book

Theory and Calculation of Alternating Current Phenomena (1900) — part 12 of 19

1 January 1900

inductance will be more than 90° behind the induced E.M.F., and therefore in partial opposition, and will tend to reduce the terminal voltage. On the other hand, if the armature current leads, the E.M.F. of self-inductance will be less than 90° behind the induced E.M.F., or in partial conjunc- tion therewith, and increase the terminal voltage. This means that the E.M.F. of self -inductance increases the ter- minal voltage with a leading, and decreases it with a lagging current, or, in other words, acts in the same manner as the armature reaction. For this reason both actions can be combined in one, and represented by what is called the syn- cJironous reactance of the alternator. In the following, we shall represent the total reaction of the armature of the alternator by the one term, synchronous reactance. While this is not exact, as stated above, since the reactance should be resolved into the magnetic reaction due to the magnet- izing action of the armature current, and the electric reac- tion due to the self-induction of the armature current, it is in general sufficiently near for practical purposes, and well suited to explain the phenomena taking place under the various conditions of load. This synchronous reactance, x, Is frequently not constant, but is pulsating, owing to the synchronously varying reluctance of the armature magnetic circuit, and the field magnetic circuit ; it may, however, be considered in what follows as constant ; that is, the E.M.Fs. induced thereby may be represented by their equivalent sine waves. A specific discussion of the distortions of the wave shape due to the pulsation of the synchronous reactance is found in Chapter XX. The synchronous reactance, x, is not a true reactance in the ordinary sense of the word, but an equivalent or effective reactance. Sometimes the total effects taking place in the alternator armature, are repre- sented by a magnetic reaction, neglecting the self -inductance.' altogether, or rather replacing it by an increase of the arma- ture reaction or armature M.M.F. to such a value as to in- clude the self-inductance. This assumption is mostly made in the preliminary designs of alternators.

"302 ALTERNATING-CURRENT PHENOMENA.

Let E0 = induced E.M.F. of the alternator, or the E.M.F. induced in the armature coils by their rotation through the constant magnetic field produced by the cur- rent in the field spools, or the open circuit voltage, more properly called the "nominal induced E.M.F.," since in reality it does not exist, as before stated.

Then E0

where

n = total number of turns in series on the armature,

JV = frequency,

M = total magnetic flux per field pole.

Let x0 = synchronous reactance,

r0 = internal resistance of alternator ; then Z0 — r0 — j x0 = internal impedance.

If the circuit of the alternator is closed by the external impedance,

Z = r-jx, the current is

E0 E0

or, /=

and, terminal voltage,

or,

+x-

ALTERNA TING-CURRENT GENERA TOR.

303

or, expanded in a series,

As shown, the terminal voltage varies with the condi- tions of the external circuit.

As an instance, in Figs. 129-134, at constant induced E.M.F.,

Eo = 2500 ;

. ^

*- —

\ \ \

\ i

***>.

X^o

I 1

4S .

(

\ \

Si &'

2°'

ELD

CHA

MCI

ERIS

TIC

1 1

E0=

250( R =

, Zo-MOj, E, xko

I , 1

1 1 1

20 10 60 80 100 180 140 160 18P 2

X) 2

0 210 2

Fig. 129. Field Characteristic of Alternator on Non-inductive Load.

' +

and the values of the internal impedance,

z0 = r0 -jXo = i - ioy.

With the current / as abscissae, the terminal voltages E as ordinates in drawn line, and the kilowatts output, = /2 r, in dotted lines, the kilovolt-amperes output, = / £, in dash-

304

AL TEKNA TING-CURRENT PHENOMENA.

dotted lines, we have, for the following conditions of external

circuit :

In Fig. 129, non-inductive external circuit, x = 0.

In Fig. 130, inductive external circuit, of the condition, r / x = -f .75, with a power factor, .6.

In Fig. 131, inductive external circuit, of the condition, r= <>, with a power factor, 0.

