Theory and Calculation of Alternating Current Phenomena (1900) — part 10 of 19

l^-n^be = total maximum M.M.F., as resultant of the M.M.Fs. of the /0-phases, combined by the parallelogram of M.M.Fs.*

If (R = reluctance of magnetic circuit per pole, as dis- cussed in Chapter X., it is

A^^ft*.

• Complete discussion hereof, see Chapter XXV.

INDUCTION MOTOR. 241

Thus, from the hysteretic loss, and the reluctance, the constants, g and b, and thus the admittance, Fare derived.

Let rQ = resistance per primary circuit ; XQ = reactance per primary circuit ; thus,

•^o = ro — j XQ = impedance per primary circuit;

rv = resistance per secondary circuit reduced to pri- mary system ;

xv = reactance per secondary circuit reduced to primary system, at full frequency, .A7";

hence,

sx! = reactance per secondary circuit at slip s; and

= secondary internal impedance.

154. We now have, Primary induced E.M.F.,

E = -e. Secondary induced E.M.F.,

Hence, Secondary current,

*-$—

Component of primary current, corresponding thereto, primary load current,

7" --/, =

Primary exciting current,

/0 =eY=e(g+jfy; hence,

242 ALTERNATING-CURRENT PHENOMENA.

Total primary current,

E.M.F. consumed by primary impedance,

E.M.F. required to overcome the primary induced E.M.F.,

• E = e; hence,

Primary terminal voltage, E. = e + Ez

We get thus, in an induction motor, at slip s and active E.M.F. e,

Primary terminal voltage,

Primary current,

or, in complex expression, Primary terminal voltage,

Primary current,

INDUCTION MOTOR. 243

To eliminate e, we divide, and get,

Primary current, at slip s, and impressed E.M.F., £0;

f=^—

or,

/= _ j + (>i-yji _ E

" (

Neglecting, in the denominator, the small quantity F, it is

Z, F

0 + r\

or, expanded,

[(j^ + A'0) + r^ -f s^ (rog -

+/ [J3 (jfo+^O + r^+JT! (xtg+r^+fx^ (xj>+ xj-

Hence, displacement of phase between current and E.M.F., tan , = ^(^o+^

Neglecting the exciting current, /<„ altogether, that is, setting Y = 0, We have

7= sEn^-

„ S

tan <D0 =

244

AL TEKNA TING-CURRENT PHENOMENA.

155. In graphic representation, the induction motor dia- gram appears as follows : —

Denoting the magnetism by the vertical vector O<b in Fig. 114, the M.M.F. in ampere-turns per circuit is repre- sented by vector OF, leading the magnetism O<& by the angle of hysteretic advance a. The E.M.F. induced in the secondary is proportional to the slip s, and represented by ~OEl at the amplitude of 180°. Dividing ~OEl by a in the proportion of rt -*- sxv and connecting a with the middle b of the upper arc of the circle OEV this line intersects the lower arc of the circle at the point 7X rr Thus, OIj\ is the E.M.F. consumed by the secondary resistance, and OI^ equal and parallel to EJ^ is the E.M.F. consumed by the secondary reactance. The angle, E^OI^\ = ^ is the angle of secondary lag.

\

The secondary M.M.F. OGl is in the direction of the vector OIfv Completing the parallelogram of M.M.Fs. with OF as diagonal and OGl as one side, gives the primary M.M.F. OG as other side. The primary current and the E.M.F. consumed by the primary resistance, represented by OIry is in line with OG, the E.M.F. consumed by the pri- mary reactance 90° ahead of OG, and represented by OIxv and their resultant Ofz0 is the E.M.F. consumed by the

INDUCTION MOTOR.

245

primary impedance. The E.M.F. induced in the primary circuit is OE', and the E.M.F. required to overcome this counter E.M.F. is OE equal and opposite to OE1. Com- bining OE with OIzQ gives the primary terminal voltage represented by vector OEy and the angle of primary lag, EOG

Fig. 115.

