CHAPTER XVII THE ALTERNATING-CURRENT TRANSFORMER 141. The simplest alternating-current apparatus is the trans- former. It consists of a magnetic circuit interlinked with two electric circuits, a primary and a secondary. The primary circuit is excited by an impressed e.m.f., while in the secondary circuit an e.m.f. is generated. Thus, in the primary circuit power is consumed, and in the secondary a corresponding amount of power is produced. Since the same magnetic circuit is interlinked with both electric circuits, the e.m.f. generated per turn must be the same in the secondary as in the primary circuit; hence, the primary generated e.m.f. being approximately equal to the impressed e.m.f., the e.m.fs. at primary and at secondary terminals have approximately the ratio of their respective turns. Since the power produced in the secondary is approximately the same as that consumed in the primary, the primary and secondary currents are approximately in inverse ratio to the turns. 142. Besides the magnetic flux interlinked with both electric circuits — which flux, in a closed magnetic circuit transformer, has a circuit of low reluctance — a magnetic cross-flux passes between the primary and secondarj^ coils, surrounding one coil only, without being interlinked with the other. This magnetic cross-flux is proportional to the current in the electric circuit, or rather, the ampere-turns or m.m.f., and so increases with the increasing load on the transformer, and constitutes what is called the self-inductive or leakage reactance of the trans- former; while the flux surrounding both coils may be con- sidered as mutual inductive reactance. This cross-flux of self-induction does not generate e.m.f. in the secondary circuit, 187 188 ALTERNATING-CURRENT PHENOMENA and is thus, in general, objectionable, by causing a drop of voltage and a decrease of output. It is this cross-flux, how- ever, or flux of self-inductive reactance, which is utilized in special transformers, to secure automatic regulation, for con- stant power, or for constant current, and in this case is exagger- ated by separating primary and secondary coils. In the con- stant potential transformer, however, the primary and secondary coils are brought as near together as possible, or even inter- spersed, to reduce the cross-flux. There is, however, a limit, to which it is safe to reduce the cross-flux, as at short-circuit at the secondary terminals, it is the e.m.f. of self-induction of this cross-flux which limits the current, and with very low self-induction, these currents may become destructive by their mechanical forces. Therefore experience shows that in large power transformers it is not safe to go below 4 to 6 per cent, cross-flux. As will be seen, by the self-inductive reactance of a circuit, not the total flux produced by, and interlinked with, the circuit is understood, but only that (usually small) part of the flux which surrounds one circuit without interlinking with the other circuit. 143. The alternating magnetic flux of the magnetic circuit surrounding both electric circuits is produced by the combined magnetizing action of the primary and of the secondary current. This magnetic flux is determined by the e.m.f. of the trans- former, by the number of turns, and by the frequency. If $ = maximum magnetic flux, / = frequency, n = number of turns of the coil, the e.m.f. generated in this coil is E = V27r/n$ 10-8 = 4A4fn^ IQ-^ volts; hence, if the e.m.f., frequency, and number of turns are de- termined, the maximum magnetic flux is E108 $ = — — V27r/n To produce the magnetism, $, of the transformer, a m.m.f. of F ampere-turns is required, which is determined by the shape and the magnetic characteristic of the iron, in the manner dis- cussed in Chapter XII. ALTERNATING-CURRENT TRANSFORMER 189 144. Consider as instance, a closed magnetic circuit transformer. The maximum magnetic induction is B = -r, where A = the cross-section of magnetic circuit. To induce a magnetic density, B, a magnetizing force of / / ampere-turns maximum is required, or —y= ampere-turns effect- ive, per unit length of the magnetic circuit; hence, for