CHAPTER XVIil. SYNCHRONOUS MOTOR. 177. In the chapter on synchronizing alternators we have seen that when an alternator running in synchronism is connected with a system of given E.M.F., 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 investigation 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 symbolic method, we may in the following, as an instance of the graphical method, treat the action of the synchronous motor diagrammatically. Let an alternator of the E.M.F., E^, be connected as synchronous motor w^ith a supply circuit of E.M.F., E^y by a circuit of the impedance Z, If E^ is the E.M.F. impressed upon the motor termi- nals, Z is the impedance of the motor of induced E.M.F., E^. If E^ is the E.M.F. at the generator terminals, Z is the impedance of motor and line, including transformers and other intermediate apparatus. If E^ is the induced E.M.F. of J:he generator, Z is the sum of the impedances of motor, line, and generator, and thus we have the prob- lem, generator of induced E.M.F. E^y and motor of induced E.M.F. E^\ or, more general, two alternators of induced E.M.Fs., E^y E^y connected together into a circuit of total impedance, Z. Since in this case several E.M.Fs. are acting in circuit § 177] sy.\ci//^ONOUs motor. 259 with the same current, it is convenient to use the current, /, as zero line 01 of the polar diagram. If I = i z= current, and Z = impedance, r = effective resistance, x = effective reactance, and s = Vr*-^ + ^2 __ absolute value of impedance, then the E.M.F. consumed by the resistance is i?i = r/, and in phase with the cur- rent, hence represented by vector 0E^\ and the E.M.F. consumed by the reactance is E^ = xi^ and 90° ahead of the current, hence the E.M.F. consumed by the impedance is E = \/'{E^f + (E^f, or = /" V/-^ -|- x^ = /-c, and ahead of the current by the angle 8, where tan 8 = x / r. We have now acting in circuit the E.M.Fs., E, E^, E^; or E^ anc^ E are components of E^y; that is, E^^ is the diagonal of a parallelogram, with E^ and E as sides. Since the E.M.Fs. E^y E^, E, are represented in the diagram, Fig. 122, by the vectors OE^, OE^, OE, to get the parallelogram of E^, E^, E, we draw arc of circle around with E^y and around E with Ey^, Their point of intersection gives the impressed E.M.F., OE^ = Eq, and completing the parallelogram OEy E^y E^, we get OEy^ = Ely the induced E.M.F. of the motor. ^^ /OEo is the difference of phase between current and im- pressed E.M.F., or induced E.M.F. of the generator. ^y^ lOEx is the difference of phase between current and in- duced E.M.F. of the motor. And the power is the current /times the projection of the E.M.F. upon the current, or the zero line OL Hence, dropping perpendiculars, E^E^ and E^E^y from E^ and E^ upon Oly it is — I\ = / X OE} = power supplied by induced E.M.F. of gen- erator. I\ = / X OE^ = electric power transformed in mechanical power by the motor. P =: i X 0E\ =? power consumed in the circuit by effective resistance. 260 AL TEKNA TING-CURRENT PHENOMENA. [§ 1 78 Obviously P^ = P^-\- P, Since the circles drawn with E^ and E^ around O and E respectively intersect twice, two diagrams exist. In gen- eral, in one of these diagrams shown in Fig. 122 in drawn Fig. 122. lines, current and E.M.F. are in the same direction, repre- senting mechanical work done by the machine as motor. In the other, shown in dotted lines, current and E.M.F. are in opposite direction, representing mechanical work con- sumed by the machine as generator. Under certain conditions, however, E^ is in the same, E^ in opposite direction, with the current ; that is, both ma- chines are generators. 