156. Thus far the diagram is essentially the same as the diagram of the stationary alternating-current trans- former. Regarding dependence upon the slip of the motor, the locus of the different quantities for different values of the slip s is determined thus,

246 ALTERNATING-CURRENT PHENOMENA.

Let £l = s£f

Assume in opposition to O&, a point A, such that

O A -r- 7X rx = Ev -• /! J.!, then

/ir, x .£", /ir, x sE r, _,

= - ^ = constant.

That is, /^ lies on a half-circle with OA = — E' as diameter.

That means Gl lies on a half-circle ^ in Fig. 115 with OC as diameter. In consequence hereof, G0 lies on half- circle^ with FB equal and parallel to OCas diameter.

Thus Ir0 lies on a half -circle with DH as diameter, which circle is perspective to the circle FB, and Ix0 lies on a half- circle with IK as diameter, and IzQ on a half-circle with LN as diameter, which circle is derived by the combination of the circles Ir0 and Ixv

The primary terminal voltage EQ lies thus on a half- circle e0 equal to the half-circle Iz9 and having to point E the same relative position as the half-circle Iz^ has to point 0.

This diagram corresponds to constant intensity of the maximum magnetism, O®. If the primary impressed volt- age EQ is kept constant, the circle e0 of the primary im- pressed voltage changes to an arc with O as center, and all the corresponding points of the other circles have to be reduced in accordance herewith, thus giving as locus of the other quantities curves of higher order which most con- veniently are constructed point for point by reduction from the circle of the loci in Fig. 115.

Torque and Power.

157. The torque developed per pole by an electric motor equals the product of effective magnetism, ® / V2, times ef- fective armature M.M.F., F / V2, times the sine of the angle between both,

INDUCTION MOTOR. 247

If «! = number of turns, 7t = current, per circuit, with /rarmature circuits, the total maximum current polarization, or M.M.F. of the armature, is

Hence the torque per pole,

If q = the number of poles of the motor, the total torque of the motor is,

The secondary induced E.M.F., Ev lags 90° behind the inducing magnetism, hence reaches a maximum displaced in space by 90° from the position of maximum magnetization. Thus, if the secondary current, Iv lags behind its E.M.F., Ev by angle, <av the space displacement between armature current and field magnetism is

hence sin (4> fj) = cos o^

We have, however,

thus, «! <$

substituting these values in the equation of the torque, it is T.

248 ALTERNATING-CURRENT PHENOMENA.

or, in practical (C.G.S.) units,

is the Torque of the Induction Motor.

At the slip s, the frequency N, and the number of poles q, the linear speed at unit radius is

hence the output of the motor, P= TV or, substituted,

is the Power of the Induction Motor.

158. We can arrive at the same results in a different way :

By the counter E.M.F. e of the primary circuit with current / ' = f0 + 7X the power is consumed, e I = e I0 + e 7r The power e I0 is that consumed by the primary hysteresis and eddys. The power e 1^ disappears in the primary circuit by being transmitted to the secondary system.

Thus the total power impressed upon the .secondary system, per circuit, is

Pi-tf,

Of this power a part, £1fl, is consumed in the secondary circuit by resistance. The remainder,

P' = fl(e-£1),

disappears as electrical power altogether ; hence, by the law of conservation of energy, must reappear as some other form of energy, in this case as mechanical power, or as the output of the motor (including friction).

Thus the mechanical output per motor circuit is

INDUCTION MOTOR. 249

Substituting,

se; se

it is

hence, since the imaginary part has no meaning as power,

and the total power of the motor,

At the linear speed, at unit radius the torque is

In the foregoing, we found

£0 = e\ 1 + j|? + Z, Y or, approximately,

or, expanded,

or, eliminating imaginary quantities,

250 ALTERNATING-CURRENT PHENOMENA.

Substituting this value in the equations of torque and of power, they become,

torque, T =

Maximum Torque.

159.  The  torque  of  the  induction  motor  is  a  maximum 
     

for that value of slip s, where

qpi r^ Eg s or, since T = -. — .T, .

