Chapter 8 alternative forms of machine equations

299
8.1. INTRODUCTION
There are alternative formulations of induction and synchronous machine equations
that warrant consideration since each has a specifi c useful purpose. In particular, (1)
linearized or small-displacement formulation for operating point stability issues; (2)
neglecting stator electric transients for large-excursion transient stability studies; and
(3) voltage-behind reactance s ( VBR s) formulation convenient for machine-converter
analysis and simulation. These special formulations are considered in this chapter.
Although standard computer algorithms may be used to automatically linearize
machine equations, it is important to be aware of the steps necessary to perform lineariza-
tion. This procedure is set forth by applying Taylor expansion about an operating point.
The resulting set of linear differential equations describe the dynamic behavior during
small displacements or small excursions about an operating point, whereupon basic linear
system theory can be used to calculate eigenvalues. In the fi rst sections of this chapter,
the nonlinear equations of induction and synchronous machines are linearized and the
eigenvalues are calculated. Although these equations are valid for operation with stator
voltages of any frequency, only rated frequency operation is considered in detail.
Over the years, there has been considerable attention given to the development of
simplifi ed models primarily for the purpose of predicting the dynamic behavior of
electric machines during large excursions in some or all of the machine variables.
Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk,
Scott Sudhoff, and Steven Pekarek.
© 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc.
ALTERNATIVE FORMS OF
MACHINE EQUATIONS
8
300 ALTERNATIVE FORMS OF MACHINE EQUATIONS
Before the 1960s, the dynamic behavior of induction machines was generally predicted
using the steady-state voltage equations and the dynamic relationship between rotor
speed and torque. Similarly, the large-excursion behavior of synchronous machines was
predicted using a set of steady-state voltage equations with modifi cations to account

for transient conditions, as presented in Chapter 5 , along with the dynamic relationship
between rotor angle and torque. With the advent of the computer, these models have
given way to more accurate representations. In some cases, the machine equations are
programmed in detail; however, in the vast majority of cases, a reduced-order model
is used in computer simulations of power systems. In particular, it is standard to neglect
the electric transients in the stator voltage equations of all machines and in the voltage
equations of all power system components connected to the stator (transformers, trans-
mission lines, etc.). By using a static representation of the power grid, the required
number of integrations is drastically reduced. Since “neglecting stator electric tran-
sients” is an important aspect of machine analysis especially for the power system
engineer, the theory of neglecting electric transients is established and the voltage equa-
tions for induction and synchronous machines are given with the stator electric tran-
sients neglected. The large-excursion behavior of these machines as predicted by these
reduced-order models is compared with the behavior predicted by the complete equa-
tions given in Chapter 5 and Chapter 6 . From these comparisons, not only do we
become aware of the inaccuracies involved when using the reduced-order models, but
we are also able to observe the infl uence that the electric transients have on the dynamic
behavior of induction and synchronous machines.
Finally, in an increasing number of applications, electric machines are coupled to
power electronic circuits. In Chapter 4 , Chapter 5 , and Chapter 6 , a great deal of the
focus was placed upon utilizing reference-frame theory to eliminate rotor-dependent
inductances (or fl ux linkage in the case of the permanent magnet machine). Although
reference-frame theory enables analytical evaluation of steady-state performance and
provides the basis for most modern electric drive controls, it can be diffi cult to apply
a transformation to some power system components, particularly power electronic
converters. In such cases, one is forced to establish a coupling between a machine
modeled in a reference frame and a power converter modeled in terms of physical
variables. As an alternative, it can be convenient to represent a machine in terms of
physical variables using a VBR model. In this chapter, the derivation of a physical
variable VBR model of the synchronous machine is provided, along with explanation

of its potential application and advantages over alternative model structures. In addi-
tion, approximate forms of the VBR model are described in which rotor position-
dependent inductances are eliminated, which greatly simplifi es the modeling of
machines in physical variables.
8.2. MACHINE EQUATIONS TO BE LINEARIZED
The linearized machine equations are conveniently derived from voltage equations
expressed in terms of constant parameters with constant driving forces, independent of
MACHINE EQUATIONS TO BE LINEARIZED 301
time. During steady-state balanced conditions, these requirements are satisfi ed, in the
case of the induction machine, by the voltage equations expressed in the synchronously
rotating reference frame, and by the voltage equations in the rotor reference frame in
the case of the synchronous machine. Since the currents and fl ux linkages are not
independent variables, the machine equations can be written using either currents or
fl ux linkages, or fl ux linkages per second, as state variables. The choice is generally
determined by the application. Currents are selected here. Formulating the small-
displacement equations in terms of fl ux linkages per second is left as an exercise for
the reader.
Induction Machine
The voltage equations for the induction machine with currents as state variables may
be written in the synchronously rotating reference frame from (6.5-34) by setting
ω = ω
e
as

v
v
v
v
r
p

XX
p
qs
e
ds
e
qr
e
dr
e
s
b
ss
e
b
ss
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤

⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
+
ω
ω
ω ωω
ω
ω
ω
ωω
ω
ωω
ω
ω
ωω
b
M
e
b
M
e

b
ss s
b
ss
e
b
M
b
M
b
M
e
b
Mr
XX
Xr
p
XX
p
X
p
XsXr
p
−+−
′
+
bb
rr
e
b

rr
e
b
M
b
M
e
b
rr r
b
rr
XsX
sX
p
XsXr
p
X
′′
−−
′′
+
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢

⎢
ω
ω
ω
ωω
ω
ωω
⎢⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
′
′
⎡
⎣
⎢
⎢
⎢
⎢

⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
i
i
i
i
qs
e
ds
e
qr
e
dr
e
(8.2-1)
where s is the slip defi ned by (6.9-13) and the zero quantities have been omitted since

only balanced conditions are considered. The reactances X
ss
and
′
X
rr
are defi ned by
(6.5-35) and (6.5-36) , respectively.
Since we have selected currents as state variables, the electromagnetic torque is
most conveniently expressed as

TXii ii
eMqs
e
dr
e
ds
e
qr
e
=
′
−
′
()
(8.2-2)
Here, the per unit version of (6.6-2) is selected for compactness. The per unit relation-
ship between torque and speed is (6.8-10) , which is written here for convenience

THp T

e
r
b
L
=+2
ω
ω
(8.2-3)
Synchronous Machine
The voltage equations for the synchronous machine in the rotor reference frame may
be written from (5.5-38) for balanced conditions as
302 ALTERNATIVE FORMS OF MACHINE EQUATIONS

v
v
v
v
e
v
qs
r
ds
r
kq
r
kq
r
xfd
r
kd

r
′
′
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥

⎥
⎥
1
2
⎥⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
+
−
r
p
XX
p
X
p
XX X
X
s
b
q
r
b
d

b
mq
b
mq
r
b
md
r
b
md
r
b
q
ω
ω
ωωω
ω
ω
ω
ω
ω
ω
rr
p
XX X
p
X
p
X
p

Xr
p
X
s
b
d
r
b
mq
r
b
mq
b
md
b
md
b
mq kq
b
kq
+− −
′
+
′
ω
ω
ω
ω
ωω ω
ωω

0
11
pp
X
p
X
p
Xr
p
X
X
r
p
X
b
mq
b
mq
b
mq kq
b
kq
md
fd b
md
ω
ωωω
ω
00
000

0
22
′
+
′
′
⎛
⎝
⎜
⎞
⎠
⎟
000
000
X
r
r
p
X
X
r
p
X
p
X
p
md
fd
fd
b

fd
md
fd b
md
b
md
′
′
+
′
⎛
⎝
⎜
⎞
⎠
⎟
′
⎛
⎝
⎜
⎞
⎠
⎟
ωω
ωω
bb
md kd
b
kd
qs

Xr
p
X
i
′
+
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
ω
rr
ds
r
kq
r
kq
r
fd
r
kd
r
i
i
i

i
i
′
′
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥

⎥
⎥
⎥
⎥
⎥
⎥
1
2
⎥⎥
⎥
⎥
⎥
⎥
⎥

(8.2-4)
where positive currents are assumed into the machine and the reactances are defi ned
by (5.5-39)–(5.5-44) .
With the currents as state variables, the per unit electromagnetic torque positive
for motor action is expressed from (5.6-2) as

