CHAPTER XVII. 

SYNCHBONIZINO AIiTEBKATOBS. 

168. All alternators, when brought to synchronism with 
each other, will operate in parallel more or less satisfactorily. 
This is due to the reversibility of the alternating-current 
machine ; that is, its ability to operate as synchronous motor. 
In consequence thereof, if the driving power of one of sev- 
eral parallel-operating generators is withdrawn, this gene- 
rator will keep revolving in synchronism as a synchronous 
motor ; and the power with which it tends to remain in 
synchronism is the maximum power which it can furnish 
as synchronous motor under the conditions of running. 

169. The principal and foremost condition of parallel 
operation of alternators is equality of frequency ; that is, 
the transmission of power from the prime movers to the 
alternators must be such as to allow them to run at the 
same frequency without slippage or excessive strains on 
the belts or transmission devices. 

Rigid mechanical connection of the alternators cannot be 
considered as synchronizing ; since it allows no flexibility or 
phase adjustment between the alternators, but makes them 
essentially one machine. If connected in parallel, a differ- 
ence in the field excitation, and thus the induced E.M.F*. of 
the machines, must cause large cross-current ; since it cannot 
be taken care of by phase adjustment of the machines. 

Thus rigid mechanical connection is not desirable for 
parallel operation of alternators. 

170. The second important condition of parallel opera- 
tion is uniformity of speed ; that is, constancy of frequency. 


§171] SYNCHRONIZING ALTERNATORS, 249 

If, for instance, two alternators are driven by independent 
single-cylinder engines, and the cranks of the engines hap- 
pen to be crossed, the one engine will pull, while the other 
is near the dead-point, and conversely. Consequently, alter- 
nately the one alternator will tend to speed up and the 
other slow down, then the other speed up and the first 
slow down. This effect, if not taken care of by fly-wheel 
capacity, causes a "hunting" or pumping action; that is, a 
fluctuation of the lights with the period of the engine revo- 
lution, due to the alternating transfer of the load from one 
engine to the other, which may even become so excessive 
as to throw the machines out of step. This difficulty does 
not exist with turbine or water-wheel driving. 

171. In synchronizing alternators, we have to distin- 
guish the phenomenon taking place when throwing the 
machines in parallel or out of parallel, and the phenome- 
non when running in synchronism. 

When connecting alternators in parallel, they are first 
brought approximately to the same frequency and same 
voltage ; and then, at the moment of approximate equality 
of phase, as shown by a phase-lamp or other device, they 
are thrown in parallel. 

Equality of voltage is much less important with modern 
alternators than equality of frequency, and equality of phase 
is usually of importance only in avoiding an instantaneous 
flickering of the lights on the system. When two alter- 
nators are thrown together, currents pass between the 
machines, which accelerate the one and retard the other 
machine until equal frequency and proper phase relation 
are reached. 

With modern ironclad alternators, this interchange of 
mechanical power is usually, even without very careful 
adjustment before synchronizing, sufficiently limited not 
to endanger the machines mechanically ; since the cross- 
currents, and thus the interchange of power, are limited 
by self-induction and armature reaction. 


250 AL TERNA TING-CURRENT PHENOMENA, [§ 172, IT 3 

In machines of very low armature reaction, that is, 
machines of " very good constant potential regulation,** 
much greater care has to be exerted in the adjustment 
to equality of frequency, voltage, and phase, or the inter- 
change of current may become so large as to destroy the 
machine by the mechanical shock ; and sometimes the 
machines are so sensitive in this respect that it is prefer- 
able not to operate them in parallel. The same applies 
in getting out of step. 

172. When running in synchronism, nearly all types 
of machines will operate satisfactorily ; a medium amount 
of armature reaction is preferable, however, such as is given 
by modern alternators — not too high to reduce the 
synchronizing power too much, nor too low to make the 
machine unsafe in case of accident, such as falling out of 
step, etc. 

If the armature reaction is very low, an accident, — such 
as a short circuit, falling out of step, opening of the field 
circuit, etc., — may destroy the machine. If the armature 
reaction is very high, the driving-power has to be adjusted 
very carefully to constancy ; since the synchronizing power 
of the alternators is too weak to hold them in step, and 
carry them over irregularities of the driving-power. 

