NAPS, University of Waterloo, Canada, October 23-24, 2000

Tw()._J.{eaction .Theory of ... ~ynchronous ·Machine~ ··: ·w:i·i;·:;·: ·Generalized Method of'Analysis--Part I

:.'!

BY R. H. PARK* ; :·. ~:··.:.;, ,::;Xr\;\,;;; \:· . . . ;, , Assodt>l,e, A. I. E. E. , ,

· Sgnopsi&.-Sta.rting with the basic a.3aumption of no 4aturatio•1 In addition, new and more. accurate equiua!~nt circuits are or hyat4re4i&, 'and ·With diatribu.Lion of armature phase m. m: ,·. deu~Loped for. synchronous and asynchronou4 machines operating

· effecliueltt'linv:iOid'al a.3 'jar as regards pheno~Mn.a depmdent upon in paraLlel, and the domain of validity of auch circuita ia established. ; o' rotor'_poiiti~n;··qeneral formula.3 are developed for current, vo!Lag•l, Throughout, the treatment haa bem generalized to include salient

: power,':'and torqus under steady a1id trllmient load condition~. poles and lln arbitrary number of rotor circui~. The anllly-, Special·· d;tailed 'formulas llre al4o dll11eioped ·which permit the sis ia tht111 .lldllpted to machines equipped with field pole collllrs,

· determination of current and torque on three-phase short circu•:t, or with amorti.saeur windings of llnJI arbitrary coMtruction. during starting,• and whm only smaU dsvilltiom from an avera7e It is proposed to continue the analysis in a. subsequent paper •

. ' opsrati~ angl~ ~reinuolved. ; • *I • • • .~' .. , .. ;~;·;·.-;; .. ::'.~:~~r~e;:·.~ 1 ;·;.H1:··f. 1·._·; l·r . ~:.: · ~.i~: ·., . . ~·"·"·,:}'\~~~-~~.·~J:F~~·~~~~~~:· ~:;'··~ r ·~··

~ .. , r · ..

T~~~£;E~~~~~n~~~:.~ ~~~ eg:~:~ ' :;::and' ·Nickle/ and establishes·:~new and general methods' of ,_calculating current . power and torque in salient'': andi','no.ri-Sa.lient pole s)rnch.rorio'!Ul machines, under both transient and· steady load conditions.

Attention' is restricted to symmetrical three-phaset machi:n.es ... !Vith field structure symmetrical about the axes of· the field winding and interpolar space, but salient poles and an arbitrary number of rot<:>rt circuits is considered.

Idealization is resorted to, to the extent that satu.ra­tion a:nd.hys~~resis ineVery magnetic circuit and eddy

i.; ib,. i. = per unit instantaneous phase currents e., eb, e. = per unit instantaneo'\Ul phase voltages 'f.o'·/tb, if; • ... per unit instantaneous pha.Se linkages

t == time in electrical radians

Then there is ·, e~:> .. p .P •. - r i,.

e • .. p .p,- ri• · e. • p .p;- r i.

It has been sqown previously;.that . 1{1. = I.d.cos 8- 11 sin 8 .

(1)

' ·: ; ·· Axis of Phase a Direct AXis _ _,.._~ • 2:e. (. . '] · Xd + :Z:q · [ ·• · . i• +

2 i, ]

- -T 1" + ~. + ~. - · a . ~· ~ irectlon of Rotation x,- Xq • • .

-: ·· .. : 3. . . [~ .. cos 2 8 + ~b cos (2 8,- 120) !t :.', ! I'• I • •

' ,· ''1:';..:· . + i. cos (2 8 + 1~0)] -.p. ;..: I4'cos (8- 120)- I, sin (8- 120) --- '

-~ . ,:

Fxo. 1

currents: in .. the. armature iron are neglected, an.d in .. 'tbe··~ption that, as far as concerns effects depend• >.'.

. ing.·on .the posiljplJ.'of the rotor, each armature winding may·be :regarded as, in effect, sin'!Uloidally distributed:3

· . ·.A •. Fu~amental,.Circu.it Equation&· ' · · . Consider the :ideal synchronous· machine of Fig. 1, and let . : · · . · . . · : · · ~ -

' •General. Engg. Dept., Oonero.l Electric Compa!}y, f?ehenec-

ta.dy, N.Y. · ·· ·· . tSinrle-phas~·. I»Behines mo.y be regarded u.S .. tbree-pho.se

mo.ohines with one phaSe open eirouited. iS~tor for e. machine with sto.tione.ey field structure.

· 3For numbered references sec l3ibliogra.pb.y. . ., Prtaenud aflh4 Winter Convsntion of the A. I. E. E., New York,

io . .f. i& + i. Xd + Xq [ • -.::Co. 3 - 3 1b-

. .

,;. + i .. ] 2 ..

[i .. cos (2 8 - 120) + i. cos (2 8 + 120)

+ i. cos 2 8] (2)

:z;,,- Xq

3 [i. cos (2 8 + 120) + ib cos 2 (J

+ ·i. cos (2 8- 120)) N. Y.,'Jan: !B-Feb: 1, 19S9. · ·

• .' ••• • • 0

.where, ·.:·,,

I I I I I I I I I I I I I I I I I I I I I I I I

.Jul.v 1!)2!) PARK: SYNCHRONOUS MACHINES 71i

l tt = per-unit excitation in direct axis I, = per-umt excitation in quadrature axis x. = direct synchronous reactance x,, = quadrature synchronous reactance zo - zero phase-sequence reactance

As shown in the Appendix, if normal linkages in the field circuit are defined as those obtaining at no load* there is in the case of no rotor circuits in the direct axis in addition to the field,

cfl = per-unit instantaneous field linkages - I - (X~- X/) id

where, I == per-unit instantaneous field current

2 i.~ • 3{i.,cos9+ibcos(8-120) +i,cos(9+120)l

<3) On the other . hand, if n additional rotor circuits

exist in the direct axis there is, <fl • I+ X,,d ft.,+ Xn• I~•

+ . .. . + X, •• I nd- (:td- z/) id

where, I ,d, I ~tt, . • etc., are the per-unit instantaneous cur­

rents in circuits 1, 2, etc., of the direct axis, xfld, x,~d. . . . etc., are per-unit mutual coefficients between the field and circuits 1, 2, etc., of the direct axis. ·

Similar relations exist for the linkages in each of the additional rotor circuits except x.- x.' is to be replac.ed by a term x... However, since all of these additional circuits are closed, it follows that there is an operational result

ld • I + I ld + I~.~ + . . . . + 1 "" '"' G (p) E + H (p) i4 (4)

where E is the per-unit value of the instantaneous fiHld voltage, and G (p) and H (p) are operators such that

G (o) = 1 G ( =) • 0 H (o) = 0 H ( =) • :cd- xi'

:r:/ • the sub transient reactance2 It will be convenient to write H (p) • x.,- x~ (p)

and to rewrite (4) in the form, I" = G (p) E + [x.,- :z:., (p)] id (~1,a)

If there are no additional rotor circuits, there is, as shown in Appendix I,

'lt = I - (:r:d - :r:/) i., E == Top 'lt +I

where To is the open circuit time constant of the fiold in radians.

There is then,

1

If there is one additional rotor circuit in the direct axis there is,

E-1 = I + Xtld I'"- (X"- X/) i" = T.,p

- r,., · 'ii,d = x,,d I,.+ xfld I- x .. , d i,,- -T-­Old p

which gives,

where,

[X"d- Xnnl Tnt<~ p + 1 G (p) = A (p)

To To,,, [X lid (xd- X/)- x!ld x .. ,d] p1

+ {(xd- x' n) T.,,, + x .. ,d ToJ p

A (p)

A (p) •[XurXlta2] To To,n P2+(X"a To+Tot.tl p+1

If there is more than one additional rotor circuit the operators G (p) and x.~ (p) will be more complicated but may be found in the same way. The effects of. external field resistance may be found by changing the tenn I in the field voltage equation to R I. Open circuited field corresponds to R e-qual to infinity.

