CHAPTER XIV. 

SHORT-CIRCUIT CURRENTS OF ALTERNATORS. 

112. The short-circuit current of an alternator is limited by 
armature reaction and armature self-inductance; that is, the 
current in the armature represents a m.m.f. which with lagging 
current, as at short circuit, is demagnetizing or opposing the 
impressed m.m.f. of field excitation, and by combining therewith 
to a resultant m.m.f. reduces the magnetic flux from that corre- 
sponding to the field excitation to that corresponding to the 
resultant of field excitation and armature reaction, and thus 
reduces the generated e.m.f. from the nominal generated e.m.f., 
eOJ to the virtual generated e.m.f., er The armature current 
also produces a local magnetic flux in the armature iron and pole- 
faces which does not interlink with the field coils, but is a true 
self-inductive flux, and therefore is represented by a reactance xr 
Combined with the effective resistance, rv of the armature 
winding, this gives the self-inductive impedance Zl = rl — 


or zt = Vr* + x*. Vectorially subtracted from the virtual 
generated e.m.f., ev the voltage consumed by the armature 
current in the self-inductive impedance Zl then gives the ter- 
minal voltage, e. 

At short circuit, the virtual generated e.m.f., ev is consumed 
by the armature self-inductive impedance, zr As the effective 
armature resistance, rv is very small compared with its self- 
inductive reactance, xv it can be neglected compared thereto, 
and the short-circuit current of the alternator, in permanent 
condition, thus is 


As shown in Chapter XXII, "Theory and Calculation of 
Alternating Current Phenomena," the armature reaction can be 
represented by an equivalent, or effective reactance, z2, and the 
self-inductive reactance, xv and the effective reactance of 

199 


200 TRANSIENT PHENOMENA 

armature reaction, x2J combine to form the synchronous react- 
ance, XQ = xl + x2, and the short-circuit current of the alterna- 
tor, in permanent condition, therefore can be expressed by 


where e0 = nominal generated e.m.f. 

113. The effective reactance of armature reaction, xv differs, 
however, essentially from the true self-inductive reactance, xv 
in that xl is instantaneous in its action, while the effective 
reactance of armature reaction, xv requires an appreciable time 
to develop: x2 represents the change of the magnetic field flux 
produced by the armature m.m.f. The field flux, however, can- 
not change instantaneously, as it interlinks with the field exciting 
coil, and any change of the field flux generates an e.m.f. in the 
field coils, changing the field current so as to retard the change 
of the field flux. Hence, at the first moment after a change of 
armature current, the current change meets only the reactance, 
xv but not the reactance x2. Thus, when suddenly short-cir- 
cuiting an alternator from open circuit, in the moment before 
the short circuit, the field flux is that corresponding to the 
impressed m.m.f. of field excitation and the voltage in the arma- 
ture, i.e., the nominal generated e.m.f., e0 (corrected for mag- 
netic saturation). At the moment of short circuit, a counter 
m.m.f., that of the armature reaction of the short-circuit 
current, is opposed to the impressed m.m.f. of the field excitation, 
and the magnetic flux, therefore, begins to decrease at such a 
rate that the e.m.f. generated in the field coils by the decrease 
of field flux increases the field current and therewith the m.m.f. 
so that when combined with the armature reaction it gives a 
resultant rn.m.f. producing the instantaneous value of field flux. 
Immediately after short circuit, while the field flux still has full 
value, that is, before it has appreciably decreased, the field m.m.f. 
thus must have increased by a value equal to the counter m.m.f. 
of armature reaction. As the field is still practically unchanged, 
the generated e.m.f. is the nominal generated voltage, e0, and 
the short-circuit current is 


SHORT-CIRCUIT CURRENTS OF ALTERNATORS 


201 


and from this value gradually dies down, with a decrease of the 
field flux and of the generated e.m.f., to 


Hence, approximately, when short-circuiting an alternator, 
in the first moment the short-circuit current is 


x' 


while the field current has increased from its normal value i0 to 
the value 

Field excitation + Armature reaction m 

Field excitation 
gradually the armature current decreases to 


and the field current again to the normal value i0. 

Therefore, the momentary short-circuit current of an alternator 
bears to the permanent short-circuit current the ratio 


,vo 


that is, 


Armature self-inductance + Armature reaction 
Armature self-inductance 


In machines of high self-inductance and low armature reaction, 
as uni-tooth high frecfUency alternators, this increase of the 
momentary short-circuit current over the permanent short- 
circuit current is moderate, but may reach enormous values in 
machines of low self-inductance and high armature reaction, as 
large low frequency turbo alternators. 

