book

Theory and Calculation of Alternating Current Phenomena (1900) — part 14 of 19

1 January 1900

Special provisions were made to keep the armature re- actance a minimum, and overcome the distortion of the field by the armature M.M.F., by means of a coil closely surrounding the armature and excited by a current of equal phase but opposite direction with the armature current (Eickemeyer). Thereby it was possible to operate a two- circuit, 96-turn armature in a bipolar field of 20 turns, at a ratio of

armature ampere-turns r> A

field ampere-turns

It is in this case,

100

V(.023 vVi + ,15)2 + 1.96

230 ./v;

(.023 A! + .15)2 + 1.96

368

AL TERNA TING-CURRENT PHENOMENA.

In Fig. 163 are given, with the speed Nv as abscissae, the values of current /, power P, and power factor cos o> of this motor.

SER

FOP

Vaf-

r =

03 .12

=(, x

_.y

= .5

->w

N =

0,,

TUMI

^~~~

^«

2Stt>

J23

]( ~

QjF

1.9

gem

1Z'_.

•-il(N)

'(.

[23

^, -

-•)-

' 9

•>->00

•JIHIII

).n

^ ^

'SI-'

1.9

P-

1S'K>

icon

cos

£>___

-sr

_70

•\

po*

ev ^

' —

—

•<

1000

<~~

•*-.

^_<ii

sew

• .

_Jq

ssa

GOO,

;',o

/

01 111

'oN

Fig. 163. Series Motor.

The shunt motor with laminated field will not operate satisfactorily in an alternating-current circuit. It will start with good torque, since in starting the current in armature, as well as in field, are greatly lagging, and thus approximately in phase with each other. With increasing speed, however, the armature current should come more into phase with the impressed E.M.F., to represent power. Since, however, the field current, and thus the field mag netism, lag nearly 90°, the induced E.M.F. of the armature rotation will lag nearly 90°, and thus not represent power.

COMMUTATOR MOTORS. 369

Hence, to make a shunt motor work on alternating-cur- rent circuits, the magnetism of the field should be approxi- mately in phase with the impressed E.M.F., that is, the field reactance negligible. Since the self-induction of the field is far in excess to its resistance, this requires the insertion of negative reactance, or capacity, in the field.

If the self-induction of the field circuit is balanced by capacity, the motor will operate, provided that the armature reactance is low, and that in starting sufficient resistance is inserted in the armature circuit to keep the armature current approximately in phase with the E.M.F. Under these conditions the equations of the motor will be similar to those of the series motor.

However, such motors have not been introduced, due to the difficulty of maintaining the balance between capacity and self-induction in the field circuit, which depends upon the square of the frequency, and thus is disturbed by the least change of frequency.

The main objection to both series and shunt motors is the destructive sparking at the commutator due to the in- duction of secondary currents in those armature coils which pass under the brushes. As seen in Fig. 162, with the normal position of brushes midway between the field poles, the armature coil which passes under the brush incloses the total magnetic flux. Thus, in this moment no E.M.F. is induced in the armature coil due to its rotation, but the E.M.F. induced by the alternation of the magnetic flux has a maximum at this moment, and the coil, when short- circuited by the brush, acts as a short-circuited secondary to the field coils as primary ; that is, an excessive current flows through this armature coil, which either destroys it, or at least causes vicious sparking when interrupted by the motion of the arm'ature.

To overcome this difficulty various arrangements have been proposed, but have not found an application.

370 ALTERNATING-CURRENT PHENOMENA.

Compared with the synchronous motor which has practically no lagging currents, and the induction motor which reaches very high power factors, the power factor of the series motor is low, as seen from Fig. 163, which repre- sents about the best possible design of such motors.

