Theory and Calculation of Alternating Current Phenomena (1900) — part 9 of 19

where the reactances, x0 and ^ , refer to the true self -induc- tance only, or to the cross-flux passing between primary and secondary coils ; that is, interlinked with one coil only. Let also

Y = g --jb- total admittance of secondary circuit,

including the internal impedance ; E0 = primary impressed E.M.F. ; E ' = E.M.F. consumed by primary counter E.M.F. ; £i = secondary terminal voltage ; EI = secondary induced E.M.F. ; I0 = primary current, total ; /oo = primary exciting current ; /i = secondary current.

Since the primary counter E.M.F.,-£"', and the second- ary induced E.M.F., E^, are proportional by the ratio of

turns, a,

E ' = — a E{. (1)

The secondary current is :

/i = **/, (2)

consisting of an energy component, gE^, and a reactive component, b E^.

To this secondary current corresponds the component of primary current,

•7o = ~a a*

The primary exciting current is —

I«>=YOE>. (4)

Hence, the total primary current is :

206 AL TERNA TING-CURRENT PHENOMENA.

(6)

The E.M.F. consumed in the secondary coil by the internal impedance is Z-J^.

The E.M.F. induced in the secondary coil by the mag- netic flux is EI.

Therefore, the secondary terminal voltage is

or, substituting (2), we have

£, = £,' {I- Z,Y} (7)

The E.M.F. consumed in the primary coil by the inter- nal impedance is Z0 I0.

The E.M.F. consumed in the primary coil by the counter E.M.F. is E'.

Therefore, the primary impressed E.M.F. is

E0 = E' + Z0S0, or, substituting (6),

(8)

\°/

136. We thus have,

primary E.M.F., E0 = - aE{ j 1 + Z0 Y0 + ^Z J , (8)

secondary E.M.F., E^ = E{ { 1 - Zl Y}, (7)

primary current, I0 = — — -{Y+a*Y0}, (6)

secondary current, /i = YEl't (2)

as functions of the secondary induced E.M.F., EJ, as pa- rameter.

ALTERNATING-CURRENT TRANSFORMER. 207

From the above we derive

Ratio of transformation of E.M.Fs. :

. 1-Z.K

Ratio of transformations of currents :

(10)

From this we get, at constant primary impressed E.M.F.,

E0 = constant ;

secondary induced E.M.F.,

E.M.F. induced per turn, E 1

n0 -\ \ 7 y \

secondary terminal voltage,

primary current,

^ 4- Y , . EA Y+a*Y0 _ w ^^ y°

secondary current,

Y

At constant secondary terminal voltage, -fi1! = const. ;

208 AL TERNA TING-CURRENT PHENOMENA.

secondary induced E.M.F.,

F1 - £l

1-^F' E.M.F. induced per turn,

^1-Z.F' primary impressed E.M.F.,

primary current,

/

secondary current,

136. Some interesting conclusions can be drawn from these equations.

The apparent impedance of the total transformer is

(14)

Substituting now, — = V, the total secondary admit-

tance, reduced to the primary circuit by the ratio of turns, it is

Y0--Y' is the total admittance of a divided circuit with the exciting current, of admittance Y0, and the secondary

AL TERN A TING-CURRENT TRANSFORMER.

209

current, of admittance Y1 (reduced to primary), as branches. Thus :

is the impedance of this divided circuit, and

That is :

(17)

The alternate-current transformer, of primary admittance Y0 , total secondary admittance Y, and primary impedance Z0 , is equivalent to, and can be replaced by, a divided circuit with the branches of admittance Y0 , the exciting current, and admittance Y' = Y/a2, the secondary current, fed over mains of the impedance Z0, the internal primary impedance.

This is shown diagrammatically in Fig. 106.

Yog

z

Fig. 106.