TXii ii Xii ii
emdds
r
fd
r
kd
r
qs
r

mq qs
r
kq
r
kq
r
ds
r
=+
′
+
′
−+
′
+
′
()( )
12
(8.2-5)
The per unit relationship between torque and rotor speed is given by (5.8-3) , which is

THp T
e
r
b
L
=+2
ω
ω
(8.2-6)

The rotor angle is expressed from (5.7-1) as

δ
ωωω
ω
=
−
⎛
⎝
⎜
⎞
⎠
⎟
br e
b
p
(8.2-7)
It is necessary, in the following analysis, to relate variables in the synchronously rotat-
ing reference frame to variables in the rotor reference frame. This is accomplished by
using (5.7-2) with the zero quantities omitted. Thus

f
f
f
f
qs
r
ds
r
qs

e
ds
e
⎡
⎣
⎢
⎤
⎦
⎥
=
−
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥
cos sin
sin cos


δδ
δδ
(8.2-8)

8.3. LINEARIZATION OF MACHINE EQUATIONS
There are two procedures that can be followed to obtain the linearized machine equa-
tions. One is to employ Taylor ’ s expansion about a fi xed value or operating point. That
is, any machine variable f
i
can be written in terms of a Taylor expansion about its fi xed
value, f
io
, as
LINEARIZATION OF MACHINE EQUATIONS 303

gf gf g f f
gf
f
iio ioi
io
i
() () ()
()
!
=+
′
+
′′
+ΔΔ
2
2

(8.3-1)
where

ff f
iio i
=+Δ
(8.3-2)
If only a small excursion from the fi xed point is experienced, all terms higher than the
fi rst-order may be neglected, and g ( f
i
) may be approximated by

gf gf g f f
iio ioi
() () ()≈+
′
Δ
(8.3-3)
Hence, the small-displacement characteristics of the system are given by the fi rst-order
terms of Taylor ’ s series,

ΔΔgf g f f
iioi
() ( )=
′
(8.3-4)
For functions of two variables, the same argument applies

gf f gf f
f
gf f f
f

gf f f
oo oo oo
(,)(,) (,) (,)
12 1 2
1
12 1
2
12 2
≈+
∂
∂
+
∂
∂
ΔΔ
(8.3-5)
where Δ g ( f
1
, f
2
) is the last two terms of (8.3-5) .
If, for example, we apply this method to the expression for induction machine
torque, (8.2-2) , then

Tiiii Tiiii
T
eqs
e
ds
e

qr
e
dr
e
eqso
e
dso
e
qro
e
dro
e
(, , , ) ( , , , )
′′
≈
′′
+
∂
eeqso
e
dso
e
qro
e
dro
e
qs
e
qs
e

iiii
i
i
(, , , )
.
′′
∂
+Δ etc
(8.3-6)
whereupon the small-displacement expression for torque becomes

ΔΔΔΔΔTXiiiiiiii
e M qso
e
dr
e
dro
e
qs
e
dso
e
qr
e
qro
e
ds
e
=
′

+
′
−
′
−
′
()
(8.3-7)
where the added subscript o denotes steady-state quantities.
An equivalent method of linearizing nonlinear equations is to write all variables
in the form given by (8.3-2) . If all multiplications are then performed and the steady-
state expressions cancelled from both sides of the equations and if all products of small
displacement terms ( Δ f
1
Δ f
2
, for example) are neglected, the small-displacement equa-
tions are obtained. It is left to the reader to obtain (8.3-7) by this technique.
Induction Machine
If either of the above-described methods of linearization is employed to (8.2-1)–(8.2-3) ,
the linear differential equations of an induction machine become
304 ALTERNATIVE FORMS OF MACHINE EQUATIONS

Δ
Δ
Δ
Δ
Δ
v
v

v
v
T
r
p
qs
e
ds
e
qr
e
dr
e
L
s
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
+
ωω
ω
ωω
ω
ω
ω
ωω
ω
ωω
ω
b
ss
e

b
ss
b
M
e
b
M
e
b
ss s
b
ss
e
b
M
b
M
b
XX
p
XX
Xr
p
XX
p
X
p
0
0−+−
XXsXr

p
XsX XiXi
s
Mo
e
b
Mr
b
rr o
e
b
rr M dso
e
rr dro
e
o
e
b
ω
ωω
ω
ω
ω
ω
′
+
′′
−−
′′
−

XX
p
XsXr
p
XXiXi
Xi X
M
b
Mo
e
b
rr r
b
rr M qso
e
rr qro
e
Mdro
e
ω
ω
ωω
−
′′
+
′
+
′′
′
−

MMqro
e
Mdso
e
Mqso
e
q
iXiXi Hp
i
′
−−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
2
Δ
ss
e
ds
e
qr
e
dr
e
r
b
i
i
i
Δ
Δ
Δ
Δ
′
′

⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎥
⎥
ω
ω

(8.3-8)
where

s
o
ero
e
=
−
ωω
ω
(8.3-9)
It is clear that with applied voltages of rated frequency, the ratio of ω
e
to ω
b
is unity.
However, (8.3-8) and (8.3-9) are written with ω
e
included explicitly so as to accom-
modate applied voltages of a constant frequency other than rated as would occur in
variable-speed drive systems. The frequency of the applied stator voltages in variable-
speed drive systems is varied by controlling the fi ring of the source converter. There-
fore, in some applications, the frequency of the stator voltages may be a controlled
variable. It is recalled from Chapter 3 that variable-frequency operation may be inves-

tigated in the synchronously rotating reference frame by simply changing the speed
of the reference frame corresponding to the change in frequency. Therefore, if fre-
quency is a system input variable, then a small displacement in frequency may be
taken into account by allowing the reference-frame speed to change by replacing ω
e

with ω
eo
+ Δ ω
e
.
It is convenient to separate out the derivative terms and write (8.3-8) in
the form

Ex Fx up =+
(8.3-10)
where

()x
T
qs
e
ds
e
qr
e
dr
e
r
b

iiii=
′′
⎡
⎣
⎢
⎤
⎦
⎥
ΔΔΔΔ
Δ
ω
ω
(8.3-11)

()u
T
qs
e
ds
e
qr
e
dr
e
L
vvvvT=
′′
[]
ΔΔΔΔΔ
(8.3-12)

LINEARIZATION OF MACHINE EQUATIONS 305

E =
′
′
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
1
000
00 0
000
00 0
00002
ω
ω
b
ss M
ss M
Mrr
Mrr
b

XX
XX
XX
XX
H
⎥⎥
⎥
⎥
⎥
⎥
⎥
(8.3-13)

