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Two Reaction Theory of Synchronous Machines Generalized Method of Analysis-Part I BY R. H. PARK* Associate, A. I. E. E.

Synopsis.-Starting with the basic assumption of no saturation or hysteresis, and with distribution of armature phase m. m. f. effectively sinusoidal as far as regards phenomena dependent upon rotor position, general formulas are developed for current, voltage, power, and torque under steady and transient load conditions. Special detailed formulas are also developed which permit the determination of current and torque on three-phase short circuit, during starting, and when only small deviations from an average operating angle are involved.

In addition, new and more accurate equivalent circuits are developed for synchronous and asynchronous machines operating in parallel, and the domain of validity of such circuits is established. Throughout, the treatment has been generalized to include salient poles and an arbitrary number of rotor circuits. The analysis is thus adapted to machines equipped with field pole collars, or with amortisseur windings of any arbitrary construction. It is proposed to continue the analysis in a subsequent paper. *

*

*

*

*

T HIS paper presents a generalization and extension ia, ib, i, = per unit instantaneous phase currents of the work of Blondel, Dreyfus, and Doherty eay eby e, = per unit instantaneous phase voltages and Nickle, and establishes new and general *a, 'rb, V1 = per unit instantaneous phase linkages methods of calculating current power and torque in salient and non-salient pole synchronous mac hines, d under both transient and steady load conditions. P dt Attention is restricted to symmetrical three-phaset machines with field structure symmetrical about Then there is the axes of the field winding and interpolar space, ea P 1a-r but salient poles and an arbitrary number of rotorl eb = P 4'b-r 'ib circuits is considered. e, = p Vlc - r i(1) Idealization is resorted to, to the extent that saturaIt has been shown previously' that tion and hysteresis in every magnetic circuit and eddy Axis of Phase a 21a Id COS 6- Iq sin 0

ea=nthereri

X 0~__

- 33

D \ \ Direction of

Rotatiion Xf

Xd

+ ib + icl ~~~~~~~~~~~~~~~[ia [3+b+c - Xq

+Xq

[ia cos 2 0+ ib Cos (2

a-

[

ib___i 2

1

J

6 - 120)

3 Quadrature Axis

+ i, cos (2 6 + 120)]

\1b

Id cos (6- 120)- Iq sin (6- 120) ic + Xd + ia + ib +ib 2 xq i]

xis of Phase c

Axis of Phase b FIG. 1

Xd- Xq

d 3 q[a COS (2

currents in the armature iron are neglected, and in the assumption that, as far as concerns effects depending on the position of the rotor, each armature winding may be regarded as, in effect, sinusoidally distributed.3 A. Fundamental Circuit Equations Consider the ideal synchronous machine of Fig. 1, and let *General Engg. Dept., General Electric Company, Schenectady, N. Y.L tSingle-phase machines may be regarded as three-phase machines with one phase open circuited. tStator for a machine with stationary field structure.3 3For numbered references see Bibliography. Presented at the Winter Convention of the A. I. E. E., New York,

N. Y., Jan. 28-Feb. i, 1929.

I

120) + ib COS (2 + i, cos 2 0]

-

= Id cos (6 + 120)

-Iq sin (6 + 120) -xo -

Xd + X,

_

where, 716

29-33

0

3

xd - zX 3

-

I C-

ia

a+i

9

+ ib +

ia_

0 +

ic

J

o 26-- 2)±1 2)+i [aCs(

o O

+ i.c os (2 6 - 120)]

120) (2)

PARK: SYNCHRONOUS MACHINES

July 1929

717

If there is one additional rotor circuit in the direct Id = per-unit excitation in direct axis axis there is, I, = per-unit excitation in quadrature axis Xd = direct synchronous reactance E-I I = I + Xf lId - (Xd -Xd) id x, = quadraturesynchronousreactance To p x0 = zero phase-sequence reactance As shown in the Appendix, if normal linkages in -Ild Fld = Xlld Ild + Xfld I -Xmi d id = To the field circuit are defined as those obtaining at no load* there is in the case of no rotor circuits in the which gives, direct axis in addition to the field, 4) = per-unit instantaneous field linkages [Xlld - Xfld] Told P + 1 = I- (Xd - Xd') id G (p) A (p) where, To Told [Xlld (Xd - Xd') - Xfld Xmldl p2 I = per-unit instantaneous field current

id

=

2

3 Iia cos 0 + ib cos (O - 120) + i cos (f9 +

120)1

Xd

(p)

XXd

+ [(Xd

X'd) Told + Xmld To]

P

A (p)

