CHAPTER XIII. THS ALTERNATING^CnRRENT TRAXSFOBMER. 116. The simplest alternating-current apparatus is the transformer. 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 induced. 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. induced per turn must be the same in the secondary as in the primary circuit ; hence, the primary induced E.M.F. being approximately equal to the impressed E.M.F., the E.M.Fs. at primary and at sec- ondary terminals have approximately the ratio of their respective turns. Since the power produced in the second- ary is approximately the same as that consumed in the primary, the primary and secondary currents are approxi- mately in inverse ratio to the turns. 117. 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 secondary coils, surrounding one coil only, without being interlinked with the other. This magnetic cross-flux is proportional to the current flowing in the electric circuit, or rather, the ampere- turns or M.M.F. increase with the increasing load on the transformer, and constitute what is called the self-induc- tance of the transformer; while the flux surrounding both 168 AL TERN A TING-CURRENT PHENOMENA. [§118 coils may be considered as mutual inductance. This cross- flux of self-induction does not induce E.M.F. in the second- ary circuit, and is thus, in general, objectionable, by causing a drop of voltage and a decrease of output ; and, therefore, in the constant potential transformer the primary and sec- ondary coils are brought as near together as possible, or even interspersed, to reduce the cross-flux. As will be seen, by the self-inductance 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 w^hich surrounds one circuit without interlinking with the other circuit. 118. 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 transformer, by the number of turns, and by the frequency. If 4> = maximum magnetic flux, N-= frequency, n = number of turns of the coil ; the E.M.F. induced in this coil is E = ViirNn^ 10-« volts, = 4.44 AV/* 10 -'volts; hence, if the E.M.F., frequency, and number of turns are determined, the maximum magnetic flux is E 1()» W'2.irNn To produce the magnetism, *, of the transformer, a M.M.F. of JF ampere-turns is required, which is determined by the shape and the magnetic characteristic of the iron, in the manner discussed in Chapter X. §119] ALTERNATING-CURRENT TRANSFORMER. 169 For instance, in the closed magnetic circuit transformer^ the maximum magnetic induction is (» = *, where 5 = the cross-section of magnetic circuit. 119. To induce a magnetic density, (B, a M.M.F. of 3C^ ampere-turns maximum is required, or, JC^ / V2 ampere- turns effective, per unit length of the magnetic circuit ; hence, for the total magnetic circuit, of length, /, SF = — ^ ampere-turns ; / = - = ^ amps, eff., where ;/ = number of turns. At no load, or open secondary circuit, this M.M.F., iF, is furnished by the exciting current, I^ , improperly called the leakage currenty of the transformer ; that is, that small amount of primary current which passes through the trans- former at open secondary circuit. In a transformer with open magnetic circuit, such as the "hedgehog" transformer, the M.M.F., iF, is the sum of the M.M.F. consumed in the iron and in the air part of the magnetic circuit (see Chapter X.). The energy of the exciting current is the energy con- sumed 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 transformer. It can, however, be represented by an equiv- alent sine wave, /^o, 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 170 AL TERN A TIKG-CURRENT PHENOMENA. [§ 1 20 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 represented by the equivalent sine wave. This equivalent sine wave, /^, leads the wave of mag- netism, *, by an angle, a, the angle