Chapter 6 symmetrical induction machines

215
6.1. INTRODUCTION
The induction machine is used in a wide variety of applications as a means of convert-
ing electric power to mechanical work. It is without doubt the workhorse of the electric
power industry. Pump, steel mill, and hoist drives are but a few applications of large
multiphase induction motors. On a smaller scale, induction machines are used as the
controlled drive motor in vehicles, air conditioning systems, and in wind turbines, for
example. Single-phase induction motors are widely used in household appliances, as
well as in hand and bench tools.
In the beginning of this chapter, classical techniques are used to establish the
voltage and torque equations for a symmetrical induction machine expressed in terms
of machine variables. Next, the transformation to the arbitrary reference frame pre-
sented in Chapter 3 is modifi ed to accommodate rotating circuits. Once this groundwork
has been laid, the machine voltage equations are written in the arbitrary reference
frame directly without a laborious exercise in trigonometry that one faces when
substituting the equations of transformations into the voltage equations expressed in
machine variables. The equations may then be expressed in any reference frame by
appropriate assignment of the reference-frame speed in the arbitrary reference-frame
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.
SYMMETRICAL INDUCTION
MACHINES
6
216 SYMMETRICAL INDUCTION MACHINES
voltage equations. Although the stationary reference frame, the reference frame fi xed
in the rotor, and the synchronously rotating reference frame are the most frequently
used, the arbitrary reference frame offers a direct means of obtaining the voltage equa-
tions in these and all other reference frames.
The steady-state voltage equations for an induction machine are obtained from the
voltage equations in the arbitrary reference frame by direct application of the material

presented in Chapter 3 . Computer solutions are used to illustrate the dynamic perfor-
mance of typical induction machines and to depict the variables in various reference
frames during free acceleration. Finally, the equations for an induction machine are
arranged appropriate for computer simulation. The material presented in this chapter
forms the basis for solution of more advanced problems. In particular, these basic
concepts are fundamental to the analysis of induction machines in most power system
and controlled electric drive applications.
6.2. VOLTAGE EQUATIONS IN MACHINE VARIABLES
The winding arrangement for a two-pole, three-phase, wye-connected, symmetrical
induction machine is shown in Figure 6.2-1 (which is Fig. 1.4-3 repeated here for conve-
nience). The stator windings are identical, sinusoidally distributed windings, displaced
120°, with N
s
equivalent turns and resistance r
s
. For the purpose at hand, the rotor wind-
ings will also be considered as three identical sinusoidally distributed windings, displaced
120°, with N
r
equivalent turns and resistance r
r
. The positive direction of the magnetic
axis of each winding is shown in Figure 6.2-1 . It is important to note that the positive
direction of the magnetic axes of the stator windings coincides with the direction of f
as
,
f
bs
, and f
cs

as specifi ed by the equations of transformation and shown in Figure 3.3-1 .
The voltage equations in machine variables may be expressed

vri
abcs s abcs abcs
p=+l
(6.2-1)

vri
abcr r abcr abcr
p=+l
(6.2-2)
where

f
abcs
T
as bs cs
fff
(
)
=
[]

(6.2-3)

f
abcr
T
ar br cr

fff
(
)
=
[]

(6.2-4)
In the above equations, the s subscript denotes variables and parameters associated with
the stator circuits, and the r subscript denotes variables and parameters associated with
the rotor circuits. Both r
s
and r
r
, are diagonal matrices each with equal nonzero ele-
ments. For a magnetically linear system, the fl ux linkages may be expressed as

l
l
abcs
abcr
ssr
sr
T
r
abcs
abcr
⎡
⎣
⎢
⎤

⎦
⎥
=
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥
LL
LL
i
i()
(6.2-5)
VOLTAGE EQUATIONS IN MACHINE VARIABLES 217
The winding inductances are derived Chapter 2 . Neglecting mutual leakage between
the stator windings and also between the rotor windings, they can be expressed as