the total magnetic circuit, of length, I, F = — y- ampere-turns; or F If / = - = — V^ amp. eff. n n\/2 where n = number of turns. At no-load, or open secondary circuit, this m.m.f., F, is fur- nished by the exciting current, 7oo, improperly called the leakage current, of the transformer; that is, that small amount of primary current which passes through the transformer at open secondarj'- circuit. In a transformer with open magnetic circuit, such as the "hedgehog" transformer, the m.m.f., F, is the sum of the m.m.f. consumed in the iron and in the air part of the magnetic circuit (see Chapter XII). The power component of the exciting current represents the power consumed by hysteresis and eddy currents and the small ohmic loss. The exciting current is not a sine wave, but is, at least in the closed magnetic circuit transformer, greatly distorted by hysteresis, though less so in the open magnetic circuit trans- former. It can, however, be represented by an equivalent sine wave, /oo, of equal intensity and equal power with the distorted wave, and a wattless higher harmonic, mainly of triple frequency. Since the higher harmonic is small compared with the total exciting current, and the exciting current is only a small part of the total primary current, the higher harmonic can, for most practical cases, be neglected, and the exciting current repre- sented by the equivalent sine wave. This equivalent sine wave, /oo, leads the wave of magnetism, "l>, by an angle, a, the angle of hysteretic advance of phase, and 190 ALTERNATING-CURRENT PHENOMENA consists of two components — the hysteretic power current in quadrature with the magnetic flux, and therefore in phase with the generated e.m.f. = ho sin a; and the magnetizing current, in phase with the magnetic flux, and therefore in quad- rature with the generated e.m.f., and so wattless, = Zoo cos a. The exciting current, ho, is determined from the shape and magnetic characteristic of the iron, and the number of turns; the hysteretic power current is power consumed in the iron sin a. = generated e.m.f. 145. Graphically, the polar diagram of m.m.fs., of a trans- former is constructed thus: Let, in Fig. 102, 0$ = the magnetic flux in intensity and phase (for convenience, as intensities, the effective values are Fig. 102. used throughout), assuming its phase as the downwards vertical; that is, counting the time from the moment where the rising magnetism passes its zero value. Then the resultant m.m.f. is represented by the vector, OF, leading 0$ by the angle, FO^ = a. The generated e.m.fs. have the phase 180°, that is, are plotted toward the left, and represented by the vectors, OE'o and OE'i. If, now, 6' = angle of lag in the secondary circuit, due to the total (internal and external) secondary reactance, the secondary current, 7i, and hence the secondary m.m.f., Fi = rii 7i lag behind E'l by an angle 6', and have the phase, 180° + 9', repre- ALTERNATING-CURRENT TRANSFORMER 191 sented by the vector OFi. Constructing a parallelogram of m.m.fs., with OF as the diagonal and OFi as one side, the other side or OFq is the primary m.m.f., in intensity and phase, and hence, dividing by the number of primary turns, no, the primary current is /o = no To complete the diagram of e.m.fs., we have now, In the primary circuit: e.m.f. consumed by resistance is I^ro, in phase with 7o, and represented by the vector, OEr^; e.m.f. consumed by reactance is IqXo, 90° ahead of /o, and represented by the vector, OExq', e.m.f. consumed by induced e.m.f. is E', equal and opposite to E'o, and represented by the vector, 0E\ Hence, the total primary impressed e.m.f. by combination of OErQ, OEzQ, and OE' by means of the parallelogram of e.m.fs. is E, - OE,, and the difference of phase between the primary impressed e.m.f. and the primary current is ^0 = EqOFq, In the secondary circuit: Counter e.m.f. of resistance is IiVi in opposition with /i, and represented by the vector, OE'r^; Counter e.m.f. of reactance is IiXi, 90° behind h, and repre- sented by the vector, OE'x^. Generated e.m.fs., E'l, represented by the vector, 0E\. Hence, the secondary terminal voltage, by