of hysteretic advance of phase, and consists of two components, — the hysteretic energy current, in quadrature* with the magnetic flux, and therefore in phase with the induced E.M.F. = /^ sin a; and the magnetizing current, in phase with the magnetic flux, and therefore in quadrature with the induced E.M.F., and so wattless, = /^ cos a. The exciting current, /^, is determined from the shape and magnetic characteristic of the iron, and number of turns ; the hysteretic energy current is — Power consumed in the iron A, sin a = '00 Induced E.M.F. 120. Graphically, the polar diagram of M.M.Fs. ot a transformer is constructed thus : Let, in Fig. 87, 6^* = the magnetic flux in intensity and phase (for convenience, as intensities, the effective values are used throughout), assuming its phase as the vertical ; § 120] ALTERNATINU-CUKRENT TRANSFORMER, 171 that is, counting the time from the moment where the rising magnetism passes its zero vahie. Then the resultant M.M.F. is represented by the vector <9^, leading t?* by the angle ^(7* = ea. The induced E.M.Fs. have the phase 180"^, that is, are plotted towards the left, and represented by the vectors OE;, and 0E{. If, now, Wj' = angle of lag in the secondary circuit, due to the total (internal and ewXternal) secondary reactance, the secondary current Z^, and hence the secondary M.M.F., (Fi= ;/i /p will lag behind E^ by an angle ^\ and have the phase, 180° + )S', represented by the vector O'S^, Con- structing a parallelogram of M.M.Fs., with C^$F as a diag- onal and C^tFj as one side, the other side or (9$F<, is the primary M.M.F., in intensity and phase, and hence, dividing by the number of primary turns, ;/^, the primary current is To complete the diagram of E.M.Fs., we have now, — In the primary circuit : E.M.F. consumed by resistance is lo^o^ in phase with /<,, and represented by the vector OEor ; E.M.F. consumed by reactance is IqXo^ 90° ahead of /«,, and represented by the vector OEo x \ E.M.F. consumed by induced E.M.F. is Eo^ equal and oppo- site thereto, and represented by the vector OEo . Hence, the total primary impressed E.M.F. by combina- tion of OE^f.^ OEo^t and OE^' by means of the parallelo- gram of E.M.Fs. is, ' Eo= OEo, and the difference of phase between the primary impressed E.M.F. and the primary current is /8o = Eo 0^0. ^ In the secondary circuit : Counter E.M.F. of resistance is /i^i in opposition with/i, and represented by the vector OEi / ; / 172 ALTERNATING-CURRENT PHENOMENA, [§121 Counter E.M.F. of reactance is I\Xxy 90° behind /i, and represented by the vector OEu ; Induced E.M.Fs., E{ represented by the vector 0E(. Hence, the secondary terminal voltage, by combination of OEy^, OEi^ and 0E{ by means of the parallelogram of E.M.Fs. is E^ = OE^, and the difference of phase between the secondary terminal . voltage and the secondary current is wi = EiO^i* As will be seen in the primary circuit the " components of impressed E.M.F. required to overcome the counter RM.Fs." were used for convenience, and in the secondary circuit the "counter E.M.Fs." Fig. 88. Transformer Diagram with 8Cr Lag In Secondary Circuit 121. 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 corre- spondence with the primary circuit by multiplying by the ratio of turns, — a = //^///j, for the reason that frequently primary and secondary E.M.F's., etc., are of such different §121] ALTERNATING-CURRENT TRANSFORMER, 173 magnitude as not to be easily represented on the same scale; or the primary circuit may be reduced to the sec- ondary in the same way. In either case, the vectors repre- senting the two induced E.M.Fs. coincide, or OE^=iOE^. e: ■^E. Fig. 89. Transformer Diagrcun with 50* Lag in Secondary Circuit Figs. 88 to 94 give the polar diagram of a transformer having the constants — ro = .2 ohms, Xo = .