CHAPTER XII 

FREQUENCY CONVERTER OR GENERAL ALTERNATING- 
CURRENT TRANSFORMER 

103. In general, an alternating-current transformer conafete of 
a magnetic circuit, interlinked with two electric circuits or sets 
of electric circuits, the primary circuit, in which power, sup- 
plied by the impressed voltage, is consumed, and the secondary 
circuit, in which a corresponding amount of electric power 
is produced; or in other words, power is transferred through 
space, by magnetic energy, from primary to secondary circuit. 
This power finds its mechanical equivalent in a repulsive llirusi 
acting between primary and secondary conductors. Thus, if 
the secondary is not held rigidly, with regards to the primary, 
it will be repelled and move. This repulsion is used in the 
constant-current transformer for regulating the current for 
constancy independent of the load. In the induction motor, 
this mechanical force is made use of for doing the work: the 
induction motor represents an alternating-current transformer, 
in which the secondary is mounted niovably with regards to 
the primary, in such a manner that, while set in motion, it still 
remains in the primary field of force. This requires, i hat the 
induction motor field is not constant in one direction, but that 
a magnetic field exists in every direction, in other words that 
the magnetic field successively assumes all directions, as a so- 
called rotating field. 

The induction motor and the stationary transformer thus are 
merely two applications of the same structure, the former using 
the mechanical thrust, the latter only the electrical power 
transfer, and both thus are special cases of what may be called 
the "general alternating-current transformer," in which both, 
power and mechanical motion, are utilized. 

The general alternating-current transformer thus consist* of 
a magnetic circuit interlinked with two sets of electric circuits, 
the primary and the secondary, which are mounted rotatably 
with regards to each other. It transforms between primary 
electrical and secondary electrical power, and also between 


FREQUENCY CONVERTER 177 

electrical and mechanical power. As the frequency of the re- 
volving secondary is the frequency of slip, thus differing from 
the primary, it follows, that the general alternating-current 
transformer changes not only voltages and current, but also 
frequencies, and may therefore be called "frequency converter." 
Obviously, it may also change the number of phases. 

Structurally, frequency converter and induction motor must 
contain an air gap in the magnetic circuit, to permit movability 
between primary and secondary, and thus they require a higher 
magnetizing current than the closed magnetic circuit stationary 
transformer, and this again results in general in a higher self- 
inductive impedance. Thus, the frequency converter and in- 
duction motor magnetically represent transformers of high ex- 
citing admittance and high self-inductive impedance. 

104. The mutual magnetic flux of the transformer is pro- 
duced by the resultant m.m.f. of both electric circuits. It is 
determined by the counter e.m.f., the number of turns, and the 
frequency of the electric circuit, by the equation : 

E = V2 rfnQ 10"8, 

where 

E = effective e.m.f., 
/ = frequency, 
n = number of turns, 
$ = maximum magnetic flux. 

The m.m.f. producing this flux, or the resultant m.m.f. of 
primary and secondary circuit, is determined by shape and 
magnetic characteristic of the material composing the magnetic 
circuit, and by the magnetic induction. At open secondary 
circuit, this m.m.f. is the m.m.f. of the primary current, which 
in this case is called the exciting current, and consists of a 
power component, the magnetic power current, and a reactive 
component, the magnetizing current. 

In the general alternating-current transformer, where the 
secondary is movable with regard to the primary, the rate of 
cutting of the secondary electric circuit with the mutual mag- 
netic flux is different from that of the primary. Thus, the fre- 
quencies of both circuits are different, and the generated e.m.fs. 
are not proportional to the number of turns as in the stationary 

transformer, but to the product of number of turns into frequency. 
12 


178 ELECTRICAL APPARATUS 

105. Let, in a general alternating-current transformer: 


.. secondary . tl ,. „ 

« = ratio — : - frequency, or "slip : 

primary n r 


thus, if: 


/ = primary frequency, or frequency of impressed e.m.f., 

sf = secondary frequency; 

and the e.m.f. generated per secondary turn by the mutual flux 
has to the e.m.f. generated per primary turn the ratio, «, 

s = 0 represents synchronous motion of the secondary; 

s < 0 represents motion above synchronism — driven by external 

mechanical power, as will be seen; 
8 = 1 represents standstill; 
s > 1 represents backward motion of the secondary, 

that is, motion against the mechanical force acting between 
primary and secondary (thus representing driving by external 
mechanical power). 
Let: 

n0 = number of primary turns in series per circuit; 
nx = number of secondary turns in series per circuit; 

a = = ratio of turns; 

Til 

Y = g — jb = primary exciting admittance per circuit; 

where: 

g = effective conductance; 
b = susceptance; 
Zq = r0 + jxo = internal primary self-inductive impedance 
per circuit, 

where: 

r0 = effective resistance of primary circuit; 
Xq = self-inductive reactance of primary circuit; 
Zn = n + jx\ = internal secondary self-inductive im- 
pedance per circuit at standstill, or for « = 1, 

where: 

rx = effective resistance of secondary coil; 
Xi = self-inductive reactance of secondary coil at stand- 
still, or full frequency, s = 1. 


FREQUENCY CONVERTER 179 

Since the reactance is proportional to the frequency, at the 
slip, 8, or the secondary frequency, sf, the secondary impedance 
is: 

Zi = ri + jsxi. 

Let the secondary circuit be closed by an external resistance, 
r, and an external reactance, and denote the latter by x at 
frequency, /, then at frequency, «/, or slip, s, it will be = *x, and 
thus: 

Z = r + jsx = external secondary impedance.1 
Let: 

#o = primary impressed e.m.f. per circuit, 
J$' = e.m.f. consumed by primary counter e.m.f., 
#i = secondary terminal e.m.f., 
#\ = secondary generated e.m.f., 
e = e.m.f. generated per turn by the mutual magnetic 
flux, at full frequency, /, 
/o = primary current, 
/oo = primary exciting current, 
/i = secondary current. 

It is then: 

Secondary generated e.m.f. : 

#'i = sriie. 

Total secondary impedance: 

Zi + Z = (n + r) +js(xi + x); 

hence, secondary current: 

T E\ _ snie 

/i ~ v T 7z - 


Zi + Z (n + r) + js (X! + x) 

1 This applies to the case where the secondary contains inductive react- 
ance only; or, rather, that kind of reactance which is proportional to the 
frequency. In a condenser the reactance is inversely proportional to the 
frequency, in a synchronous motor under circumstances independent of the 
frequency. Thus, in general, we have to set, x = x' -f x" + s'", where x' 
is that part of the reactance which is proportional to the frequency, x" that 
part of the reactance independent of the frequency, and x'" that part of the 
reactance which is inversely proportional to the frequency; and have thus, 

at slip, *, or frequency, */, the external secondary reactance, sx' -f x" -f 


x,n 


% 


180 ELECTRICAL APPARATUS 

Secondary terminal voltage : 

#i = #'i ~ JiZi = fiZ 

= N rt + jgxi I = sntf (r + jsx) 

1 1 0i + r) + js (xi + x) J \rx + r) + j* (xi + x) 

e.m.f. consumed by primary counter e.m.f. 

$' = n0e; 
hence, primary exciting current: 

/oo = #'Fo = no« (flf - jb). 

