CHAPTER XIX. 

SYNCHRONOUS MOTOR. 

198. 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 with a supply circuit of E.M.F., EQ, 
by a circuit of the impedance Z. 

If E0 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 E0 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 induced 
E.M.F. of the 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. EQ, and motor of induced' 
E.M.F. El; or, more general, two alternators of induced 
E.M.Fs., E0, Elf connected together into a circuit of total 
impedance, Z. 

Since in this case several E.M.Fs. are acting in circuit 


322 ALTERNATING-CURRENT PHENOMENA. 

with the same current, it is convenient to use the current, 
/, as zero line OI of the polar diagram. Fig. 188. 

If I=i= current, and Z = impedance, r = effective 
resistance, x = effective reactance, and s = Vr2 -f x2 = 
absolute value of impedance, then the E.M.F. consumed 
by the resistance is E,, = ri, and in phase with the cur- 
rent, hence represented by vector OE,, ; and the E.M.F. 
consumed by the reactance is E2 = xi, and 90° ahead of 
the current, hence the E.M.F. consumed by the impedance 
is E = V(£,,)2 + (E2f, or = i Vr2 + x* = is, 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, Elf EQ; 
or El and E are components of EQ ; that is, EQ is the 
diagonal of a parallelogram, with El and E as sides. 

Since the E.M.Fs. Elf Ez, E, are represented in the 
diagram, Fig. 138, by the vectors OE~lf OE2, OE, to get 
the parallelogram of £Q, Elt E, we draw arcs of circles 
around 0 with EQ , and around E with El . Their point of 
intersection gives the impressed E.M.F., OEQ = EQ, and 
completing the parallelogram OE EQ E± we get, OE± = E± , 
the induced E.M.F. of the motor. 


IOE0 is the difference of phase between current and im- 
pressed E.M.F., or induced E.M.F. of the generator. 

IOEi 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 OI. 

Hence, dropping perpendiculars, E^EJ and E^E^, from 
EQ and E! upon OI, it is — 

P0 = iX OE^ = power supplied by induced E.M.F. of gen- 

erator. 
PI = / X OE^ = electric power transformed in mechanical 

power by the motor. 
P = / x OEl = power consumed in the circuit by effective 

resistance. 


SYNCHRONOUS MOTOR. 


323 


Since the circles drawn with EQ and E± around O and K 
respectively intersect twice, two diagrams exist. In gen- 
eral, in one of these diagrams shown in Fig. 138 in drawn 


Fig. 138. 

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, £Q is in the same, E^ 
in opposite direction, with the current ; that is, both ma- 
chines are generators. 

199. It is seen that in these diagrams the E.M.Fs. are- 
considered from the point of view of the motor ; that is,. 


324 


ALTERNATING-CURRENT PHENOMENA. 


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 = OE, Figs. 139 to 141, con- 
sists of three components ; the E.M.F. OE£ — Ez, consumed 


Fig. 139. 

by the impedance of the motor, the E.M.F. 
consumed by the impedance of the line, and the E.M.F. 
EZ E = E± consumed by the impedance of the generator. 
Hence, dividing the opposite side of the parallelogram E1E(), 
in the same way, we have : OEl = E1 = induced E.M.F. of 
the motor, OEZ = 2?a = E.M.F. at motor terminals or at 
end of line, OE3 = E3 = E.M.F. at generator terminals, 
or at beginning of line. OEQ = EQ = induced E.M.F. of 
generator. 


SYNCHRONOUS MOTOR. 


325 


The phase relation of the current with the E.M.Fs. £lt 
, depends upon the current strength and the E.M.Fs. El 


and 


200. Figs. 139 to 141 show several such diagrams for 
different values of Elf but the same value of / and EQ. 
The motor diagram being given in drawn line, the genera- 
tor diagram in dotted line. 


Fig. 140. 

As seen, for small values of E1 the potential drops in 
the alternator and in the line. For the value of E1 = E0 
the potential rises in the generator, drops in the line, and 
rises again in the motor. For larger values of Ely thfe 
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. 


326 


ALTERNATING-CURRENT PHENOMENA, 


It is of interest now to investigate how the values of 
these quantities change with a change of the constants. 


Fig. 747. 