F =−
−−
′′
rX X
Xr X
sX r s
s
e
b
ss
e
b
M
e
b
ss s
e

b
M
o
e
b
Mr o
e
b
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
00
00
0
XXXiXi
sX sX r XiX
rr M dso
e
rr dro
e
o

e
b
Mo
e
b
rr r M qso
e
r
−−
′′
−−
′′
+
′
ω
ω
ω
ω
0
rrqro
e
Mdro
e
Mqro
e
Mdso
e
Mqso
e
i

Xi Xi Xi Xi
′
′
−
′
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
0
⎥⎥
⎥
⎥
⎥
⎥
⎥

⎥
⎥
⎥
⎥
⎥
(8.3-14)
In the analysis of linear systems, it is convenient to express the linear differential equa-
tions in the form

pxAxBu=+
(8.3-15)
Equation (8.3-15) is the fundamental form of the linear differential equations. It is
commonly referred to as the state equation.
Equation (8.3-10) may be written as

pxEFxEu=+
−−
() ()
11
(8.3-16)
which is in the form of (8.3-15) with

AEF=
−
()
1
(8.3-17)

BE=
−

()
1
(8.3-18)
Synchronous Machines
Linearizing (8.2-4)–(8.2-8) yields (8.3-19) . Since the steady-state damper winding cur-
rents (
′′
ii
kdo
r
kq o
r
,,
1
and
′
i
kq o
r
2
) are zero, they are not included in (8.3-19) . Since the synchro-
nous machine is generally connected to an electric system, such as a power system,
and since it is advantageous to linearize the system voltage equations in the synchro-
nously rotating reference frame, it is convenient to include the relationship between
Δ ω
r
and Δ δ in (8.3-19) . As in the case of linearized equations for the induction
machine, ω
e
is included explicitly in (8.3-19) so that the equations are in a form

convenient for voltages of any constant frequency. Small controlled changes in the
306 ALTERNATIVE FORMS OF MACHINE EQUATIONS
frequency of the applied stator voltages, as is possible in variable-speed drive systems,
may be taken into account analytically by replacing ω
e
with ω
eo
+ Δ ω
e
in the expression
for δ given by (8.2-7) .

(8.3-19)
In most cases, the synchronous machine is connected to a power system whereupon
the voltage
v
qs
r
and
v
ds
r
, which are functions of the state variable δ , will vary as the rotor
angle varies during a disturbance. It is of course necessary to account for the depen-
dence of the driving forces upon the state variables before expressing the linear dif-
ferential equations in fundamental form. In power system analysis, it is often assumed
that in some place in the system, there is a balanced source that can be considered a
constant amplitude, constant frequency, and zero impedance source (infi nite bus). This
would be a balanced independent driving force that would be represented as constant
voltages in the synchronously rotating reference frame. Hence, it is necessary to relate

the synchronously rotating reference-frame variables, where the independent driving
force exists, to the variables in the rotor reference frame. The transformation given by
(8.2-8) is nonlinear. In order to incorporate it into a linear set of differential equations,
it must be linearized. By employing the approximations that cos Δ δ = 1 and sin Δ δ = Δ δ ,
the linearized version of (8.2-8) is

Δ
Δ
Δ
Δ
f
f
f
f
qs
r
ds
r
oo
oo
qs
e
ds
e
⎡
⎣
⎢
⎤
⎦
⎥

=
−
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
cos sin
sin cos
δδ
δδ
⎦⎦
⎥
+
−
⎡
⎣
⎢
⎤
⎦
⎥
f
f
dso
r

qso
r
Δ
δ
(8.3-20)
Linearizing the inverse transformation yields
Δ
Δ
Δ
Δ
Δ
Δ
Δ
v
v
v
v
e
v
T
qs
r
ds
r
kq
r
kq
r
xfd
r

kd
r
L
′
′
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
1
2
0
⎥⎥
⎥
⎥

⎥
⎥
⎥
⎥
⎥
⎥
=
+ −r
p
XX
p
X
p
XX XXi
s
b
q
e
b
d
b
mq
b
mq
e
b
md
e
b
md d dso

ω
ω
ωωω
ω
ω
ω
ω
rr
md fdo
r
e
b
qs
b
d
e
b
mq
e
b
mq
b
md
b
md q
Xi
Xr
p
XX X
p

X
p
XX
+
′
−+−−
0
ω
ωω
ω
ω
ω
ωω ω
ii
p
Xr
p
X
p
X
p
X
p
Xr
p
qso
r
b
mq kq
b

kq
b
q
b
mq
b
mq kq
0
00000
0
11
2
ωωω
ωωω
′
+
′
′
+
bb
kq
md
fd b
md
md
fd
fd
b
fd
X

X
r
p
X
X
r
r
p
X
′
′
⎛
⎝
⎜
⎞
⎠
⎟
′
′
+
′
⎛
⎝
⎜
⎞
⎠
⎟
2
0000
000

ωω
XX
r
p
X
p
X
p
Xr
p
X
Xi
md
fd b
md
b
md
b
md kd
b
kd
mq ds
′
⎛
⎝
⎜
⎞
⎠
⎟
′

+
′
−
ω
ωωω
00
000 00
oo
r
md dso
r
fdo
r
md qso
r
mq qso
r
mq dso
r
mq dso
r
m
Xi i Xi Xi Xi Xi X+−
′
−− −()
ddqso
r
md qso
r
b

iXi Hp
p
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥

⎥
⎥
20
000000
ω
⎥⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
′
′
′
′
Δ
Δ
Δ
Δ
Δ

Δ
Δ
i
i
i
i
i
i
qs
r
ds
r
kq
r
kq
r
fd
r
kd
r
r
b
1
2
ω
ω
ΔΔ
δ
⎡
⎣

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
LINEARIZATION OF MACHINE EQUATIONS 307

Δ
Δ

Δ
Δ
f
f
f
f
qs
e
ds
e
oo
oo
qs
r
ds
r
⎡
⎣
⎢
⎤
⎦
⎥
=
−
⎡
⎣
⎢
⎤
⎦
⎥

⎡
⎣
⎢
⎤
cos sin
sin cos
δδ
δδ
⎦⎦
⎥
+
−
⎡
⎣
⎢
⎤
⎦
⎥
f
f
dso
e
qso
e
Δ
δ
(8.3-21)
It is convenient to write the above equations in the form

ΔΔ ΔfTfF

qds
r
qds
er
=+
δ
(8.3-22)

ΔΔΔfTfF
qds
e
qds
re
=+
−
()
1
δ
(8.3-23)
It is instructive to view the interconnections of the above relationships as shown in
Figure 8.3-1 . With the equations as shown in Figure 8.3-1 , a change in
Δv
qds
e
is refl ected
through the transformation to the voltage equations in the rotor reference frame and
fi nally back to the synchronously rotating reference-frame currents
Δi
qds
e

. The detail
shown in Figure 8.3-1 is more than is generally necessary. If, for example, the objective
is to study the small-displacement dynamics of a synchronous machine with its termi-
nals connected to an infi nite bus, then
Δv
qds
e
is zero and
Δv
qds
r
changes due only to Δ δ .
Also, in this case, it is unnecessary to transform the rotor reference-frame currents to
the synchronously rotating reference frame since the source (infi nite bus) has zero
impedance.
If the machine is connected through a transmission line to a large system (infi nite
bus), the small-displacement dynamics of the transmission system must be taken into
account. If only the machine is connected to the transmission line and if it is not
equipped with a voltage regulator, then it is convenient to transform the equations of
the transmission line to the rotor reference frame. In such a case, the machine and
transmission line can be considered in much the same way as a machine connected to
an infi nite bus. If, however, the machine is equipped with a voltage regulator or more
than one machine is connected to the same transmission line, it is generally preferable
to express the dynamics of the transmission system in the synchronously rotating refer-
ence frame and transform to and from the rotor reference frame of each machine as
depicted in Figure 8.3-1 .
Figure 8.3-1. Interconnection of small-displacement equations of a synchronous machine:
Park ’ s equations.
Δv
e