L
s
ls ms ms ms
ms ls ms ms
ms ms ls
LL L L

LLL L
LLLL
=
+− −
−+−
−− +
1
2
1
2
1
2
1
2
1
2
1
2
mms
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦

⎥
⎥
⎥
⎥
⎥
⎥
⎥
(6.2-6)
Figure 6.2-1. Two-pole, three-phase, wye-connected symmetrical induction machine.
q
r
f
s
f
r
ar-axis
as-axis
cs-axis
cr-axis
bs-axis
br-axis
bs
bs¢
br¢
cr¢
cs¢
ar¢
as¢
ar
as

br
cr
cs
w
r
v
as
v
br
v
cr
v
ar
v
bs
v
cs
i
as
i
bs
i
ar
i
cs
i
cr
i
br
r

s
r
s
r
s
r
r
r
r
r
r
N
s
N
s
N
s
N
r
N
r
N
r
+
+
+
+
+
+
218 SYMMETRICAL INDUCTION MACHINES

L
r
lr mr mr mr
mr lr mr mr
mr mr lr
LL L L
LLL L
LLLL
=
+− −
−+−
−− +
1
2
1
2
1
2
1
2
1
2
1
2
mmr
⎡
⎣
⎢
⎢

⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
(6.2-7)

L
sr sr
rr r
r
L=
+
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛

⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟
cos cos cos
cos c
θθ
π
θ
π
θ
π
2
3
2
3
2
3
oos cos
cos cos cos
θθ
π

θ
π
θ
π
θ
rr
rr r
+
⎛
⎝
⎜
⎞
⎠
⎟
+
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣

⎢
2
3
2
3
2
3
⎢⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
(6.2-8)
In the above inductance equations, L
ls
and L
ms
are, respectively, the leakage and mag-
netizing inductances of the stator windings; L
lr

and L
mr
are for the rotor windings. The
inductance L
sr
is the amplitude of the mutual inductances between stator and rotor
windings.
A majority of induction machines are not equipped with coil-wound rotor wind-
ings; instead, the current fl ows in copper or aluminum bars that are uniformly distrib-
uted and are embedded in a ferromagnetic material with all bars terminated in a
common ring at each end of the rotor. This type of rotor confi guration is referred to as
a squirrel-cage rotor. It may at fi rst appear that the mutual inductance between a uni-
formly distributed rotor winding and a sinusoidally distributed stator winding would
not be of the form given by (6.2-8) . However, in most cases, a uniformly distributed
winding is adequately described by its fundamental sinusoidal component and is rep-
resented by an equivalent three-phase winding. Generally, this representation consists
of one equivalent winding per phase; however, the rotor construction of some machines
is such that its performance is more accurately described by representing each phase
with two equivalent windings connected in parallel. This type of machine is commonly
referred to as a double-cage rotor machine.
Another consideration is that in a practical machine, the rotor conductors are often
skewed. That is, the conductors are not placed in the plane of the axis of rotation of
the rotor. Instead, the conductors are skewed slightly (typically one slot width) with
the axis of rotation. This type of conductor arrangement helps to reduce the magnitude
of harmonic torques that result from harmonics in the MMF waves. Such design fea-
tures are not considered here. Instead, it is assumed that all effects upon the amplitude
of the fundamental component of the MMF waveform due to skewing and uniformly
distributed rotor windings are accounted for in the value of N
r
. The assumption that the

induction machine is a linear (no saturation) and MMF harmonic-free device is an
oversimplifi cation that cannot describe the behavior of induction machines in all modes
of operation. However, in the majority of applications, its behavior can be adequately
predicted with this simplifi ed representation.
VOLTAGE EQUATIONS IN MACHINE VARIABLES 219
When expressing the voltage equations in machine variable form, it is convenient
to refer all rotor variables to the stator windings by appropriate turns ratios.