combination of OE'r^, OE'xi and ()E\ by means of the parallelogram of e.m.fs. is Ex = OE,, and the difference of phase between the secondary terminal voltage and the secondary current is di' = EiOFi. As seen, in the primary circuit the "components of impressed e.m.f. required to overcome the counter e.m.fs." were used for convenience, and in the secondary circuit the "counter e.m.fs." 192 ALTERNATING-CURRENT PHENOMENA 146. In the construction of the transformer diagram, it is usually preferable not to plot the secondary quantities, current and e.m.f., direct, but to reduce them to correspondence with the primary circuit by multiplying by the ratio of turns, a = — , for the reason that frequently primary and secondary e.m.fs., etc., are of such different magnitude as not to be easily repre- sented on the same scale; or the primary circuit may be reduced to the secondary in the same way. In either case, the vectors representing the two generated e.m.fs. coincide, or OE'i = OE'^. E'l \ \ Exi Exo E' E^-^ --^,^-n :~~ 0 ^ ____J22;^ -^^^^^^ A El E'ri' Ero Eo Fig. 103. Figs. 103 to 109 give the polar diagram of a transformer having the constants, reduced to the secondary circuit: ro = 0.2 ohm, 6o = 0.173 mhos, a^o = 0.33 ohm, E'l = 100 volts, Ti = 0.167 ohm, /i == 60 amp., Xi = 0.25 ohm, a = 30°. ^0 = 0.100 mhos. For the conditions of secondary circuit: d\ = 80° lag in Fig. 103 d'l = 20° lead in Fig. 107 50° lag " 104 50° lead '' 108 20° lag " 105 80° lead " 109 0, or in phase, " 106 As shown, with a change of d'l the other quantities, £"0, Ii, 7o, etc., change in intensity and direction. The loci described ALTERNATING-CURRENT TRANSFORMER 193 Fig. 104. ./ ^^0 Fig. 105. 13 Fig. 106. 194 ALTERNATING-CURRENT PHENOMENA Fig. 107. Fia. 108. ALTERNATING-CURRENT TRANSFORMER 195 196 ALTERNATING-CURRENT PHENOMENA by them are circles, and are shown in Fig. 110, with the point corresponding to non-inductive load marked. The part of the locus corresponding to a lagging secondary current is shown in thick full lines, and the part corresponding to leading current in thin full lines, 147. This diagram represents the condition of constant secondary generated e.m.f., E\, that is, corresponding to a con- stant maximum magnetic flux. By changing all the quantities proportionally from the dia- gram of Fig. 110, the diagrams for the constant primary im- FiG. 113. pressed e.m.f. (Fig. Ill), and for constant secondary terminal voltage (Fig. 112), are derived. In these cases, the locus gives curves of higher order. Fig. 113 gives the locus of the various quantities when the load is changed from full-load, /i = 60 amp. in a non-inductive secondary external circuit, to no-load or open-circuit: (a) By increase of secondary current; (6) by increase of secondary inductive resistance; (c) by increase of secondary condensive reactance. As shown in (a), the locus of the secondary terminal voltage, El, and thus of E^, etc., arc straight lines; and in (6) and (c), parts of one and the same circle; (a) is shown in full lines, {h) in heavy full lines, and (c) in light full lines. This diagram corre- sponds to constant maximum magnetic flux; that is, to constant secondary generated e.m.f. The diagrams representing constant ALTERNATING-CURRENT TRANSFORMER 197 primary impressed e.m.f. and constant secondary terminal voltage can be derived from the above by proportionality. 148. It must be understood, however, that for the purpose of making the diagrams plainer, by bringing the different values to somewhat nearer the same magnitude, the constants chosen for these diagrams represent not the magnitudes found in actual transformers, but refer to greatl}^ exaggerated internal losses. In practice, about the following magnitudes would be found: ro = 0.01 ohm; Xi = 0.00025 ohm; Xq = 0.033 ohm; go = 0.001 mho; ri = 0.00008 ohm; 6o = 0.00173 mho; that is, about one-tenth as large as assumed. Thus the changes of the values of Eo, Ei, etc., under the different conditions will be very much smaller. Symbolic Method 149. In symbolic representation by complex quantities the transformer problem appears as follows: The exciting