CHAPTER XXI ALTERNATING-CURRENT GENERATOR 185. In the alternating-current generator, e.m.f. is generated in the armature conductors by their relative motion through a constant or approximately constant magnetic field. When yielding current, two distinctly different m.m.fs. are acting upon the alternator armature — the m.m.f. of the field due to the field-exciting spools, and the m.m.f. of the armature current. The former is constant, or approximately so, while the latter is alternating, and in synchronous motion relatively to the former; hence fixed in space relative to the field m.m.f., or uni- FiG. 129. directional; but pulsating in a single-phase alternator. In the polyphase alternator, when evenly loaded or balanced, the result- ant m.m.f. of the armature current is more or less constant. The e.m.f. generated in the armature is due to the magnetic flux passing through and interhnked with the armature con- ductors. This flux is produced by the resultant of both m.m.fs., that of the field, and that of the armature. On open-circuit, the m.m.f. of the armature is zero, and the e.m.f. of the armature is due to the m.m.f. of the field-coils only. In this case the e.m.f. is, in general, a maximum at the moment when the armature coil faces the position midway between adjacent field-coils, as shown in Fig. 129, and thus incloses 259 260 ALTERNATING-CURRENT PHENOMENA no magnetism. The e.m.f. wave in this case is, in general, symmetrical. An exception to this statement may take place only in those types of alternators where the magnetic reluctance of the arma- ture is different in different directions; thereby, during the syn- chronous rotation of the armature, a pulsation of the magnetic flux passing through it is produced. This pulsation of the mag- netic flux generates e.m.f. in the field-spools, and thereby makes the field current pulsating also. Thus, we have, in this case, even on open-circuit, no rotation through a constant magnetic field, but rotation through a pulsating field, which makes the e.m.f. wave unsymmetrical, and shifts the maximum point from its theoretical position midway between the field-poles. In general this secondary reaction can be neglected, and the field m.m.f. be assumed as constant. Fig. 130. The relative position of the armature m.m.f. with respect to the field m.m.f. depends upon the phase relation existing in the electric circuit. Thus, if there is no displacement of phase be- tween current and e.m.f., the current reaches its maximum at the same moment as the e.m.f. or, in the position of the armature shown in Fig. 129, midway between the field-poles. In this case the armature current tends neither to magnetize nor demagnetize the field, but merely distorts it; that is, demagnetizes the trail- ing pole corner, a, and magnetizes the leading pole corner, h. A change of the total flux, and thereby of the resultant e.m.f., will take place in this case only when the magnetic densities are so near to saturation that the rise of density at the leading pole corner will be less than the decrease of density at the trailing pole corner. Since the internal self-inductive reactance of the alternator itself causes a certain lag of the current behind the generated e.m.f., this condition of no displacement can exist only in a circuit with external negative reactance, as capacity, etc. ALTERNATING-CURRENT GENERATOR 2G1 If the armature current lags, it reaches the maximum later than the e.m.f.; that is, in a position where the armature-coil partly faces the field-pole which it approaches, as shown in dia- gram in Fig. 130. Since the armature current is in opposite direc- tion to the current in the following field-pole (in a generator), the armature in this case will tend to demagnetize the field. If, however, the armature current leads — that is, reaches its maximum while the armature-coil still partly faces the field-pole which it leaves, as shown in diagram, Fig. 131 — it tends to magnetize this field-pole, since the armature current is in the same direction as the exciting current of the preceding field spools. Thus, with a leading current, the armature reaction of the alternator strengthens the field, and thereby, at constant field excitation, increases the voltage; with lagging current it weakens Fig. 131. the field, and thereby decreases the voltage in a generator. Ob- viously, the opposite holds for a synchronous motor, in which the armature current is in the opposite direction; and thus a lagging current tends to magnetize, a leading current to demagnetize, the field. 