(3) where, On the other hand, if n additional rotor circuits A (p)=[XI1d-X.fXd21 To Told p2+[Xlld To+Told] P+I exist in the direct axis there is, If there is more than one additional rotor circuit the 4) = I + XfId Ild + Xf2d I2d operators G (p) and Xd (p) will be more complicated but + . . + Xfnd 'nd - (Xd - Xd') id may be found in the same way. The effects of external field resistance may be found by changing the term I where, etc., are the per-unit instantaneous cur- in the field voltage equation to R I. Open circuited Ild, I2d, rents in circuits 1, 2, etc., of the direct axis, Xf1Id, Xf2d, field corresponds to R equal to infinity. Similarly, there will be . etc., are per-unit mutual coefficients between the field and circuits 1, 2, etc., of the direct axis. Iq = [Xq - Xq (p)] iq (5) Similar relations exist for the linkages in each of the where, additional rotor circuits except Xd - Xd' is to be replaced 2 by a term xm. However, since all of these additional i, =- ia sin 0 +ib sin(6- 120) +i, sin(6 +120) } (3a) circuits are closed, it follows that there is an operational result Xq (o) = Xq, X ( ) =q . + Ind Id I + Ild + I2d + So far, 10 equations have been established relating = G (p) E + H (P) id (4) the 15 quantities ea, eb, e,, ta, ib ,I 4'aya tby ,t'c lid, iqy where E is the per-unit value of the instantaneous field Id, I, E, 0 in a general way. It follows that when voltage, and G (p) and H (p) are operators such that any five of the quantities are known the remaining 10 may be determined. Their determination is very G (o) = 1 G (co) = 0 much facilitated, however, by the introduction of H (o) = 0 H (co) = Xd-Xd" = certain auxiliary quantities ed, e, eoy 'Pd, aqV 4/0' Xd' the subtransient reactance2 Thus let It will be convenient to write H (p) = Xd Xd (p) and to rewrite (4) in the form,1 (3b) to lia + ib + ic) Id = G (p) E + [Xd-Xd (P)] id (4a) i 3{ia+tb±%} If there are no additional rotor circuits, there is, as 2 shown in Appendix I, ed= {eacos 0 + ebcos (o- 120) + e,cos (6 + 120)1 'I = I - (Xd - Xd') id E = To pT +I 2 where To is the open circuit time constant of the field e - f{ea sin 6+eb sin(0-120)+e, sin(0+120)1 (6) in radians. 1 There is then, 1 eO = 3j { ea + e?b + e~} - G (p) T= -H 2 XdTo p +Xd )/d=3 {'{a COS 6+Pb cOs (6-120)±+Pcc9s (0+ 120)} =

i'o

*Thjs definition is somewhat different from that given in 'Pq =-

reference 2.

{ 'P sin O+'Ib sin (--120)

+'Pc sin(60+1l20) }

(7)

Transactions A. I. E. E.

PARK: SYNCHRONOUS MACHINES

718

1

A0

{

ed coS(6-120) - eq sin (6-120) + eO (16) ed cos (6+ 120)-eq sin (6 + 120) + eo Referring to Fig. 2, it may be seen that when there are no zero quantities, that is, when eo = i/0 = io = 0, the phase quantities may be regarded as the projection of vectors e, i/i, and i on axes lagging the direct axis by eb

i/a + Ab + 0J'

ec

then from Equation (1) there is 2 ed= 3{cos 6r la+co5(6-120) p i/4+cos(6+120) P i/}

=

=

Axis of Phase b

- rid

2 eq=-3 {sin6p a+ sin (6- 120) pi/b\

+ sin (6 + 120) p

r ieo= p io-/ but,

Pi

d

2

=

e

J}r i-

/

\

eq Direct Axis ed Axis of Phase a

I{COS 0 Pia + COS (6- 120) p ib/ + cos (6 + 120) p i/'}/

2

-u {sin 0 ed +

i/2a+sin(- 120) p ib+sin(0+120) p i/J} p 6

ri2d

p l =-

2

+

Axis of Phase c

/lq pGO

FIG. 2

I{sin 0 p i/a + sin (6 - 120) p Vlb + sin(f +120) pi/}

angles 0, 0 - 120 and 0 + 120, where taking the direct axis as the axis of reals, e

=e+Jje0

-3co6O 'Pa + COS (6- 120) 'Pb + cos (6 +120) V;Jp 6 eq + riq - i/d P 6 hence there is ed = P

Od-r id-/q p °

eq=p'pq-riq+i/ dpd

io Also it may be readily verified that i/d = Id- Xd id = G (p) E- Xd (P)d iq = I-Xq iq = - Xq (p) iq eo

ii0

=

-

Xoio

=

pi/-r

(8)

(9)

(10)

+ j ?,q =id41d +Ji q =

If we introduce in addition the vector quantity, I = Id +j Iq the circuit equations previously obtained may be lpIXiW

(11)/ (12) (13)