Δv
r
Δi
r
Δi
e
Δv
k
r
(8.3-22)
(8.3-19)
(8.3-23)
′
Δi
k
r
′
Δv
k
r
′
Δi
k
r
′
Δi
f

r
′

Δe
x
r
′
Δv
k
r
′
Δi
k
r
′
ΔT
1
Δw
r
/w
b
Δd
qds
qds
q1
q2
fd
d
qds
q1
q2
d
d

qds
308 ALTERNATIVE FORMS OF MACHINE EQUATIONS
If the machine is equipped with a voltage regulator, the dynamic behavior of the
regulator will affect the dynamic characteristics of the machine. Therefore, the small-
displacement dynamics of the regulator must be taken into account. When regulators are
employed, the change in fi eld voltage
Δ
′
e
xfd
r
is dynamically related to the change in ter-
minal voltage, which is a function of
Δv
qds
e
(or
Δv
qds
r
), the change in fi eld current
Δi
fd
r
, and
perhaps the change in rotor speed Δ ω
r
/ ω
b
if the excitation system is equipped with a

control to help damp rotor oscillations by means of fi eld voltage control. This type of
damping control is referred to as a power system stabilizer ( PSS ).
In some investigations, it is necessary to incorporate the small-displacement dynam-
ics of the prime mover system. The change of input torque (negative load torque) is a
function of the change in rotor speed Δ ω
r
/ ω
b
, which in turn is a function of the dynamics
of the masses, shafts, and damping associated with the mechanical system and, if long-
term transients are of interest, the steam or hydro dynamics and associated controls.
Although a more detailed discussion of the dynamics of the excitation and prime
mover systems would be helpful, it is clear from the earlier discussion that the equations
that describe the operation and control of a synchronous machine equipped with a volt-
age regulator and a prime mover system are very involved. This becomes readily
apparent when it is necessary to arrange the small-displacement equations of the com-
plete system into the fundamental form. Rather than performing this task by hand, it is
preferable to take advantage of analytical techniques, which involves formulating the
equations of each component (machine, excitation system, prime mover system, etc.)
in fundamental form. A computer routine can be used to arrange the small-displacement
equations along with the interconnecting transformations of the complete system into
the fundamental form.
8.4. SMALL-DISPLACEMENT STABILITY: EIGENVALUES
With the linear differential equations written in state variable form, the u vector repre-
sents the forcing functions. If u is set equal to zero, the general solution of the homo-
geneous or force-free linear differential equations becomes

xK
A
= e

t
(8.4-1)
where K is a vector formed by an arbitrary set of initial conditions. The exponential
e
A

t
represents the unforced response of the system. It is called the state transition matrix.
Small-displacement stability is assured if all elements of the transition matrix approach
zero asymptotically as time approaches infi nity. Asymptotic behavior of all elements
of the matrix occurs whenever all of the roots of the characteristic equation of A have
negative real parts where the characteristic equation of A is defi ned

det( )AI−=
λ
0
(8.4-2)
In (8.4-2) , I is the identity matrix and λ are the roots of the characteristic equation of
A referred to as characteristic roots, latent roots, or eigenvalues. Herein, we will use
EIGENVALUES OF TYPICAL INDUCTION MACHINES 309
the latter designation. One should not confuse the λ used here to denote eigenvalues
with the same notation used to denote fl ux linkages.
The eigenvalues provide a simple means of predicting the behavior of an induction
or synchronous machine at any balanced operating condition. Eigenvalues may either
be real or complex, and when complex, they occur as conjugate pairs signifying a mode
of oscillation of the state variables. Negative real parts correspond to state variables or
oscillations of state variables that decrease exponentially with time. Positive real parts
indicate an exponential increase with time, an unstable condition.
8.5. EIGENVALUES OF TYPICAL INDUCTION MACHINES
The eigenvalues of an induction machine can be obtained by using a standard eigen-

value computer routine to calculate the roots of A given by (8.3-17) . The eigenvalues
given in Table 8.5-1 are for the machines listed in Table 6.10-1 . The induction machine,
as we have perceived it, is described by fi ve state variables and hence fi ve eigenvalues.
Sets of eigenvalues for each machine at stall, rated, and no-load speeds are given in
Table 8.5-1 for rated frequency operation. Plots of the eigenvalues (real part and only
the positive imaginary part) for rotor speeds from stall to synchronous are given in
Figure 8.5-1 and Figure 8.5-2 for the 3- and 2250-hp induction motors, respectively.
At stall, the two complex conjugate pairs of eigenvalues both have a frequency
(imaginary part) corresponding to ω
b
. The frequency of one complex conjugate pair
decreases as the speed increases from stall, while the frequency of the other complex
conjugate pair remains at approximately ω
b
—in fact, nearly equal to ω
b
for the larger
horsepower machines. The eigenvalues are dependent upon the parameters of the
machine and it is diffi cult to relate analytically a change in an eigenvalue with a change
in a specifi c machine parameter. It is possible, however, to identify an association
between eigenvalues and the machine variables. For example, the complex conjugate
pair that remains at a frequency close to ω
b
is primarily associated with the transient
TABLE 8.5-1. Induction Machine Eigenvalues
Rating, hp Stall Rated Speed No Load
3
− 4.57 ± j 377 − 85.6 ± j 313 − 89.2 ± j 316
− 313 ± j 377 − 223 ± j 83.9 − 218 ± j 60.3
1.46 − 16.8 − 19.5

50
− 2.02 ± j 377 − 49.4 ± j 356 − 50.1 ± j 357
− 198 ± j 377 − 142 ± j 42.5 − 140 ± j 18.2
1.18 − 14.4 − 17.0
500
− 0.872 ± j 377 − 41.8 ±
j 374 − 41.8 ± j 374
− 70.3 ± j 377 − 15.4 ± j 41.5 − 14.3 ± j 42.8
0.397 − 27.5 − 29.6
2250
− 0.428 ± j 377 − 24.5 ± j 376 − 24.6 ± j 376
− 42.6 ± j 377 − 9.36 ± j 41.7 − 9.05 ± j 42.5
0.241 − 17.9 − 18.5
310 ALTERNATIVE FORMS OF MACHINE EQUATIONS
offset currents in the stator windings, which refl ects into the synchronously rotating
reference as a decaying 60-Hz variation. This complex conjugate pair, which is denoted
as the “stator” eigenvalues in Figure 8.5-1 and Figure 8.5-2 , is not present when the
electric transients are neglected in the stator voltage equations. It follows that the tran-
sient response of the machine is infl uenced by this complex conjugate eigenvalue pair
whenever a disturbance causes a transient offset in the stator currents. It is recalled that
in Chapter 6 , we noted a transient pulsation in the instantaneous torque of 60 Hz during
free acceleration and following a three-phase fault at the terminals with the machine
initially operating at near rated conditions. We also noted that the pulsations were more
damped in the case of the smaller horsepower machines than for the larger horsepower
machines. It is noted in Table 8.5-1 that the magnitudes of the real part of the complex
eigenvalues with a frequency corresponding to ω
b
are larger, signifying more damping,
for the smaller horsepower machine than for the larger machines.
The complex conjugate pair which changes in frequency as the rotor speed varies

is associated primarily with the electric transients in the rotor circuits and denoted in
Figure 8.5-1. Plot of eigenvalues for a 3-hp induction motor.
Real part
Real part, rad/s
Real eigenvalue
“Stator” eigenvalue
“Rotor” eigenvalue
Positive imaginary part
40
0
–40
–80
–120
–160
–200
–240
–280
–320
–360
–400
w
r
w
b
0 0.2 0.4 0.6 0.8 1.0
Imaginary part, rad/s
400
360
320
280

