Theory and Calculation of Electrical Apparatus (1917) — part 9 of 21

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

Zi = 7*1 + jx i = internal self-inductive impedance per sec- ondary circuit, at the secondary frequency.

Z 0 = r 0 + jx o = 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 + ri) + j 0 + xi) ; the secondary current:

/ 1 =

Z + Z i

= e (a i - ja 2 ) ;

where:

ai

• r + n j, s + ft!

( r q- t*i) 2 + (x + Xi ) 2 2 (r + ri) 2 + (x + Xi) 2 ’

188

ELECTRICAL APPARATUS

and the secondary terminal voltage:

E l = Zh = e Tr~ — sr \

= e(r + jx) (ai - ja 2 ) = e (61 - j& 2 );

where:

bi = (rai + xa 2 ) and & 2 = (ra 2 — zai): primary generated e.m.f. per circuit:

e

E 1

ac

primary load current per circuit :

I 1 = abj i = abe (a x - ;a 2 ) ;

primary exciting current per circuit :

r Y 0 e f N c .

f..- — -to-jt)--

thus, total primary current:

/o = 7 1 + /oo = c (c x — jc 2 );

where :

Ci = a&a x + ~ and c 2 = aba 2 +

ac ac

and the primary terminal voltage:

where :

#0 = ^ + IoZo = c (di - jd 2 )

1

di = “ + r 0 Ci + *oC2 and d 2 = r 0 c 2 — £oCi;

or the absolute value is :

Co = e / di 2 + d2 2 , e =

Co

Vdi 2 + d 2 2

substituting this value of e in the preceding equations, gives, as function of the primary impressed e.m.f., e 0 : secondary current:

co (ai - ja 2 ) r _ la^fa 2 2 .

h =

V di 2 + d 2 2 secondary terminal voltage:

Co (&i - jb 2 )

) /^”~2 jor, absolute, h = eo ■» ! , „ 2 Vai 2

• d 2 2 ’

Ex =

Vdi 2 + 4 2

^- e °Vdr + d 2 *’

FREQUENCY CONVERTER

189

primary current:

eo ( c i — jc 2 ) Vdi 2 + primary impressed e.m.f. :

h =

eo

I c i 2 + C 2 2 .

Vd 1 2 + d 2 i ’

T , eoidi-jdz) Eo — /-nr-. — r„;

Fia. 63. — Regulation curves of frequency converter.

secondary output:

-n nr r ii e o 2 (aibi + aj) 2) .

primary electrical input :

p _ rj? T 11 - go 2 (Cl^l + CA) '

Po - [Eoh\ + ^ ,

primary apparent input, volt-amperes :

P a Q = eoI Q .

PRIMARY AMP.

190

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.

Fig. 64.~Compounding curve of frequency converter.

From the secondary terminal voltage :

Ei = e (6i - jb 2 ),

it follows, absolute:

ei = e /6i 2 + & 2 2 ,

ei_

VbT'+b?’

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

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

110 . When running above synchronism: $ < 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 :

(а) If $ = — 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.

(б) If 5 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 APPARATUS

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.

(b) 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 60-cycle synchronous converter, such a 60/30-cycle 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 60-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.

(d) 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, p x and p 2} on the same shaft, four different speeds can be secured, corre- sponding respectively to the number of poles: p\ + p 2 , £> 2 , Pi , Pi — P2. That is, concatenation of both machines, operation

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.”

CHAPTER XIII

SYNCHRONOUS INDUCTION GENERATOR

113. If an induction machine is driven above synchronism, the power component of the primary current reverses, that 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, /«, by the frequency, jfj, of the secondary currents. Or, inversely, the frequency, f 1} of the secondary circuit is the frequency of slip — that is, the frequency with which the speed of mechanical 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.

As 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 been

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, f h upon the secondary circuit, the primary frequency, /, does not change with the load, but remains con- stant and equal to / = / 0 — fi. 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 source of constant low-frequency, /i, 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 jfi =0.

In the synchronous-induction generator, or induction machine with an e.m.f. of constant low frequency, fi, impressed upon the secondary circuit, by a synchronous machine, etc t , 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

196

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 case, the primary frequency equals speed minus secondary impressed frequency: / = fo — fi) in the latter case, the primary frequency equals the sum of speed and sec- ondary impressed frequency : / = /o + fi, 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 is fixed by an impressed frequency, so can also be considered as a frequency converter or general alternating-current transformer.