′
=ii
abcr
r
s
abcr
N
N
(6.2-9)

′
=vv
abcr
s
r
abcr
N
N
(6.2-10)

′
=ll

abcr
s
r
abcr
N
N
(6.2-11)
The magnetizing and mutual inductances are associated with the same magnetic fl ux
path; therefore L
ms
, L
mr
, and L
sr
are related as set forth by (1.2-21) with 1 and 2 replaced
by s and r , respectively, or by (2.8-57) – (2.8-59) . In particular

L
N
N
L
ms
s
r
sr
=
(6.2-12)
Thus, we will defi ne

′

=
=
+
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟
−
LL
sr
s
r
sr
ms
rr r
r
N
N
L
cos cos cos
cos

θθ
π
θ
π
θ
2
3
2
3
2
ππ
θθ
π
θ
π
θ
π
3
2
3
2
3
2
3
⎛
⎝
⎜
⎞
⎠
⎟

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

rr
rr
⎠⎠
⎟
⎡
⎣
⎢
⎢

⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
cos
θ
r
(6.2-13)
Also, from (1.2-18) or (2.8-57) and (2.8-58) , L
mr
may be expressed as

L
N
N
L
mr
r
s
ms

=
⎛
⎝
⎜
⎞
⎠
⎟
2
(6.2-14)
and if we let

′
=
⎛
⎝
⎜
⎞
⎠
⎟
LL
r
s
r
r
N
N
2
(6.2-15)
then, from (6.2-7)
220 SYMMETRICAL INDUCTION MACHINES

′
=
′
+− −
−
′
+−
−−
′
L
r
lr ms ms ms
ms lr ms ms
ms ms
LL L L
LLL L
LLL
1
2
1
2
1
2
1
2
1
2
1
2

llr ms
L+
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
(6.2-16)
where

′
=
⎛
⎝
⎜
⎞
⎠

⎟
L
N
N
L
lr
s
r
lr
2
(6.2-17)
The fl ux linkages may now be expressed as

l
l
abcs
abcr
sr
sr
T
r
abcs
abcr
′
⎡
⎣
⎢
⎤
⎦
⎥

=
′
′′
⎡
⎣
⎢
⎤
⎦
⎥
′
⎡
⎣
⎢
⎤
⎦
⎥
LL
LL
i
i
s
()
(6.2-18)
The voltage equations expressed in terms of machine variables referred to the stator
windings may now be written as

v
v
rL L
LrL

i
i
abcs
abcr
ss sr
sr
T
rr
abcs
pp
pp
′
⎡
⎣
⎢
⎤
⎦
⎥
=
+
′
′′
+
′
⎡
⎣
⎢
⎤
⎦
⎥

′
()
aabcr
⎡
⎣
⎢
⎤
⎦
⎥
(6.2-19)
where

′
=
⎛
⎝
⎜
⎞
⎠
⎟
rr
r
s
r
r
N
N
2
(6.2-20)
6.3. TORQUE EQUATION IN MACHINE VARIABLES

Evaluation of the energy stored in the coupling fi eld by (1.3-51) yields the familiar
expression for energy stored in a magnetically linear system. In particular, the stored
energy is the sum of the self-inductance of each winding times one-half the square of
its current and all mutual inductances, each times the currents in the two windings
coupled by the mutual inductance. Thus, the energy stored in the coupling fi eld may
be written

W
f abcs
T
s abcs abcs
T
sr abcr
abcr
T
rab
=+
′′
+
′′′
1
2
1
2
() ()
()
iLi iLi
iLi
ccr
(6.3-1)

TORQUE EQUATION IN MACHINE VARIABLES 221
Since the machine is assumed to be magnetically linear, the fi eld energy W
f
is equal to
the coenergy W
c
.
Before using the second entry of Table 1.3-1 to express the electromagnetic torque,
it is necessary to modify the expressions given in Table 1.3-1 to account for a P -pole
machine. The change of mechanical energy in a rotational system with one mechanical
input may be written from (1.3-71) as

dW T d
merm
=−
θ
(6.3-2)
where T
e
is the electromagnetic torque positive for motor action (torque output) and θ
rm

is the actual angular displacement of the rotor. The fl ux linkages, currents, W
f
, and W
c
,
are all expressed as functions of the electrical angular displacement θ
r
. Since