137. Separating now the internal secondary impedance from the external secondary impedance, or the impedance of the consumer circuit, it is

4 -£.+ *! (18)

where Z = external secondary impedance,

(19)

210 ALTERNATING-CURRENT PHENOMENA.

Reduced to primary circuit, it is

= Z/ + Z7. (20)

That is :

An alternate-current transformer, of primary admittance Y0, primary impedance Z0, secondary impedance Zv and ratio of turns a, can, when the secondary circuit is closed by an impedance Z (the impedance of the receiver circuit), be replaced, and is equivalent to a circtiit of impedance Z ' = a?Z, fed over mains of the impedance Z0-- Z^, where Z^ = a2Zlt shunted by a circuit of admittance Y0, which latter circuit branches off at the points a — b, between the impe- dances Z and Z-.

Generator I, Transformer I

Fig. 107.

This is represented diagrammatically in Fig. 107.

It is obvious therefore, that if the transformer contains several independent secondary circuits they are to be con- sidered as branched off at the points a, i, in diagram Fig. 107, as shown in diagram Fig. 108.

It therefore follows :

An alternate-current transformer, of x secondary coils, of the internal impedances Z^, Z^1, . . . Z-f, closed by external secondary circuits of the impedances Z1, Zn, . . . Zx, is equiv- alent to a divided circuit of x + 1 branches, one branch of

AL TERN A TING-CURRENT TRANSFORMER. Generator Transformer

211

Fig. 108.

admittance Y0) the exciting current, the other branches of the impedances ZJ + Z7, ZJ1 + Zn, . . . 2f + Zx, the latter impedances being reduced to the primary circuit by the ratio of turns, and the whole divided circuit being fed by the primary impressed E.M.F. £0, over -mains of the impedance Z0-

Consequently, transformation of a circuit merely changes all the quantities proportionally, introduces in the mains the impedance Z0 + Z^, and a branch circuit between Z0 and Z^, of admittance Y0.

Thus, double transformation will be represented by dia- gram, Fig. 109.

212 A L TERN A TING- CURRENT PHENOMENA .

With this the discussion of the alternate-current trans- former ends, by becoming identical with that of a divided circuit containing resistances and reactances.

Such circuits have explicitly been discussed in Chapter VIII., and the results derived there are now directly appli- cable to the transformer, giving the variation and the con- trol of secondary terminal voltage, resonance phenomena, etc.

Thus, for instance, if Z/ = Z0, and the transformer con- tains an additional secondary coil, constantly closed by a condenser reactance of such size that this auxiliary circuit, together with the exciting circuit, gives the reactance — x0, . with a non-inductive secondary circuit Z^ = rv we get the • condition of transformation from constant primary potential to constant secondary current, and inversely, as previously discussed.

Non-inductive Secondary Circuit.

138. In a non-inductive secondary circuit, the external secondary impedance is,

or, reduced to primary circuit,

Assuming the secondary impedance, reduced to primary circuit, as equal to the primary impedance,

• is> Y ' i r

Substituting these values in Equations (9), (10), and (13), we have

Ratio of E.M.Fs. :

(r0 — jx0} 4- ra—jx0

ALTERNATING-CURRENT TRANSFORMER. 213

•   r0-jx0  f      r0-jx0      Y|   .  .  .  \  . 
  

R + r0 — jx0 \ R + rn — /#„

or, expanding, and neglecting terms of higher than third order,

— jx0

^

or, expanded,

J|= - « 1 1 + 2 r° ^'^ + (r, -y^)(.%

Neglecting terms of tertiary order also,

£t

Ratio of currents :

^- = - -

/I ^

or, expanded,

~=--

/! a

Neglecting terms of tertiary order also,

Total apparent primary admittance :

R + r0— jx (r0 -jx0} + R (r0-

= {R + 2 (r0 - y x0} - & (go +jb0} -2 R (r0 - Jx0)