In Fig. 132, external circuit with leading current, of the condi- tion, r/x = — .75, with a power factor, .6.

In Fig. 133, external circuit with leading current, of the condi- tion, r = 0, with a power factor, 0.

In Fig. 134, all the volt-ampere curves are shown together as complete ellipses, giving also the negative or synchronous motor part of the curves.

E72

FIE 500,

.D CHARA Zf MOj. i

CTERIST(C

-.75jop60^P.F

II*

/

x''

X^N

(/_

20 40 60 80 1

K» 120 140 1

H) 180 200 220 glQ 20

0 Amp

Fig. 130. Field Characteristic of Alternator, at 60% Power-factor on Inductive Load.

Such a curve is called a field characteristic.

As shown, the E.M.F. curve at non-inductive load is nearly horizontal at open circuit, nearly vertical at short circuit, and is similar to an arc of an ellipse.

ALTERNATING-CURRENT GENERATOR. 305

FIELD CHARACTt :0=25OO, Z?1-10j, r =

RISTIC

o, 90° Lag

1 R =

o »

C"

-X

o 2"

X X

Fig. 131. Field Characteristic of Alternator, on Wattless Inductive Load.

5 I

li'.'U

1000

.'.X'OU

ACT

f 2

50C

), Z

1-1

3j. :

= -.75 c

r 6

3^F

.F.

iloo

«••""

fc y

ItilK

,-*

'"'

400

lain..

v£

,.»*'

800

,,*"

-*''"

-n pe

•M

/x.

**'

;-r

*•"'

**•!••

•

Fig. 732. Field Characteristic of Alternator, at 60% Power-factor on Condenser Load.

306

AL TERNA TING-CURRENT PHENOMENA.

1 I 1 1

FIE

LD CHARACTERISTIC

/ /

i /

E0-2500, Zo-1-IOj, = o. 90°Leading Current

I'R

= O

/ /

r tu

/ /

^-x

x''

xlO

3- A,

nps.

fig. 133. Field Characteristic of Alternator, on Wattless Condenser Load.

With reactive load the curves are more nearly straight lines.

The voltage drops on inductive, rises on capacity load.

The output increases from zero at open circuit to a maxi- mum, and then decreases again to zero at short circuit.

AL TERN A TING-CURRENT GENERA TOR.

307

4^z

Fig. 134. Field Characteristic of Alternator.

The dependence of the terminal voltage, E, upon the phase relation of the external circuit is shown in Fig. 135, which gives, at impressed E.M.F.,

E0 = 2,500 volts, for the currents,

1= 50, 100, 150, 200, 250 amperes,

the terminal voltages, E, as ordinates, with the inductance factor of the external circuit,

as abscissas.

If the internal impedance is negligible compared with the external impedance, then, approximately,

308

AL TERNA TING-CURRENT PHENOMENA,

' .C .5 .4 .3 .2 .1 0 -.1 -.2 -.3 -.1 -.5 -.0 -.7 -.8

Fig. 135. Regulation of Alternator on Various Loads.

that is, an alternator with small internal resistance and syn- chronous reactance tends to regulate for constant terminal voltage.

Every alternator does this near open circuit, especially on non-inductive load.

Even if the synchronous reactance, x0 , is not quite neg- ligible, this regulation takes place, to a certain extent, on non-inductive circuit, since for

= 0, E

and thus the expression of the terminal voltage, E, contains the synchronous reactance, x0, only as a term of second order in the denominator.

On inductive circuit, however, x0 appears in the denom- inator as a term of first order, and therefore constant poten- tial regulation does not take place as well.

ALTERNATING-CURRENT GENERATOR. 309

With a non-inductive external circuit, if the synchronous reactance, XQ, of the alternator is very large compared with the external resistance, r,

current /= —

-g. 1 _E,

approximately, or constant ; or, if the external circuit con- tains the reactance, x,

T=-** 1 - *

approximately, or constant.

The terminal voltage of a non-inductive circuit is

approximately, or proportional to the external resistance. In an inductive circuit,

£° x

approximately, or proportional to the external impedance.