186. The e.m.f. generated in the armature by the resultant magnetic flux, produced by the resultant m.m.f. of the field and of the armature, is not the terminal voltage of the machine; the terminal voltage is the resultant of this generated e.m.f. and the e.m.f. of self-inductive reactance and the e.m.f. representing the power loss by resistance in the alternator armature. That is, in other words, the armature current not only opposes or assists the field m.m.f. in creating the resultant magnetic flux, but sends a second magnetic flux in a local circuit through the armature, which flux does not pass through the field-spools, and is called the magnetic flux of armature self-inductive reactance. 262 ALTERNATING-CURRENT PHENOMENA Thus we have to distinguish in an alternator between armature reaction, or the magnetizing action of the armature upon the field, and armature self-inductive reactance, or the e.m.f. gener- ated in the armature conductors by the current therein. This e.m.f. of self-inductive reactance is (if the magnetic reluctance, and consequently the reactance, of the armature circuit is as- sumed as constant) in quadrature behind the armature current, and will thus combine with the generated e.m.f. in the proper phase relation. Obviously the e.m.f. of self-inductive reactance and the generated e.m.f. do not in reality combine, but their respective magnetic fluxes combine in the armature-core, where they pass through the same structure. These component e.m.fs. are therefore mathematical fictions, but their resultant is real. This means that, if the armature current lags, the e.m.f. of self- inductive reactance will be more than 90° behind the generated e.m.f., and therefore in partial opposition, and will tend to reduce the terminal voltage. On the other hand, if the armature cur- rent leads, the e.m.f. of self-inductive reactance will be less than 90° behind the generated e.m.f., or in partial conjunction there- with, and increase the terminal voltage. This means that the e.m.f. of self-inductive reactance increases the terminal voltage with a leading, and decreases it with a lagging current, or, in other words, acts in the same manner as the armature reaction. For this reason both actions can be combined in one, and repre- sented by what is called the synchronous reactance of the alter- nator. In the following, we shall represent the total reaction of the armature of the alternator by the one term, synchronous reactance. While this is not exact, as stated above, since the reactance should be resolved into the magnetic reaction due to the magnetizing action of the armature current, and the electric reaction due to the self-induction of the armature current, it is in general sufficiently near for practical purposes, and well suited to explain the phenomena taking place under the various condi- tions of load. This synchronous reactance, x, is occasionallj^ not constant, but is pulsating, owing to the synchronously varying reluctance of the armature magnetic circuit, and the field mag- netic circuit; it may, however, be considered in what follows as constant; that is, the e.m.fs. generated thereby may be repre- sented by their equivalent sine waves. A specific discussion of the distortions of the wave shape due to the pulsation of the syn- chronous reactance is found in Chapter XXVI. The synchron- ALTERNATING-CURRENT GENERATOR 263 ous reactance, x, is not a true reactance in the ordinary sense of the word, but an equivalent or effective reactance. Sometimes the total effects taking place in the alternator armature are repre- sented by a magnetic reaction, neglecting the self-inductive re- actance altogether, or rather replacing it by an increase of the armature reaction or armature m.m.f. to such a value as to include the self-inductive reactance. This assumption is often made in the preliminary designs of alternators. Further dis- cussion of the relation of armature reaction and self-induction see "Theory and Calculation of Electrical Circuits" under "Reactance and Apparatus." 