Similarly, there will be I.-rxq-x.(p)Ji. (5)

where,

2 i. -- slia sin 8+ib sin(&-120) +i. sin(& +120}1 (3a)

x. (o) • x" x. ( =) = x,' So far, 10 equations have been established relating

the 15 quantities e .. , ~b, e., i., i~, i., l{t., •/lb, l{t, i.,, i 9,

I.,, 19, E, 8 in a general way. It follows that when any five of the quantities are known the remaining 10 may be determined. Their. detennination · is very much facilitated, however, by the introduction of certain auxiliary quantities e~, 6 9, eo, io, ift.,, l{t,, ifto.

Thus, let

1 io = S {ia + i~ + i.l (3b)

2 ' . . ed- 3 {e. cos e + 6~ cos (8- 120) +e. cos (8 + 120)}

2 Bq-- 31e .. sin 9+e& sin(9-120)+e.sin(9+120)l (~)

1 eo ... 3 I e. + e~ + e.l

. ··~.' :: ~-: ·:::·:.

·':::i~ G (p) = Top + 1

. X/ ToP+ Xd 2 . ·.. . {f~f-"'d- 31 ~.cos 8 + "'• cos ( 8- 120) + "'·cos (8 +120}};:

:z;~ (p) = Top + 1

*This definition is somewhat different from tba.t given in reference 2. ·

l{t, =- : I .Y .. sin 9+•h sin(8-120)+.Y.Sin(8+120_)L .(;.)

P- 82

... :·~3';: .. :·.:, ~;.: .. -~­... ,~~·:

.. ·~.'

•' ... . ~:-:·;(~

718 PARK: SYNCHRONOUS MACHINES Trt~n:<n.nl.ion~ A. T. ~;. E.

. 1 to= 3 I t. +..;~+..;,I

then from Equation (1) there is

2 e~~·s{cos 8 p ..;.+cos(0-120) p ..;~+cos(8+120) p t.l

-rid

e1=-! {sin 8 p ..P. +sin (8- 120) p .Y•

+ sin(8 + 120) p t.l ·- ri, ·.:., • p t/lo - r io

· .. but,

p ..;.~ = ·t {cos 8 p i/1. +cos (8- 120) p ,y~ +cos (8 + 120) p ..;.J

·-~·{sin 8 ..;.+sin(8-120).p ..P•+sin(9+120) p ~.l p 8

· ... et~ + r i~ + 1/1, p 8

.·::· •.;. ,,.... ,. '2 . ' . . . '· . ' . . ;' p .. t,,• - a {sin·8'p "'·+.sin (8 -120) p ..;, ... . . '· ...•.. ·:~ . .. .. .. .... .

+sin (8 + 120) p ,Y,}

. 2 . -;-3ieos e.;.+ cos (8- 120) "'~+cos(~+ 120) 1/t,}p 8

..;, e, + ti,- t~ p 6 · · hertce there is ·

e4 ..; p t/1~- ~i~·= t~ p 8

e, - p·l(, ~ rh+ "" p 6 . . fo· • .p t/Jo.~ :r.~i~···. ·.,

· ., · .A1sO it ~Y be re&dily~v~aedthat: .'

. (8)

(9)

(10)

4• • 1., - :r:1 i., • G (p}.IJ:";;...i" (p) ~." (11)

.y, • I,- :r:, i, -·- z,•. (p) i,. (12) · · to • - :1:t io .. . .. ·: (13)

..•. .-.E:q'uatioll$ (8) to (13) establiSh six 1-elatively simple ': rilations between the 11 quantities e.,, e,, e.,. i ~. i., io,

.. • :i!.";::li., ·to. E, 8. In practise it is usually p0$sible to . :.;.. :det.ertiline five of these quantities directly from tbe termi­,\~,h~~:ll.ditions~ ·a.tter wmeb.'ttt·e:remaining"six·may be

.~·.:· · 'calCUlated with relative simplicity, After th.H direct, ; · i:. :,. ;q~adrature,, and .zero qll8ntities are~ known·. th.e phase 6E;~:~~!ci~ties n:iay be determined· from the identical ·.'.'::: .. :,:'relations . ·.·· . .. ·. .. ..

. :i•: - i., cos.8- i, sine+ io i~ .,;, i4 cos (8- 12())- i,.sin (6·- 120) + 1o (14)

. · · . .. ·i_· ··i~·cos (8 + 120)- i,sin·(8 + 120) + r:o

¥• - '/14 eos.8- .Y, sin e + .i/to f• • t/i4 C{)S (9- 120)- t/i, sin (8- 120) +to (15)

.•. · '{1. • Y,4 cos (9+ 120)- .;,sin (6 + 120) + t/io

,., • '"cos: 8 - e, sin 6 + '•

eb - ed cos ce- 120)- e~ sin (8- 120) + eo (16)

e. = erl cos (6 + 120)- e~ sin (8 + 120) + eo Referring to Fig. 2, it may be seen that when there

are no zero quantities. that is, when eo ,.. ..Po = io = 0, the phase quantities may be regarded as the projection or vectors e, J, ~nd ron axes lagging the direct axis by

Axis of Phase b

Axis of Phase c Fto. 2

Direet Axi$ ,/

" "

Axi5 of Phase a

angles e, 8- 120 and 8 + 120, where taking the direct axis as the axis of rea.ls,

e - 6d + i e,

~- 1/ltl + i "'' r - id + i 1:, If we introduce in addition the vector quantity,

I-I4+j1, the circuit equations previously obtained may be

./'

"'Direct Axi$ -----------~~----_,-...x; drop

Fta. 3

transferred into the corresponding vector forms, -e = P ;p- ri + [p eJi ~ ~-1-i?.

where, m ... :!:4 i4 + i x, i, Fig. 3 shows these relations graphically.

B. Armature PO'IDer Output The per-unit instantaneous power output from the

armature is neeeswily proportional to . the sum

P- 83

.J ul ~· l!l:.>fl PARK: SY:t-TCHRONOUS MACHINES

e. i, + e, i~ + e, i,. By eonl'>ideration of any instant during normal operation at unity power factor it may be seen that the factor of proportionality must bE! 2/3. Thnt is,

P = per-unit instantaneous power output == 2/3 I e. i. + e. i& + e, i, l

~ubstituting from Equations (14) and (16) there results the useful relation,

P = ed i11 + e9 iq + e0 io (17)

D. Constant Rotor Speed Suppose that the constant slip of the rotor is s. Then there is,

but,

Putting

ed = p 1/td- r i 4 - (1- s) .;,,

e, = p ..p,- r i, + (1- s) Y, 4

ifl• = G (p) E- :z:., (p) it~

"'' = - x, (p) i, p Ztt (p) + r ,. z., (p)

. C. Electrical Torque on Rotor there is p Xq (p) + r - z, (p)

It is possible to detennine the electrical torque on the rotor directly from the general relation, e4 .. P G (p) E- z4 (p) id + (1- s) x, (p) i, !Total power output I = e, = (1- s) [G (p) E- Zd (p) idl- Zq (p) i,

I mechanical power transferred across gap I Solving gives, .

i19

(20)

(21)

+ 1 mte of decrease of totn.l stored magnetic em1rgy l id • I (p Zq (p) + (1- 8)2 x, (p)J G (p) E - z, (p) e" - 1 total ohmic losses l (18) - (1 - s) :r, (p) e, l + D (p) (22)

However, since this torque depends uniquely only . (1 - s) r G (p) E- Z11 (p) e, + (1- s) zd (p) ed on the magnitudes of the currents in every circuit of

1' = D (p)

the machine, it follows that a general formula for torque (23) may be derived by considering any special case in which · arbitrnry conditions are imposed 88 to the way in v~hich · where, D (p) • zd (p) z, (p) + (1- s)' xd (p) z, (p) these currents are changing as the rotor moves. E. Two Machines Connects<~. Together

The simplest conditions to impose are that I,, I,, Suppose that two machines which we will designate i •. i~, and io remain constant as the rotor moves. In respectively by the subscripts g and h, are connected this case there will be no change in the stored magnetic together, but not to any other machines or circuits, energy of the machine as the rotor moves, and the and assume in addition that there are no zero quantities. power output of the rotor will be just equal in magni- In this case the voltages of each machine will be equal tude 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 gapl - {annature losses!

