CHAPTER XII REACTANCE OF INDUCTION APPARATUS 109. An electric current passing through a conductor is ac- companied by a magnetic field surrounding this conductor, and this magnetic field is as integral a part of the phenomenon, as is the energy dissipation by the resistance of the conductor. It is represented by the inductance, L, of the conductor, or the number of magnetic interlinkages with unit current in the conductor. Every circuit thus has a resistance, and an inductance, however small the latter may be in the so-called "non-inductive" circuit. With continuous current in stationary conditions, the inductance, L, has no effect on the energy flow; with alternating current of frequency, /, the inductance, L, consumes a voltage 2 x/Li, and is, therefore, represented by the reactance, x = 2x/L, which is measured in ohms, and differs from the ohmic resistance, r, merely by being wattless or reactive, that is, representing not dissipation of energy, but surging of energy. Every alternating-current circuit thus has a resistance and a reactance, the latter representing the effect of the magnetic field of the current in the conductor. When dealing with alternating-current apparatus, especially those having several circuits, it must be realized, however, that the magnetic field of the circuit may have no independent exist- ence, but may merge into and combine with other magnetic fields, so that it may become difficult what part of the magnetic field is to be assigned to each electric circuit, and circuits may exist which apparently have no reactance. In short, in such cases, the magnetic fields of the reactance of the electric circuit may be merely a more or less fictitious component of the resultant mag- netic field. The industrial importance hereof is that many phenomena, such as the loss of power by magnetic hysteresis, the m.m.f. required for field excitation, etc., are related to the resultant magnetic field, thus not equal to the sum of the corresponding effects of the components. 216 REACTANCE OF INDUCTION APPARATUS 217 As the transformer is the simplest alternating-current appara- tus, the relations are best shown thereon. Leakage Flux of Alternating-current Transformer 110. The alternating-current transformer consists of a mag- netic circuit, interlinked with two electric circuits, the primary circuit, which receives power from its impressed voltage, and the secondary circuit, which supplies power to its external circuit. For convenience, we may assune the secondary circuit as re- duced to the primary circuit by the ratio of turns, that is, assume ratio of turns 1 -^ 1. Let Fo = 17 - j6 = primary exciting admittance; Zo = ro+ jxo = primary self-inductive impedance; Zi = ri + jxi = secondary self-inductive impedance (reduced to the primary). The transformer thus comprises three magnetic fluxes: the mutual magnetic flux, $, which, being interlinked with primary and secondary, transforms the power from primary to secondary, and is due to the resultant m.m.f of primary and secondary cir- cuit ; the primary leakage flux, $'o, due to the m.m.f. of the primary circuit, Fo, and interlinked with the primary circuit only, which is represented by the self-inductive or leakage reactance, Xo; and the secondary leakage flux, f>'i, due to the m.m.f. of the secondary circuit, Fi, and interlinked with the secondary circuit only which is represented by the secondary reactance, Xi. As seen in Fig. 105o, the mutual flux, $ — ^usually — ^has a closed .iron circuit of low reluctance, p, thus low m.m.f ., F, and high intens- ity; the self-inductive flux or leakage reactance flux, f>'o and $'i, close through the air circuit between the primary and secondary electric circuits, thus meet with a high reluctance, po, respectively Pi, usually many hundred times higher than p. Their m.m.f s., Fo and Fif however, are usually many times greater than F; the lat- ter is the m.m.f. of the exciting current, the former that of full primary or secondary current. For instance, if the exciting current is 5 per cent, of full-load current, the reactance of the transformer 4 per cent., or 2 per cent, primary and 2 per cent, secondary, then the m.m.f. of the leakage flux