214 ALTERNATING-CURRENT PHENOMENA.

or,

b0}- 2 (r0 -Jx0}(

Neglecting terms of tertiary order also : Zt=R

Angle of lag in primary circuit :

tan S>0 = ^ , hence, rt

2^+Rb0 + 2r0b0-2Xogo-2 tan S>0 = a

Neglecting terms of tertiary order also : 'R

139. If, now, we represent the external resistance of the secondary circuit at full load (reduced to the primary circuit) by R0, and denote,

2 r0 _ _ . Internal resistance of transformer _ percentage

R0 ~ External resistance of secondary circuit ~ na^ resistance,

2 X0 _ __ ratjQ Internal reactance of transformer _ percentage J£ ' External resistance of secondary circuit nal reactance

X*.- h - ratio - percentage hysteresis,

,, , , . Magnetizing current percentage magnetizing cur-

•KO °o= g = -10 Totalsecondarycurrent = rent^

and if d represents the load of the transformer, as fraction of full load, we have

ALTERNATING-CURRENT TRANSFORMER. 215

and,

**.-«.

a

Substituting these values we get, as the equations of the transformer on non-inductive load, Ratio of E.M.Fs. :

or, eliminating imaginary quantities,

H"-"^)

Ratio of currents :

• (h +>

d

2 f

. ^

or, eliminating imaginary quantities,

1 f

a \

i i h i

216 ALTERNATING-CURRENT PHENOMENA.

Total apparent primary impedance : Z, =

or, eliminating imaginary quantities,

Angle of lag in primary circuit :

That is,

An alternate-current transformer, feeding into a non-induc- tive secondary circuit, is represented by the constants :

R0 = secondary external resistance at full load ;

p = percentage resistance ;

q = percentage reactance ;

h = percentage hysteresis ;

g = percentage magnetizing current ;

d = secondary percentage load.

All these qualities being considered as reduced to the primary circuit by the square of the ratio of turns, a2.

ALTERNATING-CURRENT TRANSFORMER.

217

140. As an instance, a transformer of the following constants may be given :

e0 =1,000; a = 10 ;

£0= 120;

p = .02 •

q = .06 ; h = .02 ; g = .04.

Substituting these values, gives : 100

=

"

V(i.oou + .02 </)2 + (.0002 + .06 <ty

*-^-£-

.1 ii V/Y 1.0014 + — Y + ( — - \ d J \ d

. 0002 .

• -.0004-

tan w,

^-

1.9972 + .

Fig. 110. Load Diagram of Transformer.

218 ALTERNATING-CURKENT PHENOMENA.

In diagram Fig. 110 are shown, for the values from d = 0 to d= 1.5, with the secondary current ix as abscis- sae, the values :

secondary terminal voltage, in volts,

secondary drop of voltage, in per cent,

primary current, in amps,

excess of primary current over proportionality with

secondary, in per cent, primary angle of lag.

The power-factor of the transformer, cos w0, is .45 at open secondary circuit, and is above .99 from 25 amperes, upwards, with a maximum of .995 at full load.

ALTERNATING-CURRENT TRANSFORMER. 219

CHAPTER XV.

THE GENERAL ALTERNATING-CURRENT TRANSFORMER OR FREQUENCY CONVERTER.

141. The simplest alternating-current apparatus is the alternating-current transformer. It consists of a magnetic- circuit, interlinked with two electric circuits or sets of electric circuits. The one, the primary circuit, is excited by an impressed E.M.F., while in the other, the secondary circuit, an E.M.F. is induced. Thus, in the primary circuit, power is consumed, in the secondary circuit a correspond- ing amount of power produced ; or in other words, power is transferred through space, from primary to secondary circuit. This transfer of power finds its mechanical equiv- alent in a repulsive thrust acting between primary and secondary. Thus, if the secondary coil is not held rigidly as in the stationary transformer, it will be repelled and move away from the primary. This mechanical effect is made use of in the induction motor, which represents a transformer whose secondary is mounted movably with re- gard to the primary in such a way that, while set in rota- tion, it still remains in the primary field of force. The condition that the secondary circuit, while revolving with regard to the primary, does not leave the primary field of magnetic force, requires that this field is not undirectional, but that an active field exists in every direction. One way of producing such a magnetic field is by exciting different primary circuits angularly displaced in space with each other by currents of different phase. Another way is to excite the primary field in one direction only, and get the cross magnetization, or the angularly displaced magnetic field, by the reaction of the secondary current.