115. To transform from a frequency, /i, to a frequency; / 2 , the frequency, f h is impressed upon the primary of an induction machine, and the secondary driven at such a speed, or fre- quency of rotation, fo, that the difference between primary impressed frequency, /i, and frequency of rotation, / 0 , that is, the frequency of slip, is the desired secondary frequency, / 2 .

There are two speeds, / 0 , which fulfill this condition: one below synchronism: / 0 = /i — / 2 , and one above synchronism: fo =/i+/ 2 . That is, the secondary frequency becomes / 2 , if the secondary runs slower than the primary revolving field of frequency, f x , or if the secondary runs faster than the primary field, by the slip, / 2 .

In the former case, the speed is below synchronism, that is, the machine generates electric power at the frequency, / 2 , in the secondary, and consumes electric power at the frequency, f h in the primary. If f 2 < /i, the speed fo = fi — f 2 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 -f- electric power output -s- mechanical power output = /i -r- f 2 -s- Jo- lt U > fu that is, the frequency converter increases the fre- queijcy, the rotation must be in backward direction, against the rotating field, so as to give a slip, f 2) greater than the impressed

SYNCHRONOUS INDUCTION GENERATOR 197

frequency, /i, and the speed is f 0 = / 2 - / x . In this case, the machine consumes mechanical power, since it is driven against the torque given by it as induction motor, and we have:

Electric power input -f- mechanical power input -r- electric power output = fi -s- /o U-

That is, the three powers, primary electric, secondary electric, and mechanical, are proportional to their respective frequencies.

As stated, the secondary frequency, / 2 , is also produced by driving the machine above synchronism, fi r that is, with a negative slip, / 2 , or at a speed, / 0 = fi +/ 2 . In this case, the machine is* induction generator, that is, the primary circuit generates electric power at frequency f h the secondary circuit generates electric power at frequency / 2 , and the machine con- sumes mechanical power, and the three powers again are propor- tional to their respective frequencies:

Primary electric output -s~ secondary electric output -f- mechanical input = fi f 2 fo. .

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, fi and jf 2 , 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, as a synchronous machine, etc.

That is, an induction machine driven at speed, /o, when sup- plied with reactive exciting current of the proper frequency, generates electric power in the stator as well as in the rotor, at the two respective frequencies, fi and / 2 , which are such that their sum is in synchronism with the speed, that is:

f\ +/2 = fo]

otherwise the frequencies, fi and / 2 , are entirely independent. That is, connecting the stator to a circuit of frequency, /i, the rotor generates frequency, / 2 = jfo — fh or connecting the rotor to

198

ELECTRICAL APPARATUS

a circuit of frequency, fa, the stator generates a frequency

/i — /o — fa- ns. The power generated in the stator, Pi, and the power generated in the rotor, P 2 , are proportional to their respective frequencies:

Pi :P*:Po=fi:fa:fo,

where Po 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, fa, or to the rotor circuit at frequency, fa, or a part to the stator and a part to the rotor circuit. Since this exciting current is reactive or wattless, it can be derived from a synchronous motor or con- verter, as well as from a synchronous generator, or an alter- nating commutating 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 witli the stator to a 25-cycle system, and with the rotor to a 60-cycle system, and feed into both systems power in the proportion of 25 -T- 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 frequencies generated by the machine can be reduced to one.

This is the case :

1 . If the two frequencies are the samerfa = fa = In thin

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 u 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 jf 2 = 4 cycles, and driving it at a speed of /o = 64 cycles, generates in the stator an e.m.f. at /i = 60 cycles, and the amount of power generated at 60 cycles is = 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 /i = 60 cycles. In this case, electric power has to be put into the machine by a generator at jf 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 fi = 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, jf 2 = 4 cycles, accordingly is so small that it can be considered as an exciter.

