CHAPTER XXIV SYNCHRONOUS MOTOR 212. In the chapter on synchronizing alternators we have seen that when an alternator running in synchronism is connected with a system of given voltage, the work done by the alternator can be either positive or negative. In the latter case the alternator consumes electrical, and consequently produces mechanical, power; that is, runs as a synchronous motor, so that the investi- gation of the synchronous motor is already contained essentially in the equations of parallel-running alternators. Since in the foregoing we have made use mostly of the sym- bolic method, we may in the following, as an example of the graphical method, treat the action of the synchronous motor graphically. Let an alternator of the e.m.f., Ei, be connected as synchron- ous motor with a supply circuit of e.m.f., Eo, by a circuit of the impedance, Z. If £"0 is the e.m.f. impressed upon the motor terminals, Z is the impedance of the motor of generated e.m.f., Ei. If Eq is the e.m.f. at the generator terminals, Z is the impedance of motor and line, including transformers and other intermediate apparatus. If Eq is the generated e.m.f. of the generator, Z is the sum of the impedances of motor, line, and generator, and thus we have the problem, generator of generated e.m.f., Eo, and motor of generated e.m.f., El] or, more general, two alternators of generated e.m.fs., Eo, El, connected together into a circuit of total impedance, Z. Since in this case several e.m.fs. are acting in circuit with the same current, it is convenient to use the current, /, as zero line 01 of the polar diagram. (Fig. 145.) Ji I = i = current, and Z = impedance, r = effective resist- ance, X = effective reactance, and z = \/r^ -{- x^ = absolute value of impedance, then the e.m.f. consumed by the resistance is £'11 = ri, and is in phase with the current; hence represented by vector OEn] and the e.m.f. consumed by the reactance is E2 = xi, and 90° ahead of the current; hence the e.m.f. consumed 301 302 ALTERNATING-CURRENT PHENOMENA by the impedance hE = ViEuY" + (£'2)^ or = i -s/r"^ -\- x- = iz, X and ahead of the current by the angle 8, where tan 8 = ~. We have now acting in circuit the e.m.fs., E, Ei, E^; or Ei and E are components of E^, that is, E^i is the diagonal of a parallelo- gram, with El and E as sides. Since the e.m.fs. Ei, Eo, E, are represented in the diagram. Fig. 145, by the vectors OEi, OEo, OE, to get the parallelogram of ^0, El, E, we draw arcs of circles around 0 with Eo, and around E with El. Their point of intersection gives the impressed e.m.f., OEq = Eo, and completing the parallelogram, OEEqEi, we get, OEi = El, the generated e.m.f. of the motor. < lOEo is the difference of phase between current and impressed e.m.f., or generated e.m.f. of the generator. < lOEi is the difference of phase between current and generated e.m.f. of the motor. And the power is the current, i, times the projection of the e.m.f. upon the current, or the zero line, 01. Hence, dropping perpendiculars, EqEo^ and EiEi^, from Eo and El upon 01, it is — Po = i y. OEo^ = power supplied bj^ generator e.m.f. of gen- erator ; Pi = ? X OEi^ = electric power transformed into mechanical power by the motor; P = i X OE\i — power consumed in the circuit by effective resistance. Obviously Po - Pi + P. Since the circles drawn with Eo and Ei around 0 and E, re- spectively, intersect twice, two diagrams exist. In general, in one of these diagrams shown in Fig. 145 in full lines, current and e.m.f. are in the same direction, representing mechanical work done by the machine as motor. In the other, shown in dotted lines, current and e.m.f. are in opposite direction, repre- senting mechanical work consumed by the machine as generator. Under certain conditions, however, £"0 is in the same, Ei in opposite direction, with the current; that is, both machines are generators. 213. It is seen that in these diagrams the e.m.fs. are considered from the point of view of the motor; that is, work done as syn- chronous motor is considered as positive, work done as generator SYNCHRONOUS MOTOR 303 is negative. In the chapter on synchronizing generators we took the opposite view, from the generator side. In a single unit-power transmission, that is, one generator supplying one synchronous motor over a line, the e.m.f. con- sumed by the impedance, E = OE, Figs. 146 to 148, consists three components; the e.m.f., OEo^ = E2, consumed by the im- pedance of the motor, the e.m.f., E2^Es'- = Ez consumed by the impedance of the line, and the e.m.f., Es^E = E4, consumed by Fig. 145. the impedance of the generator. Hence, dividing the opposite side of the parallelogram, EiEo, in the same way, we have: OEi = El = generated e.m.f. of the motor, OE2 = E2 = e.m.f. at motor terminals or at end of line, OE3 = Ez = e.m.f. at generator terminals, or at beginning of line. OEo = Eq = generated e.m.f. of generator. The phase relation of the current with the e.m.f s., Ei, Eq, de- pends upon the current strength and the e.m.f s., Ei and Eq. 214. Figs. 146 to 148 show several such diagrams for different values of Ei, but the same value of I and £"0. The motor diagram being given in drawn line, the generator diagram in dotted line. As seen, for small values of Ei the potential drops in the alter- nator and in the line. For the value of Ei = Eq the potential rises in the generator, drops in the line, and rises again in the 304 ALTERNATING-CURRENT PHENOMENA Fig. 146. Fig. 147. SYNCHRONOUS MOTOR 305 Fig. 148. Fig. 149. 