187. Let Eo = generated e.m.f. of the alternator, or the e.m.f. generated in the armature-coils by their rotation through the constant magnetic field produced by the current in the field- spools, or the open-circuit voltage, more properly called the "nominal generated e.m.f.," since in reality it does not exist as before stated. Then Eo = V2 7r/j/4> 10-8; where n = total number of turns in series on the armature, / = frequency, $ = total magnetic flux per field-pole. Let Xo = synchronous reactance, ro = internal resistance of the alternator; then Zo ^ To -{- jxo = internal impedance. If the circuit of the alternator is closed by the external im- pedance, Z = r + jx, the current j_ En En or, / = Zo + Z (ro -^r)-hj {xn + x) Eo Viro + r)2 -h {xo + xr' and, the terminal voltage, Eo{r + jx) E = IZ = En - IZn = {ro+ r) +i(a;o+ x) 264 or. ALTERNATING-CURRENT PHENOMENA EoVr"" + x^ E = = E, 1 r- -\- X- r--\-x^ or, expanded in a series, F =. F \^ - ^0^ + ^0^ , (xxo + rroY + 4 (xtq - rxp)^ "1 r2 + a;2 "^ 8 (r^ -\- x'')^ -" °5 'i II _, , 26 24 22 20 18 16 11 /' y N \ > ■~^, / / / \ \ \ / / \ \ / / \w ''^A P^s \ \ / / N % V \ \ \ \ I ^ / \ \ \ \ 12 10 8 6 4 2 ^'' \ \ \ \ V \ \ f' \ / \ / ^ FIELD CHARACTERISTIC Eo=2500, 2(r1+l0j I''/*=I E, x = o \ \i / / \ / / 1 "0 20 40 60 80 100 120 140 160 180 200 220 240 260 AMPS. Fig. 132. — Field characteristic of alternator on non-inductive load. As shown, the terminal voltage varies with the conditions of the external circuit. 188. As an example are shown in Figs. 132-137, at constant generated e.m.f., Eo = 2500; ALTERNATING-CURRENT GENERATOR 265 and the values of the internal impedance, Zo = To + jXo = 1 + 10 J, with the current, I, as abscissas, the terminal voltages, E, as ordinates in full line, and the kilowatts output, = Pr, in dotted lines, the kilovolt-amperes output, = IE, in dash-dotted lines, for the following conditions of external circuit: '0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 AMPS. Fig. 133. — Field characteristic of alternator at 60 per cent, power-factor on inductive load. In Fig. 132, non-inductive external circuit, x = 0. T In Fig. 133, inductive external circuit, of the condition, — = + 0.75, or a power-factor, O.G. In Fig. 134, inductive external circuit, of the condition, r = 0, or a power-factor, 0. In Fig. 135, external circuit with leading current, of the condi- tion, X = — 0.75, or a power-factor, 0.6. In Fig. 136, external circuit with leading current, of the condi- tion, r = 0, or a power-factor, 0. In Fig. 137, all the volt-ampere curves are shown together as 266 ALTERNATING-CURRENT PHENOMENA complete ellipses, giving also the negative or syn- chronous motor part of the curves. Such a curve is called a field characteristic. As shown, the e.m.f. curve at non-inductive load is nearly horizontal at open-circuit, nearly vertical at short-circuit, and is similar to an arc of an ellipse. With reactive load the curves are more nearly straight lines. The voltage drops on inductive load and rises on capacity load. 26 24 22 20 3^u :10 \ \ \ FIELD CHARACTERISTIC Eo=2500, Zo=1+10j,r=o. 90°LAG l2r=0 \ \, \ \ \ \, > \ N •s_ / \ \% \ ■. vV / ^w V. ft-:^ \ / ^ \ \ / \ K, N S, / \ \ \ / \ S.' \ \ A 0 20 40 60 80 100 120 140 ICO 180 200 220 240 260 280 AMPS. Fig. 134. — Field characteristic of alternator on wattless inductive load. The output increases from zero at open-circuit to a maximum, and then decreases again to zero at short-circuit. 189. The dependence of the terminal voltage, £', upon the phase relation of the external circuit is shown in Fig. 138, which gives, at impressed e.m.f., £'0 = 2500 volts, and the currents, / = 50, 100, 150, 200, 250 amp., the terminal voltages, E, as ordinates, with the inductance factor of the external circuit X ■y/r^ -{■ x^ as abscissas. ALTERNATING-CURRENT GENERATOR 267 190. If the internal impedance is negligible compared with the external impedance, then, approximately, E Eq Vr^ + x^ = Eo; that is, an alternator with small internal resistance and syn- chronous reactance tends to regulate for constant-terminal voltage. VOLTS 3600 3200 O 2800 1200 2400 1000 2000 800 1600 600 1200 400 800 200 400 100 0 0 s --- -- — *3S7( KWiT — ^ ^ -^ N \ V ^ \ ^ \ /I y /f v FIELD CHARACTERISTIC / Eo=2500, Zo=1+10j,Y=r75or 60^rP.F- / ,y ,»'' ~~> I 1 y y / / y / / /' /, / / 1 - / v^. y .