2r or, P = T p 8 - 3 I i.2 + i•' + i.= I

.. T p 8- r I id2 + i,2 + io2 I Then, T = per-unit instantaneous electrical torque

ed i~ + e, i, + eo io + r Iii + i,2 + io=l = p8

but subject to the conditions imposed, ed = - Y,, p 8- r i11

e, = ifl« p 8- r i, eo "" - rio

It therefore follows that,

T "" i. if!d- i4 .;,, (19) ... vector product of "f and {

= f X ; (19a)

a res~t which c?uld have been established directly by phys1cal . re~rung. Formula (19) is employed by Dreyfus m h1s treatment of self-excited oseillatio,ns of synchronous machines. u

Axis Phase b \

Axis Phase c

Fxa. 4

Direct Axis of a Machine

Direct Axis of h·Machine

Axis Phase a

phase for phase, and it therefore follows that the voltage vectors of each machine must coincide, as shoWn iD: 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,

6ld - e,d cos 0 - e,. sin a . ehq .. ,, .. sin 0 + e,, cos a (24)

6qd == eA<I COS 0 + 8h sin a e,. = - el., sino + e,, coso (25)

P- 84

il9

720 PARK: SYNCHRONOUS MACHINES Tra.n•aetionK A. I. F.. r~.

On the other band, for currents there will be i •• =- {i,.coso- iusinol i~, = - {i,. sin o + i,. coso l (26)

i,. • - { i~., cos o + ih, sin o I iu-- {- i ... sin 0 + i., coso} (27)

F. One Machine on an Infinite BWJ In (E), if machine k has zero impedance, it follows

from (20) and (21) that e • ., • 0, e., = bus voltage say • e.

Then for machine g there is,

'"-,sin a e;,- e coso

G~ .Torqut~ Angle Relations­From Equations (11), (12), and (19), there is,

(28)

X/ Top+ X0

+ Top + 1 x1

X/ x, To p3

+ (:r:/ r To + (x. + r To) x,J p2

+ [r (:tc~ + x, + r To) + z,/ x, To] p

+ r2 + x4 x~ Top+ 1

d (p)

==Top+ 1

By the expansion theorem there is, finally, . x,E \; = r2 + x. x,

3

(31)

T . 1,1/1• 1~1/1, x.,-x, ------ "'""'" · · :t 4 :1!4 X<~ X,

~ (To o:. + 1) ((x, a. + r) e11o + x, e,o) E-"'•1

+ ~ a. d' (a.) Then if the rotor leads the veetor if.. by an Sll.gle a

· there is

"'· -->/!sino .. . . .. . "'• .- ,Ycoso

1 3

rE ~ . i, • r* + :. x, + ~

1

':··i·\ .·1~~~·~: · '· . • I~ Vi sino:· x" - x, ·2":··:-;-.,~·o+ x. _+ .zx,x 1/l'sin2l>(29)

... . . · .... . _:::::· ··!~··.~ ...: .. ;.~~;_·.:~. ·, . . . '

(:c/ To a...'+ (:Cc~· +r To) a.+ r)e,o- (To a. :c'• + x.,) e•o a. d' (a,.)

(32) . 'A. derivation of'~this formula. for steady load eon-

.. ~·;:.~=:~;~~~;~~;;·~jj~~·;.~J~ ~Y _Do.~~· and

E-a,.t

where the summation is extended d

over the roots of

d (~) = .o and d'. (p) • d p d (p) · ... : ~ ll:.'.:;:. Tk1;.tf:P,h,46-:Short'CiieU.it:iaitli.:COt£Stant Rotor Speed

£I··-~?~ . , · .. values..ote~•a.ild«·•·befof.e .. iM•tihoit:cireuit; .The initial

;~<;~;··:::~.=~=~~~~~~~;i;~_::···.~ . : '~'obtain: the r:911ultant current:aftedbe.ihort eircui t. .

, ...•. ·~:With;;·· oa.nd.:.teoriatant:t:!ierei&iiidetail,

IJJ~i~;~~~~:{~;;~s~:~~:;:::".. . ... ~ -~---,.-"Ai;iii--.:.':.<>:·: ... _; ; .· .... : .. ' .: . ,: .· {30). 0.5. 1.0 .. LS 2.0 .1.5 3.0 :.\' / ::·:~.: ~~km.i: ou~ otthe. formulas may:-~.illw1:trated ·· . · v- o1'

rl•~~~~~L~~~:;.:~;:l;;;~~;: ·~ .. ·:. · · .. ··.··.·.:··;_.·.::·· ... ~ ... vt · . -·.··.ra.p.·:+-..1 -f~undto<be.~;~~wnm·~gs~·5r6, .. ~d7,:where

''·'' .. •"•i:.O' . .. •·. • ·• ••• , • • •• • ,. •• . · ••.

: .:.~;;•;(~·~:;rt~;'~+r)<~,p +;). .· . It. will be nMI ~ .. ~sd ..,.....;Iy be the

The phase currents may, of course, be found from

·0.0018

·0.0010

P- 85

July 1029 PARK: SYNCERONOUS· MACHINES

721 I •

case, where r • 0, a, is equal to the reciprocal of the 11"

short circuit time constant of the .machine, i. e., for o = 2- s t, ~nd refemng to Equation (28), r "" 0,

X<1 1 a1 - - -, -T = - 0.001667 x. 0

while for r = tD

1 . a, = - T; = - 0.000500

·8

·7

·5

·5

•4 Values of a;,

·3

·2

·I

o.s 1.0 1.5 2.0 . 2.5 3.0 Values Of,

Fto. 6

The root a. is found to be almost exactly equal to the value which it would have were To == =,i.e.,

r (x./ + x.) a. == 2 x,/ x, approxima~ly .

; 1.0 ~ 0.8 ';.o.6 .~ 0.4

iQ.2 - o H-!--H++-+++-1-+-+-i~-:t--

02 '3 0.4 0.6

o.a 11.0 "' 1.2 !u

1.6 1.8 2.0 2.2 2.4

V•tuesolr

FIG. 1

Thus, in the special case considered this approximatH formula gives

(0.30 + 0.60) r a. = 2 X 0.30 X 0.60 • 2•50 r

which checks the result found by the exact solution of the cubic. · I. Starting Torque · .

On infinite bwi and with slip s, there will be, choosing·

p- 86

e~ = cos s 1

e9 = sin 8 t If we now introduce a system of vectors rot:.:ting

at s per-unit angular velocity there is . · ed == 1.0 ' :\ ·. · ·

e, = - J' . p == J' s (33)

Then from ·(22) and (23), ..

ia = I f 8 x, (;' s) + r- J' (1 - 8) :t9

(j 8) l ...... . I [is Xtt (J' s) + rJ [.is Xq (f s) + rl.

+ (1- s)Z zd (;' s) :t9 Us) ·1·

j (1 - 2 s)x. (} s) - r

""; r2+(1-2 8) x,/(f s) x,'(f s)+j 8 r(xd'(f s)+xq'(i 8)].

I "I

- : ( j z, ~ s) ~ 1:2' l + I z, (is) z, (j s) . .

+ 1 : 2 8 lr +i s (%, (i s) +. ~. (1' s) ) ) } · • (34) I .• ·'

. . .• :j ..... . . [j s :t" (i 8) + r] (- j) - (1' ...:; 8) x~ (J' 8) · ~o==-r2 +(1-2 8) Xc1 (J' 8)X9 (J''8)+j s r[xtl (;' 8)+x

0(js)]

. - ~ • ' (j s) + 1: '2'} .+ I "' ~ •):• (j •) i 1 _' 2 , [r + j ~ (z, (i s) :H,.(i s)) ]} . (35)

. .".