240
200
160
120
80
40
0
EIGENVALUES OF TYPICAL INDUCTION MACHINES 311
Figure 8.5-1 and Figure 8.5-2 as the “rotor” eigenvalue. This complex conjugate pair
is not present when the rotor electric transients are neglected. The damping associated
with this complex conjugate pair is less for the larger horsepower machines than the
smaller machines. It is recalled that during free acceleration, the 3- and 50-hp machines
approached synchronous speed in a well-damped manner, while the 500- and the
2250-hp machines demonstrated damped oscillations about synchronous speed. Similar
behavior was noted as the machines approached their fi nal operating point following a
load torque change or a three-phase terminal fault. This behavior corresponds to that
predicted by this eigenvalue. It is interesting to note that this eigenvalue is refl ected
noticeably into the rotor speed, whereas the higher-frequency “stator” eigenvalue is
not. This, of course, is due to the fact that for a given inertia and torque amplitude, a
low-frequency torque component will cause a larger amplitude variation in rotor speed
than a high-frequency component.
Figure 8.5-2. Plot of eigenvalues for a 2250-hp induction motor.
Real part
Real part, rad/s
Real eigenvalue
“Stator” eigenvalue
“Rotor” eigenvalue
Positive imaginary part
10
0

–10
–20
–30
–40
–50
w
r
w
b
0 0.2 0.4 0.6 0.8 1.0
Imaginary part, rad/s
400
360
320
280
240
200
160
120
80
40
0
312 ALTERNATIVE FORMS OF MACHINE EQUATIONS
The real eigenvalue signifi es an exponential response. It would characterize the
behavior of the induction machine equations if all electric transients are mathematically
neglected or if, in the actual machine, the electric transients are highly damped, as in
the case of the smaller horsepower machines. Perhaps the most interesting feature of
this eigenvalue, which is denoted as the real eigenvalue in Figure 8.5-1 and Figure
8.5-2 , is that it can be related to the steady-state torque-speed curve. If we think for a
moment about the torque-speed characteristics, we realize that an induction machine

can operate stably only in the negative-slope portion of the torque-speed curve. If we
were to assume an operating point on the positive-slope portion of the torque-speed
curve we would fi nd that a small disturbance would cause the machine to move away
from this operating point, either accelerating to the negative-slope region or decelerating
to stall and perhaps reversing direction of rotation depending upon the nature of the
load torque. A positive eigenvalue signifi es a system that would move away from an
assumed operating point. Note that this eigenvalue is positive over the positive-slope
region of the torque-speed curve, becoming negative after maximum steady-state torque.
8.6. EIGENVALUES OF TYPICAL SYNCHRONOUS MACHINES
The linearized transformations, (8.3-22) and (8.3-23) , and the machine equations (8.3-
19) may each be considered as components as shown in Figure 8.3-1 . The eigenvalues
of the two synchronous machines, each connected to an infi nite bus, studied in Section
5.10 , are given in Table 8.6-1 for rated operation.
The complex conjugate pair with the frequency (imaginary part) approximately
equal to ω
b
is associated with the transient offset currents in the stator windings, which
cause the 60 Hz pulsation in electromagnetic torque. This pulsation in torque is evident
in the computer traces of a three-phase fault at the machine terminals shown in Figure
5.10-8 and Figure 5.10-10 . Although operation therein is initially at rated conditions,
the three-phase fault and subsequent switching causes the operating condition to change
signifi cantly from rated conditions. Nevertheless, we note that the 60 Hz pulsation is
damped slightly more in the case of the steam turbine generator than in the case of the
hydro turbine generator. Correspondingly, the relative values of the real parts of the
“stator” eigenvalues given in Table 8.6-1 indicate that the stator electric transients of
the steam unit are damped more than the stator transients of the hydro unit.
TABLE 8.6-1. Synchronous Machine Eigenvalues for Rated Conditions
Hydro Turbine Generator Steam Turbine Generator
− 3.58 ± j 377 − 4.45 ± j 377
− 133 ± j 8.68 − 1.70 ± j 10.5

− 24.4 − 32.2
− 22.9 − 11.1
− 0.453 − 0.349
− 0.855
NEGLECTING ELECTRIC TRANSIENTS OF STATOR VOLTAGE EQUATIONS 313
The remaining complex conjugate pair is similar to the “rotor” eigenvalue in
the case of the induction machine. However, in the case of the synchronous machine,
this mode of oscillation is commonly referred to as the hunting or swing mode,
which is the principal mode of oscillation of the rotor of the machine relative to
the electrical angular velocity of the electrical system (the infi nite bus in the case
of studies made in Chapter 5 ). This mode of oscillation is apparent in the machine
variables, especially the rotor speed, in Figure 5.10-8 and Figure 5.10-10 during the
“settling out” period following reclosing. As indicated by this complex conjugate
eigenvalue, the “settling out” rotor oscillation of the steam unit (Fig. 5.10-10 ) is
more damped and of higher frequency than the corresponding rotor oscillation of
the hydro unit.
The real eigenvalues are associated with the decay of the offset currents in the rotor
circuits and therefore associated with the inverse of the effective time constant of these
circuits. It follows that since the fi eld winding has the largest time constant it gives rise
to the smallest of the real eigenvalues. In Reference 1 , it is shown that the “stator”
eigenvalue and the real eigenvalues do not change signifi cantly in value as the real and
reactive power loading conditions change.
8.7. NEGLECTING ELECTRIC TRANSIENTS OF STATOR
VOLTAGE EQUATIONS
In the case of the induction machine, there are two reduced-order models commonly
employed to calculate the electromagnetic torque during large transient excursions.
The most elementary of these is the one wherein the electric transients are neglected
in both the stator and rotor circuits. We are familiar with this steady-state model from
the information presented in Chapter 6 . The reduced-order model of present interest
is the one wherein the electric transients are neglected only in the stator voltage

equations.
In the case of the synchronous machine, there are a number of reduced-order
models used to predict its large-excursion dynamic behavior. Perhaps the best known
is the voltage behind transient reactance reduced-order model that was discussed in
Chapter 5 . The reduced-order model that is widely used for power grid studies is the
one wherein the electric transients of the stator voltage equations are neglected.
The theory of neglecting electric transients is set forth in Reference 2 . To establish
this theory, let us return for a moment to the work in Section 3.4 , where the variables
associated with stationary resistive, inductive, and capacitive elements were trans-
formed to the arbitrary reference frame. It is obvious that the instantaneous voltage
equations for the three-phase resistive circuit are the same form for either transient or
steady-state conditions. However, it is not obvious that the equations describing the
behavior of linear symmetrical inductive and capacitive elements with the electric
transients neglected (steady-state behavior) may be arranged so that the instantaneous
voltages and currents are related algebraically without the operator d/dt . Since the deri-
vation to establish these equations is analogous for inductive and capacitive elements,
it will be carried out only for an inductive circuit.
314 ALTERNATIVE FORMS OF MACHINE EQUATIONS
First, let us express the voltage equations of the three-phase inductive circuit in
the synchronously rotating reference frame. From (3.4-11) and (3.4-12) with ω = ω
e