220 ALTERNATING-CURRENT PHENOMENA.

We see, consequently, that the stationary transformer and the induction motor are merely different applications of the same apparatus, comprising a magnetic circuit in- terlinked with two electric circuits. Such an apparatus can properly be called a "general alternating- current trans- former" The equations of the stationary transformer and those of the induction motor are merely specializations of the general alternating-current transformer equations.

Quantitatively the main differences between induction motor and stationary transformer are those produced by the air-gap between primary and secondary, which is re- quired to give the secondary mechanical movability. This air-gap greatly increases the magnetizing current over that in the closed magnetic circuit transformer, and requires an ironclad construction of primary and secondary to keep the magnetizing current within reasonable limits. An iron- clad construction again greatly increases the self-induction of primary and secondary circuit. Thus the induction motor is a transformer of large magnetizing current and large self-induction; that is, comparatively large primary exciting susceptance and large reactance.

The general alternating-current transformer transforms between electrical and mechanical power, and changes not only E.M.Fs. and currents, but frequencies also, and may therefore be called a "frequency converter." Obviously, it also may change the number of phases.

142. Besides the magnetic flux interlinked with both primary and secondary electric circuit, a magnetic cross- flux passes in the transformer between primary and second- ary, surrounding one coil only, without being interlinked with the other. This magnetic cross-flux is proportional to the current flowing in the electric circuit, and constitutes what is called the self-induction of the transformer. As seen, as self-induction of a transformer circuit, not the total flux produced by and interlinked with this circuit is under- stood, but only that — usually small — part of the flux

AL TERN A TING-CURRENT TRA NSFORMER. 221

which surrounds the one circuit without interlinking with the other, and is thus produced by the M.M.F. of one circuit only.

143. The mutual magnetic flux of the transformer is produced by the resultant M.M.F. of both electric circuits. It is determined by the counter E.M.F., the number of turns, and the frequency of the electric circuit, by the. equation:

Where E = effective E.M.F.

JV= frequency. n = number of turns. <£ == maximum magnetic flux.

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

144. In the general alternating-current transformer, where the secondary is movable with regard to the primary, the rate of cutting of the secondary electric circuit with the mutual magnetic flux is different from that of the primary. Thus, the frequencies of both circuits are different, and the induced E.M.Fs. are not proportional to the number of turns as in the stationary transformer, but to the product of number of turns into frequency.

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

• = ratio iS^ frequency, or « slip » ; thus, if

N '= primary frequency, or frequency of impressed E.M.F., s JV = secondary frequency ;

222 ALTERNATING-CURRENT PHENOMENA.

and the E.M.F. induced per secondary turn by the mutual flux has to the E.M.F. induced per primary turn the ratio s,

s = 0 represents synchronous motion of the secondary ;

s < 0 represents motion above synchronism — driven by external

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

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

«0 = number of primary turns in series per circuit ;

/?! = number of secondary turns in series per circuit ;

a = — = ratio of turns ; «i

Y0 =£"0 H~./A) = primary exciting admittance per circuit;

where

gQ = effective conductance ;

b0 = susceptance ;

Z0 = r0 —jx0 = internal primary self-inductive impedance

per circuit, where

r0 = effective resistance of primary circuit ;

jr0 = reactance of primary circuit ;

Zu = TI — jxv = internal secondary self -inductive impedance

per circuit at standstill, or for s = 1, where

rj = effective resistance of secondary coil ; Xl — reactance of secondary coil at standstill, or full fre- quency, s = 1.

