CHAPTER IV. GRAPHIC BEFRISXINTATION. 14. While alternating waves can be, and frequently are, represented graphically in rectangular coordinates, with the time as abscissae, and the instantaneous values of the wave as ordinates, the best insight with regard to the mutual relation of different alternate waves is given by their repre- sentation in polar coordinates, with the time as an angle or the amplitude, — one complete period being represented by one revolution, — and the instantaneous values as radii vectores. Fiq, 8, Thus the two waves of Figs. 2 and 3 are represented in polar coordinates in Figs. 8 and 9 as closed characteristic curves, which, by their intersection with the radius vector, give the instantaneous value of the wave, corresponding to the time represented by the amplitude of the radius vector. These instantaneous values are positive if in the direction of the radius vector, and negative if in opposition. Hence the two half-waves in Fig. 1 are represented by the same / 20 AL TERN A TING-CURRENT PHENOMENA. [§15 polar characteristic curve, which is traversed by the point of intersection of the radius vector twice per period, — once in the direction of the vector, giving the positive half-wave, Fig. 9. and once in opposition to the vector, giving the negative half-wave. In Figs. 3 and 9, where the two half-waves are different, they give different polar characteristics. Fig. 10. 15. The sine wave, Fig. 1, is represented in polar coordinates by one circle, as shown in Fig. 10. The diameter of the characteristic curve of the sine wave, /= OC, represents the intensity of the wave; and the am- plitude of the diameter, OCy Z w = AOC, is ih^ phase of the 516] GRAPHIC REPRESENTATION. 21 wave, which, therefore, is represented analytically by the function : — i = I cos ( =^t j T is the instantaneous value of the ampli- tude corresponding to the instantaneous value, /, of the wave. The instantaneous values are cut out on the movable ra- dius vector by its intersection with the characteristic circle. Thus, for instance, at the amplitude AOB^ == j = 2ir/j/ T' (Fig. 10), the instantaneous value is 0B'\ at the amplitude AOB2 == 2 = 2ir/2 / T, the instantaneous value is (?jff", and negative, since in opposition to the vector OB^. The characteristic circle of the alternating sine wave is determined by the length of its diameter — the intensity of the wave, and by the amplitude of the diameter — the phase of the wave. Hence, wherever the integral value of the wave is con- sidered alone, and not the instantaneous values, the charac- teristic circle may be omitted altogether, and the wave represented in intensity and in phase by the diameter of the characteristic circle. Thus, in polar coordinates, the alternate wave is repre- sented in intensity and phase by the length and direction of a vector, OQ Fig. 10. Instead of the maximum value of the wave, the effective value, or square root of mean square, may be used as the vector, which is more convenient ; and#the maximum value is then V2 times the vector OC^ so that the instantaneous values, when taken from the diagram, have to be increased by the factor V2. 16. To combine different sine waves, their graphical representations, or vectors, are combined by the parallelo- gram law. If, for instance, two sine waves, OB and OC (Fig. 11), are superposed, — as, for instance, two E.M.Fs. acting in the same circuit, — their resultant wave is represented by / 22 AL TERNA TING-CURRENT PHENOMENA. [§16 OD^ the diagonal of a parallelogram with OB and OC as sides. For at any time, /, represented by angle <^ = AOX^ the instantaneous values of the three waves, OB^ OCy OD, are their projections upon OX^ and the sum of the projections of OB and 0C\% equal to the projection of 0D\ that is, the instantaneous values of the wave OD are equal to the sum of the instantaneous values of waves OB and OC. From the foregoing considerations we have the con- clusions : The sine wave is represented graphically in polar coordi- nates by a vector, which by its length, OC, denotes tlie in- Fig. 11. tensity, and by its amplitude, AOC, the phase, of the sine wave. Sine waves are combined or resolved graphically, in polar coordifiatcs, by the laiv of parallelogram or the polygon of sine waves. Kirchhoff' s laws now assume, for alternating sine waves, the form : — a.) The resultant of all the E.M.Fs. in a closed circuit, as found by thq parallelogram of sine waves, is zero if the counter E.M.Fs. of resistance and of reactance are included. b.) The resultant of all the currents flowing towards a §17] GRAPHIC REPRESENTATION. 