201. A. — Constant impressed E.M.F. Ev, constant current 
strength I = i, variable motor excitation Ev (Fig. 142.) 

If the current is constant, = z; 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 EQ is 
constant, EQ lies on a circle eQ with EQ as radius. From 
the parallelogram, OE EQ El follows, since E1 EQ parallel 
and = OE, that El lies on a circle el congruent to the circle 
eQ, but with Ei} the image of E, as center : OEi = OE. 

We can construct now the variation of the diagram with 
the variation of El ; in the parallelogram OE EQ E1 , O and 
E are fixed, and E0 and El move on the circles <?0 el so that 
EQ E^ is parallel to OE. 


SYNCHRONOUS MOTOR. 


327 


The smallest value of El consistent with current strength 
/ is Olj = E^, 01 = EQ. In this case the power of the 
motor is Olj1 x /, hence already considerable. Increasing 
El to 02"^ OSj, etc., the impressed E.M.Fs. move to 02, 03, 
etc., the power is / x 02^, I x 03^, etc., increases first, 


Fig. 142. 

reaches the maximum at the point 3j, 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 E^ increases and becomes = £Q at 
4,, 4. At 515 5, the power becomes zero, and further on 
negative ; that is, the motor has changed to a dynamo, and 


328 AL TERNA TING-CURRENT PHENOMENA. 

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 1l, 7, the maximum value 
of Elt consistent with the current /, has been reached, and 
passing still further the E.M.F. El decreases again, while 
the power still increases up to the maximum at Slt 8, and 
then decreases again, but still El remaining generator, EQ 
motor, until at 11^ 11, the power of EQ becomes zero; that 
is, EQ changes again to a generator, and both machines are 
generators, up to 12lf 12, where the power of El is zero, El 
changes from generator to motor, and we come again to 
the motor side of the diagram, and while El still decreases, 
the power of the motor increases until lu 1, is reached. 

Hence, there are two regions, for very large El from 
5 to 6, and for very small El from 11 to 12, where both 
machines are generators ; otherwise the one is generator, 
the other motor. 

For small values of El the current is lagging, begins, 
however, at 2 to lead the induced E.M.F. of the motor Elf 
at 3 the induced E.M.F. of the generator E0. 

It is of interest to note that at the smallest possible 
value of EI} lj, 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 


SYNCHRONOUS MOTOR. 


329 


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 
the Chapter on Inductance and Capacity. 


202. B. EQ and El constant, I variable. 

Obviously EQ lies again on the circle eQ with EQ as radius 
and O as center. 


Fig. 143. 


E lies on a straight line e, passing throtigh the origin; 

Since in the parallelogram OE E0 Ev EEQ = E^ we 
derive EQ by laying a line EEQ = E± from any point E 
in the circle eQ, and complete the parallelogram. 

All these lines EEQ envelop a certain curve elt which 


030 ALTERNATING-CURRENT PHENOMENA. 

can be considered as the characteristic curve of this prob- 
lem, just as circle e^ in the former problem. 

These curves are drawn in Figs. 143, 144, 145, for the 
three cases : 1st, El = EQ ; 2d, El < EQ ; 3d, £1>£Q. 

In the first case, El = EQ (Fig. 127), we see that at 


Fig. 144. 


very small current, that is very small OE, the current / 
leads the impressed E.M.F. EQ by an angle EQOf = WQ. 
This lead decreases with increasing current, becomes zero, 
and afterwards for larger current, the current lags. Taking 
now any pair of corresponding points E, EQ, and producing 
EEQ until it intersects eit in Eif we have ^^ Ei OE — 90°, 
El = EQ , thus : OE1 = EEQ=OEQ = EQEt ; that is, EE{ = 


SYNCHRONOUS MOTOR. 


331 


2EQ. That means the characteristic curve el is the enve- 
lope of lines EEiy of constant lengths 2EQ, sliding between 
the legs of the right angle Et OE; hence, it is the sextic 
hypocyloid osculating circle <?0, which has the general equa- 
tion, with e, ei as axes of coordinates : 


In the next case, E1 < EQ (Fig. 144) we see first, that 
the current can never become zero like in the first case, 


V 


Fig. 145. 


EI = EQ, but has a minimum value corresponding to the 
minimum value of OEl : I{ = — — , and a maximum 

value : //' = — — . Furthermore, the current can never 

lead the impressed E.M.F. E^, but always lags. The mini- 


332 ALTERNATING-CURRENT PHENOMENA. 

mum lag is at the point H. The locus ev as envelope of the 
lines EEty is a finite sextic curve, shown in Fig. 144. 

