CHAPTER XV. THE GENERAL ALTERNATING-CURRENT TRANSFORMER OR FREQUENCY CONVERTER. 141. The simplest alternating-current apparatus is the alternating-current transformer. It consists of a magnetic- circuit, interlinked with two electric circuits or sets of electric circuits. The one, the primary circuit, is excited by an impressed E.M.F., while in the other, the secondary circuit, an E.M.F. is induced. Thus, in the primary circuit, power is consumed, in the secondary circuit a correspond- ing amount of power produced ; or in other words, power is transferred through space, from primary to secondary circuit. This transfer of power finds its mechanical equiv- alent in a repulsive thrust acting between primary and secondary. Thus, if the secondary coil is not held rigidly as in the stationary transformer, it will be repelled and move away from the primary. This mechanical effect is made use of in the induction motor, which represents a transformer whose secondary is mounted movably with re- gard to the primary in such a way that, while set in rota- tion, it still remains in the primary field of force. The condition that the secondary circuit, while revolving with regard to the primary, does not leave the primary field of magnetic force, requires that this field is not undirectional, but that an active field exists in every direction. One way of producing such a magnetic field is by exciting different primary circuits angularly displaced in space with each other by currents of different phase. Another way is to excite the primary field in one direction only, and get the cross magnetization, or the angularly displaced magnetic field, by the reaction of the secondary current. 220 ALTERNATING-CURRENT PHENOMENA. We see, consequently, that the stationary transformer and the induction motor are merely different applications of the same apparatus, comprising a magnetic circuit in- terlinked with two electric circuits. Such an apparatus can properly be called a "general alternating- current trans- former" The equations of the stationary transformer and those of the induction motor are merely specializations of the general alternating-current transformer equations. Quantitatively the main differences between induction motor and stationary transformer are those produced by the air-gap between primary and secondary, which is re- quired to give the secondary mechanical movability. This air-gap greatly increases the magnetizing current over that in the closed magnetic circuit transformer, and requires an ironclad construction of primary and secondary to keep the magnetizing current within reasonable limits. An iron- clad construction again greatly increases the self-induction of primary and secondary circuit. Thus the induction motor is a transformer of large magnetizing current and large self-induction; that is, comparatively large primary exciting susceptance and large reactance. The general alternating-current transformer transforms between electrical and mechanical power, and changes not only E.M.Fs. and currents, but frequencies also, and may therefore be called a "frequency converter." Obviously, it also may change the number of phases. 142. Besides the magnetic flux interlinked with both primary and secondary electric circuit, a magnetic cross- flux passes in the transformer between primary and second- ary, surrounding one coil only, without being interlinked with the other. This magnetic cross-flux is proportional to the current flowing in the electric circuit, and constitutes what is called the self-induction of the transformer. As seen, as self-induction of a transformer circuit, not the total flux produced by and interlinked with this circuit is under- stood, but only that — usually small — part of the flux AL TERN A TING-CURRENT TRA NSFORMER. 221 which surrounds the one circuit without interlinking with the other, and is thus produced by the M.M.F. of one circuit only. 143. The mutual magnetic flux of the transformer is produced by the resultant M.M.F. of both electric circuits. It is determined by the counter E.M.F., the number of turns, and the frequency of the electric circuit, by the. equation: Where E = effective E.M.F. JV= frequency. n = number of turns. <£ == maximum magnetic flux. The M.M.F. producing this flux, or the resultant M.M.F. of primary and secondary circuit, is determined by shape and magnetic characteristic of the material composing the magnetic circuit, and by the magnetic induction. At open secondary circuit, this M.M.F. is the M.M.F. of the primary current, which in this case is called the exciting current, and consists of an energy component, the magnetic energy current, and a reactive component, the magnetizing current. 144. In the general alternating-current transformer, where the secondary is movable with regard to the primary, the rate of cutting of the secondary electric circuit with the mutual magnetic flux is different from that of the primary. Thus, the frequencies of both circuits are different, and the induced E.M.Fs. are not proportional to the number of turns as in the stationary transformer, but to the product of number of turns into frequency. 