^fS ohms, ri = .00167 ohms, Xi = .0025 ohms, £-0 = .0100 mhos, fi^ = .0173 mhos, E,' = 100 volts, /j = 60 amperes, a = 10. for the conditions of secondary circuit, ^/ = 80"^ lag in Fig. 88. ^/ = 20° lead in Fig. 92. 50"^ lag '' 89. 50** lead " 93. 20° lag •' 90. 80° lead " 94. O, or in phase, ** 91. As shown with a change of (3^, Eo^Siygo^ ^^c., change in intensity and direction. The locus described by them are circles, and are shown in Fig. 95, with the point corre- sponding to non-inductive load marked. The part of the . locus corresponding to a lagging secondary current is 174 AL TEKXA TING-CURRENT PHENOMENA. [§121 f /g. 90, Tnuitfwnwr Diagram with 2Cr Lag In Secondary Circuit ^ Fig. 91. Transfornwr Diagram with Secondary Current In Phase with E.M.F. fig, 92. Traneformer Diagram with 2Cr Lead In Secondary Current, §121] ALTEXA'AT/A'G-CURKEA'T TKA.VSFORMER. 175 Fig. 03. Tnmaformtr Diagram with S(f ImuI in Secondary Circuit Fig, 04, Transformer Diagram with SCT Lead in Secondary Circuit Fig, 05, 176 ALTERNATING-CURRENT PHENOMENA, [§ 122 shown in thick full lines, and the part corresponding to leading current in thin full lines. 122. This diagram represents the condition of con- stant secondary induced E.M.F., E^y that is, corresponding to a constant maximum magnetic flux. By changing all the quantities proportionally from the diagram of Fig. 95, the diagram for the constant primary impressed E.M.F. (Fig. 96), and for constant secondary ter- minal voltage (Fig. 97), are derived. In these cases, the locus gives curves of higher order. Flq. 06. Fig. 98 gives the locus of the. various quantities when the load is changed from full load, /j = 60 amperes in a non-inductive secondary external circuit to no load or open circuit. a.) By increase of secondary resistance ; 6.) by increase of secondary inductive reactance ; c.) by increase of sec- ondary capacity reactance. As shown in a.), the locus of the secondary terminal vol- tage, ^j-, and thus of E^y etc., are straight lines; and in d.) and c), parts of one and the same circle a.) is shown i 123] ALTERNATING-CURRENT TRANSFORMER. 177 in full lines, b,) in heavy full lines, and c.) in light full lines. This diagram corresponds to constant maximum magnetic flux; that is, to constant secondary induced E.M.F. The diagrams representing constant primary impressed E.M.F. and constant secondary terminal voltage can be derived from the above by proportionality. Pig. 97. 123. It must be understood, however, that for the pur- jx)se of making the diagrams plainer, by bringing the dif- ferent values to somewhat nearer the same magnitude, the constants chosen for these diagrams represent, not the mag- nitudes found in actual transformers, but refer to greatly exaggerated internal losses. In practice, about the following magnitudes would be found : ro = .01 ohms; Xo = .033 ohms ; ri = .00008 ohms ; x^ = .00025 ohms ; g^ = .001 ohms ; do = .00173 ohms ; that is, about one-tenth as large as assumed. Thus the changes of the values of E^y E^, etc., under the different conditions will be very much smaller. A/. TERAA TINC-CVRRENT PHENOMKAA Symbolic Method. 124- In symbolic representation by complex quantities the transformer problem appears as follows : The exciting current, /„, of the transformer depends upon the primary K.M.K., which dcpendance can be rc|> resented by an admittance, the " primary admittance," Y^=^ g^ ■\- j b^, of the transformer. rig. 9B. The resistance and reactance of the primary and the secondary circuit are represented in the impedance by ^u = r^ —j^ut and Z| = r, —Jx\- Within the limited range of variation of the magnetic density in a constant [iotential transformer, admittance and impedance can usually, and with sufficient exactness, be considered as constant. Let «„ = number of primary turns in series; t, = number of secondary turns in series; a ^ -^ = ratio of turns; ^« =K" — /''.. = primary admittance ~ i'riiMIJ cuunltF K.M.k!' § 124] ALTERNATING-CURRENT TRANSFORMER. 