Theory and Calculation of Electrical Apparatus (1917) — part 4 of 21

If the induction machine has n times as many poles as the synchronous machine, the frequency of rotation of the synchro- 1

nous machine is ~ that of the induction machine, or How-

n ’ n

ever, the frequency generated by the synchronous machine must

be the frequency of the induction-machine secondary currents,

that is, the frequency of slip s.

Hence :

1—5

that is, the concatenated couple is synchronous, that is, runs at constant speed at all loads, but not at synchronous speed, but at

constant slip — 7 - 7 * r n + 1

4. Concatenation with a low-frequency commutating machine.

If a commutating machine is mounted on the induction-motor

shaft, and connected in series into the induction-motor secondary, the commutating machine generates an alternating voltage of the frequency of the currents which excite its field, and if the field is excited in series or shunt with the armature, in the circuit of the induction machine secondary, it generates voltage at the frequency of slip, whatever the latter may be. That is, the induction motor remains asynchronous, increases in slip with increase of load.

5. Excitation by a condenser in the secondary circuit of the induction motor.

As the magnetizing current required by the induction motor is a reactive, that is, wattless lagging current, it does not require a generator for its production, but any apparatus consuming lead- ing, that is, generating lagging currents, such as a condenser, can be used to supply the magnetizing current.

40 . However, condenser, or synchronous or commutating machine, etc., in the secondary of the induction motor do not merely give the magnetizing current and thereby permit power- factor control, but they may, depending on their design or appli- cation, change the characteristics of the induction machine, as regards to speed and speed regulation, the capacity, etc.

56'

ELECTRICAL APPARATUS

If by synchronous or commutating machine a voltage is inserted into the secondary of the induction machine, this vol- tage may be constant, or varied with the speed, the load, the slip, etc., and thereby give various motor characteristics. Further- more, such voltage may be inserted at any phase relation from zero to 360°. If this voltage is inserted 90° behind the secondary current, it makes this current leading or magnetizing and so in- creases the power-factor. If, however, the voltage is inserted in phase with the secondary induced voltage of the induction machine, it has no effect on the power-factor, but merely lowers the speed of the motor if in phase, raises it if in opposition to the secondary induced voltage of the induction machine, and hereby permits speed control, if derived from a commutating machine. For instance, by a voltage in phase with and proportional to the secondary current, the drop of speed of the motor can be increased and series-motor characteristics secured, in the same manner as by the insertion of resistance in the induction-motor secondary. The difference however is, that resistance in the induction-motor secondary reduces the efficiency in the same proportion as it lowers the speed, and thus is inefficient for speed control. The insertion of an e.m.f., however, while lowering the speed, does not lower the efficiency, as the power corresponding to the lowered speed is taken up by the inserted voltage and returned as output of the synchronous or commutating machine. Or, by inserting a voltage proportional to the load and in opposition to the induced secondary voltage, the motor speed can be maintained constant, or increased with the load, etc.

If then a voltage is inserted by a commutating machine in the induction-motor secondary, which is displaced in phase by angle a from the secondary induced voltage, a component of this vol- tage: sin a, acts magnetizing or demagnetizing, the other com- ponent: cos a, acts increasing or decreasing the speed, and thus various effects can be produced.

As the current consumed by a condenser is proportional to the frequency, while that passing through an inductive reactance is inverse 'proportional to the frequency, when using a condenser in the secondary circuit of the induction motor, its effective im- pedance at the varying frequency of slip is:

Zi‘ = n + j (sx 1 — >

where x 2 is the capacity reactance at full frequency.

INDUCTION MOTOR

57

For s = 0, Z\ a = oo, that is, the motor has no power at or near synchronism.

For:

it is:

Zi s = r u

and the current taken by the motor is a maximum. The power output thus is a maximum not when approaching synchronism, as in the typical induction motor, but at a speed depending on the slip,

and by varying the capacity reactance, x 2 , various values of reson- ance slip, So, thus can be produced, and thereby speed control of the motor secured. However, for most purposes, this is uneco- nomical, due to the very large values of capacity required.

Induction Motor Converted to Synchronous

41 . If, when an induction motor has reached full speed, a direct current is sent through its secondary circuit, unless heavily loaded and of high secondary resistance and thus great slip, it drops into synchronism and runs as synchronous motor.

The starting operations of such an induction motor in conver- sion to synchronous motor thus are (Fig. 21) :

First step: secondary closed through resistance: A.

Second step: resistance partly cut out: B.

Third step: resistance all cut out: C.

Fourth step: direct current passed through the secondary: D.