That is, on a non-inductive external circuit, an alternator with very low synchronous reactance regulates for constant terminal voltage, as a constant-potential ma- chine ; an alternator with a very high synchronous reac- tance regulates for a terminal voltage proportional to the external resistance, as a constant-current machine.

Thus, every alternator acts as a constant-potential ma- chine near open circuit, and as a constant-current machine near short circuit. Between these conditions, there is a range where the alternator regulates approximately as a constant power machine, that is current and E.M.F. vary in inverse proportion, as between 130 and 200 amperes in Fig. 129.

The modern alternators are generally more or less ma-

310 ALTERNATING-CURRENT PHENOMENA.

chines of the first class ; the old alternators, as built by Jablockkoff, Gramme, etc., were machines of the second class, used for arc lighting, where constant-current regula- tion is an advantage.

Obviously, large external reactances cause the same reg- ulation for constant current independently of the resistance, r, as a large internal reactance, .r0.

On non-inductive circuit, if

theoutputis

hence, if or

then

That is, the power is a maximum, and

and

7 =

V2 So {so + r0)

Therefore, with an external resistance equal to the inter- nal impedance, or, r — ^0 = VV02 + x^ , the output of an alternator is a maximum, and near this point it regulates for constant output ; that is, an mcrease of current causes a proportional decrease of terminal voltage, and inversely.

The field characteristic of the alternator shows this effect plainly.

SYNCHRONIZING ALTERNATORS. 311

CHAPTER XVIII.

SYNCHRONIZING ALTERNATORS.

All alternators, when brought to synchronism with each other, will operate in parallel more or less satisfactorily. This is due to the reversibility of the alternating-current machine ; that is, its ability to operate as synchronous motor. In consequence thereof, if the driving power of one of sev- eral parallel-operating generators is withdrawn, this gene- rator will keep revolving in synchronism as a synchronous motor ; and the power with which it tends to remain in synchronism is the maximum power which it can furnish as synchronous motor under the conditions of running.
The principal and foremost condition of parallel operation of alternators is equality of frequency ; that is, the transmission of power from the prime movers to the alternators must be such as to allow them to run at the same frequency without slippage or excessive strains on the belts or transmission devices.

Rigid mechanical connection of the alternators cannot be considered as synchronizing ; since it allows no flexibility or phase adjustment between the alternators, but makes them essentially one machine. If connected in parallel, a differ- ence in the field excitation, and thus the induced E.M.F. of the machines, must cause large cross-current ; since it cannot be taken care of by phase adjustment of the machines.

Thus rigid mechanical connection is not desirable for parallel operation of alternators.

The second important condition of parallel opera- tion is uniformity of speed ; that is, constancy of frequency.

312 ALTERNATING-CURRENT PHENOMENA.

If, for instance, two alternators are driven by independent single-cylinder engines, and the cranks of the engines hap- pen to be crossed, the one engine will pull, while the other is near the dead-point, and conversely. Consequently, alter- nately the one alternator will tend to speed up and the other slow down, then the other speed up and the first slow down. This effect, if not taken care of by fly-wheel capacity, causes a "hunting" or pumping action; that is, a fluctuation of the lights with the period of the engine revo- lution, due to the alternating transfer of the load from one engine to the other, which may even become so excessive as to throw the machines out of step, especially when by an approximate coincidence of the period of engine impulses (or a multiple thereof), with the natural period of oscillation of the revolving structure, the effect is made cumulative. This difficulty as a rule does not exist with turbine or water- wheel driving.

In synchronizing alternators, we have to distin- guish the phenomena taking place when throwing the ma- chines in parallel or out of parallel, and the phenomena when running in synchronism.

When connecting alternators in parallel, they are first brought approximately to the same frequency and same voltage ; and then, at the moment of approximate equality of phase, as shown by a phase-lamp or other device, they are thrown in parallel.