20 306 ALTERNATING-CURRENT PHENOMENA motor. For larger values of E\, the potential rises in the alter- nator as well as in the line, so that the highest potential is the generated e.m.f. of the motor, the lowest potential the generated e.m.f. of the generator. It is of interest now to investigate how the values of these quantities change with a change of the constants. 215. A. Constant impressed e.m.f., Eq, constant-current strength I = i, variable motor excitation, Ei. (Fig. 149.) If the current is constant, = i; OE, the e.m.f. consumed by the impedance, and therefore point, E, are constant. Since the intensity, but not the phase of £"0 is constant, Eo lies on a circle Co with Eo as radius. From the parallelogram, OEEoEi follows, since EiEo parallel and = OE, that Ei lies on a circle, ei, con- gruent to the circle, eo, but with Ei, the image of E, as center; Wi = 0E. We can construct now the variation of the diagram with the va- riation of El] in the parallelogram, OEEqEi, 0, and E are fixed, and Eo and Ei move on the circles, eo and ei, so that EqEi is parallel to OE. The smallest value of Ei consistent with current strength, /, is Oil = El, 01 = £"0. In this case the power of the motor is 01 1^ X I, hence already considerable. Increasing Ei to 02i, 03i, etc., the impressed e.m.fs. move to 02, 03, etc., the power is / X 02i\ / X 03i^, etc., increases first, reaches the maximum at the point 3i, 3, the most extreme point at the right, with the im- pressed e.m.f. in phase with the current, and then decreases again, while the generated e.m.f. of the motor, Ei, increases and becomes = Eo at 4i, 4. At 5i, 5, the power becomes zero, and further on negative; that is, the motor has changed to a generator, and produces electrical energy, while the impressed e.m.f., Co, still furnishes electrical energy — that is, both machines as gen- erators feed into the line, imtil at 61, 6, the power of the impressed e.m.f., Eo, becomes zero, and further on energy begins to flow back; that is, the motor is changed to a generator and the genera- tor to a motor, and we are on the generator side of the diagram. At 7i, 7, the maximum value of E^, consistent with the current, /, has been reached, and passing still further the e.m.f., Ei de- creases again, while the power still increases up to the maximum at 81, 8, and then decreases again, but still Ei remaining generator, Eo motor, until at lli, 11, the power of Eo becomes zero; that is, Eo changes again to a generator, and both machines are generators, SYNCHRONOUS MOTOR 307 up to 12i, 12, where the power of Ei is zero, Ei, changes from generator to motor, and we come again to the motor side of the diagram, and the power of the motor increases while Ei still decreases, until li, 1, is reached. Hence, there are two regions, for very large Ei from 5 to 6 and for very small Ei from 11 to 12, where both machines are genera- tors; otherwise the one is generator, the other motor. For small values of Ei the current is lagging, begins, however, at 2 to lead the generated e.m.f. of the motor, Ei, at 3 the gener- ated e.m.f. of the generator, Eq. It is of interest to note that at the smallest possible value of El, li, the power is already considerable. Hence, the motor can run under these conditions only at a certain load. If this load is thrown off, the motor cannot run with the same current, but the current must increase. We have here the curious con- dition that loading the motor reduces, unloading increases, the current within the range between 1 and 12. The condition of maximum output is 3, current in phase with impressed e.m.f. Since at constant current the loss is constant, this is at the same time the condition of maximum efficiency; no displacement of phase of the impressed e.m.f., or self-induction of the circuit compensated by the effect of the lead of the motor current. This condition of maximum efficiency of a circuit we have found already in Chapter XL 216. B. £"0 and Ei constant, I variable. Obviously Eq lies again on the circle eo with Eo as radius and 0 as center. E lies on a straight line, e, passing through the origin. Since in the parallelogram, OEEqEi, EEq = Ei, we derive Eo by laying a. line, EEo = Ei, from any point, E, in the circle, Co, and complete the parallelogram. All these lines, EEo, envelop a certain curve, ei, which can be considered as the characteristic curve of this problem, just as circle, ei, in the former problem. These curves are drawn in Figs. 150, 151, 152, for the three cases: 1st, Ei = Eo', 2d, EiEo. In the first case, Ei = £"0 (Fig. 150), we see that at very small current, that is very small OE, the current, 7, leads the impressed e.m.f., Eo, by an angle, E^oOI = do. This lead decreases with increasing current, becomes