Chapter 7 machine equations in operational impedances and time constants

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OPERATIONAL IMPEDANCES AND G(p) 275
Figure 7.3-2. Calculation of X
d
( s ) and G ( s ) for two rotor windings in direct axis. (a) Calcula-
tion of X
d
( s );
′
=v
fd
r
0
; (b) calculation of G ( s );
i
ds
r
= 0
.
+
–
(a)
(b)
r
fd
¢
r
kd
¢
w
b
X
ls
s
¢
w
b
X
lfd
s
¢
w
b
X
lkd
s
w
b
X
md
s
i
ds
r
Z
dr
(s)
w
b
s
y
ds
r
+
+
–
–
r
fd
¢ r
kd
¢
w
b
X
ls
s
¢
w
b
X
lfd
s
¢
w
b
X
lkd
s
w
b
X
md
s
i
ds
= 0
r
Z
dr
(s)
w
b
s
y
ds
r
v
fd
¢
r
i
fd
¢
r

τ
ω
ττ
Da
lfd lkd
bfd kd
ed
da
kd
db
fd
XX
rr
R
rr
=
′
+
′
′
+
′
=
′
+
′
⎛
⎝
⎜
⎞
⎠
⎟
()
(7.3-18)
The operational impedance for a fi eld and damper winding in the d -axis can be obtained
by setting
′
v
fd
r
to zero and following the same procedure, as in the case of the q -axis.
The fi nal expression is

Xs X
ss
ss
dd
dd dd
dd dd
()
()
()
=
++ +
++ +
1
1
45 46
2
12 13
2
ττ ττ
ττ ττ
(7.3-19)
where

τ
ω
d
bfd
lfd md
r
XX
1
1
=
′
′
+()
(7.3-20)
276 MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES AND TIME CONSTANTS

τ
ω
d
bkd
lkd md
r
XX
2
1
=
′
′
+()
(7.3-21)

τ
ω
d
bkd
lkd
md lfd
lfd md
r
X
XX
XX
3
1
=
′
′
+
′
′
+
⎛
⎝
⎜
⎞
⎠
⎟
(7.3-22)

τ
ω
d
bfd
lfd
md ls
ls md
r
X
XX
XX
4
1
=
′
′
+
+
⎛
⎝
⎜
⎞
⎠
⎟
(7.3-23)

τ
ω
d
bkd
lkd
md ls
ls md
r
X
XX
XX
5
1
=
′
′
+
+
⎛
⎝
⎜
⎞
⎠
⎟
(7.3-24)

τ
ω
d
bkd
lkd
md ls lfd
md ls md lfd ls lfd
r
X
XXX
XX XX XX
6
1
=
′
′
+
′
+
′
+
′
⎛
⎝
⎜
⎞
⎠
⎟
(7.3-25)
The transfer function G ( s ) may be evaluated by expressing the relationship between stator
fl ux linkages per second to fi eld voltage,
′
v
fd
r
, with
i
ds
r
equal to zero. Hence, from (7.2-5)

Gs
v
ds
r
fd
r
i
ds
r
()=
′
=
ψ

0
(7.3-26)
From Figure 7.3-2 b, this yields

Gs
X
r
s
ss
md
fd
db
dd dd
()
()
=
′
+
++ +
1
1
12 13
2
τ
ττ ττ
(7.3-27)
where τ
db
is defi ned by (7.3-17) .
7.4. STANDARD SYNCHRONOUS MACHINE REACTANCES
It is instructive to set forth the commonly used reactances for the four-winding rotor
synchronous machine and to relate these reactances to the operational impedances
whenever appropriate. The q - and d -axis reactances are

XXX
qlsmq
=+
(7.4-1)

XXX
dlsmd
=+
(7.4-2)
These reactances were defi ned in Section 5.5. They characterize the machine during
balanced steady-state operation whereupon variables in the rotor reference frame are
constants. The zero frequency value of X
q
( s ) or X
d
( s ) is found by replacing the operator
s with zero. Hence, the operational impedances for balanced steady-state operation are