In this case, for the last or synchronous-motor step, usually the direct-current supply will be connected between one phase and the other two phases, the latter remaining short-circuited to each other, as shown in Fig. 21, D. This arrangement retains a short-circuit in the rotor — now the field — in quadrature with the excitation, which acts as damper against hunting (Danielson motor).

58

ELECTRICAL APPARATUS

In the synchronous motor, Fig. 21, D, produced from the induc- tion motor, Fig. 21, C, it is:

Let:

Y 0 = g — jb = primary exciting admittance of the induction machine,

Z Q = r 0 + jx o = primary self-inductive impe- dance,

Z Y = ri + jx i = secondary self-inductive im- pedance.

Fig. 21. — Starting of induction motor and conversion to synchronous.

The secondary resistance, r x , is that of the field exciting winding, thus does not further come into consideration in calculating the motor curves, except in the efficiency, as i x 2 r x is the loss of power in the field, if i\ = field exciting current. Xi is of little further importance, as the frequency is zero. It represents the magnetic leakage between the synchronous motor poles.

To is the armature resistance and x 0 the armature self-inductive reactance of the synchronous machine.

However, x 0 is not the synchronous impedance, which enters the equation of the synchronous machine, but is only the self- inductive part of it, or the true armature self-inductance. The

INDUCTION MOTOR

59

mutual inductive part of the synchronous impedance, or the effective reactance of armature reaction, x f , is not contained in x Q .

The effective reactance of armature reaction of the synchro- nous machine, x\ represents the field excitation consumed by the armature m.m.f., and is the voltage corresponding to this field excitation, divided by the armature current which consumes this field excitation.

6, the exciting susceptance, is the magnetizing armature current, divided by the voltage induced by it, thus, x\ the effect- ive reactance of synchronous-motor armature reaction, is the reciprocal of the exciting susceptance of the induction machine.

The total or synchronous reactance of the induction machine as synchronous motor thus is:

x = x 0 + x f

= + r

The exciting conductance, g, represents the loss by hysteresis, etc., in the iron of the machine. As synchronous machine, this loss is supplied by the mechanical power, and not electrically, and the hysteresis loss in the induction machine as synchronous motor thus is : e 2 g .

We thus have:

The induction motor of the constants, per phase:

Exciting admittance : Yq = g — jb,

Primary self-inductive impedance: Z 0 = r 0 + jx 0 ,

Secondary self-inductive impedance: Zi = 7*! + jx i,

by passing direct current through the secondary or rotor, be- comes a synchronous motor of the constants, per phase:

Armature resistance : r 0 , -

Synchronous impedance: x = x Q +

. (D

Total power consumed in field excitation:

P = 2

(2)

where i = field exciting current.

Power consumed by hysteresis :

P = e 2 g .

( 3 )

60

ELECTRICAL APPARATUS

42. Let, in a synchronous motor:

Eo = impressed voltage,

E = counter e.m.f., or nominal induced voltage,

Z = r + jx = synchronous impedance,

I = i\ — ji 2 = current,

it is then :

Eq — e zj

= E + (n\ + xi 2 ) + j Gcii - n 2 ), (4)

or:

E = E 0 - ZI

— E o — (rii + ®i 2 ) - j (xii - n 2 ), (5)

or, reduced to absolute values, and choosing:

E — e = real axis in equation (4),

Eq = e Q = real axis in equation (5),

e 0 2 = (e + rii + xi 2 ) 2 + (xii — ri 2 ) 2 [e = real axis], (6)

ejf = (e 0 — rii + 2 ^ 2 ) 2 + (xii — ri 2 ) 2 [eo = real axis]. (7)

Equations (6) and (7) are the two forms of the fundamental equation of the synchronous motor, in the form most convenient for the calculation of load and speed curves.

In (7), i\ is the energy component, and i 2 the reactive com- ponent of the current with respect to the impressed voltage, but not with respect to the induced voltage; in (6), ii is the energy component and i 2 the reactive component of the current with respect to the induced voltage, but not with respect to the impressed voltage.

The condition of motor operation at unity power-factor is :

Thus:

i 2 = 0 in equation (7). e 2 = (e 0 — n^ 2 + xHi 2

at no-load, for ii = 0, this gives: e = e 0 , as was to be expected.

Equation (8) gives the variation of the induced voltage and thus of the field excitation, required to maintain unity power- factor at all loads, that is, currents, i.

From (8) follows:

INDUCTION MOTOR

61

Thus, the minimum possible value of the counter e.m.f., e, is given by equating the square root to zero, as:

x

e = ~6o.

