser (see p. 83). It is not necessary to spend much thought on the question in order to see that a periodically varying current may well be in advance of a periodically varying E.M.F., without such a state of things being in any way equivalent to a flow of current occurring in a circuit before the existence of a difference of potential between its terminals; and, in any case, since it is a fact that the condenser current is in advance of the impressed alternating E.M.F., discussions as to whether or not it is correct to say so cannot rightly be considered as serving any useful purpose. Before leaving this section on Capacity, in order to deal with a few general problems concerning the flow of currents in circuits possessing self-induction in addition to capacity and ohmic resistance, it should be stated that if this question of capacity has been treated 92 CAPACITY less fully than the subject of self-induction, it is mainly due to the fact that the latter quantity is of very much more relative importance than capacity in alternating current work. With regard to what has been called * dielectric hysteresis,' and the other and more important causes of losses in condensers when used in connection with alternating currents, it is hardly necessary to concern ourselves with these ; but it may be stated that these losses, in condensers with solid dielectrics, are approxi- mately proportional to the square of the applied volts at the terminals. In fact, a commercial condenser behaves almost exactly as if a small non-inductive resistance (which may be termed the spurious resistance of the condenser) were joined in series with a theoreti- cally perfect condenser, i.e. one in which there is no loss of energy when connected to a source of alternating E.M.F. It has also been found that the losses in a condenser are less after the latter has been thoroughly dried. 93 SELF-INDUCTION AND CAPACITY 34. Resistance, Choking Coil, and Con- denser in Series. The arrangement shown in fig. 28 consists of an alternator generating e volts at its terminals, and causing a current c to flow through a circuit of resistance = r ; inductance =pL ; and reac- tance (due to the condenser) = . FIG. 28 The vector diagram, which gives the relation between the various component E.M.F.s and the current, is drawn in fig. 29. Here oc represents the current, and oe r the resultant or effective E.M.F., equal to c x r, and in phase with the 94 SELF-INDUCTION AND CAPACITY current ; oe^ at right angles to oc in the direction of retardation, is drawn to the same scale as e n and repre- sents the total E.M.F. of self-induction; oe^ also at right angles to oc, but in advance, is the condenser E.M.F. If the latter were equal to zero which would be the case if the condenser were of infinite capacity, or if it FIG. 29 were entirely removed the total volts required at the alternator terminals would be equal to om, obtained by adding to e r an E.M.F. exactly equal and opposite to e L . But since the condenser E.M.F. is equal to oe K , the volts actually required at the alternator terminals will be equal to oe, obtained by compounding om and a force CHOKING COIL AND CONDENSER IN SERIES 95 exactly equal and opposite to e K . Thus we clearly see how the effect of capacity is to counteract the effect of self-induction. The alternator E.M.F., instead of being in advance of the current by the amount indicated by the angle moe n now lags behind the current, as shown by the angle eoe r . Also, since the distance e r e is equal to the difference between the condenser E.M.F. and the E.M.F. of self-induction, it follows that when these two forces are equal they will balance each other ; the alternator volts will be equal to oe r , and the circuit will behave in all respects as if it possessed ohmic resistance only. It is hardly necessary to point out that the slightest change in the frequency, or the capacity, or the coefficient of self-induction would instantly upset this balance. This is more especially the case with regard to the frequency, since the ratio of e L to e & is dependent upon the square of the frequency. It will be evident from a glance at the diagram that either the condenser E.M.F. or the choking coil E.M.F. may be greater than the impressed E.M.F. applied to the circuit at the alternator end. That is to say, if we connect a voltmeter across the terminals of the condenser, or of the choking coil, the reading on this voltmeter may be higher than that which would be read on a voltmeter connected across the alternator terminals. Let us, for simplicity, suppose that r, L^ and n (the frequency) remain constant, but that the capacity, K, 96 SELF-INDUCTEON AND CAPACITY of the condenser can be varied at will ; and let us plot a curve showing the relation between