178. It is seen that in these diagrams the E.M.Fs. are considered from the point of view of the motor ; that is, •§178] SYNCHRONOUS MOTOR, 2(51 work done as synchronous motor is considered as positive, work done as generator is negative. In the chapter on syn- chronizing 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. consumed by the impedance, E = OEy Figs. 123 to 125, con- sists of three components ; the E.M.F. OE^ = E^y consumed Flq. 123, by the impedance of the motor, the E.M.F. E^ E^ = E^ consumed by the impedance of the line, and the E.M.F. E^ E = E^ consumed by the impedance of the generator. Hence, dividing the opposite side of the parallelogram E^Eq, in the same way, we have : OE^ = E^= induced E.M.F. of the motor, OE^ = E^ = E.M.F. at motor terminals or at end of line, OE^ = E^ = E.M.F. at generator terminals, or at beginning of line. OEq = Eq = induced E.M.F. of generator. 262 ALTERNATING-CURRENT PHENOMENA. 179^ The phase relation of the current with the E.M.Fs. E^y Eqj depends upon the current strength and the E.M.Fs. E^ and Eq, 179. Figs. 123 to 125 show several such diagrams for different values of E^t but the same value of / and Eq, The motor diagram being given in drawn line, the genera- tor diagram in dotted line. Fig. 124. As seen, for small values of E^ the potential drops in the alternator and in the line. For the value of E^ = Eq the potential rises in the generator, drops in the line, and rises again in the motor. For larger values of E^, the potential rises in the alternator as well as in the line, so that the highest potential is the induced E.M.F. of the motor, the lowest potential the induced E.M.F. of the gen- erator. §180] SYxVC/IJiOxVOUS MOTOR. 263 It is of interest now to investigate how the values of these quantities change with a change of the constants. ^^-. «« •^'. \ X Fig. 125. 180. A. — Constant impressed E.M.F, E^y constant current strength I = /, variable motor excitation E^. (Fig. 126.) If the current is constant, = /; OE, the E.M.F. con- sumed by the impedance, and therefore point E, are con- stant. Since the intensity, but not the phase of E^ is constant, E^ lies on a circle c^ with E^ as radius. From the parallelogram, OE^ E^ E^ follows, since E^E^ parallel and = OE, that E^ lies on a circle c^ congruent to the circle e^, but with E^y the image of E, as center : OE^ = OE, Wc can construct now the variation of the diagram with the variation of E^ ; in the parallelogram OE Eq E^, O and E arc fixed, and E^ and E^ move on the circles e^ e^ so that ~E^ ^j is parallel to OE. 264 AL TERNA TING-CURRENT PHENOMENA. [§ 180 The smallest value of E^ consistent with current strength /is Olj = ^j, 01 = -Sq. In this case the power of the motor is 01 ^^ x /, hence already considerable. Increasing E^ to 02^, 03j, etc., the impressed E.M.Fs. move to 02, 03, etc., the power is / x 02,^, / x 03i^, etc., increases first, Fig. 12e. reaches the maximum at the point 3,, 3, the most extreme point at the right, with the impressed E.M.F. in phase with the current, and then decreases again, while the induced E.M.F. of the motor Ei increases and becomes = Eq at 4,, 4. At 5i, 5, the power becomes zero, and further on negative ; that is, the motor has changed to a dynamo, and § 180] SYNCHRONOUS MOTOR, 265 produces electrical energy, while the impressed E.M.F. E^ still furnishes electrical energy, that is, both machines as generators feed into the line, until at 61, 6, the power of the impressed E.M.F. E^ becomes zero, and further on power begins to flow back ; that is, the motor is changed to a gen- erator and the generator to a motor, and we are on the generator side of the diagram. At Tj , 7, the maximum value of E^y consistent with the current C, has been reached, and passing still further the E.M.F*. E^ decreases again, while the power still increases up to the maximum at 81, 8, and then decreases again, but still Ey^ remaining generator, E^ motor, until at 11,, 11, the power of E^ becomes zero ; that is, E^ changes again to a generator, and both machines are generators, until at ISj, 