£

For a given value of the counter e.m.f., e, that is, constant field excitation, it is, from (7) :

( 10 )

or, if the synchronous impedance, is very large compared with r, and thus, approximately:

z = x:

*' 2 = i x ± VI - (11)

The maximum value, which the energy current, i \ , can have, at a given counter e.m.f., e, is given by equating the square root to zero, as:

( 12 )

For:

it = 0, or at no-load, it is, by (11) :

ep + e x

Equations (9) and (12) give two values of the currents i x and i 2) of which one is very large, corresponds to the upper or unstable part of the synchronous motor-power characteristics shown on page 325 of “ Theory and Calculation of Alternating- current Phenomena,” 5th edition.

43. Denoting, in equation (5) :

U = e' - je", (13)

and again choosing $o = e 0 , as the real axis, (5) becomes :

e f — je" — (e 0 — rii — xi 2 ) — j (xii — n 2 ), (14)

and the electric power input into the motor then is :

Po = /#>, ir

= eoii,

the power output at the armature conductor is :

p : = /&, ir

— e'ii -f- 6 n i‘if

( 15 )

62

ELECTRICAL APPARATUS

hence by (14) :

Pi = h (e 0 — n*i — xi 2 ) + i 2 (%ii — n 2 ), (16)

expanded, this gives :

Pi = Coil - T (ip + ip)

= Po - n 2 , (17)

where: ^ = total current. That is, the power out-

put at the armature conductors is the power input minus the i 2 r loss.

The curfent in the field is :

io — eb, (18)

hence, the i 2 r loss in the field; of resistance, r x .

ipri = e 2 b 2 ri. (19)

The hysteresis loss in the induction motor of mutual induced voltage, e, is: e 2 g, or approximately:

P' = ePg, (20)

in the synchronous motor, the nominal induced voltage, e, does not correspond to any flux, but may be very much higher, than corresponds to the magnetic flux, which gives the hysteresis loss, as it includes the effect of armature reaction, and the hys- teresis loss thus is more nearly represented by epg (20). The difference, however, is that in the synchronous motor the hys- teresis loss is supplied by the mechanical power, and not the electric power, as in the induction motor.

The net mechanical output of the motor thus is:

P = Pi - iPr, - P'

= P 0 — i 2 r — ipri — e 2 g = e Q ii — i 2 r — e 2 b 2 r i — e 2 g, (21)

and herefrom follow efficiency, power-factor and apparent efficiency.

44 * Considering, as instance, a typical good induction motor, of the constants:

6q = 500 volts;

Y 0 = 0.01 - 0.1 j;

Zq = 0.1 -f” 0.3 jj Zi = 0.1 + 0.3

INDUCTION MOTOR

63

The load curves of this motor, as induction motor, calculated in the customary way, are given in Fig. 22.

Converted into a synchronous motor, it gives the constants:

Synchronous impedance (1) :

Z = T + jx ==: 0.1 + 10.3 j.

Fig. 23 gives the load characteristics of the motor, with the power output as abscissae, with the direct-current excitation, and thereby the counter e.m.f., e, varied with the load, so as to maintain unity power-factor.

The calculation is made in tabular form, by calculating for various successive values of the energy current (here also the total current) ii, input, the counter e.m.f., e, by equation (8):

e 2 = (5oo - 0.1 *‘i) 2 + 100.61

the power input, which also is the volt-ampere input, the power- factor being unity, is:

P o — e 0 ii = 500 ii.

From e follow the losses, by (17), (19) and (20):

in armature resistance:

0.1 iS;

in field resistance:

0.001 e 2 ;

hysteresis loss :

2.5 kw.;

and thus the power output :

p * 500 ii -

2.5 - 0.1

and herefrom the efficiency.

Fig. 23 gives the total current as i, the nominal induced voltage as e , and the apparent efficiency which here is the true efficiency, as y.

As seen, the nominal induced voltage has to be varied very greatly with the load, indeed, almost proportional thereto. That is, to maintain unity power-factor in this motor, the field excita- tion has to be increased almost proportional to the load.

It is interesting to investigate what load characteristics are given by operating at constant field excitation, that is, constant nominal induced voltage, e, as this would usually represent the operating conditions,

64

ELECTRICAL APPARATUS

Fig. 22. — Load curves of standard induction motor.

Fig. 23.-r-Load curves at unity power-factor excitation, of standard induc- tion motor converted to synchronous motor.

INDUCTION MOTOR

65

Figs. 24 and 25 thus give the load characteristics of the motor, at constant field excitation, corresponding to:

in Fig. 24: e = 2 e 0 ;

in Fig. 25: e = 5 e 0 .

