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Damper circuits and rotor leakage in the transient performance of saturated synchronous machines
D A M P E R CIRCUITS A N D R O T O R LEAKAGE I N THE T R A N S I E N T P E R F O R M A N C E OF S A T U R A T E D
SYNCHRONOUS MACHINES. BY R E I N H O L D RODENBERG,
Dr. I n g . ,
Graduate School of Engineering, Harvard University, Cambridge, Mass. 1. INTRODUCTION.
Several important phenomena in the operation of electric power systems are due to the action of the stator and rotor leakage of the feeding synchronous generators and particularly to the linkage of the rotor leakage fluxes with the damper circuits in the poles of the machines. In a previous paper,1 these effects in saturated synchronous machines were treated by a generalized method. In this p a p e r we shall consider the interaction of rotor leakage and damper circuits in greater detail, with full regard to the magnetic saturation of the main flux, and we shall include the reaction of the armature winding on the d a m p e r circuits. This detailed analysis may be justified by the fact that the development of the dangerous short-circuit currents in both magnitude and time, as well as the development of the recovery voltage in circuit breakers, limiting the interruption of short-circuit currents, which constitute two of the dominating problems in electric power engineering, depend preponderantly on these very effects. 2. FUNDAMENTAL CONDITIONS OF THE ELECTRIC AND MAGNETIC CIRCUITS.
We consider the main circuits of a synchronous machine as in Fig. I, and shown in more detail in the cross-section of Fig. 2. Actually, the various magnetic fields overlap and influence one another, but for the sake of simplicity we assume them to be separated into main and leakage fluxes, as is common in the theory of machines. In the stator circuit, the resistance of which we may neglect, 1 R. Rfidenberg, "Saturated Synchronous Machines u n d e r T r a n s i e n t Cond i t i o n s i n the Pole A x i s , " Transactions A.I.E.E., I 9 4 2 , Vol. 61, p. 297, contains a list of references. 39
4 °
REINHOLD RiJDENBERG.
[J. F. I.
the i n t e r n a l e l e c t r o m o t i v e f o r c e E is g i v e n by the t e r m i n a l v o l t a g e V a n d the l e a k a g e v o l t a g e s c a u s e d by the s t a t o r a n d r o t o r leakage fluxes ~ and ~ , E = V + o~No~,. + o~N,~r,
(~
¢0 b e i n g the c o n s t a n t a n g u l a r f r e q u e n c y a n d No the effective n u m b e r of t u r n s of the s t a t o r w i n d i n g . If the t e r m i n a l s a r e l o a d e d by a c u r r e n t I in a c o n s t a n t i m p e d a n c e X , the f i r s t
T
Fro. I.
Equivalent I
0
N
i r l ',~N i 0 01010~0,'0 J,i, ,i I
il
IITI
)lotOLO__.o)olololo
'i
o oX'o 0 o
t,
1111111
--II~llll\l ~.11~Iltlt --~LI UIII '~liilll
FIG. 2.
electric circuits of a synchronous machine.
-"
k : - - -N - z_ i I|i I Ill
Illl
--
!-
---'-
=1, I
I= --
=!: i--
I
--
Linkage of magnetic and electric circuits in the cross-section of a synchronous machine.
t e r m on the r i g h t - h a n d side of e q u a t i o n ([) is X I . We c o n s i d e r p r i m a r i l y the case w h e r e X is a p u r e r e a c t a n c e , so t h a t e q u a t i o n (I) r e m a i n s a l g e b r a i c . Iron l o s s e s m a y be n e g l e c t e d . If the s t a t o r l e a k a g e flux is p r o p o r t i o n a l t o the c u r r e n t , we can e x p r e s s the s e c o n d t e r m on the r i g h t - h a n d side of e q u a t i o n (I) by x , I w h e r e x, is the s t a t o r l e a k a g e r e a c t a n c e . T h e flux of the t h i r d t e r m , h o w e v e r , is l i n k e d , a c c o r d i n g t o Fig. 2, n o t only w i t h the s t a t o r c u r r e n t s but also with the d a m p e r c u r r e n t s a n d t h e r e f o r e this t e r m c a n n o t be e x p r e s s e d so s i m p l y .
July, 1942.]
~ATURATED SYNCHRONOUS
~IACItlNI.;S.
41
T h e excitation circuit of the r o t o r is fed by an e x t e r n a l v o l t a g e e w h i c h is g i v e n e i t h e r a s c o n s t a n t or a s a k n o w n f u n c t i o n of t i m e . W l t h r e s i s t a n c e r, n u m b e r of field t u r n s N, and r o t o r pole flux ~, the e x c i t i n g c u r r e n t i is d e t e r m i n e d by d~
e(t) = ri + N d~'
(2)
the last t e r m g i v i n g the i n f l u e n c e of the c h a n g e of flux a n d b e i n g p r e d o m i n a n t d u r i n g the t r a n s i e n t s t a t e . In the m a i n magnetic c i r c u i t of Fig. 2, the excitation c u r r e n t i p r o d u c e s the m a g n e t i c flux q~ in the pole c o r e , w h i c h r e q u i r e s a m a g n e t i z i n g c u r r e n t ix d e p e n d e n t on • a n d t h u s on the
I /U [ I- A I R ', 0
Fro. 3.
io
Constant magnetic and electric characteristics of a synchronous machine in transient state.
c o r r e s p o n d i n g e l e c t r o m o t i v e f o r c e E a c c o r d i n g t o the m a g n e t i c s a t u r a t i o n c u r v e s h o w n in Fig. 3- In a d d i t i o n , the e x c i t a t i o n c u r r e n t b a l a n c e s the a r m a t u r e r e a c t i o n of the load c u r r e n t I , w h i c h m a y be p u r e l y r e a c t i v e , a n d also the r e a c t i o n of the d a m p e r c u r r e n t id, f l o w i n g in the n u m b e r of t u r n s Ne. H e n c e the e x c i t a t i o n of the m a i n flux is d e t e r m i n e d by the r e l a t i o n
No
Nd
i = i , ( E ) + W X - W i,,.
(3)
In this e q u a t i o n we c o u n t a s p o s i t i v e : a m a g n e t i z i n g d a m p e r c u r r e n t ia, b u t a d e m a g n e t i z i n g i n d u c t i v e s t a t o r c u r r e n t I. T h e d a m p e r circuit a c c o r d i n g t o Fig. 2 is l i n k e d c o m p l e t e l y with the a r m a t u r e flux ¢~, this b e i n g the difference of the r o t o r
42
[J. F. I.
REINHOLD RUDENBERG.
core flux • a n d the r o t o r l e a k a g e flux Or. T h e r e f o r e the c u r r e n t in the s h o r t - c i r c u i t e d d a m p e r with r e s i s t a n c e r~ is g i v e n by :V d o = ~ ~dt (O - Or) + rdi~. (4) In the rotor leakage circuit the flux Or can be c o n s i d e r e d , a s is s h o w n in Fig. 2, a s e x c i t e d by the c o m b i n e d e f f e c t of s t a t o r c u r r e n t a n d d a m p e r c u r r e n t , t h e s e two h a v i n g o p p o s i t e p o s i t i v e d i r e c t i o n . T h e effective r o t o r l e a k a g e flux is t h e r e fore O , - ~0No
~id
,
(5)
if we d e f n e the r o t o r l e a k a g e r e a c t a n c e a t the s t e a d y s t a t e , r e l a t e d t o the s t a t o r c u r r e n t a n d w i t h o u t the i n f l u e n c e of the d a m p e r current, as N~0~o Xr ~ CO
/'0
(6)
T h e r o t o r pole flux • a n d the e . m . f . E are a l w a y s proportional, • O0 - E0' (7) if O0 a n d E0 d e n o t e the r a t e d v a l u e s of flux a n d s t a t o r v o l t a g e in the m a c h i n e a t no l o a d . W i t h r a t e d r o t o r v o l t a g e e0 the l a s t term of e q u a t i o n (2) t h e r e f o r e can be w r i t t e n in e i t h e r of the two f o l l o w i n g f o r m s : dO dE eor, d ( E / E o ) N - ~ = T,. d-~ = dt '
(8)
where T., -
N¢o E,
(9)
is a time c o n s t a n t of the c o m p l e t e m a c h i n e w h i c h is e n t i r e l y i n d e p e n d e n t of the m a g n e t i c s a t u r a t i o n , w h i l e T~
-
N¢o
(IO)
e0
is a time c o n s t a n t of the r o t o r p o l e s with t h e i r field w i n d i n g s
July, 2942.]
SATURATED SYNCHRONOUS MACHINES.
43
only, which is constant so long as the rated rotor values remain unchanged. For machines of the usual design the numerical value of this time constant is of the order of Tp = 3-6-I2 sec. for generated powers of IOO-IOOO-IO,OOO kva. per pole of the synchronous machine. 3, SOLUTION OF T H E CIRCUIT EQUATIONS.
Equations (I) to (5) constitute the fundamental equations of our problem which shall be solved for the transient state and with variable magnetic saturation of the machine. Although it is not absolutely necessary, we assume, for the sake of simplicity, that the important saturation of the main magnetic circuit is concentrated a t the pole cores, where it has maximum effect, and we also assume that the saturation of the magnetic leakage circuits is only low. In the given set of equations we eliminate the subordinate magnitudes, particularly those of linear dependence, and we relate the remaining terms to those parameters which contain the saturation of the main magnetic circuit, particularly to the magnetic no-load characteristic i , ( E ) . If we substitute in equation (2) the flux given by equation (8) and insert in equation (2) the value of the exciting current i of equation (3), we obtain the main equation for the change with time of the e.m.f, of the machine dE T,~ dt - e(t) -- r E i , ( E ) + m I - nidJ,
(1I)
where Na N'
Nd N
n "= - -
(I2)
express the turn ratios of armature and damper windings to the field coils. Equation (II) could be solved if the content of the square bracket were dependent only on the e.m.f. E, as is the no-load magnetizing current i~. By use of equation (5), however, and by expressing voltage and fluxes in equation (I) by reactances and currents, we obtain ( N~ ) E = X I + x d + x r I - ~ i d , (~3)
44
R E I N H O L D R/JDENBERG.
[J. |:. 1.
showing that the rotor leakage drop in the transient state depends upon the value of the damper current. Hence the stator current, expressed by e.m.f, and damper current, is E x~ n. = -I X + x . + x + X + x . + x ~ m ~d,
(I4)
and the last two terms in the bracket of equation (I I) become mI-
mE X+x. nid = X + x , + xr - X + x, + x~ nid.
(15)
Thus we have substituted the e.m.f. E for the stator current on the right-hand side of equation (II), E being the main variable of this differential equation, and we obtain Tm
[ mE dE - e(t) - r | i ~ ( E ) + X + x ~ + x r dt L X + x,~ ni~ ]. J X + x , + x r
(I6)
In order to express also the damper current id by the e.m.f. E, we introduce the rotor leakage flux ~r of equation (5) in equation (4) and obtain d~
N d - ~ + r,,id =
xr
d(
N d ~ t
n.)
I -- -*dm
(17)
"
Substituting equation (15) on the right-hand side of equation (17) and introducing E rather than ¢ on the left-hand side by use of equation (7), gives N
Go d E Koo-d + rdi
=
x~ [ dE/dr LX + + xr n X + x, did ]J - m X + x, + x r d t
"
(IS)
For the first term on the right-hand side of this reiation we can express the fraction before the bracket by equation (6), giving the r a t i o of rotor leakage flux ~r0 to stator current I0 under steady-state condition. On the other hand, the sum of all the reactances in the first denominator within the bracket is given by the quotient of e.m.f. E and current I0 for the steady state. There results, therefore, neglecting the differ.
July, i942.]
SATURATED SYNCHRONOUS ~'IACHINES.
