CHAPTER XXII ARMATURE REACTIONS OF ALTERNATORS 192. The change of the terminal voltage of an alternating current generator, resulting from a change of load at constant field excitation, is due to the combined effect of armature reaction and armature self-induction. The counter m.m.f. of the armature current, or armature reaction, combines with the impressed m.m.f. or field excitation to the resultant m.m.f., which produces the resultant magnetic field in the field poles and generates in the armature an e.m.f. called the "virtual generated e.m.f.," since it has no actual existence, but is merely a mathematical fiction. The counter e.m.f. of self-induction of the armature current, that is, e.m.f. generated by the armature current by a local magnetic flux, combines with the virtual generated e.m.f. to the actual generated e.m.f. of the armature, which corresponds to the magnetic flux in the armature core. This combined with the e.m.f. consumed by the armature resist- ance gives the terminal voltage. In most cases the effect of armature reaction and of self- induction are the same in character, and so both effects usually are contracted in one constant; for purposes of design, frequently the self-induction is represented by an increase of the armature reaction, that is, an effective armature reaction used which com- bines the effect of the true armature reaction and the armature self-induction. That is, instead of the counter e.m.f. of self- induction, a counter m.m.f. is used, which would produce the magnetic flux which would generate the e.m.f. of self-induction. For theoretical investigations usually the armature reaction is represented by an effective self-induction, that is, instead of the counter m.m.f. of the armature reaction, the e.m.f. considered, which would be generated by the magnetic flux, which the arma- ture reaction would produce. That is, both effects are com- bined in an effective reactance, the "synchronous reactance." While armature reaction and self-inductance are similar in ARMATURE REACTIONS OF ALTERNATORS 273 effect, in some cases they differ in their action; the e.m.f. of self-inductance is instantaneous, that is, appears and disappears with the current to which it is due. The effect of the armature reaction, however, requires time; the change of the magnetic field resulting from the combination of the counter m.m.f. of arma- ture reaction with the impressed m.m.f. of field excitation occurs gradually, since the magnetic field flux interlinks with the field winding, and any sudden change of the field generates an e.m.f. in the field circuit, which temporarily increases or decreases the field current, and so retards the change of the field flux. So, for instance, a sudden increase of load results in a simultaneous increase of the counter e.m.f. of self-induction and counter m.m.f. of armature reaction. With the armature reaction demagnetizing the field, the field flux begins to decrease, and thus generates an e.m.f. in the field-exciting circuit, which increases the field current and retards the decrease of field flux, so that the field flux adjusts itself only gradually to the change of circuit conditions, at a rate of speed depending upon the constants of the field-exciting circuit, etc. The extreme case hereof takes place when suddenly short- circuiting an alternator; at the first moment the short-circuit current is limited only by the self-inductance, and the magnetic field still has full strength, the field-exciting current has greatly increased by the e.m.f. generated in the field circuit by the arma- ture reaction. Gradually the field-exciting current and there- with the field magnetism die down to the values corresponding to the short-circuit condition. Thus the momentary short- circuit current of an alternator is far greater than the perma- nent short-circuit current; many times in a machine of low self-induction and high armature reaction, as a low-frequency, high-speed alternator