4 7T JV^ (>1

for,

ds

expanded, this gives,

r2 "7

or, st =

Substituting this in the equation of torque, we get the value of maximum torque,

That is, independent of the secondary resistance, rr The power corresponding hereto is, by substitution of st in P,

Pt = ;

This power is not the maximum output of the motor, but already below the maximum output. The maximum output is found at a lesser slip, or higher speed, while at the maximum torque point the output is already on the decrease, due to the decrease of speed.

INDUCTION MOTOR. 251

With increasing slip, or decreasing speed, the torque of the induction motor increases ; or inversely, with increasing load, the speed of the motor decreases, and thereby the torque increases, so as to carry the load down to the slip st, corresponding to the maximum torque. At this point of load and slip the torque begins to decrease again ; that is, as soon as with increasing load, and thus increasing slip, the motor passes the maximum torque point st, it " falls out of step," and comes to a standstill.

Inversely, the torque of the motor, when starting from rest, will increase with increasing speed, until the maximum torque point is reached. From there towards synchronism the torque decreases again.

In consequence hereof, the part of the torque-speed curve below the maximum torque point is in general un- stable, and can be observed only by loading the motor with an apparatus, whose countertorque increases with the speed faster than the torque of the induction motor.

In general, the maximum torque point, st, is between synchronism and standstill, rather nearer to synchronism. Only in motors of very large armature resistance, that is low efficiency, st > 1, that is, the maximum torque falls below standstill, and the torque constantly increases from synchronism down to standstill.

It is evident that the position of the maximum torque point, st can be varied by varying the resistance of the secondary circuit, or the motor armature. Since the slip of the maximum torque point, st, is directly proportional to the armature resistance, rlf it follows that very constant speed and high efficiency will bring the maximum torque point near synchronism, and give small starting torque, while good starting torque means a maximum torque point at low speed ; that is, a motor with poor speed regulation* and low efficiency.

Thus, to combine high efficiency and close speed regula- tion with large starting torque, the armature resistance has

252 ALTERNATING-CURRENT PHENOMENA.

to be varied during the operation of the motor, and the motor started with high armature resistance, and with in- creasing speed this armature resistance cut out as far as possible.

160. If *=:1,__

it is ^ = Vr02 + (xl + *0)2.

In this case the motor starts with maximum torque, and when overloaded does not drop out of step, but gradually slows down more and more, until it comes to rest.

If, st>l,

then ^ > Vr02 + (^ + *0)2.

In this case, the maximum torque point is reached only by driving the motor backwards, as countertorque.

As seen above, the maximum torque Tt, is entirely in- dependent of the armature resistance, and likewise is the current corresponding thereto, independent of the armature resistance. Only the speed of the motor depends upon the armature resistance.

Hence the insertion of resistance into the motor arma- ture does not change the maximum torque, and the current corresponding thereto, but merely lowers the speed at which the maximum torque is reached.

The effect of resistance inserted into the induction motor is merely to consume the E.M.F., which otherwise would find its mechanical equivalent in an increased speed, analo- gous as resistance in the armature circuit of a continuous- current shunt motor.

Further discussion on the effect of armature resistance is found under " Starting Torque."

Maximum Power.

161. The power of an induction motor is a maximum for that slip, sv, where

INDUCTION MOTOR. 253

expanded, this gives

sn — -

substituted in P, we get the maximum power,

2 {('i + ''o) + (^ + r0)2 + (^i + *o)2}

This result has a simple physical meaning : (i\ + r0) = r is the total resistance of the motor, primary plus secondary (the latter reduced to the primary), (x^ + x^ is the total reactance, and thus Vrx + r0)2 + (x^ + x0}z = z is the total impedance of the motor. Hence

is the maximum output of the induction motor, at the slip,

The same value has been derived in Chapter IX., as the maximum power which can be transmitted into a non- inductive receiver circuit over a line of resistance r, and impedance z, or as the maximum output of a generator, or of a stationary transformer. Hence :

The maximum output of an induction motor is expressed by the same formula as the maximum output of a generator, or of a stationary transformer, or the maximum output which can be transmitted over an inductive line into a non-inductive- receiver circuit.