For different values of the energy current, ij, from zero up to the maximum value possible under the given field excitation,

Fig. 24. — Load curves at constant excitation 2e, of standard induction motor converted to synchronous motor.

as given by equation (12), the reactive current, i 2 , is calculated by equation (11) :

Fig. 24: i, = 48.5 - V9410 - <?;

Fig. 25: U = 48.5 - a/58,800 - i?-

The total current then is :

i = \Zi^+ if;

the volt-ampere input : the power input:

5

Q = e 0 i;

Pq = Gails

66

ELECTRICAL APPARATUS

the power output given by (21), and herefrom efficiency ij, power-factor p and apparent efficient, y, calculated and plotted.

Figs. 24 and 25 give, with the power output as absciss®, the total current input, efficiency, power-factor and apparent efficiency.

As seen from Figs.' 24 and 25, the constants of the motor as synchronous motor with constant excitation, are very bad: the no-load current is nearly equal to full-load current, and power-

Fig. 25. — Load curves at constant excitation 5 e, of standard induction motor converted to synchronous motor.

factor and apparent efficiency are very low except in a narrow range just below the maximum output point, at which the motor drops out. of step.

Thus this motor, and in general any reasonably good induction motor, would be spoiled in its characteristics, by converting it into a synchronous motor with constant field excitation.

In Fig. 23 are shown, for comparison, in dotted lines, the apparent efficiency taken from Figs. 24 and 25, and the apparent efficiency of the machine as induction motor, taken from Fig. 22.

INDUCTION MOTOR

67

45. As further instance, consider the conversion into a syn- chronous motor of a poor induction motor: a slow-speed motor of very high exciting current, of the constants:

e 0 = 500;

Y 0 = 0.02 - 0.6 j;

Z 0 = 0.1 + 0.3 j;

Zt = 0.1 + 0.3 j.

The load curves of this machine as induction motor are given in Pig. 20.

Fig. 26. — Load curves of low-speed high-excitation induction, motor con- verted to synchronous motor, at unity power-factor excitation.

Converted to a synchronous motor, it has the constants: Synchronous impedance:

Z = 0.1 + 1.97 j.

Calculated in the same manner, the load curves, when vary- ing the field excitation with changes of load so as to maintain unity power-factor, are given in Fig. 26, and the load curves for constant field excitation giving a nominal induced voltage:

e = 1.5 Co

are given in Fig. 27.

As seen, the increase of field excitation required to maintain

68

ELECTRICAL APPARATUS

unity power-factor, as shown by curve e in Fig. 26, while still considerable, is very much less in this poor induction motor, than it was in the good induction motor Figs. 22 to 25.

The constant-excitation load curves, Fig. 27, give character- istics, which are very much superior to those of the motor as in- duction motor. The efficiency is not materially changed, as was to be expected, but the power-factor, p, is very greatly improved at all loads, is 96 per cent, at full-load, rises to unity above full-

Fig. 27. — Load curve of low-speed high-excitation induction motor con- verted to synchronous motor, at constant field excitation.

load (assumed as 75 kw.) and is given at quarter-load already higher than the maximum reached by this machine as straight induction motor.

For comparison, in Fig. 28 are shown the curves of apparent efficiency, with the power output as abscissae, of this slow-speed motor, as:

I as induction motor (from Fig. 20) ;

So as synchronous motor with the field excitation varying to maintain unity power-factor (from Fig. 26);

S as synchronous motor with constant field excitation (from

INDUCTION MOTOR

69

As seen, in the constants at load, constant excitation, S, is prac- tically as good as varying unity power-factor excitation, S 0 , drops below it only at partial load, though even there it is very greatly superior to the induction-motor characteristic, I.

It thus follows:

By converting it into a synchronous motor, by passing a direct current through the rotor, a good induction motor is spoiled, but a poor induction motor, that is, one with very high exciting current, is greatly improved.

Fig. 28. — Comparison, of apparent efficiency and speed curves of high- excitation induction motor with various forms of secondary excitation.

46. The reason for the unsatisfactory behavior of a good induc- tion motor, when operated as synchronous motor, is found in the excessive value of its synchronous impedance.

Exciting admittance in the induction motor, and synchronous impedance in the synchronous motor, are corresponding quanti- ties, representing the magnetizing action of the armature cur- rents. In the induction motor, in which the magnetic field is produced by the magnetizing action of the armature currents, very high magnetizing action of the armature current is desirable, so as to produce the magnetic field with as little magnetizing cur- rent as possible, as this current is lagging, and spoils the power- factor. In the synchronous motor, where the magnetic field is produced by the direct current in the field coils, the magnetizing action of the armature currents changes the resultant field excita- tion, and thus requires a corresponding change of the field current to overcome it, and the higher the armature reaction, the more

70

ELECTRICAL APPARATUS

has the field current to be changed with the load, to maintain proper excitation. That is, low armature reaction is necessary.