the capacity, K, and the volts across the condenser terminals, on the assumption that the alternator volts remain constant. This is easily done by merely altering the length oe K in the diagram, fig. 29, and calculating the ratio - for various values of K. e An inspection of the diagram will, however, lead to one or two interesting conclusions. Firstly, oe K _ me _ sin moe oe oe sin ome but since r, L and n are assumed to be constant, the angle ome will remain unaltered, however much we may vary K. Hence ocsin moe, e and this ratio will, therefore, be a maximum when moe is a right angle. Secondly, if r is expressed in ohms, K in farads, and L (the coefficient of self-induction) in ( practical ' units i.e. in henrys the condenser volts will be the same as the generator volts, or oe K =oe, when 2L ~ This condition, it is evident, will also be more and CHOKING COIL AND CONDENSER IN SEEIES 97 more nearly fulfilled as K is made smaller and smaller without limit ; for the lines me and oc will then both become very great, and sensibly equal in length. 2 7" If K is greater than - ., e m wn< l ^ e ^ ess ^ nan e - If K is less than this amount, e K will be greater than e. When K = - - , the ratio is a maximum, r 2 +p 2 L 2 e for this is the condition which makes moe equal to 90. Thirdly, for a given impressed E.M.F., the current flowing in a circuit of constant resistance will evidently be a maximum when oe K =oe^ for the whole of the impressed volts e will then be available for producing the current. This condition is fulfilled when : 1 1 =pL or p= _... In fig. 30 the curve already referred to has been plotted. The dotted line is the curve obtained by assuming r to be relatively large and L small ; whereas the full line curve which shows more clearly what is known as the resonance effect, i.e. the considerable rise in pressure at the end of a circuit containing capacity and self-induction for certain values of the capacity is the result of considerably reducing the ohmic resistance of the circuit. It should be noted that both curves rise from the value 1, when K=0, to a maximum when K = 3, and H 98 SELF-INDUCTION AND CAPACITY that they will again pass through the value 1 when K=6 i.e. twice the critical value which gave the maximum ' resonance ' effect. 2-5 1-5 Capacity of Condenser* 3 FIG. 30 Although we have considered the result of varying the capacity only, it will not be necessary to dwell further upon the behaviour of a circuit containing capacity and self-induction in series. CHOKING COIL AND CONDENSER IN SERIES 99 The diagram, fig. 29, is so very simple that the effect of varying L while K is kept constant, or of keeping both L and K constant and varying the frequency, may be studied without much difficulty ; and if curves are afterwards plotted in order to show the manner in which the volts (or the amperes) depend upon the variable quantity, these curves will be found to be of the same general appearance as those drawn in fig. 30. FIG. 31 35. Choking Coils and Resistance in Series ; Condenser as Shunt. In fig. 31 the circuit, which contains both resistance and self-induction, is shown bridged across by the con- denser K. Let c 2 , r 2 , and Z/ 2 denote the current, resistance, and coefficient ofself-induction of the remote, or condenser section, N ; while c,, r n and L^ represent the value of these quantities in the alternator section M. Assume the current c 2 to be known. Then, in the diagram, fig. 32, draw oe 2 (=c 2 r 2 ) to represent the H 2 100 SELF-INDUCTION AND CAPACITY effective E.M.F. in section N. Erect a perpendicular at e 2 , and make e%e n =_pL 2 c 2 ; then join oe n . Evi- dently, since (see fig. 12, p. 53) oe n is the impressed E.M.F. in the section N, it will be exactly equal to the condenser E.M.F., and we can therefore calculate the condenser current, c 3 =pKe n . Now draw oc 3 at right angles with oe n in the direction of advance, and FIG. 32 add the current vectors oc 2 and oc 3 to obtain the total current c l in the section M- Proceed to calculate the necessary impressed volts in section M in order to produce this current c n on the assumption that the condenser is short-circuited. A similar construction to that used in connection with section N is applicable here; it is merely necessary to make oe l = c l r l , and CONDENSER AS SHUNT 101 e { e m (at right angles to oCj) = pL^. By compounding the vectors e n and e. m in the usual way, the resultant oe is obtained, which is equal to the total impressed E.M.F., or the required potential difference at the alternator terminals. It is interesting to note that the current c 2 in the remote section may very well be greater than the total current c l in the alternator section, and that the relation between the currents c l and c 2 depends upon the capacity of the condenser, and the frequency, and the amount of self-induction in the section N . 