of large capacity; relatively little in a machine of low armature reaction and high self-induction, as a high-frequency unitooth alternator, 193. Graphically, the internal reactions of the alternating- current generator can be represented as follows: Let the impressed m.m.f., or field excitation, Fo, be repre- sented by the vector OFo, in Fig. 139, chosen for convenience as vertical axis. Let the armature current, I, be represented by vector 01. This current, /, gives armature reaction Fi = nl, where ?i = number of effective turns of the armature, and is repre- sented by the vector, OFi, with the two quadrature components, 274 ALTERNATING-CURRENT PHENOMENA OF' i, in line with the field m.m.f., Oh\ — and usually opposite thereto — and OF", in quadrature with OF^. OFo combined with OFi gives the resultant m.m.f., OF, with the quadrature components, OF' = OFo — 0F\, and OF". The m.m.f., OF, produces a magnetic flux, 0$, and this gener- ates an e.m.f., OE2, in the armature circuit, 90° behind OF in phase, the virtual generated e.m.f. Fig. 139. The armature self-induction consumes an e.m.f., OE3, 90° ahead of the current, thus, subtracted vectorially from OE2, gives the actual generated e.m.f., OEi. The armature resistance, r, consumes an e.m.f., OEi, in phase with the current, which subtracts vectorially from the actual generated e.m.f., and thus gives the terminal voltage, OE. 194. Analytically, these reactions are best calculated by the symbolic method. ARMATURE REACTIONS OF ALTERNATORS 275 Let the impressed m.in.f., or field-excitation, Fo, be chosen as the imaginary axis, hence represented by ^ Fo = + jfo (1) Let / = u — ji2 = armature current. (2) The m.m.f. of the armature then is Fi =nl = nC/i - ji2) (3) where ji = number of effective armature turns, and the resultant m.m.f. then is F = Fo + Fi^ jifo - ni^) + mi. (4) If, then, (P = magnetic permeance of the structure, that is, magnetic flux divided by the ampere-turns m.m.f. producing it, (P = ^, or, ^ = (9F = j10-8, (6) where / = frequency. Denoting 2 irfn 10 ~ ^ by a we have, (7) 62 = a $ (8) and since the generated e.m.f. is 90° behind the generating flux, in symbolic expression, E2= - ja^; (9) hence, substituting (5) in (9), E2 = a(P{fo - ni2) - jaiPnii, . (10) the virtual generated e.m.f. The e.m.f. consumed by the self-inductive reactance of the armature circuit is, E3 = jxl = jxii + xi2; (11) and therefore, the actual generated e.m.f. El = E2 — Es = {a(P/o - {a(Pn -{- x)i2} - jii{a(?n + x). (12) 276 ALTERNATING-CURRENT PHENOMENA The e.m.f. consumed by the armature resistance, r, is Ei = rl ^ rii — jrii) hence, the terminal voltage, E = El — Ei = ja(P/o - (a(Pn-\-x)i2 — rii\ — j\ii{a(S>n + x) — Hi]. (14) 195. It is /n = field m.m.f. ; hence ^0 = (P/o = magnetic flux, which would be produced by the field excitation, /o, if the magnetic permeance at this m.m.f., /o, were the same, (P, as at the m.m.f., F — that is, if the magnetic characteristic would not bend between /o and F, due to mag- netic saturation, or in other words, when neglecting saturation, and therefore eo = a(P/o(15) = e.m.f. generated in the armature by the field excitation, when neglecting magnetic saturation, or assuming a straight line saturation curve. eo is called the ^^ nominal generated e.m.f. of the machine." ni = armature m.m.f.; therefore, (9ni = magnetic flux produced thereby, and, a(9n.i = e.m.f. generated in the armature bj'^ the magnetic flux of armature reaction, hence, a(Pn = Xi = effective reactance, representing the armature reaction, and Xo = a(Pn + x (16) = synchronous reactance, that is, the effective reactance representing the combined effect of armature self-induction and armature reaction. Substituting (15) and (16) in (14) gives, £' = (eo — a:o^2 — rii) — jix^ii — rio) (17) It follows herefrom: In an alternating-current generator, the combined effect of armature reaction and self-induction can be represented by an effective reactance, the sijnchronous reactance, Xo, which