In other words, in the induction motor, the armature reaction magnetizes, thus should be large, that is, the synchronous react- ance high or the exciting admittance low; in the synchronous motor the armature reaction interferes with the impressed field excitation, thus should be low, that is, the synchronous imped- ance low or the exciting admittance high.

Therefore, a good synchronous motor makes a poor induction motor, and a good induction motor makes a poor synchronous motor, but a poor induction motor — one of high exciting admit- tance, as Fig. 20 — makes a fairly good synchronous motor.

Here a misunderstanding must be guarded against: in the theory of the synchronous motor, it is explained, that high synchronous reactance is necessary for good and stable synchro- nous-motor operation, and for securing good power-factors at all loads, at constant field excitation. A synchronous motor of low synchronous impedance is liable to be unstable, tending to hunt and give poor power-factors due to excessive reactive currents.

This apparently contradicts the conclusions drawn above in the comparison of induction and synchronous motor.

However, the explanation is found in the meaning of high and low synchronous reactance, as seen by expressing the synchro- nous reactance in per cent. : the percentage synchronous reactance is the voltage consumed by full-load current in the synchronous reactance, as percentage of the terminal voltage.

. When discussing synchronous motors, we consider a synchro- nous reactance of 10 to 20 per cent, as low, and a synchronous reactance of 50 to 100 per cent, as high.

In the motor, Figs. 22 to 25, full-load current — at 75 kw. out- put — is about 180 amp. At a synchronous reactance of x = 10.3, this gives a synchronous reactance voltage at full-load current, of 1850, or a synchronous reactance of 370 per cent.

In the poor motor, Figs. 20, 26 and 27, full-load current is about 200 amp., the synchronous reactance x = 1.97, thus the react- ance voltage 394, or 79 per cent., or of the magnitude of good synchronous-motor operation.

That is, the motor, which as induction motor would be consid- ered as of very high exciting admittance, giving a low synchro- nous impedance when converted into a synchronous motor, would as synchronous motor, and from the viewpoint of synchronous-

INDUCTION MOTOR

71

motor design, be considered as a high synchronous impedance motor, while the good induction motor gives as synchronous motor a synchronous impedance of several hundred per cent., that is far beyond any value which ever would be considered in syn- chronous-motor design.

Induction Motor Concatenated with Synchronous

47. Let an induction machine have the constants :

Y 0 = g — jb = primary exciting admittance,

Zq = ro + jx o = primary self-inductive im- pedance,

Zi = n + jx i = secondary self-inductive im- pedance at full frequency, reduced to primary,

and let the secondary circuit of this induction machine be con- nected to the armature terminals of a synchronous machine mounted on the induction-machine shaft, so that the induction- motor secondary currents traverse the synchronous-motor arma- ture, and let :

Z% = r 2 + jx 2 = synchronous impedance of the synchronous machine, at the full frequency im- pressed upon the induction machine.

The frequency of the synchronous machine then is the fre- quency of the induction-motor secondary, that is, the frequency of the induction-motor slip. The synchronous-motor frequency

also is the frequency of synchronous-motor rotation, or - times

To

the frequency of induction-motor rotation, if the induction motor has n times as many poles as the synchronous motor. ’

Herefrom follows:

1—5

72 ELECTRICAL APPARATUS

. Thus the machine couple has synchronous-motor character- istics, and runs at a speed corresponding to synchronous speed of a motor having the sum of the induction-motor and syn- chronous-motor poles as number of poles.

If n = 1, that is, the synchronous motor has the same number of poles as the induction motor,

5 = 0.5,

1 - s = 0.5,

that is, the concatenated couple operates at half synchronous speed, and shares approximately equally in the power output.

If the induction motor has 76 poles, the synchronous motor four poles, n = 19, and:

s = 0.05,

1 - s = 0.95,

that is, the couple runs at 95 per cent, of the synchronous speed of a 76-polar machine, thus at synchronous speed of an 80-polar machine, and thus can be substituted for an 80-polar induction motor. In this case, the synchronous motor gives about 5 per cent., the induction motor 95 per cent, of the output; the synchronous motor thus is a small machine, which could be con- sidered .as a synchronous exciter of the induction machine.

48 . Let:

$o = e'o + je " o = voltage impressed upon in- duction motor.

Jpi = e\ + je" i = voltage induced in induc- tion motor, by mutual magnetic flux, reduced to full frequency.

#2 = e' 2 + je" 2 = nominal induced voltage of synchronous motor, re- duced to full frequency.