current, /oo, of the transformer depends upon the primary e.m.f., which dependence can be represented by an admittance, the "primary admittance," Fo = g^i — jbo, of the transformer. The resistance and reactance of the primary and the secondary circuit are represented in the impedance by Zo = To + jxo, and Zi = ri + jxi. Within the limited range of variation of the magnetic density in a constant-potential transformer, admittance and impedance can usually, and with sufficient exactness, be considered as constant. Let no = number of primary turns in series; Hi = number of secondary turns in series; a = — = ratio of turns; ni ' Fo = ^0 — jho = primary admittance Exciting current Primary induced e.m.f. ' 198 ALTERNATING-CURRENT PHENOMENA Zo = To -{■ jxo = primary impedance e.m.f. consumed in primary coil by resistance and reactance "~ Primary current ' Zi = Ti -\- jxi = secondary impedance e.m.f. consumed in secondary coil by resistance and reactance Secondary current ' where the reactances, Xo and Xi, refer to the true self-induction only, or to the cross-flux passing between primary and second- ary coils; that is, interlinked with one coil only. Let also Y = g —jb = total admittance of secondary circuit, in- cluding the internal impedance; £"0 = primary impressed e.m.f.; E' = e.m.f. consumed by primary counter e.m.f.; El = secondary terminal voltage; E'l = secondary generated e.m.f.; 7o = primary current, total; Zoo = primary exciting current; 1 1 = secondary current. Since the primary counter e.m.f., Eo', and the secondary generated e.m.f., E\, are proportional by the ratio of turns, a, E'o = + aE'i. (1) 771/ 771/ The secondary current is f 1 = YE\. (2) consisting of a power component, gEi , and a reactive component, hE\. To this secondary current corresponds the component of primary current. - YE\ YE'_ ^.'- a ■ - a' ^^^ The primary exciting current is /oo = YoE\ (4) ALTERNATING-CURRENT TRANSFORMER 199 Hence, the total primary current is h = f 0 + /oo (5) YE' 7o = ^{F+a^Fo} (6) = -^{Y + a^Y.]. The e.m.f. consumed in the secondary coil by the internal impedance is ZJi. The e.m.f. generated in the secondary coil by the magnetic flux is E\. Therefore, the secondary terminal voltage is El = E'l — Zili] or, substituting (2), we have El = E\{1 - ZiY}' (7) The e.m.f. consumed in the primary coil by the internal im- pedance is Zq/o. The e.m.f. consumed in the primary coil by the counter e.m.f. isE'. Therefore, the primary impressed e.m.f. is Eq = E -j- Zolof or, substituting (6), ZoY] Eo=^ E' jl +ZoFo + ^} = - aE\ 1 + ZoFo + ZoF (8) 150. We thus have, primary e.m.f., E^ = -ai'i{l+ ZqFo + ^}' (8) secondary e.m.f., Ex = E'l {1 - ZiY}, (7) E' primary current, 7o = ^ {F + a^Fo}, (6) d 200 ALTERNATING-CURRENT PHENOMENA secondary current, /i = FE'/, (2) as functions of the secondary generated e.m.f., Ei , as parameter. From the above we derive Ratio of transformation of e.m.fs.: E^ 1 + ZoFo + Z,Y E, ~ 1 -ZiK Ratio of transformations of currents : b M 1 _U 2 ^0 (9) (10) From this we get, at constant primary impressed e.m.f., Eq = constant; secondary generated e.m.f., ^0 1 E\ = - « l+ZoFo + ^' e.m.f. generated per turn, E, 1 dE = - ''' l+ZoFo + ^' secondary terminal voltage, Eo 1 - ZiY E, ^ - ^ l+ZoFo + ^' primary current. /o = E(,) F + a^Fo ZoF = £"0 -2 + ^0 l+ZuFo + ^ • 1+ZoFo-f ZoF a- secondary current, ^0 F /i = - I+Z0F0 + At constant secondary terminal voltage, El = const.; (11) ALTERNATING-CURRENT TRANSFORMER 201 secondary generated e.m.f., e.m.f. generated per turn, primary impressed e.m.f., primary current, secondary current, E\ = 8E = E, 1 -ZiF' E\ 1 ni 1 -ZiF' l+Zoro + Z,Y Eq = — aEi h = /i = Ex aiii ' 1-ZiY E, F + a^Fo. a 1 -ZiF' F 1 -ZiF (12) 151. Some interesting conclusions can be drawn from these equations. The apparent impedance of the total transformer is ZoY j. I+Z0F0 + 17 _ j^ _ o (13) + Zo(f. + -) Z, = 1 ^^« + ^ + Zo (14) Substituting now, -7 = F', the total secondary admittance, reduced to the primary circuit by the ratio of turns, it is 1 Zt = V + Zc (15) Fo+F' ' ^'• Fo + F' is the total admittance of a divided circuit with the exciting current of admittance, Fo, and the