Equations (8) to (13) establish six reelatively simple irectAxis relations between the 11 quantities ed, e., eo, id, iqy i0, to i/d, i'q, i/, E, 0. In practise it is usually possible xi d p determine five of these quantities directly from the terminal conditions, after which the remaining six may be r calculated with relative simplicity, After the direct, quadrature, and zero quantities are known the phase FIG. 3 quantities may be determined from the identical relations transferred into the corresponding vector forms, 'ta ='td cos G-iq sinG0+ io e = pi/i=ri +[p0] ji/i ib= td COS(0- 120)- i0zsin (6- 120) + io (14) _o = Ix iC= id COS (6 + 120)-iq sin (6 + 120) + io where, zi =XZdid+ jXqiq {Pa = Pd cos 60-Pqsin 6 + {0 lAb = i/d cos (6 -120) - 'P sin (60-120) + {0 (15) BFig. 3 shows these relations graphically. The per-unit instantaneous power output from the C= 'Pd COS (6 + 120)- i/q sin (6 + 120) + 'Ps 6 is necessarily proportional to the sum armature + e5 ed cos sin 60-eq e=

July 1929

PARK: SYNCHRONOUS MACHINES

ea ia + eb ib + e, i. By consideration of any instant during normal operation at unity power factor it may be seen that the factor of proportionality must be 2/3. That is, P = per-unit instantaneous power output = 2/3 { ea ia + eb ib + e,i, } Substituting from Equations (14) and (16) there results the useful relation, P = edid + eqiq + eO i0 (17) C. Electrical Torque on Rotor It is possible to determine the electrical torque on the rotor directly from the general relation, {Total power output} = {mechanical power transferred across gap} + {rate of decrease of total stored magnetic energy} - {total ohmic losses} (18) However, since this torque depends uniquely only on the magnitudes of the currents in every circuit of the machine, it follows that a general formula for torque may be derived by considering any special case in which arbitrary conditions are imposed as to the way in which these currents are changing as the rotor moves. The simplest conditions to impose are that Id, Iq, id, iq, and io remain constant as the rotor moves. In this case there will be no change in the stored magnetic energy of the machine as the rotor moves, and the power output of the rotor will be just equal in magnitude and opposite in sign to the rotor losses. It follows that under the special conditions assumed, Equation (18) becomes simply, {armature power output} = {mechanical power across gap} - {armature losses} or, P = T p

0-

2r

= Tp G- r { id2

{a2+

719

D. Constant Rotor Speed Suppose that the constant slip of the rotor is s. Then there is, ed = P ld-rid- (1-S) 'q eq =p V1,q-r iq + (1-s) {d but, 4I'd = G (p) E-Xd (p) id q = -xq (p) iq p Xd (p) + r = Zd (P) Putting PXq (p) + r = zq (p) there is + (1-s) Xq (p) iq (20) e = p G (p) E -Zd (p) Pd eq = (1-s) [G (p) E-Xd (p) idl-Zq (p) iq (21) Solving gives, id = Zq (p) ± Xq (p)] C (p) E-Zq (p) ed -(1-s) xq (p) eq } . D (p) (22) (1-s) r G (p) E-2d (p) eq - (1-s) Xd (p) ed iq D (p) (23)

{[s

(1x-)2

where, D (p) Zd (p) 2q (P) + (1 - S)2 Xd (p) Xq (p) E. Two Machines Connected Together Suppose that two machines which we will designate respectively by the subscripts g and h, are connected together, but not to any other machines or circuits, and assume in addition that there are no zero quantities. In this case the voltages of each machine will be equal Axis Phase b gq e

2 r

\

ib2 + ic2}oh

/

egd

Machine Xogvh~~~~~~~~~~~~ Axis Phase a

+ ig2 + i2 }hd

Then, T = per-unit instantaneous electrical torque ± ed id + eqiq + eO io + r {td2 ± iq2 ji2}q pO but subject to the conditions imposed, ed p 0G - rid = e, Ad PO0-r iq eo = r io It therefore follows that, T = q Atd - id (6q (19) = vector product of s6 and z = if X i (19a) a result which could have been established directly by physical reasoning. Formula (19) is employed by Dreyfus in his treatment of self-excited oscillations of synchronous machines.'4

Direct Axis of g Machine Direct Axis of

/ Axis Phase c

=

FIG. 4

phase for phase, and it therefore follows that the voltage vectors of each machine must coincide, as shown in

Fig. 4.

Referring to the figure it will be seen that the direct and quadrature components of voltage of the two machines are subject to the mutual relations, ehd = e,d cos 6- eq sin 6 (24) ehql = e7d sinA + egq, cos 6 e0d = ehd cos 6 + ehq sin 6 (25) eg,q = -ehd sin 6 + ehqcos a

720

PARK: SYNCHRONOUS MACHINES

On the other hand, for currents there will be ihd - - {i%d COS 6- igq sin ±} ihq = - {igdsin 6 + iq cos5} (26) igd = - [ihd COS 6 + ihq sin 6} 6 igq = (27) COS } F. One Machine on an Infinite Bus In (E), if machine h has zero impedance, it follows from (20) and (21) that ehd = 0, ehq = bus voltage -

say

=

I-ihdsin

Transactions A. I. E. E.