Equality of voltage is much less important with modern alternators than equality of frequency, and equality of phase is usually of importance only in avoiding an instantaneous flickering of the lights on the system. When two alter- nators are thrown together, currents pass between the machines, which accelerate the one and retard the other machine until equal frequency and proper phase relation are reached.

With modern ironclad alternators, this interchange of mechanical power is usually, even without very careful

SYNCHRONIZING ALTERNATORS. 313

adjustment before synchronizing, sufficiently limited net to endanger the machines mechanically ; since the cross- currents, and thus the interchange of power, are limited by self-induction and armature reaction1.

In machines of very low armature reaction, that is, machines of " very good constant potential regulation," much greater care has to be exerted in the adjustment to equality of frequency, voltage, and phase, or the inter- change of current may become so large as to destroy the machine by the mechanical shock ; and sometimes the machines are so sensitive in this respect that it is prefer- able not to operate them in parallel. The same applies in getting out of step.

When running in synchronism, nearly all types of machines will operate satisfactorily ; a medium amount of armature reaction is preferable, however, such as is given by modern alternators — not too high to reduce the synchronizing power too much, nor too low to make the machine unsafe in case of accident, such as falling out of step, etc.

If the armature reaction is very low, an accident, — such as a short circuit, falling out of step, opening of the field circuit, etc., — may destroy the machine. If the armature reaction is very high, the driving-power has to be adjusted very carefully to constancy ; since the synchronizing power of the alternators is too weak to hold them in step, and carry them over irregularities of the driving-power.

Series operation of alternators is possible only by rigid mechanical connection, or by some means whereby the machines, with regard to their synchronizing power, act essentially in parallel ; as, for instance, by the arrange- ment shown in Fig. 120, where the two alternators, Al} A2, are connected in series, but interlinked by the two coils of a large transformer, T, of which the one is connected

314

AL TERNA TING-CURRENT PHENOMENA.

across the terminals of one alternator, and the other across the terminals of the other alternator in such a way that, when operating in series, the coils of the transformer will

Fig. 136.

be without current. In this case, by interchange of power through the transformers, the series connection will be maintained stable.

In two parallel operating alternators, as shown in Fig. 137, let the voltage at the common bus bars be assumed

Fig. 137.

as zero line, or real axis of coordinates of the complex representation ; and let —

SYNCHRONIZING ALTERNATORS. 315

e = difference of potential at the common bus bars of

the two alternators,

Z = r — jx = impedance of external circuit, Y = g --jb = admittance of external circuit ;

hence, the current in external circuit is

Let

J?i = e-i — je\ = #2 (cos u>1 — j sin £>i) = induced E.M.F. of first

machine ; £2 = e.2 — _/>•/ = a2 (cos w2 — j sin w2) = induced E.M.F. of sec-

ond machine ;

/! = /! -f-//i' = current of first machine ; /2 = /2 -j-yY2' = current of second machine ; Z^ = T! — jxi = internal impedance, and Yv = gi -- jbl = inter-

nal admittance, of first machine ; Z2 = r2 — jxz = internal impedance, and K2 =gz ~~ jb<i = inter-

nal admittance, of second machine.

Then,

i^! , or ^ —je^= (e 2Z2, or <?2 —jej= (e 72 , or

This gives the equations —

4* + *"-**;

or eight equations with nine variables: ^, ^', ^2, ^/, /lf

316 ALTERNATING-CURRENT PHENOMENA.

Combining these equations by twos,

elrl -f eSxj. = er^ + t\2l2- e*r9 + ^/^2 = e substituted in

'i + H = we have

and analogously,

'1^1 — ^iVi + 'a *a — <?aVa = ' (^ + ^2 + dividing,

b + ^i + ^2 ^i ^;i + <?a ^ — ^iVi — ^a' ^2 ' substituting

g = V COS a Cl = tfj COS Wj ^2 = ^2 COS d)2

^ = z/ sin a ^/ = ^ sin oJj ^2' = a2 sin <o2 gives

a\ v\ cos (en — aQ + a2z>2 cos (a2 — a2) tfj z/! sin (ai — w^) -- a^Vs sin (a2 — a>2)

as the equation between the phase displacement angles and oi2 in parallel operation.