23 distributing point, as found by the parallelogram of sine waves, is zero. The energy equation expressed graphically is as follows : The power of an alternating-current circuit is repre- sented in polar coordinates by the product of the current ; /, into the projection of the E.M.F., Ey upon the current, or by the E.M.F., E, into the projection of the current, /, upon the E.M.F., or IE cos (/j^). 17. Suppose, as an instance, that over a line having the resistance, r, and the reactance, x = 2irNLy — where A^ = frequency and L = inductance, — a current of / amperes be sent into a non-inductive circuit at an E.M.F. of E ,Er E • ^^ J • \ 1 Vi \ E. "e. — ~^^E. Fig. 12. volts. What will be the E.M.F. required at the generator end of the line ? In the polar diagram. Fig. 12, let the phase of the cur- rent be assumed as the initial or zero line. Of. Since the receiving circuit is non-inductive, the current is in phase with its E.M.F. Hence the E.M.F., E, at the end of the line, impressed upon the receiving circuit, is represented by a vector, OE. To overcome the resistance, r, of the line, an E.M.F., /r, is required in phase with the current, repre- sented by OE^ in the diagram. The self-inductance of the line induces an E.M.F. which is proportional to the current / and reactance -r, and lags a quarter of a period, or 90°, behind the current. To overcome this counter E.M.F. /' 24 ALTERNATING-CURRENT PHENOMENA. [§18 of self-induction, an E.M.F. of the value Ix is required, in phase 90® ahead of the current, hence represented by- vector OEj^. Thus resistance consumes E.M.F. in phase,, and reactance an E.M.F. 90° ahead of the current. The E.M.F. of the generator, E^y has to give the three RM.Fs., Ey E^f and E^y hence it is determined as their resultant. Combining by the parallelogram law, OE,. and OEj^, give OEgy the E.M.F. required to overcome the impedance of the line, and similarly OE^ and OE give OE^, the E.M.F. required at the generator side of the line, to yield the E.M.F. E at the receiving end of the line. Algebraically, we get from Fig. 12 — or, E = -s/EJ" — {IxY - Ir. In this instance we have considered the E.M.F. con- sumed by the resistance (in phase with the current) and the E.M.F. consumed by the reactance (90? ahead of the current) as parts, or components, of the impressed E.M.F., E^t and have derived E^ by combining E^., E^, and E, 18. We may, howeyer, introduce the effect of the induc- tance directly as an E.M.F., E^ , the counter E.M.F. of self-induction = I,j and lagging 90° behind the current ; and the E.M.F. consumed by the resistance as a counter E.M.F., -£*/ = /r, but in opposition to the current, as is done in Fig. 18 ; and combine the three E.M.Fs. E^j if/, EJ, to form a resultant E.M.F., /:, which is left at the end of the line. 118] GRAPHIC REPRESENTATIO^r, 2& E^ and E^ combine to form E^y the counter E.M.F. of impedance; and since E^ and E^ must combine to form Ey E^ is found as the side of a parallelogram, OE^EJ, whose other side, OEJy and diagonal, OE, are given. Or we may say (Fig. 14), that to overcome the counter E.M.F. of impedance, OE^y of the line, the component, OE^y of the impressed E.M.F., together with the other component OEy must give the impressed E.M.F., OE^, As shown, we can represent the E.M.Fs. produced in a circuit in two ways — either as counter E.M.Fs., which com- bine with the impressed E.M.F., or as parts, or components. Ftg, 14, of the impressed E.M.F., in the latter case being of opposite phase. According to the nature of the problem, either the one or the other way may be preferable. As an example, the E.M.F. consumed by the resistance is /r, and in phase with the current ; the counter E.M.F. of resistance is in opposition to the current. The E.M.F. consumed by the reactance is /r, and 90° ahead of the cur- rent, while the counter E.M.F. of reactance is 90° behind the current ; so that, if, in Fig. 15, (?