The torque corresponding to the maximum output Pp is,.

254 ALTERNATING-CURRENT PHENOMENA.

This is not the maximum torque ; but the maximum torque, Tt, takes place at a lower speed, that is, greater slip,

• since,

-that is, st > sp.

It is obvious from these equations, that, to reach as large an output as possible, r and z should be as small as possible ; that is, the resistances ^ + r0, and the impedances, z, and thus the reactances, x± + x0, should be small. Since r± + r0 is usually small compared with x^ -f- x0 it follows, that the problem of induction motor design consists in con- structing the motor so as to give the minimum possible reactances, x^ + x0.

Starting Torque.

162. In the moment of starting an induction motor, the slip is

hence, starting current,

Oo -

or, expanded, with the rejection of the last term in the denominator, as insignificant,

T _io11 010,io1 .

• 8

and, displacement of phase, or angle of lag,

fi + r0] + *! [Jfx 4- Jf0]) - jf (r0 ^ - *0 rt)

„ _ 1 W°

r0)

INDUCTION MOTOR. 255

Neglecting the exciting current, g = 0 = b, these equa- tions assume the form,

or, eliminating imaginary quantities,

and tan w0 =

• 'o

That means, that in starting the induction motor without additional resistance in the armature circuit, — in which case ^ + x0 is large compared with t\ •+• r0, and the total impe- dance, z, small, — the motor takes excessive and greatly lagging currents.

The starting torque is

T0=

That is, the starting torque is proportional to the armature resistance, and inversely proportional to the square of the total impedance of the motor.

It is obvious thus, that, to secure large starting torque, the impedance should be as small, and the armature resis- tance as large, as possible. The former condition is the condition of large maximum output and good efficiency and speed regulation ; the latter condition, however, means inefficiency and poor regulation, and thus cannot properly be fulfilled by the internal resistance of the motor, but only by an additional resistance which is short-circuited while the motor is in operation.

256 ALTERNATING-CURRENT PHENOMENA.

Since, necessarily,

ri<*,

''<•<

and since the starting current is, approximately,

7 =f , we have, Ta <

would be the theoretical torque developed at 100 per cent efficiency and power factor, by E.M.F., E0, and current, /, at synchronous speed.

Thus, T0<T00,

and the ratio between the starting torque T0, and the theo- retical maximum torque, T^, gives a means to judge the perfection of a motor regarding its starting torque.

This ratio, T0 / Tw, exceeds .9 in the best motors.

Substituting 7 = E0 / z in the equation of starting torque, it assumes the form,

7V,.

Since 4 IT N / q = synchronous speed, it is :

The starting torque of the induction motor is equal to the resistance loss in the motor armature, divided by the synchro- nous speed.

The armature resistance which gives maximum starting torque is

INDUCTION MOTOR. 257

dr, expanded, this gives,

the same value as derived in the paragraph on "maximum torque."

Thus, adding to the internal armature resistance, r/ in starting the additional resistance,

makes the motor start with maximum torque, while with in- creasing speed the torque constantly decreases, and reaches zero at synchronism. Under these conditions, the induc- tion motor behaves similarly to the continuous-current series motor, varying in the speed with the load, the difference being, however, that the induction motor approaches a definite speed at no load, while with the series motor the speed indefinitely increases with decreasing load.

The additional armature resistance, t", required to give a certain starting torque, if found from the equation of starting torque :

Denoting the internal armature resistance by rj, the total armature resistance is ^ = r^ + r".

and thus, ?A Eg rj + r"

4 TT N (r^ + r^ + r0)2 + (Xl + *0)2 ' hence,

This gives two values, one above, the other below, the maximum torque point.

258 ALTERNATING-CURRENT PHENOMENA.

Choosing the positive sign of the root, we get a larger armature resistance, a small current in starting, but the torque constantly decreases with the speed.