The power supplied to the external circuit is

of which that supplied by the first machine is,

/i = «\ ; by the second machine,

/2 = «a •

The total electrical work done by both machines is,

P = Pl + P*, of. which that done by the first machine is,

PI = '! h - e,' // ; by the second machine,

SYNCHRONIZING ALTERNATORS. . 317

The difference of output of the two machines is,

denoting

£>! -f- 0)2 <QI — o>2 s

2 2

A^>/AS may be called the synchronizing power of the machines, or the power which is transferred from one ma- chine to. the other by a change of the relative phase angle.

SPECIAL CASE. — Two equal alternators of equaL excitation.

Substituting this in the eight initial equations, these assume the form, —

e- = t x0 — t r0 e2' = /2 .r0 — // r0 . *g=i\ +'a eb = i{ + /a'

4* + 4" -</ + *"-<

Combining these equations by twos,

substituting el = a cos o^

e{ = a sin o^ ^2 = a cos 0)2 e2f = a sin o)2,

we have a (cos wx + cos wa) = * (2 + r0^ + a (sin Si + sin w2) = e (x^g — r0 li)

expanding and substituting —

8 =

318 AL TERN A TING-CURRENT PHENOMENA.

a cos e cos 8 = e ( 1 +

rQg --Xzb

a sin e cos 8 = ^^ ^

hence

That is

-and cos 8 = -

tan e = — ^ ^ — = constant.

A2 = constant;

-M1*5

z±aiy , /^o^-^o^\2.

cos 8

at no-phase displacement between the alternators, or, -we have e = ^ — .

V/('