179 Zo = ro — jxo = primary impedance K^I .F. cons umed in primary coil by resistance and reactance . Primary current Zi = ri — yji*, = secondary impedance E.M.F. consumed in secondary coil by resistance a nd reacta nce . Secondary current where the reactances, x^ and Xy^, refer to the true self-induc- tance only, or to the cross-flux passing between primary and secondary coils ; that is, interlinked with one coil only. Let also Y =^4-y^ = total admittance of secondary circuit, including the internal impedance ; Eo = primary impressed E.M.F. ; EJ = E.M.F. consumed by primary counter E.M.F. ; El = secondary terminal voltage ; E{ = secondary induced E.M.F. ; lo = primary current, total ; /^ = primary exciting current ; /i = secondary current. Since the primary' counter E.M.F., E^, and the second- ary induced E.M.F., E^y are proportional by the ratio of turns, a, • , e: == -aE/. (1) The secondary current is : A = VEi', (2) consisting of an energy component, ^E^', and a reactive component, gE^. To this secondary current corresponds the component of primary current, , ^ ^ /; ^^XE^^L^, (3) a o The primary exciting current is — J^=Y,e:. (4) Hence, the total primary current is : /„ = /;+ /« (5) YE.: a" + YoE:, 180 ALTERNATING-CURRENT PHENOMENA. [§125 or, /. =^'{F+a«Ko} (6) a The E.M.F. consumed in the secondary coil by the internal impedance is Z^J^, The E.M.F. induced in the secondary coil by the mag- netic flux is E^, Therefore, the secondary terminal voltage is Ex = Ex — Zxl\ , or, substituting (2), we have Ex = E({\^ZxY^ (7) The E.M.F. consumed in the primary coil by the inter- nal impedance is Z^ /^. The E.M.F. consumed in the primary coil by the counter E.M.F. is E^, Therefore, the primary impressed E.M.F. is Eo = E^ + Zoloy or, substituting (6), Z.Y Eo = EJ)l + ZoVo+^ Z,Y (8) = _a^, i + z,n + a" 125. We thus have, primary E.M.F., E, = - aR( | 1 + Z^n + ^ | , (8) secondary E.M.F., E^ = E( { 1 — Z, K}, (7) p ' primary current, /„ = — {1" + ^/*}',}, (6) a secondary current, ly — IVT,', (2) as functions of the secondary induced E.M.F., E^^ as pa- rameter. 1126] AL TERN A TING-CURRENT TRANSFORMER, 181 F*rom the above we derive Ratio of transformation, of E.M.Fs. : 7 Y — - = — rt 1 -ZiF (^) Ratio of transformations of currents : -^= -Ml + tf>ii'l. A a\ ^ Y) (10) From this we get, at constant primary impressed E.M.F., E^ = constant secondary induced E.M.F., E E^ = - £p a 7 K' E.M.F. induced per turn, IE = -^^^ ^ ''^ i + z,>; + :?i2:' secondary terminal voltage, l+ZoY,+ a' primary current, /o = E a y+a^v, _ o _ i 7 V = £. ^+ y. l+ZoY,+ z.y' a' secondary current. T __ ■'^O a 7 Y (11) At constant secondary termmal voltage, El = const. ; 182 ALTERNATING-CURRENT PHENOMENA, [§ 12© secondary induced E.M.F., E.M.F. induced per turn, (VI) //i 1 - Zj }' ' primary impressed E.M.F., 7 V Eo = — iiEi ^ ; ' 1-ZiK ' primary current, " ~ a i-z,y' secondary current, /^ = E^ - — ? . 'i-z,y 126. Some interesting conclusions can be drawn from these equations. The apparent impedance of the total transformer is ^- = t = - y+..y: '''^ ~ y ' Z,^ L_^ + Z,. (14) n + 4- y Substituting now, -— = K', the total secondary admit- tance, reduced to the primary circuit by the ratio of turns, it is Z, = ^ -, + ^0. (15) V^+y is the total admittance of a divided circuit with the exciting current, of admittance F^, and the secondary §127] ALTERNATING-CURRENT TRANSFORMER. 