Since the reactance is proportional to the frequency, at the slip s, or the secondary frequency s N, the secondary

impedance is :

Zl = r1-jsxl.

Let the secondary circuit be closed by an external re- sistance r, and an external reactance, and denote the latter

ALTERNATING-CURRENT TRANSFORMER, 223

by x at frequency N, then at frequency s N, or slip s, it will be = s x, and thus :

Z = r — jsx = external secondary impedance.* Let

£0 = primary impressed E.M.F. per circuit, E ' = E.M.F. consumed by primary counter E.M.F., £1 = secondary terminal E.M.F., EI = secondary induced E.M.F., e = E.M.F. induced per turn by the mutual magnetic flux,

at full frequency JY, IQ = primary current, ^0 = primary exciting current, 7i = secondary current.

It is then :

Secondary induced E.M.F.

EI = sn^e.

Total secondary impedance

Z, + Z= (r, + r) hence, secondary current

Secondary terminal voltage

• This applies to the case where the secondary contains inductive reac- tance only ; or, rather, that kind of reactance which is proportional to the fre- quency. In a condenser the reactance is inversely proportional to the frequency, in a synchronous motor under circumstances independent of the frequency. Thus, in general, we have to set, x = x' + x" -\ x"\ where x' is that part of the reactance which is proportional to the frequency, x" that part of the reac- tance independent of the frequency, and x'" that part of the reactance which is inversely proportional t6 the frequency ; and have thus, at slip s, or frequency sN, the external secondary reactance sx' + x" -f- — — .

224 AL TERNA TING-CURRENT PHENOMENA,

E.M.F. consumed by primary counter E.M.F.

£'= -«<>';

hence, primary exciting current :

700 = E ' YQ = — «0 e (g0 + /£<))•

Component of primary current corresponding to second- ary current 7X :

hence, total primary current,

// 1

Primary impressed E.M.F.,

We get thus, as the Equations of the General Alternating-Current Transformer:

Of ratio of turns, a ; and ratio of frequencies, s ; with the E.M.F. induced per turn at full frequency, e, as parameter, the values :

Primary impressed E.M.F.,

Secondary terminal voltage,

Primary current,

\ 1

ALTERNATING-CURRENT TRANSFORMER. 225

Secondary current,

II =7— -7-

Therefrom, we get : Ratio of currents,

Ratio of E.M.Fs.,

Total apparent primary impedance,

, , . x" . x'"

where x—x-\ --- - —

s s2

in the general secondary circuit as discussed in foot-note, page 221.

Substituting in these equations :

*-l,

gives the

General Equations of the Stationary Alternating-Current Transformer :

z*+z\ z, + z

'* = -»•< \ .,;,* +IU-

»* (Zj + Z)

ALTERNA TING-CURRENT PHENOMENA.

r nte

yi =

Z, + Z /! a

P f1 + f7^ + Z'Y ^o_= _ a } a (Z-j + 2}

& I- Z*

( Z, + Z

1+ 2//° x+^oKo]

a2 (Zj + Z) _ I

l + ^Fo^ + Z) J

Substituting in the equations of the general alternating- current transformer,

Z = 0, gives the

General Eqtiations of tJie Induction Motor:

a'r^-jsx^ ^ = 0.

1 i ^o +y^o

70 = _ s «0 f ] -T, . .

1 «•(>-!— y**o

r j«,^

A = —

—5 "^ : — ~ + (ro — y^o)(^b +/

«2^i — JSXi

Returning now to the general alternating-current trans^ former, we have, by substituting

(ri + r? + ^2 (*i + *)2 = **f, and separating the real and imaginary quantities,

-±- (r0 (r, + r)+sx9(Xl + x)) 22

ALTERNATING-CURRENT TRANSFORMER, 227

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

Substituting this in the equations of the general trans- former, we get,

£,= - «0 e\ I + - fr fa + r)

146. The true power is, in symbolic representation (see Chapter XII.) :

228 ALTERNATING-CURRENT PHENOMENA.

denoting,

safe*

-7F = W

gives :