is 20 times that of the mutual flux, and the mutual flux 50 times the leakage flux, hence the reluctance of leakage flux 50 X 20 = 1000 times that of the mutual or main flux: pi = 1000 p. 218 ELECTRIC CIRCUITS REACTANCE OF INDUCTION APPARATUS 219 111. Usually, as stated, the leakage fluxes are not considered as such, but represented by their reactances, in the transformer diagram. Thus, at non-inductive load, it is, Fig. 106, 0* = mutual, or main magnetic flux, chosen as negative ver- tical. OF = m.m.f. required to produce flux, 0*, and leading it by the angle of hysteretic advance of phase, FO*. OE'i = e.m.f. induced in the secondary circuit by the mutual flux, and 90° behind it. TRANSFORMER DIAGRAM NON-INDUCTIVE LOAD Q MAGNETIC FiQ. 106. IiXi = secondary reactance voltage, 90° behind the secondary current, and combining with OE'i to OEi = true secondary induced voltage. From this subtracts the secondary resistance voltage, Ziri, leaving the sec- ondary terminal voltage, and, in phase with it at non- inductive load, the secondary current and secondary m.m.f., OFi. From component, OFi, and resultant, OF, follows the other com- ponent. 220 ELECTRIC CIRCUITS OFo = primary m.m.f. and in phase with it the primary current. OE'o = primary voltage consumed by mutual flux, equal and opposite to 0E\. hxo = primary reactance voltage, 90*^ ahead of the primary current OFo. From IqXo as component and E'q as resultant follows the other component, OEo, and adding thereto the primary resistance vol- tage, Zoro, gives primary supply voltage. In this diagram. Fig. 106, the primary leakage flux is represented by O^'o, in phase with the primary current, OFo, and the secondary leakage flux is represented by O^'i, in phase with the secondary current, OFi. As shown in Fig. 105o, the primary leakage flux, 'o, passes through the iron core inside of the primary coil, together with the resultant flux, f>, and the secondary leakage flux, $'i, passes through the secondary core, together with the mutual flux, $. However, at the moment shown in Fig. 105o, $'i and $ in the secondary core are opposite in direction. This obviously is not possible, and the flux in the secondary core in this moment is $ — $'i, that is, the magnetic disposition shown in Fig. 105o is merely nominal, but the actual magnetic distribution is as shown in Fig. 105a; the flux in the primary core, $o = ^ + ^'o, the flux in the secondary core, $i = $ — $'i. As seen, at the moment shown in Fig. 105o and 105a, all the leakage flux comes from and interlinks with the primary winding, none with the secondary winding, and it thus would appear, that all the self-inductive reactance is in the primary circuit, none in the secondary circuit, or, in other words, that the secondary circuit of the transformer has no reactance. However, at a later moment of the cycle, shown in Fig. 105c, all the leakage flux comes from and interlinks with the secondary, and this figure thus would give the impression, that all the leakage reactance of the transformer is in the secondary, none in the primary winding. In other words, the leakage fluxes of the transformer and the mutual or main flux are not independent fluxes, but partly tra- verse the same magnetic circuit, so that each of them during a part of the cycle is a part of any other of the fluxes. Thus, the react- ance voltage and the mutual inductive voltage of the transformer REACTANCE OF INDUCTION APPARATUS 221 • • are not separate e.m.fs., but merely mathematical fictions, com- ponents of the resultant induced voltage, OEi and OEo, induced by the resultant fluxes, O^o in the primary, and 0¥[ in the sec- ondary core. 112. In Fig. 107 are plotted, in rectangular coordinates, the magnetic fluxes: The mutual or main magnetic flux, $; The primary leakage flux, $'o; The resultant primary flux, f>o = ^ + ^'o; The secondary leakage flux, ^\; The resultant secondary flux, $i = $ — $'i; MAGNETIC FLUXES OF TRANSFORMER ^ - 6.2 01 "1.5 0i' - 1.06**: #i« 6 ^i-1.9 00 - 60°; ^0=7.6. Fig. 107. and the magnetic distribution in the transformer, during the moments marked as a, 6, c, d, e, /, g, in Fig. 107, is shown in Fig. 105. In Fig. 105a, the primary flux is larger than the