183 current, of admittance V (reduced to primary), as branches. Thus : ^ — 7 == Zl (16) is the impedance of this divided circuit, and Z — Z' A- 7 That is : (17) T/ie alternate-current transformery of primary admittance Y^ , total secondary admittance V, and primary impedance Z^ , is equivalent tOy and can be replaced by^ a divided circuit with the branches of admittance V^, the exciting current^ and admittance V = Y/d^, the secondary current, fed over mains of the impedance Z^, the internal primary impedance. This is shown diagram m at ically in Fig. 99. Qanerator r E. Transformer J3 Receiving Circuit Fig. 09. 127. Separating now the internal secondary impedance from the external secondary impedance, or the impedance of the consumer circuit, it is y = ^. + ^; where Z = external secondary impedance, h (18) (19) 184 ALTERKATIXG-CURRENT PHENOMENA. [8 127 Reduced to primary circuit, it is ^ = ^ = a»Z, + a*Z Y' Y ^ That is : = Z{ + Z'. (20) An alteniate-current transformer^ of primary admittance Y^, prifnary impedance Z^, secondary impedaftce Z^y and ratio of turns a^ can^ when the secondary circuit is closed by an impedaftce Z (the impedaftce of the receiver circuit) ^ be replaced^ and is equivaleftt to a circuit of impedance Z* = a^Zy fed oi'cr mairts of the impedaftce Z^+ Z^, zvhere Z^ = d^Z^y shunted by a circuit of admittafice 1^, which latter circuit branclies off at t/te points a — by betwecft the impe* dafues Z^ and Z^. Q«n«pator Tntn«fbrm«p ^ R«o«iving O Cirouii- Z I z. Y.: a 1 Zfa«Z, Z^«^Z Ftq. 100. This is represented diagrammatically in Fig. 100. It is obvious therefore, that if the transformer contains several independent secondary circuits they are to be con- sidered as branched off at the points a — by in diagram Fig. 100, as shown in diagram Fig. 101. It therefore follows : An alterftate-currcftt transformer y of x secondary coils y of the ititemal ifftpedaficcs Z^y Z^^y . . . Z^y closed by external secoftdary circuits of the impcdafices Z^y Z^^y . . . Z'y is cquii^ alctit to a divided circuit of x + 1 braftches, one brafich of 4127] ALTERNATING-CURRENT TRANSFORMER. 185 Q«n9rfttor E. Tran«fbrm«r R«e«!vinf Circuits -^m^ Fig, 101. admittatuc V^, the exciting current ^ the other branches of the impedances Z/.+ Z^, Z^" + Z"y . . . Z/ + Z', the latter impedances being reduced to the primary circuit by the ratio of turftSf and the whole divided circuit being fed by the primary impressed E.M,F, E^^ over mains of the impedance Consequently, transformation of a circuit merely changes all the quantities proportionally, introduces in the mains the impedance Z^+ Zj', and a branch circuit between Z^ and Z(y of admittance Y^, Thus, double transformation will be represented by dia- gram. Fig. 102. TrfineformT Trmnsform«p R«o«!ving Circuits FI9, 102. 18G ALTERNATING-CCRKEXT PHENOMEXA. [§ 120 With this the discussion of the alternate-current trans- former ends, by becoming identical with that of a divided circuit containing resistances and reactances. Such circuits have explicitly been discussed in Chapter VIII., and the results derived there are now directly appli- cable to the transformer, giving the variation and the con- trol of secondary terminal voltage, resonance phenomena, etc. Thus, for instance, if Z( = Z^, and the transformer con- tains a secondary coil, constantly closed by a condenser reactance of such size that this auxiliary circuit, together with the exciting circuit, gives the reactance —x^y with a non-inductive secondary circuit Z^ = /-p we get the condi- tion of transformation from constant primary potential to constant secondary current, and inversely, as previously discussed. Non-indiictivc Secondary Circuit, 128. In a non-inductive secondary circuit, the external secondary impedance is, or, reduced to primary circuit, a' a' Assuming the secondary impedance, reduced to primary circuit, as equal to the primary impedance, Q ^1 = Z,0 = To — J Xq j It IS, V 1 1 Substituting these values in Equations (9), (10), and (13), we have Ratio of E.M.Fs. : R + r^— jXo S 128] ALTERNATING-CURRENr TRANSFORMER. 