173. Series operation of alternators is possible only by 
rigid mechanical connection, or by some means whereby 
the machines, with regard to their synchronizing power, 
act essentially in parallel ; as, for instance, by the arrange- 
ment shown in Fig. 120, where the two alternators, .-i,, W.^, 
are connected in series, but interlinked by the two coils 
of a large transformer, 7", of which the one is connected 
across the terminals of one alternator, and the other across 
the terminals of the other alternator in such a way that, 
when operating in series, the coils of the transformer will 
be without current. In this case, by interchange of power 


§ 174] 


SYNCIiROxWIZING ALTERNATORS. 


251 


through the transformers, the series connection will be 
maintained stable. 


yjUUlMW 


/ » V K9 9 999n9 


3 


r 


Ftg. 120. 


174. In two parallel operating alternators, as shown in 
Fig. 121, let the voltage at the common bus bars be assumed 


Fig, 121. 

as zero line, or real axis of coordinates of the complex 
method ; and let — 


252 AL TERN A TING-CURRENT PHENOMENA, [§ 1 74 

e = difference of potential at the common bus bars of 

the two alternators, 
Z ^= r — jx = impedance of external circuit, 
K=s^»--|-y^ = admittance of external circuit; 

hence, the current in external circuit is 

/- —JX 

Let 

£i = fi —/<?/ = ^a (cos wj — jsm wi) = induced E.M.F. of first 
machine ; 

£^ = <f2 — yVj' = a^ (cos «2 — y sin wa) =*: induced E.M.F. of sec- 
ond machine ; 

/, = t\ +jtY = current of first machine ; 

/j = /j +7/2' = current of second machine ; 

Zi = ri —jxi = internal impedance, and I'l =^i + J^\ = inter- 
nal admittance, of first machine ; 

Z2 = ra —jxn = internal impedance, and Fa =^2 +7^2 = inter- 
nal admittance, of second machine. 

Then, 

, 2 I ^. ' 2 ., 2 . 

^2 ^ '2 — <^2 > 

^1 = <f + /iZi , or ^-1 — yVi'= (^ + Ari + //jfi) — y (/'i.Ti — //ri) ; 
^2 = ^ + /3Z2, or ^2 — yV= (^ + ^*2'2 + 'V-^s) —J{hX'i — /'a'/'s) ; 

/ = /i + /a , or ^^ +yV^ = (/i + /a) +y (// + i/). 

This gives the equations — 

<fi = ^ + /i ri + // Jfi ; 

^j =^+/«r2 + i2X2\ 

Cx — t\Xx — h^\\ 

^a — '2*^2 — '2 ''a > 

^<?^ = A + h ; 
^^ = fV + ^2' ; 

V + ^'2"'' = ^^2^ 

or eight equations with nine variables: e^, e^, e^, r./, /\, /'i', 
^2 » '2 » ^' 


§174] SYNCHRONIZING ALTERNATORS, »25S 

Combining these equations by twos, 

C\rx + e{xx = ery^ + iy^z^ ; 
^jTj + ^2' ^2 == er^ + '2^2^; 
substituted in 

we have 

^\g\ + ^i'^ + ^2^8 + ^«'^« = ^(^1 + ^» + ^) ; 

and analogously, 

^xh — ^iVi + ^s^a — ^2^2 = ^ (^1 + ^2 + ^) ; 
dividing, 

^+;ri +^2 ^ e\K \ + <^2^2 + ^/ h + ^a' ^2 . 

^ + ^l + ^2 ^1 ^1 + ^2 ^2 — ^\ g\ — ^2 g2 * 

substituting 

g ■= V cos a ^1 = <Zi cos Wi ^2 = ^2 cos a>2 

b •=^ V sin a e( = </i sin a>i e^ = 02 sin (02 
gives 

^ + ^ 1 + ^ 2 ^ <7x g/| COS (ai — ctf|) + <72 7'2 cos (ag — a»2) 

^ + ^1 + ^2 ''i ^1 sin (tti — a>i) + <i2^2 sin (02 — W2) 

as the equation between the phase displacement angles wj 
and Wj in parallel operation. 

The power supplied to the external circuit is 

of which that supplied by the first machine is, 

P\ = ^h ; 
by the second machine, 

pi = ik . 
The total electrical work done by both machines is, 

^=7^1 + ^, 
of which that done by the first machine is, 

Py = ^i /i — ^1 t\ ; 
by the second machine, 

1 2 "^^ ^2 '2 ^~ ^2 '2 • 


254 : ALTERNATING-CURREXT PHEXOMEXA. [§176 

The difference of output of the two machines is, 
denoting 

0)1 4" <«>• *^\ Wg « 

— ' •= ^ = 0. 

