IV. Regulation 115. As primary and secondary winding of the transformer can- not occupy the same space, and in addition some insulation — more or less depending on the voltage — must be between them, there is thus a space between primary and secondary through which the primary current can send magnetic flux which does not interlink with the secondary winding, but is a self-induc- tive or leakage flux and in the same manner the secondary current sends self-inductive or leakage flux through the space between primary and secondary winding. These fluxes give rise to the self -inductive or leakage reactances x\ and Xz of the transformer. Or in other words, two paths exist for magnetic flux in the transformer: the path surrounding primary and secondary coils, through which flows the mutual magnetic flux of the transformer, which is the useful flux, that is, the flux which transfers the power from primary to secondary circuit; and the space between pri- mary and secondary winding through which the self-inductive or leakage flux passes, that is, the flux interlinked with one wind- ing only, but not the other one. The latter flux thus does not transmit power, but consumes reactive voltage and thereby pro- duces a voltage drop and a lag of the current behind the voltage, that is, is in general objectionable. The mutual magnetic flux passes through a closed magnetic circuit, with the (vector) difference between primary and second- ary current, that is, the exciting current J0 = /i as m.m.f. The self-inductive flux passes through an open magnetic circuit of high reluctance, the narrow space between primary and secondary windings, but it is due to the full m.m.f. of primary or secondary current and, therefore, in spite of the high reluctance of the leakage flux path due to the high m.m.f. (20 times as great as that of the mutual flux at 5 per cent, exciting current), this flux and the reactance voltages caused by it are appreci- 286 ELEMENTS OF ELECTRICAL ENGINEERING able, usually between 2 per cent, and 8 per cent, in modern transformers. The distribution of the leakage flux between primary and secondary winding, that is, between primary reactance x\ and secondary Xz, is to some extent arbitrary (see discussion in "Theory and Calculation of Electric Circuits'')) and the methods of test give only the sum of the primary and the secondary re- actance, the latter reduced to the primary by the ratio of trans- formation : Xi + a2x2. 116. The total reactance of primary and secondary, and also TRANSFORMER I mpedance and Short Circuit Losses 7 .1 .2 .3 .1 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 l.i 1.5 FIG. 156. — Impedance and short circuit losses of transformer. the total (effective) resistance of primary and secondary winding are measured by impressing voltage on the primary coil, with the secondary winding short-circuited, and measuring volts, amperes and watts. In this test the voltage usually is impressed upon the high voltage winding, as the impedance voltage is only a small part of the operating voltage of the transformer. Such "impedance curves" and "short-circuit loss curves" for the transformers in Figs. 154 and 155 are shown in Fig. 156. If the short-circuit loss is greater than the sum of primary and ALTERNA TING-CURRENT TRANSFORMER 287 secondary izr losses, the difference represents load losses caused by eddy currents in the conductors, etc. * The reactance of the transformer is often given as percentage. Six per cent, reactance thus means that the primary ix, as per cent, of the primary impressed voltage, plus the secondary ix as per cent, of the secondary voltage, is 6 per cent. Or: *'*' + **'*'= 0.06. e\ Especially since x\ and xz cannot be separated experimentally, but the impedance test gives the sum of primary reactance x\, and secondary reactance x2 reduced to the primary by the ratio of transformation a, that is this is permissible. The foremost effects of the leakage reactance of the trans- former are, to affect the voltage