Secondary output of the transformer

Internal loss in secondary circuit,

m -2 t s n\ ^\2

-Pi = 'i2 n = ( — — }

V ** /

Total secondary power,

**

Internal loss in primary circuit,

r»i -9 -9o

^o = V'o = 4 rt<r

Total electrical output, plus loss,

2

Total electrical input of primary,

Hence, mechanical output of transformer,

P=P»-P* = w(l-s)(r E.atio,

mechanical

output _ P 1 — S _ speed

total secondary power P -- P l

147. Thus,

In a general alternating transformer of ratio of turns, a, and ratio of frequencies, s, neglecting exciting current, it is :

Electrical input in primary, P

ALTERNATING-CURRENT TRANSFORMER. 229

Mechanical output,

P - jg-j)«iV(r+rO '

Electrical output of secondary,

Losses in transformer,

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

148. At

s = 0, synchronism, PQ = 0, P = 0, Pl = 0. At 0 < s < 1, between synchronism and standstill.

Pl , P and PQ are positive ; that is, the apparatus con- sumes electrical power PQ in the primary, and produces mechanical power P and electrical power Pl -j- P^ in the secondary, which is partly, P-^, consumed by the internal secondary resistance, partly, Pl , available at the secondary terminals.

In this case it is :

•Pi + ^i1 _ J P ~l-s>

that is, of the electrical power consumed in the primary circuit, P0, a part P^ is consumed by the internal pri- mary resistance, the remainder transmitted to the secon- dary, and divides between electrical power, P1 + P^1, and mechanical power, P, in the proportion of the slip, or drop below synchronism, s, to the speed : 1 — s.

230 ALTERNATING-CURRENT PHENOMENA.

In this range, the apparatus is a motor. At s > 1 ; or, backwards driving,

P < 0, or negative ; that is, the apparatus requires mechanical power for driving.

It is then : P0 - A1 - A1 < PI ;

that is : the secondary electrical power is produced partly by the primary electrical power, partly by the mechanical power, and the apparatus acts simultaneously as trans- former and as alternating-current generator, with the sec- ondary as armature.

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

In this case, the secondary frequency is higher than the primary.

At s < 0, beyond synchronism,

P < 0 ; that is, the apparatus has to be driven by mechanical

power. /o<0; that is, the primary circuit produces electrical power

from the mechanical input.

At r+r! + srj. = 0, or, s < — ^±^J ;

rt

the electrical power produced in the primary becomes less than required to cover the losses of power, and />0 becomes positive again. We have thus :

K-£±fl r\

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

• r-±^ < s < 0 ?i

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

ALTERNATING-CURRENT TRANSFORMER. 231

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

consumes mechanical and primary electrical power ; pro- duces secondary electrical power.

T

GENERAL ALTERNATE CURRENT TRANSFORMER

A

648

Fig

H

149. As an instance, in Fig. Ill are plotted, with the slip s as abscissae, the values of :

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

Mechanical output as Curve III. ;

Primary electrical input as Curve IV. ;

for the values :

n,e = 100.0 ; r = A ;

r» — 4. i x = .3;

232 ALTERNATING-CURRENT PHENOMENA.

hence, p = 16,000 ^2.

pl, Pi _ 8,000 j«.

0 """ l -i , — j" ?

„ _ 4,000 s + (5 + J) .

~ 1 I 2 '

p = 20,000 s (1 - j)

150. Since the most common practical application of the general alternating current transformer is that of fre- quency converter, that is to change from one frequency to another, either with or without change of the number of phases, the following characteristic curves of this apparatus are of great interest.

151. The regulation curve ; that is, the change of second- ary terminal voltage as function of the load at constant im- pressed primary voltage.

152. The compounding curve ; that is, the change of pri- mary impressed voltage required to maintain constant sec- ondary terminal voltage.