θθ
rrm
P
=
⎛
⎝
⎜
⎞
⎠
⎟
2
(6.3-3)
where P is the number of poles in the machine, then

dW T
P
d
me r
=−
⎛
⎝
⎜
⎞
⎠
⎟
2
θ
(6.3-4)
Therefore, to account for a P -pole machine, all terms on the right-hand side of Table

1.3-1 should be multiplied by P /2. Therefore, the electromagnetic torque may be evalu-
ated from

T
PW
er
cr
r
(, )
(, )
i
i
θ
θ
θ
=
⎛
⎝
⎜
⎞
⎠
⎟
∂
∂2
(6.3-5)
The abbreviated functional notation, as used in Table 1.3-1 , is also used here for the
currents. Since L
s
and
′

L
r
are not functions of θ
r
, substituting W
f
from (6.3-1) into (6.3-
5) yields the electromagnetic torque in Newton meters (N·m)

T
P
e abcs
T
r
sr abcr
=
⎛
⎝
⎜
⎞
⎠
⎟
∂
∂
′′
2
() []iLi
θ
(6.3-6)
In expanded form, (6.3-6) becomes

T
P
Lii i i ii i
e msasarbrcrbsbrar
=−
⎛
⎝
⎜
⎞
⎠
⎟
′
−
′
−
′
⎛
⎝
⎜
⎞
⎠
⎟
+
′
−
′
−
2
1

2
1
2
1
2
11
2
1
2
1
2
3
2
′
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
{
+
′
−
′
−
′

⎛
⎝
⎜
⎞
⎠
⎟
⎤
⎦
⎥
+
i
ii i i i
cr
cs cr br ar r
sin [
θ
aas br cr bs cr ar cs ar br r
ii iii iii()()()]cos
′
−
′
+
′
−
′
+
′
−
′
⎫

⎬
⎭
θ

(6.3-7)
222 SYMMETRICAL INDUCTION MACHINES
The torque and rotor speed are related by

TJ
P
pT
erL
=
⎛
⎝
⎜
⎞
⎠
⎟
+
2
ω
(6.3-8)
where J is the inertia of the rotor and in some cases the connected load. The fi rst term
on the right-hand side is the inertial torque. In (6.3-8) , the units of J are kilogram·meter
2

(kg·m
2
) or Joules·second

2
(J·s
2
). Often the inertia is given as a quantity called WR
2
,
expressed in units of pound mass·feet
2
(lbm·ft
2
). The load torque T
L
is positive for a
torque load on the shaft of the induction machine.
6.4. EQUATIONS OF TRANSFORMATION FOR ROTOR CIRCUITS
In Chapter 3 , the concept of the arbitrary reference frame was introduced and applied
to stationary circuits. However, in the analysis of induction machines, it is necessary
to transform the variables associated with the symmetrical rotor windings to the arbi-
trary reference frame. A change of variables that formulates a transformation of the
three-phase variables of the rotor circuits to the arbitrary reference frame is

′
=
′
fKf
qd r r abcr0
(6.4-1)
where

()

′
=
′′′
[]
f
qd r
T
qr dr r
fff
00

(6.4-2)

()
′
=
′′′
[]
f
abcr
T
ar br cr
fff
(6.4-3)

K
r
=
−
⎛

⎝
⎜
⎞
⎠
⎟
+
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟
2
3
2
3
2
3
2
3
cos cos cos
sin sin sin
ββ

π
β
π
ββ
π
ββ
π
+
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥

⎥
⎥
2
3
1
2
1
2
1
2
(6.4-4)

βθθ
=−
r
(6.4-5)
The angular displacement θ is defi ned by (3.3-5) – (3.3-8) , and θ
r
is defi ned by

d
dt
r
r
θ
ω
=
(6.4-6)
The inverse of (6.4-4) is
EQUATIONS OF TRANSFORMATION FOR ROTOR CIRCUITS 223