XX
qq
()0 =
(7.4-3)

XX
dd
()0 =
(7.4-4)
STANDARD SYNCHRONOUS MACHINE REACTANCES 277
Similarly, the steady-state value of the transfer function is

G
X
r
md
fd
()0 =
′
(7.4-5)
The q - and d -axis transient reactances are defi ned as

′
=+
′
′
+
XX
XX
XX
qls
mq lkq
lkq mq
1
1
(7.4-6)

′
=+
′
′
+
XX
XX
XX
dls
md lfd
lfd md
(7.4-7)
Although
′
X
q
has not been defi ned previously, we did encounter the d -axis transient
reactance in the derivation of the approximate transient torque-angle characteristic in
Chapter 5 .
The q - and d -axis subtransient reactances are defi ned as

′′
=+
′′
′
+
′
+
′′
XX
XX X
XX XX X X
qls
mq lkq lkq
mq lkq mq lkq lkq lkq
12
1212
(7.4-8)

′′
=+
′′
′
+
′
+
′′
XX
XXX
XX XX XX
dls
md lfd lkd
md lfd md lkd lfd lkd
(7.4-9)
These reactances are the high-frequency asymptotes of the operational impedances.
That is

XX
qq
()∞=
′′
(7.4-10)

XX
dd
()∞=
′′
(7.4-11)
The high-frequency response of the machine is characterized by these reactances. It is
interesting that G ( ∞ ) is zero, which indicates that the stator fl ux linkages are essentially
insensitive to high frequency changes in fi eld voltage. Primes are used to denote transient
and subtransient quantities, which can be confused with rotor quantities referred to the
stator windings by a turns ratio. Hopefully, this confusion is minimized by the fact that

′
X
d
and
′
X
q
are the only single-primed parameters that are not referred impedances.
Although the steady-state and subtransient reactances can be related to the opera-
tional impedances, this is not the case with the transient reactances. It appears that the
d -axis transient reactance evolved from Doherty and Nickle ’ s [3] development of an
approximate transient torque-angle characteristic where the effects of d -axis damper
windings are neglected. The q -axis transient reactance has come into use when it
became desirable to portray more accurately the dynamic characteristics of the
solid iron rotor machine in transient stability studies. In many of the early studies,
only one damper winding was used to describe the electrical characteristics of the
q -axis, which is generally adequate in the case of salient-pole machines. In our earlier
278 MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES AND TIME CONSTANTS
development, we implied a notational correspondence between the kq 1 and the fd
windings and between the kq 2 and the kd windings. In this chapter, we have associated
the kq 1 winding with the transient reactance (7.4-6) , and the kq 2 winding with the
subtransient reactance (7.4-8) . Therefore, it seems logical to use only the kq 2 winding
when one damper winding is deemed adequate to portray the electrical characteristics
of the q axis. It is recalled that in Chapter 5 , we chose to use the kq 2 winding rather
than the kq 1 winding in the case of the salient-pole hydro turbine generator.
It is perhaps apparent that the subtransient reactances characterize the equivalent
reactances of the machine during a very short period of time following an electrical
disturbance. After a period, of perhaps a few milliseconds, the machine equivalent
reactances approach the values of the transient reactances, and even though they are
not directly related to X
q
( s ) and X
d
( s ), their values lie between the subtransient and
steady-state values. As more time elapses after a disturbance, the transient reactances
give way to the steady state reactances. In Chapter 5 , we observed the impedance of
the machine “changing” from transient to steady state following a system disturbance.
Clearly, the use of the transient and subtransient quantities to portray the behavior of
the machine over specifi c time intervals was a direct result of the need to simplify the
machine equations so that precomputer computational techniques could be used.
7.5. STANDARD SYNCHRONOUS MACHINE TIME CONSTANTS
The standard time constants associated with a four-rotor winding synchronous machine
are given in Table 7.5-1 . These time constants are defi ned as

′
τ
qo
and
′
τ
do
are the q - and d -axis transient open-circuit time constants.