n>^ + -o^\2 , fxn £— rb

From the eight initial equations we get, by combina-

(''o2

subtracted and expanded —

.or, since

<?! — <?2 = ^ (cos wj — cos G2) = — 2 tf2 sin c sin 8 ^/ — <?/ = a (sin wx — sin w2) = 2 a cos e sin 8 ; we have 2 a s*n 8 - r0sin c} — 2 ay0 sin 8 cos (c -f a), where tan d = ±2- . /"o SYNCHRONIZING ALTERNATORS. 319 The difference of output of the two alternators is A/ =/! — /2 = e (/i — /2) ; hence, substituting, substituting, 2ggsin8{jfrcos £ - r0 sin c}; , H XQ£ — r*b 2 i 'of, T •*<) " \ i 2 J + V 2 we have, 2a2 sin 8 cos 8 j *0( 1 + r°* + x°*\ — r0 expanding, A/ = nr Hence, the transfer of power between the alternators, A pt is a maximum, if 8 = 45° ; or Wj — w2 = 90° ; that is, when the alternators are in quadrature. 320 ALTERNATING-CURRENT PHENOMENA. The synchronizing power, A p / A 8, is a maximum if 8 = 0 ; that is, the alternators are in phase with each other. 197. As an instance, curves may be plotted for, a =2500, with the angle 8 = U)l a>2 as abscissae, giving the value of terminal voltage, e • the value of current in the external circuit, / = ey ; the value of interchange of current between the alternators, *i-*2; the value of interchange of power between the alternators, A p =A-/2; the value of synchronizing power, — ^ . A o For the condition of external circuit, g = 0, b = 0, y = 0, .05, 0, .05, .08, 0, .08, .03, + .04, .05, .03, - .04, .05. SYNCHRONOUS MOTOR. 321 CHAPTER XIX. SYNCHRONOUS MOTOR. 198. In the chapter on synchronizing alternators we have seen that when an alternator running in synchronism is connected with a system of given E.M.F., the work done by the alternator can be either positive or negative. In the latter case the alternator consumes electrical, and consequently produces mechanical, power ; that is, runs as a synchronous motor, so that the investigation of the synchronous motor is already contained essentially in the equations of parallel-running alternators. Since in the foregoing we have made use mostly of the symbolic method, we may in the following, as an instance of the graphical method, treat the action of the synchronous motor diagrammatically. Let an alternator of the E.M.F., E±, be connected as synchronous motor with a supply circuit of E.M.F., EQ, by a circuit of the impedance Z. If E0 is the E.M.F. impressed upon the motor termi- nals, Z is the impedance of the motor of induced E.M.F., E±. If E0 is the E.M.F. at the generator terminals, Z is the impedance of motor and line, including transformers and other intermediate apparatus. If EQ is the induced E.M.F. of the generator, Z is the sum of the impedances of motor, line, and generator, and thus we have the prob- lem, generator of induced E.M.F. EQ, and motor of induced' E.M.F. El; or, more general, two alternators of induced E.M.Fs., E0, Elf connected together into a circuit of total impedance, Z. Since in this case several E.M.Fs. are acting in circuit 322 ALTERNATING-CURRENT PHENOMENA. with the same current, it is convenient to use the current, /, as zero line OI of the polar diagram. Fig. 188. If I=i= current, and Z = impedance, r = effective resistance, x = effective reactance, and s = Vr2 -f x2 = absolute value of impedance, then the E.M.F. consumed by the resistance is E,, = ri, and in phase with the cur- rent, hence represented by vector OE,, ; and the E.M.F. consumed by the reactance is E2 = xi, and 90° ahead of the current, hence the E.M.F. consumed by the impedance is E = V(£,,)2 + (E2f, or = i Vr2 + x* = is, and ahead of the current by the angle 8, where tan 8 = x / r. We have now acting in circuit the E.M.Fs., E, Elf EQ; or El and E are components of EQ ; that is, EQ is the diagonal of a parallelogram, with El and E as sides. Since the E.M.Fs. Elf Ez, E, are represented in the diagram, Fig. 138, by the vectors OE~lf OE2, OE, to get the parallelogram of £Q, Elt E, we draw arcs of circles around 0 with EQ , and around E with El . Their point of intersection gives the impressed E.M.F., OEQ = EQ, and completing the parallelogram OE EQ E± we get, OE± = E± , the induced E.M.F. of the motor. IOE0 is the difference of phase between current and im- pressed E.M.F., or induced E.M.F. of the generator. IOEi is the difference of phase between current and in- duced E.M.F. of the motor. And the power is the current /times the projection of the E.M.F. upon the current, or the zero line OI. Hence, dropping perpendiculars, E^EJ and E^E^, from EQ and E! upon OI, it is — P0 = iX OE^ = power supplied by induced E.M.F. of gen- erator. PI = / X OE^ = electric power transformed in mechanical power by the motor. P = / x OEl = power consumed in the circuit by effective resistance. SYNCHRONOUS MOTOR. 