45
ence of E a n d E0, w h i c h is i n s i g n i f i c a n t h e r e , x~ ~NaX
dE~dr + xs + xr
~ro Io
Io Eo
dE dt
(I9)
H o w e v e r , we c a n c o m b i n e this t e r m with the f i r s t term on the l e f t - h a n d side of e q u a t i o n ( I 8 ) , o b t a i n i n g • o
Nd
-
¢ro
Eo
dE a~a dE dt - Xd E~o d t '
(20)
w h e r e ~ is the a r m a t u r e flux a s d i f f e r e n c e of pole a n d r o t o r l e a k a g e flux in the s t e a d y s t a t e . T h e l a s t term on the r i g h t h a n d side of e q u a t i o n (I8) can be w r i t t e n xS
Ne
X + x. +
x.
+
xr
di~ = L ~ - - dt dt
(21) '
'
so t h a t the e n t i r e e q u a t i o n (I8) for the d a m p e r c u r r e n t m a y be e x p r e s s e d a s ¢a dE did N~ Eoo-dt + raid + Lr dt - o.
(22)
W e see t h e r e f o r e t h a t L~ is the s e l f - i n d u c t a n c e of the d a m p e r w i n d i n g with the r o t o r l e a k a g e flux. I t is e x p r e s s e d in e q u a t i o n (2I) by the r o t o r l e a k a g e i n d u c t a n c e x./w v i e w e d from the a r m a t u r e , r e d u c e d by the s q u a r e of the t u r n r a t i o of d a m p e r a n d a r m a t u r e c i r c u i t s , a n d m u l t i p l i e d by a d i m i n u tion f a c t o r P
X+x. X + x , + x r '
(23)
w h i c h e x p r e s s e s the e f f e c t of the a r m a t u r e r e a c t i o n on the d a m p e r circuit. Our p r o b l e m now c o n s i s t s in s o l v i n g s i m u l t a n e o u s l y the differential e q u a t i o n s (I6) a n d (22), both c o n t a i n i n g as dep e n d e n t v a r i a b l e s E a n d i~ o n l y , b u t with a n o n - l i n e a r t e r m i , ( E ) c o n s t i t u t i n g the m a g n e t i c c h a r a c t e r i s t i c of the m a c h i n e . In o r d e r t o s o l v e t h e s e e q u a t i o n s by a s i m p l e g r a p h i c a l m e t h o d , we n o t e t h a t in the m a i n e q u a t i o n (I6) the d a m p e r c u r r e n t L1 p l a y s m e r e l y the p a r t of a c o r r e c t i o n term for the
46
R E I N H O L D Rf)DENBERG.
[J. F. I.
variation of the main flux and e.m.f, with time. It may be sufficient therefore if we satisfy equation (22) by a good approximation for the general variation of ie and introduce that damper current into the main equation (16). Thus the assumption may be made, derived from the form of equation (22), that the d a m p e r current changes nearly exponentially with time, so that did
dt
ia -
(24)
T '
still allowing for a slow variation of the time constant T, the magnitude of which is left undetermined. By introducing equation (24) in equation (22) and denoting the time constant of the damper with the rotor leakage flux Lr Tr = - , (25) r4 we obtain, with use of equation (9),
Nda~aTdE ( Tr) N ' b o " dt + r d I -- T id = o,
(26)
by which id is determined as a function of d E / d t . Inserting the damper current id in the last term of equation (16), this term becomes
rpnid = p
Nd2cba r Tm d E ~ d r rd N2¢o I -- T r / T
T~ T,. d E / d t Tp I -- T~/T
(27)
The central expression is so written that we see immediately that the first two fractions give the r a t i o of two time constants. These are: a d a m p e r time constant Td, determined by resistance and number of turns of the d a m p e r circuit and by the armature flux linked with this circuit and therefore giving a value different from Tr in equation (25), and the main pole time constant Tp, determined by the corresponding constants of the excitation winding and already given by equation (Io). By combining the last term of equation (I6), g i v e n by equation (27), with the left-hand side of equation
July, i942.1
SATURATED SYNCHRONOUS MACHINES.
47
( I 6 ) , we now o b t a i n
v., i+i2
T~/Tp
g/r = e(t) -- r [ i~(E) q - X q _ mx ,Eq _ x , ] . ( 2 8 )
H e r e i n t h e r i g h t - h a n d side c o n t a i n s the q u o t i e n t of e.m.f, a n d t o t a l r e a c t a n c e of the c i r c u i t c o n s t i t u t i n g the s t e a d y s t a t e n e t w o r k c h a r a c t e r i s t i c , as s h o w n in Fig. 3. If we now i n t r o d u c e on the l e f t - h a n d side the r o t o r pole time c o n s t a n t T~ of e q u a t i o n 0 o ) , r a t h e r t h a n the m a c h i n e time c o n s t a n t T ~ of e q u a t i o n (9), we o b t a i n the differential e q u a t i o n for the e . m . f . of a s a t u r a t e d m a c h i n e with d a m p e r c i r c u i t s , in the f i n a l form
[
T~
]d(E/Eo)
Tp + I -- ~I'r/T
e(t) =
dt
_ _
eo
~i __
_ _ .
~o
( 2 9 )
T h e last t e r m Z i on the r i g h t - h a n d side d e n o t e s the sum of the i n t e r n a l a n d e x t e r n a l s t e a d y - s t a t e m a g n e t i z i n g c u r r e n t u n d e r t h e e . m . f . E , this sum b e i n g a l w a y s a g i v e n f u n c t i o n of E , see Fig. 3. T h e e x p r e s s i o n in the b r a c k e t on the l e f t - h a n d side of e q u a t i o n (29) is a r e s u l t a n t time c o n s t a n t a c c o r d i n g t o w h i c h the e . m . f . E , a n d s i m u l t a n e o u s l y the d a m p e r c u r r e n t id, c a n v a r y . H e n c e we h a v e the c o n d i t i o n a l e q u a t i o n
T = Tp -t-
Td I
--
T~/T'
(30)
w h i c h c a n be w r i t t e n a s a q u a d r a t i c e q u a t i o n
T2 -- T(Tp + rid + Tr) + TpT~ = o.
(3 I)
T h u s two p o s s i b l e d i f f e r e n t time c o n s t a n t s a p p e a r , c o r r e s p o n d i n g t o the two r o t o r c i r c u i t s . W i t h m o d e r a t e r o t o r l e a k a g e , a s in a c t u a l m a c h i n e s , the r o t o r l e a k a g e t i m e c o n s t a n t T~ is s m a l l c o m p a r e d with the o t h e r time c o n s t a n t s . T h e n the two s o l u t i o n s of e q u a t i o n (31 ) are in v e r y good a p p r o x i m a t i o n T t = T~ + Td;
T, T~ + T~ ~ T~,
T~ = I+
T.
(32)
48
R E I N H O L D R/JDENBERG.
[J- t;. I.
where the last term on the right-hand side is a further approximation for relatively small Ta. The large main flux time constant T1 is given by the sum of the time constants of field winding and d a m p e r circuit, both related to the main fluxes in armature and poles. The small leakage flux time constant T~, on the o t h e r hand, is given chiefly by the rotor leakage flux from pole to pole closing itself through the d a m p e r circuit. 4. G ~ P m C A L ~Pm~SS~TAZION OF THE SOLVTION. W e see n o w that our differential equation (29) has two solutions, a forced solution belonging to the main flux in the pole core, and a free solution belonging to the rotor leakage flux b e t w e e n the poles. T h e forced solution is given by
TI d ( E / E o ) dt
-
e(t) eo
i ~ ( E ) + rail io
=
Ai, - ~o
(33)
and determines by the large v a l u e of T1 a slow transient
~ J ~ ~ - - - - ~i~(E)
E ® T ~ I.--~->
0
• Ir
FIG. 4:" Graphical representation of the slow and the rapid transient
changes of state.
change with time of the e.m.f. E and of the corresponding main pole flux. This flux is excited by an ohmic current elf determined by the excitation voltage e(t), from which the magnetization current i, for the main circuit and the steady state armature reaction current rnI, both corresponding to E, are to be deducted. For constant excitation voltage e, the right-hand side of Fig. 4 gives a graphical representation of
July, 1942.]
~ATURATED SYNCHRONOUS MACHINES.
49
the change of e.m.f. E with time w h i c h is for e v e r y E proportional to the difference current Ail, closely cross-hatched in Fig. 4. The free solution is given by the remaining differential equation 7"2 d ( E / E o ) dt
-
mI2 io
-
A,i,,. io
(34)
and determines by the small value of T2 a r a p i d transient change with time of the rotor leakage flux. This flux is excited by the d a m p e r current but is not influenced either by the excitation voltage or by the iron magnetizing current i~. The change of this leakage flux and its currents is represented graphically by the left-hand side of Fig. 4, which is widely cross-hatched and shows as exciting current only that of the rotor leakage reactance xr. With no saturation in the rotor leakage circuit, the leakage flux and the corresponding part of the d a m p e r current decay exponentially, since ki2 is then proportional to E. With saturation of the leakage flux we can a p p l y equation (34) graphically in the same way as described for equation (33). The total stator current is always given as the sum of the forced and free solutions, i.e., I~ + I2, as already indicated in equation (I4). While in Fig. 4 the forced Ail decreases towards zero at the intersection of the saturation and the network characteristics, yielding a steady state voltage E~ and a sustained current I , in the stator, the free ki2 decays towards zero at the zero line, leaving no current or voltage in the d a m p e r winding. The rate of decrease is slow for the forced current Ai~ with the large time constant T1 but r a p i d for the free current Ai2 with the small time constant T> Hence Ai2 will have nearly vanished when ki~ begins to move. In Fig. 5 the development of the short-circuit current with time is shown when the terminals of a saturated generator are suddenly s h o r t circuited and the excitation voltage is kept constant, the generator previously having been w i t h o u t load. The magnetic no-load characteristic of the machine, i,(E), in conjunction with the stator leakage characteristic x . J and the
5°
R E I N H O L D RUDENBERG.
[J. F. I.
t o t a l l e a k a g e c h a r a c t e r i s t i c (x, q- xr)I d e t e r m i n e s c o m p l e t e l y the t r a n s i e n t s t a t e , i n c l u d i n g the i n i t i a l c o n d i t i o n s . T h e c u r r e n t of u n b a l a n c e Ail as d e f i n e d by e q u a t i o n s (28) a n d (33) is g i v e n by the h o r i z o n t a l w i d t h of the c l o s e l y c r o s s h a t c h e d area of Fig. 5a a s the difference b e t w e e n the p r e v i o u s e x c i t a t i o n c u r r e n t il = e/r a n d the sum of the n o - l o a d m a g n e t i z i n g c u r r e n t i~(E) a n d the s t e a d y - s t a t e s h o r t - c i r c u i t c u r r e n t I1 = E / ( x , Jr xr), the l a t t e r b e i n g p l o t t e d in the r o t o r s c a l e b a c k w a r d s from i l a s a s t r a i g h t l i n e . S i n c e /~il is dep e n d e n t only on E , the c u r v e E(t) c a n be d r a w n as in Fig. 5b, e i t h e r in s t e p - b y - s t e p c o n s t r u c t i o n w i t h the a l w a y s k n o w n v a l u e of dE/dt or by a q u a d r a t u r e y i e l d i n g t a s a f u n c t i o n of E
o ,..
¢
b
! ,]E
O
k
iI
I
0
FIG. 5. Development of slow and rapid transient currents at sudden short circuit of the terminals: (a) Diagram of characteristics; (b) Change with time.
a f t e r s e p a r a t i o n of the v a r i a b l e s in e q u a t i o n ( 3 3 ) . T o e v e r y E the c u r r e n t 11 can be c o r r e l a t e d by m e a n s of the t o t a l l e a k age characteristic. T h i s slow t r a n s i e n t s t a t o r s h o r t - c i r c u i t c u r r e n t c o r r e s p o n d s t o the slow v a r i a t i o n of the m a i n pole flux w i t h t h e s a t u r a t e d time c o n s t a n t T1. In a d d i t i o n t o t h i s , a r a p i d l y v a r y i n g flux is p r o d u c e d by the c u r r e n t axis s h o w n a s w i d e l y c r o s s - h a t c h e d b e t w e e n the two l e a k a g e c h a r a c t e r i s t i c s in Fig. 5a a n d a c t i n g on the r o t o r l e a k a g e p a t h o n l y . T h i s d a m p e r c u r r e n t is b a l a n c e d by a n a d d i t i o n a l s t a t o r c u r r e n t Is, both d e c a y i n g r a p i d l y with a n e x p o n e n t i a l t i m e c o n s t a n t T2 a c c o r d i n g t o e q u a t i o n ( 3 4 ) . I n s t e a d of a s t e p - b y - s t e p d e c r e a s e of axis down t o z e r o , or of a q u a d r a t u r e of e q u a t i o n (34), we m a y t u r n the s t a t o r l e a k a g e
July, 1942.]