The expressions for average power and torque then become, ·

P"" • 1/2 (ec~ • id -+: e9 : i~J To• == 1/2 (i0 • Yttl...:_ ic~', lfq]

where the dot indicates the scalar product, or, P 0 ., :_ 1/2 '[1 • id - i • ioJ .

.. ·1/2 [Real of ic~ .,.. Imaginary of i ~] (36) There is in general,

ed + r ~ .. = p f.~- (1- s) ~' , e9 + t ~ 9 • (1- 8) f~ + p Y,,

· ·j e.~ + r ~tl -. (1 ·- s) I e9 + .r ~, . p . :

. ""d = I p - (1- 8) I (1- s) p

(37)

p (e 0 + r i 0 ) - (1 - s) (ed + r.it~) lfq • pt + (1- s)' (38)

(33)

8)1.

6)

722 PARK: SYNCHRONOUS MACHINES Tra.nsa.otions A. I. E. E.

·'·~ = is (en + r i.~) + (1- s) (e. + r i.) '~" .. 1-2s

. j s (e.j • + r i 1) .- ·(1- s) (e~ + r i~) V'o == . :·.' . . 1- 2 s .

with e.t .. 1.0; ~~ .== -:- j i

·'· .. j s + j s 'r i~~ + (1 - s) (- j) + (1 - s) ri, 'I'~ . ' " 1- 2 8 I

,·,

-.. r .

= - .i + 1

_ 2 8 [.i 8 i~ + (1 - s) ,i,] (39)

.t y;, •. fs (~ i +r i.)- (l- 8) - r (1- s) i.· . . . . . 1-2 s ' .

,. ,;.. ... .. ~ . ; . : . . : . ·.·: - (1- 2 8) + ;. [is iq- (1 ~ s) i.] -· · · 1-28

r • - 1 ~ 1='2S [.is i,- (1 - 8)ic~] (40)

Thus, r .

i, ~:.<.- .i) + i, ·1-28 (i 8 i" + (1 ~ 8) i,) ··,, ·• I ~

T..,==1/2 .i:. · r · ~. -it~.,·<:- 1)- ia . 1 _ 2 8

(i 8 i,- (1- s) ic~) . :

' '; ::;••"r [(1-8)(i,'+ii) ]: . ... p,;.,+---

.. , .. :·.: .. , .. 2(1-2 8) + 2 . . . : ' ~- :~::~··'!l!: :~~~~.:.;·.•, I·: 81.! • J ttl. .

: !.·,;: ·i·:!' :::!r; i:· ' T8 ["•'+i.l J ' •I .. ~ ·p ._:+.:-:-··ci,2 "+ ii) + • . " •

. , .... · : : , .. ;':2 : . i · 2(1 ':""i,2 s) + 2 . . . . . ;::··:: .. : f :\':r1r.·· -: . · ;,: . "· .·] ·tc1

•. · .. •·:.:'·t'+t 2 ' r8 · ·

. :- P ~=+r· '. 2 •. + 2(1-2 8) (i, + .i i:~>2 (41)

Mr: Ralph Hammar, who has been engaged in the application of the general method of calculaticrn out­lined above, ·to the predetermination . of the starting torque of practical synchronous motors,· haa suggested an interesting .modification of formulas (36) and (41), based upon the fact that, since the total in •. m. f. con­sists of direCt and quadrature components pul.sa.ting at slip frequency, it may be resolved into tw~ components, one moving'forward at a per-unit speed 1 - 8 + ~· - 1.0, and the other moving backward at a per-unit speed 1 - 8 - 8 =· l ...:. 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 a.S con­cerns the for\vard component the quadrature current i, is equivalent to a d-e~ j i 1, while as regards backward component it is ertuivalent t6 a 4irect component

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

. 1 ... forward current = i 1 == 2 (ia + i i,)

1 backward current == i& • 2 (ia- .i i,)

,If we define by analogy,

forward voltage

. '

· backwa~d voltage = ; (e~- i eq)

There is,

i,=; { 1-_2;~ +i [~~(.is) :x,(js)]·} +

.-( •• (j •) "• (j s) + 1.:2 • (r + i' [•• 0' s)

+ x. (.is)]) }

(42)

(43)

i. == ~ { / [x, (.is) - Xc~ (j 8)] } + { x~ (j s) x, (.is) ·

.,_.. 1

:2 8

(r +is [zr~ (J" s) + x, 0"_8)] ) }

'e, - 1.0 (44) e& ... 0 (45)

P.,. = e1 • i1 -= real of i1 ( 46)

T To. • P •• + df' + 1=28 ib: (47)

J. Zero Armature Resistance, One Machine Connected to a.n I nji.nite 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 So. At the instant t ... 0, let o and E change.

There is,

. Eo·- t/tao 1 A G (p) A E ~d ·= ~~ - LXrt (p) "'d + Xci (p)

Y,,o 1 i =----At/t ' x, .. x~ (p) ' .. .P<~ =~e coso

y,,--esino

From which there is, by obvious re-arrangement,

P- 87

July 192!> PARK: SYNCHRONOUS MACHINES 723

E-ecoso x.,-x.(p) +e () (C'OSOo-COSO)

x. ' x. x. p

esino· x~-x,(p) i, = _ _;___ + e (sino- sin Oo)

X9 Xq X9 (p) (48)

Then,

E e sin o x.- x, T ... · + e2 sin 2 o

Xa 2 Xa X~

. · .X9 - Xq (p) , , + e2 coso ( ) (sm o- sm Oo)

XqXq p . (49)

' x. - x. (p) + e2 sino ( )' (cos oo- coso) XaXd p

But quantities aa ... a,,., a.,., ex,,., b,., /1',. may be found such that ·

xq"==x,(=)

x/ - Xa ( "") L: a ... == 1.0 L: a 9,. ,.. 1.0

L: b,. - 1.0

(50)

It therefore follows from the operational rule that, I

J (p) F (t) • F (o) q, (t) + f q, (t- u) F.' (u) d u (51)

where,

that if ¢ (t) ... f (p) . 1

0 = 0 (t) p o .... o' (t)

A E ==A E (t) p A E =A E' (t)

Equations ( 48) and ( 49) may be rewritten in the form,

E:.... e coso i.-----x.

x x"~ J1

· · I e d - d a c-"'dnl .: • > e"'d'IU sin.'o .. (~) 01 (u) d ••

T X X 11 ..;:;;;...J <In • . , "" d d . 0 / .

(48a)

e sino i, =- ---x, . ·, '·'

E e sin o . · e2 (xa - x~) • · • · T = + 2 sm2o

Xa . Xd X~

, ' • 1: . ', . . . '

+e x~~x:,' coso~ aq,. C01f"

1 f :e"'t"" ~os o(u) o' (1t) du.:

esino 1 '· ·

Xa . :::S b,. e-A,.I f 111" A E' (u)· d u (49a)

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

1 Xa - xi' ~ ex ... s T •• ""'2 XtJ xi' ~a ... cxa .. 2 + s'

Since ex~. . .

-;---+ 2 is never greater than 72, and , ex s

L: atJ .. = L: a,,. =·l.o · · it follows that T ... is never greater than

1 { x.- xl x, .:_ z/ · ) 1

-4 ~ + II Xu x. x, x9

(52)

(53)

Equation (53) thus provides a very simple criterion of the maximum possible start~ng torque of a syn.,. chronous motor of given dimensions, when· armature resistance is neglected. .

The · same formula may also be Used to obtain o. simple expression for the damping and synchronizing comp9nents of pulsating torque due to a given small angular pulsation of the rotor.

Thus if the angular pulsation is A o - [A 8} sin (s t)

and if the pulsation of torque is expressed in the form

P- 88

.· r II

724 ·PARK: SYNCHRONOUS lt.lACHINES Tra.~\iou A. I. E. E. ,:,·. , • •. • L., , ..... . ·. :· . '·'

i ·::.• .. :;:AT== T,·Ao + Td ddt Ao ..

there results,·

' · T, :~. T.· .J t e2 ,~in=·o~ x4- ~~~ ~ ·~· ·a;; 82 ~

.. .. · . , . , X a X a. 0:dn + S -: : ·: .. ;· ·: .. ;

+ ... r, ; !