Choosing the negative sign, we get a smaller resistance, a large starting current, and with increasing speed the torque first increases, reaches a maximum, and then de- creases again towards synchronism.

These two points correspond to the two points of the speed-torque curve of the induction motor, in Fig. 116, giving the desired torque T0.

The smaller value of r1" will give fairly good speed regu- lation, and thus in small motors, where the comparatively large starting current is no objection, the permanent arma- ture resistance may be chosen to represent this value.

The larger value of rj' allows to start with minimum current, but requires cutting out of the resistance after the start, to secure speed regulation and efficiency.

Synchronism. 163. At synchronism, s = 0, we have,

or,

0, T=Q;

that is, power and torque are zero. Hence, the induction motor can never reach complete synchronism, but must slip sufficiently to give the torque consumed by friction.

Running near Synchronism.

164. When running near synchronism, at a slip s above the maximum output point, where s is small, from .02 to .05 at full load, the equations can be simplified by neglect- ing terms with s, as of higher order.

INDUCTION MOTOR. 25 £

We then have, current,

or, eliminating imaginary quantities,

angle of lag, o*i + *o ,

c2 (r_ -I- <r_\ -4- r.2 h r.

tan w0

T =

or, inversely,

A A

that is,

Near sychronism, the slip, s, of an induction motor, or its drop in speed, is proportional to the armature resistance> i\ and to the power, P, or torque, T.

Example.

165. As an instance are shown, in Fig. 116, character- istic curves of a 20 horse-power three-phase induction motor, of 900 revolutions synchronous speed, 8 poles, frequency of 60 cycles.

The impressed E.M.F. is 110 volts between lines, and the motor star connected, hence the E.M.F. impressed per circuit :

~ = 63.5 ; or EQ = 63.5.

260

AL TERN A TING-CURRENT PHENOMENA.

The constants of the motor are :

Primary admittance, Y = .1 + .4 j. Primary impedance, Z = .03 — .09 j. Secondary impedance, Zx = .02 — .085/.

In Fig. 116 is shown, with the speed in per cent of

•synchronism, as abscissae, the torque in kilogrammetres,

as ordinates, in drawn lines, for the values of armature resistance :

116. Speed Characteristics of Induction Motor.

rt = .02 : short circuit of armature, full speed.

^ = .045 : .025 ohms additional resistance.

^ = .18 : .16 ohms additional, maximum starting torque.

^ = .75 : .73 ohms additional, same starting torque as rt == .045.

On the same Figure is shown the current per line, in dotted lines, with the verticals or torque as abscissae, and the horizontals or amperes as ordinates. To the same torque always corresponds the same current, no matter what the speed be.

INDUCTION MOTOR.

261

On Fig. 117 is shown, with the current input per line as abscissae, the torque in kilogrammetres and the output in horse-power as ordinates in drawn lines, and the speed and the magnetism, in per cent of their synchronous values, as ordinates in dotted lines, for the armature resistance ^ = .02 or short circuit.

20

lase Induotio Motor.

. 60Cyc

110V

Jiagram

=.03-.09j z£0=J&B

\

\

\

12

-1

Amperes 150 1 200

2,50

300

Fig. 117. Current Characteristics of Induction Motor.

In Fig. 118 is shown, with the speed, in per cent of synchronism, as abscissae, the torque in drawn line, and the output in dotted line, for the value of armature resist- ance ?i = .045, for the whole range of speed from 120 per

262

ALTERNA TING-CURRENT PHENOMENA.

cent backwards speed to 220 per cent beyond synchronism, showing the two maxima, the motor maximum at s = .25, and the generator maximum at s = — .25.

166. As  seen  in  the  preceding,  the  induction  motor  is 
     

characterized by the three complex imaginary constants,

Y0 = g0 +jbw the primary exciting admittance, Z0 = r0 —jx0, the primary self-inductive impedance, and Zi = r± — jx^ the secondary self-inductive impedance,

Fig. 1 18. Speed Characteristics of Induction Motor.

reduced to the primary by the ratio of secondary to pri- mary turns.