9 T When K= - - 2 these two currents will be equal. If K is greater than this amount, c 2 is less than c l9 If K is less than this amount, c 2 is greater than c p Although many other combinations of circuits containing capacity and self-induction may occur in practice, it should not be necessary to consider any more special cases ; the method of constructing the vector diagrams, from which can be determined, by actual measurement, the values of the various alternat- ing quantities, has been made, it is hoped, sufficiently clear to enable anyone to predetermine the relation between current flow and impressed E.M.F. in any circuit, even should the conditions differ somewhat from those in the problems already worked out. The 102 SELF-INDUCTION AND CAPACITY fact that we have generally for simplicity of con- struction assumed to be known one of those quantities which it is the object of the diagram to determine, is of little consequence, seeing that the finished diagram gives the proper relations between the various E.M.F.s and currents, and that a simple proportion sum on the slide rule enables us to express the lengths scaled off the diagram in the proper units. There is, perhaps, one more problem which may, with advantage, be considered before taking up the question of mutual induction, and that is the case of a number of circuits joined in parallel across the terminals of a constant pressure alternating current source of supply. We will assume the various branches of the circuit to have ohmic resistance and self-induction only, in order that the diagrams may remain clear, and simple of construction ; but it should be understood that the introduction of capacity in the circuits does not really increase the difficulties of the problem ; and the ex- ceedingly neat geometrical solution given below, and published by Mr. Alex. Russell in the ' Electrician ' of January 13, 1893, is equally applicable to the case of branched circuits containing resistance, capacity, and self-induction, jointly or otherwise, in any or all of the branches of the divided circuit. 36, Current Flow in divided Circuit. Let r. ,r 2 . . . be the resistances of the various branches ; CURRENT FLOW IN DIVIDED CIRCUIT 103 L 15 L 2 . . . the coefficients of self-induction, and c p c 2 . . . the currents. Then, if e is the potential difference between the points a and l> (see fig. 33), the relation between the current Cj in any one branch and the im- pressed E.M.F. e will be given by the expression : e = c 1 Vv+P 2L i 2 (29) where the quantity /v/r^+p 2 !^ 2 is what has been called the impedance (see p. 54). It should, however, be pointed out that the above relation is only true on the assumption that there is no FIG. 33 < mutual induction ' between the branches of the circuit ; that is to say, that the magnetism due to the current in any one branch does not induce any appreciable E.M.F. in any other branch. In order that we may graphically determine the various currents and their phase differences, let us draw, in fig. 34, two lines, ox and OY, at right-angles to each other, and mark off on ox the distances or n Or 2 . . . equal to r 1} r 2 . . .; and OL n OL 2 . . . on OY equal to >L 15 j?L 2 . . . Join r^, r 2 L 2 . . . . Then the length of the lines ^Lj, .r 2 L 2 ... is a measure of the impe- dances of the various branches, and the angles d. Trigonometry (Prac.), Adams, 2s. 6.i. Tuit, Tower Bridge, $s. [net. Turbines and Pressure Engines, 145. Turning, &c., Holtzapffel, 5 vols., Turning Lathes, Lukin, 35. [5/. ys. Turning, Metal, 4-r. Typography Questions, 6d. see Jacobi. Wagstaff s Metric System, is. 6J. Walker's Coal Mining, zs. 6d. - Colliery Lighting, 2s. 6d. . Dynamo Building, 2s. '" Electricity, 6s. Electric Engineer's Tables, 2s. Electric Lighting for Marine gineers, 55. , s's Dynamo, IDJ-. 6d. s-Tayler's Sugar Machine, 55-. r Supply, Godfrey. I'sTele ,,,,. lephone Handbook, 4.?. 6d. ster's Practical Forestry, 3^. 6d. *es' Transformer Design, 2s. loven's German Technological tionary, $s. .ofen's Forth Bridge, 5-r. j mouth's Drum Armatures, 7s.6d. . nittaker's Library of Popular Science, 6 vols. , zs. 6d. each. Library of Arts, &c. Specialists' Series. Wiley's Yosemite, 15*. Wilkinson's Electrical Notes, 6s. 6d. Cable Laying, 12s. 6d. Wire, by Bucknall Smith, 7*. 6d. Table, Boult, 5*. Wood's Discount Tables, is. Light, zs. 6d. Wood Carving, Leland, $s. Woodward's Manual Training, 5^. Woodwork, Barter, 7 s - &d. Joints, Adams, is. Woven Design, Beaumont, 2is. Wyatt's Diff. and Integ. Calculus, 3.. 6d. Yosemite, Alaska, &c., Wiley, 155-. FULL CATALOGUE, Post Free, ON APPLICATION TO WHITTAKER & CO., PATERNOSTER SQUARE, LONDON, E.C.