()
cos sin
cos sin
cos
K
r
−
=−
⎛
⎝
⎜
⎞
⎠
⎟
−
⎛
⎝
⎜
⎞
⎠
⎟
+
⎛
⎝
⎜
⎞
⎠
1
1

2
3
2
3
1
2
3
ββ
β
π
β
π
β
π
⎟⎟
+
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢

⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
sin
β
π
2
3
1
(6.4-7)
The r subscript indicates the variables, parameters, and transformation associated
with rotating circuits. Although this change of variables needs no physical interpreta-
tion, it is convenient, as in the case of stationary circuits, to visualize these transforma-
tion equations as trigonometric relationships between vector quantities, as shown in
Figure 6.4-1 .
It is clear that the above transformation equations for rotor circuits are the trans-
formation equations for stationary circuits, with β used as the angular displacement of
the arbitrary reference frame rather than θ . In fact, the equations of transformation for
stationary and rotor circuits are special cases of a transformation for all circuits, station-
ary or rotating. In particular, if in β , θ
r
is replaced by θ
c

, where

d
dt
c
c
θ
ω
=
(6.4-8)
then ω
c
, the angular velocity of the circuits, may be selected to correspond to the
circuits being transformed, that is, ω
c
= 0 for stationary circuits and ω
c
= ω
r
for rotor
circuits. Although this more general approach could have been used in Chapter 3 , it
does add to the complexity of the transformation, making it somewhat more diffi cult
to follow without deriving any advantage from the generality of the approach, since
only two types of circuits, stationary or fi xed in the rotor, are considered in
this chapter.
Figure 6.4-1. Transformation for rotating circuits portrayed by trigonometric relationships.
f
br
¢
f

cr
¢
f
dr
¢
f
ar
¢
f
qr
¢
w
r
q
r
w
q
b
224 SYMMETRICAL INDUCTION MACHINES
It follows that all equations for stationary circuits in Section 3.3 and Section 3.4
are valid for rotor circuits if θ is replaced by β and ω by ω − ω
r
. The phasor and steady-
state relations for stationary circuits, given in Section 3.7 , Section 3.8 , and Section 3.9 ,
also apply to rotor circuits of an induction machine if we realize that the rotor variables,
during balanced, steady-state operation are of the form

′
=
′

−+
[]
FF t
ar r e r erf
20cos ( ) ( )
ωω θ
(6.4-9)

′
=
′
−+ −
⎡
⎣
⎢
⎤
⎦
⎥
FF t
br r e r erf
20
2
3
cos ( ) ( )
ωω θ
π
(6.4-10)

′
=

′
−+ +
⎡
⎣
⎢
⎤
⎦
⎥
FF t
cr r e r erf
20
2
3
cos ( ) ( )
ωω θ
π
(6.4-11)
where θ
erf
(0) is the phase angle of
′
F
ar
at time zero.
6.5. VOLTAGE EQUATIONS IN ARBITRARY
REFERENCE-FRAME VARIABLES
Using the information set forth in Chapter 3 and in the previous section, we know the
form of the voltage equations in the arbitrary reference frame without any further
analysis [1] . In particular

vri
qd s s qd s dqs qd s
p
00 0
=++
ω
ll
(6.5-1)

′
=
′′
+−
′
+
′
vri
qd r r qd r r dqr qd r
p
00 0
()
ωω
ll
(6.5-2)
where

()l
dqs
T
ds qs

=−
[]
λλ
0
(6.5-3)

()
′
=
′
−
′
[]
l
dqr
T
dr qr
λλ
0
(6.5-4)
The set of equations is complete once the expressions for the fl ux linkages are deter-
mined. Substituting the equations of transformation, (3.3-1) and (6.4-1) , into the fl ux
linkage equations expressed in abc variables (6.2-18) , yields the fl ux linkage equations
for a magnetically linear system

l
l
qd s
qd r
ss s ssr r

rsr
T
sr
0
0
11
1
′
⎡
⎣
⎢
⎤
⎦
⎥
=
′
′′
−−
−
KL K KL K
KL K K
() ()
()() LLK
i
i
rr
qd s
qd r
()
−