114. Superimposed upon this transient term, resulting from 
the gradual adjustment of the field flux to a change of m.m.f., is 
the transient term of armature reaction. In a polyphase 
alternator, the resultant m.m.f. of the .armature in permanent 
conditions is constant in intensity and revolves with regard to 
the armature at uniform synchronous speed, hence is stationary 


202 


TRANSIENT PHENOMENA 


with regard to the field. In the first moment, however, the 
resultant armature m.m.f . is changing in intensity and in velocity, 
approaching its constant value by a series of oscillations, as 
discussed in Chapter XIII. Hence, with regard to the field, the 
transient term of armature reaction is pulsating in intensity and 
oscillating in position, and therefore generates in the field coils 


Field Current 


Armature Current 


Fig. 50. Three-phase short-circuit current of a turbo-alternator. 

an e.m.f. and causes a corresponding pulsation in the field 
current and field terminal voltage, of the same frequency as 
the armature current, as shown by the oscillogram of such a 
three-phase short-circuit, in Fig. 50. This pulsation of field 
current is independent of the point in the wave, at which the 
short-circuit occurs, and dies out gradually, with the dying out 
of the transient term of the rotating m.m.f. 

In a single-phase alternator, the armature reaction is alter- 
nating with regard to the armature, hence pulsating, with double 
frequency, with regard to the field, varying between zero and its 


SHORT-CIRCUIT CURRENTS OF ALTERNATORS 203 

maximum value, and therefore generates in the field coils a 
double frequency e.m.f., producing a pulsation of field current 
of double frequency. This double-frequency pulsation of the 
field current and voltage at single-phase short-circuit is pro- 
portional to the armature current, and does not disappear 
with the disappearance of the transient term, but persists also 
after the permanent condition of short-circuit has been reached, 


Armature 
current 


"Field 
current 


' BHBnSSMMttHKattMMMNMBflUffiHMHMMHBB 
Fig. 61. Single-phase short-circuit current of a three-phase turbo-alternator. 

merely decreasing with the decrease of the armature current. 
It is shown in the oscillogram of a single-phase short-circuit on 
a three-phase alternator, Fig. 51. 

Superimposed on this double frequency pulsation is a single- 
frequency pulsation due to the transient term of the armature 
current, that is, the same as on polyphase short-circuit. With 
single-phase short-circuit, however, this normal frequency pul- 
sation of the field depends on the point of the wave at which 
the short-circuit occurs, and is zero, if the circuit is closed at 
the moment when the short-circuit current is zero, as in Fig. 51, 
and a maximum when the short-circuit starts at the maximum 
point of the current wave. As this normal frequency pulsation 
gradually disappears, it causes the successive waves of the 
double frequency pulsation to be unequal in size at the 
beginning of the transient term, and gradually become equal, 
as shown in the oscillogram, Fig. 52. 

The calculation of the transient term of the short-circuit 
current of alternators thus involves the transient term of the 


204 


TRANSIENT PHENOMENA 


armature and the field current, as determined by the self- 
inductance of armature and of field circuit, and the mutual 
inductance between the armature circuits and the field circuit, 
and the impressed or generated voltage; therefore is rather 
complicated; but a simpler approximate calculation can be 


A 


Armature 
current 


Field 
current 


62.5 amp. 


Fig. 52. Single-phase short-circuit current of a three-phase turbo-alternator. 

given by considering that the duration of the transient term is 
short compared with that of the armature reaction on the field. 

(A) Polyphase alternator. 

115. Let np = number of phases; 6 = 2nft = time-phase 
angle; n0 = number of field turns in series per pole; n^ number 
of armature turns in series per pole; Z0 = r0 — jx0 = self-inductive 
impedance of field circuit; Zl = rt — jxl = self-inductive impe- 
dance of armature circuit; p = permeance of field magnetic cir- 
cuit; a = 2 7ifn1 10~8 = induction coefficient of armature; E0 = 

Tji 

exciter voltage; / 0 = — - = field exciting current, in permanent 

*• 

condition; i0 = field exciting current at time 0; i'0° = field 
exciting current immediately after short-circuit; i = armature 

2 re- 
current at time 6, and kt = - — = transformation ratio of field 


SHORT-CIRCUIT CURRENTS OF ALTERNATORS 205 

to resultant armature. Counting the time angle 6 from the 
moment of short circuit, 6 = 0, and letting 6' = time-phase 
angle of one of the generator circuits at the moment of short 
circuit, we have, 

SF0 = n0/0 = field excitation, in permanent or stationary con- 
dition, (1) 

^o = P&o = PnJo = magnetic flux corresponding thereto, 
and 

C " aP*o = aapnJ0. (2) 

= nominal generated voltage, maximum value, at 0 = 0. 