regulation, and to determine the short-circuit current and the mechanical forces resulting from it. 117. The exciting current, being a small and practically con- stant component of the primary current, does not affect the regu- lation of the transformer appreciably, and thus can be neglected in the calculation of the regulation curve. If this is done, the secondary quantities can be reduced to the primary by the ratio of transformation (or inversely), that is, by multiplying all secondary voltages and dividing all secondary currents by a, and multiplying all secondary impedances by a2, or inversely when reducing from primary to secondary.1 Or, primary and secondary impedances can be given in per cent., that is, the primary ir and ix in per cent, of the primary voltage, the secondary ir and ix in per cent, of the secondary voltage, and in this case, primary and secondary quantities can be directly added. This usually is the most convenient way, at least for approximate calculation. Thus in the transformer shown in Fig. 154, let £ = 0.02 be the total reactance (2 per cent.), at full non- inductive load. 1 As the transformation ratio of the voltage is a, that of the current is -i the transformation ratio of the impedances (resistance and reactance), is volts a2, as impedance = — — amperes 288 ELEMENTS OF ELECTRICAL ENGINEERING p = 0.02 is the total resistance, primary and secondary com- bined. At the percentage p of the non-inductive load, the voltage consumed by reactance is p% = 0.02 p and in quadrature with the current and thus with the voltage at non-inductive load, hence subtracts by ^/difference of squares: while the voltage consumed by the resistance is pp — 0.02 p and in phase with the voltage, hence directly subtracts, leaving: - P2? ~ PP = V 1 - 0.0004 p2- 0.02 p as the voltage at percentage p of load, given as per cent, of the open-circuit or no-load voltage. The voltage drop at-frac- REGULATION of TRANSFORMER Non-inductive Load; I Z = .02 + .02j II .01 + .04 j III .01 + .08 j 3.5 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 FIG. 157. — Regulation curve of transformer: non-inductive load. tional load p, as fraction of full-load voltage, that is, the regula- tion of the transformer at non-inductiVe load, then is R = 1 - VI - p2^ + PP = 1 - V 1 - 0.0004 p2 + 0.02 p or, resolved by the binomial, and dropping the higher terms: R = PP + \ P2? = 0.02 p + 0.0002 p2 = P (P + \ P?} = 0.02 p (1 + 0.01 p) As curves I, II, III in Fig. 157 are shown the regulation curves of three transformers: ALTERNATING-CURRENT TRANSFORMER 289 I: 2 per cent, resistance and 2 per cent, reactance. II : 1 per cent, resistance and 4 per cent, reactance. Ill : 1 per cent, resistance and 8 per cent, reactance. FIG. 158. — Vector diagram of transformer regulation. % 6.5, REGULATION of TRANSFORMER Inductive Load: 20 Lag I Z = .02 + .02 3 II .01 + .04 3 III .01 + .08 3 / <<* / / 5^ / / 5.0 / / 4^ / S 4.^ 17 ^^f ^ / x^ s^ 3.0 / / Ix ^ s^ 2.5 / / <> ^1 2.0 / ^ t> ^ 1^ S / /? ^ 1.0 / £ ^ ^ X X Lc iad :— — »- .1 ^ .3 .4 ^ .6 .7 .8 .9 1.0 LI LI U 1.4 U FIG. 159. — Regulation of transformer, moderately inductive load. Calculated respectively by the equations given at end of next paragraph. 118. At inductive load of power-factor cos o>, that is, the lag of the current behind the voltage by angle w, the regulation 290 ELEMENTS OF ELECTRICAL ENGINEERING curve is derived from the vector diagram Fig. 158. The ir voltage is in phase with the current, the ix voltage 90 deg. ahead of the current. Resolving both of these voltages into components in phase and in quadrature with the terminal voltage, gives (Fig. 158): 16.0 / % / 10.0 REGULATION of TRANSFORMER Inductive Load: 60 Lag I Z = .02 -f .02 j II .01+.04J III .01+ .08 j / / 9.5 / 9.0 / 8.5 / / 8.0 / 7.5 / 7.0 / / 6.5 III / ^ / / 5.5 / / / / 5.0 / / / 4.5 / ^ _x ^^O / / / X x^ 3.5 / / / I X x 3.0 / / / «x X 2.5 / / /I x / 2.0 / / .X X 1.5 / / / ^ 1.0 / 'x S x^ Loa d:_ — >- .5 ^ X .1 .2 .3 .4 .6 