In the alternating-series motor, as well as in the shunt motor, no position of an armature coil exists wherein the coil is dead; but in every position E.M.F. is induced in the armature coil : in the position parallel with the field flux an E.M.F. in phase with the current, in the position at right angles with the field flux an E.M.F. in quadrature with the current, intermediate E.M.Fs. in intermediate positions. At the speed irJV/2 the two induced E.M.Fs. in phase and in quadrature with the current are equal, and the armature coils are the seat of a complete system of symmetrical and balanced polyphase E.M.Fs. Thus, by means of stationary brushes, from such a commutator polyphase currents could be derived.

REACTION MACHINES. 371

CHAPTER XXI.

REACTION MACHINES.

In the chapters on Alternating-Current Genera- tors and on Induction Motors, the assumption has been made that the reactance x of the machine is a constant. While this is more or less approximately the case in many alternators, in others, especially in machines of large arma- ture reaction, the reactance x is variable, and is different in the different positions of the armature coils in the magnetic circuit. This variation of the reactance causes phenomena which do not find their explanation by the theoretical cal- culations made under the assumption of constant reactance.

It is known that synchronous motors of large and variable reactance keep in synchronism, and are able to do a considerable amount of work, and even carry under circumstances full load, if the field-exciting circuit is broken, and thereby the counter E.M.F. E± reduced to zero, and sometimes even if the field circuit is reversed and the counter E.M.F. E± made negative.

Inversely, under certain conditions of load, the current and the E.M.F. of a generator do not disappear if the gene- rator field is broken, or even reversed to a small negative value, in which latter case the current flows against the E.M.F. EQ of the generator.

Furthermore, a shuttle armature without any winding will in an alternating magnetic field revolve when once brought up to synchronism, and do considerable work as a motor.

These phenomena are not due to remanent magnetism nor to the magnetizing effect of Foucault currents, because

372 AL TERNA TING-CURRENT PHENOMENA.

they exist also in machines with laminated fields, and exist if the alternator is brought up to synchronism by external means and the remanent magnetism of the field poles de- stroyed beforehand by application of an alternating current.

These phenomena cannot be explained under the assumption of a constant synchronous reactance; because in this case, at no-field excitation, the E.M.F. or counter E.M.F. of the machine is zero, and the only E.M.F. exist- ing in the alternator is the E.M.F. of self-induction; that is, the E.M.F. induced by the alternating current upon itself. If, however, the synchronous reactance is constant, the counter E.M.F. of self-induction is in quadrature with the current and wattless; that is, can neither produce nor consume energy.

' In the synchronous motor running without field excita- tion, always a large lag of the current behind the impressed E.M.F. exists; and an alternating generator will yield an E.M.F. without field excitation, only when closed by an external circuit of large negative reactance ; that is, a circuit in which the current leads the E.M.F., as a condenser, or an over-excited synchronous motor, etc.

Self-excitation of the alternator by armature reaction can be explained by the fact that the counter E.M.F. of self-induction is not wattless or in quadrature with the cur- rent, but contains an energy component ; that is, that the reactance is of the form X = h —jx, where x is the wattless component of reactance and h the energy component of reactance, and h is positive if the reactance consumes power, — in which case the counter E.M.F. of self-induc- tion lags more than 90° behind the current, — while h is negative if the reactance produces power, — in which case the counter E.M.F. of self-induction lags less than 90° behind the current.

A case of this nature has been discussed already in the chapter on Hysteresis, from a different point of view.

REACTION MACHINES. 373

There the effect of magnetic hysteresis was found to distort the current wave in such a way that the equivalent sine wave, that is, the sine wave of equal effective strength and equal power with the distorted wave, is in advance of the wave of magnetism by what is called the angle of hysteretic advance of phase a. Since the E.M.F. induced by the magnetism, or counter E.M.F. of self-induction, lags 90° behind the magnetism, it lags 90 -f- a behind the current ; that is, the self-induction in a circuit containing iron is not in quadrature with the current and thereby wattless, but lags more than 90° and thereby consumes power, so that the reactance has to be represented by X = Ji —jx, where h is what has been called the " effective hysteretic resis- tance."