/, is the current, — OEr = OE/ = OE^ = OE^' = OE, = OE/ = E.M.F. consumed by resistance, counter E.M.F. of resistance, E.M.F. consumed by inductance, counter E.M.F. of inductance, E.M.F. consumed by impedance, counter E.M.F. of impedance. 26 ALTERNATING-CURRENT PHENOMENA. [§§19,20 Obviously, these counter E.M.Fs. are different from, for instance, the counter E.M.F. of a synchronous motor, in so far as they have no independent existence, but exist only through, and as long as, the current flows. In this respect they are analogous to the opposing force of friction in mechanics. Fiq, 15. 19. Coming back to the equation found for the E.M.F. at the generator end of the line, — we find, as the drop of potential in the line — This is different from, and less than, the E.M.F. of impedance — Hence it is wrong to calculate the drop of potential in a circuit by multiplying the current by the impedance ; and the drop of potential in the line depends, with a given current fed over the line into a non-inductive circuit, not only upon the constants of the line, r and x, but also upon the E.M.F., -£", at end of line, as can readily be seen from the diagrams. « 20. If the receiver circuit is inductive, that is, if the current, /, lags behind the E.M.F., E, by an angle w, and we choose again as the zero line, the current 01 (Fig. 16), the E.M.F., OE is ahead of the current by angle «. The §20] GRAPHIC REPRESENTAT/O.V, 27 E.M.F. consumed by the resistance, Ir, is in phase with the current, and represented by OE^) the E.M.F. consumed by the reactance, 7^, is 90° ahead of the current, and re- presented by OEj.. Combining OEy OE^, and OE^y we get OE^, the E.M.F. required at the generator end of the line. Comparing Fig. 16 with Fig. 13, we see that in the former OEo is larger ; or conversely, if E^ is the same, E will be less with an inductive load. In other words, the drop of potential in an inductive line is greater, if the receiving circuit is inductive, than if it is non-inductive. From ¥\g, 16, — Eo = V(^ cos a> + Jry + {E^m u> -f Jx)\ Fig. 16. If, however, the current in the receiving circuit is leading, as is the case when feeding condensers or syn- chronous motors whose counter E.M.F. is larger than the -impressed E.M.F., then the E.M.F. will be represented, in Fig. 17, by a vector, OEy lagging behind the current, Oly by the angle of lead w; and in this case we get, by combining OE with OE^y in phase with the current, and OEj^y 90° ahead of the current, the generator E.M.F., OE^y which in this case is not only less than in Fig. 16 and in Fig. 13, but may be even less than E ; that is, the poten- tial rises in the line. In other words, in a circuit with .leading current, the self-induction of the line raises the .potential, so that the drop of potential is less than with 28 AL TERN A TING- CURRENT PHENOMENA. [§21 a non-inductive load, or may even be negative, and the voltage at the generator lower than at the other end of the line. These same diagrams, Figs. 13 to 17, can be considered as polar diagrams of an alternating-current generator of an E.M.F., E^y a resistance E.M.F., E^ = /r, a reactance, Ejc = Zr, and a difference of potential, E, at the alternator terminals; and we see, in this case, that with an inductive load the potential difference at the alternator terminals will be lower than with a non-inductive load, and that with a non-inductive load it will be lower than when feeding into Fig. 77. a circuit with leading current, as, for instance, a synchro- nous motor circuit under the circumstances stated above. 21. As a further example, we may consider the dia- gram of an alternating-current transformer, feeding through its secondary circuit an inductive load. For simplicity, we may neglect here the magnetic hysteresis, the effect of which will be fully treated in a separate chapter on this subject. Let the time be counted from the moment when the magnetic flux is zero. The phase of the flux, that is, the amplitude of its maximum value, is 90° in this case, and, consequently, the phase of the induced E.M.F., is 180% §21] GRAPHIC REPRESENTATION. 