′′
τ
qo
and
′′
τ
do
are the q - and d -axis subtransient open-circuit time constants.

′
τ
q
and
′
τ
d
are the q - and d -axis transient short-circuit time constants.

′′
τ
q
and
′′
τ
d
are the q - and d -axis subtransient short-circuit time constants.
In the above defi nitions, open and short circuit refers to the conditions of the stator
circuits. All of these time constants are approximations of the actual time constants,
and when used to determine the machine parameters, they can lead to substantial errors
in predicting the dynamic behavior of a synchronous machine. More accurate expres-
sions for the time constants are derived in the following section.
7.6. DERIVED SYNCHRONOUS MACHINE TIME CONSTANTS
The open-circuit time constants, which characterize the duration of transient changes
of machine variables during open-circuit conditions, are the reciprocals of the roots
of the characteristic equation associated with the operational impedances, which, of
course, are the poles of the operational impedances. The roots of the denominators
of X
q
( s ) and X
d
( s ) can be found by setting these second-order polynomials equal to zero.
From X
q
( s ), (7.3-7)
DERIVED SYNCHRONOUS MACHINE TIME CONSTANTS 279

ss
qq
qq qq
2
12
13 13
1
0+
+
+=
ττ
ττ ττ
(7.6-1)
From X
d
( s ), (7.3-19)

ss
dd
dd dd
2
12
13 13
1
0+
+
+=
ττ
ττ ττ
(7.6-2)
The roots are of the form

s
bb c
b
=− ± −
22
1
4
2
(7.6-3)
The exact solution of (7.6-3) is quite involved. It can be simplifi ed, however, if the
quantity 4 c / b
2
is much less than unity [4] . In the case of the q -axis

4
4
2
13
12
2
c
b
qq
qq
=
+
ττ
ττ
()
(7.6-4)
TABLE 7.5-1. Standard Synchronous Machine Time Constants
Open-Circuit Time Constants

′
=
′
′
+
τ
ω
qo
bkq
lkq mq
r
XX
1
1
1
()

′
=
′
′
+
τ
ω
do
bfd
lfd md
r
XX
1
()

′′
=
′
′
+
′
+
′
⎛
⎝
⎜
⎞
⎠
⎟
τ
ω
qo
bkq
lkq
mq lkq
mq lkq
r
X
XX
XX
1
2
2
1
1

′′
=
′
′
+
′
+
′
⎛
⎝
⎜
⎞
⎠
⎟
τ
ω
do
bkd
lkd
md lfd
md lfd
r
X
XX
XX
1
Short-Circuit Time Constants

′
=
′
′
+
+
⎛
⎝
⎜
⎞
⎠
⎟
τ
ω
q
bkq
lkq
mq ls
mq ls
r
X
XX
XX
1
1
1

′
=
′
′
+
+
⎛
⎝
⎜
⎞
⎠
⎟
τ
ω
d
bfd
lfd
md ls
md ls
r
X
XX
XX
1

′′
=
′
′
+
′
+
′
+
′
τ
ω
q
bkq
lkq
mq ls lkq
mq ls mq lkq ls lkq
r
X
XXX
XX XX XX
1
2
2
1
11
⎛⎛
⎝
⎜
⎞
⎠
⎟

′′
=
′
′
+
′
+
′
+
′
⎛
⎝
⎜
⎞
⎠
τ
ω
d
bkd
lkd
md ls lfd
md ls md lfd ls lfd
r
X
XXX
XX XX XX
1
⎟⎟
280 MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES AND TIME CONSTANTS
It can be shown that