SATURATED SYNCHRONOUS MACHINES.
5I
characteristic x, towards the complete leakage characteristic x~ -t- xr with a velocity corresponding to the time constant Y2. Then any intermediate position of this line always determines the left-hand end up to which the total stator current I t -t- I2 has to be measured. Fig. 5b shows the rapidly decaying additional current I s which forms a large initial increase of the short-circuit current. While the r a p i d transient current changes exponentially with time, so long as no saturation prevails in the leakage paths, the slow transient current deviates widely from an exponential variation due to the effect of saturation in the main magnetic circuit. In Fig. 6 the sudden interruption of a terminal short circuit of a saturated generator is shown, the excitation voltage IE . _ , , ~ - - E . ~ . . . . . . . . . .= - ~ - ~ - -
INN 0
f il
Q
0
t b
FIG. 6. Development of slow and rapid transient recovery voltages at sudden interruption of a s h o r t circuit a t the terminals: (a) Diagram of characteristics; (b) Change w i t h t i m e .
a g a i n being kept constant. Here, according to Fig. 6a, both leakage characteristics determine the previous steady-state condition of the fluxes by intersection with the no-load characteristic. The characteristic x~ gives the stator main flux, the characteristic x , H-Xr gives the rotor pole flux. At the instant of interruption of the short circuit, the rotor pole flux is kept constant by the field winding but the stator flux is also kept constant by the additional action of the d a m p e r winding. Thus the e.m.f., corresponding to the rotor pole flux, starts with the value E B and increases slowly according to the current of unbalance Air which lies between the characteristic i~,(E) and the vertical line through il, the l a t t e r constituting the open terminal characteristic. The stator voltage, on the o t h e r hand, j u m p s at the instant of interruption from zero to the value Ea, equal to the
52
R E I N H O L D RU'DENBERG.
[J. F. I.
previous stator leakage voltage, and increases rapidly towards the higher value of the e.m.f. E, as in Fig. 6b, due to the decaying rotor leakage flux. Thus with interruption of the circuit the voltage at the terminals of the generator increases instantaneously due to the stator leakage; it then increases rapidly due to the rotor leakage, and it further increases slowly due to the main flux in the machine, finally attaining the no-load voltage. The r a p i d transient voltage is nearly exponential with the time constant T2; the slow transient voltage is distorted by saturation and is to be derived by the shaded area in Fig. 6a with the time constant T1. It may be noted that the scales of the characteristics and times in Figs. 5 and 6 are chosen the same so that the rates of change of the voltages are directly comparable. 5. M A G ~ P , T I C T I ~ - C O N S T A N T S .
The development of many of the transient conditions which may occur in the operation of saturated synchronous machines has been shown in a previous paper,1 in which, in analogy to non-saturated circuits, there was assumed a wellknown quadratic equation for the time constants. In the present p a p e r a more detailed analysis of the magnetic linkages has led us to a similar quadratic equation (3~), except that we have now arrived at relations determining numerically all the time constants, and the damper time constants a p p e a r here separated into the magnitudes Te, belonging to the main armature flux, and TT, belonging to the rotor leakage flux. Furthermore, we have shown here by equation (23) to what extent the d a m p e r time constants are influenced by armature reaction. The damper time constant of the main flux is, according to equation (27),
Td = p T , ( - ~ ) 2r ~a Nd~, r,~ c~o - o reie
(35)
The last expression on the right-hand side is obtained by substituting equation (IO) for Tp and introducing as the exciting transient the d a m p e r current rather than the field current, these two being in the r a t i o of their turn numbers.
July, 1942.]
SATURATED SYNCHRONOUS MACHINES.
53
F o r n u m e r i c a l c o m p u t a t i o n s the c e n t r a l e x p r e s s i o n of e q u a t i o n (35) is c o n v e n i e n t a l s o . I t s h o w s t h a t this d a m p e r time c o n s t a n t Td is only a f r a c t i o n of the m a i n flux t i m e c o n s t a n t Tp, s i n c e the d a m p e r c r o s s section, d e t e r m i n e d by Nd2/rd, is a l w a y s m u c h s m a l l e r t h a n the field coil c r o s s section, d e t e r m i n e d by N2/r. F u r t h e r m o r e , the d a m p e r time c o n s t a n t is r e d u c e d by the f a c t o r p r e p r e s e n t i n g by e q u a t i o n (23) the i n f l u e n c e of a r m a t u r e r e a c t i o n . S i m i l a r l y , we o b t a i n for the d a m p e r t i m e c o n s t a n t of the r o t o r l e a k a g e f l u x , a c c o r d i n g t o e q u a t i o n s (25) a n d (21), a n d u s i n g e q u a t i o n s (6) a n d ( I O ) ,
Tr = pTp
(N~)~r~o Nd¢~____2o -~ r--~ ¢--~ = p rdid '
(36)
w h i c h a g a i n is a s m a l l f r a c t i o n of T~t, in v i e w of the a d d i t i o n a l e f f e c t of the r a t i o of the r o t o r l e a k a g e flux Cr0 t o the m a i n pole flux q0. By e q u a t i o n s (IO), (35), (36), we c a n c o m p u t e r e a d i l y the r e s u l t a n t time c o n s t a n t s of e q u a t i o n (32) for a n y g i v e n m a c h i n e . W e h a v e t o n o t e , h o w e v e r , t h a t due t o the f a c t o r p both d a m p e r time c o n s t a n t s are d e p e n d e n t on the l o a d , w h i c h in e q u a t i o n (23) is e x p r e s s e d in t e r m s of the e x t e r n a l r e a c t a n c e X. A t no load we h a v e X = oo and t h e r e f o r e p = I, i n d i c a t i n g no a r m a t u r e r e a c t i o n a n d full v a l u e of the d a m p e r t i m e c o n s t a n t s . A t t e r m i n a l s h o r t circuit, we h a v e X = o, a n d t h u s , with e q u a l r o t o r a n d s t a t o r r e a c t a n c e s , for e x a m p l e , p = 1. This s h o w s t h a t the a d d i t i o n a l a r m a t u r e c u r r e n t s , d e v e l o p e d s i m u l t a n e o u s l y with the d a m p e r c u r r e n t s , r e d u c e the v a l u e of both d a m p e r time c o n s t a n t s with inc r e a s i n g load t o w a r d s a b o u t o n e - h a l f a t s h o r t circuit. Howe v e r , a t a n y a c t u a l load this e f f e c t is slight, s i n c e X then outw e i g h s the l e a k a g e r e a c t a n c e s in e q u a t i o n (23). 6. VARIOUS DAMPER CIRCUITS.
A c t u a l d a m p e r c o n s t r u c t i o n s of s y n c h r o n o u s m a c h i n e s d i f f e r t o s o m e e x t e n t from o u r s i m p l i f y i n g a s s u m p t i o n s . In p a r t i c u l a r , the r o t o r l e a k a g e field is n o t the only flux l i n k e d with the d a m p e r c i r c u i t . Fig. 7 s h o w s t h a t a f u r t h e r flux 4)~ m a y s u r r o u n d the d a m p e r c o n d u c t o r s , c l o s i n g i t s e l f p a r t i a l l y t h r o u g h the d a m p e r s l o t s , p a r t i a l l y a r o u n d the end r i n g s of
54
R E I N H O L D R/)DENBERG.
[J. F. I.
the damper. This additional flux is produced by the damper currents id only, but not by the stator currents I . If we include this additional leakage flux in the fundamental conditions, equations (I), (2), (3), (5) remain unchanged. In equation (4), however, we have to add on the right-hand side a small term L~did/dt, giving the induced voltage by the independent d a m p e r leakage inductance L~. Correspondingly we have to add to the time constant Tr, related to the rotor leakage flux by equation (36), a small time constant Te due to this inherent d a m p e r leakage flux. This magnitude is not influenced by armature reaction and therefore does not contain the factor p. The time constant Td in equation (35) remains
IliJlf
ollJll
• I I I I J ~ l l l J l l I
Illi/Jl
Fro. 7.
il
liJ
rl
III ITI iililll[ lilfffli Ilfiili
PI il li II
T h e magnetic fluxes i n space linked w i t h a d a m p e r w i n d i n g i n the pole shoes of salient p o l e s .
unchanged since it is determined by the main armature flux only. Actual dampers, as shown in Fig. 8, are b u i l t for the most part in the form of squirrel cages, with spatially distributed damper bars. Such circuits are very closely linked with the distributed armature currents but much less closely with the lumped field winding. Hereby, a further leakage between d a m p e r and field winding occurs, increasing the independent time constant T~. On the o t h e r hand, the distributed currents of squirrel-cage dampers are not completely linked with the main flux. Hence the d a m p e r time constant T~ becomes somewhat smaller than in equation (35). This can easily be
July, 1942.]
SATURATED SYNCHRONOUS MACHINES.
55
t a k e n i n t o a c c o u n t by i n c l u d i n g in the n u m b e r of t u r n s Nd of the d a m p e r a n a d d i t i o n a l l i n k a g e f a c t o r a c c o r d i n g t o f a m i l i a r rules. S i n c e the a c t u a l d i s t r i b u t i o n of the d a m p e r c u r r e n t s over the p a r a l l e l d a m p e r b a r s of a s q u i r r e l c a g e is free, the e n t i r e d a m p e r flux c o n t a i n s in s p a c e f u n d a m e n t a l as well as h a r m o n i c f i e l d s . T h e r e f o r e , t h e r e m a y a p p e a r here a w h o l e s e r i e s of time c o n s t a n t s r a t h e r t h a n the two d e r i v e d for l u m p e d c i r c u i t s . H o w e v e r , the a m p l i t u d e s of the h a r m o n i c fields, v o l t a g e s a n d c u r r e n t s will be s m a l l , s i n c e they are d e t e r m i n e d m e r e l y by difference effects m o d e r a t e in m a g n i t u d e . E v e n if t h e r e is no a c t u a l d a m p e r w i n d i n g , but the pole c o r e s or pole s h o e s are m a d e of s o l i d steel, a s in Fig. 9, eddy c u r r e n t s can d e v e l o p in s u c h s o l i d p a r t s a c t i n g q u a l i t a t i v e l y O?
0 0 0 0 0 0 0
I
FIG. 8.
Squirrel-cage d a m p e r w i n d i n g i n contrast t o the l u m p e d circuit of Fig. 7.
a s d a m p e r c u r r e n t s . Q u a n t i t a t i v e l y t h e i r time c o n s t a n t s Te a n d T r are s o m e t i m e s s m a l l e r b e c a u s e of the h i g h e l e c t r i c r e s i s t a n c e of the s t e e l . T h e y c a n be c o m p u t e d ~ r e a d i l y for a n y c r o s s s e c t i o n of the s t e e l p o l e s a n d m a y c o n t r i b u t e part i c u l a r l y t o Te. In Fig. 9 the s p a t i a l d i s t r i b u t i o n of t h e s e d a m p i n g eddy c u r r e n t s is s h o w n in c o m p a r i s o n t o the d i s t r i b u tion of the f i e l d c u r r e n t s . S i n c e the d i s t r i b u t i o n s d i f f e r from each o t h e r , r e s i d u a l l e a k a g e is c a u s e d a n d here also a s m a l l i n d e p e n d e n t time c o n s t a n t T~ is d e v e l o p e d in a d d i t i o n t o 7", A s o l i d r o t o r y o k e h a s the s a m e e f f e c t in p r i n c i p l e ; h o w e v e r , the l i n k a g e with both s t a t o r a n d r o t o r w i n d i n g s is very s m a l l a n d t h e r e f o r e the i n d e p e n d e n t l e a k a g e is l a r g e , w h i l e the c o n t r i b u t i o n t o Tr v a n i s h e s . 2 R. Riidenberg, "Elektrlsche Schaltvorg~inge" (a book), 3rd e d i t i o n , Berlin, x933, p. 7o.