secondary current of admittance, F' (reduced to primary), as branches. Thus, 1 [7/ — ^'0 (16) 202 ALTERNATING-CURRENT PHENOMENA is the impedance of this divided circuit, and Zt = Z'o + Zq. (17) That is, The alternate-current transformer, of primary admittance Yo, total secondary admittance Y, and primary impedance Zq, is equivalent to, and can he replaced by, a divided circuit with ■ the branches of admittance Fo, the exciting current, and admittance Y Y' = —^, the secondary current, fed over mains of the impedance Zo, the internal primary impedance. This is shown diagrammatically in Fig. 114. Generator A ^ Yog UVi E» o P i z„§ ^ Transformer Receiving Circuit Fig. 114. 152. Separating now the internal secondary impedance from the external secondary impedance, or the impedance of the consumer circuit, it is 1 ^ = Zi-\- Z; where Z = external secondary impedance, Z = Ii (18) (19) Reduced to primary circuit, it is 1 jr> = y = a'Zi + a'Z = Z\ + Z'. (20) That is. ALTERNATING-CURRENT TRANSFORMER 203 An alternate-current transformer, of primary admittance Yq, primary impedance Zo, secondary impedance Zi, and ratio of turns a, can, when the secondary circuit is closed by an impedance, Z {the impedance of the receiver circuit) , be replaced, and is equiva- lent to a circuit of impedance, Z' — a^Z, fed over mains of the impedance, Zo + Z'i, where Z'l = a^Zi, shunted by a circuit of admittance, Yo, which latter circuit branches off at the points, a, b, between the impedances, Zo and Z'x. This is represented diagrammatically in Fig. 115. Generator T Transformer I "El Fig. 116. It is obvious, therefore, that if the transformer contains sev- eral independent secondary circuits, they are to be considered as branched off at the points a, b, in diagram, Fig. 115, as shown in diagram. Fig. 116. It therefore follows: An alternate-current transformer, of s secondary coils, of the 204 ALTERNATING-CURRENT PHENOMENA internal impedances, Zi , Zi ,. . .Zi% closed by external secondary circuits of the impedances, Z , Z ,.. .Z', is equivalent to a divided circuit of s -{- 1 branches, one branch of admittance, Yo, the excit- ing current, the other branches of the impedances, Z\ + Z^ , Zx^ + Z^^ , . . . Zi* + Z^, the latter impedances being reduced to the primar'y circuit by the ratio of turns, and the whole divided circuit being fed by the primary impressed e.m.f., Eo, over mains of the impedance, Zq. Consequently, transformation of a circuit merely changes all the quantities proportionally, introduces in the mains the impedance, Zq + Z'l, and a branch circuit between Zo and Z'l, of admittance Yq. Thus, double transformation will be represented by diagram. Fig. 117. With this the discussion of the alternate-current transformer ends, by becoming identical with that of a divided circuit con- taining resistances and reactances. Transformer Transformer Receiving Circuit Fig. 117. Such circuits have been discussed in detail in Chapter IX, and the results derived there are now directly applicable to the transformer, giving the variation and the control of secondary terminal voltage, resonance phenomena, etc. Thus, for instance, if Z'l = Zo, and the transformer contains an additional secondary coil, constantly closed by a condensive reactance of such size that this auxiliary circuit, together with the exciting circuit, gives the reactance, ~ Xo, with a non-inductive secondary circuit, Zi = ri, we get the condition of transformation from constant primary potential to constant secondary current, and inversely. ALTERNATING-CURRENT TRANSFORMER 205 153. As seen, the alternating-current transformer is charac- terized by the constants: Ratio of turns: a = — 71 1 Exciting admittance: Yo = go — jbo. Self-inductive impedances: Zo = ro -{- jxo. Zi = ri + jxi. Since the effect of the secondary impedance is essentially the same as that of the primary impedance (the only difference being, that no voltage is consumed by the exciting current in the secondary impedance, but voltage is consumed in the primary impedance, though very small in a constant-potential trans- former), the individual values of the two impedances, Zo and