2 r i Xq- :t/ ~· aqn $2 . '.

e cos vo . · . , . . . ' .... . .. x, :r:, : (a,,.)z + sz ·. · ..... : ' '.· .

. , . ;~ '.,i· ,·• ;.·. I .. '·I ''

(54)

oo ... average . ~ngular displacement, i. e., total

. x.- x/ """' a~, p + :r; x • ez cos o .LJ + . sin o • ' Jl p a,.9

Therefore in the case under consideration there is for machine a,

T [ ~ + 2 (Xao - x •• ) ] • .. = · e o . Xdo Xda :2: 90

4

(55)

where.: e . == per-unit bus voltage .. ;. I .. • per-unit ex~H:ation of machine a, etc ..

This ,equation can be represented by Fig. 8, in which the charge through the circuit :epresents (o.,) and the

.··--1.~----~ .. · C1a Cz. . ·c ..

F1o. 8

. angle. •.o ;=o.oo +·A o. · It can be shown that for the ca$e of no addition.al voltage across the circuit represents the electrical

rotor circuits, Equations (54) are exactly equivalent to t~rque of the .machine (T .. ) •. Equations (24) and (25) in Doherty ~nd Nickle's pap1!1', . ··The capacitances and resistances must be chosen Synchronous Machines III. The n$W formulas here:in so that 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 tlie machine, expresse~ in radians. :; t·<·.~ ·, ·. . · . '· . . ·

. K. : ·The' Equivalent Circuit of Sy'IZckronous M achi1~es · . Operating in ParalleL at No Loa.d, Neglecting r:he

1 R"" = _c __ _ ""a,., ..

: Effect of Armature Resistance · ·. ' · .. ,; ,: . '1 ··j, . • ·.•

Let, :: ,;:'::·.o .. • angle of rotor a ;i!.nd bus The equation for the mechanical torque is

T, .. -T.+M.ps .. · · ... ··;•,. >:;: 8o • angle of rotor a ip space ' · · ... · where:.

Iri general,' the shaft torque of a machine depeD.dS on its acceleration and speed in spaice, and the magJli­tude and rate of change of the bus itoltage a.S a vector. If all of the· machines are opera tint at no load and if there is no ·armature resistance, a small displaceme•:nt

M .. - inertia factor of machine a in radians 2 X stored mech. energy at normal speed - base power

0.462 w RZ cev. i:Oo min. r . = 2 1r' f _____ .,:_ ____ ...;_ base kw.

of ·any one machine will change the magnitude of the bus voltage only by a second ord~ quantity; con:se­quently for .small displacements the magnitude of the bus voltage may be regarded as hed, and only the s .. per-unit speed of machine a angle of the bus and rotor need be considered.. Further- "

·more, the .. electrical torque may 'he found in terms ' .... time in seconds ( p ... ·ddt. ) of (o) by employing an infinite ·bbs formula. :But Equation· (49a) implies the alternative general ope.ra-tional form, · · But, s. - P e.

(56)

(57)

· . :, Thus there is T,., = T .. + M .. p2 e.. (S7a)

which corresponds to the equivalent: circuit of Fig. 9, in which change = e"

L. = M. The machine operating on an infiiJ.te bus can be

P- 89

li i

PARK: SYNCHRONOUS MACHINES 725 July 1!12'J

represented)y the equivalent circuit of Fig. 10, since the condition

B~ = <>~ = o is fulfilled.

Several machines in parallel on the same bus may be

· La

Fxa. 9

represented by the diagram of Fig. 11, since the con­ditions '.

8 .. - o. ~ 8b- ob • . . . ( = bus angle in . space) T. + T6 + T.,·.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

. X c--­e2 (58)

Shaft torques are, of course, represented by voltages.

L.~ ·J:~ =LJCla

I

I I

Fxa. 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 wit~ increase in speed.

Governors and other prime mover characteristics may also be represented by connecting their circuits

P-90

in the b.i.ductive branch of the:;circuit ... Thus a gover­nor which acts through a single time constant may be· represented by the circuit of Fig. 14, where · ·· '·

. ·~

Fxa. 11 : l,.

'·

:; .:

c Fxa. 12 ~ . ' . ,.

" ·;I

,,

L : '

·' : :

R : ' I i i

·.; i

Tsa

Fxa. 13

Fxa. 14

1 ' R,·= 1 . regu atton

time constant of governor in elec. radians C, • R, (59)

........ --------------·--------====

J : '

(59

PARK: SYNCHRONOUS MACHINES Tr~DS&etious A. !. E .

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 Cc. = = on account of I == 0. .

Results similar to these have been previou~1ly shown by Arnold, .Nickle,t0 and others,. but simpler and more approximat·e circuits were used, the branch'as of the several circuits were not directly evaluated in terms of machine constants, and the derivation was in.complete in that the limitation to no load and zer.o resistance was not appreciated. · . · L. Torque Angle Relations of a· Synchronous Machine · ·Connected to an Infinite Bus, for Small Angular

. · Dev:iatirms./rom ~n Average Operating Anqle There is,':in' general, .. .l ' ....

T =io T 0 + c. T ... ("'dO + c. "'.) ( i qO + c. i q)

' . . - (ido + C. io) (,Y,c. + C. ,Y,) For small angular deviations;

c. T =. i ~0 l "'.· + "'dO c. i q - i dO c. "'q ;.... 1ft qO c. 1 d

== f f~:+\~~~x~ (p) l C. i.- { Vtqo + iqo x. CP) l t.. i• (60)

L

c I~

T

FIG. 15

e•o+C. ~~.;_,p C. f•-r(ioo+A i.)-(f,o+A fi)(l+p A o)

eqo+C. e,.;_,p A f,-r(i,o+A i,)+(f•o+A l/l•)(l+p A o)

c.·e. =: p A f•- r C. i•- tf.',o p A o- A vi, C. e, = p C. f,- r C. i, + tf.'oo P A o + A \'~"

from which there i~:~·

z• (p) A·i~~.- x, (p) A i, == - A ec1- f,o p C. o z, (p) A i, + X<1 (p) C. io = - C. e, + if;.oo p A o

'I . '

c. i~~. = .. z, (p)(- A e~~.- tf.o.o p C. o) + x, (p)(- C. e, + ~·•o p A o)

D (p).

(61)

p- 91

c. i, = z~(p)(- c.e.+.J;dopC.o)-xd(p)(- C.ed- .J;.op ~

D (p)

where,

D (p) = Za (p) Z1 (p) + Xa (p) X1

(p) but from Equations (28),

edo + c. e. = e sin!(oo + c. o) e.o + c. e. == e cos (oo + c. o)

c. ea == e cos .So c. o c. eq = - e sin So c. o

c. id == . i . I

- (e cos Oo + ..Y.o p) z. (p) + (e sin Oo + !/lao p) x. (p) · D (p)

c. i, = (e sin Oo + Vtdo p) Z.t (p) + (e cos Oo + 1/;qo p) x. (p)

D (p)

[tfi•o+i•oXq (p)] \ .. -- ( (e sin oo + 1/l•o p) z. (p) }

(f

. ·.. . · l +Ce cos So + f.o p) x. (p) AT= ((

·, . { (e cos Oo + .P~o p) z. (p) } +[.Pqo+i,oxd (p)J

· - (e sin Oo + 1/;do p) Xq (p) D (p).

say,

c. T = f (p) . c. o From (57a) the eq~ation for shaft torque become>

c. T. = (M p2 + f (p) ) . c. o Thus,

1 c. o = M p2 + f (p) . AT,

Appendix Formula for Linkages and Voltage in Field Circuit w

no Additional Rotor. Circuits In this case the per-unit field ·li.¥ages will depe

linearly on the armature and field cUITents. That in general,

'M' = a I- b i. Then if normal linkages are defined as those existi

ai: no load there must be a = 1.0. The quantity b may be found by suddenly impressi

terminal linkages tf-• with no initial currents in t

machines and E = 0. By definition there is, initially

. l/ld ~.~ =- x./

July l!J2!J PARK: SYNCHRONOUS ::V1ACHINES 727

but also there must be. from the definition of xi

I- 1/1,1 id = __ ..:...;..:_

hence there must be an initial induced field current of amount

I= 1/1• ( 1- ). But, initially the field linkages are zero, thus

'I' == Vtu 1 - [4 + ~] = 0 · Xu X,;

hence · r~ =~ ~·-.-:- x> 1

Similarly, there will be · · · E = per-unit field voltage

=cp'I'+di Normal field voltage will be here defined as those

existing at no load and normal voltage. This requires 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 ·

(Top+ 1) I= 0 cp'I'+I ;=0

o/ =I c = To = time ·constant of field with armature

open circuited.