consists of the two components: Xo = X -\- xi (18) where, X = true self-inductive reactance of the armature circuit. Xi = a(9n = effective reactance of armature reaction, (19) I ARMATURE REACTIONS OF ALTERNATORS 277 and the nominal generated e.m.f., €o = a(Pfo; (15) where, n = number of armature turns, effective, /"o = field excitation, in ampere-turns, a = 2TfnlO-\ (7) (P = magnetic permeance of the field structure at a magnetic flux in the field-poles corresponding to the virtual generated e.m.f., E2. The introduction of the term "synchronous reactance," Xo, and "nominal generated e.m.f.," eo, is hereby justified, when dealing with the permanent condition of the electric circuit. The case of the transient phenomena of momentary short- circuit currents, etc., is discussed in a chapter on "Transient Phenomena and Oscillations," section I. It must be understood that the "nominal generated e.m.f.," Co, in an actual machine, in which the magnetic characteristic bends due to the approach to magnetic saturation, is not the voltage generated by the field excitation /o at open-circuit, but is the voltage which would be generated, if at excitation, /o, the magnetic permeance, (P = -^ were the same as at the actual flux existing in the machine — that is, if the magnetic characteristic would continue in a straight line passing through the origin when prolonged. The equation (17) may also be written E = 60- Zol; (20) where, Zo = r + jxo = synchronous impedance of the alternator. / = ii - jii, or, more generally E = Eo- Zol, (22) and so is the equation of a circuit, supplied by the e.m.f., Eo, with the current, /, over the impedance, Zo, as has been discussed in the chapter on resistance, inductive reactance and conden- sive reactance. 278 ALTERNATING-CURRENT PHENOMENA An alternator so is equivalent to an e.m.f., Ea, the nominal generated e.m.f., supplying current over an impedance, Zo, the synchronous impedance. 196. In theoretical investigations of alternators, the syn- chronous reactance, Xo, is usually assumed as constant, and has been assumed so in the preceding. In reality, however, this is not exactly, and frequently not even approximately correct, but the synchronous reactance is different in different positions of the armature with regard to the field. Since the relative position of the armature to the field varies with the armature current, and with the phase angle of the armature current, the regulation curve of the alternator, and other characteristic curves, when calculated under the assump- tion of constant synchronous reactance, Ta&y differ considerably from the observed curves, in machines in which the synchronous reactance varies with the position of the armature. The two components of the synchronous reactance are the self- inductive reactance, and the effective reactance of armature reaction. The self-inductive reactance represents the e.m.f. generated in the armature by the local field produced in the armature by the armature current. The magnetic reluctance of the self-inductive field of the armature coil, however, is, in general, different when this coil stands in front of a field-pole, and when it stands midway between two field-poles, and the self-inductive reactance so periodically varies, between two extreme values, representing respectively the positions of the armature coils in front of, and midway between the field-poles, that is, pulsates with double frequency, between a value, x' , corresponding to the position in front, and a value, x", corre- sponding to a position midway between the field-poles. Depend- ing upon the structure of the machine, as the angle of the pole arc, that is, the angle covered by the pole face, either x' or x" may be the larger one. The effective reactance of armature reaction, X\, corresponds to the magnetic flux, which the armature would produce in the field-circuit. With the armature coil facing the field-pole, that is, in a nearly closed magnetic