/ o = i'o — ji " o = primary current in induc- tion motor.

1 1 = i' i — ji" i = secondary current of in- duction motor and cur- rent in synchronous motor.

Denoting by Z 8 the impedance, Z , at frequency, s, it is:

Total impedance of secondary circuit, at frequency, s :

. Z 8 = Z x s + z % *

= {ri + r 2 ) + f(x 1 -f x 2 ),

( 3 )

INDUCTION MOTOR

73

and the equations are : in primary circuit :

E q = Ei + Zolo', in secondary circuit :

sEi = sE 2 + Z 8 Ii)

and, current:

Jo = h + ye,.

From (6) follows:

h = Io ~ YEh and, substituting (7) into (5) :

= sE 2 + Z 8 Iq — Z 8 YE 1}

hence :

$E 2 -j- Z*I o

s + Z 8 Y 7

substituting (8) into (4) gives:

sE 2 + (Z* + sZq + Z 8 Z Q Y) i 0

E o

and, transposed:

s + Z S Y

or:

Denoting :

and:

r.i.+ r * = v >

s

Xi + x 2 = x'

Z a

— = r' + jV = Z'

it is, substituting into (9) and (10) :

#0 (1 + Z’Y) = $2 + (Z' + Z 0 + Z'ZoY) Jo,

r+zT = - (r+iPF + 2o ) ^

.(4)

(5)

( 6 )

(7)

( 8 )

*(i + T r )-

— + Zo ^

1 + *

< s

y) ] Jo,

(9)

E*

p 7 - ( -

• Zoj

Io •

(10)

Z s

1 L V

■° \s + Z’Y

(ID

( 12 )

(13)

74

ELECTRICAL APPARATUS

Denoting :

Y^zy = £ = (W)

us a voltage which is proportional to the nominal induced voltage of the synchronous motor, and :

TTWY + Zo = Z= r +jx (15)

and substituting (14) and (15) into (13), gives:

E = E 0 - ZIq. (16)

This is the standard synchronous-motor equation, with im- pressed voltage, E 0 , current, / 0 , synchronous impedance, Z ) and nominal induced voltage, E .

Choosing the impressed voltage, E 0 = c 0 as base line, and substituting into (16), gives:

e! + je" = (e 0 - ri' 0 - xi " 0 ) — j (xi' 0 - n" 0 ), (17)

and, absolute:

e 2 = (e 0 — — x 0 i"o) 2 + (^ v o — r 0 i" o) 2 . (18)

• From this equation (18) the load and speed curves of the concatenated couple can now be calculated in the same manner as in any synchronous motor.

That is, the concatenated couple, of induction and synchronous motor, can be replaced by an equivalent synchronous motor of the constants, e, eo, Z and / 0 .

49 . The power output of the synchronous machine is :

= «#*/',

where:

/a + jb , c + jd/'

denotes the effective component of the double-frequency prod- uct: ( ac-pbd ); see “Theory and Calculation of Alternating- current Phenomena,” Chapter XVI, 5th edition.

The power output of the induction machine is:

Pi = //i, (1 - s) (20)

thus, the total power output of the concatenated couple:

P = Pi + P2

= /Ii,sE 2 +(1 — $)Ei/';

( 21 ) •

INDUCTION MOTOR

75

substituting (7) into (21):

P = /Jo ~ Fifr, sP 2 + (1 - *)#i/'; (22)

from (8) follows:

sP 2 = #i (s + ZF) - Z/ 0 , and substituting this into (22), gives:

P = /Io - FP X , (1 + Z*F) - Z s / 0 /'; (23)

from (4) follows:

^1 — Dq — Z 0 Jo,

and substituting this into (23) gives :

P - //o (1 + ZoF) - FP 0 , Po (1 + Z*Y) -

/o (Z* + Z o + Z 0 Z*F)/'. (24)

Equation (24) gives the power output, as function of impressed voltage, E 0 , and supply current, 7 0 .

The power input into the concatenated couple is given by:

Po = /E 0j Io/', (25)

or, choosing i?o = e 0 as base line :

Po = e Q i' 0 . e (26)

The apparent power, or volt-ampere input is given by:

Q = e 0 i 0 , (27)

where :

io = v/ 2’ , o 2 + t"o 2 is the total primary current.

From P, P 0 and Q now follow efficiency, power-factor and apparent efficiency.

50 . As an instance may be considered the power-factor control of the slow-speed 80-polar induction motor of Fig. 20, by a small synchronous motor concatenated into its secondary circuit. Impressed voltage:

Co = 500 volts.