323 Since the circles drawn with EQ and E± around O and K respectively intersect twice, two diagrams exist. In gen- eral, in one of these diagrams shown in Fig. 138 in drawn Fig. 138. lines, current and E.M.F. are in the same direction, repre- senting mechanical work done by the machine as motor- In the other, shown in dotted lines, current and E.M.F. are in opposite direction, representing mechanical work con- sumed by the machine as generator. Under certain conditions, however, £Q is in the same, E^ in opposite direction, with the current ; that is, both ma- chines are generators. 199. It is seen that in these diagrams the E.M.Fs. are- considered from the point of view of the motor ; that is,. 324 ALTERNATING-CURRENT PHENOMENA. work done as synchronous motor is considered as positive, work done as generator is negative. In the chapter on syn- chronizing generators we took the opposite view, from the generator side. In a single unit-power transmission, that is, one generator supplying one synchronous motor over a line, the E.M.F. consumed by the impedance, E = OE, Figs. 139 to 141, con- sists of three components ; the E.M.F. OE£ — Ez, consumed Fig. 139. by the impedance of the motor, the E.M.F. consumed by the impedance of the line, and the E.M.F. EZ E = E± consumed by the impedance of the generator. Hence, dividing the opposite side of the parallelogram E1E(), in the same way, we have : OEl = E1 = induced E.M.F. of the motor, OEZ = 2?a = E.M.F. at motor terminals or at end of line, OE3 = E3 = E.M.F. at generator terminals, or at beginning of line. OEQ = EQ = induced E.M.F. of generator. SYNCHRONOUS MOTOR. 325 The phase relation of the current with the E.M.Fs. £lt , depends upon the current strength and the E.M.Fs. El and 200. Figs. 139 to 141 show several such diagrams for different values of Elf but the same value of / and EQ. The motor diagram being given in drawn line, the genera- tor diagram in dotted line. Fig. 140. As seen, for small values of E1 the potential drops in the alternator and in the line. For the value of E1 = E0 the potential rises in the generator, drops in the line, and rises again in the motor. For larger values of Ely thfe potential rises in the alternator as well as in the line, so that the highest potential is the induced E.M.F. of the motor, the lowest potential the induced E.M.F. of the gen- erator. 326 ALTERNATING-CURRENT PHENOMENA, It is of interest now to investigate how the values of these quantities change with a change of the constants. Fig. 747. 201. A. — Constant impressed E.M.F. Ev, constant current strength I = i, variable motor excitation Ev (Fig. 142.) If the current is constant, = z; OE, the E.M.F. con- sumed by the impedance, and therefore point E, are con- stant. Since the intensity, but not the phase of EQ is constant, EQ lies on a circle eQ with EQ as radius. From the parallelogram, OE EQ El follows, since E1 EQ parallel and = OE, that El lies on a circle el congruent to the circle eQ, but with Ei} the image of E, as center : OEi = OE. We can construct now the variation of the diagram with the variation of El ; in the parallelogram OE EQ E1 , O and E are fixed, and E0 and El move on the circles <?0 el so that EQ E^ is parallel to OE. SYNCHRONOUS MOTOR. 327 The smallest value of El consistent with current strength / is Olj = E^, 01 = EQ. In this case the power of the motor is Olj1 x /, hence already considerable. Increasing El to 02"^ OSj, etc., the impressed E.M.Fs. move to 02, 03, etc., the power is / x 02^, I x 03^, etc., increases first, Fig. 142. reaches the maximum at the point 3j, 3, the most extreme point at the right, with the impressed E.M.F. in phase with the current, and then decreases again, while the induced E.M.F. of the motor E^ increases and becomes = £Q at 4,, 4. At 515 5, the power becomes zero, and further on negative ; that is, the motor has changed to a dynamo, and 328 AL TERNA TING-CURRENT PHENOMENA. produces electrical energy, while the impressed E.M.F. E^ still furnishes electrical energy, that is, both machines as generators feed into the line, until at 61} 6, the power of the impressed E.M.F. E§ becomes zero, and further on power begins to flow back ; that is, the motor is changed to a gen- erator and the generator to a motor, and we are on the generator side of the diagram. At 1l, 7, the maximum value of Elt consistent with the current /, has been reached, and passing still further the E.M.F. El decreases again, while the power still increases up to the maximum at Slt 8, and then decreases again, but still El remaining generator, EQ motor, until at 11^ 11, the power of EQ becomes zero; that is, EQ changes again to a generator, and both machines are generators, up to 12lf 12, where the power of El is zero, El changes from generator to motor, and we come again to the motor side of the diagram, and