field-current, Xi, therefore is usually far greater than with the armature coil facing midway between the field-poles, in a more or less open magnetic circuit. Hence, Xi, also varies between two extreme values, Xi and Xi" , corresponding respectively to the position in Hne with, and in ARMATURE REACTIONS OF ALTERNATORS 279 quadrature with, the field-poles. In this case, usually Xi is larger than x/'. Since Xi = a(Pn, where (P = magnetic permeance, (P varies between (P', corresponding to the position of the armature coil opposite the field-poles, and (P", corresponding to the position of the armature coil midway between the field-poles. Usually (P' is far larger. This means that the two components of the resultant m.m.f. F: Fi, in line with, and F" in quadrature with, the field-poles, act upon magnetic circuits of very different permeance, (P' and (P", and the components of magnetic flux, due to F' and F" respectively, are $' = (P'F' $" z= (p"F". The two components of magnetic flux, ^' and $", therefore are in general, not proportional to their respective m.m.fs. F' and F", and the resultant flux, 4>, accordingly is not in fine with the resultant m.m.f., F, but differs therefrom in direction, being usually nearer to the center line of the field-poles. That is, the resultant magnetic flux, , is more nearly in fine with the impressed m.m.f. of field excitation, Fo, than the resultant m.m.f., F, is — or in other words — the magnetic flux is shifted by the armature reaction less than the resultant m.m.f. is shifted. 197. To consider, in the investigation of the armature reactions of an alternator, the difference of the magnetic reluctance of the structure in the different directions with regard to the field, that is, the effect of the polar construction of the field, or the use of definite polar projections in the magnetic field, the reactions of the machine must be resolved into two components, one in line and the other in quadrature with the center line of the field- poles, or the direction of the impressed m.m.f. or field-excitation, Fo. Denoting then the components in line with the field-poles or parallel with the field-excitation, Fo, by prime, as /', F', etc., and the components facing midway between the field-poles, or in quadrature position with the field-excitation, Fo, by second, as /", F", the diagram of the alternator reactions is modified from that given in Fig. 139. Choosing again, in Fig. 140, the impressed m.m.f. or field- excitation, Fo, as vertical vector OFo, the current, 01, consists 280 ALTERNATING-CURRENT PHENOMENA of the component, 01' , in line with Fq, or vertical, and 01" in quadrature with F^, or horizontal. The armature reaction, 0F\, gives the components, 0F\ and OFi", and the resultant m.m.f. therefore consists of two components, OF' = OFq — OFi', and OF" = OFi". Fig. 140. Let now (P' = permeance of the field magnetic circuit; (23) (?" = permeance of the magnetic circuit through the armature in quadrature position to the field-poles; (24) the components of the resultant magnetic flux are, $' = (p'F'^ represented by 0^'; and $" = (P"F", represented by Oi", and the resultant magnetic flux, by combination of 0$' and 0^", is 0$, and is not in line with OF, but differs therefrom, being usually nearer to OFo- ARMATURE REACTIONS OF ALTERNATORS 281 The virtual generated e.m.f. is E2 = a4>, and represented by OE2, 90° behind 0$. Let now x' = self-inductive reactance of the armature when facing the field-poles, and thus corresponding to the compo- nent, /', of the current, (25) and x" = self-inductive reactance of the armature when facing midway between the field-poles, and thus corresponding to the component, /", of the current. (26) Then £"3 = xT = e.m.f. consumed by the self-induction of the current component, /', and £"'3 = x"I" = e.m.f. consumed by the self-induction of the current component, /". £'3 is represented by vector OE's, 90° ahead of 01', and £"'3 is represented by vector 0E"3, 90° ahead of 