187 2 ^ +/-,.— 7 .v„ \^ A' + r^ — y A„y ) or, expanding, and neglecting terms of higher than third order, y^i ) ^ + r^ —J^'^'o \^ + ro — Jxo) (ro —jXo){go+jK)\ ; -or, expanded, f^= - ^ j 1 + 2 ^" ~/'^'- + (r, - y.r j(^, + jb;) \ Neglecting terms of tertiary order also, :^' = - .7 t 1 + 2 ^-"—.-^^X . E, \ ^ R S Ratio of currents : 4-= --{1 ^{go^Jbo\R^r,^jXo^Y or, expanded, Neglecting terms of tertiary order also, ^=-^ {l+R(go+jbo)y Total apparent primary admittance : 1 + '^- •^'^<-. + (r, - A„) {go +JK) y Eo i^+ ^u — JXo '^ 7 ~ 1 A + r^ — yj;, = {R+ (•/-„ ->.T„) + J? (r. - yr.) (i', +>*.)} {1- (^. +/*.) 188 AL TERNA TING-CURRENT PHENOMENA. [§ 1 29 or, Neglecting terms of tertiary order also : Angle of lag in primary circuit : tan u)o = — , hence, 2 ^ +Rb, + 2 robo -2xogo -2J^g,b, tan u)o = i^ . Neglecting terms of tertiary order also : 2^ + Kb, tan u)o = . 1 + 2^-^^, 129. If, now, we represent the external resistance of the secondary circuit at full load (reduced to the primary circuit) by R^^ and denote, ^^ fo _ ratin Internal resistance o f transformer percentage intCf- '~rI "" P "" Extemaf^istance of secondiry'cir^t ~ ^al resistance, ** Xq _. q __ x^\AC\ I"* ^"*'* ^ reactance of tr ansformer ___ percentage mtcr- J^ * Kxtemai resistance of secondary circuit ||^| reaCtance R..,.= h = ratio ^.!^"j::^::^™r:' = percentage hysteresis, R b = a = ratio — -*'^*ii"*l*-*="'^l^- = Percentage magnetizing cur- ** ** ^ Toul secondary current rent and if d represents the load of the transformer, as fraction of full load, we have JO Ra § 1 29 ] AL TERN A TING-CURRENT TRANSFORMER, 189 and, 2r. R = pd. 2x, = <\d, Rgo h ~~d' lib. d' Substituting these values we get, as the equations of the transformer on non-inductive load, Ratio of E.M.Fs. : I = - . { 1 + j or, eliminating imaginary quantities, Ratio of currents : /._ 1 li . (h+yg) , (P-yq)(t.+yg) ) /i « ( ^ ^ ^ 2 ) or, eliminating imaginary quantities. ^Mi I h ph + qg+g« t a\ ^ d^ 2d* S 190 AL TERNA TING-CURRENT PHENOMENA. L§ 1 29 Total apparent primary impedance : or, eliminating imaginary quantities, ^t 2 ^Ii//«) .fji + .p-hj Angle of lag in primary circuit : tan u>o = 1 + ^p _ ^ _ ^ _ hg hMijg! ^ ^ d That is, ^ An alternate-current transformery feeding into a non-induc- tive secondary circuit, is represented by the constants : Ro = secondary external resistance at full load; p = percentage resistance ; q = percentage reactance ; h = percentage hysteresis ; g = percentage magnetizing current ; d = secondary percentage load. All these qualities being considered as reduced to the primary circuit by the square of the ratio of turns, c?. I 130] ALTERNATINO-CURKENT TRAKiSFORMER. 191 130. As an instance, a transformer of the following constants may be given; R„ = 120 ; = 1,000; 10; . p =.0L'- Substituting these values, gives : 100 h = .0-J ; g = .04. -v^i.iMH4 + .o:i ,// + (.0002 + .oil rf/ - ''>'^''* . = i\!L ■ R, 1.2 ' = .1 ^1 Ji 1.0014 + - -^y + / ■'**- - .0002 Y; 1.9073 + .02 a" + - Tr. rf=rn •r. 1 / / i 5 ; 7- ' 1 4 , ^ i^ T-n — ^ < ei :^ — 1 w ^ ^ A . "iry l«- lUO ~m} — S.CO .^nV fff rOS. UMt Diagram ef Tnaufermir. 192 ALTERNATING-CURRENT PHENOMENA, [§130 In diagram Fig. 103 are shown, for the values from d =0 to rf= 1.5, with the secondary current c^ as abscis- sae, the values : secondary terminal voltage, in volts, secondary drop of voltage, in per cent, primary current, in amps, excess of primary current over proportionality with secondary, in per cent, primary angle of lag. The power-factor of the transformer, cos w^, is .45 at open secondary circuit, and is above .99 from 25 amperes, upwards, with a maximum of .995 at full load. ALTERNATING-CURRENT TRANSFORMER, 193