Component of primary current corresponding to secondary 
current, /\: 


aMCri + O+^Cxi + x)}' 
hence, total primary current: 

/o = /oo + / 0 

f 1 1 , g - jb 

lfl2 (n + r) + js(xi + x) « 

Primary impressed e.m.f. : 

$o = E' + /oZo 

I a2 (ri + r) + js (xi + x) J J 

We get thus, as the 
Equations of the General Alternating-current Transformer, of 
ratio of turns, a; and ratio of frequencies, s; with the e.m.f. 
generated per turn at full frequency, e, as parameter, the values: 

Primary impressed e.m.f. : 

ft = «oe { 1 + £ (-T-^ £L-_ + (r. + ,*„) (, - jb) } . 

Secondary terminal voltage: 
et ft n + jsxi l r + jsx 

1 (n + r) + js (xi + x) J Oi+r) +js fo + x) 

Primary current : 

io = Sttoe ^ - 2 7 , — r 7 — -7 . r + - r 

1 a2 (ri + r) + js (xi + x) s J 


FREQUENCY CONVERTER 181 

Secondary current: 

T stiie 


(ri + r) + js (xi + x) 

Therefrom, we get: 
Ratio of currents: 

r - \ I l + t ^ " #> l<ri + r) + i* (atl + *>' r 

Ratio of e.m.fs. : 

* *fl+-2 ? ■ ?? w° . % + (r° + J*>) (g - jb) 
Eo a* a2 (ri + r) + js (xi + x) 

#1 "~ * - _ rt + jsxi_ 

(ri + r) + js (xi + x) 

Total apparent primary impedance: 

Z« = ^0 = ?8{(ri + r)+i*(ae1+*)} 

/ n # 

1 + -it -L ?-^~w v— n + (ro + J*o) (ff - j"6) 
o^ (r, + r) + js (ji + x) > f 

1 + v (9 ~ Jb) l(ri + r) + js (*, + *)] 

o 


where: 


, , x" x'" 

x = x' + — + 


8 9 


2 


in the general secondary circuit as discussed in footnote, page 179. 
Substituting in these equations : 

* = 1, 

gives the 

General Equations of the Stationary Alternating-current Transformer 
Substituting in the equations of the general alternating-current 
transformer : 

Z = 0, • 

gives the 

General Equations of the Induction Motor 

Substituting: 

(ri + r)2 + s2 (Xi + x)2 = zk\ 


182 ELECTRICAL APPARATUS 

and separating the real and imaginary quantities: 

#o = no6 J [l + -*—; (r0 (ri + r) + *x0 (xi + x)) + (rtf + xjb) J 
- 3 \J^i W*i + *) - * (ri + r)) + (rob - Xotf)] J , 

/§ - ^ I Lis^ + aJ - 4~iv +iJr 

/i = ^ {(ri + r) - j* (xi + x) 

Neglecting the exciting current, or rather considering it as 
a separate and independent shunt circuit outside of the trans- 
former, as can approximately be done, and assuming the primary 
impedance reduced to the secondary circuit as equal to the 
secondary impedance: 

Yo = 0, - « = Z\. 

Substituting this in the equations of the general transformer 
we get: 

#o = no6 { 1 + \ \t\ (n + r) + sxx (xi + x)] 


- J 2 [srx (xi + x) - Xi (r! + r)] 

Zk 


$i = *"-e \[r (n + r) + «2x (xi + x)] - js[rxi -xri]}, 

Zk 

Zk 

106. The true power is, in symbolic representation: 

P = WY, 

denoting: 

srti2e2 
-. - = w 

Zk2 

gives: 

Secondary output of the transformer: 


FREQUENCY CONVERTER 183 

Internal loss in secondary circuit: 

Total secondary power: 

Pi + Pi1 = (-"*) % (r + n) =sw(r + r,) ; 

Internal loss in primary circuit: 

Po1 = to*r0 = toVid* = ( M ri = 9TiW\ 
Total electrical output, plus loss: 

Pl = Pi+ Pi1 + Pol = (8U£) \r + 2ri) = 8w(r + 2 r,) ; 
Total electrical input of primary: 

Po = [tfo/o]1 = s (**) 2 (r + n + «r0 = tp (r + n + Wi); 

Hence, mechanical output of transformer: 

P = Po-Pl = w(l -8)(r + ri); 
Ratio: 

mechanical output _- P _ 1—8 _ speed 
total secondary power Pi + Pi1 « slip 

Thus, 

In a general alternating transformer of ratio of turns, a, and 
ratio of frequencies, «, neglecting exciting current, it is: 
Electrical input in primary: 


p = sni2e* (r + rt + r^) . 
0 (rV+r^ + ^tei+'i)1' 


(r, + r)* + s* (xx + x) 
Mechanical output: 


P _ J!iJ_z *)l|i^!_(L.+i!1) • 


Electrical output of secondary : 


D 82ni2e*r 

*i — v 


(n + ry + sHxi + x)*' 

Losses in transformer: 

2 sViiVri 


P0i + pti = pi = 


(ri + r)2 + 8* (xi + x)2 


184 BLECTRIl 'AI. APPA It A TVS 

Of these quantities, P1 and Pi are always positive; P0 and P 
can be positive or negative, according to the value of s. Thus 
the apparatus can either produce mechanical power, acting U 
a motor, or consume mechanical power; and it can either con- 
sume electrical power or produce electrical power, as a generator, 

107. AI: 

s = 0, synchronism, Pa = 0, P - 0, Pi = 0. 

At I) < s < 1, between synchronism and standstill. 

Pi, P and Pa are positive; that is, the apparatus consumes 
electrical power, P„, in the primary, and produces mechanical 
power, P, and electrical power, Pi 4- Pi1, in the secondary, which 
is partly, Pi', consumed by the internal secondary resistance, 
partly, Pi, available at the secondary terminals. 

In this case: 

P, + fY _s_ 
P 1 -a' 

that is, of the electrical power consumed in the primary circuit, 
Po, a part Pul is consumed by the internal primary resistance, 
the remainder transmitted to the secondary, and divides between 
electrical power, Pi + Pi1, and mechanical power, P, in the 
proportion of the slip, or drop below synchronism, s, to the 
speed: 1 — s. 

In this range, the apparatus is a motor. 

At 8 > 1; or backward driving, P < 0, or negative; that is, 
the apparatus requires mechanical power for driving. 

Then : 

Po - P,' - Pi' < P,; 
that is, the secondary electrical power is produced partly by 
the primary electrical power, partly by the mechanical power, 
and the apparatus acts simultaneously as transformer nnd as 
alternating-current generator, with the secondary as armature. 

The ralio of mechanical input to electrical input is the ratio 
of speed to synchronism. 

In this case, the secondary frequency is higher than the 
p ill nary. 

At: 

a < 0, beyond synchronism, 
P < 0; that is, the apparatus has to be driven by mechanical 
power. 


FREQUENCY CONVERTER 185 

Po < 0; that is, the primary circuit produces electrical power 
from the mechanical input. 
At: 

r + ri + sri = 0, or, s = - , 

the electrical power produced in the primary becomes less than 
required to cover the losses of power, and Po becomes positive 
again. 

We have thus: 

. r + ri 

8 < 

consumes mechanical and primary electric power; produces 
secondary electric power. 

_ r+ft <S<Q 

consumes mechanical, and produces electrical power in primary 
and in secondary circuit. 

0 < s < 1 

consumes primary electric power, and produces mechanical and 
secondary electrical power 

1 < s 

consumes mechanical and primary electrical power; produces 
secondary electrical power. 