12, where the power of E^ is zero, Ex changes from generator to motor, and we come again to the motor side of the diagram, and while E^ still decreases, the power of the motor increases until 1,, 1, is reached. Hence, there arc two regions, for very large E^ from G to 7, and for very small E^ from 11 to 12, where both machines are generators ; otherwise the one is generator, the other motor. For small values of E^ the current is lagging, begins, however, at 2 to lead the induced E.M.F. of the motor E^y at 3 the induced E.M.F. of the generator E^, It is of interest to note that at the smallest possible value of E^y Ij, 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 condition 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 max- imum efficiency : no displacement of phase of the impressed 2iW A/. TKHA-A rti\G-CURRE.VT P//F..VO.VKXA. [| 181 Iv.M.I"'., or Kclf-induction of the circuit compensated by the effect of the lead of the motor current. This condition of iiiiiximum t-fficiency of a circuit we have found already in Chapter VIII. on Inductance and Capacity. 181. B. r.g aiiel J-\ constant, I variable. < >l)vi(iitEq. In the first case, Ey^ = E^ (Fig. 127), we see that at Fig. 128. very small curren^, that is very small OE, the current / leads the impressed E.M.F. Eq by an angle £*q(?C=%. This lead decreases with increasing current, becomes zero, and alterwards for larger current, the current lags. Taking now any pair of corresponding points E, E^, and producing EEq until it intersects ^,, in E^ we have ^^ E^ OE = 90®, E^ = Eq, thus : OE^ = EE^^OE^ = ^^^i \ that is, EE^ = 268 AL TERN A TING-CURRENT PHENOMENA. 181 ^Eq, That means the characteristic curve e^ is the enve- lope of lines EE^y of constant lengths 2^^, sliding between the legs of the right angle E^ 0E\ hence, it is the sextic hypocyloid osculating circle e^^ which has the general equa- tion, with c, Ci as axes of coordinates : VTl^ + -y/'f = -s/TE^ In the next case, E^ < Eq (Fig. 128) we see first, that the current can never become zero like in the first case^ Fig. 129. El = E^y but has a minimum value corresponding to the E — E minimum value of OE^-. //= ^^ ^ "\ and a maximum F A- F " . 1^ 1 value : I^ = . Furthermore, the current can never -J lead the impressed E.AI.F. is^, but always lags. The mini- ^ § 182] SYNCIl/^OXOUS MOTOR, 269 mum lag is at the point H, The locus c^, as envelope of the lines EE^y is a finite sextic curve, shown in F*ig. 128. In the case E^ > Eq (Fig. 129) the current cannot equal zero either, but begins at a finite value C\', corresponding to the minimum value of OE^ : // = ;;; . At this value however, the alternator E^ is still generator and changes to a motor, its power passing through zero, at the point corresponding to the vertical tangent, onto r^ with a very large lead of the impressed E.M.F. against the cur- rent. At /T the lead changes to lag. The minimum and maximum value of current in the three conditions are given by : o p 1st. /=(), 7=:=^^. , *» 2d. I = ^" "~ ^^ 7 = -^0 H- E^ \ z z 3d. 7 = ^'""^% l^^l±Ml^ z z Since the current passing over the line at ^, = (?, that is, when the motor stands still, is I^ = E^j s^ we see that in such a synchronous motor-plant, when running at syn- chronism, the current can rise far beyond the .value it has at standstill of the motor, to twice this value at 1, some- what less at 2, but more at 3. Hence in such a case, if the synchronous motor drops out of step, the current passing over the line goes down to one-half or less ; or, in other words, in such a system the motor, under certain conditions of running, is more liable to burn up than when getting out of step. 182. C. E^ = constant^ E^ varied so t/iat the efficiency is a maximum for all currents. (Fig. 130.) Since we have seen that the output at a given current strength, that is, a given loss, is