secondary, and all leakage fluxes (xo and Xi) come from the primary flux, that is, there is no secondary leakage flux. In Fig. 1056, primary and secondary flux equal, and primary and secondary leakage flux equal and opposite, though small. In Fig. 105c, the secondary flux is larger, all leakage flux (xo and xi) comes from the secondary flux, that is, there is no primary leakage flux. 222 ELECTRIC CIRCUITS In Fig. 105d, there is no primary flux, and all the secondary flux is leakage flux. In Fig. 105e, there is no mutual flux, all primary flux is primary leakage flux, and all secondary flux is secondary leakage flux. In Fig. 105/, there is no secondary flux, and all primary flux ts leakage flux. In Fig. 105g, the primary flux is larger than the secondary, and all leakage flux comes from the primary, the same as in 105a. Figs. 105a to 105/, thus show the complete cycle, corresponding to diagrams. Figs. 106 and 107. These figures are drawn with the proportions. p -^ po -5- Pi = 1 -5- 12.5 H - 12.5 F ^Fo ^Fx = 1 -i- 3.8 H - 3 H-$'o H-$', = 1 4- 0.317 H r- 0.25. thus are greatly exaggerated, to show the effect more plainly. Actually, the relations are usually of the magnitude, P -4- Po -5- Pi = 1 -^ 1000 -r- 1000 F -^ Fo -^ Fi = 1 -T- 20.6 -5- 20 $ -5- $'o -^ ^'i = 1 -^ 0.02 4- 0.02 113. In symbolic representation, denoting, f» = mutual magnetic flux. fJ = mutual induced voltage. f>o = resultant primary flux. f>'o = primary leakage flux. fJo = primary terminal voltage, /o = primary current. Zo = ro + jxo = primary self-inductive imped- ance, f 1 = resultant secondary flux. f>'i = secondary leakage flux. ^1 = secondary terminal voltage, /i = secondary current. Zi = ri + jxi = secondary self-inductive im- pedance, and c = 2irfn where n = number of turns. REACTANCE OF INDUCTANCE APPARATUS 223 It then is Cf' 1 = jxolo c^\ = jxji cf = ^ = ^0 - Zolo = ^1 + Zili cf = ^0 — r-o/o = ^ + jXolo cf 1 = ^1 + ri/i = ^ - jxili f'o = f — f f 'i = f + f 1, thus, the total leakage flux ¥ = ¥o + f 'i = f - f 1. 114. One of the important conclusions from the study of the actual flux distribution of the transformer is that the distinction between primary and secondary leakage flux, $'o and $'i, is really an arbitrary one. There is no distinct primary and secondary leakage flux, but merely one leakage flux, $', which is the flux passing between primary and secondary circuit, and which during a part of the cycle interlinks with the primary, during another part of the cycle interhnks with the secondary circuit Thus the corresponding electrical quantities, the reactances, Xo and Xi, are not independent quantities, that is, it can not be stated that there is a definite primary reactance, Xo, and a definite secondary react- ance, Xi, but merely that the transformer has a definite reactance, X, which is more or less arbitrarily divided into two parts; x = Xo + Xi, and the one assigned to the primary, the other to the second- ary circuit. As the result hereof, "mutual magnetic flux" f>, and the mutual induced voltage, E, are not actual quantities, but rather mathe- matical fictions, and not definite but dependent upon the distri- bution of the total reactance between the primary and the sec- ondary circuit. This explains why all methods of determining the transformer reactance give the total reactance Xq + Xi. However, the subdivision of the total transformer reactance into a primary and a secondary reactance is not entirely arbitrary. Assuming we assign all the reactance to the primary, and consider the secondary as having no reactance. Then the mutual mag- netic flux and mutual induced voltage would be cf = jP = jPo - [ro + i (xo + xi)] /o and the hysteresis loss in the transformer would correspond hereto, by the usual assumption in transformer calculations. 