44
13
12
2
12 1 2
1
ττ
ττ
qq
qq
kq kq lkq lkq
mq kq kq
rr X X
Xr r()
()
(+
≈
′′ ′
+
′
′
+
′
22
2
)
(7.6-5)
In the case of the d -axis

4
4
13
12
22
ττ
ττ
dd
dd
fd kd lfd lkd
md fd kd
rr X X
Xr r()
()
()+
≈
′′ ′
+
′
′
+
′
(7.6-6)
In most cases, the right-hand side of (7.6-5) and (7.6-6) is much less than unity. Hence,
the solution of (7.6-3) with 4 c / b
2
≪ 1 and c / b ≪ b is obtained by employing the
binomial expansion, from which

s
c
b
1
=−
(7.6-7)

sb
2
=−
(7.6-8)
Now, the reciprocals of the roots are the time constants, and if we defi ne the transient
open-circuit time constant as the largest time constant and the subtransient open-circuit
time constant as the smallest, then

′
=
=+
τ
ττ
qo
qq
b
c
12
(7.6-9)
and

′′
=
=
+
τ
τ
ττ
qo
q
qq
b
1
1
3
21
/
(7.6-10)
Similarly, the d -axis open-circuit time constants are

′
=+
τττ
do d d12
(7.6-11)

′′
=
+
τ
τ
ττ
do
d
dd
3
21
1/
(7.6-12)
The above derived open-circuit time constants are expressed in terms of machine
parameters in Table 7.6-1 .
DERIVED SYNCHRONOUS MACHINE TIME CONSTANTS 281
TABLE 7.6-1. Derived Synchronous Machine Time Constants
Open-Circuit Time Constants

′
=
′
′
++
′
′
+
τ
ωω
qo
bkq
lkq mq
bkq
lkq mq
r
XX
r
XX
11
1
1
2
2
()()

′
=
′
′
++
′
′
+
τ
ωω
do
bfd
lfd md
bkd
lkd md
r
XX
r
XX
11
()()

′′
=
′
′
+
′
′
+
⎛
⎝
⎜
⎞
⎠
⎟
+
′
′
τ
ω
ω
qo
bkq
lkq
mq lkq
lkq mq
bkq
r
X
XX
XX
r
1
1
1
2
2
1
1
2
(XXX
r
XX
lkq mq
bkq
lkq mq
2
1
1
1
+
′
′
+
)
()
ω

′′
=
′
′
+
′
′
+
⎛
⎝
⎜
⎞
⎠
⎟
+
′
′
+
τ
ω
ω
do
bkd
lkd
md lfd
lfd md
bkd
lkd
r
X
XX
XX
r
X
1
1
1
(
XX
r
XX
md
bfd
lfd md
)
()
1
ω
′
′
+
Short-Circuit Time Constants

′
=
′
′
+
+
⎛
⎝
⎜
⎞
⎠
⎟
+
′
′
+
τ
ωω
q
bkq
lkq
mq ls
ls mq b kq
lkq
mq
r
X
XX
XX r
X
XX
11
1
1
2
2
lls
ls mq
XX+
⎛
⎝
⎜
⎞
⎠
⎟

′
=
′
′
+
+
⎛
⎝
⎜
⎞
⎠
⎟
+
′
′
+
τ
ωω
d
bfd
lfd
md ls
ls md b kd
lkd
md ls
l
r
X
XX
XX r
X
XX
X
11
ssmd
X+
⎛
⎝
⎜
⎞
⎠
⎟

′′
=
′
′
+
′
+
′
+
′
τ
ω
q
bkq
lkq
mq ls lkq
mq ls mq lkq ls lkq
r
X
XXX
XX XX XX
1
2
2
1
11
⎛⎛
⎝
⎜
⎞
⎠
⎟
+
′
′
+
+
⎛
⎝
⎜
⎞
⎠
⎟
′
′
+
1
1
1
2
2
1
1
ω
ω
bkq
lkq
mq ls
ls mq
bkq
lkq
r
X
XX
XX
r
X
XXX
XX
mq ls
ls mq
+
⎛
⎝
⎜
⎞
⎠
⎟