⎡
⎣
⎢
⎤
⎦
⎥
′
⎡
⎣
⎢
⎤
⎦
⎥
1
0
0
(6.5-5)
We know from Chapter 3 that for L
s
of the form given by (6.2-6)
VOLTAGE EQUATIONS IN ARBITRARY REFERENCE-FRAME VARIABLES 225

KL K
ss s
ls M
ls M
ls
LL
LL
L

()
−
=
+
+
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1
00
00
00
(6.5-6)
where

LL
Mms
=
3
2
(6.5-7)
Since
′

L
r
is similar in form to L
s
, it follows that

KL K
rr r
lr M
lr M
lr
LL
LL
L
′
=
′
+
′
+
′
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥

⎥
−
()
1
00
00
00
(6.5-8)
It can be shown that

KL K K L K
ssr r r sr
T
s
M
M
L
L
′
=
′
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥

⎥
⎥
−−
() ()()
11
00
00
000
(6.5-9)
Similar forms to (6.5-6) and (6.5-8) are obtained if the mutual leakage terms shown in
Chapter 2 are included in the stator and rotor inductance matrices in (6.5-5) . Proving
this is left as a homework exercise at the end of the chapter. The voltage equations are
often written in expanded form. From (6.5-1) and (6.5-2)

vri p
qs s qs ds qs
=+ +
ωλ λ
(6.5-10)

vri p
ds s ds qs ds
=− +
ωλ λ
(6.5-11)

vrip
sss s00 0
=+
λ

(6.5-12)

′
=
′′
+−
′
+
′
vri p
qr r qr r dr qr
()
ωωλ λ
(6.5-13)

′
=
′′
−−
′
+
′
vri p
dr r dr r qr dr
()
ωωλ λ
(6.5-14)

′
=

′′
+
′
vrip
rrr r00 0
λ
(6.5-15)
Substituting (6.5-6) , (6.5-8) , and (6.5-9) into (6.5-5) yields the expressions for the fl ux
linkages. In expanded form

λ
qs ls qs M qs qr
Li L i i=+ +
′
()
(6.5-16)

λ
ds ls ds M ds dr
Li L i i=+ +
′
()
(6.5-17)

λ
00slss
Li=
(6.5-18)
226 SYMMETRICAL INDUCTION MACHINES

′
=
′′
++
′
λ
qr lr qr M qs qr
Li L i i()
(6.5-19)

′
=
′′
++
′
λ
dr lr dr M ds dr
Li L i i()
(6.5-20)

′
=
′′
λ
00rlrr
Li
(6.5-21)
The voltage and fl ux linkage equations suggest the equivalent circuits shown in Figure
6.5-1 .
Since machine and power system parameters are generally given in ohms or in per

unit of a base impedance, it is convenient to express the voltage and fl ux linkage equa-
tions in terms of reactances rather than inductances. Hence, (6.5-10) – (6.5-15) are often
written as
Figure 6.5-1. Arbitrary reference-frame equivalent circuits for a three-phase, symmetrical
induction machine.
+
+
–
+
+
–
––
r
r
¢
i
qs
v
qs
¢
¢
v
qr
r
s
L
ls
L
M
L

lr
¢
L
lr
¢
wl
ds
¢
(w – w
r
) l
dr
i
qr
+–
+
–
+
–
+
+
–
–
r
r
¢
i
ds
i
0s

v
0s
v
ds
¢
¢
v
dr
+
–
¢
¢v
0r
r
s
r
s
L
ls
L
ls
L
M
L
lr
¢
wl
qs
¢
(w – w

r
) l
qr
i
dr
r
r
¢
i
0r
VOLTAGE EQUATIONS IN ARBITRARY REFERENCE-FRAME VARIABLES 227

vri
p
qs s qs
b
ds
b
qs
=+ +
ω
ω
ψ
ω
ψ
(6.5-22)

vri
p
ds s ds

b
qs
b
ds
=− +
ω
ω
ψ
ω
ψ
(6.5-23)

vri
p
sss
b
s00 0
=+
ω
ψ
(6.5-24)