56
REINHOLD RUDENBERG.
[J. F. I.
If a copper d a m p e r and solid steel parts are acting together, as in many actual machines, the effects of both merely add, increasing all the main and leakage time constants. In cylindrical rotors for turbo-generators, as shown unrolled in Fig. Ioa, the linkage between field currents and eddy currents in the solid steel parts is relatively large. In Fig. Iob the distribution of the field currents and of the damping eddy currents along the circumferential direction is shown for both the currents in the solid iron and in the b e t t e r conducting
I I I 1
I
I I Ii~~
llPOleshoe ,
I
I
I
I polecore
I
i ~IHIIJ
-JBHHJ
N FIG. 9.
Distribution of damping eddy currents in the solid steel parts of salient poles.
metal wedges closing the slots. We see that the distribution of the entire d a m p e r currents iu is similar to that of the field currents i. Hence t h e i r linkage with the main flux is nearly perfect and a relatively large main flux d a m p e r time constant T~ is developed by the large steel cross section, according to equation (35), while the additional leakage and therefore T~ are only slight. Since the currents in the metal wedges are completely linked with the armature flux, as is the case with an ideal damper, the quotient ~a/~0 remains in equation (35) for this part of the d a m p e r circuit. However, the eddy
J u l y , 1942.]
SATURATED
SYNCHRONOUS
MACHINES.
57
c u r r e n t s in the s o l i d t e e t h a n d p a r t i c u l a r l y in the core a r e l i n k e d with the t o t a l pole flux • a n d t h e r e f o r e we m a y c a n c e l the q u o t i e n t q~o/qs, in e q u a t i o n (35) for t h e i r p a r t . All t h r e e p a r t s of the c u r r e n t s , in wedges, t e e t h a n d c o r e , are s u b j e c t e d t o a r m a t u r e r e a c t i o n by the f a c t o r o. F i g u r e I oc s h o w s the d i s t r i b u t i o n of field c u r r e n t a n d eddy c u r r e n t s a l o n g the r a d i a l d i r e c t i o n of the s o l i d r o t o r . T h e l a r g e c u r r e n t s in the m e t a l l i c w e d g e s flow a b o v e the f i e l d c u r r e n t s , s u r r o u n d i n g the a r m a t u r e flux only a n d l e a v i n g the r o t o r l e a k a g e flux b e t w e e n field w i n d i n g s a n d w e d g e s b e l o w . This c o r r e s p o n d s e x a c t l y t o o u r a s s u m p t i o n in Fig. 2 a n d
o
:oooo
o
o
= o
o
=
o
o
+o
~.
- -- - - ' - -- -- =--~
, •
a
]
wedges
] '
~cor •
I
"solid steel FIG. IO. D i s t r i b u t i o n of damping eddy currents in the solid m e t a l parts ol cylindrical rotors: (a) Unrolled rotor surface w i t h w i n d i n g , wedges and teeth; (b) Circumferential d i s t r i b u t i o n of exciting and eddy currents; (c) Radial dist r i b u t i o n of exciting and eddy currents.
therefore the w e d g e c u r r e n t s give a l e a k a g e time c o n s t a n t T,. a s in e q u a t i o n ( 3 6 ) . T h e eddy c u r r e n t s in t e e t h a n d c o r e , h o w e v e r , d i f f e r in t h e i r r a d i a l d i s t r i b u t i o n from t h a t of the field c u r r e n t s a n d s u r r o u n d the e n t i r e pole flux. H e n c e they g i v e , b e s i d e s t h e i r p a r t i c i p a t i o n in Td, a n a d d i t i o n a l t i m e c o n s t a n t T~ w i t h no a r m a t u r e r e a c t i o n f a c t o r , due t o the inh e r e n t l e a k a g e of the s o l i d s t e e l d a m p e r c i r c u i t s . A l t h o u g h the s p a t i a l d e v e l o p m e n t of the eddy c u r r e n t s in the s o l i d c r o s s s e c t i o n of the r o t o r is q u i t e f r e e , Fig. IO i n d i c a t e s t h a t the d i s t r i b u t i o n f o l l o w s a line w h i c h m a y be b e t w e e n the n e a r l y s i n u s o i d a l d i s t r i b u t i o n of the a r m a t u r e c u r r e n t s a n d
58
REINHOLD RUDENBERG.
[J. F. I.
the more trapezoidal distribution of the field currents. Hence the difference from both of them is fairly small and thus only very slight higher harmonics of the d a m p e r currents are developed, and the corresponding additional time constants will be insignificant. As to the numerical values of the d a m p e r time constants for machines of the usual design, the copper cross sections of d a m p e r cage winding and field winding, determinative for the middle expression in equation (35), are mostly in the r a t i o 1/5-1/20. Thus the main flux d a m p e r time constant T~ may be of the order of i/IO of Tp, which means a b o u t o.3-1.2 sec. By eddy current formation in the solid steel circuits this v a l u e may be doubled or tripled. The rotor leakage flux, on the o t h e r hand, is mostly in the r a t i o 1/5-1/10 of the main flux. Its time constant Tr therefore, with regard to the last fraction in the middle expression of equation (36), is of the order of o.I see., including a slight increase by eddy-current effects. 7. RAPID TRANSIENT EFFECTS.
The fundamental equations ( I ) and (I3) for the stator circuit represent merely the amplitudes of voltages and currents which vary harmonically with frequency ~0. It is well known, however, that at switching processes in the stator circuit, superposed transients with different variation with time may a p p e a r which prevent any discontinuity in the momentary values of the stator voltages and currents. Such additional currents flow without impressed voltage and suffer a r a p i d decay. They are determined in t h e i r initial values by the instantaneous variations of the amplitudes in our previous time diagrams. With sudden inductive loading or shortcircuiting of the terminals, there a p p e a r direct currents in the stator circuit, decaying with the time constant of this circuit. With capacitive loading or sudden interruption of the circuit, there a p p e a r damped oscillations, the natural frequency of which is given by self-inductance and capacitance of machine and load. A further question in the same direction may be considered, namely, whether the stator equations (I) and (13), containing the amplitudes of sinusoidal voltages, remain sufficiently
.July, 1942.]
SATURATED SYNCHRONOUS MACHINFS.
59
a c c u r a t e although our s o l u t i o n s y i e l d e x p o n e n t i a l o r s a t u r a t e d v a r i a t i o n s of t h e s e a m p l i t u d e s with t i m e . In o r d e r t o p r o v e the a d m i s s i b i l i t y of e q u a t i o n (I), we go b a c k t o the differential e q u a t i o n of the s t a t o r c i r c u i t e = Ri + L di
dt'
(37)
in w h i c h L d e n o t e s a c o n s t a n t t o t a l s e l f - i n d u c t a n c e a n d R the r e s i s t a n c e of the s t a t o r circuit. T h e c u r r e n t i m a y follow a d a m p e d sinusoidal variation l
i = I~ - ~ . ~ i %
(38)
c o n t a i n i n g a n e x p o n e n t i a l l y v a r y i n g a m p l i t u d e , the time cons t a n t r of w h i c h is k e p t a r b i t r a r y in o r d e r t o a p p r o x i m a t e the form of o u r p r e v i o u s s o l u t i o n s . I n s e r t i n g e q u a t i o n (38) in (37) g i v e s the a m p l i t u d e e q u a t i o n
H e n c e the i n d u c t i v e c o m p o n e n t of the i m p e d a n c e is n o t a l t e r e d for a n y v a l u e of r, only the o h m i c c o m p o n e n t is dec r e a s e d by the a c t i o n of a d e c a y i n g c u r r e n t , the d e c r e a s e b e i n g g r e a t e r t h e s m a l l e r the time c o n s t a n t r. So l o n g , h o w e v e r , as r ~-
I
¢0
-
I
377
sec.
=
2.6 5. IO-a s e c . ,
(4o)
the a d d i t i o n a l o h m i c t e r m L / r is n e g l i g i b l y s m a l l c o m p a r e d with the i n d u c t i v e t e r m of e q u a t i o n (39), a s is also the t r u e r e s i s t a n c e t e r m R in all m a c h i n e s of c o n s i d e r a b l e s i z e . S i n c e the s m a l l e s t a c t u a l time c o n s t a n t of the r a p i d t r a n s i e n t v a r i a tion is of the o r d e r of o.I s e c . , i.e., n e a r l y 40 t i m e s the n u m e r i c a l v a l u e in e q u a t i o n (40), we see t h a t e q u a t i o n (I), c o n t a i n i n g i n d u c t i v e c o m p o n e n t s o n l y , is sufficiently a c c u r a t e for the v a r y i n g a m p l i t u d e s as w e l l . I t is well k n o w n t h a t the e x p e r i m e n t a l d e t e r m i n a t i o n of the l e a k a g e of s y n c h r o n o u s m a c h i n e s is a r a t h e r intricate p r o b l e m a n d t h a t the P o t i e r m e t h o d , a n d o t h e r s t e a d y s t a t e m e a s u r e m e n t s a s w e l l , do n o t s e c u r e a c c u r a t e r e § u l t s . T h i s rOE. 234, NO. I399--- 3
60
R E I N H O L D R/.JDENBERG.
[J. F. 1.
is due p a r t l y to saturation effects and p a r t l y to the unknown subdivision into stator and rotor leakage. In the light of the present investigation the sudden interruption of a terminal short circuit presents itself as a useful method for the determination of both stator and rotor leakages. For usual machines, the active periods of the three voltage ascents in Fig. 6b are very different from one another, namely: zero, a b o u t a t e n t h of a second, and several seconds. In consequence, not only can the voltages EA and E ~ be measured with accuracy from an osciIIogram but both parts can be separated easily, giving directly the rated stator leakage and rotor leakage voltages if the machine was previously excited for a short-circuit current equal to rated current. With o t h e r prior currents the effect of saturation on the p a t h s of both leakage fluxes can be measured separately. This complete test can be made, of course, only with machines having material d a m p e r effects, since otherwise the point E a disappears in the oscilIogram and merely the total leakage E ~ can be determined. The large time of ascent of the slow transient voltage, present with open-circuited terminals of the machine, favors decisively the accurate determination of point F~B in contrast to any o t h e r test with shortcircuited terminals which gives a much smaller duration of the slow transient state. Since the machine with rated shortcircuit current operates u n d e r nearly unsaturated condition of the main flux, the slow transient ascent is exponential in the beginning and this fact may be used for an even more exact determination of EB. The r a p i d transient phenomena produced by the rotor leakage d a m p e r time constant Tr have significance only with sudden switching processes in the stator, such as making or breaking a circuit, occurring in times which are small compared with the value of Tr. All the many o t h e r transient procedures in the operation of synchronous machines, however, such as voltage control by a field regulator, or mechanical oscillations of the rotor during disturbances of stability, have periods of fluctuation of the order of seconds. Since this time is very large compared with Yr but not with Y~, such procedures initiate only slow transient phenomena of the main flux, but ,never produce rapid transient disturbances of
July, I942.]
SATURATED ~YNCHRONOUS MACItINES.
(~I
balance of the leakage flux. R a t h e r the rotor leakage flux follows all slower fluctuations without any d e l a y just as the stator leakage flux does, both acting in complete unison, so that no separation of t h e i r effects is necessary. 8. SUMMARY.
The sudden development of short-circuit currents, and of recovery voltages at interruption of currents, depends on the effects of stator and rotor leakages as well as of saturation in the generating synchronous machines. The non-linear problem of the interaction of these features with the performance of the main windings and auxiliary d a m p e r circuits is solved analytically. This leads to a simple graphical representation which is numerically determined by the well-known magnetic and electric characteristics of machine and network and by two time constants of very different magnitudes which are derived from the ordinary data of the machines. Hence the variation with time, of short-circuit currents and of recovery voltages, can easily and rigorously be computed for every given example. The deviations of actual d a m p e r circuits from the ideal form and the effects of eddy currents in solid steel parts of the rotor are discussed so far that numerical results can be derived readily. A new transient test for synchronous machines is suggested by which rotor and stator leakage voltages can be measured separately, avoiding the uncertainties of the usual steady state leakage tests.