01". The resultant e.m.f. of self-induction then is given by the combination of OE'3 and OE"^, as OEz. It is not 90° ahead of 01, but either more or less. In the former case, the self-induction consumes power, in the latter case, it produces power. That is, in such an arma- ture revolving in the structure of non-uniform reluctance, the e.m.f. of self-induction is not wattless, but may represent con- sumption, or production of power, as "reaction machine." (See "Calculation of Electrical Apparatus.") Subtracting vectorially OE3 from the virtual generated e.m.f. OEi, gives the actual generated e.m.f., 0E\, and subtracting therefrom the e.m.f. consumed by the armature resistance, OEi, in phase with the current, 01, gives the terminal voltage, OE. 198. Here the diagram has been constructed graphically, by starting with the field-excitation, Fq, the armature current, /, and the phase angle between the armature current, /, and the field-excitation, F^ — that is, the angle between the position in which the armature current reaches its maximum, and the direc- tion of the field-poles. This angle, however, is unknown. Usu- ally the terminal voltage, OE, the current, 01, and the angle, 282 AL TERN A TING-C URRENT PHENOMENA EOI, between current and terminal voltage are given. From these latter quantities, however, the diagram cannot be con- structed, since the position of the field-excitation, Fo, and so the directions, in which the electric quantities have to be resolved into components, are still unknown, when starting the construc- tion of the diagram. That is, as usually, the graphical representation affords an insight into the inner relations of the phenomena, but not a method for their numerical cal- culations, and for the latter purpose, the symbolic method is required. Let Eo = nominal generated e.m.f., or e.m.f. corresponding to the field-excitation, Fo, on a straight line continuation of the magnetic characteristic from the actual value of the field onward Fig. 141. -as shown by Fig, 141. The impressed m.m.f., or field excitation, is then given by jFo. (27) Let I = I' -\- I" = armature current, (28) where the component, /', is in line, the component, /", in quad- rature with: jFo. If n = number of effective armature turns, the m,m.f. of the armature current, /, or the armature reaction, then is f 1 = nl, (29) with its components, in phase and in quadrature with the field; Fx' = nl', 1 F^." = nI"•,\ and the components of the resultant m.m.f. then are F' =jFo-^nr, F" = nl"; (30) (31) ARMATURE REACTIONS OF ALTERNATORS 283 and the resultant F = jFo + nl' + nl". (32) The components of the magnetic flux, in line and in quadrature with jFo, then are «!>' = (?'F' = (P'iJFo-^nr); (33) = (P"nr'; (34) hence, the resultant magnetic flux $=$' + $" = (P'ijFo + nl') + (P"n7" (35) The e.m.f. generated by this magnetic flux, 4>, or the virtual generated e.m.f. is El = — ja^ = - a(P'(Fo + jnl') + - 3a(9"nl". (36) The e.m.f. consumed by the self-inductive reactance, x' , of the current component, /', is, E'^ = jxT, (37) the e.m.f. consumed by the self-inductive reactance, x" , of the current component, I" , is E'\ = jx"!", (38) and the total e.m.f. consumed by self-induction thus is E, =j{x'r+x"n', (39) hence, the actual generated e.m.f. El = E2 — E3 = a(P'Fo - jl'{a(9'n + x') - jl"{a(9"n -\- x"). (40) The e.m.f. consumed by the resistance, r, is £4 = rl = rV -F rZ"; (41) 284 ALTERNATING-CURRENT PHENOMENA hence, the terminal voltage of the machine is E = El — Ei = a(P'Fo - r{r -\- j{a(?'n -\- x')\ - I" {r -}- j {a(P"n + x")}. (42) In this equation of the terminal voltage, x'o = a(P'n + x', I x"o = a(P"n + x", J (43) are effective reactances, corresponding to the two quadrature positions; that is x'o = synchronous reactance corresponding to the position of the armature circuit parallel to the field circuit; (44a) x"o = synchronous reactance corresponding to the position of the armature circuit in quadrature with the field circuit; (446) a(P'Fo