Zi, are of less importance than the resultant or total impedance of the transformer, that is, the sum of the primary impedance plus the secondary impedance reduced to the primary circuit: Z' = Zo + cx'Zu and the transformer accordingly is characterized by the two constants: Exciting admittance, Fo = !7o — jbo. Total self-inductive impedance, Z' = r' -\- jx'. Especially in constant-potential transformers with closed magnetic circuit — as usually built — the combination of both impedances into one, Z', is permissible as well within the errors of observation. Experimentally, the exciting admittance, Yo = go — jbo, and the total self-inductive impedance, Z^ = r' -]r jx', are deter- mined by operating the transformer at its normal frequency: 1. With open secondary circuit, and measuring volts eo, amperes to, and watts Wo, input — excitation test. 2. With the secondary short-circuited, and measuring volts ei, amperes z'l, and watts pi, input. (In this case, usually a far lower impressed voltage is required — impedance test.) It is then: io yo = — ' eo Po bo = Vz/o^ + Sfo^ ei z 2l r u' c = ^z — r ' 206 ALTERNATING-CURRENT PHENOMENA If a separation of the total impedance Z' into the primary impedance and the secondary impedance is desired, as a rule the secondary reactance reduced to the primary can be assumed as equal to the primary reactance: Ci'Xi = Xq, except if from the construction of the transformer it can be seen that one of the circuits has far more reactance than the other, and then judgment or approximate calculation must guide in the division of the total reactance between the two circuits. If the total effective resistance, r', as derived by wattmeter, equals the sum of the ohmic resistances of primary and of secondary reduced to the primary: r' = ro + aV, the ohmic resistances, ro and ri, as measured by Wheatstone bridge or by direct current, are used. If the effective resistance is greater than the resultant of the ohmic resistances: r' > ro + aVi, the difference: r" ^ r' — (ro + aVi) may be divided between the two circuits in proportion to the ohmic resistances, that is, the effective resistance distributed between the two circuits in the proportion of their ohmic resist- ances, so giving the effective resistances of the two circuits, r'o and r'l, by: r'o -^ r'l = ro -^ ri; or, if from the construction of the transformer as the use of large solid conductors, it can be seen that the one circuit is entirely or mainly the seat of the power loss by hysteresis, eddies, etc., which is represented by the additional effective resistance, r" , this resistance, r", is entirely or mainly assigned to this circuit. In general, it therefore may be assumed: r 0 = ro X a^o x' Xi — 2a2 ro + « ri r 1 = ri — ; — ro + aVi ALTERNATING-CURRENT TRANSFORMER 207 Usually, the excitation test is made on the low-voltage coil, the impedance test on the high-voltage coil, and then reduced to the same coil as primary. Hereby the currents and voltages are more nearly of the same magnitude in both tests. 154. In the calculation of the transformer: The exciting admittance, Yq, is derived by calculating the total exciting current from the ampere-turns excitation, the mag- netic characteristic of the iron and the dimensions of the main magnetic circuit, that is the magnetic circuit interlinked with primary and secondary coils. The conductance, go, is derived from the hysteresis loss in the iron, as given by magnetic density, hysteresis coefficient and dimensions of magnetic circuit, allow- ance being made for eddy currents in the iron. The ohmic resistances, ro and ri, are found from the dimen- sions of the electric circuit, and, where required, allowance made for the additional effective resistance, r". The reactances, xq and Xi, are calculated by calculating the leakage flux, that is the magnetic flux produced by the total primary respectively secondary ampere-turns, and passing be- tween primary and secondary coils, and within the primary respectively secondary coil, in a magnetic circuit consisting largely of air. In this case, the iron part of the magnetic leakage circuit can as a rule be neglected.