In this case the impressed frequency and the speed are constant, and consequently the secondary frequency. Gen- erally the frequency converter is used to change from a low frequency, as 25 cycles, to a higher frequency, as 62.5 cycles, and is then driven backward, that is, against its torque, by mechanical power. Mostly a synchronous motor is employed, connected to the primary mains, which by over-excitation compensates also for the lagging current of the frequency converter.

Let,

Y0 = g0 +j&0 = primary exciting admittance per circuit of the frequency converter.

Z^ = rt —jx^— internal self inductive impedance per secondary circuit, at the secondary frequency.

ALTERNATING-CURRENT TRANSFORMER. 233

Z^ = r0 — jx^ = 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 = induced 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 noninductive lead.

We then have,

total secondary impedance,

Z + Z1 = (r-^rl)-j(x + x1) secondary current,

where,

r + r. x + Xl

(r + 0>2 + (* + ^)2 (r +^i)2 + (* +

secondary terminal voltage,

Ei = IiZ = e ^4-T

— e(r —jx) (at where,

primary induced E.M.F. per circuit,

primary load current per circuit,

71 = abli = abe (a{ primary exciting current per circuit,

234 ALTERNATING-CURRENT PHENOMENA.

thus, total primary current,

70 = 71 + /oo

= e (fi

where,

<. = •+£ <•.=«+!

primary terminal voltage :

where,

d -— re x d -re -x

ac

or absolute,

e0 = e vX2 + 42

. = e° -

V^2 + 4«

substituting this value of e in the preceding equations, gives, as function of the primary impressed E.M.F., e0: secondary current,

7 = > absolu 7 = vi

V4» + 42 v ^i2 +

secondary terminal voltage,

primary current,

, _

primary impressed E.M.F.

^ _ ^0 (4

" V4 secondary output,

gl^ +

AL TERNA TING-CURRENT TRANSFORMER.

235

primary electrical input,

i +

Lr°:oj </• + </.*

primary apparent input, voltamperes, <2o = 4/o

Substituting thus different values for the secondary in- ternal impedance Z gives the regulation curve of the fre- quency converter.

REGULATION CURVES

VOLTS CONSTANT! 25 CYCLES

DARY 62.5 CYCLE QUARTER-PHASE

TDARY 20

CURRENT PER PHASE, AMP.

Fig. 112,

Such a curve, taken from tests of a 20"0 KW frequency converter changing from 6300 volts 25 cycles three-phase, to 2500 volts 62.5 cycles quarter-phase, is given in Fig. 112.

236 AL TERN A TING-CURRENT PHENOMENA.

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it follows, absolute,

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TDAR 2

r CUR }

1ENT |

PER P 0

HASE, 4

AMP.

t

.',

I

|

)

Fig. 113.

Substituting these values in tne above equation gives the quantities as functions of the secondary terminal vol- tage, that is at constant el, or the compounding curve.

The compounding curve of the frequency converter above mentioned is given in Fig. 113.

INDUCTION MOTOR. 237

CHAPTER XVI.

INDUCTION MOTOR.

151. A specialization of the general alternating-current transformer is the induction motor. It differs from the stationary alternating-current transformer, which is also a specialization of the general transformer, in so far as in the stationary transformer only the transfer of electrical energy from primary to secondary is used, but not the mechanical force acting between the two, and therefore primary and secondary coils are held rigidly in position with regard to each other. In the induction motor, only the mechanical force between primary and secondary is used, but not the transfer of electrical energy, and thus the secondary circuits closed upon themselves. Transformer and induction motor thus are the two limiting cases of the general alternating- current transformer. Hence the induction motor consists of a magnetic circuit interlinked with two electric circuits or sets of circuits, the primary and the secondary circuit, which are movable with regard to each other. In general a num- ber of primary and a number of secondary circuits are used, angularly displaced around the periphery of the motor, and containing E.M.Fs. displaced in phase by the same angle. This multi-circuit arrangement has the object always to retain secondary circuits in inductive relation to primary circuits and vice versa, in spite of their relative motion.