^■' ''" / / \ ^ ^ ^' / 1 / 1 y < 22 _i . >j 20 oo OO 18 X X 16 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 xlOO=AMPS. Fig. 13G. — Field characteristic of alternator on wattless condenser load. as a term of first order, and therefore constant-potential regu- lation does not take place as well. With a non-inductive external circuit, if the synchronous 1 f/ 1 j / FIELD CHARACTERISTIC Eo=2500, Zo=110j, r=o, 90°Leading Current i-r=o 1 i / i i / 1 ' 1 i 1 1 i 1 1 / ) 1 i 1 i 1 i 1 i 1 1 9 .'^1 I 1 ^ 'OyA y / 3"/ / ,"'= ^ / / / / r / / J / // / / /, // / / / / y // / / / / / !/ / / / 1 / \4 A / / ''/ / A / // r / ^■^ ALTERNATING-CURRENT GENERATOR 269 ^ -^ \ STK) / / / \ 300 ^ -^ s \ ( \ ^ E«= 2500, zri+ic \ N \ ^^ ^0 N, / f \ \ \ / / 150 S \ j \ / \ \ v 100 ^ / N s y/ 50 \ V 2aX 2.5^ i jO 11 W 3jO 300 y kaoo 1 )0 1 JO s 0 0 50 1 0 1 JO IJO i 00^ v. 300 350 400 450 500 // v^ 100 // ^ / / \ \ ^ 150 A { , \ \ / 1 r s^ ^ 200 // / \ y \ /I s. ^ / / \ ^ i ^ ^ S ^ y 350 \] \ k / 1 / 400 \ ^ / 450 \ / \ Fig. 137. — Field characteristic of alternator. 34 32 30 28 26 24 22 20 |l8 Tl6 Mill 1/ / Eo-= 2500, Z„= 1+10 j 1 = 50 Amps. 1=100 " 1=150 " 1=200 " 1=250 " / / 'y / I ^ ^ yf T ___^ ^ ?^ '/ / ^ tk ■ops __^ "^ ^ / y / 3ii- ^ ^ y [/ 1 i. ,. ^ ^ /> y / / 1 ^oo^ ^ y ^ / / 1 ^ rt>V* '^ y^ / ./ X12 10 k^ r /■ f ■^ ^^■^ / oV ,o^^^ y .j; t 6 4 a< / — " / / / / 1 .9 .7 .6 .5 .4 .3 .2 .1 0 -.1 -.2 -.3 -.4 -.5 ■ X .7 -.8 -.9 -1 Fig. 138. — Regulation of alternator on various loads. 270 ALTERNATING-CURRENT PHENOMENA reactance, Xo, of the alternator is very large compared with the external resistance, r, current J Eo 1 Eo Xq / /f^ -^ f\2 Xq ^M^0 approximately, or constant; or, if the external circuit contains the reactance, x, Eo 1 Eo I = ^|^^^(m approximately, or constant. In this case, the terminal voltage of a non-inductive circuit is E = ^r, Xq approximately proportional to the external resistance. In an inductive circuit, E = — -j-— -y/r"^ -\- x^' Xo "T X approximately proportional to the external impedance. 191. That is, on a non-inductive external circuit, an alter- nator with very low synchronous reactance regulates for con- stant-terminal voltage, as a constant-potential machine, an alternator with a very high synchronous reactance regulates for a terminal voltage proportional to the external resistance as a constant-current machine. Thus, every alternator acts as a constant-potential machine near open-circuit, and as a constant-current machine near short- circuit. Between these conditions, there is a range where the alternator regulates approximately as a constant-power machine, that is, current and e.m.f. vary in inverse proportion, as between 130 and 200 amp. in Fig. 132. The modern alternators are generally more or less machines of the first class; the old alternators, as built by Jablockkoff, Gramme, etc., were machines of the second class, used for arc lighting, where constant-current regulation is an advantage. Very high-power steam-turbine alternators are now again built with fairly high reactance, for reasons of safety. Obviously, large external reactances cause the same regula- ALTERNATING-CURRENT GENERATOR 271 tion for constant current independently of the resistance, r, as a large internal reactance, Xo. On non-inductive circuit, if Eq ana / = E the output is and Hence, if or E,r E,h- P ^ IE = (r + ro)2 + xo^' dP ^ a:o^ - r'^ + rp^ ^ a^o = V'r^ — ro^, dr the power is a maximum, and and 2 {20 + ro} ^0 E V2zo{zq + ro} Therefore, with an external resistance equal to the internal impedance, or, r = Zo = y/ro'^ + Xq^, the output of an alternator is a maximum, and near this point it regulates for constant output; that is, an increase of current causes a proportional decrease of terminal voltage, and inversely. The field characteristic of the alternator shows this effect plainly.