A similar phenomenon takes place in alternators of vari- able reactance, or what is the same, variable magnetic reluctance.

Obviously, if the reactance or reluctance is vari- able, it will perform a complete cycle during the time the armature coil moves from one field pole to the next field pole, that is, during one-half wave of the main current. That is, in other words, the reluctance and reactance vary with twice the frequency of the alternating main current. Such a case is shown in Figs.. 164 and 165. The impressed E.M.F., and thus at negligible resistance, the counter E.M.F., is represented by the sine wave E, thus the magnetism pro- duced thereby is a sine wave 4>, 90° ahead of E. The reactance is represented by the sine wave x, varying with the double frequency of E, and shown in Fig. 164 to reach the maximum value during the rise of magnetism, in Fig. 165 during the decrease of magnetism. The current / re- quired to produce the magnetism <l> is found from 3> and-^r in combination with the cycle of molecular magnetic friction of the material, and the power P is the product IE As seen in Fig. 164, the positive part of P is larger than the

374 AL TERNA TING-CURRENT PHENOMENA.

^ —

s~~~

•^

•^ /

\ ^

—

1 —

•*

'^^

_^-

— '

k*s'

x^^

b:S

Fig. 164, Variable Reactance, Reaction Machine.

Fig. 165. Variable Reactance, Reaction Machine.

REACTION MACHINES.

375

negative part ; that is, the machine produces electrical energy as generator. In Fig. 165 the negative part of P is larger than the positive ; that is, the machine consumes electrical energy and produces mechanical energy as synchronous mqtor. In Figs. 166 and 167 are given the two hysteretic cycles or looped curves <J>, / under the two conditions. They show that, due to the variation of reactance x, in the first case the hysteretic cycle has been overturned so as to represent not consumption, but production of electrical

Fig. 166. Hysteretic Loop of Reaction Machine.

energy, while in the second case the hysteretic cycle has been widened, representing not only the electrical energy consumed by molecular magnetic friction, but also the me- chanical output.

It is evident that the variation of reluctance must be symmetrical with regard to the field poles ; that is, that the two extreme values of reluctance, maximum and mini- mum, will take place at the moment where the armature

J76

ALTERNA TING-CURRENT PHENOMENA.

coil stands in front of the field pole, and at the moment where it stands midway between the field poles.

The effect of this periodic variation of reluctance is a distortion of the wave of E.M.F., or of the wave of current, or of both. Here again, as before, the distorted wave can be replaced by the equivalent sine wave, or sine wave of equal effective intensity and equal power.

The instantaneous value of magnetism produced by the

Fig. 167. Hysteretic Loop of Reaction Machine.

armature current — which magnetism induces in the arma- ture conductor the E.M.F. of self-induction — is propor- tional to the instantaneous value of the current, divided by the instantaneous value of the reluctance. Since the extreme values of the reluctance coincide with the sym- metrical positions of the armature with regard to the field poles, — that is, with zero and maximum value of the in- duced E.M.F., EQ, of the machine, — it follows that, if the current is in phase or in quadrature with the E.M.F. EQ, the reluctance wave is symmetrical to the current wave, and the wave of magnetism therefore symmetrical to the

REACTION MACHINES. 377

current wave also. Hence the equivalent sine wave of magnetism is of equal phase with the current wave ; that is, the E.M.F. of self-induction lags 90° behind the cur- rent, or is wattless.

Thus at no-phase displacement, and at 90° phase dis- placement, a reaction machine can neither produce electri- cal power nor mechanical power.

If, however, the current wave differs in phase from the wave of E.M.F. by less than 90°, but more than zero degrees, it is unsymmetrical with regard to the reluctance wave, and the reluctance will be higher for ris- ing current than for decreasing current, or it will be higher for decreasing than for rising current, according to the phase relation of current with regard to induced E.M.F., £Q.