29 since the induced E.M.F. lags 90° behind the inducing flux. Thus the secondary induced E.M.F., E^^ will be represented by a vector, OE^, in Fig. 18, at the phase 180°. The secondary current, /j, lags behind the E.M.F. E^ by an angle wj, which is determined by the resistance and inductance of the secondary circuit ; that is, by the load in the secondary circuit, and is represented in the diagram by the vector OF^y of phase 180 + «. ef_l. Flq. 18. Instead of the secondary current, /j, we plot, however, the secondary M.M.F., F^ = f^i A> where n^ is the number of secondary turns, and F^ is given in ampere-turns. This makes us independent of the ratio of transformation. From the secondary induced E.M.F., E^, we get the flux, 4», required to induce this E.M.F., from the equation — ^i = V2 7r«iA^*10-«. where — El = secondary induced E.M.F., in effective volts, jV — frequency, in cycles per second. f/i = number of secondary turns. 4> = maximum value of magnetic flux, in webers. The derivation of this equation has been given in a preceding chapter. This magnetic flux, 4>, is represented by a vector, O^, at 30 AL TERNA TING-CURRENT PHENOMENA, [§ 22 the phase 90^ and to induce it an M.M.F., $F, is required, which is determined by the magnetic characteristic of the iron, and the section and length of the magnetic circuit of the transformer ; it is in phase with the flux 4>, and repre- sented by the vector OFy in effective ampere-turns. The effect of hysteresis, neglected at present, is to shift OF ahead of OMy by an angle a, the angle of hysteretic lead. (See Chapter on Hysteresis.) This M.M.F., $F, is the resultant of the secondary M.M.F., IFj, and the primary M.M.F., IF^; or graphically, OF is the diagonal of a parallelogram with OF^ and OF^ as sides. OF^ and OF being known, we find OF^^ the primary ampere- turns, n^ , and therefrom, the primary current, I^ = if^ln^y which corresponds to the secondary current, /j. To overcome the resistance, r^, of the primary coil, an E.M.F., E^ = /r^, is required, in phase with the current, /^, and represented by the vector OE^, To overcome the reactance, x^ = 2ir n^ L^ , of the pri- mary coil, an E.M.F. JE*, = I^x^ is required, 90° ahead of the current /^, and represented by vector, OE^. The resultant magnetic flux, 4>, which in the secondary coil induces the E.M.F., -f^, induces in the primary coil an E.M.F, proportional to E^ by the ratio of turns n^l n-^t and in phase with E^ , or, — which is represented by the vector OEJ. To overcome this counter E.M.F., E^\ a primary E.M.F., E^y is required, equal but opposite to EJ ^ and represented by the vector, OE^. The primary impressed E.M.F., E^^ must thus consist of the three components, OE^^ OE^y and OE^^ and is, there- fore, a resultant OE^^ while the difference of phase in the primary circuit is found to be >^ a»^ = EjOA, 22. Thus, in Figs. 18, 19, and 20, the diagram of an ^temating-current transformer is drawn for the same sec- 122] GRAPHIC REPRESENTATION. 31 •ondary E.M.F., E^^ and secondary current, /j, but with dif- ferent conditions of secondary displacement : -. — In Fig. 18, the secondary current, f^ , lags 60° behind the sec- ondary E.M.F., Ex, In Fig. 19, the secondary current, /i, is in phase with the secondary E.M.F., Ey. In Fig. 20, the secondary current, I^ , leads by 60** the second- ary E.M.F., E^, Ef« — -« — These diagrams show that lag in the secondary circuit in- creases and lead decreases, the primary current and primary E.M.F. required to produce in the secondary circuit the ^ame E.M.F. and current ; or conversely, at a given primary >A Flq, 20. impressed E.M.F., E^^ the secondary E.M.F., E^, will be smaller with an inductive, and larger with a condenser (leading current) load, than with a non-inductive load. At the same time we see that a difference of phase existing in the secondary circuit of a transformer reappears / V 82 AL TERNA TING-CURRENT PHENOMENA, [§ 22 in the primary circuit, somewhat decreased if leading, and slightly increased if lagging. Later we shall see that hysteresis reduces the displacement in the primary circuit, so that, with an excessive lag in the secondary circuit, the lag in the primary circuit may be less than in the secondary. A conclusion from the foregoing is that the transformer is not suitable for producing currents of displaced phase ; since primary and secondary current are, except at very light loads, very nearly in phase, or rather, in opposition,, to each other. i 23] SYMBOLIC METHOD. 88