If El < EQ , at small EQ — El , H can be above the zero 
line, and a range of leading current exist between two ranges 
of lagging current. 

In the case E1 > EQ (Fig. 145) the current cannot equal 
zero either, but begins at a finite value C±, corresponding 

to the minimum value of OEQ : // = * — -. At this 

value however, the alternator E1 is still generator and 
changes to a motor, its power passing through zero, at the 
point corresponding to the vertical tangent, onto elf with 
a very large lead of the impressed E.M.F. against the cur- 
rent. At H the lead changes to lag. 

The minimum and maximum value of current in the 
three conditions are given by : 

Minimum: Maximum: 

1st. 7=0, 7=^. 


Since tfie current passing over the line at El = O, that 
is, when the motor stands still, is 70 = EQj z, 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. 

203. C. EQ = constant, El varied so that the efficiency is a 
maximum for all currents. • (Fig. 146.) 

Since we have seen that the output at a given current 
strength, that is, a given loss, is a maximum, and therefore 


SYNCHRONOUS MOTOR. 


333 


the efficiency a maximum, when the current is in phase 
with the induced E.M.F. EQ of the generator, we have as 
the locus of EQ the point EQ (Fig. 146), and when E with 
increasing current varies on <?, E± must vary on the straight 
line ev parallel to c. 

Hence, at no-load or zero current, El = E0, decreases 
with increasing load, reaches a minimum at OE^ perpen- 
dicular to clt and then increases again, reaches once more 


Fig. 146. 


El = EQ at E?, and then increases beyond E0. The cur- 
rent is always ahead of the induced E.M.F. El 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 Ef, where OEf = EfEQ = 
1/2 x ~OE^ and is then = / x "^7/2. Hence, since OEf = 


EJ2,f=E()/2randP 


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. 


-334 ALTERNATING-CURRENT PHENOMENA. 

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. 
El here acting in the same way as the condenser capacity 
in Chapter IX. 


204. 


Fig. 147. 


D. En = constant ; P = constant. 


If the power of a synchronous motor remains constant, 
we have (Fig. 147) / x OE^ = constant, or, since OE1 — 


SYNCHRONOUS MOTOR. 


335 


Ir, I = OE1/ r, and: OE1 x OE? = O£l X E1EJ = 
constant. 

Hence we get the diagram for any value of the current 
/, at constant power Plt by making OE1 = I r, E1E01 = Pl j I 
erecting in EQl a perpendicular, which gives two points of 
intersection with circle eQ, EQ, one leading, the other lagging. 
Hence, at a given impressed E.M.F. EQ, the same power P± 


E, 


1250 7 

1100/1580 31/16.7 

1480 32 

1050/1840 2/25 

2120 
2170 


37.5 
40 


45.5 


16.7 


Fig. U8. 

can be transmitted by the same current I with two different 
induced E.M.Fs. E} of the motor; one, OEl = EEQ small, 
corresponding to a lagging current ; and the other, OEl = 
EEQ large, corresponding to a leading current. The former 
is shown in dotted lines, the latter in drawn lines, in the 
diagram, Fig. 147. 

Hence a synchronous motor can work with a given out- 
put, at the same current with two different counter E.M.Fs. 


336 


ALTERNATING-CURRENT PHENOMENA. 


E1. In one of the cases the current is leading, in the 
Dther lagging. 

In Figs. 148 to 151 are shown diagrams, giving the points 

E0 = impressed E.M.F., assumed as constant = 1000 volts, 
E = E.M.F. consumed by impedance, 
E' = E.M.F. consumed by resistance. 


EflOOO 

P=6000 

34O< E,<1920 

7< I < 43 


Fig. 149. 


I 

1450 17.3 

1170/1910 10/30 
1040/1930 8/37.5 


10/30 
17.3 


of the motor, Elt is OElt equal and 
shown in the diagrams, to avoid 


The counter E.M.F. 
parallel EEQ, but not 
complication. 

The four diagrams correspond to the values of power, 
or motor output, 
P = 1,000, 6,000, 


9,000, 


12,000 watts, and give : 
1 < I < 49 Fig. 132. 


P = 1,000 46 < El < 2,200, 

P = 6,000 340 < £, < 1,920, 7 < I < 43 Fig. 133. 

P = 9,000 540 < El < 1,750, 11.8 < / < 38.2 Fig. 134. 

P = 12,000 920 < El < 1,320, 20 < I < 30 Fig. 153. 


SYNCHRONOUS MOTOR. 