Choosing a four-polar synchronous motor, the induction machine would have to be redesigned with 76 poles, giving:

n = 19 ,

s = 0.05.

7(5

ELECTRICAL APPARATUS

With the same rotor diameter of the induction machine, the pole pitch would be increased inverse proportional to the number of poles, and the exciting susceptance decreased with the square thereof, thus giving the constants :

Yo — g — jb = 0.02 - 0.54 j;

Zq = ro +jx 0 = 0.1 + 0.3 j;

Z i = n + jxi = 0.1 + 0.3 j.

Assuming as synchronous motor synchronous impedance, reduced to full frequency:

Z 2 = 4" jx 2 = 0.02 0.2 j

this gives, for 5 = 0.05 :

Z 8 = (fi + r 2 ) +js (xi + x 2 ) = 0.12 + 0.025 j,

and:

Z' = r' +jx' = ^ = 2.4 + 0.5 j,

o

Z = r+ jx = 0.84 + 1.4 j,

and from (14) :

F = &

‘ 1.32 - 1.29 f

thus:

~ = (50 ° " °- 84lV ° “ lA ^o) 2 + (1-4 i'o - 0.84 i*'o) 2 , (28)

and the power output :

p = // 0 (0.83C + 0.048 j) - (10 - 270 j),

(508 - 32 j) - h (0.241 + 0.326 j)/'. (29)

61 . Fig. 29 shows the load curves of the concatenated couple, under the condition that the synchronous-motor excitation and thus its nominal induced voltage, e 2 , is varied so as to maintain unity power-factor at all loads, that is:

i" o = 0;

this gives from equation (28) : e 2 2

3.39

= (500 - 0.84 »'„)* + 1.96 *' 0 *

78

ELECTRICAL APPARATUS

P = / (0.8361 f'o — 10) + j (0.048 i'o + 270),

(508 = 0.241 i'o) - j (32 + 0.326 i' Q /'

• (0.836 i'o - 10) (508 - 0.241 i'o) - (0.048 i'o + 270)

(32 + 0.326 i).

As seen from the curve, e 2j of the nominal induced voltage, the synchronous motor has to be overexcited at all loads. However, e 2 first decreases, reaches a minimum and then increases again, thus is fairly constant over a wide range of load, so that with this type of motor, constant excitation should give good results.

Fig. 30 then shows the load curves of the concatenated couple for constant excitation, on overexcitation of the synchronous motor of 70 per cent., or

e 2 = 850 volts.

(It must be kept in mind, that e 2 is the voltage reduced to full frequency and turn ratio 1 :1 in the induction machine: At the slip, s == 0.05, the actual voltage of the synchronous motor would be se 2 = 42.5 volts, even if the number of secondary turns of the induction motor equals that of the primary turns, and if, as usual, the induction motor is wound for less turns in the secondary than in the primary, the actual voltage at the synchronous motor terminals is still lower.)

As seen from Fig. 30:

the power-factor is practically unity over the entire range of load, from less than one-tenth load up to the maximum output point, and the current input into the motor thus is practically proportional to the load.

The load curves of this concatenated couple thus are superior to those, which can be produced in a synchronous motor at con- stant excitation.

For comparison, the curve of apparent efficiency, from Fig. 30, is plotted as CS in Fig. 28. It merges indistinguishably into the unity power-factor curve, So, except at its maximum output point.

Induction Motor Concatenated with Commutating Machine

62 . While the alternating-current commutating machine, espe- cially of the polyphase type, is rather poor at higher frequencies, it becomes better at lower frequencies, and at the extremely low frequency of the induction-motor secondary, it is practically as

INDUCTION MOTOR

79

good as the direct-current commutating machine, and thus can be used to insert low-frequency voltage into the induction-motor secondary.

With series excitation, the voltage of the commutating machine is approximately proportional to the secondary current, and the speed characteristic of the induction motor remains essentially the same: a speed decreasing from synchronism at no-load, by a slip, s, which increases with the load.

With shunt excitation, the voltage of the commutating machine is approximately constant, and the concatenated couple thus tends toward a speed differing from synchronism.

In either case, however, the slip, s, is not constant and independ- ent of the load, and the motor couple not synchronous, as when using a synchronous machine as second motor, but the motor couple is asynchronous, decreasing in speed with increase of load.

The phase relation of the voltage produced by the commutating machine, with regards to the secondary current which traverses it, depends on the relation of the commutator brush position with regards to the field excitation of the respective phases, and thereby can be made anything between 0 and 2 t, that is, the voltage inserted by the commutating machine can be energy voltage in phase — reducing the speed — or in opposition to the induction-motor induced voltage — increasing the speed; or it may be a reactive voltage, lagging and thereby supplying the induction-motor magnetizing current, or leading and thereby still further lowering the power-factor. Or the commutating machine voltage may be partly in phase — modifying the speed — and partly in quadrature — modifying the power-factor.