SYNCHRONOUS MACHINES. BY R E I N H O L D RODENBERG,
Dr. I n g . ,
Graduate School of Engineering, Harvard University, Cambridge, Mass. 1. INTRODUCTION.
Several important phenomena in the operation of electric power systems are due to the action of the stator and rotor leakage of the feeding synchronous generators and particularly to the linkage of the rotor leakage fluxes with the damper circuits in the poles of the machines. In a previous paper,1 these effects in saturated synchronous machines were treated by a generalized method. In this p a p e r we shall consider the interaction of rotor leakage and damper circuits in greater detail, with full regard to the magnetic saturation of the main flux, and we shall include the reaction of the armature winding on the d a m p e r circuits. This detailed analysis may be justified by the fact that the development of the dangerous short-circuit currents in both magnitude and time, as well as the development of the recovery voltage in circuit breakers, limiting the interruption of short-circuit currents, which constitute two of the dominating problems in electric power engineering, depend preponderantly on these very effects. 2. FUNDAMENTAL CONDITIONS OF THE ELECTRIC AND MAGNETIC CIRCUITS.
We consider the main circuits of a synchronous machine as in Fig. I, and shown in more detail in the cross-section of Fig. 2. Actually, the various magnetic fields overlap and influence one another, but for the sake of simplicity we assume them to be separated into main and leakage fluxes, as is common in the theory of machines. In the stator circuit, the resistance of which we may neglect, 1 R. Rfidenberg, "Saturated Synchronous Machines u n d e r T r a n s i e n t Cond i t i o n s i n the Pole A x i s , " Transactions A.I.E.E., I 9 4 2 , Vol. 61, p. 297, contains a list of references. 39
4 °
REINHOLD RiJDENBERG.
[J. F. I.
the i n t e r n a l e l e c t r o m o t i v e f o r c e E is g i v e n by the t e r m i n a l v o l t a g e V a n d the l e a k a g e v o l t a g e s c a u s e d by the s t a t o r a n d r o t o r leakage fluxes ~ and ~ , E = V + o~No~,. + o~N,~r,
(~
¢0 b e i n g the c o n s t a n t a n g u l a r f r e q u e n c y a n d No the effective n u m b e r of t u r n s of the s t a t o r w i n d i n g . If the t e r m i n a l s a r e l o a d e d by a c u r r e n t I in a c o n s t a n t i m p e d a n c e X , the f i r s t
T
Fro. I.
Equivalent I
0
N
i r l ',~N i 0 01010~0,'0 J,i, ,i I
il
IITI
)lotOLO__.o)olololo
'i
o oX'o 0 o
t,
1111111
--II~llll\l ~.11~Iltlt --~LI UIII '~liilll
FIG. 2.
electric circuits of a synchronous machine.
-"
k : - - -N - z_ i I|i I Ill
Illl
--
!-
---'-
=1, I
I= --
=!: i--
I
--
Linkage of magnetic and electric circuits in the cross-section of a synchronous machine.
t e r m on the r i g h t - h a n d side of e q u a t i o n ([) is X I . We c o n s i d e r p r i m a r i l y the case w h e r e X is a p u r e r e a c t a n c e , so t h a t e q u a t i o n (I) r e m a i n s a l g e b r a i c . Iron l o s s e s m a y be n e g l e c t e d . If the s t a t o r l e a k a g e flux is p r o p o r t i o n a l t o the c u r r e n t , we can e x p r e s s the s e c o n d t e r m on the r i g h t - h a n d side of e q u a t i o n (I) by x , I w h e r e x, is the s t a t o r l e a k a g e r e a c t a n c e . T h e flux of the t h i r d t e r m , h o w e v e r , is l i n k e d , a c c o r d i n g t o Fig. 2, n o t only w i t h the s t a t o r c u r r e n t s but also with the d a m p e r c u r r e n t s a n d t h e r e f o r e this t e r m c a n n o t be e x p r e s s e d so s i m p l y .
July, 1942.]
~ATURATED SYNCHRONOUS
~IACItlNI.;S.
41
T h e excitation circuit of the r o t o r is fed by an e x t e r n a l v o l t a g e e w h i c h is g i v e n e i t h e r a s c o n s t a n t or a s a k n o w n f u n c t i o n of t i m e . W l t h r e s i s t a n c e r, n u m b e r of field t u r n s N, and r o t o r pole flux ~, the e x c i t i n g c u r r e n t i is d e t e r m i n e d by d~
e(t) = ri + N d~'
(2)
the last t e r m g i v i n g the i n f l u e n c e of the c h a n g e of flux a n d b e i n g p r e d o m i n a n t d u r i n g the t r a n s i e n t s t a t e . In the m a i n magnetic c i r c u i t of Fig. 2, the excitation c u r r e n t i p r o d u c e s the m a g n e t i c flux q~ in the pole c o r e , w h i c h r e q u i r e s a m a g n e t i z i n g c u r r e n t ix d e p e n d e n t on • a n d t h u s on the
I /U [ I- A I R ', 0
Fro. 3.
io
Constant magnetic and electric characteristics of a synchronous machine in transient state.
c o r r e s p o n d i n g e l e c t r o m o t i v e f o r c e E a c c o r d i n g t o the m a g n e t i c s a t u r a t i o n c u r v e s h o w n in Fig. 3- In a d d i t i o n , the e x c i t a t i o n c u r r e n t b a l a n c e s the a r m a t u r e r e a c t i o n of the load c u r r e n t I , w h i c h m a y be p u r e l y r e a c t i v e , a n d also the r e a c t i o n of the d a m p e r c u r r e n t id, f l o w i n g in the n u m b e r of t u r n s Ne. H e n c e the e x c i t a t i o n of the m a i n flux is d e t e r m i n e d by the r e l a t i o n
No
Nd
i = i , ( E ) + W X - W i,,.
(3)
In this e q u a t i o n we c o u n t a s p o s i t i v e : a m a g n e t i z i n g d a m p e r c u r r e n t ia, b u t a d e m a g n e t i z i n g i n d u c t i v e s t a t o r c u r r e n t I. T h e d a m p e r circuit a c c o r d i n g t o Fig. 2 is l i n k e d c o m p l e t e l y with the a r m a t u r e flux ¢~, this b e i n g the difference of the r o t o r
42
[J. F. I.
REINHOLD RUDENBERG.
core flux • a n d the r o t o r l e a k a g e flux Or. T h e r e f o r e the c u r r e n t in the s h o r t - c i r c u i t e d d a m p e r with r e s i s t a n c e r~ is g i v e n by :V d o = ~ ~dt (O - Or) + rdi~. (4) In the rotor leakage circuit the flux Or can be c o n s i d e r e d , a s is s h o w n in Fig. 2, a s e x c i t e d by the c o m b i n e d e f f e c t of s t a t o r c u r r e n t a n d d a m p e r c u r r e n t , t h e s e two h a v i n g o p p o s i t e p o s i t i v e d i r e c t i o n . T h e effective r o t o r l e a k a g e flux is t h e r e fore O , - ~0No
~id
,
(5)
if we d e f n e the r o t o r l e a k a g e r e a c t a n c e a t the s t e a d y s t a t e , r e l a t e d t o the s t a t o r c u r r e n t a n d w i t h o u t the i n f l u e n c e of the d a m p e r current, as N~0~o Xr ~ CO
/'0
(6)
T h e r o t o r pole flux • a n d the e . m . f . E are a l w a y s proportional, • O0 - E0' (7) if O0 a n d E0 d e n o t e the r a t e d v a l u e s of flux a n d s t a t o r v o l t a g e in the m a c h i n e a t no l o a d . W i t h r a t e d r o t o r v o l t a g e e0 the l a s t term of e q u a t i o n (2) t h e r e f o r e can be w r i t t e n in e i t h e r of the two f o l l o w i n g f o r m s : dO dE eor, d ( E / E o ) N - ~ = T,. d-~ = dt '
(8)
where T., -
N¢o E,
(9)
is a time c o n s t a n t of the c o m p l e t e m a c h i n e w h i c h is e n t i r e l y i n d e p e n d e n t of the m a g n e t i c s a t u r a t i o n , w h i l e T~
-
N¢o
(IO)
e0
is a time c o n s t a n t of the r o t o r p o l e s with t h e i r field w i n d i n g s
July, 2942.]
SATURATED SYNCHRONOUS MACHINES.
43
only, which is constant so long as the rated rotor values remain unchanged. For machines of the usual design the numerical value of this time constant is of the order of Tp = 3-6-I2 sec. for generated powers of IOO-IOOO-IO,OOO kva. per pole of the synchronous machine. 3, SOLUTION OF T H E CIRCUIT EQUATIONS.
Equations (I) to (5) constitute the fundamental equations of our problem which shall be solved for the transient state and with variable magnetic saturation of the machine. Although it is not absolutely necessary, we assume, for the sake of simplicity, that the important saturation of the main magnetic circuit is concentrated a t the pole cores, where it has maximum effect, and we also assume that the saturation of the magnetic leakage circuits is only low. In the given set of equations we eliminate the subordinate magnitudes, particularly those of linear dependence, and we relate the remaining terms to those parameters which contain the saturation of the main magnetic circuit, particularly to the magnetic no-load characteristic i , ( E ) . If we substitute in equation (2) the flux given by equation (8) and insert in equation (2) the value of the exciting current i of equation (3), we obtain the main equation for the change with time of the e.m.f, of the machine dE T,~ dt - e(t) -- r E i , ( E ) + m I - nidJ,
(1I)
where Na N'
Nd N
n "= - -
(I2)
express the turn ratios of armature and damper windings to the field coils. Equation (II) could be solved if the content of the square bracket were dependent only on the e.m.f. E, as is the no-load magnetizing current i~. By use of equation (5), however, and by expressing voltage and fluxes in equation (I) by reactances and currents, we obtain ( N~ ) E = X I + x d + x r I - ~ i d , (~3)
44
R E I N H O L D R/JDENBERG.
[J. |:. 1.
showing that the rotor leakage drop in the transient state depends upon the value of the damper current. Hence the stator current, expressed by e.m.f, and damper current, is E x~ n. = -I X + x . + x + X + x . + x ~ m ~d,
(I4)
and the last two terms in the bracket of equation (I I) become mI-
mE X+x. nid = X + x , + xr - X + x, + x~ nid.
(15)
Thus we have substituted the e.m.f. E for the stator current on the right-hand side of equation (II), E being the main variable of this differential equation, and we obtain Tm
[ mE dE - e(t) - r | i ~ ( E ) + X + x ~ + x r dt L X + x,~ ni~ ]. J X + x , + x r
(I6)
In order to express also the damper current id by the e.m.f. E, we introduce the rotor leakage flux ~r of equation (5) in equation (4) and obtain d~
N d - ~ + r,,id =
xr
d(
N d ~ t
n.)
I -- -*dm
(17)
"
Substituting equation (15) on the right-hand side of equation (17) and introducing E rather than ¢ on the left-hand side by use of equation (7), gives N
Go d E Koo-d + rdi
=
x~ [ dE/dr LX + + xr n X + x, did ]J - m X + x, + x r d t
"
(IS)
For the first term on the right-hand side of this reiation we can express the fraction before the bracket by equation (6), giving the r a t i o of rotor leakage flux ~r0 to stator current I0 under steady-state condition. On the other hand, the sum of all the reactances in the first denominator within the bracket is given by the quotient of e.m.f. E and current I0 for the steady state. There results, therefore, neglecting the differ.
July, i942.]
SATURATED SYNCHRONOUS ~'IACHINES.
45
ence of E a n d E0, w h i c h is i n s i g n i f i c a n t h e r e , x~ ~NaX
dE~dr + xs + xr
~ro Io
Io Eo
dE dt
(I9)
H o w e v e r , we c a n c o m b i n e this t e r m with the f i r s t term on the l e f t - h a n d side of e q u a t i o n ( I 8 ) , o b t a i n i n g • o
Nd
-
¢ro
Eo
dE a~a dE dt - Xd E~o d t '
(20)
w h e r e ~ is the a r m a t u r e flux a s d i f f e r e n c e of pole a n d r o t o r l e a k a g e flux in the s t e a d y s t a t e . T h e l a s t term on the r i g h t h a n d side of e q u a t i o n (I8) can be w r i t t e n xS
Ne
X + x. +
x.