To p + Xd

Xd'

Xd Xq

Top + 1

+ [Xd r To + (Xd + r To) Xj + [r (Xd + (d++

ihq

To)

rT+

[

Xq

Xd

d

Xd Xq

p2 q

To] p

'XT

d (p)&

(31)

Top-Fl

By the expansion theorem there is, finally, Xq E r2 1d = + Xd Xq

(28)

q

+

Then if the rotor leads the vector s6, by an angle 8 there is

1

4tq --J,sin6 6 d =

X

Xd

+ r2 + Xd Xq To p + 1

e.

Then for machine g there is, ed = e sin 6 eq = e cos 6 G. Torque Angle Relations From Equations (11), (12), and (19), there is, T

Xq

To P3

1q

COS

Xd(Toa,+ 1) ((Zqan + r) ed_ + X.eq_) a, d' (a,)

rE + r2 Xd

X,

1

Xd' To a .2 + (Xd +-r To) a,, + r)eqo- (To an X d + Xd) edO Xd F 8 (29) (q + i/'2sin 2 a, d' (a,,) Xq Xd 2Xd Xq E--ant (32) A derivation of this formula for steady load con(32) end ditions has been previously given by Doherty and where the summation is extended over the roots of Nickle.9 d H. Three-Phase Short Circuit with Constant Rotor Speed d (a) = 0 and d' (P) = dp d (p) T=

Iq '

cos 6 +

Id sin 6 1

Maintained Since a three-phase short circuit causes ed and eq to vanish suddenly, its effect with constant rotor speed maintained may be found by impressing ed = - edo, eq = - ego in (22) and (23) where edo and eqo are the values of ed and e, before the short circuit. The initial currents existing before the short circuit must be added to the currents found in this way in order to obtain the resultant current after the short circuit. With s = 0 and E constant there is in detail.I Zq (p) edo + XQ (p) eq o xq E - r edo- xq eqo D (p) r2 + Xd xq iq

-

Zd (p) eqO- Xd (p) edO

1

D (p)

r E

-

r eqo

r2 +

The phase currents may, of course, be found from -0.0018

-0.0016

-0.0014 Values of o~

-0.0012--

-0.0010-0.0008

-0.0006-_

+ Xd edO

-00004A

xdxq

-0.0002-

(30)I

The working out of the formulas may be illustrated by consideration of the simple case of a machine with no rotor circuits in addition to the field. In this case there is X0 (p) = T0 p Xd _Xd' +1 To p +F 1 Xd P

0.5

1.0


0.6

Xd )S) (+

2

J- 2

3.0

FIG. 6

0.2-

xq (i s)- r + (1-2 s) Xd'(j S) xq'(j s) +j s r[Xd'C( S) +X,'(j S)]

{

/

o k

}

1 j (1-2 s)

lVaues of cor

-4

i

=

e,=p ijs

-8

E

2 - s t, and referring to Equation (28),

&d

-

(1

-

s)

eq p e+riq e A-1 s) (1-s) P

p (ed A- rid) A- (1- s)

i

-(

(q A- r i,,)

(7

p(e,, A- r iq)-(1- s) (ed A- r id) )1/ = 1S2(38) 2 P' -(

Transactions A. I. E. E.

PARK: SYNCHRONOUS MACHINES

722

41d =

j s (ed + r ?td) + (1- s) (eq ± r iq) 12 1 2s1

Y'2

j s (eq + r iq)- (1- s) (ed + rid) 1-2 s

-j .i It follows that the vector amounts of forward and backward m. m. f. or current are (ic

+i q)

ib =2 (id j iq)

backward current =

with ed = 1.0, eq = jjs +jsrid + (1- s) (-j) + (1- s) r iq Wd 1-2s

(42)

If we define by analogy,

-(1- 2s) j + r [ S id + (1-s) iq] 1- 2s

forward voltage =

-i + 1- 2 [jsid + (1-s) i]q

backward voltage

(39)

j s (-j + r iq) - (1- s) - r (1-s) id

There is, 1 {

1- 2s

=

= 2 2f

forward current =

=

2 (ed + j 2

eq)

(ed-jel)

-2r

(43)

)

-(1 - 2s) + r[jsiq- (1-s) id]

s) + 12(r + js [Xd Q s)

Xd (js) X

r

s1i-2 (1

=r+ 1 2

S) id]

(40)

r + Xq

Thus,

~~

pi~~

Tav=1/2

2 s(is --i

+(

)i)|

r

-

id (1) id -js is (l (1 S)i)

=Pav+2(1 pa

r

(1

r

-

rq2+id =Pav +r 2

2) +

Pa,

+

=2 i[xq (j s) -Xd (j s)]

1

r

s

221-2 2(1-2 s) (iq

d] ±

~

+iiid)2

(44)

(45)

eb =0

d

[

Xd (i s) z' (i s

1.0

=

ef

id

;.