zero, and afterward for larger cur- rent, the current lags. Taking now any pair of corresponding 308 ALTERNATING-CURRENT PHENOMENA points, E, Eo, and producing EEo until it intersects d, in £',, we have < EiOE = 90°, Ei = Eo, thus: OEi = EEo = OEo = E^r, Fig. 150. Fig. 151. that is, EEi = 2 £"0. That means the characteristic curve, Ci, is the envelope of lines EEi, of constant lengths, 2 Eo, sliding between the legs of the right angle, EiOE; hence, it is the sextic hypocy- SYNCHRONOUS MOTOR 309 cloid osculating cirele, eo, which has the general equation, with e, Bi as axes of coordinates, In the next case, Ei < Eo (Fig. 151), we see first, that the current can never become zero like in the first case, Ei = Eo, but has a minimum value corresponding to the minimum value e 7771/ Tf Eo — El J . . Eo — El of OE : / 1 = , and a maximum value: / i = Furthermore, the current may never lead the impressed e.m.f., Eo, but always lags. The minimum lag is at the point, H. The locus, d, as envelope of the lines, £'£'0, is a finite sextic curve, shown in Fig. 151. Fig. 152. If El < Eo, at small Eo — Ei, H can be below the zero line, and a range of leading current exists between two ranges of lag- ging currents. In the case, Ei > Eo (Fig. 152), the current cannot equal zero either, but begins at a finite value, I'l, corresponding to the mini- mi — Eo mum value of OE, I'l = At this value, however, the alternator. Ex, is still generator and changes to a motor, its power passing through zero, at the point corresponding to the vertical tangent, upon Ci, with a very large lead of the impressed e.m.f. against the current. At H the lead changes to lag. 310 AL TERN A TING-C URREN T PHENOMENA The minimum and maximum values of current in the three conditions are given by: Minimum Maximum 1st. 7 = 0, J. 2Eo z 2d. i_^^-^\ z Eo -{- El z 3.1. 7 = ^'-^». ^ Eo + Ei Since the current in the hne at Ex = 0, that is, when the motor stands still, is /o = — ' , we see that in such a synchronous motor- plant, when running at synchronism, the current can rise far be- yond the value "it has at standstill of the motor, to twice this value at 1, somewhat less at 2, but more at 3. 217. C. Eo = constant, E\ varied so that the efficiency is a maximum for all currents. (Fig. 153.) Since we have seen that the output at a given current strength, that is, a given loss, is a maximum, and therefore the efficiency a maximum, when the current is in phase with the generated e.m.f., Eo, of the generator, we have as the locus of £"0 the point, Eq (Fig. 153), and when E with increasing current varies on e, E\ must vary on the straight line, ei, parallel to e. SYNCHRONOUS MOTOR 311 Hence, at no-load or zero current, Ei = Eq, decreases with increasing load, reaches a minimum at OEi^ perpendicular to ei, and then increases again, reaches once more Ei = Eo at Ei^, and then increases beyond Eq. The current is always ahead of the generated e.m.f., Ei, of the motor, and by its lead compensates for the self-induction of the system, making the total circuit non- inductive. The power is a maximum at Ei^, where OEi* = Ei'^Eo = 0.5 X OWo, and is then = 7 X ^"- Since OE^' = Ir = ^, I = ^^ Eo^ and P = -r~, hence = the maximum power which, over a non- 4 r' inductive line of resistance ?• can be transmitted, at 50 per cent. efficiency, into a non-inductive circuit. In this case, 7-1 ■> ^ -^0 ^ "^/^e) In general, it is, taken from the diagram, at the condition of maximum efficiency, El = V{Eo - Iry-\- Px^- Comparing these results with those in Chapter XI on Induct- ive and Condensive Reactance, we see that the condition of maximum efficiency of the synchronous motor system is the same as in a system containing resistance and condensive reactance, fed over an inductive line, the lead of the current against the generated e.m.f., Ei, here acting in the same way as the con- denser capacity in Chapter XI. 218. D. Eo = constant; Pi = constant. If the power of a synchronous motor remains constant, we have (Fig. 154) I X OEi^ = constant, or, since OE^ = Ir, I = ^^^ and OE' X OE^^ = OE' X E'E,' = constant, r Hence we get the diagram for any value of the current, I, at p constant power, Pi, by making OE^ = Ir, E^Eo'^ = -j erecting in £"0^ a perpendicular, which gives two points of intersection with circle, eo, Eo, one leading, the other lagging. Hence, at a given impressed e.m.f., Eo, the same power. Pi, can be trans- mitted by the same current, /, with two different generated e.m.fs., El, of the motor; one, OEi = £'£'0 small, corresponding 312 ALTERNATING-CURRENT PHENOMENA to a lagging current; and the other, OEi = EEq large, corre- sponding to a leading current. The former is shown in dotted lines, the latter in full lines, in the diagram, Fig. 154. Hence a synchronovis motor can work with a given output, at the same current with two different counter e.m.fs., Ei. In one of the cases the current is leading, in the other lagging. Fig. 154. In Figs. 155 to 158 are shown diagrams, giving the points £"0 = impressed e.m.f., assumed as constant = 1000 volts, E = e.m.f. consumed by impedance, E^ = e.m.f. consumed by resistance (not