′′
=
′
′
+
′
+
′
+
′
⎛
⎝
⎜
⎞
⎠
τ
ω
d
bkd
lkd
md ls lfd
md ls md lfd ls lfd
r
X
XXX
XX XX XX
1
⎟⎟
+
′
′
+
+
⎛
⎝
⎜
⎞
⎠
⎟
′
′
+
1
1
1
ω
ω
bkd
lkd
md ls
ls md
bfd
lfd
md ls
ls
r
X
XX
XX
r
X
XX
X
++
⎛
⎝
⎜
⎞
⎠
⎟
X
md
282 MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES AND TIME CONSTANTS
The short-circuit time constants are defi ned as the reciprocals of the roots of the
numerator of the operational impedances. Although the stator resistance should be
included in the calculation of the short-circuit time constants; its infl uence is generally
small. From X
q
( s ), (7.3-7)

ss
qq
qq qq
2
45
46 46
1
0+
+
+=
ττ
ττ ττ
(7.6-13)
From X
d
( s ), given by (7.3-19)

ss
dd
dd dd
2
45
46 46
1
0+
+
+=
ττ
ττ ττ
(7.6-14)
The roots are of the form given by (7.6-3) and, as in the case of the open-circuit time
constants, 4 c / b
2
≪ 1 and c / b ≪ b . Hence

′
=+
ττ τ
qq q45
(7.6-15)

′′
=
+
τ
τ
ττ
q
q
qq
6
54
1/
(7.6-16)

′
=+
ττ τ
dd d45
(7.6-17)

′′
=
+
τ
τ
ττ
d
d
dd
6
54
1/
(7.6-18)
The above derived synchronous machine time constants are given in Table 7.6-1 in
terms of machine parameters. It is important to note that the standard machine time
constants given in Table 7.5-1 are considerably different from the more accurate derived
time constants. The standard time constants are acceptable approximations of the
derived time constants if

′
>>
′
rr
kq kq21
(7.6-19)
and

′
>>
′
rr
kd fd
(7.6-20)
In the lumped parameter approximation of the rotor circuits,
′
r
kd
is generally much larger
than
′
r
fd
, and therefore the standard d -axis time constants are often good approximations
of the derived time constants. This is not the case for the q -axis lumped parameter
approximation of the rotor circuits. That is,
′
r
kq2
is seldom if ever larger than
′
r
kq1
, hence
the standard q -axis time constants are generally poor approximations of the derived
time constants.
PARAMETERS FROM SHORT-CIRCUIT CHARACTERISTICS 283
7.7. PARAMETERS FROM SHORT-CIRCUIT CHARACTERISTICS
For much of the twentieth century, results from a short-circuit test performed on an
unloaded synchronous machine were used to establish the d -axis parameters [5] . Alter-
native techniques have for the most part replaced short-circuit characterization. Despite
being replaced, many of the terms, such as the short-circuit time-constants, have roots
in the analytical derivation of the short circuit response of a machine. Thus, it is useful
to briefl y describe the test herein.
If the speed of the machine is constant, then (7.2-1) – (7.2-6) form a set of linear
differential equations that can be solved using linear system theory. Prior to the short
circuit of the stator terminals, the machine variables are in the steady state and the stator
terminals are open-circuited. If the fi eld voltage is held fi xed at its prefault value, then
the Laplace transform of the change in
′
v
fd
r
is zero. Hence, if the terms involving
r
s
2
are
neglected, the Laplace transform of the fault currents (defi ned positive out of the
machine), for the constant speed operation ( ω
r
= ω
b
), may be expressed

is
Xs
ss
rv s
Xs
sv s
qs
r
q
b
bs qs
r
d
bqs
r
b
()
/() ()
()
()=−
++
+−
1
2
22
2
2
αω
ω
ωω
vvs
ds
r
()
⎡
⎣
⎢
⎤
⎦
⎥
(7.7-1)

is
Xs
ss
rv s
Xs
sv s
ds
r
d
b
bs ds
r
q
bds
r
b
()
/() ()
()
()=−
++
++
1
2
22
2
2
αω
ω
ωω
vvs
qs
r
()
⎡
⎣
⎢
⎤
⎦
⎥
(7.7-2)
where