r

s) (iq2 + id2)-

2 s) L+25stq

(iq2

ib

(iS)]) }

7Tav

(41)

Mr. Ralph Hammar, who has been engaged in the application of the general method of calculation outlined above, to the predetermination of the starting torque of practical synchronous motors, has suggested an interesting modification of formulas (36) and (41), based upon the fact that, since the total m. m. f. consists of direct and quadrature components pulsating at slip frequency, it may be resolved into two components, one moving forward at a per-unit speed 1-s+s=1.0, and the other moving backward at a per-unit speed 1- s -s = 1 -2 s. Thus from this standpoint half of both the direct and quadrature components will move forward, and half backward. Since the quadrature axis is ahead of the direct it follows that as far as concerns the for-ward component the quadrature current iq is equivalent to a d-c. j iq, while as regards backward component it is equivalent to a direct component

PaV

=

ef . Pav

if +

=

real of if

rijf2

+

1-2si2

(46) (47)

J. Zero Armature Resistance, One Machine Connected to an Infinite Bus Assume that a machine of negligible armature resistance is operating from an infinite bus of per-unit voltage e, at synchronous speed, with a steady excitation voltage Eo, and displacement angle 8o. At the instant t = 0, let a and E change. There is, 1 G 1G+ () A E- AdO Xd (P) Xd (P) Xdi 1 , t __ qO 1' Xq -Xq (p) Ad = e cos 8 ~ i A - l From which there is, by obvious re-arrangement,

PARK: SYNCHRONOUS MACHINES

July 1929 E- ecosa

Xd

id

Xd

Xd (P)

+ e Xd

(Cos

Cos°

+

)

c Y

(p) - G(p) Xd AE

e sina5

=

Xd - Xd"flau

e

e Xq

Xq (P)

X(?

x (P))-e (sina- sin8o) XrjXq

(48)

e

+

Xd

(p) e2cosa Xq XqXq (P) (sin Xq

-

sin0o)

a-

+ e2 sin a Xd - Xd (p) ) Xd Xd (p) (co , d(p e sin

(49)

(P).E+2zc'Cos6 ag0_gt ea8Cos()o')

(p) Xdinx(P)XdXd-XdG (P)

T+

Xq (p)

AE-e

Xq- X

-

Xd

Xd

(p)

(Y}

Xd

-Xdfl

a,2n

* 1 ~~~~~

Xd

Xd

1=

(p)

Xif

Xq

()+

at

E-qn,

bn

=

where,

2Xd Xq

saUZ

Xqt

sin 28

eadnu sin 8(u) &'(u)

du

qrE

azqntf ECqucs5()5()d

aq

bnbE1 ,, 4'r- A E' (u) d u

(49a)

(49a) may be used to determine starting torque and current with zero armature resistance, by introducing a (t) = s t, a' (t) = s. Thus the average component of torque is found to be,

I

=2

-

Xd

1 Xq

adn S adn2 + S2

adn

Xd'

Xq"_

Xq Xq y Since

a

Xd- Xd

ad- 6F-nt a~(qn S

a0qn

aq 2 + s2

(52)

aso

as2 - s2 is never greater than 1 2^ and

E adn =

=

0

E

Xd

1.0

+-Ff)

a

e2(xd-Xq)

2

1.0 It therefore follows from the operational rule that,

J (p) F (t) = F (o) 4 (t)

J naqu cos 5 (u) a' (u) d u tLqnICi 0

Eesin6

b,En

(cx) Xd"=Xd = 1.0

E ad,, adn

a.,

e sina

Tav

Xd (p)-XdG(P)

-,

cxd t Formula

e

(50) Xd

-

Xqe2

Xqt

Xq

Xq(p)

Xq" Xq X,

Xq

Xd X

But quantities adn aqn, ad, aqqn bn,,, [,, may be found such thatXq

(48a)

sin a

+e2 Xd Xd sin 32adn E-adnI

a)

-

En A E (u) d u

Xq

-F+ e

Xd-Xq ~~+22 Xd X1e2 X,i sin 2a

sin 8 (u) 5 (u) dd u

E

0°

iq

Then, T

t4'

bn e

Xd

+e

E e sin a

adn Ead

-1

Xd Xd (P)

723

aqn = 1.0

it follows that Tav is never greater than 1

(t- u) F' (u) d u (51)

4

Xd Xd

Xd Xd -Ft

Xq, Xq" Xq Xq"

53

Equation (53) thus provides a very simple criterion of the maximum possible starting torque of a synthat if chronous motor of given dimensions, when armature a = a (t) resistance is neglected. a' The same formula may also be used to obtain p a = (t) A E = A\ E (t) simple expression for the damping and synchronizing ap A\ E = A\ E' (t) components of pulsating torque due to a given small Equations (48) and (49) may be rewritten in the form, angular pulsation of the rotor. if the E 8- ecos iS (s t) E-e cosangular a /\ pulsation = [Ab] sin 4