In the first case, if the reluctance is higher for rising, lower for decreasing, current, the magnetism, which is pro- portional to current divided by reluctance, is higher for decreasing than for rising current ; that is, its equivalent sine wave lags behind the sine wave of current, and the E.M.F. or self-induction will lag more than 90° behind the current ; that is, it will consume electrical power, and thereby deliver mechanical power, and do work as syn- chronous motor.

In the second case, if the reluctance is lower for rising, and higher for decreasing, current, the magnetism is higher for rising than for decreasing current, or the equivalent sine wave of magnetism leads the sine wave of the current, and the counter E.M.F. at self-induction lags less than 90° be- hind the current ; that is, yields electric power as generator, and thereby consumes mechanical power.

In the first case the reactance will be represented by X = h — jx, similar as in the case of hysteresis ; while in the second case the reactance will be represented by X = - h- jx.

378 ALTERNATING-CURRENT PHENOMENA.

The influence of the periodical variation of reac- tance will obviously depend upon the nature of the variation, that is, upon the shape of the reactance curve. Since, however, no matter what shape the wave has, it can always be dissolved in a series of sine waves of double frequency, and its higher harmonics, in first approximation the assump- tion can be made that the reactance or the reluctance vary with double frequency of the main current ; that is, are represented in the form,

x = a + b cos 2 /8.

Let the inductance, or the coefficient of self-induction, be represented by —

L = I + <£ cos 2 /3

= /(I + y COS 2 0)

where y = amplitude of variation of inductance.

Let

u> = angle of lag of zero value of current behind maximum value of inductance L.

It is then, assuming the current as sine wave, or repla- cing it by the equivalent sine wave of effective intensity /,

Current,

= I V2 sin (/? - £).

The magnetism produced by this current is,

where n = number of turns. Hence, substituted,

sin (/? - 5) (1 + y cos 2 0), or, expanded,

n when neglecting the term of triple frequency, as wattless.

REACTION MACHINES, 379

Thus the E.M.F. induced by this magnetism is,

hence, expanded —

e = - 2 TT 7W7 V2 !7 1 - 2\ cos £ cos /3 + /I + sn sn

IV ZJ \ 2

and the effective value of E.M.F.,

l + 2

= 2 TT NII\ + - 7 cos 2 a. ^ Hence, the apparent power, or the voltamperes —

-J2 — y COS 2 u>

The instantaneous value of power is

2sin(/? — c(,)f/l — |\ cos w cos y3 +

sin eo sin /3 [. . 7

and, expanded —

sin 2 eo cos2 /3 + sin 2 /3 ( cos 2 w — 2 \ 1 V 2/J

Integrated, the effective value of power is

380 AL TERNA TING-CURRENT PHENOMENA.

hence, negative, that is, the machine consumes electrical, and produces mechanical, power, as synchronous motor, if o> > 0 ; that is, with lagging current; positive, that is, the machine produces electrical, and consumes mechanical, power, as generator, if to > 0 ; that is, with leading current. The power factor is

r j_ P_ _ y sin 2 ai

hence, a maximum, if,

or, expanded, 1

cos2£ = i

The power, P, is a maximum at given current, /, if

sin 2 w = 1 ; that is,

to = 45°

at given E.M.F., E, the power is p= __

hence, a maximum at or, expanded,

1 + 1T

We have thus, at impressed E.M.F., E, and negli- gible resistance, if we denote the mean value of reactance,

x=lTtNl. Current

REACTION MACHINES. 381

Voltamperes,

Power,

^g2 y sin 2 £

2^fl+^--ycos2

Power factor,

,. / 77 T-N y sin 2 to f = cos (E, /) = '

2 y/l + J^ _ y cos 2 A Maximum power at

*+i

Maximum power factor at

to > 0 : synchronous motor, with lagging current, w < 0 : generator, with leading current.