numbered). The counter e.m.f. of the motor, Ei, is OEi, equal and parallel -£'£'0, but not shown in the diagrams, to avoid complication. The four diagrams correspond to the values of power, or motor output. P = 1,000, P = 1,000 P = 6,000 P = 9,000 P = 12,000 6,000, 9,000, 12,000 watts, and give: 46 < £1 < 2,200, 1 < / < 49 340 < El < 1,920, 540 < El < 1,750, 920 < El < 1,320, 7 < / < 43 11.8 < / < 38.2 20 < / < 30 Fig. 155. Fig. 156. Fig. 157. Fig. 158. As seen, the permissible value of counter e.m.f., Ei, and of current, /, becomes narrower with increasing output. SYNCHRONOUS MOTOR 313 Eo=lOOO P = 1000 46 lei sin a, eO = ^1 - (|-j (2) The displacement of phase between current i, and e.m.f. e = zi consumed by the impedance, z, is r cos {i, e) = ~ . X sm (i, e) - - (3) Since the three e.m.fs. acting in the closed circuit, ep = e.m.f. of generator, ei = counter e.m.f. of synchronous motor, e = zi = e.m.f. consumed by impedance, ^ If Co = e.m.f. at motor terminals, z = internal impedance of the motor; if So = terminal voltage of the generator, z = total impedance of line and motor; if eo = e.m.f. of generator, that is, e.m.f. generated in generator armature by its rotation through the magnetic field, z includes the generator impedance also. 316 ALTERNATING-CURRENT PHENOMENA form a triangle, that is, ei and e are components of eo, it is (Figs. 159 and 160), eo' e-^ + 6^ + 2 ee\ cos (ei, e), hence, cos (ei, e) = eo 2-ei2 2 — p,2 — j'2i'2 eo" — ei 2''^^ 2 ziex (4) (5) 2eie since, however, by diagram, cos (ci, e) = cos (?', e — i, e\) = cos (^, e) cos (z, ei) + sin {i, e) sin (/, d) (6) substitution of (2), (3) and (5) in (6) gives, after some trans- position, eo- - er - zH'' - 2rp = 2xViW - V^ 0) the fundamental equation of the synchronous motor, relating im- pressed e.m.f., eo; counter e.m.f., ei; current, i\ power, p, and re- sistance, r; reactance, x; impedance, z. e=*iie— -i Fig. 159. Fig. 160. This equation shows that, at given impressed e.m.f., eo, and given, impedance, z = \^r^ -\- x^, three variables are left, ei, i, p, of which two are independent. Hence, at given eo and z, the current, ^, is not determined by the load, p, only, but also by the excitation, and thus the same current, i, can represent widely different loads, p, according to the excitation ; and with the same load, the current, i, can be varied in a wide range, by varying the field-excitation, ei. The meaning of equation (7) is made more perspicuous by some transformations, which separate ei and i, as function of p and of an angular parameter, 0. Substituting in (7) the new coordinates; /9 = et^ — z~'P or. ei' zH^ = vi"' . (8) SYNCHRONOUS MOTOR we get Co' - a\/2-2rp = 2 r 13' z \ 2 substituting again, en" = a 2 zp = b r = ez, hence, x = z\/l — e- 2 rp = th, we get — z^p^ a - aV2 - eb = V(l -e2)(2Q:2_2,32-62); and, squared, 2 2 . /1 2^«2 /o^ ^, , b^{l - 6^) (ff - 66)^ e^a^ + (1 - e2)/32 - aV2(a - e6) H ^ ' 2 substituting (a-€6)\/2 ea ^- = V, /SVl - e' = t^, gives, after some transposition, 2e hence, if j;2 _|_ ^2 _ ^^^ ^ ' a{a — 2tb), R ' it is (1 - e^) (a -2 eb)a 2e2 y2 _|_ 1^2 ^ 2^2 317 (9) (10) (11) = 0; (12) (13) (14) (15) (16) the equation of a circle with radius, R. Substituting now backward, we get, with some transpositions, {r2(ei2 + zV) — 2^(eo^ — 2rp)\" + {rx(er + zH")Y- = x'^z^eo^ieo^ — 4 rp) (17) the fundamental equation of the synchronous motor in a modified form. The separation of d and i can be effected by the introduction of a parameter, 0, by the equations r^(ei^ + zH^) — z^{eo^ — 2 rp) = a^^eoA/eo^ — 4 rp cos rx{ei^ — zH"^) = xze^-s/ e^ = 4 rp sin ^] These equations (18), transposed, give ei = ^2 { ^^^°^ - 2 rp) + ^ (^ cos (^ + sin 0 j \/eo^ - 4 rp | (18) 318 ALTERNATING-CURRENT PHENOMENA The parameter, , has no direct physical meaning, apparently. These equations (19) and (20), by giving the values of d and i as functions of p and the parameter, 0, enable us to construct the 'power characteristics of the synchronous motor, as the curves relating Ci and i, for a given power, p, by attributing to ^ all different values. Since the variables, v and w, in the equation of the circle (16) are quadratic functions of e\ and i, the power characteristics of the synchronous motor are quartic curves. They represent the action of the synchronous motor under all conditions of load and excitation, as an element of power trans- mission even including the line, etc. Before discussing further these power characteristics, some special conditions may be considered. 