337 


E, I 

* 1440 21.2 

3 1200/1660 15/30 

1080/1750 13/34.7 


900/1590 11.8/38.2. 


720/1100 13/34.7 
620/820 15/30 
/3 540 21.2 


3 1280 24.5 

2 1120/1320 21/28.6 
all— l-QQO/1260 30/30 


920/1100 
020 


21/28.6 
24.5 


P=I200O 

920< E,< 1320 

20<l<30 


Fig. 151. 

As seen, the permissible value of counter E.M.F. Ev and 
of current /, becomes narrower with increasing output. 


338 ALTERNATING-CURRENT PHENOMENA. 

In the diagrams, different points of EQ are marked with 
1, 2, 3 . . . , when corresponding to leading current, with 
21, 31, . . . , when corresponding to lagging current. 

The values of counter E.M.F. Ev and of current 7 are 
noted on the diagrams, opposite to the corresponding points 

*o- 

In this condition it is interesting to plot the current as 

function of the induced E.M.F. El of the motor, for con- 
stant power /V Such curves are given in Fig. 155 and 
explained in the following on page 345. 

205. While the graphic method is very convenient to 
get a clear insight into the interdependence of the different 
quantities, for numerical calculation it is preferable to ex- 
press the diagrams analytically. 

For this purpose, 

Let z = Vr2 -j- x2 = impedance of the circuit of (equivalent) 
resistance r and (equivalent) reactance x = 2 TT NL, containing 
the impressed E.M.F. e0* and the counter E.M.F. et of the syn- 
chronous motor; that is, the E.M.F. induced in the motor arma- 
ture by its rotation through the (resultant) magnetic field. 

Let i = current in the circuit (effective values). 

The mechanical power delivered by the synchronous 
motor (including friction and core loss) is the electric 
power consumed by the C. E.M.F. e1; hence — 

p = *>! cos ft,^), (1) 

thus, — 


* If f0 = E.M.F. at motor terminals, z = internal impedance of the 
motor; if eo= terminal voltage of the generator, z = total impedance of line 
and motor; if t0= 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. 


SYNCHRONOUS MOTOR. 339 

The displacement of phase between current i and E.M.F. 
= z i consumed by the impedance z is : 


cos (ie) = - 


sin (/<?) 


x 


(3) 


Since the three E.M.Fs. acting in the closed circuit : 

e0 = E.M.F. of generator, 

fi = C.E.M.F. of synchronous motor, 

e = zi = E.M.F. consumed by impedance, 

form a triangle, that is, c^ and e are components of ^0, it is 
(Fig. 152) : 

e1 „ 2 eZ .1 „?. ^2 ,'2 

hence, cos (*,.#) = •*- — — = -0 — - — . (5) 

2 e^e '2,zie^ 

since, however, by diagram : 

cos (el , e) = cos (/, e — /', e^) 

= cos (/, e) cos (/, ^i) + sin (t, e) sin (/, ^) (6) 

substitution of (2), (3) and (5) in (6) gives, after some trans- 
position : 

the Fundamental Equation of tJie Synchronous Motor, relat- 
ing impressed E.M.F., <?0 ; C. E.M.F., ^ ; current z; power, 
/, and resistance, r ; reactance, x ; impedance s. 

This equation shows that, at given impressed E.M.F. e$f 
and given impedance s = Vr2 + x*, three variables are left, 
ev i,p, of which two are independent. Hence, at given ^ 
and s, the current i is not determined by the load / only, 
but also by the excitation, and thus the same current i can 
represent widely different loads p, according to the excita- 
tion ; and with the same load, the current i can be varied 
in a wide range, by varying the field excitation e1. 

The meaning of equation (7) is made more perspicuous 


340 ALTERNATING-CURRENT PHENOMENA. 


by some transformations, which separate ev and i, as func- 
tion of/ and of an angular parameter <£. 
Substituting in (7) the new coordinates : 


V2 


V2 


or, 


_ 

V2 


we get 


substituting again, e<f = a 
Izp = b 

r = €Z 
hence, x = z Vl — e2 


jr. 753. 


we jret 


a — a V2 — e b = V(l — e2) (2 a2 — 2 £2 - 
and, squared, 

substituting 

gives, after some transposition, 

v* -f ze/2 = (-1 ~ *") a (a — 2 tb\ 


(9) 


)» (11) 

— 0, (12) 

(13) 
(14) 


SYNCHRONOUS MOTOR. 341 

hence'if 


i* + w* = £* (16) 

the equation of a circle with radius R. 