Bibliography 1. Doherty, R. E. a.nd Nickle, C . .A.., Synchronou.1 Machine11

Y, .A.. I. E. E. Quarterly TnA.Ns., Vol. 48, No.2, April, 1929. 2. Pa.rk, R. H. a.nd Robertson, B. L., Tha Raactane~ of

Synchronou.11 Machina1, .A.. I. E. E. Qun.rterly TRANS., Vol. 47, No. 2, April, 1928, p. Sl4.

3. Park, R. H., "Definition ol an Idea.! Synchronous Machine o.nd Formulo. for the Arm~~oture Flux Linkages," General Elac. R6ll., June, 1928, Vol. 31, pp. 332-334.

4. · Alger, P. L., Tha· Calculation of the Reactance of Synchrono"IU Machinu, A. I. ·E. E. Quarterly TRANS., Vol. 47, No.2, April, 1928, p. 493. .

5. Doherty, R: E. and Niclcle, C. A., Synchronou.~ Machine' IV, A. I. E. E. Quo.rterly TnANS., Vol. 47, No. 2, April, 1!>28, p. 457, Discussion p. 487.

6. Wiesemn.n, R. W., Graphical Determination of Magnetic fi'ield4; Practical Application to Sali~nt-Pole Synchronous Machine Delion, A • .r. E. E. TRANS., Vol. xr.vr. 1!>27, p. 141.

7. Doherty, R. E. o.nd Nickle, C. A., Synchronous Machines III, Torqu.,..Angle CharacteristiC~ Under Tranlient Conditiom, A. I. E. E. TuANS., Vol. XLVI, 1927, p.l.

8. BekkU, S., "Sudden Short Circuit of Alterno.tor," Resea.rches of the Electrotechnica.l La.bora.tory No. 203, June, 1927.

9. Doherty, R. E. a.nd Nickle, C. A., Synchronou.1 Machines I and II, An E:z:lension of Blondel's Two-Reaction Theory, A. I. E. E. TRANs., Vol. XLV, pp 912-47.

10. Nickle, C. A., Oscil!ograph·ic Solution of Electro­mechanical Sy$te711s, A .. I. E. E. TitANS., Vol. XLIV, 1925, pp. 844-856.

11. Dreyfus, L., "Ansgleichvorga.nge Boim Plotzlichen Kurzsohluas von Synohron Genero.toren," Archiv f. Elecirotech., 5 s 103, 1916.

12. Dreyfu£, L., "Ausg!eichvorga.nge in c.ler Symmetrischen Mehrpho.senmo.schine,'' Elektrotech. u..'!\M aschinenbau. 30 S 25 121, 13!J, 1912. . . •, •

13. Dreyfus, 1., "Freie Mo.gnetisohe Energie . zwis~hen Verketteten Mehrpho.sensystemen," Elektrotech U, M aschinenbau; 29 s. 801, 1911.

'' 14. Dreyfus, L., "Einfuhrung in die Theorie der Selbster­regten Schwingungen Synchroner Mo.sohinen," Elektrotech. u. Masch-inenbau., 29 S. 323,345, 1911.

Discussion · U. C. Specht: I should think Mr. Pu.rk's th~ori could be

o.pplied just as well to the so-called synchronous induotion . motor, ~ha.t is an induction motor in which the rotor teeth .

. between the poles are cut out for lL disto.nce of a.bo.ut one-third of the pole pitch. Such a. motor runs at synchronous speed; However, the pull-out torque is much less than that of n.n induc­tion motor with the full number of teeth.

C. MacMillana There was one sta.tement in the' first page of Mr. Pa.rk's paper to, the effect tha.t "Idealization is resorted to, to the extent that saturation n.nd hystere"is in every ma.linetio · circuit a.nd eddy currents in the armature iron are neglected ..• ". And with rega.rd to Fig, 5, Mr. Park rema.rked tha.t itl'epresented a. rigorous solution. Perhaps Mr. Park oould give us &little more insight into the effect of taking into account saturation, a.nd give other cases in which oerta.in elements ha.ve been neglected with more or less effect upon the tina.! results. ·

W. J. Lyon: In a. paper of this description, oerta.fu premises should be chosen a.nd, with these o.lways in mind, the ma.th~ matioal development should be rigorous. The paper ma.y then. be criticized· either beea.use of insufficient premises or beco.use of incorrect ; ma.thematical development. I believe that the former is. the kinder method; it is the one I sha.ll employ.

The premises tho.t Mr. 'Pa.rk chooses a.re that ·the :field a.nd a.rmo.ture windings a.re symmetrica.l, tho.t saturation and hystere­sis are neglected a.nd that the armature windings are in effect sinusoido.lly distributed. I ta.ke this last to mean that the a.ir­gap flux due to the a.rma.ture currents is sinusoida.lly distributed, fori! the armature windings themselves were sinusoida.lly distrib­uted, there would be produced spo.oe harmonics in the o.i.r-go.p flux distribution due to the so.liency of the poles, which, as we all know, would complicate the problem tremendously. In order tha.t the mathematicn.I method used by Mr. Park shall be rigor­ous, I believe it is necessa.ry to make one further a.ssumption. I think I can best expla.in this by asking you to consider the result of supplying the field winding with a. sinusoida.l <1urrent while· the a.rma.ture rotates a.t some speed which may be ca.lled synchronous. Under these conditions, there will :first be pro.:. duced in the a.rmo.tur11 windings two sets of ba.lo.nced currents eo.oh of which will produce 3 component flux distributions in the jmp. The tirst of these is wha.t would be produced if the o.ir­go.p were uniform, o.nd is proportiono.l to 1/2 (:z:d + :z:9 - z4), whero :z:a equa.ls tbe ·arino.ture lea.ka.ge rea.ctn.nce. . The second ot these components is proportiona.l to 1/2 (:z:tl - : 9). · The third component is ol tbe sru:ne size as the second. Using the vo.lues tha.t Mr. Pa.rk gives under Sec.tion H of his pa.per, the relo.tive ma.gnjtudes of these components would be (0.8 - : .. ) a.nd 0.2. The first o.nd second components rea.ct on the field, n.nd produce in it o. current of the impressed :field lrequenoy. These a.re the components tho.t Mr. Park ha.s recognized, but the third component produces a.n entirely different frequency in the field, which will then be reflected into the a.rmature and the process will be repeated. Tho.t is, in this respect, it ie simila.r to the condition that exists in o. single-phase a.lterna.tor. As tar o.s I a.m a.wa.re, the Heo.viside opera.tiona.I·method cannot be used to obtll.in u. rigorous solution for the single--phase a.Iterna.tor. In spite of 'this, I think the objection that I ha.ve raised is of no more importo.nce tha.n the etfect of neglecting . sa.turo.tion or

P-92

len au;

be )ll

th ~d d; c-

re td io ,, d ·e ·e h

s t-

il. e a

f l l

i28 PARK: SYNCHRONOUS MACHINES Transa.ctions A. I. E. E.

hysteresis or the space harmonics in the air-gap flux that are .c~uallY· produced, but I do think that Mr. Park should ha.vo mentioned this point in order that the mathematicn.l work which he builds up should be rigorous. I discovered the necessHy of this additional premise about a year ago' while working on the problem of the ,transient short-circuit currents in a salient-pole alternator. · · ·. · ·

There is· a~other. problem of considerable interest that Mr. Pa.rk has not mentioned. . About two years ago I became aware that 'Mr. Fortescue's historic 1918 paper could be extended to

. the transient :case by using an operational method. That is to ,ay, a.ny: of, the problems involving unbalanced circuits thn.t Mr. Fortescue rigorously solved in the steady state can also he solved in the transient sta.te by an operationo.l method.