220. A. Maxiinum Output. Since the expression of ei and i [equations (19) and (20)] con- tain the square root, -s/cq" — 4 rp, it is obvious that the maximum value of p corresponds to the moment where this square root disappears by passing from real to imaginary; that is, Co" — 4 rp = 0, V = '£ (21) This is the same v-alue which represents the maximum power transmissible by e.m.f., eo, over a non-inductive line of resistance, r; or, more generally, the maximum power which can be trans- mitted over a line of impedance. into any circuit, shunted by a condenser of suitable capacity. Substituting (21) in (19) and (20), we get, Co 2r (22) SYNCHRONOUS MOTOR 319 and the displacement of phase in the synchronous motor, hence, cos (ei, i) =-:— = -; lei z tan (ei, i) = - -, (23) that is, the angle of internal displacement in the synchronous motor is equal, but opposite to, the angle of displacement of line impedance, (ei, i) = - (e, i), - - (2, r), (24) and consequently, (eo, i) =0; (25) that is, the current, ?, is in phase with the impressed e.m.f., eo- If 2 < 2 r, ei < eo; that is, motor e.m.f. < generator e.m.f. If 2 = 2 r, ei = eo; that is, motor e.m.f. = generator e.m.f. If 2 > 2 r, ei > go; that is, motor e.m.f. > generator e.m.f. In either case, the current in the synchronous motor is leading. 221. B. Running Light, p = 0. When running light, or for p = 0, we get, by substituting in (19) and (20), eoz /l ^^ = TV2 ?o /r z-\2 ^ j 1 + ^ cos eo or < eo. 225. F. Numerical Example. Figs. 161 and 162 show the characteristics of a 100-kw, motor supplied from a 2500-volt generator over a distance of 5 miles, the line consisting of two wires, No. 2 B. & S., 18 in. apart. In this case we have: eo = 2500 volts constant at generator terminals; r — 10 ohms, including line and motor; X = 20 ohms, including line and motor; hence z = 22.36 ohms. (43) Substituting these values, we get: 25002 - ei2 - 500 i^ - 20 p = 40 ViH{' - p^ (7) {ei2 + 500 z2 - 31.25 X 10« + 100 pp + {2 e^" - 1000 z^p - 7.8125 X 1014 - 5 X 109 p. (17) ex = 5590 X (19) ^fi^(l - 3.2 X 10-« p) + (0.894 cos cb + 0.447 sin 0) Vl -6.4 X 10-" p|. (20) i = 250 X VM(1 - 3.2 X 10-«p) + (0.894 cos<^ - 0.447 sin «/>) Vl6.4Xl0-«p}. Maximum output, p = 156.25 kw. (21) at ei = 2795 volts i = 125 amp. Running light, (22) ei2 + 500 i^ - 6.25 X 10* + 40 iei = 0 ei = 20i ± V6.25 X 10^ - 100 i^ (28) SYNCHRONOUS MOTOR 325 At the minimum value of counter e.m.f., ei= 0 is z = 112 (29) At the minimum value of current, i= 0 is ei = 2500 (30) At the maximum value of counter e.m.f., ei = 5590 is i = 223.5 (31) At the maximum value of current, i = 250 is ei = 5000. (32) Curve of zero displacement of phase, ei = 10 V(250 - f)' + 4 i^ (34) = 10 V6.25 X 10* - 500 i + 5 i\ MO ma 'i&bo ~ 2000 2500 -^000 saoo iooo taoo eooo ewo Fig. 162. Minimum counter e.m.f. point of this curve, i = 50, ei = 2240. (35) Curve of maximum displacement of phase, p = 10 r- (40) (6.25 X 10« - ei2)2 + 0.65 X 10« i^ - 10^" i^ = 0 (42) 326 ALTERNATING-CURRENT PHENOMENA Fig. 161 gives the two ellipses of zero power in full lines, with the curves of zero displacement in dotted, the curves of maximum displacement in dash-dotted lines, and the points of maximum power as crosses. Fig. 162 gives the motor-power characteristics for p = 10 kw.; p = 50 kw.; p = 100 kw.; p = 150 kw., and p = 156.25 kw., together with the curves of zero displacement and of maximum displacement. 226. G. Discussion of Results. The characteristic curves of the synchronous motor, as shown in Fig. 162, have been observed frequently, with their essential features, the V-shaped curve of no-load, with the point rounded off and the two legs slightly curved, the one concave, the other 140 120 100 80 2.60 40 20 1 / /// / \ / [/ \ ^\ ^ // y \ \ // ^ N N. [/^ / 1000 2000 2500 Volts Fig. 163. 3000 4000 4500 5000 convex; the increased rounding off and contraction of the curves with increasing load; and the gradual shifting of the point of minimum current with increasing load, first toward lower, then toward higher, values of counter e.m.f., e\. The upper parts of the curves, however, I have never been able to observe completely and consider it as probable that they correspond to a condition of synchronous motor running, which is unstable. The experimental observations usually SYNCHRONOUS MOTOR 327 extend about over that part of the curves of Fig. 162 which is reproduced in Fig, 163, and in trying to extend the curves further to either side, the motor is thrown out of synchronism. It must be understood, however, that these power charac- teristics of the synchronous motor in Fig. 162 can be considered as approximations only, since a number of assumptions are made which are not, or only partly, fulfilled in practice. The fore- most of these are: 1. It is assumed that ei can be varied unrestrictedly, while in reality the possible increase of ei is limited by magnetic saturation. Thus in Fig. 162, at an impressed e.m.f., eo = 2500 volts, ei rises up to 5590 volts, which may or may not be beyond that which can be produced by the motor, but certainly is beyond that which can be constantly given by the motor. 2. The reactance, x, is assumed as constant. While the reactance of the line is practically constant, that of the motor is not, but varies more or less with the saturation, decreasing for higher values. This decrease of x increases the current, i, corresponding to higher values of ei, and thereby bends the curves upward at a lower value of ei than represented in Fig. 162. It must be understood that the motor reactance is not a simple quantity, but represents the combined effect of self- induction, that is, the e.m.f. generated in the armature con- ductor by the current therein and armature reaction, or the variation of the counter e.m.f. of the motor by the change of the resultant field, due to the superposition of the m.m.f. of the armature current upon the field-excitation; that is, it is the "synchronous reactance." 