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 FIG. 160. — Regulation of transformer, highly inductive load. ir cos co and ix sin co in phase with e, ix cos co and —ir sin co in quadrature with e. The former thus directly subtract, and the latter subtract by A/difference of squares, thus giving as resultant voltage : — (ix cos co — ir sin co) 2 — (ir cos co + ix sin co) ALTERNATING-CURRENT TRANSFORMER 291 or, since ir at full load as fraction of e is p, and ix as fraction of 6 is £; at the fraction p of the load: ir = pp, ix = p£, the re- sultant voltage is: \/l — p2 (^ cos co — p sin co)2 — P (p cos co + £ sin co) and the regulation of the transformer, at inductive load of angle of lag co, thus is- R = 1 — A/1 — p2 (£ cos co — p sin co)2 + p (p cos co + £ sin co). Resolving again the square root by the binomial, and arrang- ing, gives, by dropping out terms of higher order: v2 R — p (p cos co + £ sin co) + ~ (£ cos co — p sin co)2 In Figs. 159 and 160 are shown, for the angles of lag co = 20° (moderately inductive load, 94 per cent, power-factor), and co = 60° (highly inductive load, 50 per cent, power-factor), the regulation of the same three transformers as in Fig. 157, cal- culated respectively from the expression: REGULATION OF TRANSFORMERS Per cent, resistance, p = 0.02 0.01 0.01 Per cent, reactance, £ = 0.02 0.04 0.08 ^ w of lag: Curve I Curve II Curve III Fig. 157, 0° R = 0.02p O.Olp O.Olp (1 +0.01p) (1 +0.08p) (1 +0.32p) Fig. 159, 20° R = 0.0256p 0.0231p 0.0368p (1 + 0.0027p) (1 + 0.025p) (1 +0.07?) Fig. 160, 60° R = 0.0273p 0.0396p 0.0743p (!4+0.001p) (1 + 0.0016p) (1+0. 0066 p) Non-inductive 20° lag. 60° lag. P I II III I II III I II III 0.2 0.40 0.20 0.21 0.51 0.46 0.75 0.55 0.79 1.49 0.4 0.80 0.41 0.45 1.02 0.93 1.52 1.09 1.58 2.99 0.6 1.20 0.63 0.71 1.54 1.40 2.30 1.64 2.38 4.48 0.8 1.61 0.85 1.00 2.05 1.88 3.13 2.18 3.17 5.98 1.0 2.02 1.08 1.32 2.56 2.37 3.94 2.73 3.96 7.48 1.2 2.43 1.31 1.66 3.08 2.85 4.79 3.28 4.76 9.00 1.4 2.84 1.56 2.03 3.59 3.34 5.67 3.83 5.56 10.50 1.6 3.25 1.80 2.42 4.10 3.83 6.56 4.38 6.35 12.00 i 292 ELEMENTS OF ELECTRICAL ENGINEERING 119. As seen, at non-inductive load, Fig. 157, the reactance of the transformer, even if fairly high, has practically no effect, but the resistance controls the regulation. At moderately inductive load reactance as well as resistance affect the regulation; doubling the reactance while halving the resistance, gives practically the same regulation. FIG. 161. — High reactance transformer construction. At highly inductive load the reactance of the transformer be- gins to predominate over the resistance in affecting the regulation. Thus, where close regulation is required, as in lighting and general distribution transformers, low reactance is of impor- tance. This is given by reducing the section of the leakage path — that is, bringing primary and secondary windings as close together FIG. 162. — Low reactance transformer construction. as possible — and by reducing the m.m.f. which produces the leak- age flux, by subdividing primary and secondary winding into a number of coils and intermixing these coils, so that the leakage flux of each path is due to a small part of the total m.m.f. of primary or secondary only, as shown in Figs. 161 and 162. In Fig. 162 the m.m.f. of each of the four leakage paths is due to ALTERNATING-CURRENT TRANSFORMER 293 one-fourth of the m.m.f. as in Fig. 161, and the leakage flux density thus reduced to one-fourth of what it is in Fig. 161. As furthermore the section of each leakage flux in Fig. 162 is materially less than in Fig. 161, due to the lesser thickness of the coils, it follows that in Fig. 162 the leakage flux interlinked with each turn of each winding, and thus the reactance of the transformer, is materially less than one-quarter of what it is in Fig. 161. The regulation of the transformer at anti-inductive load, that is, for leading secondary current, obviously is given by the same equation as that for lagging current, by merely substituting — co for co.