(t) = f (P) . 1

-

~~~Thus

Xd

and if the punlscation6~~~~Aof torquel

is! expresose

in

the form

724

PARK: SYNCHRONOUS MACHINES

'A T

=

TsA6±+ Td

d

dt A

d

there results,

T= Tso ± e2 sin20 + elcs6 S

Xq X

Xd -

Xqll

X

Xd"

aqn 2S

-Xd' sTd=e2in26oXd e2sin2 Td= Xd Xd"

ad,

(ad

+

82

(54)

scadn n+ 82

aqns8 cqn +el cos2 o Xq - Xq" + e2 COS2 6 Xq Xq Pt (aq,)2 + s2 where, c eIdO COS60 +± e2 (Xd Xq) cos 2 6o X,j ~~Xd XqCl bo = average angular displacement, i. e., total angle = 6 = 5o + A 6. It can be shown that for the case of no additional rotor circuits, Equations (54) are exactly equivalent to Equations (24) and (25) in Doherty and Nickle's paper, Synchronous Machines III. The new formulas herein developed are, however, very much simpler in form, especially since in the case which Doherty and Nickle have treated, there is only one term in the summation; that is, n = 1, and a is merely the reciprocal of the short circuit time constant of the machine, expressed in radians. K. The Equivalent Circuit of Synchronous Machines Operating in Parallel at No Load, Neglecting the Effect of Armature Resistance Let, ba = angle of rotor a and bus Oo = angle of rotor a in space In general, the shaft torque of a machine depends on its acceleration and speed in space, and the magnitude and rate of change of the bus voltage as a vector. If all of the machines are operating at no load and if there is no armature resistance, a small displacement of any one machine will change the magnitude of the bus voltage only by a second order quantity; consequently for small displacements the magnitude of the bus voltage may be regarded as fixed, and only the angle of the bus and rotor need be considered. Furthermore, the electrical torque may be found in terms of (6) by employing an infinite bus formula. But Equation (49a) implies the alternative general operational form,

T

6 =e Idosin Xd Xdd'P

sin26 +e2(Xd-2 Xq) Xd Xq

+ ,,

anq p

, e2 cos

~~~~~~~~XqXq'p+al

sin

Therefore in the case under consideration there is for machine a, T = eIa (Xda - Xqa) IXda ±e Xda Xqa J6

adn 82 2 (cd )2 + s2

XqXq" ~~~ (a~~n)2

Xq -Xq"-

+

t

Transactions A. I. E. E.

+

Xqa

-

eqa

+Xqa where: e

P

Ea

2

2

nqa

p +

anqa .6a

(55)

per-unit bus voltage per-unit excitation of machine a, etc. This equation can be represented by Fig. 8, in which the charge through the circuit represents (da) and the -

Ia =

Rla Co

i

R2a C

Rna

-a-Cn

C2Cn

FIG. 8 voltage across the circuit represents the electrical torque of the machine (Ta). The capacitances and resistances must be chosen so that Xda Xqa Coa e Ia Xqa + e2 (Xda Xqa)

Cna

Xqa Xqa" e2anQa (Xqa-Xqa")

1 Cna anqa The equation for the mechanical torque is Tsa Ta ± MaPSa where: Ma = inertia factor of machine a in radians

na

(57)

2_X_stored_mech._energy_at_normal_spee base power

-2 r f

0.462 W R2 (rev, per mm. 1000 / base kw.

per-unit speed of machine a

5a t

time in seconds

But, 5a = p Ga

~ ~ ~ ~ Thus there is

d = dt J

Ta+±Map2

Tsa = (57a) Ga which corresponds to the equivalent circuit of Fig. 9, which change - Ga ~~~~~~~in The machine opeoratingr on axn infinite bus can be

PARK: SYNCHRONOUS MACHINES

July 1929

725

represented by the equivalent circuit of Fig. 10, since in the inductive branch of the circuit. Thus a governor which acts through a single time constant may be the condition represented by the circuit of Fig. 14, where Oa = fa = 0 is fulfilled. Rna CRa Cob Several machines in parallel on the same bus may be

La

La

Tsa

Tsb

Lb FIG. 11

Ta

Tsa FIG. 9 C

represented by the diagram of Fig. 11, since the conditions = . . . (= bus angle in space) Oa -ba = Ob Ta + Tb + TC, etc. = bus power output = 0 A transmission line may be represented by a, condenser. Thus two machines connected by a line of reactance (x) would be represented by the circuit of Fig. 12, where

C=

FIG. 12

L

x

(58)

e2

R

Shaft torques are, of course, represented by voltages.