3. Furthermore, this synchronous reactance usually is not a constant quantity even at constant induced e.m.f., but varies with the position of the armature with regard to the field; that is, varies with the current and its phase angle, as discussed in the chapter on the armature reactions of alternators. While in most cases the synchronous reactance can be assumed as con- stant, with sufficient approximation, sometimes a more com- plete investigation is necessary, consisting in a resolution of the synchronous impedance in two components, in phase and in quadrature respectively with the field-poles. Especially is this the case at low power-factors. So by gradually decreasing the excitation and thereby the e.m.f., e, the curves may, especially at light load, occasionally be extended 328 ALTERNATING-CURRENT PHENOMENA below zero, into negative values of e, or onto the part of the curve, B, in Fig. 161, while the power still remains constant and positive, as synchronous motor. In other words, the motor keeps in step even if the field-excitation is reversed; the lagging component of the armature reaction magnetizes the field, in opposition to the demagnetizing action of the reversed field excitation. 4. These curves in Fig. 162 represent the conditions of con- stant electric power of the motor, thus including the mechan- ical and the magnetic friction (core loss). While the mechanical friction can be considered as approximately constant, the mag- netic friction is not, but increases with the magnetic induction; that is, with ei, and the same holds for the power consumed for field excitation. Hence the useful mechanical output of the motor will on the same curve, p = const., be larger at points of lower counter e.m.f., ei, than at points of higher ei; and if the curves are plotted for constant useful mechanical output, the whole system of curves will be shifted somewhat toward lower values of ei; hence the points of maximum output of the motor correspond to a lower e.m.f. also. It is obvious that the true mechanical power characteristics of the synchronous motor can be determined only in the case of the particular conditions of the installation under consideration. 227. H. Phase Characteristics of the Synchronous Motor. While an induction motor at constant impressed voltage is fully determined as regards to current, power-factor, efficiency, etc., by one independent variable, the load or output; in the synchronous motor two independent variables exist, load and field-excitation. That is, at constant impressed voltage the current, power-factor, etc., of a synchronous motor can at the same power output be varied over a wide range bj^ varying the field-excitation, that is, the counter e.m.f. or "nominal gener- ated e.m.f." Hence the synchronous motor can be utilized to fulfill two independent functions: to carry a certain load and to produce a certain wattless current, lagging by under-excitation, leading by over-excitation. Synchronous motors are, therefore, to a considerable extent used to control the phase relation and thereby the voltage, in addition to producing mechanical power. The same applies to synchronous converters. With given impressed e.m.f., field-excitation or nominal gener- SYNCHRONOUS MOTOR 329 ated e.m.f. corresponding thereto, and load, determine all the quantities of the synchronous motor, as current, power-factor, etc. Thus if in diagram Fig. 164, OE = e = e.m.f. consumed by the counter e.m.f. or nominal generated e.m.f. of the synchronous motor, and if Po = output of motor (exclusive of friction and core loss and, if the exciter is driven by the motor, power consumed P by the exciter), ii = ~ = power component of current, repre- sented by Oil, and the current vector therefore must terminate on a line, i, perpendicular to 01 1. If, then, r = resistance and X = reactance of the circuit between counter e.m.f., e, and im- FiG. 164. pressed e.m.f., eo, OEr = iiv = e.m.f. consumed by resistance, OEj: = iix = e.m.f. consumed by reactance of the power com- ponent of the current, i\, hence OE'i = e.m.f. consumed by impedance of the power component of the current, ii, and the impedance voltage of the total current lies on the perpendicular e' on OE'i. Producing OEi = OE, and drawing an arc with the impressed e.m.f., eo, as radius and E\ as center, the point of intersection with e' gives the impedance voltage, OE' , and corresponding thereto the current 01 = i; and completing the parallelogram, OEEqE', gives the impressed e.m.f., OEq. Hence, by impressed e.m.f., ^o, counter e.m.f., e, and load, Po, the vector diagram is determined, and thereby the vectors, 01 = 330 ALTERNATING-CURRENT PHENOMENA current, OEq = impressed e.m.f., OE = counter e.m.f., and their phase relation. Or, in symboUc representation, let Ed = e'o — je"o = impressed e.m.f.; eo = VV^ + eo"^ (1) E = e' — je" = e.m.f. consumed by counter e.m.f.; e = \/6'2 + e"2; (2) I = i = current, assumed