CHAPTER XIV. THE OSNI!RAIj AIiTEBNATINa-CUBBENT TBAKBFOBMSB. 131. 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 indAiction 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 moving with regard to the primary, does not leave the primary field of magnetic force, requires that this field is not undirec- tional, 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 mag- netic field, by the reaction of the secondary current. 194 AL TERNA TING-CURRENT PHENOMENA. [§131 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 altertiating- 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 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. 132. 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 §§133-136] ALTERNATING-CURRENT TRANSFORMER. 195 which surrounds the one circuit without interlinking with the other, and is thus produced by the M.M.F. of one circuit only. 133. The common 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. iV= 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. 134. 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. 135. Let, in a general alternating-current transformer: s = ratio ??^°i?^1 frequency, or " slip " ; - .- pnmary ^ •'' ft thus, if JV= primary frequency, or frequency of impressed E.M.F., s JV=i secondary frequency ; 196 AL TERN A TING-CURRENT PHENOMENA. [f 1 36 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^ J =r represents synchronous motion of the secondary ; J < represents motion above synchronism — driven by external mechanical power, as will be seen ; J = 1 represents standstill ; J > 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 Wo = number of primary turns in series per circuit ; fix = number of secondary turns in series per circuit ; a = — = ratio of turns ; Vq = go +y^o = primary admittance per circuit ; where go '= effective conductance ; do = susceptance ; Zq == Tq — jXo = internal primary impedance per circuit, where To = effective resistance of primary circuit ; Xo = reactance of primary circuit ; Zii = ri — jxi = internal secondary impedance per circuit at standstill, or for x = 1, where ri = effective resistance of secondary coil ; ^1 = reactance of secondary coil at standstill, or full fre- quency, s = 1, Since the reactance is proportional to the frequency, at the slip Sf or the secondary frequency sA^, the secondary impedance is : Zi = r, —j'sxi. Let the secondary circuit be closed by an external re- sistance r, and an external reactance, and denote the latter I 135] ALTERNATING-CURRENT TRANSFORMER. 197 by X at frequency N, then at frequency sN, or slip j, it will be = J ;r, and thus : Z ■= r — jsx = external secondary impedance.* Let -5*0 = primary impressed E.M.F. per circuit, E^ = E.M.F. consumed by primary counter E.M.F., Ex = secondary terminal E.M.F., E( = secondary induced E.M.F., e = E.M.F. induced per turn by the mutual magnetic flux, at full frequency N^ I^ = primary current, J^^ = primary exciting current, /i = secondary current. It is then : Secondary induced E.M.F. E( = sr/if. Total secondary impedance Zi + Z = (ri + r) — yj (Xi + x) ; hence, secondary current y El k Hye ^ • ' "■ Zi + Z " (ri + r) - yx (x^ + x) ' Secondary terminal voltage E\ = El — -^Zi = /yZ = snie \ 1 ri-jsxi ) _ snie{r^jsx) ^ \ {ri + r)-'js{xi + x)) (ri+r) ^js{xi + x) ♦ 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, jt" that part of the reac- tance independent of the frequency, and jr'" that part of the reactance which is inversely proportional to the frequency ; and have thus, at slip J, or frequency sN^ the external secondary reactance jjt' -|- jc" -|- X*" 198 AL TERN A TING-CURRENT PHENOMENA. [§ 1 35^ E.M.F. consumed by primary counter E.M.F, hence, primary exciting current : Component of primary current corresponding to second- ary current /^ : j' ^ — Ix a n^se a^{{r^^ + ^) - js (^1 + X)) ' hence, total primary current, = «,«,,jl 1 ^ioJUlh I ai(ri + r) —js(xi + x) s Primary impressed E.M.F., { (i'{ri + r)-js{x^+x) ) We get thus, as the Equations of the General Alternating-Current Transfortner : 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., E,= -n,e \ 1 + 1- "-^^-y^y +(r,^jx,){g,+jbo) \ . Secondary terminal voltage, E, = sn,e \ 1 ^JLZiZfjIi ] = sn'e ^^^11^ ^ \ {r,+ r)--js{x,+x) S {r,+r)^js(x^+x) Primary current, a^{n + r)-js{x^ + x) s §136] AL TERN A TING-CURRENT TRANSFORMER. \m Secondary current, /i = sn^e (n + ^) -js^x^^rx) Therefrom, we get : Ratio of currents, Er Ratio of KM.Fs., fl + - El a — ■< s ro — jx^ ^^(n + r)-js(x^ + x) + (ro-JXo)Uo+J^o) I n-zj^^j_ Total apparent primary impedance, Z, = f? = -i'- {(V, + r) - /(.r, + X)} fl + A ro —JXo a^(^i+ '•) +J^(Xi + x) + (''o-y-^o)c^o+yM a' s where " x"" x==x^+±-+--- s j" in the general secondary circuit as discussed in foot-note> page 4. Substituting in these equations : gives the General Equations of the Stationary Alternating-Current Transformer : E, = -n,e jl + l^^+Zoii ^ Z, + Z) z, + z Jn = — nae 1 <^' {Z, + Z) + n . 200 ALTERNATING-CURRENT PHENOMENA. [S 13d 1 + £i =z — a a' (Z, + Z) + Z,K, 1- Zi Z, = :|»=a«(Z, + Z)- 1 + Zi + Z z, a« (Z, + Z) + Z,F. l + a»Ko(-^i + -^) Substituting in the equations of the general alternating- current transformer, Z = 0, gives the General Equations of the Induction Motor: = 0. I. = {a^{ri—jsxi) s ) A Sftit n—jsxi ^ — ^ h + 7 ^^^ + ^"^"^ ("■' "•'•''"^ I • z, = <7 1+ — ^ ^0 —7-^0 tfTi — y^^i + (^0 -JXo)(go +J^o) /Returning now to the general alternating-current trans- former, we have, by substituting (n + r)« -t- ^ (x, + xy = V, and separating the real and imaginary quantities, £,= -noe\ h + -^^(ro(r, + r)-^sxo(x, + x)) 1136] ALTERNATING'CURRENT TRANSFORMER. 201 + (^0^0 +^o^o) + y -/-iC-f '•o(-^i +^)-^o(n + r)) V\77. + (^0^0 - ^o--^o^o)1 I • 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, y, = 0, ^, = ^11. a* Substituting this in the equations of the general trans- former, we get, ^0= - ''o^ I 1 + -^ [n (^1 + '•) + J^i (^1 + x)\ + ^ [-f ''i (-^1 + •^) - ^1 (n + '•)]}• ^k 136. If -£■ = « + y)3 = KM.F., in complex quantities, and I ^=^ c '■\- J d =^ current in complex quantities, the power is, P^\E, I \^ EI cos {E, I) ^ac+ pd. 202 AL TERNA TING-CURRENT PHENOMENA. [% 1 37 Making use of this, and denoting, gives: Secondary output of the transformer Internal loss in secondary circuit, Total secondary power, P, + P^ = f^Y^'' + n) = -f w (r + r,). Internal loss in primary circuit, Total electrical output, plus loss, P'^P,^- P,' + Pi = /fJfV (r + 2rO = sw{r^2r,\ Total electrical input of primary, P^ = I ^0^0 I = -f [ ^ j ('' + n + ^n) = w' C'' + n + J^i). Hence, mechanical output of transformer, 7^= T'o - ^' = o' (1 - j) (r + r,). Ratio, mechanical outpat P 1 — S »peed total lecondary power p \ p \ - slip 137. Thus, In a general alternating transformer of ratio of turns, a^ and ratio of frequencies, j, neglecting exciting current, it is : Electrical input in primary, -ft a §138] ALTERNATING-CURRENT TRANSFORMER, 203 Mechanical output, F = s. (1 _ s) fi.^e^ (r + + Electrical output of secondary, P. = . s^ rt^ ^ r Losses in transformer, C'-i + ry + s' (x, + x)^ Of these quantities, P^ and P^ are always positive ; Pq and P can be positive or negative, according to the value of s. Thus the apparatus can either produce mechanical power, acting as a motor, or consume mechanical power; and it can either consume electrical power or produce electrical power, as a generator. 138. At s = Oy synchronism, Fq = 0, F = 0, Fi = 0. At < .f < 1, between synchronism and standstill. P^ , P and Pq are positive ; that is, the apparatus con- sumes electrical power P^ in the primary, and produces mechanical power P and electrical power P^ + P^^ in the secondares which is partly, P^^y consumed by the internal secondary resistance, partly, P^ , available at the secondary terminals. In this case it is : Pi + Pi' _ s . F "l^s' that is, of the electrical power consumed in the primar}' circuit, Pq , a part Pq^ is consumed by the internal pri- mary resistance, the remainder transmitted to the secon- dary, and divides between electrical power, P^ + P^^, and mechanical power, P, in the proportion of the slip, or drop below synchronism, s, to the speed : 1 — j. 204 AL TERN A TING-CURRENT PHENOMENA, [§138 In this range, the apparatus is a motor. At J > 1 ; or, backwards driving, -P < 0, or negative ; that is, the apparatus requires mechanical power for driving. It is then : P^ - Po^ - P^^ < 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 /* < 0, beyond synchronism, P