145. Let, in a general alternating-current transformer : * = ratio iS^ frequency, or « slip » ; thus, if N '= primary frequency, or frequency of impressed E.M.F., s JV = secondary frequency ; 222 ALTERNATING-CURRENT PHENOMENA. and the E.M.F. induced per secondary turn by the mutual flux has to the E.M.F. induced per primary turn the ratio s, s = 0 represents synchronous motion of the secondary ; s < 0 represents motion above synchronism — driven by external mechanical power, as will be seen ; s = 1 represents standstill ; s > 1 represents backward motion of the secondary that is, motion against the mechanical force acting between primary and secondary (thus representing driving by ex- ternal mechanical power). Let «0 = number of primary turns in series per circuit ; /?! = number of secondary turns in series per circuit ; a = — = ratio of turns ; «i Y0 =£"0 H~./A) = primary exciting admittance per circuit; where gQ = effective conductance ; b0 = susceptance ; Z0 = r0 —jx0 = internal primary self-inductive impedance per circuit, where r0 = effective resistance of primary circuit ; jr0 = reactance of primary circuit ; Zu = TI — jxv = internal secondary self -inductive impedance per circuit at standstill, or for s = 1, where rj = effective resistance of secondary coil ; Xl — reactance of secondary coil at standstill, or full fre- quency, s = 1. Since the reactance is proportional to the frequency, at the slip s, or the secondary frequency s N, the secondary impedance is : Zl = r1-jsxl. Let the secondary circuit be closed by an external re- sistance r, and an external reactance, and denote the latter ALTERNATING-CURRENT TRANSFORMER, 223 by x at frequency N, then at frequency s N, or slip s, it will be = s x, and thus : Z = r — jsx = external secondary impedance.* Let £0 = primary impressed E.M.F. per circuit, E ' = E.M.F. consumed by primary counter E.M.F., £1 = secondary terminal E.M.F., EI = secondary induced E.M.F., e = E.M.F. induced per turn by the mutual magnetic flux, at full frequency JY, IQ = primary current, ^0 = primary exciting current, 7i = secondary current. It is then : Secondary induced E.M.F. EI = sn^e. Total secondary impedance Z, + Z= (r, + r) hence, secondary current Secondary terminal voltage * This applies to the case where the secondary contains inductive reac- tance only ; or, rather, that kind of reactance which is proportional to the fre- quency. In a condenser the reactance is inversely proportional to the frequency, in a synchronous motor under circumstances independent of the frequency. Thus, in general, we have to set, x = x' + x" -\ x"\ where x' is that part of the reactance which is proportional to the frequency, x" that part of the reac- tance independent of the frequency, and x'" that part of the reactance which is inversely proportional t6 the frequency ; and have thus, at slip s, or frequency sN, the external secondary reactance sx' + x" -f- — — . 224 AL TERNA TING-CURRENT PHENOMENA, E.M.F. consumed by primary counter E.M.F. £'= -«<>'; hence, primary exciting current : 700 = E ' YQ = — «0 e (g0 + /£<))• Component of primary current corresponding to second- ary current 7X : hence, total primary current, // 1 Primary impressed E.M.F., We get thus, as the Equations of the General Alternating-Current Transformer: Of ratio of turns, a ; and ratio of frequencies, s ; with the E.M.F. induced per turn at full frequency, e, as parameter, the values : Primary impressed E.M.F., Secondary terminal voltage, Primary current, \ 1 ALTERNATING-CURRENT TRANSFORMER. 225 Secondary current, II =7— -7- Therefrom, we get : Ratio of currents, Ratio of E.M.Fs., Total apparent primary impedance, , , . x" . x'" where x—x-\ --- \- — s s2 in the general secondary circuit as discussed in foot-note, page 221. Substituting in these equations : *-l, gives the General Equations of the Stationary Alternating-Current Transformer : z*+z\ z, + z '* = -»•< \ .,;,* +IU- »* (Zj + Z) ALTERNA TING-CURRENT PHENOMENA. r nte yi = Z, + Z /! a P f1 + *f7\^ + Z'Y* ^o_= _ a } a (Z-j + 2} & I- Z* ( Z, + Z 1+ 2//° x+^oKo] a2 (Zj + Z) _ I l + ^Fo^ + Z) J Substituting in the equations of the general alternating- current transformer, Z = 0, gives the General Eqtiations of tJie Induction Motor: a'r^-jsx^ ^ = 0. 1 i ^o +y^o 70 = _ s «0 f ] -T, . . 1 «•(>-!— y**o r j«,^ A = — —5 "^ : — ~ + (ro — y^o)(^b +/ «2^i — JSXi Returning now to the general alternating-current trans^ former, we have, by substituting (ri + r? + ^2 (*i + *)2 = **f, and separating the real and imaginary quantities, -±- (r0 (r, + r)+sx9(Xl + x)) 22 ALTERNATING-CURRENT TRANSFORMER, 227 Neglecting the exciting current, or rather considering it as a separate and independent shunt circuit outside of the transformer, as can approximately be done, and assum- ing the primary impedance reduced to the secondary circuit as equal to the secondary impedance, Substituting this in the equations of the general trans- former, we get, £,= - «0 e\ I + - fr fa + r) 146. The true power is, in symbolic representation (see Chapter XII.) : 228 ALTERNATING-CURRENT PHENOMENA. denoting, safe* -7F = W gives : Secondary output of the transformer Internal loss in secondary circuit, m -2 t s n\ ^\2 -Pi = 'i2 n = ( — — } V ** / Total secondary power, ** Internal loss in primary circuit, r»i -9 -9o ^o = V'o = 4 rt that is, of the electrical power consumed in the primary circuit, P0, a part P^ is consumed by the internal pri- mary resistance, the remainder transmitted to the secon- dary, and divides between electrical power, P1 + P^1, and mechanical power, P, in the proportion of the slip, or drop below synchronism, s, to the speed : 1 — s. 230 ALTERNATING-CURRENT PHENOMENA. In this range, the apparatus is a motor. At s > 1 ; or, backwards driving, P < 0, or negative ; that is, the apparatus requires mechanical power for driving. It is then : P0 - A1 - A1 < PI ; that is : the secondary electrical power is produced partly by the primary electrical power, partly by the mechanical power, and the apparatus acts simultaneously as trans- former and as alternating-current generator, with the sec- ondary as armature. The ratio of mechanical input to electrical input is the ratio of speed to synchronism. In this case, the secondary frequency is higher than the primary. At s < 0, beyond synchronism, P < 0 ; that is, the apparatus has to be driven by mechanical power. /o<0; that is, the primary circuit produces electrical power from the mechanical input. At r+r! + srj. = 0, or, s < — ^±^J ; rt the electrical power produced in the primary becomes less than required to cover the losses of power, and />0 becomes positive again. We have thus : K-£±fl r\ consumes mechanical and primary electric power ; produces secondary electric power. - r-±^ < s < 0 ?i consumes mechanical, and produces electrical power in primary and in secondary circuit. ALTERNATING-CURRENT TRANSFORMER. 231 consumes primary electric power, and produces mechanical and secondary electrical power. consumes mechanical and primary electrical power ; pro- duces secondary electrical power. T GENERAL ALTERNATE CURRENT TRANSFORMER A 648 Fig H 149. As an instance, in Fig. Ill are plotted, with the slip s as abscissae, the values of : Secondary electrical output as Curve I. ; Total internal loss as Curve II. ; Mechanical output as Curve III. ; Primary electrical input as Curve IV. ; for the values : n,e = 100.0 ; r = A ; r» — 4. i x = .3; 232 ALTERNATING-CURRENT PHENOMENA. hence, p = 16,000 ^2. pl, Pi _ 8,000 j«. 0 """ l -i , — j" ? „ _ 4,000 s + (5 + J) . ~ 1 I 2 ' p = 20,000 s (1 - j) 150. Since the most common practical application of the general alternating current transformer is that of fre- quency converter, that is to change from one frequency to another, either with or without change of the number of phases, the following characteristic curves of this apparatus are of great interest. 1. The regulation curve ; that is, the change of second- ary terminal voltage as function of the load at constant im- pressed primary voltage. 2. The compounding curve ; that is, the change of pri- mary impressed voltage required to maintain constant sec- ondary terminal voltage. In this case the impressed frequency and the speed are constant, and consequently the secondary frequency. Gen- erally the frequency converter is used to change from a low frequency, as 25 cycles, to a higher frequency, as 62.5 cycles, and is then driven backward, that is, against its torque, by mechanical power. Mostly a synchronous motor is employed, connected to the primary mains, which by over-excitation compensates also for the lagging current of the frequency converter. Let, Y0 = g0 +j&0 = primary exciting admittance per circuit of the frequency converter. Z^ = rt —jx^— internal self inductive impedance per secondary circuit, at the secondary frequency. ALTERNATING-CURRENT TRANSFORMER. 233 Z^ = r0 — jx^ = internal self inductive impedance per primary circuit at the primary frequency. a = ratio of secondary to primary turns per circuit. b = ratio of number of secondary to number of primary circuits. c = ratio of secondary to primary frequencies. Let, e = induced E.M.F. per secondary circuit at secondary frequency. Z = r — jx = external impedance per secondary circuit at secondary frequency, that is load on secondary system, where x — 0 for noninductive lead. We then have, total secondary impedance, Z + Z1 = (r-^rl)-j(x + x1) secondary current, where, r + r. x + Xl (r + 0>2 + (* + ^)2 (r +^i)2 + (* + secondary terminal voltage, Ei = IiZ = e ^4-T — e(r —jx) (at where, primary induced E.M.F. per circuit, primary load current per circuit, 71 = abli = abe (a{ primary exciting current per circuit, 234 ALTERNATING-CURRENT PHENOMENA. thus, total primary current, 70 = 71 + /oo = e (fi where, <. = •**+£ <•.=«**+! primary terminal voltage : where, d -— re x d -re -x ac or absolute, e0 = e vX2 + 42 . = e° - V^2 + 4« substituting this value of e in the preceding equations, gives, as function of the primary impressed E.M.F., e0: secondary current, 7 = > absolu 7 = vi V4» + 42 v ^i2 + secondary terminal voltage, primary current, , _ primary impressed E.M.F. ^ _ ^0 (4 " V4 secondary output, gl^ + AL TERNA TING-CURRENT TRANSFORMER. 235 primary electrical input, i + Lr°:oj