+
xr
di~ = L ~ - - dt dt
(21) '
'
so t h a t the e n t i r e e q u a t i o n (I8) for the d a m p e r c u r r e n t m a y be e x p r e s s e d a s ¢a dE did N~ Eoo-dt + raid + Lr dt - o.
(22)
W e see t h e r e f o r e t h a t L~ is the s e l f - i n d u c t a n c e of the d a m p e r w i n d i n g with the r o t o r l e a k a g e flux. I t is e x p r e s s e d in e q u a t i o n (2I) by the r o t o r l e a k a g e i n d u c t a n c e x./w v i e w e d from the a r m a t u r e , r e d u c e d by the s q u a r e of the t u r n r a t i o of d a m p e r a n d a r m a t u r e c i r c u i t s , a n d m u l t i p l i e d by a d i m i n u tion f a c t o r P
X+x. X + x , + x r '
(23)
w h i c h e x p r e s s e s the e f f e c t of the a r m a t u r e r e a c t i o n on the d a m p e r circuit. Our p r o b l e m now c o n s i s t s in s o l v i n g s i m u l t a n e o u s l y the differential e q u a t i o n s (I6) a n d (22), both c o n t a i n i n g as dep e n d e n t v a r i a b l e s E a n d i~ o n l y , b u t with a n o n - l i n e a r t e r m i , ( E ) c o n s t i t u t i n g the m a g n e t i c c h a r a c t e r i s t i c of the m a c h i n e . In o r d e r t o s o l v e t h e s e e q u a t i o n s by a s i m p l e g r a p h i c a l m e t h o d , we n o t e t h a t in the m a i n e q u a t i o n (I6) the d a m p e r c u r r e n t L1 p l a y s m e r e l y the p a r t of a c o r r e c t i o n term for the
46
R E I N H O L D Rf)DENBERG.
[J. F. I.
variation of the main flux and e.m.f, with time. It may be sufficient therefore if we satisfy equation (22) by a good approximation for the general variation of ie and introduce that damper current into the main equation (16). Thus the assumption may be made, derived from the form of equation (22), that the d a m p e r current changes nearly exponentially with time, so that did
dt
ia -
(24)
T '
still allowing for a slow variation of the time constant T, the magnitude of which is left undetermined. By introducing equation (24) in equation (22) and denoting the time constant of the damper with the rotor leakage flux Lr Tr = - , (25) r4 we obtain, with use of equation (9),
Nda~aTdE ( Tr) N ' b o " dt + r d I -- T id = o,
(26)
by which id is determined as a function of d E / d t . Inserting the damper current id in the last term of equation (16), this term becomes
rpnid = p
Nd2cba r Tm d E ~ d r rd N2¢o I -- T r / T
T~ T,. d E / d t Tp I -- T~/T
(27)
The central expression is so written that we see immediately that the first two fractions give the r a t i o of two time constants. These are: a d a m p e r time constant Td, determined by resistance and number of turns of the d a m p e r circuit and by the armature flux linked with this circuit and therefore giving a value different from Tr in equation (25), and the main pole time constant Tp, determined by the corresponding constants of the excitation winding and already given by equation (Io). By combining the last term of equation (I6), g i v e n by equation (27), with the left-hand side of equation
July, i942.1
SATURATED SYNCHRONOUS MACHINES.
47
( I 6 ) , we now o b t a i n
v., i+i2
T~/Tp
g/r = e(t) -- r [ i~(E) q - X q _ mx ,Eq _ x , ] . ( 2 8 )
H e r e i n t h e r i g h t - h a n d side c o n t a i n s the q u o t i e n t of e.m.f, a n d t o t a l r e a c t a n c e of the c i r c u i t c o n s t i t u t i n g the s t e a d y s t a t e n e t w o r k c h a r a c t e r i s t i c , as s h o w n in Fig. 3. If we now i n t r o d u c e on the l e f t - h a n d side the r o t o r pole time c o n s t a n t T~ of e q u a t i o n 0 o ) , r a t h e r t h a n the m a c h i n e time c o n s t a n t T ~ of e q u a t i o n (9), we o b t a i n the differential e q u a t i o n for the e . m . f . of a s a t u r a t e d m a c h i n e with d a m p e r c i r c u i t s , in the f i n a l form
[
T~
]d(E/Eo)
Tp + I -- ~I'r/T
e(t) =
dt
_ _
eo
~i __
_ _ .
~o
( 2 9 )
T h e last t e r m Z i on the r i g h t - h a n d side d e n o t e s the sum of the i n t e r n a l a n d e x t e r n a l s t e a d y - s t a t e m a g n e t i z i n g c u r r e n t u n d e r t h e e . m . f . E , this sum b e i n g a l w a y s a g i v e n f u n c t i o n of E , see Fig. 3. T h e e x p r e s s i o n in the b r a c k e t on the l e f t - h a n d side of e q u a t i o n (29) is a r e s u l t a n t time c o n s t a n t a c c o r d i n g t o w h i c h the e . m . f . E , a n d s i m u l t a n e o u s l y the d a m p e r c u r r e n t id, c a n v a r y . H e n c e we h a v e the c o n d i t i o n a l e q u a t i o n
T = Tp -t-
Td I
--
T~/T'
(30)
w h i c h c a n be w r i t t e n a s a q u a d r a t i c e q u a t i o n
T2 -- T(Tp + rid + Tr) + TpT~ = o.
(3 I)
T h u s two p o s s i b l e d i f f e r e n t time c o n s t a n t s a p p e a r , c o r r e s p o n d i n g t o the two r o t o r c i r c u i t s . W i t h m o d e r a t e r o t o r l e a k a g e , a s in a c t u a l m a c h i n e s , the r o t o r l e a k a g e t i m e c o n s t a n t T~ is s m a l l c o m p a r e d with the o t h e r time c o n s t a n t s . T h e n the two s o l u t i o n s of e q u a t i o n (31 ) are in v e r y good a p p r o x i m a t i o n T t = T~ + Td;
T, T~ + T~ ~ T~,
T~ = I+
T.
(32)
48
R E I N H O L D R/JDENBERG.
[J- t;. I.
where the last term on the right-hand side is a further approximation for relatively small Ta. The large main flux time constant T1 is given by the sum of the time constants of field winding and d a m p e r circuit, both related to the main fluxes in armature and poles. The small leakage flux time constant T~, on the o t h e r hand, is given chiefly by the rotor leakage flux from pole to pole closing itself through the d a m p e r circuit. 4. G ~ P m C A L ~Pm~SS~TAZION OF THE SOLVTION. W e see n o w that our differential equation (29) has two solutions, a forced solution belonging to the main flux in the pole core, and a free solution belonging to the rotor leakage flux b e t w e e n the poles. T h e forced solution is given by
TI d ( E / E o ) dt
-
e(t) eo
i ~ ( E ) + rail io
=
Ai, - ~o
(33)
and determines by the large v a l u e of T1 a slow transient
~ J ~ ~ - - - - ~i~(E)
E ® T ~ I.--~->
0
• Ir
FIG. 4:" Graphical representation of the slow and the rapid transient
changes of state.
change with time of the e.m.f. E and of the corresponding main pole flux. This flux is excited by an ohmic current elf determined by the excitation voltage e(t), from which the magnetization current i, for the main circuit and the steady state armature reaction current rnI, both corresponding to E, are to be deducted. For constant excitation voltage e, the right-hand side of Fig. 4 gives a graphical representation of
July, 1942.]
~ATURATED SYNCHRONOUS MACHINES.
49
the change of e.m.f. E with time w h i c h is for e v e r y E proportional to the difference current Ail, closely cross-hatched in Fig. 4. The free solution is given by the remaining differential equation 7"2 d ( E / E o ) dt
-
mI2 io
-
A,i,,. io
(34)
and determines by the small value of T2 a r a p i d transient change with time of the rotor leakage flux. This flux is excited by the d a m p e r current but is not influenced either by the excitation voltage or by the iron magnetizing current i~. The change of this leakage flux and its currents is represented graphically by the left-hand side of Fig. 4, which is widely cross-hatched and shows as exciting current only that of the rotor leakage reactance xr. With no saturation in the rotor leakage circuit, the leakage flux and the corresponding part of the d a m p e r current decay exponentially, since ki2 is then proportional to E. With saturation of the leakage flux we can a p p l y equation (34) graphically in the same way as described for equation (33). The total stator current is always given as the sum of the forced and free solutions, i.e., I~ + I2, as already indicated in equation (I4). While in Fig. 4 the forced Ail decreases towards zero at the intersection of the saturation and the network characteristics, yielding a steady state voltage E~ and a sustained current I , in the stator, the free ki2 decays towards zero at the zero line, leaving no current or voltage in the d a m p e r winding. The rate of decrease is slow for the forced current Ai~ with the large time constant T1 but r a p i d for the free current Ai2 with the small time constant T> Hence Ai2 will have nearly vanished when ki~ begins to move. In Fig. 5 the development of the short-circuit current with time is shown when the terminals of a saturated generator are suddenly s h o r t circuited and the excitation voltage is kept constant, the generator previously having been w i t h o u t load. The magnetic no-load characteristic of the machine, i,(E), in conjunction with the stator leakage characteristic x . J and the
5°
R E I N H O L D RUDENBERG.
[J. F. I.
t o t a l l e a k a g e c h a r a c t e r i s t i c (x, q- xr)I d e t e r m i n e s c o m p l e t e l y the t r a n s i e n t s t a t e , i n c l u d i n g the i n i t i a l c o n d i t i o n s . T h e c u r r e n t of u n b a l a n c e Ail as d e f i n e d by e q u a t i o n s (28) a n d (33) is g i v e n by the h o r i z o n t a l w i d t h of the c l o s e l y c r o s s h a t c h e d area of Fig. 5a a s the difference b e t w e e n the p r e v i o u s e x c i t a t i o n c u r r e n t il = e/r a n d the sum of the n o - l o a d m a g n e t i z i n g c u r r e n t i~(E) a n d the s t e a d y - s t a t e s h o r t - c i r c u i t c u r r e n t I1 = E / ( x , Jr xr), the l a t t e r b e i n g p l o t t e d in the r o t o r s c a l e b a c k w a r d s from i l a s a s t r a i g h t l i n e . S i n c e /~il is dep e n d e n t only on E , the c u r v e E(t) c a n be d r a w n as in Fig. 5b, e i t h e r in s t e p - b y - s t e p c o n s t r u c t i o n w i t h the a l w a y s k n o w n v a l u e of dE/dt or by a q u a d r a t u r e y i e l d i n g t a s a f u n c t i o n of E
o ,..
¢
b
! ,]E
O
k
iI
I
0
FIG. 5. Development of slow and rapid transient currents at sudden short circuit of the terminals: (a) Diagram of characteristics; (b) Change with time.
a f t e r s e p a r a t i o n of the v a r i a b l e s in e q u a t i o n ( 3 3 ) . T o e v e r y E the c u r r e n t 11 can be c o r r e l a t e d by m e a n s of the t o t a l l e a k age characteristic. T h i s slow t r a n s i e n t s t a t o r s h o r t - c i r c u i t c u r r e n t c o r r e s p o n d s t o the slow v a r i a t i o n of the m a i n pole flux w i t h t h e s a t u r a t e d time c o n s t a n t T1. In a d d i t i o n t o t h i s , a r a p i d l y v a r y i n g flux is p r o d u c e d by the c u r r e n t axis s h o w n a s w i d e l y c r o s s - h a t c h e d b e t w e e n the two l e a k a g e c h a r a c t e r i s t i c s in Fig. 5a a n d a c t i n g on the r o t o r l e a k a g e p a t h o n l y . T h i s d a m p e r c u r r e n t is b a l a n c e d by a n a d d i t i o n a l s t a t o r c u r r e n t Is, both d e c a y i n g r a p i d l y with a n e x p o n e n t i a l t i m e c o n s t a n t T2 a c c o r d i n g t o e q u a t i o n ( 3 4 ) . I n s t e a d of a s t e p - b y - s t e p d e c r e a s e of axis down t o z e r o , or of a q u a d r a t u r e of e q u a t i o n (34), we m a y t u r n the s t a t o r l e a k a g e
July, 1942.]