as zero vector; Z = r -\- jx = impedance of circuit between eo and e. Z is the synchronous impedance of the motor, if eo is its ter- minal voltage. It is the impedance of transmission line with transformers and motor, if eo is terminal voltage of generator, and Z is synchronous impedance of motor and generator, plus impe- dance of line and transformers, if eo is the nominal generated e.m.f. of the generator (corresponding to its field-excitation). It is, then, Eo = E -\- iZ, (3) or, e'o — ie"o = e' — je" + ir -f- jix, (4) and, resolved, je'o = e' -\- ir-, (5) |e"o = e" - ix. (6) The power output of the motor (inclusive of friction and core loss, and if the exciter is driven by the motor, power consumed by exciter) is current times power component of generated e.m.f., or Po = e'i. (7) Hence, the calculation of the motor, of supply voltage eo from power output, Pq, occurs by the equations: Chosen: i = current. p. ' (7) .' , (5) e'o = e' -\- ir, (1) e"o = ±V< eo^ — e.)' (8) (6) e" = e"o + ix (2) e = Ve'^ + e"^. That is, at given power, Po, to every value of current, i, corre- spond two values of the counter e.m.f., e (and hence the field - excitation). SYNCHRONOUS MOTOR 331 Solving equations (8) for i and Po, that is, eliminating e', e'o, e"o, e", gives as the nominal generated e.m.f., e = Jeo' - rH'- + xH^ - 2 rPo ± 2 xi Jeo' ~ (^ + ^^') '' ^^^ and the power-factor of the motor is, - e' Po cos y = — = The power-factor of the supply is cos do = Po, . e'o I Po + ri^ (10) (11) eo eo eoi From equation (9), by solving for i, i can now be expressed as function of Po and e, that is, of power output and field-excitation. K '. / / / [-' • TTH 20 V 700 y <. / / / / / rSi ^- / / .w /' / / / / / / / .4 t^ \ / / ^ r^ / / / / / / 1 o** f ^V 5^ 9P'> / / / / 1/ / / 1 ^t- \ c f ^ 90/ y / / A eoo ^'C — -- -- — ' \ s. ^' ^ \y f < 1 ^ -- —- ~^ I 1 i/ / N \ 1 ^7 / > i ff500 1 / ^/ 1 / 1 A if/ / 7 400 1 1 J / 1 / // / /' /// ,^ ^ / 1^ ^^€ f^ Po =2200 1 V / J > e r1600 V A / _^ -^ ^ z =1+4 j 100 '7 1 / P' = 2 3KV /. (/ 0 XOO 200 800 400 500 COO 700 800 KILOWATTS Fig. 166. 1 1 ^ ^ POWER FAC1 roR — / > / / — 1) A r /^ ""^^ '^ C^^^l^^c. // / "f/V '■^ -> i ^ '-'«■ vc> N "*s /' ji / \ ^{ \i L \ /■/ ^ /' / / / 1 / / 1 1 ^0^2200 V 1 / / e = 2000 V / z =l + 4i / 1 F' - tKV /. / y ^ : .^^' i^ ^' J^ ^ ■^ 1 1 00 2 )0 a 00 F 4 t IG. )0 tlLOV 1( u VATT 57. 30 8 6 30 7 )0 s )0 334 ALTERNATING-CURRENT PHENOMENA The curve of unity power-factor (12) divides the synchronous motor-phase characteristics into two sections, one, for lower e, with lagging, the other with leading current. The study of these ''phase characteristics," Fig. 165, gives the best insight into the behavior of the synchronous motor under conditions of steady operation. 7^ ■^ \ / ^ "^ ^-^k ' 20 / ^ too ^ ^ ^^ ■^ 10 0 3U0 400 SOO GOO 700 800 300 KILOWATTS Fig. 169. For low values of e (e = 1600, under excitation, Fig. 166), the load curves are similar to those of an induction motor. The current is lagging, the power-factor rises from a low initial value to a maximum, and then falls again. With increasing excitation (e = 2000, Fig. 167) the power-factor curve rises to values beyond those available in induction motors, approaches and ultimately touches unity, and with still higher excitation (e = 2180, Fig. 168) two points of unity power-factor exist, at P = 20 and P = 450 kw. output, which are separated by a range with leading current, while at very low and very high load the current is lagging. The first point of unity power-factor 336 AL TERN A TING-C URREN T PHENOMENA moves toward P = 0, and then disappears, that is, the current becomes leading ah'eady at no-load, and the second point of unity power-factor moves with increasing excitation toward higher loads, from P = 450 kw. at e = 2180 in Fig. 168, to P = 700 kw. at e = 2400, Fig. 169, and P = 900 kw. at e = 2800, Fig. 170, while the power-factor and thereby the apparent efficiency decrease at light loads. The maximum output in- creases with the increase of excitation and almost proportionally thereto. n ^ ■-' ■ ' ) ^ 1 A .^ EF :IC;£ Nov \ ^ ' z / y- ' ~" -- _ y ■^ .^ ■^ 1 / y ^ y. 1 1 / / / / \ 1 A ' 4 V / / 1 1 / "^ / / boo 1 1 / W yT MO 1 / r/ / ^ y 1 ,o1 ^t- / ^ / ^ ^ f eo-- = 2: 100 V J , -^ e.= = 2800 V 1 J z = =i+4i P' 0 -^ ■^ ^ 8000 -"^ ^ ^ 2800 \ \ s s. > ^, \ 2400 \ \ 2200 c ON ST ^NT CU RR ;ni SY NC^ IRC NO JS MO- roF e )=i 20 3 V 1 = 300 AM 3 y z = 1 + 4 J A P' =2 0+ e'' 10 8 K W y '/ y 1/ 1600 / ^ y /^ -^ 7 ^^ "/ ^ y / X 1100 ,^ -' y r z' y /' r\ eT'"* ^' / y e X' y 1200 t^ "'^ ' r 'A -^ ^ y / /^ 1000 LA 3 ' . -- / ;^ rH / f y / .^^r 800 / 4 =* ^T 1 / V ^ ^ .^1 1 / y^o / / / y y / /" / / /^^ y / / /' / y / '^ 800 KILOWATTS Fig. 171. that at the same impressed voltage, with the same current input the power output of the synchronous motor can vary over a wide range, and also that for each value of power output two points exist, one with lagging, the other with leading current. 