108. As an example, in Fig. 62 are plotted, with the slip, s, as 
abscissae, the values of: 

Secondary electrical output as Curve I. ; 
total internal loss as Curve II.; 

mechanical output as Curve III.; 

primary electrical output as Curve IV. ; 


for the values : 

nie = 100.0; 

r = 0.4; 

n = 0.1; 

x = 0.3; 

xi = 0.2; 

186 
hence 

ELEC 

TR1 

Pi - 
Pi1 = 
Po- 

P - 

.ALTS 

CAL 

16 
1 
Bfl 

1 

40C 

20, 

SSATE 

APPj 

O00s=. 

+ H= ' 

JOs*. 

M*' 
0 s (5 
1 + s" 
KW«{1 

l + « 

CURHEN 

84 

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Tfii 

rus 

SFOS 

FR 

-i 

5C 

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> 

il 

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X 

DU 

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5 

sa 

as 

K 

.1 

,^* 

■4 

ii 

\ 

fZ 

'- 

(9 

'"r 

\.^' 

if 

/ 

;# 

V 

f 

/ 

\ 

<* 

¥ 

/ 

\ 

^ 

y 

" 

A 

^ 

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Fio. 6 
109 
gener 
conve 
either 
follow 
intere 

1. 
terraij 
prima 

2.— Mpeed-power curves of general alternating- current, transfo 
Since the most common practical application ol 
il alternating-current transformer is that of frequ 
rter, that is, to change from one frequency to ano 
with or without change of the number of phases 
ing characteristic curves of this apparatus are of 
st: 

'he regulation curve; that is, the change of secon 
lal voltage as function of the load at constant imprt 
ry voltage. 

mm. 
the 
erjcy 

her, 
the 
Toat 

lary 
ssed 

FREQUENCY CONVERTER 187 

2. The compounding curve; that is, the change of primary 
impressed voltage required to maintain constant secondary 
terminal voltage. 

In this case the impressed frequency and the speed are con- 
stant, and consequently the secondary frequency is also constant. 
Generally the frequency converter is used to change from a low 
frequency, as 25 cycles, to a higher frequency, as 60 or 62.5 
cycles, and is then driven backward, that is, against its torque, 
by mechanical power. Mostly a synchronous motor is em- 
ployed, connected to the primary mains, which by overexcitation 
compensates also for the lagging current of the frequency 
converter. 

Let: 

Y = g — jb = primary exciting admittance per circuit of 
the frequency converter. 

Z\ = fi + j%\ = internal self-inductive impedance per sec- 
ondary circuit, at the secondary frequency. 

Zo = r0 + jxo = internal self-inductive impedance per primary 
circuit at the primary frequency. 

a = ratio of secondary to primary turns per circuit. 

b = ratio of number of secondary to number of primary 
circuits. 

c = ratio of secondary to primary frequencies. 

Let: 

e = generated e.m.f. per. secondary circuit at secondary 
frequency. 

Z = r + jx = external impedance per secondary circuit at 
secondary frequency, that is load on secondary system, where 
x — 0 for non-inductive load. 

To calculate the characteristics of the frequency converter, 
we then have: 
the total secondary impedance : 

Z + Zi = (r + rx) +j(x + xx); 
the secondary current: 


/i = z + z = e(fli -i***); 


where: 


r + ri , x + X] 

0l _ — — ancj fl2 = 


(r + nY +(x + xxy m" "2 (r + rxy + (x + xtf' 


188 ELECTRICAL APPARATUS 

and the secondary terminal voltage: 

*-*'*- -ST*5 

= e (r + jx) (ai - ja2) = e (61 - j7>2) ; 
where: 

61 = (rai + xa2) and 62 = (ra2 — xa\) : 
primary generated e.m.f. per circuit: 

ac 
primary load current per circuit: 

/l = abji = abe (ai — ja2); 
primary exciting current per circuit: 

loo = -°- = (g — jb) : 
ac ac 

thus, total primary current: 

/o = Z1 + /oo = e (ci - jc2); 


where: 


Ci = abai + and c2 = aba2 H 

ac ac 


and the primary terminal voltage: 

= e (di - jd2) 
where: 

di = + r0Ci + XoC2 and d2 = r0c2 — ZoCi; 
ac 

or the absolute value is: 

substituting this value of 6 in the preceding equations, gives, 
as function of the primary impressed e.m.f., e0: 
secondary current: 


/. = 


eo (<Ji - joj) /o? + a24 

v^ + ^or'absolutc'/l = eoV^ + ^; 

secondary terminal voltage: 

„ _ e0 (6, - jfr2) /67 + 6,2 . 


FREQUENCY CONVERTER 


primary current: 

i _ e5 (c' ~ic») 

primary impressed e.m.f. : 


'--<&W 


i 

s 

1 

fe 

2.-« 

-■.ill 

„ 

REGULATION CURVES 
PRIMARY. 6350 VOLTS CONSTANT 

sec 

25 CYCLES THREE-PHASE 
DNDARY, 62.5CVCLES Q.UARTER-PH 

SE 

1 1 1 1 

fl 30 « M 

Flo. 63.— Regulation c 
secondary output : 

Pi = [AVt]' 
primary electrical input: 

P. = W.W 


u of frequency converter. 


«.' (oioi + o=4e) 
<fi" + oV 


eus (cidi + c2'fs) 

'if + oV ; 


primary apparent input, volt-amperes: 
P., - ejo. 


itw 


ELECTRICAL APPARATUS 


Substituting thus different values for the secondary external 
impedance, Z, gives the regulation curve of the frequency 
converter. 

Such a curve, taken from tests of a 200-kw. frequency converter 
changing from 6300 volts, 25 cycles, three-phase, to 2500 volts, 
62.5 cycles, quarter-phase, is given in Fig. 63. 


*■;.; 

/ 

/ 

/ 

/ 

C0MP0UND1NQ CURVES 
ECQNOARY, 2500 VOLTS CON5TAN1 

6Z, S CYCLES QUARTER- PHASE 
PRIMARY, 25 CYCLES TMREE-PMAS 

° 

0 

1 

" 

n 

" T'1 

r 

i 

0 

Fio. 6*.-— Compounding curve »l frequency converter. 
From the secondary terminal voltage: 
£i = e(bt -jfe), 
it follows, absolute: 


3 Vfcl1 + 6j* 


e — 


vV + W 


Substituting these values in the above equation gives the 
quantities as functions of the secondary terminal voltage, that 
is, at constant, e-i, or the compounding curve. 

The compounding curve of the frequency converter above 
mentioned is given in Fig. 64, 

110. When running above synchronism: a < 0, the general 
alternating-current transformer consumes mechanical power and 


FREQUENCY CONVERTER 191 

produces electric power in both circuits, primary and secondary, 
thus can not be called a frequency converter, and the distinc- 
tion between primary and secondary circuits ceases, but both 
circuits are generator circuits. The machine then is a two-fre- 
quency induction generator. As the electric power generated 
at the two frequencies is proportional to the frequencies, this 
gives a limitation to the usefulness of the machine, and it appears 
suitable only in two cases: 

(a) If s = —1, both frequencies are the same, and stator 
and rotor circuits can be connected- together, in parallel or in 
series, giving the "double synchronous-induction generator." 
Such machines have been proposed for steam-turbine alternators 
of small and moderate sizes, as they permit, with bipolar con- 
struction, to operate at twice the maximum speed available for 
the synchronous machine, which is 1500 revolutions for 25 cycles, 
and 3600 revolutions for 60 cycles. 