As an instance is shown in Fig. 168, with angle to as abscissae, the values of current, power, and power factor, for the constants, —

E = 110

x = 3

y =.8

hence, j 41

Vl.45 — cos 2 £

2017 sin 2w

P =

f= cos (E,I)

1.45 — cos 2 w

.447 sin 2 G>

As seen from Fig. 152, the power factor / of such a machine is very low — does not exceed 40 per cent in this instance.

382

ALTERNA TING-CURRENT PHENOMENA.

Fig. 188. Reaction Machine.

DISTORTION OF WAVE-SHAPE. 383

CHAPTER XXII.

DISTORTION OF WAVE-SHAPE AND ITS CAUSES.

In the preceding chapters we have considered the alternating currents and alternating E.M.Fs. as sine waves or as replaced by their equivalent sine waves.

While this is sufficiently exact in most cases, under certain circumstances the deviation of the wave from sine shape becomes of importance, and with certain distortions it may not be possible to replace the distorted wave by an equivalent sine wave, since the angle of phase displacement of the equivalent sine wave becomes indefinite. Thus it becomes desirable to investigate the distortion of the wave, its causes and its effects.

Since, as stated before, any alternating wave can be represented by a series of sine functions of odd orders, the investigation of distortion of wave-shape resolves itself in the investigation of the higher harmonics of the alternating wave.

In general we have to distinguish between higher har- monics of E.M.F. and higher harmonics of current. Both depend upon each other in so far as with a sine wave of impressed E.M.F. a distorting effect will cause distortion of the current wave, while with a sine wave of current passing through the circuit, a distorting effect will cause higher harmonics of E.M.F.

In a conductor revolving with uniform velocity through a uniform and constant magnetic field, a sine wave of E.M.F. is induced. In a circuit with constant resistance and constant reactance, this sine wave of E.M.F. produces

384 ALTERNATING-CURRENT PHENOMENA.

a sine wave of current. Thus distortion of the wave-shape or higher harmonics may be due to : lack of uniformity of the velocity of the revolving conductor ; lack of uniformity or pulsation of the magnetic field ; pulsation of the resis- tance ; or pulsation of the reactance.

The first two cases, lack of uniformity of the rotation or of the magnetic field, cause higher harmonics of E.M.F. at open circuit. The last, pulsation of resistance and reac- tance, causes higher harmonics only with a current flowing in the circuit, that is, under load.

Lack of uniformity of the rotation is of no practical in- terest as cause of distortion, since in alternators, due to mechanical momentum, the speed is always very nearly uniform during the period.

Thus as causes of higher harmonics remain :

1st. Lack of uniformity and pulsation of the magnetic field, causing a distortion of the induced E.M.F. at open circuit as well as under load.

2d. Pulsation of the reactance, causing higher harmonics under load.

3d. Pulsation of the resistance, causing higher harmonics under load also.

Taking up the different causes of higher harmonics we have : —

Lack of Uniformity and Pulsation of tJie Magnetic Field.

Since most of the alternating-current generators contain definite and sharply defined field poles covering in different types different proportions of the pitch, in general the magnetic flux interlinked with the armature coil will not vary as simply sine wave, of the form :

$ cos /?,

but as a complex harmonic function, depending on the shape and the pitch of the field poles, and the arrangement of the armature conductors. In this case, the magnetic flux issu-

DISTORTION OF WAVE-SHAPE. 385

ing from the field pole of the alternator can be represented by the general equation,

4> = A0 + A, cos /8 + A* cos 2(3 + Az cos 3/8 + . . .

^ sin £ + -#2 sin 2 0 + .#, sin 3 ft + . . .

If the reluctance of the armature is uniform in all directions, so that the distribution of the magnetic flux at the field-pole face does not change by the rotation of the armature, the rate of cutting magnetic flux by an armature conductor is <£, and the E.M.F. induced in the conductor thus equal thereto in wave shape. As a rule A0, Az, At . . . By B± equal zero ; that is, successive field poles are equal in strength and dis- tribution of magnetism, but of opposite polarity. In some types of machines, however, especially induction alternators, this is not the case.