, ' : .;,.r . ' . • .

. : ·:•:' :: .. /•'

'· f···

l ·~: .• 1 ~~~,_~-r~-+,_~-+-~-+4-Hf~.~

., .. ''~ ;- 't&HH--l-+++tfhH-+++4-Hi-+-+11t.;.·~

\ ' ·;.;:;~{t ~·· l=--b-'"l:r:.lll·,ro::!-.;._vH-+1'-+-,-M-11-+--H--l--++-h'-.!-

1

t-1,

· ... J of-+...I-+-IH-1-+-+-+-+++++-¥-¥-H-H

.AH-+~4-+-~-++~~~~~~+-~-+~

., ' aH-++++-H-+=-t--+1-1+-!Hfl-++-H-+-1 ·'

.l2H-+-+++4-+-H-+~4'<//++-H-+-l-+-l .....

' FIG.' "1-ELFJCTRICA.L TORQUE OF A H I r~ ;· ; '

;:~:: · .. _':(:·:j) ~'i.i · ~~~'!!~=-~=r~enw ,

Note: The alx pointe plotted ~bove were t&ken experlntentall:r at a temper&ture of 26 dec. cent. · · '. · The balance of the dotted curve was t&ken at wrying higher tempero.tures · The eoUd curve was c:a.lc:ulated for & temperature of 25 deg. cent., and the pointe In oquaree for 100 deg. cent.

~ ~ :; :. ! ~ :i ._:;;. :: : . .

W. R. flel~~~ (communicated after adjournment) Since the completion of Mr. Park's paper as presented, experimental work has been. don~ .. which illustra.tes the worth of the new theory developed. . : '' . . . A .sma.ll sy~chronous motor rn.ted 15 hp., 220 volts, 1800 rev. per min. was selected for study. Its squirrel-oa.ge winding was removed in order to simplify the mecho.nioal work of calculation . . ·This motor;-yvas set up with an electric dyna.mometer, a.n<1 its .torque-slip curve w&s ta.ken at 110 -volts over the entire range of slip from 100. per cent to zero. Power wa.s supplied to it by o. speoia.l 220-volt sine-we.ve generator of 375-kv-a.. capo.oity. The

• constants of :,this machine· a.s well a.s of. the connecting line and switches were,determined, and these constants were included in the caloulation of torque in order to simulate as closely as possible infinite-bus conditions. · ,.

The torque as indicated by the dynamometer was corrected for -winda.ge and friction and for dynamometer errors, the net elec­.trical torque exerted on the rotor thus being obtained. ··By the use of meo.sured constants of the test mo.ohine o.ntl its

circuit, .a mathematica.l evaluation of torque was made for o. temperature of 25 deg. cent. .These constants a.s determined, on & per uxUt ba.ae of 24 .. 15 kv-o.. at 220 volts, are as follows:

X~ • 1.0 xl = O.li8 :::1 - 0.442 X/ = 0.4.42 r "' 0.0325 T, = 159 radians

Normal torque • 96.8 lb-ft. The ~al~ula.ted curve appears as a solid line in Fig. 1 herewith and near 1t.1s the dotted curve obtained experimentally.

Dunng the progress of the test run with the dynamometer no control or regular measurement of the synchronous motor temperature wo.s made. It fluctuated 'widely with changes in slip, a.nd the only points taken at a. known temperature are the ones near the b~ginning of the run. For the first six, those surrounded by c~rcles, the tempera.ture probably had not risen far from a.mbient of 25 deg. ;cent. They check quite closelv the calculated curve.· · •

In order to evaluate the effect of temperature change eight points were ca.lculated on the basis of 100 deg. cent~ These are shown within squares. They show a. better coincidence of slips at which ma.ximums occur, but little improvement in the maxi­mum error. Preyious theories have usually neglected. the effect of salie!l,t poles. The solid line in Fig. 2. shows the calculated curve in this ca.se, that is, for x9 • :::.:· ~A comparison of the solid curves in Figs. 1 a.nd 2 shows the large differences between tho. two calculations. · ' • · · · .. 1.-'H.· Summers• (communica.ted after adjournment) One