(b) If 8 is very small, so that the power produced in the low- 
frequency circuit is very small and may be absorbed by a small 
"low-frequency exciter." 

Further discussion of both of these types is given in the 
Chapter XIII on the "Synchronous Induction Generator." 

111. The use of the general alternating-current transformer as 
frequency converter is always accompanied by the production 
of mechanical power when lowering, and by the consumption 
of mechanical power when raising the frequency. Thus a second 
machine, either induction or synchronous, would be placed on the 
frequency converter shaft to supply the mechanical power as 
motor when raising the frequency, or absorb the power as 
generator, when lowering the frequency. This machine may be 
of either of the two frequencies, but would naturally, for eco- 
nomical reasons, be built for the supply frequency, when motor, 
and for the generated or secondary frequency, when generator. 

Such a couple of frequency converter and driving motor and 
auxiliary generator has over a motor-generator set the advan- 
tage, that it requires a total machine capacity only equal to the 
output, while with a motor-generator set the total machine 
capacity equals twice the output. It has, however, the dis- 
advantage not to be as standard as the motor and the generator. 

If a synchronous machine is used, the frequency is constant ; 
if an induction machine is used, there is a slip, increasing with 
the load, that is, the ratio of the two frequencies slightly varies 


192 


ELECTRICAL API' A HA TVS 


with the load, so that the latter arrangement is less suitable when 
tying together two systems of constant frequencies. 

112. Frequency converters may be used: 

(a) For producing a moderate amount of power of a higher or 
a lower frequency, from a large alternating-current system. 

(6) For tying together two alternating-current systems of 
different frequencies, and interchange power between them, so 
that either acts as reserve to the other. In this case, electrical 
power transfer may be either way. 

(c) For local frequency reduction for commutating machines, 
by having the general alternating-current transformer lower the 
frequency, for instance from 60 to 30 cycles, and take up the 
lower frequency, as well as the mechanical power in a commu- 
tating machine on the frequency converter shaft. Such a 
combination has been called a "Motor Converter." 

Thus, instead of a BO-cycle synchronous converter, such a 
6Q/30-cyc!e motor converter would offer the advantage of the 
lower frequency of 30 cycles in the commutating machine. The 
commutating machine then would receive half its input electric- 
ally, as synchronous converter, half mechanically, as direct- 
current generator, and thus would be half converter and half 
generator; the induction machine on the same shaft would change 
half of its (iO-cycle power input into mechanical power, half into 
30-cycle electric power. 

Such motor converter is smaller and more efficient than a 
motor-generator set, but larger and less efficient than a syn- 
chronous converter. 

Where phase control of the direct-current voltage is desired, 
the motor converter as a rule does not require reactors, as the 
induction machine has sufficient internal reactance. 

(rf) For supplying low frequency to a second machine on the 
same shaft, for speed control, as "concatenated motor couple." 
That is, two. induction motors on the same shaft, operating 
in parallel, give full speed, and half speed is produced, 
at full efficiency, by concatenating the two induction ma- 
chines, that is, using the one as frequency converter for feeding 
the other. 

By using two machines of different number of poles, /i, and 
pi, on the same shaft, four different speeds can lie secured, corre- 
sponding respectively to the number of poles: », + p\, »2, ph 
pi — pa. That is, concatenation of both machines, opentJoq 


FREQUENCY CONVERTER 193 

of one machine only, either the one or the other, and differential 
concatenation. 

Further discussion hereof see under "Concatenation." 
In some forms of secondary excitation of induction machines, 
as by low-frequency synchronous or commutating machine in 
the secondary, the induction machine may also be considered 
as frequency converter. Regarding hereto see "Induction 
Motors with Secondary Excitation." 


13 


CHAPTKR XIII 
SYNCHRONOUS INDUCTION GENERATOR 

113. If an induction machine is driven above synchronism, 

the power component of the primary current reverses, thai is, 
energy flows outward, and the machine becomes an induction 
generator. The component of current required for magnetiza- 
tion remains, however, the same; that is, the induction generator 
requires the supply of a reactive current for excitation, just as 
the induction motor, and so must be connected to some apparatus 
which gives a lagging, or, what is the same, consumes a leading 
current. 

The frequency of the e.m.f. generated by the induction gen- 
erator, /, is lower than the frequency of rotation or speed, /0. 
by the frequency, ft, of the secondary currents. Or, inversely, 
the frequency, ft, of the secondary circuit is the frequency of 
slip — that is, the frequency with which the speed of mechaoioal 
rotation slips behind the speed of the rotating field, in the induc- 
tion motor, or the speed of the rotating field slips behind the 
speed of mechanical rotation, in the induction generator. 

A3 in every transformer, so in the induction machine, the 
secondary current must have the same ampere-turns as the 
primary current less the exciting current, that is, the secondary 
current is approximately proportional to the primary current, 
or to the load of the induction generator. 

In an induction generator with short-circuited secondary, 
the secondary currents are proportional, approximately, to the 
e.m.f. generated in the secondary circuit, and this e.m.f. is pro- 
portional to the frequency of the secondary circuit, that is, 
the slip of frequency behind speed. It so follows that the slip 
of frequency in the induction generator with short-circuited 
secondary is approximately proportional to the load, that is, 
such an induction generator does not produce constant syn- 
chronous frequency, but a frequency which decreases slightly 
with increasing load, just as the speed of the induction motor 
decreases slightly with increase of load. 

Induction generator and induction motor so have also l>eeu 


SYNCHRONOUS INDUCTION GENERATOR 195 

called asynchronous generator and asynchronous motor, but 
these names are wrong, since the induction machine is not 
independent of the frequency, but depends upon it just as much 
as a synchronous machine — the difference being, that the 
synchronous machine runs exactly in synchronism, while the 
induction machine approaches synchronism. The real asyn- 
chronous machine is the commutating machine. 

114. Since the slip of frequency with increasing load on the 
induction generator with short-circuited secondary is due to 
the increase of secondary frequency required to produce the 
secondary e.m.f. and therewith the secondary currents, it follows: 
if these secondary currents are produced by impressing an e.m.f. 
of constant frequency, flf upon the secondary circuit, the primary 
frequency, /, does not change with the load, but remains con- 
stant and equal to / = /0 — /i. The machine then is a syn- 
chronous-induction machine — that is, a machine in which the 
speed and frequency are rigid with regard to each other, just as 
in the synchronous machine, except that in the synchronous- 
induction machine, speed and frequency have a constant dif- 
ference, while in the synchronous machine this difference is zero, 
that is, the speed equals the frequency. 

By thus connecting the secondary of the induction machine 
with a 8010*06 of constant low-frequency, fl9 as a synchronous 
machine, or a commutating machine with low-frequency field 
excitation, the primary of the induction machine at constant 
speed, /o, generates electric power at constant frequency, /, 
independent of the load. If the secondary /i = 0, that is, a 
continuous current is supplied to the secondary circuit, the 
primary frequency is the frequency of rotation and the machine 
an ordinary synchronous machine. The synchronous machine so 
appears as a special case of the synchronous-induction machine 
and corresponds to /i = 0. 