The E.M.F. induced in a full-pitch armature turn — that is, armature conductor and return conductor distant from former by the pitch of the armature pole (corresponding to the distance from field pole center to pole center) is, 8 = $0 - 3>180

= 2 \Ai cos /3 + Aa cos 3 (3 + A6 cos 5 0 + . . .

BI sin j3 + Bz sin 3 ft + jB6 sin 5 ft + . . . \

Even with an unsymmetrical distribution of the magnetic flux in the air-gap, the E.M.F. wave induced in a full-pitch armature coil is symmetrical ; the positive and negative half waves equal, and correspond to the mean flux distribution of adjacent poles. With fractional pitch windings — that is, windings whose turns cover less than the armature pole pitch — the induced E.M.F. can be unsymmetrical with unsymmetrical magnetic field, but as a rule is symmetrical also. In unitooth alternators the total induced E.M.F. has the same shape as that induced in a single turn.

With the conductors more or less distributed over the surface of the armature, the total induced E.M.F. is the resultant of several E.M.Fs. of different phases, and is thus more uniformly varying ; that is, more sinusoidal, approaching

386 ALTERNATING-CURRENT PHENOMENA.

sine shape, to within 3% or less, as for instance the curves Fig. 169 and Fig. 170 show, which represent the no-load and full-load wave of E.M.F. of a three-phase multitooth alternator. The principal term of these harmonics is the third harmonic, which consequently appears more or less in all alternator waves. As a rule these harmonics can be considered together with the harmonics due to the varying reluctance of the magnetic circuit. In ironclad alternators with few slots and teeth per pole, the passage of slots across the field poles causes a pulsation of the magnetic reluc- tance, or its reciprocal, the magnetic inductance of the circuit. In consequence thereof the magnetism per field pole, or at least that part of the magnetism passing through the armature, will pulsate with a frequency 2 y if y = num- ber of slots per pole.

Thus, in a machine with one slot per pole, the instanta- neous magnetic flux interlinked with the armature con- ductors can be expressed by the equation :

<£ = $ cos /? [1 + e cos [2 (3 — o>] j where, ® = average magnetic flux,

c = amplitude of pulsation, and to = phase of pulsation.

In a machine with y slots per pole, the instantaneous flux interlinked with the armature conductors will be :

<f> = & cos /8 { 1 + c cos [2 y ft — o>] | ,

if the assumption is made that the pulsation of the magnetic flux follows a simple sine law, as first approximation.

In general the instantaneous magnetic flux interlinked with the armature conductors will be :

^ = * cos 0 {1 + 6! cos (2 0 - SO + e, cos (4 £ - oV,) + . . . f , where the term ey is predominating if y = number of arma- ture slots per pole. This general equation includes also the effect of lack of uniformity of the magnetic flux.

DISTORTION OF WAVE-SHAPE.

387

Nil LoLd

,"14 .5 y,

Fig. 169. No-load

of E.M.F. of Multitooth Three-phaser.

130

JMtfl I

oad

120 ''

= 12

7.0

»=

3 ;

'--

s,

110

100

•'/

"--v^

1 — ^.^

— V

f '

(50

100

120

13(1

140

150

100

170

ISO

Fig. 170. Full-Load Waue of E.M.F. of Multitooth Three-phaser.

388 ALTERNATING-CURRENT PHENOMENA.

In case of a pulsation of the magnetic flux with the frequency 2y, due to an existence of y slots per pole in the armature, the instantaneous value of magnetism interlinked with the armature coil is :

<£ = $ COS ft {1 + e COS [2 y ft — £]}.

Hence the E.M.F. induced thereby :

e = — n — — dt

*»

And, expanded :

e= V27rA^<fc{sin/?+e-^=— sin[(2y — 1) 0 - «J]

Hence, the pulsation of the magnetic flux with the frequency 2 y, as due to the existence of y slots per pole, introduces two harmonics, of the orders (2 y — 1) and (2 7+1).