•> •> 

A //AS may be called the synchronizing power of the 
machines, or the power which is transferred from one ma- 
chine to the other by a change of the relative phase angle. 

175. Special Case. — Tivo equal alternators of equal 

excitation, 

(j^ = ao = a 

Z,\ = i&2 ^^ ^0 • 

Substituting this in the eight initial equations, these 
assume the form, — 

e^ =e + i^f\, + /,\r„ 
f • • • , 

^2 '2 ^U — '3 '0 • 

eh = // + // 
Combining these equations by twos, 

^I' + ^s' = ^(^0^^— ''o^); 

substituting e^ = a cos a»i 

^•/ = f7 sin «! 
^*2 = <7 cos <i>2 
e^ = (1 sin <i>2, 

we have a (cos W| + cos wo) = ^' (2 + Tq^" + x^ b) 
a (sin 0*1 + sin «2) = € (x^g — ro^) ; 

expanding and substituting — 

(111 -(- a>2 
*> 


^ (Ui — a>2 


§175] SYNCHRONIZING ALTERNATORS, 265 

or a COS € cos 8 = ^(l + ''jiSAj^\ 

a sm c cos o = ^ ~ ^^^~7> ' 

hence tan c = "^Q*^ "~ ^Q = constant. 

That is wi + aij = constant; 

and cos 8 = fy/^l + ^oZii^V + ^-»Xr:i<^J; 

rtr cos 8 
or, <? = 


at no-phase displacement between the alternators, or, 

8 = ^' - ^^ = ; 
2 

we have e = 


s 


From the eight initial equations we get, by combina- 
tion N / O »N 

^i''o + 'i ^0 = <? ('o + ^o) + h ('o + ^oV 

subtracted and expanded — 

/ _ r^(f\ — ej) + Xs,{e{ — eO . 

or, since 

^j — <f J = rtT (cos w, — cos wj) = — 2 d^ sin c sin 8 
e( — c^ = a (sin u>i — sin 0)2)= 2 <? cos c sin 8 ; 

we have 

2 a sin 8 ^ • 1 

/i — /j = r — {Xo cos € — r^, sm c} 

= 2 /i^'o sin 8 cos (c + a), 
where 

tan a = ^ . 


266 ALTERNATING-CURJ^ENT PHENOMENA, [§ 176 

The difference of output of the two alternators is 

^P =/i —pi = ^ O'l — h) ; 
hence, substituting, 

A/ = {xo cos c — /'o sin c }; 

substituting, 

a cos 8 


^ = — ■ 


sm € = 


^0^ — ^0^^ 

2 


v/(i + '-^^^^^ J+ ("«-^----'^«-'' J 


cos € = 


we have, 

expanding, 

a«sin2s|xo+^| 


A/ = 


2» 


or 


A / = — — , 

a»sin28|^„+|l 

. ^^_ 2.^cos2a|.o + || 

Hence, the transfer of power between the alternators, 
A/, is a maximum, if 8 = 45'' ; or w, — w^ = 90° ; that is, 
when the alternators are in quadrature. 


§176] S YNCHRONIZING A L TERN A TORS, 257 ' 

The synchronizing power, A// A 8, is a maximum if 
3=0; that is, the alternators are in phase with each other. 

176. As an instance, curves may be plotted 
for, 

a =2500, 

Zo --= ri -jx^ = 1 - lOy ; or Y^ = g^ +jb^ = .01 + .1/ 

with the angle 8 = ^^ ^ ^' as abscissae, giving 

the value of terminal voltage, e ; 

the value of current in the external circuit, i ^ ey\ 

the value of interchange of current between the alternators, 

i\ — k ; 
the value of interchange of power between the alternators, A / 

the value of synchronizing power, — ^ , in dash-dot line, Curve V. 

A 

For the condition of external circuit, 


g= 0, 


/>= 0, 


,1' = 0, 


.05, 


0, 


.05, 


.08, 


0, 


.08, 


.03, 


+ .04, 


.05, 


.03, 


-.04, 


.05. 


258 ' ALTERNATING-CURRENT PHENOMENA, [§177