In the synchronous-induction generator, or induction machine 
with an e.m.f. of constant low frequency, fh impressed upon the 
secondary circuit, by a synchronous machine, etc., with increas- 
ing load, the primary and so the secondary currents change, and 
the synchronous machine so receives more power as synchronous 
motor, if the rotating field produced in the secondary circuit 
revolves in the same direction as the mechanical rotation — 
that is, if the machine is driven above synchronism of the 
e.m.f. impressed upon the secondary circuit — or the synchronous 


190 • ELECTRICAL APPARATUS 

machine generates more power as alternator, if the direction of 
rotation of the secondary revolving field is in opposition to the 
speed. In the former ease, the primary frequency equals speed 
minus secondary impressed frequency: / = fn — j\\ in the latter 
case, the primary frequency equals the sum of speed and sec- 
ondary impressed frequency:/ = f<, + /i, and the machine is a 
frequency converter or general alternating-current transformer, 
with the frequency, /i, as primary, and the frequency, /,as 
secondary, transforming up in frequency to a frequency, /, 
which is very high compared with the impressed frequency, 
so that the mechanical power input into the frequency con- 
verter is very large compared with the electrical power input. 

The synchronous-induction generator, that is, induction gen- 
erator in which the secondary frequency or frequency of slip h 
fixed by an impressed frequency, so can also be considered as a 
frequency converter or general alternating-current transformer. 

116. To transform from a frequency, /(l to a frequency; ft, the 
frequency, f%, is impressed upon the primary of an induction 
machine, and the secondary driven at such a speed, or fre- 
quency of rotation, /«. that the difference between primary 
impressed frequency, /,, and frequency of rotation, /0, that, is, 
the frequency of slip, is the desired secondary frequency,/!, 

There are two speeds, /„, which fulfill this condition: one 
below synchronism: /u = f\ —ft, and one above synchronism: 
/« = /i + /=■ That is", the secondary frequency beoomaH f$, 
if the secondary runs slower than the primary revolving field 
of frequency,/,, or if the secondary runs faster than the primary 
field, by the slip, /s. 

In the former case, the speed is below synchronism, that is, 
the machine generates electric power at. the frequency, /=, in the 
secondary, and consumes electric power at the frequency, /,, 
in the primary. If /3 < fu the speed /0 = f, — /5 is between 
standstill and synchronism, and the machine, in addition to 
electric power, generates mechanical power, as induction motor, 
and as has been seen in the chapter on the "General Alternating- 
current Transformer," it is, approximately: 

Electric power input ■*■ electric power output -=- mechanical 
power output ■• f\ ■*■ ft + ft- 

If ft > !'• I hat is, the frequency converter increases the hc- 
quency, the rotation must be in backward direction, against the 
rotating field, so as to give a slip, /;, greater than the tmpnmd 


SYNCHRONOUS INDUCTION GENERATOR 107 

frequency, /i, and the speed is /n = f* — ft. In this ease, the 
iiim-tiine consumes mechanical power, since it is driven against 

I the torque given by it as induction motor, and we have: 
Klectric power input ■*■ mechanical power input + electric 
power output - f% -+■ ff + fa. 
That is, the three powers, primary electric, secondary electric, 
and mechanical, are proportional to their respective frequencies. 
As stated, the secondary ■ frequency, St, is also produced by 
driving the machine above synchronism, /,, that is, with a 
negative slip, St, or at a speed, /0 = /i + Si- In this case, the 
machine is induction generator, that is, the primary circuit 
generates electric power at frequency Si, the secondary circuit 
generates electric power at frequency St- and the machine con- 
sumes mechanical power, and the three powers again arc proper- 
Itional to their respective frequencies: 
Primary electric output + secondary electric output +■ 
mechanical input = /t -s- /» -*- /o- 
Since in this case of oversynchronous rotation, both electric 
circuits of the machine generate, it can not be called a frequency 
converter, but is an electric generator, converting mechanical 
power into electric power at two different frequencies, /L and 
and so is called a synchronous-induction machine, since 
the sum of the two frequencies generated by it equals the fre- 
quency of rotation or speed — that, is, the machine revolves in 
synchronism with the sum of the two frequencies generated 
by it. 

It is obvious that like all induction machines, this synchro- 
nous-induction generator requires a reactive lagging current for 
excitation, which has to be supplied to it by some outside source, 
s a synchronous machine, etc. 

That is, an induction machine driven at speed, /«, when sup- 

ilied with reactive exciting current of the proper frequency, 

;enerates electric power in the stator as well as in the rotor, at 

he two respective frequencies, /■ and/-, which are such that their 

in synchronism with the speed, that is: 


A + />-/.; 


otherwise the Frequencies, /, and /-, are entirely independent, 
i connecting the stator to a circuit of frequency, Si, the 
•otor generates frequency, /» = /o - /i, or connecting the rotor to 


198 


ELECTRIC A], APPARATUS 


a circuit of frequency, />, the stator generates a frequency 

116. The power generated in the stator, Pu and the power 
generated in the rotor, Pi, are proportional to their respective 
frequencies : 

P,:P,:P, -/•:/•:/* 

where P0 is the mechanical input (approximately, that is, neg- 
lecting losses). 

As seen here the difference between the two circuits, stator 
and rotor, disappears — that is, either can be primary or sec- 
ondary, that is, the reactive lagging current required for excita- 
tion can be supplied to the stator circuit at frequency, ft, or to 
the rotor circuit at frequency, ft, or a part to the stator and a pan 
to the rotor circuit. Since this exciting current is reactive or 
wattless, it can bo derived from a synchronous motor or con- 
verter, as well as from a synchronous generator, or an alter- 
nating comimitating machine. 

As the voltage required by the exciting current is proportional 
to the frequency, it also follows that the reactive power input or 
the volt-amperes excitation, is proportional to the frequency 
of the exciting circuit. Hence, using the low-frequency circuit 
for excitation, the exciting volt-amperes are small. 

Such a synchronous-induction generator therefore is a two- 
frequency generator, producing electric power simultaneously 
at two frequencies, and in amounts proportional to these fre- 
quencies. For instance, driven at 85 cycles, it can connect with 
the stator to :i 25-eycle system, and with the rotor to a 60-cycle 
system, and feed into both systems power in the proportion of 
25 + 60, as is obvious from the equations of the general alter- 
nating-current transformer in the preceding chapter 

117. Since the amounts of electric power at the two fre- 
quencies are always proportional to each other, such a machine 
is hardly of much value for feeding into two different systems, 
but of importance are only the cases where the two frequence 
generated by the machine can he reduced to one. 

This is the case: 


1. If the two frequencies are the same:/! —ft 


s* 


In this 


case, stator and rotor can be connected together, in parallel 
or in series, and the induction machine then generates electric 
power at half the frequency of its speed, that is, runs at double 


SYNCHRONOUS INDUCTION GENERATOR 199 

synchronism of its generated frequency. Such a " double syn- 
chronous alternator" so consists of an induction machine, in 
which the stator and the rotor are connected with each other in 
parallel or in series, supplied with the reactive exciting current 
by a synchronous machine — for instance, by using synchronous 
converters with overexcited field as load — and driven at a speed 
equal to twice the frequency required. This type of machine 
may be useful for prime movers of very high speeds, such as 
steam turbines, as it permits a speed equal to twice that of the 
bipolar synchronous machine (3000 revolutions at 25, and 7200 
revolutions at 60 cycles). 