3. 3. If the rotor of frequency, jf 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

ELECTRICAL APPARATUS

external circuit is/ 0 — / 2 ; hence equals /i, or the stator frequency. Stator and rotor so give the same effective frequency, /i, and irrespective of the frequency, / 2 generated in the rotor, and the frequencies, f x and / 2 , accordingly become indefinite, that is, fi may be any frequency. / 2 then becomes / 0 — f h 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 e.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 2 fz = nif h

where : nj = number of effective stator turns,

n 2 = number of effective rotor turns, and fi

and J 2 are the respective frequencies.

Herefrom follows :

fi -5- /2 = n 2 - 7 - n x ;

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

Or, since fo = fi + f 2 is the frequency of rotation:

fi fo = -5-(ni + n 2 ,)

h n x + n 2 jfo *

That is, the frequency, /i, generated by the synchronous- induction machine with commutator, is the frequency of rotation, fo, times the ratio of rotor turns, w 2 , to total turns, n x + n 2 .

Thus, it can be 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* = fi + h = ni;

if n 3 and n 2 , and therewith the two frequencies, are very different, the two powers, P 3 and P 2 , 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, / x , but with best economy only if f x = 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 x is either nearly equal to syn- chronism, f 0j 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:

Y i = g — jb = exciting admittance of the stator,

Zi = rj + jx i = self-inductive impedance of the stator,

Z% = r 2 + jx 2 = self-inductive impedance of the rotor,

202

ELECTRICAL APPARATUS

and:

e = 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 3 ; and :

I = total current, or current supplied to the external circuit,

I I = stator current,

1 2 = rotor current.

With series connection of stator and rotor:

1 = 1 , =

with parallel connection of stator and rotor:

I = Jj + Z 2 .

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

s = -1;

the exciting admittance of the rotor is:

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

E' 2 = 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 :

1 2 = V 2 + #"a,

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 2 , on the stator circuit accordingly corresponds to:

r 2 = i ' 2 - if",;

hence, the total magnetic effect is:

h — I'z = (i'i — i'l) + j + %" 2 );

SYNCHRONOUS INDUCTION GENERATOR 203

and since the total effect must be the exciting current:

Io = i'o + j\

it follows that :

i'x — i f 2 = ■i'o and i n i + i’\ — i" 0 .

Hence, the stator power current and rotor power current, i'l and i' 2 , 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.

( b ) 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 be changed with a 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 between the nominal generated e.m.f. of the exciter, and the terminal voltage of the stator — that is, corresponds to :

Z = Zq + Z 2 + Z.

Here Zq — synchronous impedance of the exciter, reduced to full frequency, f h

Zi = self-inductive impedance of the rotor, reduced to full frequency, f u

Zi = self-inductive impedance of the stator.

If then Eo = nominal generated e.m.f. of the exciter generator, that is, corresponding to the field excitation, and, h = i — jii = stator current or output current, the stator terminal voltage is :

Ei = Eo + ZI h or, E 0 = E + (r ■+ jx) (i — and, choosing E± = ei as real axis, and expanding:

E 0 = (ei + ri + xij) + j (xi - n), and the absolute value :

e ° 2 = ( e i + ri + xii) 2 + (xi — n*i) 2 , ei — V e 0 2 — (xi — ri\Y — (ri + xii).

121 . As an example is shown, in Fig. 65, in dotted lines, with the total current, I = /i 2 + i x 2 , as abscissae, the voltage regu- lation of such a machine, or the terminal voltage, 61, 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 /j = i. (Curve I.)

2. For inductive load of 80 per cent, power-factor, or h =

I (0.8 — 0.6 j). (Curve II.)

3. For anti-inductive load of 80 per cent, power-factor, or

h = I (0.8 + 0.6 j). (Curve III.)

SYNCHRONOUS INDUCTION GENERATOR

205

For the constants :

hence:

Then:

e 0 = 2000 volts, ^2 = 1 + 0.5 j,

Z\ = 0.1 -f 0.3 j, Zq = 0.5 + 0.5 j;

Z = 1.6 T 1.3 j.

= V4 X 10“ - (1.3 i - 1.6 k)~ - (1.6 i -j- 1.3 70; hence, for non-inductive load, = 0:

e, = VI X 10“ ~LG!)7- - l.fl i;

Fig. 65. — Synchronous induction generator regulation curves.

for inductive load of 80 per cent, power-factor i x = 0.6 1, i = 0.8 I:

ei = V4 X 10“ - 0.0064 7 2 - 2.06 7;

and for anti-inductive load of 80 per cent, power-factor i y =

• 0.6/, i = 0.8/:

ei = V4~xl[)« -T7* - 0.5 7.