If y = 1 it is : e = V2 TT Nn <i> (sin /3 + 1 sin (0 — £) + ^ sin (3 /? - £)} ;

that is : In a unitooth single-phaser a pronounced triple harmonic may be expected, but no pronounced higher harmonics.

Fig. 171 shows the wave of E.M.F. of the main coil of a monocyclic alternator at no load, represented by :

e = E (sin (3 — .242 sin ( 3 /3 — 6.3) — .046 sin (5/3- 2.6)

.068 sin (7 £ — 3.3) — .027 sin (9 ft — 10.0) — .018 sin (11 /3 - 6.6) + .029 sin (13 ft - 8.2)};

hence giving a pronounced triple harmonic only, as expected. If y = 2, it is :

e = V2 TT Nn 4> j sin £ + ^ sin (3 ft - «J) + |f sin (5 ft - Si)

DISTORTION OF WAVE-SHAPE.

389

the no-load wave of a unitooth quarter-phase machine, hav- ing pronounced triple and quintuple harmonics. If 7 = 3, it is :

in/3+ sin(5j8— fi) + sin (7 ft - S>) I .

That is : In a unitooth three-phaser, a pronounced quin- tuple and septuple harmonic may be expected, but no pro- nounced triple harmonic.

Fig. 155. No-load Wave of E.M.F. of Unitooth Monocyclic Alternator.

Fig. 156 shows the wave of E.M.F. of a unitooth three- phaser at no load, represented by :

e = E (sin /3 — .12 sin (3 £ — 2.3) — .23 sin (5 (3 — 1.5) + .134 sin (7 ft _ 6.2) - .002 sin (9 /3 + 27.7) - .046 sin (11 /? — 5.5) +.031 sin (13)8-61.5)}.

Thus giving a pronounced quintuple and septuple and a lesser triple harmonic, probably due to the deviation of, the field from uniformity, as explained above, and deviation of the pulsation of reluctance from sine shape. In some especially favorable cases, harmonics as high as the 23d and 25th have been observed, caused by pulsation of the reluc- tance.

390 ALTERNATING-CURRENT PHENOMENA.

100

50 60 70 80 90 1 00

30 140 150 160 170 180

Fig. 172. No-load Wave of E.M.F. of Unitooth Three-phase Alternator.

In general, if the pulsation of the magnetic inductance is denoted by the general expression :

l + ^"cYcos(2yj8-aY), 1

the instantaneous magnetic flux is :

= $ cos 13

ey cos (2 y ff -

cos((2y+l)

hence, the E.M.F.

2 ; sm(P —

DISTORTION OF WAVE-SHAPE. 391

Pulsation of Reactance.

The main causes of a pulsation of reactance are : magnetic saturation and hysteresis, and synchronous motion. Since in an ironclad magnetic circuit the magnetism is not proportional to the M.M.F., the wave of magnetism and thus the wave of E.M.F. will differ from the wave of cur- rent. As far as this distortion is due to the variation of permeability, the distortion is symmetrical and the wave of induced E.M.F. 'represents no power. The distortion caused by hysteresis, or the lag of the magnetism behind the M.M.F., causes an unsymmetrical distortion of the wave which makes the wave of induced E.M.F. differ by more than 90° from the current wave and thereby represents power, — the power consumed by hysteresis.

In practice both effects are always superimposed ; that is, in a ferric inductance, a distortion of wave-shape takes place due to the lack of proportionality between magnetism and M.M.F. as expressed by the variation in the hysteretic cycle.

This pulsation of reactance gives rise to a distortion consisting mainly of a triple harmonic. Such current waves distorted by hysteresis, with a sine wave of impressed E.M.F., are shown in Figs. 66 to 69, Chapter X., on Hy- steresis. Inversely, if the current is a sine wave, the mag- netism and the E.M.F. will differ from sine shape.

For further discussion of this distortion of wave-shape by hysteresis, Chapter X. may be consulted.