From these constants and the impressed E.M.F. cot the motor can be calculated as follows :

Let,

e = counter E.M.F. of motor, that is E.M.F. induced in the primary by the mutual magnetic flux.

At the slip s the E.M.F. induced in the secondary cir- cuit is, se

INDUCTION MOTOR. 263

Thus the secondary current,

where,

«l = -5T

r* + Atf r? +

The primary exciting current is,

thus, the total primary current,

/0 = /! + /oo = * (^i + A) where,

The E.M.F. consumed by the primary impedance is, ^ = /oZ0 = * (r0 ->0) (^

the primary counter E.M.F. is e, thus the primary impressed E.M.F.,

£, where,

c\ — or, absolute,

^0 =

hence,

This value substituted gives,

Secondary current,

ffi+A A = *b T7=

Primary current,

°~

Impressed E.M.F.,

264 ALTERNATING-CURRENT PHENOMENA.

Thus torque, in synchronous watts (that is, the watts output the torque would produce at synchronous speed),

tf + tf

hence, the torque in absolute units,

= =

N (f* + r22) W where N= frequency.

The power output is torque times speed, thus :

The power input is,

^•l2 +

The voltampere input,

o2 ( Vi + V,) /o2 ( Vi - V8)

hence,

efficiency,

J\ _ a, (I - s)

J? Vi + V2

power factor,

apparent efficiency,

<2o

torque efficiency, * a.

./V Vi + V.

• That 5s the ratio of actual torque to torque which would be profloced, if there were nc losses of energy in the motor, at the same power input.

INDUCTION MOTOR. 265

apparent torque efficiency,*

rrt

Q0 ~ V W+1?YT^

167. Most instructive in showing the behavior of an induction motor are the load curves and the speed curves.

The load curves are curves giving, with the power out- put as abscissae, the current imput, speed, torque, power factor, efficiency, and apparent efficiency, as ordinates.

The speed curves give, with the speed as abscissae, the torque, current input, power factor, torque efficiency, and apparent torque efficiency, as ordinates.

The load curves characterize the motor especially at its normal running speeds near synchronism, the speed curves over the whole range of speed.

In Fig. 119 are shown the load curves, and in Fig. 120 the speed curves of a motor of the constants, K0 = .01 + .!/

z* = .i -.3>

Z, = .1 - .3j

INDUCTION GENERATOR.

168. In the foregoing, the range of speed from s = 1, standstill, to s = 0, synchronism, has been discussed. In this range the motor does mechanical work.

It consumes mechanical power, that is, acts as generator or as brake outside of this range.

For, s > 1, backwards driving, P becomes negative, representing consumption of power, while T remains posi- tive ; hence, since the direction of rotation has changed, represents consumption of power also. All this power is consumed in the motor, which thus acts as brake.

For, s < 0, or negative, P and T become negative, and the machine becomes an electric generator, converting me- chanical into electric energy.

• That is the ratio of actual torque to torque which would be produced if there were neither losses of energy nor phase displacement in the motor, at the same voltampere input.

266

ALTERNA TING-CURRENT PHENOMENA.

The calculation of the induction generator at constant frequency, that is, at a speed increasing with the load by the negative slip, slt is the same as that of the induction motor except that sl has negative values, and the load curves for the machine shown as motor in Fig. 119 are shown in Fig. 121 for negative slip s{ as induction generator.

CURV

POWER 4000

"£>

Fig. 119.

Again, a maximum torque point and a maximum output point are found, and the torque and power increase from zero at synchronism up to a maximum point, and then de- crease again, while the current constantly increases.

INDUCTION MOTOR.

267

Fig. 120.