22 338 ALTERNATING-CURRENT PHENOMENA As regards phase characteristics and load characteristics, the same appHes to the synchronous converter as to the syn- chronous motor, except that in the former the continuous cur- rent output affords a means of automatically varying the excitation with the load. 230. The investigation of a variation of the armature reaction and the self-induction, that is, of the synchronous reactance, with the position of the armature in the magnetic field, and so the intensity and phase of the current in its effect on the charac- teristic curves of the synchronous motor, can be carried out in the same manner as done for the alternating-current generator in Chapter XX. In the graphical and the symbolic investigations in Chapter XX, the current, I = ii — jiz, has been considered as the output current, and chosen of such phase as to differ less than 90° from the terminal voltage, E = ex -\- je^, so representing power output. Choosing then the current vector, 01, in opposite direction from that chosen in Figs. 139 and 140, and then constructing the diagram in the same manner as done in Chapter XX, brings the output current, 01, more than 90° displaced from the terminal voltage, OE. Then the current consumes power, that is, the machine is a synchronous motor. The graphical representation in Chapter XX so applies equally well to alternating-current generator as to synchronous motor, and the former corresponds to the case Z EOI < 90°, the latter to the case: Z EOI > 90°. In the same manner, in the symbolic representation of Chapter XX, choosing the current as 7 = — t'l + JH, or, in the final equation, where the current has been assumed as zero vector, 7 = — i, that is, reversing all the signs of the current, gives the equations of the synchronous motor. Choosing the same denotations as in Chapter XX, and sub- stituting — i for -\- i vn. equation (64) so gives the general equation of the synchronous motor, _ (ei — riy + (e2 — x\d){e2 — x"oi) and for non-inductive load, _ (e — n)^ + x'o x"oi^ ^^ ~ V(e - riy- + x,"H^ SYNCHRONOUS MOTOR 339 Or, by choosing 01 in the graphic, and / = /' + /" in the syrnboUc method, as the input current, the diagram can be constructed by combining the vectors in their proper directions, that is, where they are added in Chapter XX, they are now subtracted, and inversely. For instance. El = E2 + E3, E = Ei + Ei, etc. The reversal of the sign of the current in the above equations, compared with the equations of Chapter XX, shows that in the synchronous motor, the effect of lag and of lead of the input current are the opposite of the effect of lag and lead of the output current in the generator, as discussed before. It also follows herefrom, that the representation of the internal reactions of the synchronous motor by an effective reactance, the "synchronous reactance," is theoretically justified; but that, like in the alternating-current generator, this reactance may have to be resolved in two components, x'o and x", parallel and at right angles respectively to the field-poles. 231. The phase characteristics, Fig. 165, and more particularly the no-load curve, is of special importance in the so-called syn- chronous condenser', that is, a synchronous machine running idle and producing lagging or leading current at will. As at constant impressed voltage, the reactive current taken by the synchronous machine depends upon, and varies with the field-excitation, synchronous motors offer a convenient means for producing reactive currents of varying amounts. As lagging reactive currents can more conveniently be pro- duced by stationary reactors, synchronous machines are mainly used for producing leading currents, or producing reactive cur- rents varying between lag and lead. Therefore, the name "synchronous condenser" for such machines. Their foremost use is : 1. For power-factor correction in systems of low power- factor, such as systems containing many induction motors or other reactive devices. In this case, the synchronous condenser is connected in shunt to the circuit as close to the source of the reactive lagging currents as feasible. 2. For voltage control of long-distance transmission lines. In very long lines, especially at 60 cycles, the inherent voltage regulation at the receiving end of the line becomes very poor, and then a synchronous condenser is made to "float" on the 340 ALTERNATING-CURRENT PHENOMENA receiving circuit, controlled by a voltage regulator so that its reactive current varies from lag at no-load on the line, to lead at heavy load, and thereby maintains the line voltage constant. In synchronous condensers, low armature reaction is an ad- vantage, as requiring less field regulation. As synchronous condensers must run at high leading currents, and this is the condition where the tendency to surging is greatest, synchronous condensers are usually supplied with anti-hunting devices. For this purpose, generally a squirrel-cage winding in the field-poles is used. Such a winding is desirable also to improve the self-starting character of the machine. Very large synchronous condensers are in successful operation on transmission lines of such length, that without the syn- chronous condenser, operation of the circuits would be entirely impossible. SECTION VI GENERAL WAVES