Substituting now backwards, we get, with some trans- 
positions : 

{r* (ef + z*i2) - z* (Vo2 - 2 r/)}2 + {r x (e? - z*i2)}2 = 

*2.sV(^02-4r/) (17) 

the Fundamental Rquation of the Synchronous Motor in a 
modified form. 

The separation of e± and i can be effected by the intro- 
duction of a parameter <£ by the equations : 

r3- (e? — z2 /2) - z2 (ef — 2rp)=xze() V<r0a — ±rp cos <£ 


rx (e? - z2/2) =xze» Vtf - 4 r/ sin ' l ' 
These equations (18), transposed, give 

N 

+ sin</> 


The parameter <^> has no direct physical meaning, appar- 
ently. 

These equations (19) and (20), by giving the values ef 
el and i as functions of / and the parameter <£ enable us 
to construct the Power Characteristics of the Synchronous 
Motor, as the curves relating ev and i, for a given power /, 
by attributing to <£ all different values. 


342 ALTERNATING-CURRENT PHENOMENA. 

Since the variables v and w in the equation of the circle 
(16) are quadratic functions of e1 and /', the Power Charac- 
teristics 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 transmission even including the line, etc. 

Before discussing further these Power Characteristics, 
some special conditions may be considered. 

206. A. Maximum Output. 

Since the expression of el and i [equations (19) and 
(20)] contain the square root, W02 — 4 rp, 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, 

tf _ 4 rp = 0, 

°r> 

/ = £.. (21) 

This is the same value which represents the maximum 
power transmissible by E.M.F., eQ, 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, 


and the displacement of phase in the synchronous motor. 

cor(A,0-^--i 

tc± z 

hence, 

tan fa, /) = -?, (23) 


SYNCHRONOUS MOTOR. 343 

that is, the angle of internal displacement in the synchron- 
ous motor i§ equal, but opposite to, the angle of displace- 
ment of line impedance, 

('i, 0 = - (', 0, 

= ~ <X '), (24) 

and consequently, 

(.-0,0=0; (25) 

that is, the current, z, is in phase with the impressed 
E.M.F., *0. 

If 2 < 2 r, el < <?0; that is, motor E.M.F. < generator E.M.F. 

If z = 2 r, el = e0 ; that is, motor E.M.F. = generator E.M.F. 

If z > 2 r, <?! > r0; that is, motor E.M.F. > generator E.M.F. 

In either case, the current in the synchronous motor is 
leading. 

207. B. Running Light, p = 0. 

When running light, or for / = 0, we get, by substitut- 
ing in (19) and (20), 


(26) 


Obviously this condition cannot well be fulfilled, since p 
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 / = 0 into equation (7) gives, after squar- 
ing and transposing, 

e* + e<* 4- 3*,-« - 2 ^V - 2 22rV + 2 ra*'V - 2 *2*V = 0. (27) 

This quartic equation can be resolved into the product 
of two quadratic equations, 

0. | (28) 

0. j 


344 ALTERNATING-CURRENT PHENOMENA. 

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., eit is ^ = 0 at / = ^2. (29) 
The minimum value of current, z, is / = 0 at et = e0 . (30) 
The maximum value of E.M.F., elt is given by Equation (28)', 

/= e* + 22z2 -e<?±2 xiel = 0 ; 
by the condition, 


hence, 


The maximum value of current, z, is given by equation 
(28) by 

— = 0, as 
del 

(32) 

If, as abscissas, elt and as ordinates, zi, are chosen, the 
axis of these ellipses pass through the points of maximum 
power given by equation (22). 

It is obvious thus, that in the V-shaped curves of syn- 
chronous motors running light, the two sides of the curves 
are not straight lines, as usually assumed, but arcs of ellipses, 
the one of concave, the other of convex, curvature. 

These two ellipses are shown in Fig. 154, and divide the 
whole space into six parts — the two parts A and A', whose 
areas contain the quartic curves (19) (20) of synchronous 
motor, the two parts B and B', whose areas contain the 
quartic curves of generator, and the interior space C and 
exterior space D, whose points do not represent any actual 
condition of the alternator circuit, but make el , i imaginary. 

A and A' and the same B and B' ', are identical condi- 
tions of the alternator circuit, differing merely by a simul- 


SYNCHRONOUS MOTOR. 