′
=
′′
+
−
⎛
⎝
⎜

⎞
⎠
⎟
′
+
′
vri
p
qr r qr
r
b
dr
b
qr
ωω
ω
ψ
ω
ψ
(6.5-25)

′
=
′′
−
−
⎛
⎝
⎜
⎞

⎠
⎟
′
+
′
vri
p
dr r dr
r
b
qr
b
dr
ωω
ω
ψ
ω
ψ
(6.5-26)

′
=
′′
+
′
vri
p
rrr
b
r00 0

ω
ψ
(6.5-27)
where ω
b
is the base electrical angular velocity used to calculate the inductive reac-
tances. Flux linkages (6.5-16) – (6.5-21) now become fl ux linkages per second with the
units of volts

ψ
qs ls qs M qs qr
Xi X i i=+ +
′
()
(6.5-28)

ψ
ds ls ds M ds dr
Xi X i i=+ +
′
()
(6.5-29)

ψ
00slss
Xi=
(6.5-30)

′
=

′′
++
′
ψ
qr lr qr M qs qr
Xi X i i()
(6.5-31)

′
=
′′
++
′
ψ
dr lr dr M ds dr
Xi X i i()
(6.5-32)

′
=
′′
ψ
00rlrr
Xi
(6.5-33)
In the above equations, the reactances are obtained by multiplying ω
b
times
inductance. It is left to the reader to modify the equivalent circuits shown in Figure
6.5-1 to accommodate the use of reactances rather than inductances in the voltage

equations.
The voltage equations (6.5-10) – (6.5-15) or (6.5-22) – (6.5-27) are written in terms
of currents and fl ux linkages (fl ux linkages per second). Clearly, the currents and fl ux
linkages are related and both cannot be independent or state variables. In transfer func-
tion formulation and computer simulation of induction machines, we will fi nd it desir-
able to express the voltage equations in terms of either currents or fl ux linkages (fl ux
linkages per second).
If the currents are selected as independent variables and the fl ux linkages or fl ux
linkages per second are replaced by the currents, the voltage equations become
228 SYMMETRICAL INDUCTION MACHINES

v
v
v
v
v
v
r
p
XX
p
qs
ds
s
qr
dr
r
s
b
ss

b
ss
0
0
0
′
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
+
ω
ω

ωω
bb
M
b
M
b
ss s
b
ss
b
M
b
M
s
b
ls
b
M
XX
Xr
p
XX
p
X
r
p
X
p
X
ω

ω
ω
ωω
ω
ωω
ω
ω
0
00
00 000
−+ −
+
ωωω
ωω
ωω
ω
ωω
ω
−
⎛
⎝
⎜
⎞
⎠
⎟
′
+
′
−
⎛

⎝
⎜
⎞
⎠
⎟
′
−
−
⎛
⎝
⎜
⎞
⎠
⎟
r
b
Mr
b
rr
r
b
rr
r
b
Xr
p
XX
X
00
MM

b
M
r
b
rr r
b
rr
r
b
lr
p
XXr
p
X
r
p
X
ω
ωω
ωω
ω
00
00000
−
−
⎛
⎝
⎜
⎞
⎠

⎟
′′
+
′
′
+
′
⎡
⎣
⎢
⎢
⎢
⎢⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
′
′
′
⎡
⎣
⎢
⎢
⎢
i
i
i
i
i
i
qs
ds
s

qr
dr
r
0
0
⎢⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥

(6.5-34)
where

XXX
ss ls M
=+
(6.5-35)

′
=
′

+XXX
rr lr M
(6.5-36)
The fl ux linkages per second may be expressed from (6.5-28) – (6.5-33) as