outstanding feature of this paper is tha.t the equations are expressed in "per-unit" quantities. 'l'his fea.ture presents the important advanta.ge that the quantities involved o.re those which are known directly from test or from the design of the machine. Furthermore, ma.chines o.re more readily compared with others of different rating when their constants are known in "per-unit" va.lues .

~~~r+-b~-r~~~~-1-~4-~~

g1o .. L::~--'" IJ ... ~ o~~-rr+~~~~~+-~.~-v~~ ~10~~-++4-+-+-~~~~~~~~-+~ ~ ~~-+~+-~-+4-~~~~~-+~+-~

Fro. PERCENT SUP

2-TonquE CuRvES FOR A H I Snlld Lin-Calculated Dotted Lln-Experlwent&l

However, these equations may not look fa.milia.r at first sight to those who o.ro more used to dealing in the physical quanti tie~ such a.s ohms, o.mperes, volts, henrys, and farads. It therefore seems appropria.te to point out the correla.ti•Jn between this new method and the more usual one. Consider, for exa.mple, Equa­tions (4a) in the paper, together with the two immedio.tely following. · ·

Ordinarily we should have written for a mo.ohine with no rotor circuits in addition to the field,. (R1 + L1 p) 11 - M 7' [j., cos 6 + jb cos (9 - 120")

+ j. cos (II + 120•)1 - E1 (1) In this equo.tion n, is field resista.uce, 'Lt is field inductauce, M

is mutuo.l inductance between field and any one arma.ture phase in the position of ma..'Cimum mutual inducta.nce, j., }b. and j,

P- 93

July 10:0:0

~>ro nrmo.ture pha.se current~. o.nd E; i~ fielrl voltage. All these qunntitieK nro in prn.ctical units.

Let

2 j,, • :i' [i,. cos 0 + ib co~ (o - 1:.!0°) + j, cos (0 + 120°)] (2)

Divicle through by J,. R, where J n is thu.t value of field current ~ha.t ca. uses normal voltage to be gonEjrated in the armature a.t no loa.d. ·

E, I, 3 L1 M' -- • (1 + Top) - - - -- __ P_ j I (3) 1 n R1 I. 2 R, 1" L, '

where T,- L!IR1 (4)

Let EfiJ,.RI • E ) r,;r,. - r (5) j,1/in • ia

Where i,. is normal armature current, maximum value .. Then

E• T ( [ 3 M in ) • P - -- -- ·-- ia + I. 2 L1 I,.

This equation ma.y be put into the form:

E,.. Tap'lt +l

'It !=I - (_2_ M;. ) i11 2 L1ln }

(6)

(7)

These results ma.y be compared with those immediately follow­ing (4a) in the pa.per: It is ea.sy to show by methods similar to. those use!f by Mr. Pa.rk that ,

3 . 2

j\t[ '" --·• Xd -- %c/ Lt I.

(8)

'where ::,1 and zl a.re. the per-unit diroct a.xis synchronous a.nd tra.nsient reo.ota.nce respectively. Then the equations a.re the same o.s those given by the a.uthor.

Without this la.st rela.tion we should doubtless have written the equations a.s in (2') or even a.s in (I). But both of these con­tain cumbersome qua.utities involving mutual a.nd self induc­tances. The expression used by Mr. Pn.rk is simpler and more expressive a.nrl contains quantities which n.re fa.milia.r to most engineers who have to do with synchronous ma,.chines.

Throughout this paper, a.lthough the problem is inherlmtly complex and some of the demonstro.tions a.pp.ila.r long, the results a.r.e relatively simple a.nd in terms o.da.pted for imroedia.te use.

R. H. Park: Mr. Specht is entirely correct in his tl:iof1ght that the present theory could be a.pplied to synchronous induction motors such a.s those to which he refers. In principle this type: of motor is not greatly different from a sa.! ient-pole s:Ynchronous' motor with a.n amortiaseur winding.

In reply to Mr. Ma.cMillan's question, ·r would point out that it is o.lways necessa.ry to make some assumptions; in fa,.ct, even in the simplest problems there invariably exists a.n. enormous ntimbar of assumptions most of which a.re not recognized a.s such. Thorofare a rigorous ~olution invarin.bly means only "rigorous on Lha hllol!ia of tho a.saumptiona." Tho solutions presouted in the pa.per are rigorously correct in tb.is sense. . . ·

Professor J..yon ha.s criticized the a.ssumptions ma.de .in the paper. This is due to a. very definite misconception a.s to their cha.ro.cter. Thus, referring to the a.ssumption "tha.t the a.rmature windings are in effect sinusoidally dis·tributed," Professor Lyon states tha.t it is necessa.ry' to interpret the ata.tement to mea.n something other than it says. I be-lieve, however, that this is not necessary. Thus he states tho.t wore the a.rma.ture windings themselves sinuaoida.lly distributed, there would be produced spo.oe harmonics in the o.ir-gap flux di:~tribution due to the sa.lioncy of the poles, which "would complicate the problem tremendously." The rea.son that. it. does not aomplica.te the problem is tha.t with a. sinusoidal distribution of turns only spo.ce !unda.mentDJ. tlu.x produces a.ny u.rma.ture flux linkages.

This follows from the f1l.Ct that 2~ 27 ! cos n o cos "' o J>e·.., I evs n 8 si~ :n o do = o

if n o.nd m o.re integers a.nd n - m.

Although this o.nswers. Professor Lyon's objootfon th~ impres­sion should not. ba derived tha.t the theory developed strictly presupposes o.n exa.ctly sinusoidally distributed winding. , The full sta.tement of the a.ssumption referred to is that the armature windings are in effect sil!l.usoida.lly distributed "a.s far a.s concerns effects depending on rotor position." Interpreted in tenus of self. o.nd mutual inductive coefficients this sto.tement is exa.ctly equ1valent to the two a.ssumptions,

. (a.) tha.t the self inductance of the a.rma.ture circuits is ~xpres-sible by an equa.tion of the form, . ' .

L • La + L! cos 2 o (b) tha.t the mutual inductance between the a.rma.ture and

any rotor circuit is expressible by a.n equation of tb.e form, M • M, cos 8

The assumptions which ho.ve been ma.de in ·previou~· studies, such a.s those whiah Mr. Ku in his pa.per "Tra.nsient Analysis of A-C. Machinery" ha.s referred to a.s "exact," ha.ve been precisely the same except tha.t the second ha.rmonio term in the a.rma.ture self inductance equation ha.s not been considered, while amortis­seur a.nd other circuits in a.ddition to the field ha.ve been neglected.

Actus.lly': the expressions for both a.rlna.ture a.nd rotor self inductance will involve a.ll of the even ha.rmonics of a.ngula.r position, o.nd the expression for anna.ture mutua.! inductance will involve all of the odd ha.rmonics of angula.r position. · However, tests on 1l.Ctual machines ho.ve shown tha.t in most ca.ses only the uro and second ba.rmonios of a.rmature self inductance, the first hs.rmonic of mutua.! inductance, and the zero ha.rmonic of rotor circuit self inductance a.re of prima.ry importance .. .Abo~t the only ca.se in which a· consideration of a.ny other' ha.rmonics would be of vs.lue a.ppea.rs to be in the study ot locking torques. With proper design, however, the tendenc)o to took at other tb.a.n half speed is slight.

Professor Lyon is, in my opinion, also in error ·when he en­dea.vors to point out the inadequacy of the theory by means of a.n example in which alternating current is supplied to a rota.ting field in a salient-pole ma.chine. In this connection be merely makes certain dogma.tic sta.tements without proof, which would not agree With the writer's findings. However, such mere state­ments or opinions do not constitute a proof, a.nd the writer con­tin'ues. to disagree with them.

.As I understand the exa.mple given, ·it correspouds to the application of o.lterna.ting current to a. synchronous machine field winding with a. three-phase short circuit across the a.rmo.ture terminals.. . . .

From Equations· (ll) a.nd (22) of the pa.per there is in this ca.se

Ia • [l~ <=•-~·I(P)> <pzq(p)+(l-sl)z.(p)> J·a(p)E . . .D<~ . .

- total per unit rotor excitation. Evidently this is of the se.me frequency a.s tha.t of the field voltage E. That is, there are no harmonics, Mr. C .. A. Nickle lia.s s·ub­mitted a. physical interpretation of this result. Thus he sta.tes

"Let I 'cos t. be impressed on the field which is running at a. speed S expressed a.s a fraction of synchronous speed ·tor the frequency of the current impressed on the field winding. The current J cos t produces a. spMe funda.roenta.l flux I cos t in the direct a.xis and spo.ce hurmonics alternating in time. Dut since . the a.rma.ture winding i:~ a.ssumed to ha.ve sine distribution, these ha.rmonic tiuxes can induce no voltage a.nd ma.y be neglected. The funda.ments.l flux, I coa I, ma.y be resolved into two rota.ting components I /2 rota.ting forwa.rd at normu.l speed with reforeneu

P-94

730 PARK: SYNCHRONOUS MACHINES Trn.naa.ctioos l'.. I. E. E.

to the pole structure n.nd I /2 rota.~ing ba.ckward a.t the sa.me speed. The first component roLa.tes forward with respect to the armature at a. speed 1 + S a.nd since the o.rnio.ture is short circuited polypha.se. a. positive pha.se sequence current o.f fre­quency 1 + 8 will flow in the armature winding. Similarly a. negative pha.se sequence current of frequency I - S will flow in the a.nna.ture winding due to the backward rota.ting'componellt of .field .f!u."C. The polyphase current i•+s will produce a. roto.ting Ul. m. f. o.lso i•+s nnd thiR m. m. f. rota.Les forwo.rd rela.tive to the pole structure a.t norma.! speed. Two fluxes n.re produced due to dissymmetry of the direct o.nd qua.dro.ture a.xes. These are

i•+s (. 1 + 1 ) -2- Xd Xq

rotating forward with respect to the poles a.t normal speed or 1 + S with respect to the armature, a.nd

it+s ( 1 1) --2- Zcl -. Xq

rotating ba.ckward with respect to the poles a.t normal speed or a.t 1 - 8 backward with respect to the 8ol'Ul&ture. The first will

induce a.n armature voltage of frequency 1 + S, :~.nd tho seeond a. voltage of frequency 1 - S.

The armature current of frequency i 1 • 5 produces a.n m. m. f.

i, ·s rota.tmg ba.okwa.rd with respect to the poles a.t norm:~.l speed <J.nd produces two rotating fluxes. One,

j I ·S ( - 2-- xl + :t/)

·rota.\'.es backward with respect to the a.rma.ture a.t e. spoeu 1 - S :~.nd

i, -s --2- (xi - Xq

1)

rotates f1'rward with respect to the a.rme.ture :~.t a. speed 1 + S. Thus, thi~• current n.lso prod~oes armature voltages.of frequencies 1 + S a.nd 1 - S. Therofore, no frequencies other thn.n (1 + S) a.nd (1 - H) need be 11ssumod, in thenrmnturt' winding nnd there­fore, these a.re the only frequencies thnt exist there. Moreover, this impliell thn.t only fundn.menta.l frequency ca.n exist in the rotor."