is the e.m.f. which would be generated by the field excitation, Fo, with the permeance, (?', in the direction in which the field excitation, Fg, acts, that is Eo = a(P'Fo = nominal generated e.m.f. (45) and it is: terminal voltage, E = Eo- r(r + jVo) - I"{r + jx"o). (46) That is, even with an heteroform structure, as a machine with definite polar projections, the armature reaction and armature self-induction can be combined by the introduction of the terms "nominal generated e.m.f." and "synchronous reactance," as defined above, except that in this case the syn- chronous reactance, Xo, has two different values, .t'o and x"o, corresponding respectively to the two main axes of the magnetic structure, in line and in quadrature with the field-poles. 199. In the equation (46), E, Eo, V and 7" are complex quantities, and I" is in phase with Eo, I' is in quadrature behind Eo, and so behind I": hence, I' can be represented by /' = - jtr\ (47) ARMATURE REACTIONS OF ALTERNATORS 285 where t = ratio of numerical values of /" and /', that is r t = jr, = isin d (48) and 6= angle of lag of current, /, behind nominal generated e.m.f., Eq. Then / = /' + /"= /"(I - jt), (49) or I" = rht ^'i '' = YTjt <^°' Substituting these values (50) in equation (46) gives E=E,- Y^rjt I (^ + .^^''o) - JKr + jx'o) I . (51) In this equation, Eo leads I by angle 9. Hence^ choosing the current, I, as zero vector, / = i, (52) the e.m.f., Eo, which leads i by angle, 6, can be represented bj^ Eo = Co (cos d -\-j sin 6), (53) or, since by equation (48), sin 6 = — . and cos 6 = — , > (54) vT-H" vT+T' ^ Substituting (52) and (55) in equation (51), gives eo Vl + <^ - t { (r + jx"o) - j^ (r + jx'o) } . . ^ = ^Jt (56) Let E = e,-\- jei, (57) where - = tan e', (58) ei and 6' = angle of lag of current, i, behind terminal voltage, E, (59) 286 ALTERNA TING-C URRENT PHENOMENA substituting (57) in (56) and transposing, eoVrH'-(ei']-je2){l-jt)-i{(r-\-jx"o)-jt(r-\-jx'o)} = 0, (60) or, expanded, {eoVl + t^-ei-te2-i{r-htx'o))-\-j{tei-e2 + i(tr-x'\)} =0. (61) As the left side is a complex quantity, and equals zero, the real part as well as the imaginary part must be zero, and equation (61) so resolves into the two equations Co VT+I" - e, - ^62 - 4 (r + tx'o) = 0, tei — ei -\- i {tr — x'\) = 0. From equation (63) follows t 62 + X"oi ei + ri Substituting (64) in (62), and expanding, gives (ei + ny + (62 + x',i) (62 + x"oi) 60 = That is, if V{ei + n)2 + (e2 + x"^' x'o = synchronous reactance in the direction of the field- excitation, x"o = synchronous reactance in quadrature with the field excitation, r = armature resistance, i = armature current, E = ei -\- je2 ^ e(cos 6' + j sin 6') = terminal voltage, that is, 62 tan ^' = — = angle of lag of current i behind terminal voltage, e, the nominal generated e.m.f. of the machine is (ei + n)2 -f (ea + x'oi) {e^ + x"oi) eo = V(ei + n)2+(e2 + a;'V)2 (e cos e' + ri)^ + (e sin 6' + x'oi) (e sin d' + x"oi) Vie cos e' + riy + (e sin 9', + x"o^y (62) (63) (64,^ (65) (66) (67) (68) ARMATURE REACTIONS OF ALTERNATORS 287 and the field excitation, /o, required to give terminal voltage, e, at current, i, and angle of lag, 6', is, therefore •^'' " a(P'n~27r/n2(P' ^''''^ and the position angle, 6, between the field-excitation, /o, and the armature current, i, that is, between the direction of the field-poles and the direction in which the armature current reaches its maximum, is €2 + x"oi e sin d' -f x'\i tan 9 = t = ; — - = T7-j :-• (7Uj ei -\- ri e cos 6 -{- ri 200. At non-inductive load, d = e and ^2 = 0 (71) from (68), ^ (e + n)^ + a^oVV' ^^ \/(e + n)2+a;Vi'' ^ ^ If 3^ 0 ^^ 3; 0 ^= 3^0, (7 1200 1000 800 600 400 200 0 "^ s \, ""^^ "^^ ^ ^ ^ ^ -V "^N ^ i' \ V n \ 11' V \ V \ \ \ \ \ \ V \ ^ \ \ \ \ \ \ \ \ \ \ \ ''^ \ ^ ^ 20 40 CO 80 100 120 110 100 180 200 220 240 260 AMPERES Fig. 142. an alternator of pulsating synchronous reactance, the wave-shape of the machine changes more or less with the load and the char- acter of the load.