Hence, r 


= momentary short-circuit current at time 0 — 0, and 


= resultant armature reaction thereof. 

Assume this armature reaction as opposite to the field excita- 
tion, 

37 - Vo°; (5) 

as is the case at short circuit. 

The resultant m.m.f. of the magnetic circuit at the moment 
of short-circuit is 

£F0 - £F0° - JFi0. (6) 

At this moment, however, the field flux is still <£0, and the result- 
ant m.m.f. is given by (1) as 

•!F0 = JF0 = n0/0. (7) 

Substituting (4), (5), (7) in (6) gives 


hence, ° 


, 

1 

0 = - 


206 TRANSIENT PHENOMENA 


Writing *2=«, (9) 


we have i0° = --Z-J70; (10) 

that is, at the moment of short circuit the field exciting current 
rises from 70 to i°, and then gradually dies down again to 70 at 

r° Q 

a rate depending on the field impedance Z0, that is, by e ^ , as 
discussed in preceding chapters. Hence, it can be represented 

by 


x, 

The resultant armature m.m.f., or armature reaction, is 

npnj° 

~2~ 

thus the magnetic flux which would be produced by it is 

pnpnj° - 

~T~' 

and therefore the voltage generated by this flux is 

apnpnj° 
~2~ 

hence, 


Voltage corresponding to the m.m.f. of armature current. 
Armature current 

that is, x2 is the equivalent or effective reactance of armature 
reaction. 

In equations (10) and (11) the external self-inductance of the 
field circuit, that is, the reactance of the field circuit outside of 
the machine field winding, has been neglected. This would 


SHORT-CIRCUIT CURRENTS OF ALTERNATORS 


207 


introduce a negative transient term in (11), thus giving equation 
(11) the approximate form 


0-0 _ 


(12) 


where #3 = self-inductive reactance of the field circuit outside 
of alternator field coils. 

The more complete expression requires consideration when 
x3 is very large, as when an external reactive coil is inserted in 
the field circuit. 

In reality, x2 is a mutual inductive reactance, and x3 can be 
represented approximately by a corresponding increase of xr 

116. If / = maximum value of armature current, we have 


hence, 


and 


and 


npnj 

— — — = armature m.m.f., 


. npnj 

00 o 

= resultant m.m.f., 
E = ap& 

= e.m.f. maximum generated thereby, 


(13) 


(14) 


= armature current, maximum. 

Substituting (13) in (14) gives 

npapn 
xj = 


and 


or, by (9), 


n^apn^ x^ + x2 
2 " ' 


I=*n-*i, 


(15) 


(16) 


208 


TRANSIENT PHENOMENA 


where — — = kt = transformation ratio of field turns to 


n 


resultant armature turns; hence, 

T — I"! 
i — h,ti( 


, + 


(17) 


Substituting (11) in (17) thus gives the maximum value of the 
armature current as 


•TO 


9xt + x2 xl 

the instantaneous value of the armature current as 


(18) 


i = ktl{ 


cos (6 -.*)>-« ** cos V , (19) 


X. (X. + £,) 

1 v 1 ' 2' 

and by equation (10) of Chapter XIII, the armature reaction as 


f 


-*9\ 

. + xjs x° )( . -^e n) 
? ?-r L < I - e *' cos 0 f , 


(20) 


where x^ + x2 = x0 is the synchronous reactance of the alter- 
nator. 

For 6 = oo, or in permanent condition, equations (18), (19), 
(20) assume the usual form: 


and 


i = kt!0 -2 cos (0 

x 


f 

f 


(21) 


117. As an example is shown, in Fig. 53, the instantaneous 
value of the transient short-circuit current of a three-phase 
alternator, with the time angle 6 as abscissas, and for the con- 
stants: the field turns, n0 = 100; the normal field current, 
70 = 200 amp.; the field impedance, Z0 = rQ - jx0 = 1.28 - 
160 j ohms; the armature turns, n^ = 25, and the armature 


SHORT-CIRCUIT CURRENTS OF ALTERNATORS 209 

impedance, Zl = rl - jx^ - 0.4 - 5 / ohms. For the phase 
angle, 0' = 0, the transformation ratio then is 

np nl 3 
and the equivalent impedance of armature reaction is 


= 15, 
and we have 

7= 400(1 + 3s-0'008'), (18) 

i = 400 (1 + 3 £-°-008') (cos 0 - £-°-08'), (19) 

and / = 15,000 (1 + 3 e-°-008') (1 - e'0'080 cos 0). (20) 


1600 

800 

0 

-800 
-1600 


4Q06 + 3C 


\y 


!OS 0 


Fig. 53. Short-circuit current of a three-phase alternator. 