268 ALTERNATING-CURRENT PHENOMENA.

169. The induction generator differs essentially from the ordinary synchronous alternator in so far as the induc- tion generator has a definite power factor, while the syn- chronous alternator has not. That is, in the synchronous alternator the phase relation between current and terminal voltage entirely depends upon the condition of the external circuit. The induction generator, however, can operate only if the phase relation of current and E.M.F., that is, the power factor required by the external circuit, exactly coin- cides with the internal power factor of the induction gen- erator. This requires that the power factor either of the external circuit or of the induction generator varies with the voltage, so as to permit the generator and the external circuit to adjust themselves to equality of power factor.

Beyond magnetic saturation the power factor decreases ; that is, the lead of current increases in the induction ma- chine. Thus, when connected to an external circuit of con- stant power factor the induction generator will either not generate at all, if its power factor is lower than that of the external circuit, or, if its power factor is higher than that of the external circuit, the voltage will rise until by magnetic saturation in the induction generator its power factor has fallen to equality with that of the external circuit. This, however, requires magnetic saturation in the induction gen- erator, which is objectionable, due to excessive hysteresis losses in the alternating field.

To operate below saturation, — that is, at constant inter- nal power factor, — the induction generator requires an exter- nal circuit with leading current, whose power factor varies with the voltage, as a circuit containing synchronous motors or synchronous converters. In such a circuit, the voltage of the induction generator remains just as much below the counter E.M.F. of the synchronous motor as necessary to give the required leading exciting current of the induction generator, and the synchronous motor can thus to a certain extent be called the exciter of the induction generator.

INDUCTION MOTOR. 269

When operating self-exciting, that is shunt-wound, con- verters from the induction generator, below saturation of both the converter and the induction generator, the condi- tions are unstable also, and the voltage of one of the two machines must rise beyond saturation of its magnetic field.

When operating in parallel with synchronous alternat- ing generators, the induction generator obviously takes its leading exciting current from the synchronous alternator, which thus carries a lagging wattless current.

170. To generate constant frequency, the speed of the induction generator must increase with the load. Inversely, when driven at constant speed, with increasing load on the induction generator, the frequency of the current generated thereby decreases. Thus, when calculating the character- istic curves of the constant speed induction generator, due regard has to be taken of the decrease of frequency with increase of load, or what may be called the slip of fre- quency, s.

Let in an induction generator,

Y0 = gQ + j\ — primary exciting admittance,

Z0 = r0 — jxQ = primary self-inductive impedance,

Zi = r^ — jXj_ = secondary self-inductive impedance,

reduced to primary, all these quantities being reduced to the frequency of synchronism with the speed of the ma- chine, N.

Let e — induced E.M.F., reduced to full frequency.

s = slip of frequency, thus : (1-j) N = frequency gener- ated by machine.

We then have

Secondary induced E.M.F. se thus, secondary current,

r in r\ — Jsx\

270 ALTERNATING-CURRENT PHENOMENA.

where,

primary exciting current,

In = EY0 = e thus, total primary current,

/0 = /i + foo where,

^1 = <*\ + £b

primary impedance voltage, & = S0(r0-

primary induced E.M.F.,

thus, primary terminal voltage,

£0 = e(l-s) -S0(r0-j[l- s] x0) = e where,

fi = ! - s ~ rA - (1 - s hence, absolute,

e0 = e V^ and,

Thus,

Secondary current,

T eO (ai

Primary current,

j _ eo (A + A)

Primary terminal voltage,

j-. ^0 ^"l

£« = —T-,

INDUCTION MOTOR. Torque and mechanical power input,

T— P —\f nl — e°ai r* ~ -e ^ ~ 7^+^

Electrical output,

271

ELECTRICAL OUTPUT P , WATTS 1000 2000 3COO 4000 fiOOO fiOOO 7000 8000

Fig. 122.

Voltampere output, G, = <

Efficiency,

j

power factor,

272 AL TERNA TING-CURRENT PHENOMENA.

or,

p,j b* - V, = ^- = ^T^

In Fig. 122 is plotted the load characteristic of a con- stant speed induction generator, at constant terminal vol- tage e 0 = 110, and the constants,

K0 = .01 + .!/

171. As instance may be considered a power trans- mission from an induction generator of constants Y0, Z0, Zj, over a line of impedance Z = r —jx, into a synchron- ous motor of synchronous impedance Zz = rz — jxz, operat- ing at constant field excitation.