a maximum, and therefore 270 AL TERN A TING-CURRENT PHENOMENA. [§ 1 82 the efficiency a maximum, when the current is in phase with the induced E.M.F. E^ of the generator, we have as the locus of Eq the point E^ (Fig- 130), and when E with increasing current varies on ^, E^ must vary on the straight line Cx parallel to c. Hence, at no-load or zero current, E^ = E^, decreases with increasing load, reaches a minimum at ^E^ perpen- dicular to ^1, and then increases again, reaches once more :^ Fig. 130. El = Eq at Ei^y and then increases beyond Eq, The cur- rent is always ahead of the induced 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 E^\ where OEi* = E^^Eq = 1/2 X OEf^, and is then = / x EO/2. Hence, since OE^* = /r = iE'o/ 2, / = E^/ 2 r and /^ = Eq^/ 4 r, hence = the maxi- mum power which, over a non-inductive line of resistance r can be transmitted, at 50 per cent, efficiency, into a non- inductive circuit. § 183] SYNCHRONOUS MOTOR. 271 In this case, In general, it is, taken from the- diagram, at the condi- tion of maximum efficiency : Comparing these results with those in Chapter IX. on Self-induction and Capacity, we see that the condition of maximum efficiency of the synchronous motor system is the same as in a system containing only inductance and capacity, the lead of the current against the induced E.M.F. E^ here acting in the same way as the condenser capacity in Chapter IX. Fig. 131. 183. D. E^ =^ constant ; P ^ constant. If the power of a synchronous motor remains constant, wc have (Fig. 131) / x OE^ = constant, or, since OE^ = 272 ALTERNATING-CURRENT PHENOMENA. 18a /r, I = OE^jr, and: OE^ x OE^ = OE"^ x E'E^' = constant. Hence we get the diagram for any value of the current /, at constant power Pi , by making OE^ = /^, E^Eq^ = P^ j C erecting in E^ a perpendicular, which gives two points of intersection with circle e^, E^, one leading, the other lagging. Hence, at a given impressed E.M.F. E^, the same power Pi 1250 1100/1580 1480 1060/1840 S120 2170 2300 I 7 8116.7 92 2/2& 87.5 40 45.5 qSO/lWO 7/49 .^ ^ 1440 lOM) 800 Ej^lOOO P=IOOO 46 « E^2200 1sv,^^ ^ Sy^ EflOOO P=6000 340i cos (/'i ^,), (1) thus, — cos (f\ dy) = ^ sm i„(/...) = v/i-(-4)-J (2) ♦ If i\y = E.M.F. at motor terminals, z = internal impedances of the motor; if eo= terminal voltage of the generator, s = total impedance of line and motor; if ^f^ = E.M.F. of generator, that is, E.M.F. induced in generator armature by its rotation through the magnetic field, z includes the generator impedance also. f6 ALTERNATJNG-CURRENT PHENOMENA. [S 184 The displacement of phase between current i and E.M.F. = si consumed by the impedance s is : cos (*>) = sin (»>) = Since the three E.M.Fs. acting in the closed circuit: e^ = E.M.F. of generator, tx = C.E.M.F. of synchronous motor, £ =3 SI = E.M.F. consumed by impedance, form a triangle, that is, Ci and ^ are components of c^, it is {Fig. 136) : ^,* = ^,' + ^» + 2^^,cos(* -J- x\ three variables arc left, ^\, i, p< of which two are independent. Hence, at given <■(, and z, the current /' is not determined by the load / only, but also by the excitation, and thus the same current i can represent widely different loads /, according to the excita- tion ; and with the same load, the current / can be varied in a wide range, by varying the field excitation c^. The meaning of equation (7) is made more perspicuous § 184] SYNCHRONOUS MOTOR. 