2. If of the two frequencies, one is chosen so low that the 
amount of power generated at this frequency is very small, and 
can be taken up by a synchronous machine or other low-fre- 
quency machine, the latter then may also be called an exciter. 
For instance, connecting the rotor of an induction machine to a 
synchronous motor of /2 = 4 cycles, and driving it at a speed 
of /o = 64 cycles, generates in the stator an e.m.f. at f\ = 60 
cycles, and the amount of power generated at 60 cycles is 6pj[ = 
15 times the power generated by 4 cycles. The machine then 
is an induction generator driven at 15 times its synchronous 
speed. Where the power at frequency, /2, is very small, it would 
be no serious objection if this power were not generated, but con- 
sumed. That is, by impressing /2 = 4 cycles upon the rotor, 
and driving it at /0 = 56 cycles, in opposite direction to the rotat- 
ing field produced in it by the impressed frequency of 4 cycles,, 
the stator also generates an e.m.f. at f\ = 60 cycles. In this 
case, electric power has to be put into the machine by a generator 
at /2 = 4 cycles, and mechanical power at a speed of /0 = 56 
cycles, and electric power is produced as output at /i = 60 cycles. 
The machine thus operated is an ordinary frequency converter, 
which transforms from a very low frequency, /2 = 4 cycles, to 
frequency /i = 60 cycles or 15 times the impressed frequency, 
and the electric power input so is only one-fifteenth of the electric 
power output, the other fourteen-fifteenths are given by the 
mechanical power input, and the generator supplying the im- 
pressed frequency, /2 = 4 cycles, accordingly is so small that it 
can be considered as an exciter. 

118. 3. If the rotor of frequency, /2, driven at speed, /0, is 
connected to the external circuit through a commutator, the 
effective frequency supplied by the commutator brushes to the 


200 


ELEi TR1CAL APPARATUS 


external circuit is/„ — /s; hence equals/,, or the atator frequency. 
Stator and rotor so give the same effective frequency, /,, and 
irrespective of the frequency, /s generated in the rotor, and the 
frequencies, /[ and /s, accordingly become indefinite, that is, 
jx may lie any frequency, /i then becomes f„ — /,, but. by the 
commutator is transformed to the same frequency, /i. If the 
stator and rotor were used on entirely independent electric 
circuits, the frequency would remain indeterminate. As soon, 
however, as stator and rotor are connected together, a relation 
appears due to the transformer law, that the secondary ampere- 
turns must equal the primary ampere-turns (when neglecting 
the exciting ampere-turns). This makes the frequency dependent 
upon the number of turns of stator and rotor circuit. 

Assuming the rotor circuit is connected in multiple with the 
stator circuit— as it always can be, since by the commutator 
brushes it has been brought to the same frequency. The rotor 
c.m.f. then must be equal to the stator e.m.f. The e.m.f., how- 
ever, is proportional to the frequency times number of turns, 
and it is therefore: 

n-J, - ■»,/,, 

where: /i] = number of effective stator turns, 

>H = number of effective rotor turns, and f\ 

and/* are the respective frequencies. 
Herefrom follows: 

/,+/,-«, + »,; 

that is, the frequencies are inversely proportional to the number 
of effective turns in stator and in rotor. 

Or, since /o = A + /a is the frequency of rotation : 


I +«! 


+ »., 
ft 


That is, the frequency, /,, generated by the synchronous- 
induction machine with commutator, is the frequency of imntinn. 
/o, times the ratio of rotor turns, m, to total turns, n, + n». 

Thus, it can lie made anything by properly choosing the 
number of turns in the rotor and in the stator, or, what amounts 
to the same, interposing between rotor and stator a transformer 
of the proper ratio of transformation. 


SYNCHRONOUS INDUCTION GENERATOR 201 

The powers generated by the stator and by the rotor, how- 
ever, are proportional to their respective frequencies, and so are 
inversely proportional to their respective turns. 

• 

Pi -§-P» =/i +h = n2 -*- m; 

if n\ and n2, and therewith the two frequencies, are very different, 
the two powers, Pj and P2, are very different, that is, one of the 
elements generates very much less power than the other, and 
since both elements, stator and rotor, have the same active 
surface, and so can generate approximately the same power, the 
machine is less economical. 

That is, the commutator permits the generation of any de- 

sired frequency, /1, but with best economy only if f\ = w, or 

half-synchronous frequency, and the greater the deviation from 
this frequency, the less is the economy. If one of the fre- 
quencies is very small, that is, f\ is either nearly equal to syn- 
chronism, /o, or very low, the low-frequency structure generates 
very little power. 

By shifting the commutator brushes, a component of the rotor 
current can be made to magnetize and the machine becomes a 
self-exciting, alternating-current generator. 

The use of a commutator on alternating-current machines is 
in general undesirable, as it imposes limitations on the design, 
for the purpose of eliminating destructive sparking, as discussed 
in the chapter on "Alternating-Current Commutating Machines." 

The synchronous-induction machines have not yet reached a 
sufficient importance to require a detailed investigation, so only 
two examples may be considered. 

119. 1. Double Synchronous Alternator, 

Assume the stator and rotor of an induction machine to be 
wound for the same number of effective turns and phases, and 
connected in multiple or in series with each other, or, if wound 
for different number of turns, connected through transformers 
of such ratios as to give the same effective turns when reduced 
the same circuit by the transformer ratio of turns. . 

Let: 

Yi — Q — jb — exciting admittance of the stator, 

Z\ = ri + jx\ = self-inductive impedance of the stator, 

Z2 = r2 + jx* = self-inductive impedance of the rotor, 


202 ELECTRICAL APPARATUS 

and: 

6 = e.m.f. generated in the stator by the mutual inductive 
magnetic field, that is, by the magnetic flux corresponding to 
the exciting admittance, Y\\ 
and: 

/ = total current, or current supplied to the external circuit, 
I\ = stator current, 
I2 = rotor current. 

With series connection of stator and rotor: 

/ = /, = h, 
with parallel connection of stator and rotor: 

/ = /i + /2. 

Using the equations of the general induction machine, the 
slip of the secondary circuit or rotor is : 

« = -1; 
the exciting admittance of the rotor is: 

Yt = g - jsb = g + jb, 
and the rotor generated e.m.f.: 

E\ = se = — e; 

that is, the rotor must be connected to the stator in the opposite 
direction to that in which it would be connected at standstill, 
or in a stationary transformer. 

That is, magnetically, the power components of stator and 
rotor current neutralize each other. Not so, however, the 
reactive components, since the reactive component of the rotor 
current: 

U = i\ + ji\ 

in its reaction on the stator is reversed, by the reversed direction 
of relative rotation, or the slip, s = — 1, and the effect of the 
rotor current, I*} on the stator circuit accordingly corresponds 
to: 

i 2 — I 2 — 3l 2, 

hence, the total magnetic effect is : 

/i-/'2 = (*\-*'t)+j(i"i + t"t); 


SYNCHRONOUS INDUCTION GENERATOR 203 

and since the total effect must be the exciting current: 

i o — to tj 0, 

it follows that : 

i'x — i't = i'o and i"\ + i#/t = t 


99 


Hence, the stator power current and rotor power current, 
i'x and i\y are equal to each other (when neglecting the small 
hysteresis power current). The synchronous exciter of the 
machine must supply in addition to the magnetizing current, 
the total reactive current of the load. Or in other words, such 
a machine requires a synchronous exciter of a volt-ampere 
capacity equal to the volt-ampere. excitation plus the reactive 
volt-amperes of the load, that is, with an inductive load, a large 
exciter machine. In this respect, the double-synchronous 
generator is analogous to the induction generator, and is there- 
fore suited mainly to a load with leading current, as over- 
excited converters and synchronous motors, in which the reactive 
component of the load is negative and so compensates for the 
reactive component of excitation, and thereby reduces the size 
of the exciter. 