Let, e0 = counter E.M.F. or nominal induced E.M.F. of synchronous motor at full frequency ; that is, frequency of synchronism with the speed of the induction generator. By the preceding paragraph the primary current of the induction generator was,

primary terminal voltage, E0 = e thus, terminal voltage at synchronous motor terminals,

where,

4 = fi ~ rA ~ C1 - J) *A 4 =

Counter E.M.F. of synchronous motor,

E2

'

where,

/ = 4 - r& - (1 or absolute,

INDUCTION MOTOR.

since, however,

Z=.0|4-6j

ULL F EQUE EXCIT/ 5 VOL'

OUTPUT OF SYNCHRONOUS, WATTS 1000 2000 I 8000 4000 5000

274 ALTERNATING-CURRENT PHENOMENA.

Thus,

Current, _ e2 (1 - j) (^ +y7;2)

'

Terminal voltage at induction generator,

Terminal voltage at synchronous motor,

and herefrom in the usual way the efficiencies, power fac- tor, etc. are derived.

When operated from an induction generator, a syn- chronous motor gives a load characteristic very similar to that of an induction motor operated from a synchronous generator, but in the former case the current is leading, in the latter lagging.

In either case, the speed gradually falls off with increas- ing load (in the synchronous motor, due to the falling off of the frequency of the induction generator), up to a maxi- mum output point, where the motor drops out of step and comes to standstill.

Such a load characteristic of the induction generator in Fig. 121, feeding a synchronous motor of counter E.M.F. eQ = 125 volts (at full frequency) and synchronous impe- dance Z2 = .04 — Gj, over a line of negligible impedance is shown in Fig. 123.

CONCATENATION, OR TANDEM CONTROL OF INDUCTION MOTORS.

172. If of two induction motors the secondary of the first motor is connected to the primary of the second motor, the second machine operates as motor with the E.M.F. and frequency impressed upon it by the secondary of the first machine, which acts as general alternating-current trans- former, converting a part of the primary impressed power

INDUCTION MOTOR. 275

into secondary electrical power for the supply of the second machine, and a part into mechanical work.

The frequency of the secondary E.M.F. of the first motor, and thus the frequency impressed upon the second motor, is the frequency of slip below complete synchronism, s. The frequency of the secondary induced E.M.F. of the second motor is the difference between its impressed frequency, s, and its speed ; thus, if both motors are connected together mechanically to turn at the same speed, 1 — s, the secondary frequency of the second motor is 2^—1, hence equal to zero at s = .5. That is, the second motor reaches its syn- chronism at half speed. At this speed its torque becomes equal to zero, the energy current flowing into it, and conse- quently the energy component of the secondary current of the first "motor, and thus the torque of the first motor be- comes equal to zero also, when neglecting the hysteresis energy current of the second motor. That is, a system of concatenated motors with short-circuited secondary of the second motor approaches half synchronism, in the same manner as the ordinary induction motor approaches syn- chronism. With increasing load, its slip below half syn- chronism increases.

More generally, any pair of induction motors connected in concatenation divide the speed so that the sum of their two respective speeds approaches synchronism at no load ; or, still more generally, any number of concatenated motors run at such speeds that the sum of the speeds approaches synchronism at no load.

With mechanical connection between the two motors, concatenation thus offers a means to operate a pair of induction motors at full efficiency at half speed in tandem, as well as at full speed in parallel, and thus gives the same advantage as the series-parallel control of the continuous- current motor.

In starting, a concatenated system is controlled by re- sistance in the armature of the second motor.

276 ALTERNATING-CURRENT PHENOMENA.

Since, with increasing speed, the frequency impressed upon the second motor decreases proportionally to the de- crease of voltage, when neglecting internal losses in the first motor, the magnetic density of the second motor re- mains practically constant, and thus its torque the same as when operated at full voltage and full frequency under the same conditions.