277 by some transformations, which separate e^ and /, as func- tion of / and of an angular parameter <^. Substituting in (7) the new coordinates : ys _ V + y* /•' 1 Vii J or, K V2 V2 J (8) we get -aV2-2r/ = 2f^/^^'-,V*; (9) «-^iC- — Fig. 136. Fig. 137. we get ^ _ a V2 - c/; = V(l - €-) (^a-^ - 2 fi'^ - /^, (11) and, squared, ,2^2_^(1_,2^ )3^-a V2 (rt -€/.) + ^''^^"''') + (^7/^)' = 0, (12) substituting (/i-c/0 V2 2c ' P VI - c^ = w, gives, after some transposition, (13) (14) 278 AL TERN A TING-CURRENT PHENOMENA, [§ 1 84 hence, if , R = « /(^ - «^) ( '^ - ^^^ (15) * . * ^^ ^'^ z-^ + w^ = i?2 (16) the equation of a circle with radius R. Substituting now backwards, we get, with some trans- positions : {r^ (e,^ + z'i") - z' (e,^ - 2 rp)Y + {rx (r,' - z^i'^)y = ^2Vu^(V-'iA-/) (17) the Fundamental Equation of the Synchronous Motor in a modified form. The separation of ^i and / can be effected by the intro- duction of a parameter ^ by the equations : f^(e^ — z^i'^) ^z'^{e^ -2.rp) = x z e^Vcf-T7p co^ ^ rxXe^ — z^i'^ =^ xze^^ ^c^ — A^rp sin These equations (18), transposed, give ^1 = y/|{^ (V- '^ ^/) + y ('^cos ^ + sin \ VV-4 rp | The parameter <^ has no direct physical meaning, appar- ently. These equations (19) and (20), by giving the values of ex and / as functions of p and the parameter 4> enable us to construct the Power Characteristics of the Synchronous Motor, as the curves relating e^ and /, for a given power /, by attributing to all different values.. § 185] SVA'C/fA'CAOrS MOTOR, 270 Since the variables v and %v in the equation of the circle (16) are quadratic functions of e^ and /, the Power Charac- teristics of the Synchronous Motor are Qnartic Cnn'cs, They represent the action of the synchronous motor under all conditions of load and excitation, as an element of power transmission even including the line, etc. Before discussing further these Power Characteristics^ some special conditions may be considered. 185. A. Maximum Output. Since the expression of c^ and /- [equations (19) and (20)] contain the square root, vVq- — 4;-/, it is obvious that the maximum value of / corresponds to the moment where this square root disappears by passing from real to imaginary ; that is, ..^^ - 4 r/ = 0, / = -/- . (21) 4;- This is the same value which represents the maximum power transmissible by E.M.F., r^, over a non-inductive line of resistance, r; or, more generally, the maximum power which can be transmitted over a line of impedance, into any circuit, shunted by a condenser of suitable capacity^ Substituting (21) in (19) and (20), we get, 2r J (22) and the displacement of phase in the synchronous motor. COs{c\J) = -J^ = -; tCi ' z hence, tan(^„;) = --!^, (23) r 280 ALTERNATING-CURRENT PHENOMENA, [§ 186 that is, the angle of internal displacement in the synchron- ous motor is equal, but opposite to, the angle of displace- ment of line impedance, (^1, = - (^» 0» = - (^, r\ (24) and consequently, (^0, /•) - ; (25) that is, the current, /, is in phase with the impressed E.M.F., ^Q. If ;? < 2 r, ^1 < ^o; that is, motor KM.F. < generator E.M.F. If 2r = 2 r, ^1 = ^q; that is, motor E.M.F. = generator E.M.F. If ^ > 2 r, ^1 > ^o; that is, motor E.M.F. > generator E.M.F. In either case, the current in the synchronous motor is leading. 186. B. Running Lights / = 0. When running light, or for / = 0, we get, by substitut- ing in (19) and (20), (26) Obviously this condition can never be fulfilled absolutely, since/ must at least equal the power consumed by friction, etc. ; and thus the true no-load curve merely approaches the curve/ = 0, being, however, rounded off, where curve (26) gives sharp corners. Substituting / = into equation (7) gives, after squar- ing and transposing, e^ -f ro^ + zU"^ - 2 V^o' - 2 r»/2^o' + 2 r'^i'^c^^ - 2 at^/V = 0. (27) This quartic equation can be resolved into the product of two quadratic equations, Cy + 2*/^ — <'o^ + 2 xici = generator. | ^no\ e^ + z^i'i _ e^"- — 2 AT/V, = motor. ) ^ ^ § 186] SYNCHRONOUS MOTOR. 