SATURATED SYNCHRONOUS MACHINES.
5I
characteristic x, towards the complete leakage characteristic x~ -t- xr with a velocity corresponding to the time constant Y2. Then any intermediate position of this line always determines the left-hand end up to which the total stator current I t -t- I2 has to be measured. Fig. 5b shows the rapidly decaying additional current I s which forms a large initial increase of the short-circuit current. While the r a p i d transient current changes exponentially with time, so long as no saturation prevails in the leakage paths, the slow transient current deviates widely from an exponential variation due to the effect of saturation in the main magnetic circuit. In Fig. 6 the sudden interruption of a terminal short circuit of a saturated generator is shown, the excitation voltage IE . _ , , ~ - - E . ~ . . . . . . . . . .= - ~ - ~ - -
INN 0
f il
Q
0
t b
FIG. 6. Development of slow and rapid transient recovery voltages at sudden interruption of a s h o r t circuit a t the terminals: (a) Diagram of characteristics; (b) Change w i t h t i m e .
a g a i n being kept constant. Here, according to Fig. 6a, both leakage characteristics determine the previous steady-state condition of the fluxes by intersection with the no-load characteristic. The characteristic x~ gives the stator main flux, the characteristic x , H-Xr gives the rotor pole flux. At the instant of interruption of the short circuit, the rotor pole flux is kept constant by the field winding but the stator flux is also kept constant by the additional action of the d a m p e r winding. Thus the e.m.f., corresponding to the rotor pole flux, starts with the value E B and increases slowly according to the current of unbalance Air which lies between the characteristic i~,(E) and the vertical line through il, the l a t t e r constituting the open terminal characteristic. The stator voltage, on the o t h e r hand, j u m p s at the instant of interruption from zero to the value Ea, equal to the
52
R E I N H O L D RU'DENBERG.
[J. F. I.
previous stator leakage voltage, and increases rapidly towards the higher value of the e.m.f. E, as in Fig. 6b, due to the decaying rotor leakage flux. Thus with interruption of the circuit the voltage at the terminals of the generator increases instantaneously due to the stator leakage; it then increases rapidly due to the rotor leakage, and it further increases slowly due to the main flux in the machine, finally attaining the no-load voltage. The r a p i d transient voltage is nearly exponential with the time constant T2; the slow transient voltage is distorted by saturation and is to be derived by the shaded area in Fig. 6a with the time constant T1. It may be noted that the scales of the characteristics and times in Figs. 5 and 6 are chosen the same so that the rates of change of the voltages are directly comparable. 5. M A G ~ P , T I C T I ~ - C O N S T A N T S .
The development of many of the transient conditions which may occur in the operation of saturated synchronous machines has been shown in a previous paper,1 in which, in analogy to non-saturated circuits, there was assumed a wellknown quadratic equation for the time constants. In the present p a p e r a more detailed analysis of the magnetic linkages has led us to a similar quadratic equation (3~), except that we have now arrived at relations determining numerically all the time constants, and the damper time constants a p p e a r here separated into the magnitudes Te, belonging to the main armature flux, and TT, belonging to the rotor leakage flux. Furthermore, we have shown here by equation (23) to what extent the d a m p e r time constants are influenced by armature reaction. The damper time constant of the main flux is, according to equation (27),
Td = p T , ( - ~ ) 2r ~a Nd~, r,~ c~o - o reie
(35)
The last expression on the right-hand side is obtained by substituting equation (IO) for Tp and introducing as the exciting transient the d a m p e r current rather than the field current, these two being in the r a t i o of their turn numbers.
July, 1942.]
SATURATED SYNCHRONOUS MACHINES.
53
F o r n u m e r i c a l c o m p u t a t i o n s the c e n t r a l e x p r e s s i o n of e q u a t i o n (35) is c o n v e n i e n t a l s o . I t s h o w s t h a t this d a m p e r time c o n s t a n t Td is only a f r a c t i o n of the m a i n flux t i m e c o n s t a n t Tp, s i n c e the d a m p e r c r o s s section, d e t e r m i n e d by Nd2/rd, is a l w a y s m u c h s m a l l e r t h a n the field coil c r o s s section, d e t e r m i n e d by N2/r. F u r t h e r m o r e , the d a m p e r time c o n s t a n t is r e d u c e d by the f a c t o r p r e p r e s e n t i n g by e q u a t i o n (23) the i n f l u e n c e of a r m a t u r e r e a c t i o n . S i m i l a r l y , we o b t a i n for the d a m p e r t i m e c o n s t a n t of the r o t o r l e a k a g e f l u x , a c c o r d i n g t o e q u a t i o n s (25) a n d (21), a n d u s i n g e q u a t i o n s (6) a n d ( I O ) ,
Tr = pTp
(N~)~r~o Nd¢~____2o -~ r--~ ¢--~ = p rdid '
(36)
w h i c h a g a i n is a s m a l l f r a c t i o n of T~t, in v i e w of the a d d i t i o n a l e f f e c t of the r a t i o of the r o t o r l e a k a g e flux Cr0 t o the m a i n pole flux q0. By e q u a t i o n s (IO), (35), (36), we c a n c o m p u t e r e a d i l y the r e s u l t a n t time c o n s t a n t s of e q u a t i o n (32) for a n y g i v e n m a c h i n e . W e h a v e t o n o t e , h o w e v e r , t h a t due t o the f a c t o r p both d a m p e r time c o n s t a n t s are d e p e n d e n t on the l o a d , w h i c h in e q u a t i o n (23) is e x p r e s s e d in t e r m s of the e x t e r n a l r e a c t a n c e X. A t no load we h a v e X = oo and t h e r e f o r e p = I, i n d i c a t i n g no a r m a t u r e r e a c t i o n a n d full v a l u e of the d a m p e r t i m e c o n s t a n t s . A t t e r m i n a l s h o r t circuit, we h a v e X = o, a n d t h u s , with e q u a l r o t o r a n d s t a t o r r e a c t a n c e s , for e x a m p l e , p = 1. This s h o w s t h a t the a d d i t i o n a l a r m a t u r e c u r r e n t s , d e v e l o p e d s i m u l t a n e o u s l y with the d a m p e r c u r r e n t s , r e d u c e the v a l u e of both d a m p e r time c o n s t a n t s with inc r e a s i n g load t o w a r d s a b o u t o n e - h a l f a t s h o r t circuit. Howe v e r , a t a n y a c t u a l load this e f f e c t is slight, s i n c e X then outw e i g h s the l e a k a g e r e a c t a n c e s in e q u a t i o n (23). 6. VARIOUS DAMPER CIRCUITS.
A c t u a l d a m p e r c o n s t r u c t i o n s of s y n c h r o n o u s m a c h i n e s d i f f e r t o s o m e e x t e n t from o u r s i m p l i f y i n g a s s u m p t i o n s . In p a r t i c u l a r , the r o t o r l e a k a g e field is n o t the only flux l i n k e d with the d a m p e r c i r c u i t . Fig. 7 s h o w s t h a t a f u r t h e r flux 4)~ m a y s u r r o u n d the d a m p e r c o n d u c t o r s , c l o s i n g i t s e l f p a r t i a l l y t h r o u g h the d a m p e r s l o t s , p a r t i a l l y a r o u n d the end r i n g s of
54
R E I N H O L D R/)DENBERG.
[J. F. I.
the damper. This additional flux is produced by the damper currents id only, but not by the stator currents I . If we include this additional leakage flux in the fundamental conditions, equations (I), (2), (3), (5) remain unchanged. In equation (4), however, we have to add on the right-hand side a small term L~did/dt, giving the induced voltage by the independent d a m p e r leakage inductance L~. Correspondingly we have to add to the time constant Tr, related to the rotor leakage flux by equation (36), a small time constant Te due to this inherent d a m p e r leakage flux. This magnitude is not influenced by armature reaction and therefore does not contain the factor p. The time constant Td in equation (35) remains
IliJlf
ollJll
• I I I I J ~ l l l J l l I
Illi/Jl
Fro. 7.
il
liJ
rl
III ITI iililll[ lilfffli Ilfiili
PI il li II
T h e magnetic fluxes i n space linked w i t h a d a m p e r w i n d i n g i n the pole shoes of salient p o l e s .
unchanged since it is determined by the main armature flux only. Actual dampers, as shown in Fig. 8, are b u i l t for the most part in the form of squirrel cages, with spatially distributed damper bars. Such circuits are very closely linked with the distributed armature currents but much less closely with the lumped field winding. Hereby, a further leakage between d a m p e r and field winding occurs, increasing the independent time constant T~. On the o t h e r hand, the distributed currents of squirrel-cage dampers are not completely linked with the main flux. Hence the d a m p e r time constant T~ becomes somewhat smaller than in equation (35). This can easily be
July, 1942.]
SATURATED SYNCHRONOUS MACHINES.
55
t a k e n i n t o a c c o u n t by i n c l u d i n g in the n u m b e r of t u r n s Nd of the d a m p e r a n a d d i t i o n a l l i n k a g e f a c t o r a c c o r d i n g t o f a m i l i a r rules. S i n c e the a c t u a l d i s t r i b u t i o n of the d a m p e r c u r r e n t s over the p a r a l l e l d a m p e r b a r s of a s q u i r r e l c a g e is free, the e n t i r e d a m p e r flux c o n t a i n s in s p a c e f u n d a m e n t a l as well as h a r m o n i c f i e l d s . T h e r e f o r e , t h e r e m a y a p p e a r here a w h o l e s e r i e s of time c o n s t a n t s r a t h e r t h a n the two d e r i v e d for l u m p e d c i r c u i t s . H o w e v e r , the a m p l i t u d e s of the h a r m o n i c fields, v o l t a g e s a n d c u r r e n t s will be s m a l l , s i n c e they are d e t e r m i n e d m e r e l y by difference effects m o d e r a t e in m a g n i t u d e . E v e n if t h e r e is no a c t u a l d a m p e r w i n d i n g , but the pole c o r e s or pole s h o e s are m a d e of s o l i d steel, a s in Fig. 9, eddy c u r r e n t s can d e v e l o p in s u c h s o l i d p a r t s a c t i n g q u a l i t a t i v e l y O?
0 0 0 0 0 0 0
I
FIG. 8.
Squirrel-cage d a m p e r w i n d i n g i n contrast t o the l u m p e d circuit of Fig. 7.
a s d a m p e r c u r r e n t s . Q u a n t i t a t i v e l y t h e i r time c o n s t a n t s Te a n d T r are s o m e t i m e s s m a l l e r b e c a u s e of the h i g h e l e c t r i c r e s i s t a n c e of the s t e e l . T h e y c a n be c o m p u t e d ~ r e a d i l y for a n y c r o s s s e c t i o n of the s t e e l p o l e s a n d m a y c o n t r i b u t e part i c u l a r l y t o Te. In Fig. 9 the s p a t i a l d i s t r i b u t i o n of t h e s e d a m p i n g eddy c u r r e n t s is s h o w n in c o m p a r i s o n t o the d i s t r i b u tion of the f i e l d c u r r e n t s . S i n c e the d i s t r i b u t i o n s d i f f e r from each o t h e r , r e s i d u a l l e a k a g e is c a u s e d a n d here also a s m a l l i n d e p e n d e n t time c o n s t a n t T~ is d e v e l o p e d in a d d i t i o n t o 7", A s o l i d r o t o r y o k e h a s the s a m e e f f e c t in p r i n c i p l e ; h o w e v e r , the l i n k a g e with both s t a t o r a n d r o t o r w i n d i n g s is very s m a l l a n d t h e r e f o r e the i n d e p e n d e n t l e a k a g e is l a r g e , w h i l e the c o n t r i b u t i o n t o Tr v a n i s h e s . 2 R. Riidenberg, "Elektrlsche Schaltvorg~inge" (a book), 3rd e d i t i o n , Berlin, x933, p. 7o.