The result of the relative motion between primary and secondary is, that the E.M.Fs. induced in the secondary or the motor armature are not of the same frequency as the E.M.Fs. impressed upon the primary, but of a frequency which is the difference between the impressed frequency

238 ALTERNATING-CURRENT PHENOMENA.

and the frequency of rotation, or equal to the "slip," that is, the difference between synchronism and speed (in cycles). Hence, if

N = frequency of main or primary E.M.F.,

and s = percentage slip ;

sJV = frequency of armature or secondary E.M.F.,

and (1 — s) N= frequency of rotation of armature.

In its reaction upon the primary circuit, however, the armature current is of the same frequency as the primary current, since it is carried around mechanically, with a fre- quency equal to the difference between its own frequency and that of the primary. Or rather, since the reaction of the secondary on the primary must be of primary frequency — whatever the speed of rotation — the secondary frequency is always such as to give at the existing speed of rotation a reaction of primary frequency.

152. Let the primary system consist of /0 equal circuits, displaced angulary in space by 1 //0 of a period, that is, 1 //„ of the width of two poles, and excited by /»0 E.M.Fs. displaced in phase by 1 //0 of a period ; that is, in other words, let the field circuits consist of a symmetrical /0-phase system. Analogously, let the armature or secondary circuits consist of a symmetrical /rphase system.

Let

n0 = number of primary turns per circuit or phase ; «a = number of secondary turns per circuit or phase ;

a = -^ = ratio of total primary turns to total secondary turns n\P\ or ratio of transformation.

Since the number of secondary circuits and number of turns of the secondary circuits, in the induction motor — as in the stationary transformer — is entirely unessential, it is preferable to reduce all secondary quantities to the primary system, by the ratio of transformation, a ; thus

INDUCTION MOTOR. 239

if E{ = secondary E.M.F. per circuit, El = aE{

= secondary E.M.F. per circuit reduced to primary system;

if // = secondary current per circuit, fl= —

= secondary current per circuit reduced to primary system ; if r^ = secondary resistance per circuit, rt = a2 r{

= secondary resistance per circuit reduced to primary system ; if x± = secondary reactance per circuit, xt = a2 x\

= secondary reactance per circuit reduced to primary system ; if £/ = secondary impedance per circuit, z1 = azz\

= secondary impedance per circuit reduced to primary system ;

that is, the number of secondary circuits and of turns per secondary circuit is assumed the same as in the primary system.

In the following discussion, as secondary quantities, the values reduced to the primary system shall be exclusively used, so that, to derive the true secondary values, these quantities have to be reduced backwards again by the factor

a = ?*£-. «iA 153. Let

$ = total maximum flux of the magnetic field per motor pole, We then have

E— V2 77-72 TV^ 10 ~8 = effective E.M.F. induced by the mag- netic field per primary circuit.

Counting the time from the moment where the rising magnetic flux of mutual induction & (flux interlinked with both electric circuits, primary and secondary) passes through zero, in complex quantities, the magnetic flux is denoted by

and the primary induced E.M.F.,

240 ALTERNATING-CURRENT PHENOMENA.

where

e= V2irrt7V<I>10-8 maybe considered as the "Active E.M.F. of the motor," or " Counter E.M.F."

Since the secondary frequency is s N, the secondary in- duced E.M.F. (reduced to primary system) is El = — se. Let

I0 = exciting current, or current passing through the motor, per primary circuit, when doing no work (at synchronism),

and

K= g -j- j 'b = orimary admittance per circuit = — .

We thus have,

ge = magnetic energy current, ge* = loss of power oy hysteresis (and eddy currents) per primary coil.

Hence

= total loss of energy by hysteresis and eddys,

as calculated according to Chapter X. be = magnetizing current, and n0be = effective M.M.F. per primary circuit;

hence ^n0be = total effective M.M.F. ;

z

and