This means that the double-synchronous alternator has zero 
armature reaction for non-inductive load, but a demagnetizing 
armature reaction for inductive, a magnetizing armature reac- 
tion for anti-inductive load, and the excitation, by alternating- 
reactive current, so has to be varied with the character of the 
load, in general in a far higher degree than with the synchronous 
alternator. 

120. 2. Synchronous-induction Generator with Low-frequency 
Excitation. 

Here two cases exist: 

(a) If the magnetic field of excitation revolves in opposite 
direction to the mechanical rotation. 

(6) If it revolves in the same direction. 

In the first case (a) the exciter is a low-frequency generator 
and the machine a frequency converter, calculated by the same 
equations. 

Its voltage regulation is essentially that of a synchronous 
alternator: with increasing load, at constant voltage impressed 
upon the rotor or exciter circuit, the voltage drops moderately 
at non-inductive load, greatly at inductive load, and rises at 


204 


ELECTRICAL APPARATUS 


anti-inductive load. To maintain constant terminal voltage, 
the excitation has to lie changed with ;i change of load and 
character of load. With a low-frequency synchronous machine 
as exciter, this is done by varying the field excitation of the 
exciter. 

At constant field excitation of the synchronous exciter, the 
regulation is that due to the impedance lietween the nominal 
generated e.m.f. of the exciter, and the terminal voltage of the 
stator — that, is, corresponds to: 

Z = Z0 + Zi + Z„ 
Here 7.t, = synchronous impedance of the exciter, reduced to full 
frequency, /i, 
Zj = self-inductive impedance of the rotor, reduced to full 

frequency. /i, 
Zi = self-inductive impedance of the stator. 
If then ED = nominal generated e.m.f. of the exciter generator, 
that is, corresponding to the field excitation, and, 
/i = i — jti = stator current or output current, the stator 
terminal voltage is: 

E, - E0 + ZI„ or, E» = E + (r + jx) {i - ji\); 
and, choosing Ei = p| as real axis, and expanding: 

Bo = («i + « + Mf) + j (ri - ri>), 
and the absolute value: 


- (g. + ri 


■ xn) 


(«■ 


nO* — (n + j/i). 


121. As an example is shown, in Fig. 6.5, in dotted lines, with 
the total current, / = y/i- + f"i!, iis abscissa?, the voltage regu- 
lation of such a machine, or the terminal voltage, tu with a 
four-cycle synchronous generator as exciter of the 60-cycle 
synchronous-induction generator, driven as frequency converter 
at 56 cycles. 

1. For non-inductive load, or I, - i. (Curve I.) 

2. For inductive load of 80 per cent, power-factor, or /i * 
7(0.8 - 0.6 j). (Curve U.J 

3. For anti-inductive load of 80 per cent, power-factor, or 
/, = 7(0.8 + O.Gj). (Curve III.) 


SYNCHRONOUS INDUCTION GENERATOR 205 

For the constants: 

e„ - 2000 volts, Z, = 1 + 0.5 j, 

Zx = 0.1 + 0.3 j, Z„ = 0.5 + 0.5 j; 

ence: 

Z = 1.6 + 1.3 j. 


e, = Vi X 10' - (1.3 i - 1.6 ii)1 - (1.6 i + 1.3 *'i); 
hence, for non-inductive load, ii = 0: 

c, = V4 X 10« - 1.69f* - 1.6 i; 


; «* --'' 

^^ -"" 

^ ^ -- ^ ^ 

&&*** 

"""' ^sz~— --. ~~ ~~ — -_ 

1800 "s^. --...in " ■"•-^ 

S --1 "^ iu 

11 ^"i>^ ^^ 

2 ^*, V 

wo "^-J 

Z ^is 

300 ^ 

Fio. 65. — Synchronous induction generator regulation curves. 

for inductive load of 80 per cent, power-factor j'i = 0.6 /, i = 0 

e, = Vi X 10' - 00064/* - 2.06/; 

and for anti-inductive load of 80 per cent, power-factor 1 
- 0.6/,* = 0.8/: 


-Vi 


X 10« -4/1 - 0.5/. 


As seen, due to the internal impedance, anil especially the 
resistance of this machine, the regulation is very poor, and even 
at the chosen anti-inductive load no rise of voltage occurs. 

122. Of more theoretical interest is the case (b), where the 


206 ELECTRICAL APPARATUS 

exciter is a synchronous motor, and the synchronous-induction 
generator produces power in the stator and in the rotor circuit. 
In this case, the power is produced by the generated e.m.f., E 
(e.m.f. of mutual induction, or of the rotating magnetic field}, 
of the induction machine, and energy flows outward in both 
circuits, in the stator into the receiving circuit, of terminal 
voltage, #i, in the rotor against the impressed e.m.f. of the 
synchronous motor exciter, En. The voltage of one receiving 
circuit, the stator, therefore, is controlled by a voltage impressed 
upon another receiving circuit, the rotor, and this results in some 
interesting effects in voltage regulation. 

Assume the voltage, Ea, impressed upon the rotor circuit as 
the nominal generated e.m.f. of the synchronous-motor exciter, 
that is, the field corresponding to the exciter field excitation, 
and assume the field excitation of the exciter, and therewith 
the voltage, Ea, to be maintained constant. 

Reducing all the voltages to the stator circuit by the ratio of 
their effective turns and the ratio of their respective frequencies, 
the same e.m.f., E, is generated in the rotor circuit as in the 
stator circuit of the induction machine. 

At no-load, neglecting the exciting current of the induction 
machine, that is, with no current, we have En = E = E\. 

If a load is put on the stator circuit by taking a current, /, 
from the same, the terminal voltage, Ex, drops below I he gene- 
rated e.m.f., E, by the drop of voltage in the impedance, Zu of 
the stator circuit. Corresponding to the stator current, I,, a 
current, /a, then exists in the rotor circuit, giving the same 
ampere-turns as Ii, in opposite direction, and so neutralizing the 
m.m.f. of the stator (as in any transformer). This current, It, 
exists in the synchronous motor, and the synchronous motor 
e.m.f., Eo, accordingly drops below the generated e.m.f., E, of 
the rotor, or, since Ea is maintained constant, E rises above Ea 
with increasing load, by the drop of volt age in the rotor impedance, 
2V, and the synchronous impedance, Z«, of the exciter. 

That is, the stator terminal voltage, E,, drops with increasing 
load, by the stator impedance drop, and rises with increasing 
load by the rotor and exciter impedance drop, since the latter 
causes the generated e.m.f., E, to rise. 

If then the impedance drop in the rotor circuit is greater than 
thai in the stator, with increasing load the terminal voltage, 
Ei, of the machine rises, that is, the machine automatically 


SYNCHRONOUS INDUCTION GENERATOR 207 

overcompounds, at constant-exciter field excitation, and if the 
stator and the rotor impedance drops are equal, the machine 
compounds for constant voltage. 

In such a machine, by properly choosing the stator and rotor 
impedances, automatic rise, decrease or constancy of the terminal 
voltage with the load can be produced. 