345 


\ 


r 


\ 


I 


\ 


4000 3000 \^ 2000 1000 


Volts 1000 2000\/3000 4000 5000 


\ 


/A' 


\ 


\ 


Fig. 154. 

taneous reversal of current and E.M.F. ; that is, differing 
by the time of a half period. 

Each of the spaces A and B contains one point of equa- 
tion (22), representing the condition of maximum output 
of generator, viz., synchronous motor. 

208. C. Minimum Current at Given Power. 

The condition of minimum current, t, at given power, /, 
is determined by the absence of a phase displacement at the 
impressed E.M.F. eQ, 


346 AL TERNA TING-CURRENT PHENOMENA. 

This gives from diagram Fig. 153, 

e1* = e(? + i*z*-2ie0r, (33) 

or, transposed, 

This quadratic curve passes through the point of zero 
current and zero power, 

through the point of maximum power (22), 


and through the point of maximum current and zero power, 


enx 


r 


(35) 


and divides each of the quartic curves or power character- 
istics into two sections, one with leading, the 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, 

/ = 'J2, (36) 

while at no-load, the current can reach the maximum value, 
/=^, (35) 

the same value as would exist in a non-inductive circuit of 
the same resistance. 

The minimum value at C.E.M.F. el} at which coincidence 


SYNCHRONOUS MOTOR. 347 

of phase (eQ , -i) = 0, can still be reached, is determined from 
equation (34) by, 


as 

i — e - — - (37} 

The curve of no-displacement, or of minimum current, is 
shown in Figs. 138 and 139 in dotted lines.* 

209. D. Maximum Displacement of Phase. 

(e%, i} = maximum. 
At a given power/ the input is, 

A =P + i*r = e,i cos (*0, *) ; (38) 

hence, 

cosfo, 0 = /+/V. (39) 

At a given power /, this value, as function of the current 
i, is a maximum when 

d_(p + 

di\ 
this gives, 


(40) 
or, 

(41) 

That is, the displacement of phase, lead or lag, is a 
maximum, when the power of the motor equals the power 

* It is interesting to note that the equation (34) is similar to the value, 
<?! = \/(^0 — 2 r)2 — z'2jr2, which represents the output transmitted over an 
inductive line of impedance, z = vV2 + jr2 into a non-inductive circuit. 

Equation (34) is identical with the equation giving the maximum voltage, 
e± , at current, i, which can be produced by shunting the receiving circuit with a 

condenser; that is, the condition of " complete resonance " of the line, z = 

x 
Vr'2 + x'2, with current, ». Hence, referring to equation (35), el = t0 ~ is 

the maximum resonance voltage of the line, reached when closed by a con- 
denser of reactance, — x. 


348 


ALTERNATING-CURRENT PHENOMENA. 


consumed by the resistance ; that is, at the electrical effi- 
ciency of 50 per cent. 

Substituting (40) in equation (7) gives, after squaring 


/ N 


TSOO 8000^ #WU 3000 3uOO 
Fig. 155. 

and transposing, the Ouartic Equation of Maximum Dis- 
placement, 

<>02 - e*y + **z2 (s2 + 8 r2) + 2 j*e* (5 r2 - 22) - 2 / V 

(32 + 3 ^ = Oi (42) 

The curve of maximum displacement is shown in dash- 
dotted lines in Figs. 154 and 155. It passes through the 


SYNCHRONOUS MOTOR. 349 

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. 

210. E. Constant Counter E.M.F. 

At constant C.E.M.F., el = constant, 

If 

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 


the current is lagging at no load ; with increasing load the 
lag decreases, the current comes into coincidence of phase 
with eQ , then becomes leading, reaches a maximum lead ; 
then the lead decreases again, the current comes again into 
coincidence of phase, and becomes lagging, until the motor 
falls out of step. 

If eQ < <?! , the current is leading at no load, and the 
lead first increases, reaches a maximum, then decreases ; 
and whether the current ever comes into coincidence of 
phase, and then becomes lagging, or whether the motor 
falls out of step while the current is still leading, depends, 
whether the C.E.M.F. at the point of maximum output is 
> <?0 or < *0. 

211. F. Numerical Instance. 

Figs. 154 and 155 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. 


350 ALTERNATING-CURRENT PHENOMENA. 

In this case we have, 

<?0 = 2500 volts constant at generator terminals; ^| 
r — 10 ohms, including line and motor ; /^gs 

x = 20 ohms, including line and motor ; j 

hence z = 22.36 ohms. 