56
REINHOLD RUDENBERG.
[J. F. I.
If a copper d a m p e r and solid steel parts are acting together, as in many actual machines, the effects of both merely add, increasing all the main and leakage time constants. In cylindrical rotors for turbo-generators, as shown unrolled in Fig. Ioa, the linkage between field currents and eddy currents in the solid steel parts is relatively large. In Fig. Iob the distribution of the field currents and of the damping eddy currents along the circumferential direction is shown for both the currents in the solid iron and in the b e t t e r conducting
I I I 1
I
I I Ii~~
llPOleshoe ,
I
I
I
I polecore
I
i ~IHIIJ
-JBHHJ
N FIG. 9.
Distribution of damping eddy currents in the solid steel parts of salient poles.
metal wedges closing the slots. We see that the distribution of the entire d a m p e r currents iu is similar to that of the field currents i. Hence t h e i r linkage with the main flux is nearly perfect and a relatively large main flux d a m p e r time constant T~ is developed by the large steel cross section, according to equation (35), while the additional leakage and therefore T~ are only slight. Since the currents in the metal wedges are completely linked with the armature flux, as is the case with an ideal damper, the quotient ~a/~0 remains in equation (35) for this part of the d a m p e r circuit. However, the eddy
J u l y , 1942.]
SATURATED
SYNCHRONOUS
MACHINES.
57
c u r r e n t s in the s o l i d t e e t h a n d p a r t i c u l a r l y in the core a r e l i n k e d with the t o t a l pole flux • a n d t h e r e f o r e we m a y c a n c e l the q u o t i e n t q~o/qs, in e q u a t i o n (35) for t h e i r p a r t . All t h r e e p a r t s of the c u r r e n t s , in wedges, t e e t h a n d c o r e , are s u b j e c t e d t o a r m a t u r e r e a c t i o n by the f a c t o r o. F i g u r e I oc s h o w s the d i s t r i b u t i o n of field c u r r e n t a n d eddy c u r r e n t s a l o n g the r a d i a l d i r e c t i o n of the s o l i d r o t o r . T h e l a r g e c u r r e n t s in the m e t a l l i c w e d g e s flow a b o v e the f i e l d c u r r e n t s , s u r r o u n d i n g the a r m a t u r e flux only a n d l e a v i n g the r o t o r l e a k a g e flux b e t w e e n field w i n d i n g s a n d w e d g e s b e l o w . This c o r r e s p o n d s e x a c t l y t o o u r a s s u m p t i o n in Fig. 2 a n d
o
:oooo
o
o
= o
o
=
o
o
+o
~.
- -- - - ' - -- -- =--~
, •
a
]
wedges
] '
~cor •
I
"solid steel FIG. IO. D i s t r i b u t i o n of damping eddy currents in the solid m e t a l parts ol cylindrical rotors: (a) Unrolled rotor surface w i t h w i n d i n g , wedges and teeth; (b) Circumferential d i s t r i b u t i o n of exciting and eddy currents; (c) Radial dist r i b u t i o n of exciting and eddy currents.
therefore the w e d g e c u r r e n t s give a l e a k a g e time c o n s t a n t T,. a s in e q u a t i o n ( 3 6 ) . T h e eddy c u r r e n t s in t e e t h a n d c o r e , h o w e v e r , d i f f e r in t h e i r r a d i a l d i s t r i b u t i o n from t h a t of the field c u r r e n t s a n d s u r r o u n d the e n t i r e pole flux. H e n c e they g i v e , b e s i d e s t h e i r p a r t i c i p a t i o n in Td, a n a d d i t i o n a l t i m e c o n s t a n t T~ w i t h no a r m a t u r e r e a c t i o n f a c t o r , due t o the inh e r e n t l e a k a g e of the s o l i d s t e e l d a m p e r c i r c u i t s . A l t h o u g h the s p a t i a l d e v e l o p m e n t of the eddy c u r r e n t s in the s o l i d c r o s s s e c t i o n of the r o t o r is q u i t e f r e e , Fig. IO i n d i c a t e s t h a t the d i s t r i b u t i o n f o l l o w s a line w h i c h m a y be b e t w e e n the n e a r l y s i n u s o i d a l d i s t r i b u t i o n of the a r m a t u r e c u r r e n t s a n d
58
REINHOLD RUDENBERG.
[J. F. I.
the more trapezoidal distribution of the field currents. Hence the difference from both of them is fairly small and thus only very slight higher harmonics of the d a m p e r currents are developed, and the corresponding additional time constants will be insignificant. As to the numerical values of the d a m p e r time constants for machines of the usual design, the copper cross sections of d a m p e r cage winding and field winding, determinative for the middle expression in equation (35), are mostly in the r a t i o 1/5-1/20. Thus the main flux d a m p e r time constant T~ may be of the order of i/IO of Tp, which means a b o u t o.3-1.2 sec. By eddy current formation in the solid steel circuits this v a l u e may be doubled or tripled. The rotor leakage flux, on the o t h e r hand, is mostly in the r a t i o 1/5-1/10 of the main flux. Its time constant Tr therefore, with regard to the last fraction in the middle expression of equation (36), is of the order of o.I see., including a slight increase by eddy-current effects. 7. RAPID TRANSIENT EFFECTS.
The fundamental equations ( I ) and (I3) for the stator circuit represent merely the amplitudes of voltages and currents which vary harmonically with frequency ~0. It is well known, however, that at switching processes in the stator circuit, superposed transients with different variation with time may a p p e a r which prevent any discontinuity in the momentary values of the stator voltages and currents. Such additional currents flow without impressed voltage and suffer a r a p i d decay. They are determined in t h e i r initial values by the instantaneous variations of the amplitudes in our previous time diagrams. With sudden inductive loading or shortcircuiting of the terminals, there a p p e a r direct currents in the stator circuit, decaying with the time constant of this circuit. With capacitive loading or sudden interruption of the circuit, there a p p e a r damped oscillations, the natural frequency of which is given by self-inductance and capacitance of machine and load. A further question in the same direction may be considered, namely, whether the stator equations (I) and (13), containing the amplitudes of sinusoidal voltages, remain sufficiently
.July, 1942.]
SATURATED SYNCHRONOUS MACHINFS.
59
a c c u r a t e although our s o l u t i o n s y i e l d e x p o n e n t i a l o r s a t u r a t e d v a r i a t i o n s of t h e s e a m p l i t u d e s with t i m e . In o r d e r t o p r o v e the a d m i s s i b i l i t y of e q u a t i o n (I), we go b a c k t o the differential e q u a t i o n of the s t a t o r c i r c u i t e = Ri + L di
dt'
(37)
in w h i c h L d e n o t e s a c o n s t a n t t o t a l s e l f - i n d u c t a n c e a n d R the r e s i s t a n c e of the s t a t o r circuit. T h e c u r r e n t i m a y follow a d a m p e d sinusoidal variation l
i = I~ - ~ . ~ i %
(38)
c o n t a i n i n g a n e x p o n e n t i a l l y v a r y i n g a m p l i t u d e , the time cons t a n t r of w h i c h is k e p t a r b i t r a r y in o r d e r t o a p p r o x i m a t e the form of o u r p r e v i o u s s o l u t i o n s . I n s e r t i n g e q u a t i o n (38) in (37) g i v e s the a m p l i t u d e e q u a t i o n
H e n c e the i n d u c t i v e c o m p o n e n t of the i m p e d a n c e is n o t a l t e r e d for a n y v a l u e of r, only the o h m i c c o m p o n e n t is dec r e a s e d by the a c t i o n of a d e c a y i n g c u r r e n t , the d e c r e a s e b e i n g g r e a t e r t h e s m a l l e r the time c o n s t a n t r. So l o n g , h o w e v e r , as r ~-
I
¢0
-
I
377
sec.
=
2.6 5. IO-a s e c . ,
(4o)
the a d d i t i o n a l o h m i c t e r m L / r is n e g l i g i b l y s m a l l c o m p a r e d with the i n d u c t i v e t e r m of e q u a t i o n (39), a s is also the t r u e r e s i s t a n c e t e r m R in all m a c h i n e s of c o n s i d e r a b l e s i z e . S i n c e the s m a l l e s t a c t u a l time c o n s t a n t of the r a p i d t r a n s i e n t v a r i a tion is of the o r d e r of o.I s e c . , i.e., n e a r l y 40 t i m e s the n u m e r i c a l v a l u e in e q u a t i o n (40), we see t h a t e q u a t i o n (I), c o n t a i n i n g i n d u c t i v e c o m p o n e n t s o n l y , is sufficiently a c c u r a t e for the v a r y i n g a m p l i t u d e s as w e l l . I t is well k n o w n t h a t the e x p e r i m e n t a l d e t e r m i n a t i o n of the l e a k a g e of s y n c h r o n o u s m a c h i n e s is a r a t h e r intricate p r o b l e m a n d t h a t the P o t i e r m e t h o d , a n d o t h e r s t e a d y s t a t e m e a s u r e m e n t s a s w e l l , do n o t s e c u r e a c c u r a t e r e § u l t s . T h i s rOE. 234, NO. I399--- 3
60
R E I N H O L D R/.JDENBERG.
[J. F. 1.
is due p a r t l y to saturation effects and p a r t l y to the unknown subdivision into stator and rotor leakage. In the light of the present investigation the sudden interruption of a terminal short circuit presents itself as a useful method for the determination of both stator and rotor leakages. For usual machines, the active periods of the three voltage ascents in Fig. 6b are very different from one another, namely: zero, a b o u t a t e n t h of a second, and several seconds. In consequence, not only can the voltages EA and E ~ be measured with accuracy from an osciIIogram but both parts can be separated easily, giving directly the rated stator leakage and rotor leakage voltages if the machine was previously excited for a short-circuit current equal to rated current. With o t h e r prior currents the effect of saturation on the p a t h s of both leakage fluxes can be measured separately. This complete test can be made, of course, only with machines having material d a m p e r effects, since otherwise the point E a disappears in the oscilIogram and merely the total leakage E ~ can be determined. The large time of ascent of the slow transient voltage, present with open-circuited terminals of the machine, favors decisively the accurate determination of point F~B in contrast to any o t h e r test with shortcircuited terminals which gives a much smaller duration of the slow transient state. Since the machine with rated shortcircuit current operates u n d e r nearly unsaturated condition of the main flux, the slow transient ascent is exponential in the beginning and this fact may be used for an even more exact determination of EB. The r a p i d transient phenomena produced by the rotor leakage d a m p e r time constant Tr have significance only with sudden switching processes in the stator, such as making or breaking a circuit, occurring in times which are small compared with the value of Tr. All the many o t h e r transient procedures in the operation of synchronous machines, however, such as voltage control by a field regulator, or mechanical oscillations of the rotor during disturbances of stability, have periods of fluctuation of the order of seconds. Since this time is very large compared with Yr but not with Y~, such procedures initiate only slow transient phenomena of the main flux, but ,never produce rapid transient disturbances of
July, I942.]
SATURATED ~YNCHRONOUS MACItINES.
(~I
balance of the leakage flux. R a t h e r the rotor leakage flux follows all slower fluctuations without any d e l a y just as the stator leakage flux does, both acting in complete unison, so that no separation of t h e i r effects is necessary. 8. SUMMARY.
The sudden development of short-circuit currents, and of recovery voltages at interruption of currents, depends on the effects of stator and rotor leakages as well as of saturation in the generating synchronous machines. The non-linear problem of the interaction of these features with the performance of the main windings and auxiliary d a m p e r circuits is solved analytically. This leads to a simple graphical representation which is numerically determined by the well-known magnetic and electric characteristics of machine and network and by two time constants of very different magnitudes which are derived from the ordinary data of the machines. Hence the variation with time, of short-circuit currents and of recovery voltages, can easily and rigorously be computed for every given example. The deviations of actual d a m p e r circuits from the ideal form and the effects of eddy currents in solid steel parts of the rotor are discussed so far that numerical results can be derived readily. A new transient test for synchronous machines is suggested by which rotor and stator leakage voltages can be measured separately, avoiding the uncertainties of the usual steady state leakage tests.