281 which are the equations of two ellipses, the one the image of the other, both inclined with their axes. The minimum value of C.E.M.F., o-, (33) z or, transposed, e^ = ^(e^^irf^i'xK (34) This quadratic curve passes through the point of zero current and zero power, i = 0, e^=e^^ through the point of maximum power (22), ; ^0 ^ ^0 ^ * — — « C\ — — • 2r ' 2r' and through the point of maximum current and zero power^ i = ^\ ^i = ^iL£, (35) r r and divides each of the quartic curves or* power character- istics into two sections, one with leading, thfe other with lagging, current, which sections are separated by the two points of equation 34, the one corresponding to minimum, the other to maximum, current. It is interesting to note that at the latter point the current can be many times larger than the current which would pass through the motor while at rest, which latter current is, / = ^^ , (36) z while at no-load, the current can reach the maximum value, /=-% (35) r the same value as would exist in a non-inductive circuit of the same resistance. The minimum value at C.E.M.F. e^^ at which coincidence 284 ALTEXifATING-CURRENT PHENOMENA. \% 18S of phase {i?^, /) = 0, can still be reached, is determined from equation (34) by. ,_,.r-,,_,.?, (37) The curve of no-displacement, or of minimum current, is shown in Figs. 138 and 139 in dotted lines.* 188. D. Maximum Displacement of P/iase. (cj, /) = maximum. At a given power/ the input is, A =/ + '■''- = '«'■ cos (e^, I) i (38). . Iience, cos(<-o, /) = i-+J^. (39) At a given power /<, this value, as function of the current is a maximum when this gives, / = (V; or. (40) (41) That is, the displacement of phase, lead or lag, is a hen the power of the motor equals the power Me thai the eciuittion (:14} ia similai to the value, which repri'sdils ihc oul]iul Iransniilted ovvr an eal wUh Iho equ.ition giving ihc maximum voltage, Ik produced by shunting the receiving circuit with ■ idition ol " complete rewinance " of the line, : = . Hence, relerring to equation (35), ^ , = f ^ ~ is voltage of ihe line, reached when closed by a con- <.= VK ~.r inductive line ol mpedance Eqna on (M) is idem t, , at curr which can con.len«er that s. the CO V7' +-J , with cnrrent, i Ihc maxi, um re (onance v densci of eactnn Ce, - J. £188] SYNCMfiOJVOUS MOTOR. 285 consumed by the resistance ; that is, at the electrical effi- ciency of 50 per cent. Substituting (40) in equation (7) gives, after squaring and transposing, the Quartic Equation of Maximum Dis- placement, W - 'iT + '*=' (a' + 8 '-') + 2 iW (5 r' - H) - 2 » V,' (.= -1-30 = 0. (42) The curve of maximum displacement is shown in dash- dotted lines in Figs. 138 and 139. It passes through the 286 AL TERN A TING-CURRENT PHENOMENA, [ §§ 1 89, 1 90 point of zero current — as singular or nodal point — and through the point of maximum power, where the maximum displacement is zero, and it intersects the curve of zero displacement. 189. E. Constant Counter E,M.F. At constant C.E.M.F., e^ = constant, the current at no-load is not a minimum, and is lagging. With increasing load, the lag decreases, reaches a mini- mum, and then increases again, until the motor falls out of step, without ever coming into coincidence of phase. If e^ov < e^. 190. F. Niimcrical Instance, Figs. 138 and 139 show the characteristics of a 100- kilowatt motor, supplied from a 2500-volt generator over a distance of 5 miles, the line consisting of two wires, No. 2 B. & S.G., 18 inches apart. §190] SYNCHRONOUS MOTOR. 287 In this case we have, e^ = 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, 2500^ - e^ - 500 i^ - 20/ = 40 VTV'^^ (7) {^1^ + 500 /2 _ 31.25 X 10« + 100/}2 + {2 e^ - 1000 i'-y = 7.8125 X 10^ - 5 + lOV. (17) ^1 = 5590 (19) VH(l-^-2xl0-«/) + (.894cos<^+.447sin<^)Vr^6.4xl0"-«/}. i = 559 (20) Vi{(l-^-2xl0-V>) + (.894cos<^-.447sin<^)Vi^6.4xl0-«/}. (22) Maximum output, / = 156.25 kilowatts (21) at fi = 2,795 volts /■ =125 amperes Running light, ^j« + 500 /* - 6. 25 X 10* = F 40 /Vx = I ,^8^ ^1 = 20 /• i V6.25 X 10* - 100 n \ ^ ^ At the minimum value of C.KM.F. ^i = is / = 112 (29) At the minimum value of current, / = is ^i = 2500 (30) At the maximum value of C.KM.F.