Substituting these values, we get, 

25002 - e* - 500 i* - 20 / = 40 V*V -/2 (7) 

{^2 + 500 ?2 - 31.25 X 106 + 100 /}2 + (2 ^2 - 1000 /2}2 = 

7.8125 x 1015 - 5 + 109/. (17) 

el = 5590 (19) 

V| {(1 — 3.2 x 10~6/) + (.894 cos <£+ .447sin <£) Vl-6.4xlO-6/}. 
* = 559 (20) 


— 6.4xlO-6/}. 

Maximum output, 

p = 156.25 kilowatts (21) 

at *i = 2,795 volts 

i = 125 amperes 
Running light, 

^ + 500 /a - 6.25 x 104 =p 40 /^ = 0 
^ = 20 / ± V6.25 X 104 — 100 i* 

At the" minimum value of C.E.M.F. e1 = 0 is / = 112 (29) 
At the minimum value of current, / = 0 is el = 2500 (30) 
At the maximum value of C.E.M.F. ev = 5590 is / = 223.5 (31) 
At the maximum value of current i — 250 is el = 5000 (32) 

Curve of zero displacement of phase, 


€l = 10 V(250 - O2 + 4 *a (34) 

= 10 V6.25 x 104 — 500 / + 5 / 2 
Minimum C.E.M.F. point of this curve, 

/ = 50 ^ = 2240 (35) 

Curve of maximum displacement of phase, 

/ = 10 *'2 (40) 

(6.25 X 106-^2)2 + .65 X 106 /« - 1010/2 = 0. (42) 


SYNCHRONOUS MOTOR. 351 

Fig. 154 gives the two ellipses of zero power, in drawn 
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. 155 gives the motor-power characteristics, for, 

/ = 10 kilowatts. 
p = 50 kilowatts. 
/ = 100 kilowatts. 
p = 150 kilowatts. 
p = 156.25 kilowatts. 

together with the curves of zero displacement, and of maxi- 
mum displacement. 

212. G. Discussion of Results. 

The characteristic curves of the synchronous motor, as 
shown in Fig. 155, 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 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 towards lower, then towards higher, 
values of C.E.M.F. el. 

The upper parts of the curves, however, I have never 
been able to observe experimentally, and consider it as 
probable that they correspond to a condition of synchro- 
nous motor-running, which is unstable. The experimental 
observations usually extend about over that part of the 
curves of Fig. 155 which is reproduced in Fig. 156, 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 char- 
acteristics of the synchronous motor in Fig. 155 can be con- 
sidered as approximations only, since a number of assump- 


352 


ALTERNA TING-CURRENT PHENOMENA. 


tions are made which are not, or only partly, fulfilled in 
practice. The foremost of these are : • 

1. It is assumed that el can be varied unrestrictedly, 
while in reality the possible increase of el is limited by 
magnetic saturation. Thus in Fig. 155, at an impressed 
E.M.F., eQ = 2,500 volts, el rises up to 5,590 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. 


Fig. 156. 

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 /, corresponding to higher values of elt and 
thereby bends the curves upwards at a lower value of ^ 
than represented in Fig. 155. 

It must be understood that the motor reactance is not 
a simple quantity, but represents the combined effect of 


SYNCHRONOUS MOTOR. 353 

self-induction, that is, the E.M.F. induced in the armature 
conductor by the current flowing therein and armature 
reaction, or the variation of the C. E.M.F. of the motor 
by the change of the resultant field, due to the superposi- 
tion of the M.M.F. of the armature current upon the field 
excitation ; that is, it is the " synchronous reactance." 

3. These curves in Fig. 155 represent the conditions 
of constant electric power of the motor, thus including the 
mechanical and the magnetic friction (core loss). While 
the mechanical friction can be considered as approximately 
constant, the magnetic friction is not, but increases with 
the magnetic induction ; that is, with elf and the same holds 
for the power consumed for field excitation. 

Hence the useful mechanical output of the motor will 
on the same curve, / = const., be larger at points of lower 
C.E.M.F., elt than at points of higher e^\ and if the curves 
are plotted for constant useful mechanical output, the whole 
system of curves will be shifted somewhat towards lower 
values of ^ ; 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-character- 
istics of the synchronous motor can be determined only 
in the case of the particular conditions of the installation 
under consideration. 


354 AL TERN A TING-CURRENT PHENOMENA,