Distortion of wave-shape takes place also by the pulsation of reactance due to synchronous rotation, as dis- cussed in chapter on Reaction Machines.

In Figs. 148 and 149, at a sine wave of impressed E.M.F., the distorted current waves have been constructed. Inversely, if a sine wave of current,

/ = / cos B,

392 ALTERNATING-CURRENT PHENOMENA.

passes through a circuit of synchronously varying reac- tance ; as for instance, the armature of a unitooth alterna- tor or synchronous motor — or, more general, an alternator whose armature reluctance is different in different positions with regard to the field poles — and the reactance is ex- pressed by

or, more general,

X =

the wave of magnetism is

X = x 1 + yr ^ cos (2 y ft- &

hence the wave of induced E.M.F.

= *sin/3 + sin ()8 - fflO +

[e, sin ((2 y + 1) sin ((2y+ l)/8 -«,+!)]} ;

that is, the pulsation of reactance of frequency, 2y, intro- duces two higher harmonics of the order (2y — 1), and

(2y + l\

If ^T=^l

, =*{sin0 + |sinG8-a) + .|l sin (3/J-o,)^

Since the pulsation of reactance due to magnetic satu- ration and hysteresis is essentially of the frequency, 21V,

DISTORTION OF WAVE-SHAPE. 393

— that is, describes a complete cycle for each half -wave of current, — this shows why the distortion of wave-shape by hysteresis consists essentially of a triple harmonic.

The phase displacement between e and i, and thus the power consumed or produced in the electric circuit, depend \ipon the angle, o>, as discussed before.

In case of a distortion of the wave-shape by reactance, the distorted waves can be replaced by their equivalent sine waves, and the investigation with suffi- cient exactness for most cases be carried out under the assumption of sine waves, as done in the preceding chapters.

Similar phenomena take place in circuits containing polarization cells, leaky condensers, or other apparatus representing a synchronously varying negative reactance. Possibly dielectric hysteresis in condensers causes a dis- tortion similar to that due to magnetic hysteresis.

Pulsation of Resistance.

To a certain extent the investigation of the effect of synchronous pulsation of the resistance coincides with that of reactance ; since a pulsation of reactance, when unsymmetrical with regard to the current wave, introduces an energy component which can be represented by an " effective resistance."

Inversely, an unsymmetrical pulsation of the ohmic resistance introduces a wattless component, to be denoted by "effective reactance."

A typical case of a synchronously pulsating resistance is represented in the alternating arc.

The apparent resistance of an arc depends upon the current passing through the arc ; that is, the apparent

resistance Of the arc = Potential difference^between electrodes jg high

for small currents, low for large currents. Thus in an alternating arc the apparent resistance will vary during

304 ALTERNATING-CURRENT PHENOMENA.

every half-wave of current between a maximum value at zero current and a minimum value at maximum current, thereby describing a complete cycle per half-wave of cur- rent.

Let the effective value of current passing through the arc be represented by /.

Then the instantaneous value of current, assuming the current wave as sine wave, is represented by

/ = 7V2sin/3;

and the apparent resistance of the arc, in first approxima- tion, by

R = r (1 + e cos 2 j8) ;

thus the potential difference at the arc is

e = iR = /V2Vsin/3(l -f e cos 2/3)

Hence the effective value of potential difference,

and the apparent resistance of the arc,

r.-f-ry/t-. + f

The instantaneous power consumed in the arc is,

Hence the effective power,

DISTORTION OF WAVE-SHAPE. 395

The apparent power, or volt amperes consumed by the arc, is,

thus the power factor of the arc,

that is, less than unity.

Provenance

Shelf: Reference library
Book: Theory and Calculation of Alternating Current Phenomena (1900)

Author: Charles Proteus Steinmetz (with Ernst J. Berg)
Rights: Published in 1900, before 1929, and therefore in the public domain in the United States.
Collected By: StanBot reference library