224 ELECTRIC CIRCUITS Assigning, however, all the reactance to the secondary circuit, and assuming the primary as non-inductive, the mutual flux and mutual induced voltage would be c^ = ^ = ^o — ^o/o, hence larger, and the hysteresis loss calculated therefrom larger than under the previous assumption. The first assumption would give too low, and the last too high a calculated hysteresis loss, in most cases. By the usual transformer theory, the hysteresis loss under load is calculated as that corresponding to the mutual induced voltage, E. The proper subdivision of the total transformer reactance, a?, into primary reactance, aJo, and secondary reactance, aJi, would then be that, which gives for a uniform magnetic flux, $, corresponding to the mutual induced voltage, E, the same hysteresis loss, as exists with the actual magnetic distribution of ^o = ^ + ^'o in the primary, and $i = $ — ^\ in the secondary core. Thus, if Vo is the volume of iron carrying the primary flux, $o, at flux den- sity. Bo, Vi the volume of iron carrying the secondary flux, f>i, at flux density, Bi, the flux density of the theoretical mutual mag- netic flux would be given by VoBo^-' + ViBi^'^ jgi-e = Vo+Vi from B then follows $, E, and thus xo and Xi. This does not include consideration of eddy-current losses. For these, an approximate allowance may be made by using 1.7 as exponent, instead of 1.6. Where the magnetic stray field undejr load causes additional losses by eddy currents, these are not included in the loss assigned to the mutual magnetic flux, but appear as an energy component of the leakage reactances, that is, as an increase of the ohmic re- sistances of the electric circuits, by an effective resistance. 116. Usually, the subdivision of x into Xo and Xi, by this as- sumption of assigning the entire core loss to the mutual flux, is sufficiently close to equality, to permit this assumption. That is, the total transformer reactance is equally divided between primary and secondary circuit. This, however, is not always justified, and in some cases, the one circuit may have a higher reactance than the other. Such, for instance, is the case in some very high voltage transformers, and usually is the case in induction motors and similar apparatus. It is more commonly the case, where true self-inductive fluxes REACTANCE OF INDUCTION APPARATUS 225 exist, that is, magnetic fluxes produced by the current in one circuit, and interlinked with this circuit, closing upon themselves in a path which is entirely distinct from that of the mutual mag- netic flux, that is, has no part in common with it. Such, for in- stance, frequently is the self-inductive flux of the end connections of coils in motors, transformers, etc. To illustrate: in the high- voltage shell-type transformer, shown diagrammatically in Fig. 108, with primary coil 1, closely adjacent to the core, and high-voltage secondary coil 2 at considerable distance: The primary leakage flux consists of the flux in spaces, a, between the yokes of the transformer, closing through the iron core, C, and the flux through the spaces, 6, outside of the trans- former, which enters the faces, F, of the yokes and closes through the central core, C The secondary leakage flux contains the same two components: the flux through the spaces, a, between the yokes closing, however, through the outside shells, /S, and the flux through the spaces, 6, outside of the transformer, and entering the faces, F, but in this case closing through the shells, S. In addition to these two com- ponents, the secondary leakage flux contains a third component, passing through the spaces, 6, between the coils, but closing, through outside space, c, in a complete air circuit. This flux has no corresponding component in the primary, and the total secondary leakage reactance in this case thus is larger than the total primary reactance. Similar conditions apply to magnetic structures as in the in- duction motor, alternator, etc. In such a case as represented by Fig. 108, the total reactance of the transformer, with (2) as primary and (1) as secondary, would be greater than with (1) as primary and (2) as secondary. In this case, when subdividing the total reactance into primary reactance and secondary reactance, it would appear legitimate to divide it in proportion of the total reactances with (1) and (2) as primary, respectively. That is, if a; = total reactance, with coil (1) as primary, and (2) as secondary, and x' = total reactance, with coil (2) as primary, and (1) as secondary, then it is: With coil (1) as primary and (2) as secondary, 15 f26 ELECTRIC