Tsa

Coa

Sa POa

FIG. 13

La

Cia

Ta Cna

I

FIG. 10

Mechanical damping, such as that due to a fan on a motor shaft or that due to the prime mover, is represented by resistance in series with the inductance (L) as in Fig. 13. (R) must be chosen equal to the rate of decrease in available driving torque with increase in speed. Governors and other prime mover characteristics may also be represented by connecting their circuits

FIG. 14

1L R=regulation Cg =

time constant of governor in elec. radians R(9 Rg(9

PARK: SYNCHRONOUS MACHINES

726

An induction motor is represented by the simple circuit of Fig. 15 and is precisely the circuit of a synchronous machine with only one time constant and C = on account of I = 0. Results similar to these have been previously shown by Arnold, Nickle,10 and others, but simpler and more approximate circuits were used, the branches of the several circuits were not directly evaluated in terms of machine constants, and the derivation was incomplete in that the limitation to no load and zero resistance was not appreciated. L. Torque Angle Relations of a Synchronous Machine Connected to an Infinite Bus, for Small Angular Deviations from an Average Operating Angle There is, in general, T = To + A T = (4do + A'd) (iqO + A iq) -

(ido + A id) ('q00 + t'q)

For small angular deviations, A T = iqO A 'ld + PdO A iq - idO A 41q - 'qO A id ={dO + idO Xq (p) A ii- {{qO + iqo Xd(p)} A id

A iq = Zd (p) (-A eq +

Transactions A. I. E. E.

4'dO p 6)-Xd (p) (-

O0 P A 6)

where,

D (p) = Zd (p) Zq (P) + Xd (p) Xq (p) but from Equations (28), edo + A ed = e sin (6o + A 6) eqO +.,A eq = e cos (o + A 5) A ed = e cos 6o A 6 A eq =- e sin 6o A 6 (62) A id = -(e cos 6 + j'qp)Zq(p)±(e sino6 + IdOp)Xq(p)

Do

.A6

iq (e sin 6o -+4dO p) Zd (p) +. (e cos 6o +1-0o P) Xd (P) D (p)

*A6

(60) d

[4ldO+idO Xq (p)] ¢ AT= +

L o

cC- I

ed -

D ()

[4qO+iqO Xd

{ (P)]

gR

(e sin 6 +

ltdO

± (e cos 6o +

P)

Zd

(63)

(P)

VIqo P)

Xd

}

(P)

(64)

(e cos 6o + 4'qO p) Zq (p) -

(e sin 6o + 4'dO p) Xq (p)J

D(p) say,

A T = f (p) . -A

From (57a) the equation for shaft torque becomes A T, = (M p2+ f (p)).6

Thus,

A68

= M p2 + (p) .A T, f

(65)

Appendix Formula for Linkages and Voltage in Field Circuit with no Additional Rotor Circuits In this case the per-unit field linkages will depend ed0+A\ ed=p A 'Pd-r(ido+A id>- ('Pq9+A 'q)(l+P A 6) linearly on the armature and field currents. That is, d + A d) (l+P A 6) in general, eqo +,A eq =p A 'Pq-r(iqo+,A iq)(q) +dO± A ed= P A d -r A id- 'qO p A 6 A-'q =aIbid a ,A eq = p A V1P- r A iq + 'PdO P A 6 + A 'Pd Then if normal linkages are defined as those existing from which there is at no load there must be a = 1.0. Zd (p) A cid- Xq (p) A iq =- A ed - 'qo p A 6 The quantity b may be found by suddenly impressing Zq (p) A iq + Xd (p) A id =- A eq + 'PdO p A 6 terminal linkages 'Pd with no initial currents in the machines and E = 0. A ~~~~~~ci = ~~~~~~~~By definition there is, initially Zq (p)(- A\ed-q 'poo /6) + Xq (p)(- A eq + 'PdOP A 6) 'Pd D (p) id = FIG. 15

(61)

Xd

727

PARK: SYNCHRONOUS MACHINES

July 1929

but also there must be from the definition of Xd2

12. Dreyfus, L., "Ausgleichvorgange in der Symmetrischen

Mehrphasenmaschine," Elektrotech. u. Maschinenbau, 30 S 25,

121,139,1912.

I- d Xd

13. Dreyfus, L., "Freie Magnetische Energie zwischen Verketteten Mehrphasensystemen," Elektrotech u. Maschinenbau,

hence there must be an initial induced field current of 29 S. 891, 1911.

14. Dreyfus, L., "Einfuhrung in die Theorie der Selbsterregten Sehwingungen Synchroner Maschinen," Elektrotech. u. S. 323, 345, 1911. Maschinenbau, 29 ________

amount

Xd

I= ld

Xd'

But, initially the field linkages are zero, thus =

-F Xd

L

b

Xd I + X' Xd

0

b = Xd - Xd'

hence

Diseussion

H. C. Specht: I should think Mr. Park's theory could be applied just as well to the so-called synchronous induction motor, that is an induction motor in which the rotor teeth between the poles are cut out for a distance of about one-third of the pole pitch. Such a motor runs at synchronous speed.