and for balanced conditions

vp
qs
e
eds
e
qs

e
=+
ωλ λ
(8.7-1)

vp
ds
e
eqs
e
ds
e
=− +
ωλ λ
(8.7-2)
For balanced steady-state conditions, the variables in the synchronously rotating refer-
ence frame are constants. Hence, we can neglect the electric transients by neglecting

p
qs
e
λ
and
p
ds
e
λ
. Our purpose is to obtain algebraically related instantaneous voltage
equations in the arbitrary reference frame that may be used to portray the behavior with
the electric transients neglected (steady-state behavior). To this end, it is helpful to

determine the arbitrary reference-frame equivalent of neglecting
p
qs
e
λ
and
p
ds
e
λ
. This
may be accomplished by noting from (3.10-1) that the synchronous rotating and arbi-
trary reference-frame variables are related by

fKf
qd s
e
qd s
e
00
=
(8.7-3)
From (3.10-7)

e
ee
ee
K =
−−−
−−

⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
cos( ) sin( )
sin( ) cos( )
θθ θθ
θθ θθ
0
0
001
(8.7-4)
It is recalled that the arbitrary reference-frame variables do not carry a raised index. If
(8.7-1) and (8.7-2) are appropriately substituted in (8.7-3), the arbitrary reference-frame
voltage equations may be written as

vp
qs e ds e ds qs
=− − + +()
ωωλ ωλ λ
(8.7-5)

vp
ds e qs e qs ds

=− − +()
ωωλ ωλ λ
(8.7-6)
These equations are identical to (3.4-11) and (3.4-12) but written in a form that pre-
serves the identity of
p
qs
e
λ
and
p
ds
e
λ
. In particular, the fi rst and third terms on the right-
hand side of (8.7-5) and (8.7-6) result from transforming
p
qs
e
λ
and
p
ds
e
λ
to the arbitrary
reference frame. Thus, for balanced conditions, neglecting the electric transients in the
arbitrary reference frame is achieved by neglecting these terms. The resulting equa-
tions are

v
qs e ds
=
ωλ
(8.7-7)

v
ds e qs
=−
ωλ
(8.7-8)
These equations, taken as a set, describe the behavior of linear symmetrical inductive
circuits in any reference frame with the electric transients neglected. They could not
NEGLECTING ELECTRIC TRANSIENTS OF STATOR VOLTAGE EQUATIONS 315
be deduced from the equations written in the form of (3.4-11) and (3.4-12) , and at fi rst
glance, one might question their validity. Although one recognizes that these equations
are valid for neglecting the electric transients in the synchronously rotating reference
frame, it is more diffi cult to accept the fact that these equations are also valid in an
asynchronously rotating reference frame where the balanced steady-state variables are
sinusoidal. However, these steady-state variables form orthogonal balanced sinusoidal
sets for a symmetrical system. Therefore, the λ
ds
( λ
qs
) appearing in the ν
qs
( ν
ds
) equation
provides the reactance voltage drop. It is left to the reader to show that the linear alge-

braic equations in the arbitrary reference frame for a linear symmetrical capacitive
circuit are

iq
qs e ds
=
ω
(8.7-9)

iq
ds e qs
=−
ω
(8.7-10)
Let us consider what we have done; the arbitrary reference-frame voltage equations
have been established for inductive circuits with the electric transients neglected, by
neglecting the change of fl ux linkages in the synchronously rotating reference frame.
This is the same as neglecting the offsets that occur in the actual currents as a result of
a system disturbance. However, during unbalanced conditions, such as unbalanced
voltages applied to the stator circuits, the voltages in the synchronously rotating refer-
ence frame will vary with time. For example, 60 Hz unbalanced stator voltages give
rise to a constant and a double-frequency voltage in the synchronously rotating refer-
ence frame. Therefore, the fl ux linkages in the synchronously rotating reference frame
will also contain a double-frequency component. It follows that during unbalanced
conditions, neglecting the change in the synchronously rotating reference-frame fl ux
linkages results in neglecting something more than just the electric transients. There-
fore, the voltage equations, which have been derived by neglecting the change in the
fl ux linkages in the synchronously rotating reference frame apply for balanced or sym-
metrical conditions, such as simultaneous application of balanced voltages, a change
in either load or input torque and a three-phase fault. Consequently, the zero sequence

quantities are not included in the machine equations given in this chapter.
Induction Machine
The voltage equations written in the arbitrary reference frame for an induction machine
with the electric transients of the stator voltage equations neglected may be written
from (6.5-22)–(6.5-33) , with the zero quantities eliminated and (8.7-7) and (8.7-8)
appropriately taken into account.

vri
qs s qs
e
b
ds
=+
ω
ω
ψ
(8.7-11)

vri
ds s ds
e
b
qs
=−
ω
ω
ψ
(8.7-12)
316 ALTERNATIVE FORMS OF MACHINE EQUATIONS

′
=
′′
+
−
⎛
⎝
⎜
⎞
⎠
⎟
′
+
′
vri
p
qr r qr
r
b
dr
b
qr
ωω
ω
ψ
ω
ψ
(8.7-13)

′

=
′′
−
−
⎛
⎝
⎜
⎞
⎠
⎟
′
+
′
vri
p
dr r dr
r
b
qr
b
dr
ωω
ω
ψ
ω
ψ
(8.7-14)
where

ψ

qs ls qs M qs qr
Xi X i i=+ +
′
()
(8.7-15)

ψ
ds ls ds M ds dr
Xi X i i=+ +
′
()
(8.7-16)

′
=
′′
++
′
ψ
qr lr qr M qs qr
Xi X i i()
(8.7-17)

′
=
′′
++
′
ψ
dr lr qr M ds dr

Xi X i i()
(8.7-18)
Although the reference-frame speed appears in the speed voltages in the rotor voltage
equations, it does not appear in the stator voltage equations.
The voltage equations may be expressed in terms of currents by appropriately
replacing the fl ux linkages per second in (8.7-11)–(8.7-14) with (8.7-15)–(8.7-18) or
directly from (6.5-34) with the 0 s and 0 r quantities, and all derivatives in the ν
qs
and
ν
ds
voltage equations eliminated and with ω set equal to ω
e
. Hence

v
v
v
v
rX X
Xr
qs
ds
qr
dr
s
e
b
ss
e

b
M
e
b
ss s
e
b
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
−−
ω
ω
ω
ω
ω
ω
ω

ω
0
XX
p
XXr
p
XX
M
b
M
r
b
Mr
b
rr
r
b
rr
r
b
0
ω
ωω
ωω
ωω
ω
ωω
ω
−
⎛

⎝
⎜
⎞
⎠
⎟
′
+
′
−
⎛
⎝
⎜
⎞
⎠
⎟
′
−
−
⎛
⎝⎝
⎜
⎞
⎠
⎟
−
−
⎛
⎝
⎜
⎞

⎠
⎟
′′
+
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
X
p
XXr
p
X
M
b
M

r
b
rr r
b
rr
ω
ωω
ωω
⎥⎥
⎥
⎥
⎥
⎥
⎥
⎥
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
i

i
i
i
qs
ds
qr
dr

(8.7-19)
where X
ss
and
′
X
rr
are defi ned by (6.5-35) and (6.5-36) , respectively. It is important to
note that a derivative of i
qs
( i
ds
) appears in
′′
vv
qr dr
()
; however, i
qs
and i
ds
are algebraically

related to
′
i
qr
and
′
i
dr
by the equations for v
qs
and v
ds
. Hence, one might conclude that
′
i
qr