CIRCUITS Primary reactance, — ^ - "=* ^^ ~ JT^ ~ x + x'' Secondary reactance, _ x' _ xx' ' x + x' x + x' With coil (2) as primary and (1) as secondary, F F S :l o o Is 8 F F ff^^\ I _-M-_j \\^^ Primary reactance, Secondary reactance, t + x'-^ ^,^ xx' _ '^' x + x' x + x' 116. By test, the two total reactances, x and x*, can be derived by considering, that in Fig. 107 at the moments, / and d, the total flux is leakage flux, as more fully shown in Fig. 105/ and lOM, and the flux measured from /, gives the reactance, x, measured from d, gives the reactance, d. REACTANCE OF INDUCTION APPARATUS 227 Assuming we connect primary coil and secondary coil in series with each other, but in opposition, into an alternating-current circuit, as shown in Fig. 109, and vary the number of primary and secondary turns, until the voltage, ei, across the secondary coil, s, becomes equal to rii. Then no flux passes through the secondary coil, that is, the condition. Fig. 107/, exists, and the voltage, eo, across the primary coil, p, gives the total reactance, Xy for p as primary, eo^ = i2 (ro2 + x^). Varying now the number of turns so that the voltage across the primary coil equals its resistance drop, eo = roi, then the Fig. 109. voltage across the secondary coil, s, gives the total reactance, x^, for s as primary, It would rarely be possible to vary the turns of the two coils, p and s. However, if we short-circuit s and pass an alternating current through p, then at the very low resultant magnetic flux and thus resultant m.m.f., primary and secondary current are practically in opposition and of the same m.m.f., and the mag- netic flux in the secondary coil is that giving the resistance drop riii, that is, e\ = ri f i is the true primary voltage in the secondary, and the voltage across the primary terminals thus is that giving primary resistance drop, roio, total self-inductive reactance; xio, and the secondary induced voltage, riii. Thus, eo^ = (Wo + riii y+ xHq^, i 228 ELECTRIC CIRCUITS or, since ii practically equals io, Co' = to^ [(ro + ri)2 + x^], and inversely, impressing a voltage upon coil, «, and short-cir- cuiting the coil p, gives the leakage reactance, x', for s as primary, ei' = ii^ [(ro + ri)2 + a/")]. Thus, the so-called "impedance test" of the transformer gives the total leakage reactance Xq + xi, for that coil as primary, which is used as such in the impedance test. Where an appreciable difference of the total leakage flux is expected when using the one coil as primary, as when using the other coil, the impedance tests should be made with that coil as primary, which is intended as such. Since, however, the two leakage fluxes are usually approximately equal, it is immaterial which coil is used as primary in the impedance test, and gener- ally that coil is used, which gives a more convenient voltage and current for testing. Magnetic Circuits of. Induction Motor 117. In general, when dealing with a closed secondary winding, as an induction-motor squirrel-cage, we consider as the mutual inductive voltage, -F, the voltage induced by the mutual magnetic flux, $, that is, the magnetic flux due to the resultant of the pri- mary and the secondary m.m.f. This voltage, ^F, then is con- sumed in the closed secondary winding by the resistance, ri/i, and the reactance, jxili, thus giving, ^ = (ri + jxi) /i. The reactance voltage, jxj/i, is consumed by a self-inductive flux, f>i', that is, a magnetic leakage flux produced by the second- ary current and interlinked with the secondary circuit, and the actual or resultant magnetic flux interlinked with the secondary circuit, that is, the magnetic flux, which passes beyond the second- ary conductor through the armature core, thus is the vector dif- ference, fi = $ — if, and the actual voltage induced in the second- ary circuit by the resultant magnetic flux interlinked with it thus is, -Fi = -F — jxili. This voltage is consumed by the resistance of the secondary circuit, f!i = ri/i, and the voltage consumed by self-induction, jxi/i, is no part of -Fi, but as stated, is due to the self -inductive flux, $i', which vectorially subtracts from the mutual magnetic flux, f>, and thereby leaves the flux, ^i', which