Thus the commutating machine in the induction-motor secondary can be used for power-factor control or for speed control or for both.

It is interesting to note that the use of the commutating ma- chine in the induction- motor secondary gives two independent variables : the value of the voltage, and its phase relation to the current of its circuit, and the motor couple thus has two degrees of freedom. With the use of a synchronous machine in the induction-motor secondary this is not the case; only the voltage of the synchronous machine can be controlled, but its phase adjusts itself to the phase relation of the secondary circuit, and the synchronous-motor couple thus has only one degree of free- dom. The reason is: with a synchronous motor concatenated to

80

ELECTRICAL APPARATUS

the induction machine, the phase of the synchronous machine is fixed in space, by the synchronous-motor poles, thus has a fixed relation with regards to the induction-motor primary system. As, however, the induction-motor secondary has no fixed position relation with regards to the primary, but can have any position slip, the synchronous-motor voltage has no fixed position with regards to the induction-motor secondary voltage and current, thus can assume any position, depending on the relation in the secondary circuit. Thus if we assume that the synchronous- motor field were shifted in space by a position degrees (electrical) : this would shift the phase of the synchronous-motor voltage by a degrees, and the induction-motor secondary would slip in posi- tion by the same angle, thus keep the same phase relation with regards to the synchronous-motor voltage. In the couple with a commutating machine as secondary motor, however, the posi- tion of the brushes fixes the relation between commutating- machine voltage and secondary current, and thereby imposes a definite phase relation in the secondary circuit, irrespective of the relations between secondary and primary, and no change of relative position between primary and secondary can change this phase relation of the commutating machine.

Thus the commutating machine in the secondary of the induc- tion machine permits a far greater variation of conditions of operation, and thereby gives a far greater variety of speed and load curves of such concatenated couple, than is given by the use of a synchronous motor in the induction-motor secondary.

53 . Assuming the polyphase low-frequency commutating machine is series-excited, that is, the field coils (and compensat- ing coils, where used) in series with the armature. Assuming also that magnetic saturation is not reached within the range of its use.

The induced voltage of the commutating machine then is proportional to the secondary current and to the speed.

Thus: 1 e 2 = pii (1)

is the commutating-machine voltage at full synchronous speed, where i x is the secondary current and p a constant depending on the design.

At the slip, s } and thus the speed (1 — $), the commutating machine voltage thus is:

(1 — s) e 2 = (1 — s) pii .

( 2 )

INDUCTION MOTOR

81

As this voltage may have any phase relation with regards to the current, i ly we can put:

$2 = (pi + jp2)h

(3)

where :

v = Vv 1 2 + y>2 2

(4)

and:

tan co =

Pi

(5)

is the angle of brush shift of the commutating machine.

(pi + jp 2 ) is of the nature and dimension of an impedance, and we thus can put :

Z° = p 1 + jp 2 (6)

as the effective impedance representing the commutating machine. At the speed (1 — s),

the commutating machine is represented by the effective impedance:

(1 - s) Z° = (1 — s) pi + j (1 - s) p 2 . (7)

It must be understood, however, that in the effective impedance of the commutating machine,

Z° = Pi +jp2 ,

Pi as well as p 2 may be negative as well as positive.

That is, the energy component of the effective impedance, or the effective resistance, pi, of the commutating machine, may be negative, representing power supply. This simply means, that the commutator brushes are set so as to make the commutating machine an electric generator, while it is a motor, if pi is positive.

If pi = 0, the commutating machine is a producer of wattless or reactive power, inductive for positive, anti-inductive for negative, p 2 .

The calculation of an induction motor concatenated with a commutating machine thus becomes identical with that of the straight induction motor with short-circuited secondary, except that in place of the secondary inductive impedance of the induc- tion motor is substituted the total impedance of the secondary circuit, consisting of :

1. The secondary self-inductive impedance of the induction machine.

6

82

ELECTRICAL APPARATUS

2. The self-inductive impedance of the commutating machine comprising resistance and reactance of armature and of field, and compensating winding, where such exists.

3. The effective impedance representing the commutating machine.

It must be considered, however, that in (1) and (2) the re- sistance is constant, the reactance proportional to the slip, s, while (3) is proportional to the speed (1 — a).

54. Let:

Yo — g— jb = primary exciting admittance of the induction motor.

Z o = r 0 + jx o = primary self-inductive im- pedance of the induction

motor.