and
′
i
dr
may be selected as independent or state variables. This is not the case, since it
can be shown that all currents are both algebraically and dynamically related to the
stator voltages. Thus, if
′
i
qr
and
′
i

dr
are selected as state variables, the state equation must
be written in a nonstandard form, which is not the most convenient form for computer
simulation [1] .
Synchronous Machine
The stator voltage equations of the synchronous machine written in the arbitrary refer-
ence frame are given by (5.4-1) . As illustrated by (8.7-7) and (8.7-8) , the electric
NEGLECTING ELECTRIC TRANSIENTS OF STATOR VOLTAGE EQUATIONS 317
transients are neglected in the stator voltage equations in the arbitrary reference frame
by neglecting the derivative of fl ux linkages and setting ω = ω
e
. Thus, in terms of fl ux
linkages per second and with the electric transients neglected, the stator voltage equa-
tions of the synchronous machine expressed in the arbitrary reference frame are of the
same form as (8.7-11) and (8.7-12) with the exception that positive current is assumed
out of the terminals of the synchronous machine. This is done since reduced-order
models have traditionally been used in power system analysis, where current is usually
defi ned as positive out of the machine. It follows that the voltage equations for the
synchronous machine in the rotor reference frame with the stator electric transients
neglected are obtained by neglecting the derivative of the fl ux linkages in Park ’ s
equations and setting ω
r
= ω
e
. Thus, with the 0 s quantities omitted

vri
qs
r
sqs

r
e
b
ds
r
=− +
ω
ω
ψ
(8.7-20)

vri
ds
r
sds
r
e
b
qs
r
=− −
ω
ω
ψ
(8.7-21)

′
=
′′
+

′
vri
p
kq
r
kq kq
r
b
kq
r
111 1
ω
ψ
(8.7-22)

′
=
′′
+
′
vri
p
kq
r
kq kq
r
b
kq
r
222 2

ω
ψ
(8.7-23)

′
=
′′
+
′
vri
p
fd
r
fd fd
r
b
fd
r
ω
ψ
(8.7-24)

′
=
′′
+
′
vri
p
kd

r
kd kd
r
b
kd
r
ω
ψ
(8.7-25)
where

ψ
qs
r
ls qs
r
mq qs
r
kq
r
kq
r
Xi X i i i=− + − +
′
+
′
()
12
(8.7-26)

ψ
ds
r
ls ds
r
md ds
r
fd
r
kd
r
Xi X i i i=− + − +
′
+
′
()
(8.7-27)

′
=
′′
+−+
′
+
′
ψ
kq
r
lkq kq
r

mq qs
r
kq
r
kq
r
Xi X i i i
111 12
()
(8.7-28)

′
=
′′
+−+
′
+
′
ψ
kq
r
lkq kq
r
mq qs
r
kq
r
kq
r
Xi X i i i

222 12
()
(8.7-29)

′
=
′′
+−+
′
+
′
ψ
fd
r
lfd fd
r
md ds
r
fd
r
kd
r
Xi X i i i()
(8.7-30)

′
=
′′
+−+
′

+
′
ψ
kd
r
lkd kd
r
md ds
r
fd
r
kd
r
Xi X i i i()
(8.7-31)
As in the case of the induction machine, the voltage equations for the synchronous
machine may be written in terms of the currents. Hence, (8.7-32) results from the
substitution of (8.7-26)–(8.7-31) into (8.7-20)–(8.7-25) or directly from (5.5-38) with
the 0 s quantities and derivatives in the
v
qs
r
and
v
ds
r
voltage equations eliminated and with
ω
r
set equal to ω

e
in the
v
qs
r
and
v
ds
r
voltage equations.
318 ALTERNATIVE FORMS OF MACHINE EQUATIONS

v
v
v
v
e
v
r
qs
r
ds
r
kq
r
kq
r
xfd
r
kd

r
s
e
b
′
′
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
1

2
ω
ω
XXXX
Xr X X
p
Xr
d
e
b
md
e
b
md
e
b
qs
e
b
mq
e
b
mq
b
mq kq
00
00
0
1
ω

ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
−−−
′
++
′
′
+
′
′
p
X
p
X
p
X
p
Xr
p
X
X
r

p
b
kq
b
mq
b
mq
b
mq kq
b
kq
md
fd b
ωω
ωωω
ω
1
22
00
000
0 XX
X
r
r
p
X
X
r
p
X

p
md
md
fd
fd
b
fd
md
fd b
md
⎛
⎝
⎜
⎞
⎠
⎟
′
′
+
′
⎛
⎝
⎜
⎞
⎠
⎟
′
⎛
⎝
⎜

⎞
⎠
⎟
00
0
ωω
ωωωω
b
md
b
md kd
b
kd
X
p
Xr
p
X00
′
+
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥⎥
⎥
⎥
⎥
−
−
′

′
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
i
i
i
i
i
i
qs

r
ds
r
kq
r
kq
r
fd
r
kd
r
1
2

(8.7-32)
The reactances given in (8.7-32) are defi ned by (5.5-39)–(5.5-44) . As in the case of the
induction machine, the formulation of the voltage equations in terms of currents, with
the stator electric transients neglected, results in equations in which all machine currents
are both algebraically and dynamically related to the stator voltages. This gives rise to
an inconvenient (nonstandard) form of the state equation [1] .
8.8. INDUCTION MACHINE PERFORMANCE PREDICTED WITH
STATOR ELECTRIC TRANSIENTS NEGLECTED
It is instructive to compare the large-excursion induction machine behavior predicted
with stator electric transients neglected with that predicted by the detailed equations
given in Chapter 6 . The material presented in Chapter 6 has already made us aware of
the inaccuracies involved when both the stator and rotor transients are neglected. Addi-
tional information regarding the large-excursion accuracy of the reduced-order models
with only stator electric transients neglected and with both stator and rotor electric
transients neglected is given by T.L. Skvarenina [3] .
Free Acceleration Characteristics

The free acceleration characteristics predicted for the 2250-hp induction motor with
the electric transients neglected in the stator voltage equations are given in Figure
8.8-1 . The parameters and operating conditions are identical to those used in Chapter
6 . A comparison with Figure 6.10-6 reveals that the only signifi cant difference is in
the initial starting transient. Although a transient occurs in the instantaneous starting
torque, it is much less pronounced when the stator electric transients are neglected.
Our fi rst reaction is to assume that the transient that remains is due to the rotor
circuits. Although this is essentially the case, we must be careful with such an
interpretation since we are imposing a condition upon the voltage equations that
could not be realized in practice. We are aware from our earlier analysis that
the stator electric transient gives rise to a 60 Hz pulsating torque and a complex
INDUCTION MACHINE WITH STATOR ELECTRIC TRANSIENTS NEGLECTED 319
Figure 8.8-1. Machine variables during free acceleration of a 2250-hp induction motor pre-
dicted with stator electric transients neglected.
conjugate eigenvalue pair with a frequency (imaginary part) of approximately ω
e
.
Since we are neglecting the electric transients in the stator voltage equations, we
would expect discrepancies to occur whenever this transient is excited and whenever
it infl uences the behavior of the machine. Once the stator electric transient subsides,
the torque-speed characteristics are identical for all practical purposes. For the
machine studied, the speed is not signifi cantly infl uenced by the 60-Hz transient
torque during free acceleration. If the inertia were relatively small or if the stator
voltages were of frequency considerably less than rated, as occurs in variable speed
drive systems, the pulsating electromagnetic torque could have a signifi cant infl uence
upon the behavior of the machine.
The oscillation about synchronous speed, which is determined primarily by the
rotor circuits, is still present, as is clearly illustrated in the case of the 2250-hp machine.
This oscillation does not occur when the electric transients of the rotor circuits are
neglected. Another interesting feature regarding the transient characteristics of the