(B) Single-phase alternator. 

118. In a single-phase alternator, or in a polyphase alternator 
with one phase only short-circuited, the armature reaction is 
pulsating. 

The m.m.f. of the armature current, 

i = 7 cos (0 - 6'), (22) 

of a single-phase alternator, is, with regard to the field, 


hence, for position angle 00 = time angle 0, or synchronous 
rotation, 

^=7(1 + cos 2 (0-0')}; (23) 


210 TRANSIENT PHENOMENA 

that is, of double frequency, with the average value, 


(24) 


pulsating between 0 and twice the average value. 

The average value (24) is the same as the value of the poly- 
phase machine, for np = 1. 

Using the same denotations as in (A), we have: 

(1) JF0 = Vo, (25) 

(26) 
Denoting the effective reactance of armature reaction thus : 

x2=^JT> (27) 

and substituting (27) in (26) we obtain 

fri° =-Vofl + cos 2 (0 - 0')} = -X/o {1 + cos 20'}; (28) 
xl xl 

hence, by (6), 

Vo = Vo° - -XU1 + cos 2 6'} 
i 

and 

t'0° = — -2/0 \\ H — — cos 2 Q' i , (29) 

and the field current, 

r° a 

X -f X £~*» ( X ) 

xl x^ + a:2 

119. If / = maximum value of armature current, 

aFt =|'/ {1 + cos 2(0 -0')} (31) 

= armature m.m.f.; 


SHORT-CIRCUIT CURRENTS OF ALTERNATORS 211 

hence, 

SF = Vo - fri (32) 

= resultant m.m.f. 
Since, however, 


(33) 
*i •*•! 

and, by (27), 


we have, by (33) 


17 = 7. (34) 

ap 2 x2 


Substituting (30), (31), and (34) into (32) gives 


7 = n 
2 x 


or, substituting, 

kt = 2 — ° = transformation ratio, (35) 

ni 

and rearranging, gives 

' - *^ "j±?l!: 

as the maximum value of the armature current. 

This is the same expression as found in (18) for the poly- 
phase machine, except that now the reactances have different 
values. 


212 


TRANSIENT PHENOMENA 


Heref rom it follows that the instantaneous value of the armature 
current is 

~ *°6) ( -10 ) 

' ! cos (0 - 6') - e ** cos 6' { , (37) 


and, by (31), the armature reaction is 


(38) 


For 0 = oo, or permanent condition, equations (30), (36), (37), 
and (38) give 


v-VU* 


cos 2 (0 - 00 1 , 


X, + X, 


cos (0 - 00, 


and 


cos2(0-00}. 


(39) 


As seen, the field current i0 is pulsating even in permanent 
condition, the more so the higher the armature reaction x2 
compared with the armature self-inductive reactance xr 

120. Choosing the same example as in Fig. 52, paragraph 
117, but assuming only one phase short-circuited, that is, a single- 
phase short circuit between two terminals, we have the effective 
armature series turns, nv = 25 V3 = 43.3; the armature impe- 
dance, Zl = rl - jx^ = 0.8 — 10 /; 6' = 0; the transformation 
ratio, kt = 4.62, and the effective reactance of armature reaction 

o 

x2 = —x0 =15; herefrom, 

7 = 555(1 + 1.5e-°-op8'), (36) 

i = 555 (1 + 1.5 £-°-008') (cos 0 - r*'080), (37) 

and / - 12,000 (1 + 1.5£-°-008fl) (1 + cos 2 0); (38) 


SHORT-CIRCUIT CURRENTS OF ALTERNATORS 213 

and the field current is 

iQ = 200 (1 + 1.5 £-°-008 ' ) (1 + 0.6 cos 2 6). (30) 

In this case, in the open-circuited phase of the machine, a 
high third harmonic voltage is generated by the double frequency 
pulsation of the field, and to some extent also appears in the 
short-circuit current. 


SECTION II 
PERIODIC TRANSIENTS 


PEKIODIC TRANSIENTS