Similarly, there will be

However, the pull-out torque is much less than that of an induc-

per-unit field voltage = c p I + dI Normal field voltage will be here defined as those

C. MacMillan: There was one statement in the first page of Mr. Park's paper to the effect that "Idealization is resorted

E

existing

=

tion motor with the full number of teeth.

to, to the extent that saturation and hysteresis in every magnetic

circuit and eddy currents in the armature iron are neglected. This requires requires normal voltage. This And with regard to Fig. 5, Mr. Park remarked that it represented

no load load and and normal voltage. existing at at no

that d = 1. The quantity c may then be recognized as the time constant of the field in radians when the armature is open circuited, since with the field shorted under these conditions there is (To p + 1) I O0 = 0 cpT ± I

.

.

a rigorous solution. Perhaps Mr. Park could give us a little more insight into the effect of taking into account saturation, and give

other cases in which certain elements have been neglected with more or less effect upon the final results. W. J. Lyon: In a paper of this description, certain premises should be chosen and, with these always in mind, the mathe-

matical development should be rigorous. The paper may then criticized either because of insufficient premises or because be = of incorrect mathematical development. I believe that the c = To= time constant of field with armature former is the kinder method; it is the one I shall employ. The premises that Mr. Park chooses are that the field and open circuited. armature windings are symmetrical, that saturation and hysteresis are neglected and that the armature windings are in effect Bibliography 1. Doherty, R. E. and Nickle, C. A., Synchronous Machines sinusoidally distributed. I take this last to mean that the airgap flux due to the armature currents is sinusoidally distributed, V, A. I. E. E. Quarterly TRANS., Vol. 48, No. 2, April, 1929. 2. Park, R. H. and Robertson, B. L., The Reactance of for if the armature windings themselves were sinusoidally distribSynchronous Machines, A. I. E. E. Quarterly TRANS., Vol. 47, uted, there would be produced space harmonics in the air-gap flux distribution due to the saliency of the poles, which, as we all No. 2, April, 1928, p. 514. 3. Park, R. H., "Definition of an Ideal Synchronous know, would complicate the problem tremendously. In order Machine and Formula for the Armature Flux Linkages," General that the mathematical method used by Mr. Park shall be rigorous, I believe it is necessary to make one further assumption. Elec. Rev., JuIne, 1928, Vol. 31, pp. 332-334. 4. Alger, P. L., The Calculation of the Reactance of I think I can best explain this by asking you to consider the Synchronous Machines, A. I. E. E. Quarterly TRANS., Vol. 47, result of supplying the field winding with a sinusoidal current while the armature rotates at some speed which may be called No. 2, April, 1928, p. 493. 5. Doherty, R. E. and Nickle, C. A., Synchronous synchronous. Under these conditions, there will first be proMachines IV, A. I. E. E. Quarterly TRANS., Vol. 47, No. 2, duced in the armature windings two sets of balanced currents each of which will produce 3 component flux distributions in the April, 1928, p. 457, Discussion p. 487. 6. Wieseman, R. W., Graphical Determination of Magnetic gap. The first of these is what would be produced if the airFields; Practical Application to Salient-Pole Synchronous Machine gap were uniform, and is proportional to 1/2 (Xd + Xq - Xa), where Xa equals the armature leakage reactance. The second Design, A. I. E. E. TRANS., Vol. XLVI, 1927, p. 141. 7. Doherty, R. E. and Nickle, C. A., Synchronous of these components is proportional to 1/2 (Xd - xq). The Machines III, Torque-Angle Characteristics Under Transient third component is of the same size as the second. Using the values that Mr. Park gives under Section H of his paper, the Conditions, A. I. E. E. TRANS., Vol. XLVI, 1927, p. 1. 8. Bekku, S., "Sudden Short Circuit of Alternator," relative magnitudes of these components would be (0.8 - xa) Researches of the Electrotechnical Laboratory No. 203, June, and 0.2. The first and second components react on the field, and produce in it a current of the impressed field frequency. 1927. 9. Doherty, R. E. and Nickle, C. A., Synchronous These are the components that Mr. Park has recognized, but the Machines I and II, An Extension of Blondel's Two-Reaction third component produces an entirely different frequency in the field, which will then be reflected into the armature and the Theory, A. I. E. E. TRANS., VOl. XLV, PP 912-47. 10. Nickle, C. A., Oscillographic Solution of Electro- process will be repeated. That is, in this respect, it is similar mechanical Systems, A. I. E. E. TRANS., VOl. XLIV, 1925, PP. to the condition that exists in a single-phase alternator. As far as I am aware, the Heaviside operational method cannot be used 844-856. 11. Dreyfus, L., "Ausgleichvorgange Beim Plotzlichen to obtain a rigorous solution for the single-phase alternator. Kurzschluss von Synchron Generatoren," Archiv f. Electrotech., In spite of this, I think the objection that I have raised is of no more importance than the effect of neglecting saturation or 5 S103, 1916.