induction machine is that the varying envelope of the machine currents during free
acceleration does not occur when the stator electric transients are neglected. Therefore,
320 ALTERNATIVE FORMS OF MACHINE EQUATIONS
we must conclude that the varying current envelope depicted in Figure 6.10-5 and
Figure 6.10-6 occurs due to the interaction of stator and rotor electric transients.
A word of explanation is necessary. In the computer simulation, the terminal volt-
ages are applied in the initial condition mode. Although this has no infl uence upon the
solution which follows, we can see the ambiguity which occurs when imposing impos-
sible restrictions upon the behavior of electric circuits. Here we see that the stator
voltages are algebraically related to all machine currents since the stator and rotor cur-
rents change instantaneously when the stator voltages are applied in the initial condition
mode. It is interesting that this situation, which is impossible practically, does not give
rise to an initial torque.
Changes in Load Torque
As noted in Chapter 6 , the steady-state voltage equations, along with the dynamic
relationship between torque and speed, can generally by used to predict the dynamic
response to changes in load torque for small horsepower induction machines. However,
this reduced model, wherein both the stator and rotor electric transients are neglected,
cannot adequately predict the dynamic response of large horsepower induction machines
to load torque disturbances. On the other hand, studies reported in Reference 3 reveal
that with only the stator electric transients neglected, the predicted dynamic response
to load torque changes in the vicinity of rated torque is, for all practical purposes,
identical to that predicted by the detailed model. We would expect this since, due to
the inertia of the mechanical system, a change in load torque would normally excite a
negligibly small transient offset in the stator currents.
Three-Phase Fault at Machine Terminals
The dynamic behavior of the 2250-hp induction machine during and following a three-
phase fault at the terminals, predicted with the stator electric transients neglected, is
given in Figure 8.8-2 . An indication of the accuracy of this reduced-order model can
be ascertained by comparing these plots with those given in Figure 6.13-2 . The same

parameters and operating conditions are used. Initially, the machine is operating at base
torque. A three-phase fault at the terminals is simulated by setting v
as
, v
bs
, and v
cs
to
zero at the instant v
as
passes through zero going positive. After six cycles, the source
voltages are reapplied. Note the step change in all machine currents and torque at the
instant the fault is applied and again at the instant the fault is removed and the stator
voltages reapplied. The algebraic relationship between the machine currents and stator
voltages is clear.
With the electric transients neglected in the stator voltage equations, transient offset
currents will appear only in the rotor circuits. At the beginning of the three-phase fault
and following the reapplication of the terminal voltages, the rotor offset currents will
refl ect to the stator as balanced decaying sinusoidal currents with a frequency corre-
sponding to the rotor speed. However, since transient offsets are neglected in the stator
circuits, the rotor currents will not contain a sinusoidal component due to the refl ection
of the stator offset currents into the rotor circuits.
INDUCTION MACHINE WITH STATOR ELECTRIC TRANSIENTS NEGLECTED 321
With the stator electric transients absent, a decaying 60 Hz pulsating electromag-
netic torque does not occur. A comparison of the response predicted with the stator
electric transients neglected and that predicted by the detailed model reveals the error,
which can occur in rotor speed when the 60 Hz torque is neglected. In most power
system applications, this error would not be suffi cient to warrant the use of the detailed
model; however, for investigations of speed control of variable-speed drive systems
this simplifi ed model would probably not suffi ce. Also, if instantaneous shaft torques

are of interest, the detailed model must be used.
If the reduced-order model with both stator and rotor electric transients neglected
is used, the machine currents, and thus the torque, would instantaneously become zero
at the time of the fault. The calculated speed would decrease, obeying the dynamic
Figure 8.8-2. Dynamic performance of a 2250-hp induction motor during a three-phase fault
at the terminals predicted with stator electric transients neglected.
i
as
, kA
i
bs
, kA
i
cs
, kA
i
ar
, kA
5.97
0
–5.97
5.97
0
–5.97
5.97
0
–5.97
5.97
0
–5.97

5.97
0
–5.97
5.97
0
–5.97
′
i
br
, kA
′
i
cr
, kA
T
e
, 10
3
N•m
′
17.8
0
Speed,
r/min
1800
1620
0.05 second
3-phase
fault
Fault cleared

322 ALTERNATIVE FORMS OF MACHINE EQUATIONS
relationship between the load torque and rotor speed. The currents and torque would
instantaneously assume their steady-state values for the specifi c rotor speed at the time
the stator voltages are reapplied.
8.9. SYNCHRONOUS MACHINE PERFORMANCE PREDICTED WITH
STATOR ELECTRIC TRANSIENTS NEGLECTED
The reduced-order model of the synchronous machine obtained by neglecting the stator
electric transients is used widely in the power industry as an analysis tool. Therefore,
it is important to compare the performance of the synchronous machine predicted by
the reduced-order equations with that predicted by the detailed model (Chapter 5 ),
especially for disturbances common in transient stability studies. Since this comparison
is of most interest to the power systems engineer, positive currents are assumed out of
the machine terminals, which allows direct comparison with the performance charac-
teristics in Section 5.10 . Additional information regarding the large-excursion accuracy
of the reduced-order model is found in References 1 and 3 .
Changes in Input Torque
A change in input torque would not normally excite an appreciable transient offset in
the stator currents. Therefore, neglecting the electric transients in the stator voltage
equations has negligible effect upon the accuracy in predicting the dynamic behavior
of the typical synchronous machine during normal input torque disturbances.
Three-Phase Fault at Machine Terminals
The dynamic behavior of the steam turbine generator during and following a three-
phase fault at the terminals, predicted with the electric transients neglected in the stator
voltage equations, is shown in Figure 8.9-1 and Figure 8.9-2 . An indication of the
accuracy of this reduced-order model can be obtained by comparing the behavior
depicted in these fi gures to that shown in Figure 5.10-10 and Figure 5.10-11 . The
machine and operating conditions are identical. Initially, the machine is connected to
an infi nite bus delivering rated MVA at rated power factor. (Machine data are given in
Section 5.10 .) The input torque of the steam turbine generator is held constant at (0.85)
2.22 × 10

6
N·m and
′
=
(
)
(
)
E
xfd
r
248 2 3 26./kV
. With the machine operating in this
steady-state condition, a three-phase fault at the terminals is simulated by setting v
as
,
v
bs
, and v
cs
to zero at the instant v
as
passes through zero going positive. The instantaneous
changes in the machine currents and torque at the initiation and removal of the fault
demonstrate the algebraic relationship between stator voltages and machine currents.
With the stator electric transients neglected, the offset transients do not appear, and
consequently, the 60 Hz pulsating electromagnetic torque is not present during and
following the three-phase fault. The absence of the 60 Hz transient torque is especially
apparent in the torque versus rotor angle characteristics given Figure 8.9-2 .
SYNCHRONOUS MACHINE WITH STATOR ELECTRIC TRANSIENTS NEGLECTED 323

Figure 8.9-1. Dynamic performance of a steam turbine generator during a three-phase fault
at the terminals predicted with stator electric transients neglected.
Figure 8.9-2. Torque versus rotor angle characteristics for the study shown in Figure 8.9-1 .
8.88
4.44
–4.44
0
T
e
,10
6
N•m
90
180
d, electrical degrees