while El still decreases, the power of the motor increases until lu 1, is reached. Hence, there are two regions, for very large El from 5 to 6, and for very small El from 11 to 12, where both machines are generators ; otherwise the one is generator, the other motor. For small values of El the current is lagging, begins, however, at 2 to lead the induced E.M.F. of the motor Elf at 3 the induced E.M.F. of the generator E0. It is of interest to note that at the smallest possible value of EI} lj, the power is already considerable. Hence, the motor can run under these conditions only at a certain load. If this load is thrown off, the motor cannot run with the same current, but the current must increase. We have here the curious condition that loading the motor reduces, unloading increases, the current within the range between 1 and 12. The condition of maximum output is 3, current in phase with impressed E.M.F. Since at constant current the loss is constant, this is at the same time the condition of max- imum efficiency : no displacement of phase of the impressed SYNCHRONOUS MOTOR. 329 E.M.F., or self-induction of the circuit compensated by the effect of the lead of the motor current. This condition of maximum efficiency of a circuit we have found already in the Chapter on Inductance and Capacity. 202. B. EQ and El constant, I variable. Obviously EQ lies again on the circle eQ with EQ as radius and O as center. Fig. 143. E lies on a straight line e, passing throtigh the origin; Since in the parallelogram OE E0 Ev EEQ = E^ we derive EQ by laying a line EEQ = E± from any point E in the circle eQ, and complete the parallelogram. All these lines EEQ envelop a certain curve elt which 030 ALTERNATING-CURRENT PHENOMENA. can be considered as the characteristic curve of this prob- lem, just as circle e^ in the former problem. These curves are drawn in Figs. 143, 144, 145, for the three cases : 1st, El = EQ ; 2d, El < EQ ; 3d, £1>£Q. In the first case, El = EQ (Fig. 127), we see that at Fig. 144. very small current, that is very small OE, the current / leads the impressed E.M.F. EQ by an angle EQOf = WQ. This lead decreases with increasing current, becomes zero, and afterwards for larger current, the current lags. Taking now any pair of corresponding points E, EQ, and producing EEQ until it intersects eit in Eif we have ^^ Ei OE — 90°, El = EQ , thus : OE1 = EEQ=OEQ = EQEt ; that is, EE{ = SYNCHRONOUS MOTOR. 331 2EQ. That means the characteristic curve el is the enve- lope of lines EEiy of constant lengths 2EQ, sliding between the legs of the right angle Et OE; hence, it is the sextic hypocyloid osculating circle <?0, which has the general equa- tion, with e, ei as axes of coordinates : In the next case, E1 < EQ (Fig. 144) we see first, that the current can never become zero like in the first case, V Fig. 145. EI = EQ, but has a minimum value corresponding to the minimum value of OEl : I{ = — — , and a maximum value : //' = — — . Furthermore, the current can never lead the impressed E.M.F. E^, but always lags. The mini- 332 ALTERNATING-CURRENT PHENOMENA. mum lag is at the point H. The locus ev as envelope of the lines EEty is a finite sextic curve, shown in Fig. 144. If El < EQ , at small EQ — El , H can be above the zero line, and a range of leading current exist between two ranges of lagging current. In the case E1 > EQ (Fig. 145) the current cannot equal zero either, but begins at a finite value C±, corresponding to the minimum value of OEQ : // = * — -. At this value however, the alternator E1 is still generator and changes to a motor, its power passing through zero, at the point corresponding to the vertical tangent, onto elf with a very large lead of the impressed E.M.F. against the cur- rent. At H the lead changes to lag. The minimum and maximum value of current in the three conditions are given by : Minimum: Maximum: 1st. 7=0, 7=^. Since tfie current passing over the line at El = O, that is, when the motor stands still, is 70 = EQj z, we see that in such a synchronous motor-plant, when running at syn- chronism, the current can rise far beyond the value it has at standstill of the motor, to twice this value at 1, some- what less at 2, but more at 3. 203. C. EQ = constant, El varied so that the efficiency is a maximum for all currents. • (Fig. 146.) Since we have seen that the output at a given current strength, that is, a given loss, is a maximum, and therefore SYNCHRONOUS MOTOR. 333 the efficiency a maximum, when the current is in phase with the induced E.M.F. EQ of the generator, we have as the locus of EQ the point EQ (Fig. 146), and when E with increasing current varies on <?, E± must vary on the straight line ev parallel to c.

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