A.s seen, due to the internal impedance, and especially the resistance of this machine, the regulation is very poor, and even at the chosen anti~indu.ctive load no rise of voltage occurs.

122. Of more theoretical interest is the case (6), 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, Ex, in the rotor against the impressed e.m.f. of the synchronous motor exciter, Eo. 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, Eo , 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, E 0 , 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 Eo = E = Ei.

If a load is put on the stator circuit by taking a current, I, from the same, the terminal voltage, Ei, drops below the gene- rated e.m.f., E, by the drop of voltage in the impedance, Z h of the stator circuit. Corresponding to the stator current, h, a current, I 2 , then exists in the rotor circuit, giving the same ampere-turns as h, in opposite direction, and so neutralizing the m.m.f. of the stator (as in any transformer). This current, / 2 , 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 E 0 is maintained constant, E rises above E 0 with increasing load, by the drop of voltage in the rotor impedance, Z 2 , and the synchronous impedance, Zo, of the exciter.

That is, the stator terminal voltage, Ei, 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 that in the stator, with increasing load the terminal voltage, E h 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, J, differs in phase from the generated e.m.f., E, the corresponding current, 1 2 , also differs; but a lagging component of 1 1 corresponds to a leading component in / 2 , since the stator circuit slips behind, the rotor circuit is driven ahead of the rotating magnetic field, and inversely, a leading component of 1 1 gives a lagging component of 1 2 . 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 1 = 7*1 + jx 1 = stator self-inductive impedance,

Z 2 = r 2 + jx 2 = rotor self-inductive impedance, reduced to the

stator circuit by the ratio of the effective turns, t = n2 , and the

7 n 1

ratio of frequencies, a = £0

h.

fi

= To + jx 0 = synchronous impedance of the synchronous- motor exciter;

= terminal voltage of the stator, chosen as real axis, = ei;

Eq = 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 f = taE, and the actual nominal generated e.m.f. of the synchronous exciter is E\ = taE 0 .

Let;

E 1

h = i — ji x = 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:

1 2 = i + jii.

(If Y = exciting admittance, the exciting current is J 0 = EY , and the total rotor current then 7 0 + 1 2 -)

Then in the rotor circuit :

E — Eo (Z 0 + Z 2 ) h, (1)

and in the stator circuit :

E = Ei + ZJ X . (2)

Hence:

Ei = Eo + 1 2 (Z 0 + Zo) — IiZij (3)

or, substituting for I x and / 2 :

Ei = Eo + i (Zo + Z 2 — Zi) + jii (Zo + Z 2 + Zi). (4) Denoting now :

Zo + Zi + Z2 = Z3 = 7*3 + jX'ly /--w

Zo + Z 2 — Zi = Z.1 = 7*4 + jx 4,

and substituting :

= ft + 7Z4 + jiiZzi ( 6 )

or, since Ei = e\

Eo = 61 — zZ 4 — ji\Z\

= (ei - r 4 i + - j {xd + r 3 ii), (7)

or the absolute value :

eo 2 = (e x — r 4 i + a^i) 2 + (x 4 < + r*t’i) 2 . (8)

Hence :

ei = Ve 0 2 - (xd + r%i 1) 2 + rd — xdi- (9)

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

i = 0, ii = 0 and e\ = e 0 .

SYNCHRONOUS INDUCTION GENERATOR 209

At non-inductive load:

ii = 0 and ei = \Ze 0 2 — xdi 2 + rd. (10)

ei first increases, from its no-load value, e 0 , reaches a maximum,

and then decreases Since:

again.

r 4 = r 0 + r 2 - r h

at:

X 4 = x 0 + %2 — r h r 4 = 0 and £ 4 = 0,

or,

71 = 70 + r 2 ,

and:

%i — Xo + x 2 ,

Ci = e 0 , that is, in this case the terminal vol-

tage is constant at all non-inductive loads, at constant exciter excitation.

In general , or for h — i — ji h