This, however, applies only to non-inductive load. If the 
current, I, differs in phase from the generated e.m.f., E, the 
corresponding current, J2, also differs; but a lagging component 
of I\ corresponds to a leading component in It, since the stator 
circuit slips behind, the rotor circuit is driven ahead of the 
rotating magnetic field, and inversely, a leading component of 
7i gives a lagging component of 72. The reactance voltage of 
the lagging current in one circuit is opposite to the reactance 
voltage of the leading current in the other circuit, therefore 
does not neutralize it, but adds, that is, instead of compounding, 
regulates in the wrong direction. 

123. The automatic compounding of the synchronous induc- 
tion generator with low-frequency synchronous-motor excitation 
so fails if the load is not non-inductive. 

Let: 

Z\ = T\ + jxi = stator self-inductive impedance, 

Z2 = r2 + jx<t = rotor self-inductive impedance, reduced to the 

stator circuit by the ratio of the effective turns, t = - , and the 

J x ni' 

h 
ratio of frequencies, a = t) 

Ji 
Zo = r0 + jxo = synchronous impedance of the synchronous- 
motor exciter; 
Ei = terminal voltage of the stator, chosen as real axis, = e\\ 
Eo = nominal generated e.m.f. of the synchronous-motor 

exciter, reduced to the stator circuit; 
E = generated e.m.f. of the synchronous-induction generator 
stator circuit, or the rotor circuit reduced to the stator circuit. 
The actual e.m.f. generated in the rotor circuit then is E' = 
taE, and the actual nominal generated e.m.f. of the synchronous 
exciter is E\ = taEo. 
Let: 

I\ = i ' — jii = current in the stator circuit, or the output 
current of the machine. 


208 ELECTRICAL APPARATUS 

The current in the rotor circuit, in which the direction of 
rotation is opposite, or ahead of the revolving field, then is, 
when neglecting the exciter current : 

li = i + jii. 

. (If Y = exciting admittance, the exciting current is Io — EYf 
and the total rotor current then h + 72.) 
Then in the rotor circuit: 

E = E0 + (Z0 + Z2) A, (1) 

and in the stator circuit : 

E = Ex + Zi/i. (2) 

Hence : 

Ex = E0 + U (Zo + Z2) - 1XZU (3) 

or, substituting for 1\ and 72: 

Ex = E0 + i (Zo + Z2 - Zx) + jit (Z0 + Z2 + Z,). (4) 
Denoting now: 


Z0 + Z\ + Z2 = Z3 = r3 + jx8, 
Z0 + Z2 — Zi = Zj = r4 + jx4, 


(5) 


and substituting: 

tf, = E0 + iZA + jUZh (6) 

or, since Ex = e\\ 

E0 = ei — iZK — }i\Zz 

= (ei - rAi + x3t'i) - j (xii + riil)) (7) 

or the absolute value: 

e02 = (ex - rAi + x$ix)2 + (xAi + r3/i)2. (8) 

Hence : 

ex = vco2 - (xAi + r3ii)2 + r4i - x3ii. (9) 

That is, the terminal voltage, ci, decreases due to the decrease 
of the square root, but may increase due to the second term. 
At no-load: 

i = 0, ix = 0 and e\ = <v 


SYNCHRONOUS INDUCTION GENERATOR 209 
At non-inductive load: 

t"i = 0 and ei = Ve0* - zAH2 + rAi. (10) 

e\ first increases, from its no-load value, e0, reaches a maximum, 

and then decreases again. 

Since: 

r* = r0 + r2 - rlt 

X\ = x0 + Xt — rx, 
at : r4 = 0 and au = 0, 

or, 

ri = r0 + r2, 
Xi = Xo + x2, 

and: 

ei = ^o, that is, in this case the terminal vol- 
tage is constant at all non-inductive loads, at constant exciter 
excitation. 

In general, or for I\ = i — ji\, 

if ii is positive or inductive load, from equation (9) follows 
that the terminal voltage, eh drops with increasing load; while 

if i\ is negative or anti-inductive load, the terminal voltage, 
6i, rises with increasing load, ultimately reaches a maximum 
and then decreases again. 

From equation (9) follows, that by changing the impedances, 
the amount of compounding can be varied. For instance, at 
non-inductive load, or in equation (10) by increasing the re- 
sistance, r4, the voltage, ci, increases faster with the load. 

That is, the overcompounding of the machine can be increased 
by inserting resistance in the rotor circuit. 

124. As an example is shown, in Fig. 65, in full line, with the 
total current, / = y/i2 + ix2, as abscissae, the voltage regulation 
of such a machine, or the terminal voltage, eh with a four- 
cycle synchronous motor as exciter of a 60-cycle synchronous- 
induction generator driven at 64-cycles speed. 

1. For non-inductive load, or Ii = i. (Curve I.) 

2. For inductive load of 80 per cent, power-factor; or 
/, = / (0.8 - 0.6 j). (Curve II.) 

3. For anti-inductive load of 80 per cent, power-factor; or 
Ii - / (0.8+ 0.6 j). (Curve III.) 

14 


210 ELECTRICAL APPARATUS 

For the constants : 


eo = 2000 volts. 
Zt = 0.1 + 0.3j". 
Zt~ 1 +0.5 j. 


Then: 


Z0 = 0.5 + 0.5 j. 
a = 0.067. 

( = 1, that is, the 
same number of 
turns in stator 
and rotor. 


Z, = 1.6 + 1.3 j and Z« = 1.4 + 0.7 j. 
Hence, substituting in equation (9) : 

Cl = V4X 10'- (0.7 1 + 1.6(1)* + 1.4 1 - 1.3 i,; 


MM 

*- 

--+■ 

• 

V 

X 

\ 

V 

S 

^ 

\ 

1IUII 

"~ 

■ — 

— it« 

w— 

- 

»- 

» 

U N 

Fia. 66. — Synchronous induction generator; voltage regulation with power- 
factor of load. 

thus, for non-inductive load, n = 0: 

e, = Vi X 10* - 0.49 i* + 1.4 i; 

for inductive load of 80 per cent. power-factori'i=0.67;i = 0.87: 

«i = V4 X 10» - 2.31 7* + 0.34 7; 

and for anti-inductive load of 80 per cent, power-factor i"i = 
-0.6 1;x -0.8 7: 

e, = V4 X 10« - 0.16 /* + 1.9 7. 

Comparing the curves of this example with those of the same 
machine driven as frequency converter with exciter generator, 


SYNCHRONOUS INDUCTION GENERATOR 211 

and shown in dotted lines in the same chart (Fig. 65), it is 
seen that the voltage is maintained at load far better, and 
especially at inductive load the machine gives almost perfect 
regulation of voltage, with the constants assumed here. 

To show the variation of voltage with a change of power- 
factor, at the same output in current, in Fig. 66, the terminal 
voltage, 6i, is plotted with the phase angle as abscissae, from 
wattless anti-inductive load, or 90° lead, to wattless inductive 
load, or 90° lag, for constant current output of 400 amp. As 
seen, at wattless load both machines give the same voltage but 
for energy load the type (b) gives with the same excitation a 
higher voltage, or inversely, for the same voltage the type (a) 
requires a higher excitation. It is, however, seen that with the 
same current output, but a change of power-factor, the voltage 
of type (a) is far more constant in the range of inductive load, 
while that of type (b) is more constant on anti-inductive load, 
and on inductive load very greatly varies with a change of 
power-factor.