α
ω
=+
⎛
⎝
⎜
⎞
⎠
⎟
bs
qd
r
Xs Xs2
11
() ()
(7.7-3)
It is clear that the 0 quantities are zero for a three-phase fault at the stator terminals. It
is also clear that ω
r
, ω
b
, and ω
e
are all equal in this example.
Initially, the machine is operating open-circuited, hence

vV
qs
r
s
= 2
(7.7-4)

v
ds
r
= 0
(7.7-5)
The three-phase fault appears as a step decrease in
v
qs
r
to zero. Therefore, the Laplace
transform of the change in the voltages from the prefault to fault values are

vs
V
s
qs
r
s
()=−
2
(7.7-6)

vs
ds
r
()= 0
(7.7-7)
If (7.7-6) and (7.7-7) are substituted into (7.7-1) and (7.7-2) , and if the terms involving
r
s
are neglected except for α , wherein the operational impedances are replaced by
their high-frequency asymptotes, the Laplace transform of the short-circuit currents
becomes
284 MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES AND TIME CONSTANTS

is
Xs
ss
V
qs
r
q
b
bs
()
/()
()=
++
1
2
2
22
αω
ω
(7.7-8)

is
Xs
ss
V
s
ds
r
d
b
bs
()
/()
=
++
⎛
⎝
⎜
⎞
⎠
⎟
1
2
2
22
2
αω
ω
(7.7-9)
where

α
ω
=
∞
+
∞
⎛
⎝
⎜
⎞
⎠
⎟
bs
qd
r
XX2
11
() ()
(7.7-10)
Replacing the operational impedances with their high frequency asymptotes in α is
equivalent to neglecting the effects of the rotor resistances in α .
If we now assume that the electrical characteristics of the synchronous machine
can be portrayed by two rotor windings in each axis, then we can express the operational
impedances in terms of time constants. It is recalled that the open- and short-circuit
time constants are respectively the reciprocals of the roots of the denominator and
numerator of the operational impedances. Therefore, the reciprocals of the operational
impedances may be expressed

11
11
11Xs X
ss
ss
qq
qo qo
qq
()
()()
()()
=
+
′
+
′′
+
′
+
′′
ττ
ττ
(7.7-11)

111 1
11Xs X
ss
ss
dd
do do
dd
()
()()
()()
=
+
′
+
′′
+
′
+
′′
ττ
ττ
(7.7-12)
These expressions may be written as [6]

11
1
11Xs X
As
s
Bs
s
qq q q
()
=+
+
′
+
+
′′
⎛
⎝
⎜
⎞
⎠
⎟
ττ
(7.7-13)

11
1
11Xs X
Cs
s
Ds
s
dd d d
()
=+
+
′
+
+
′′
⎛
⎝
⎜
⎞
⎠
⎟
ττ
(7.7-14)
where

A
q qo q qo q
qq
=−
′
−
′′
−
′′ ′
−
′′ ′
τττ ττ
ττ
(/)(/)
/
11
1
(7.7-15)

B
q qo q qo q
qq
=−
′′
−
′′′
−
′′ ′′
−
′′′
τττ ττ
ττ
(/)(/)
/
11
1
(7.7-16)
The constants C and D are identical to A and B , respectively, with the q subscript
replaced by d in all time constants.
Since the subtransient time constants are considerably smaller than the transient
time constants, (7.7-13) and (7.7-14) may be approximated by