induces fJi. REACTANCE OF INDUCTION APPARATUS 229 In other words: In any closed secondary circuit, as a squirrel-cage of an induc- tion motor, the true induced e.m.f . in the circuit, that is, the e.m.f . induced by the actual magnetic flux interlinked with the circuit, is the resistance drop of the circuit, J^Ji = ri/i. This is true whether there is one or any number of closed sec- ondary circuits — or squirrel-cages in an induction motor. In each , El the current, Zi is — , where ri is the resistance of the circuit, and ]^i the voltage induced by the flux which passes through the cir- cuit. The ^1 of the different squirrel-cages then would differ from each other by the voltage induced by the leakage flux which passes between them, and which is represented by the self- inductive reactance of the next squirrel-cage: E\, where /'i = -7— is the current in the inner squirrel-cage of voltage, jF'i, and resistance, r'l, and x\ /'i, is the reactance of the flux between the two squirrel-cages. The mutual magnetic flux and the mutual induced e.m.f. of the common induction motor theory thus are mathematical fictions and not physical realities. The advantage of the introduction of the mutual magnetic flux, ^, and the mutual induced voltage, -F, in the induction-motor theory, is the ease and convenience of passing therefrom to the secondary as well as the primary circuit. Where, however, a number of secondary circuits exist, as in a multiple squirrel-cage, it is preferable to start from the innermost magnetic flux, that is, the magnetic flux passing through the innermost squirrel-cage, and the voltage induced by it in the latter, which is the resistance drop of this squirrel-cage. In the same manner, in a primary circuit, the actual or total magnetic flux interlinked with the circuit, ^0, is that due to the impressed voltage, Eqj minus the resistance drop, ro7o, -F'o = -Fo — ro/o. Of this magnetic flux, ^0, a part, ^'0, passes as primary leak- age flux between primary and secondary, without reaching the secondary, and is represented by the primary reactance voltage, jxofo, and the remainder — usually the major part — is impressed upon the secondary circuit as mutual magnetic flux, f = $0 — *'o, corresponding to the mutual inductive voltage, ^ = -F'o — jxolo- The mutual magnetic flux, $, then is impressed upon the second- 230 ELECTRIC CIRCUITS ary, and as stated above, a part of it, the secondary leakage flux, ^'i, is shunted across outside of the secondary circuit, the re- mainder, ^' = ^ — ^'i, passes through the secondary circuit and corresponds to ri/i. 118. Appljdng this to the polyphase induction motor with single squirrel-cage secondary. Let Yo — g — jb = primary exciting admittance; Zo = ro + jxo = primary self-inductive impedance; Zi = ri + jxi = secondary self-inductive impedance at full frequency, reduced to the primary. Let Pi = the true induced voltage in the secondary, at full frequency, corresponding to the magnetic flux in the armature core. The secondary current then is The mutual inductive voltage at full frequency, ^ = ^1 + jxifi Thus the exciting current, /oo = YoP = ((7-i&)(l+if)^i where sbxi Qi = g + q2 = b — ri sgxi ri and the total current, /o = /i + /oo = Pi 1^ +qi -ig2|» hence, the primary impressed voltage, Po =P + Zoh = ^1 j 1 + i ^ + (ro + jxo) [f^ + qi" 392^]^ = Pi (ci + jCi), REACTANCE OF INDUCTION APPARATUS 231 where Ci = 1 + ro(— + qi) + Xoq2 = 1 + s — + ToQi + x^q^ BXi , /S . \ s(Xi + Xo) , fi \ri / fi . choosing now the impressed voltage as zero vector, ^0 = eo gives Ci +JC2 or, absolute, ei = —7= V Cl^ + 02^ the torque of the motor is D = /^i, /i/ ^ ri ri(ci2 + 02^)' the power, P ^ g^i^l - g) ^ g(l - g)eo^ n ri(ci2 + 02^) the volt-ampere input, Q = 6oio etc. As seen, this method is if anjrthing, rather les& convenient than the conventional method, which starts with the mutual inductive voltage p. It becomes materially more advantageous, however, when dealing with double and triple squirrel-cage structures, as it permits starting with the innermost squirrel-cage, and gradually building up toward the primary circuit. See "Multiple Squirrel- cage Induction Motor," "Theory and Calculation of Electrical Apparatus."