ψ
ψ
ψ
ψ
ψ
ψ
qs
ds
s
qr
dr
r
ss M
ss M
XX
XX
0
0
00 00
0000
′
′
′
⎡
⎣

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
000 000
00 00
0000
00000
X
XX
XX
X
i
ls
Mrr
Mrr
lr

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

qqs
ds
s
qr
dr
r
i
i

i
i
i
0
0
′
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
(6.5-37)
If fl ux linkages or fl ux linkages per second are selected as independent variables, then
(6.5-37) may be solved for currents and written as

i
i
i
i
i
i
D
XX
X
qs
ds
s
qr
dr
r
rr M
rr
0
0
1
00 00
0
′
′
′
⎡
⎣
⎢
⎢
⎢

⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
′
−
′
000 0
00 000
00 00
0000
00000
−
−
−
′
⎡
⎣
⎢
⎢

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
X
D
X
XX
XX
D
X
M
ls
Mss
Mss
lr
⎤⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥

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

ds
s
qr
dr
r
0
0
(6.5-38)
TORQUE EQUATION IN ARBITRARY REFERENCE-FRAME VARIABLES 229
where

DXX X
ss rr M
=
′
−
2
(6.5-39)
Substituting (6.5-38) for the currents into (6.5-22) – (6.5-27) yields the voltage equations
in terms of fl ux linkages per second as given by (6.5-40)

v
v
v
v
v
v
rX
D
pr

qs
ds
s
qr
dr
r
srr
bb
s
0
0
0
′
′
′
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥

⎥
⎥
⎥
=
′
+−
ω
ω
ω
XX
D
rX
D
prX
D
r
X
p
rX
D
rX
M
b
srr
b
sM
s
ls b
rM rs
00

00 0
00 000
00
−
′
+−
+
−
′′
ω
ωω
ω
ss
b
r
b
rM r
b
rss
b
r
lr b
D
p
rX
D
rX
D
p
r

X
p
+
−
−
′
−
−
′
+
′
′
+
ω
ωω
ω
ωω
ωω
ω
0
00 0
0 000 0
⎡⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢

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

′
ψ
ψ
ψ
ψ
ψ
ψ
qs
ds
s
qr
dr
r
0
0
⎡⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥

⎥
⎥
⎥

(6.5-40)
It is interesting to note that each q - and d -voltage equation contains two derivatives of
current when currents are selected as independent or state variables, (6.5-34) . When
fl ux linkages are selected as independent variables, (6.5-40) , each q - and d -voltage
equation contains only one derivative of fl ux linkage. This property makes it more
convenient to implement a computer simulation of an induction machine with fl ux
linkages as state variables rather than with currents.
6.6. TORQUE EQUATION IN ARBITRARY
REFERENCE-FRAME VARIABLES
The expression for the electromagnetic torque in terms of arbitrary reference-frame vari-
ables may be obtained by substituting the equations of transformation into (6.3-6) . Thus

T
P
e s qd s
T
r
sr r qd r
=
⎛
⎝
⎜
⎞
⎠
⎟
[]

∂
∂
′
[]
′
−−
2
1
0
1
0
() ()Ki LKi
θ
(6.6-1)
This expression yields the torque expressed in terms of current as

T
P
Lii ii
e M qs dr ds qr
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜

⎞
⎠
⎟
′
−
′
3
22
()
(6.6-2)
230 SYMMETRICAL INDUCTION MACHINES
where T
e
is positive for motor action. Other equivalent expressions for the electromag-
netic torque of an induction machine are

T
P
ii
eqrdrdrqr
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜

⎞
⎠
⎟
′′
−
′′
3
22
()
λλ
(6.6-3)

T
P
ii
edsqsqsds
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
−

3
22
()
λλ
(6.6-4)
Equations (6.6-3) and (6.6-4) may be somewhat misleading since they seem to imply
that the leakage inductances are involved in the energy conversion process. This,
however, is not the case. Even though the fl ux linkages in (6.6-3) and (6.6-4)
contain the leakage inductances, they are eliminated by the algebra within the
parentheses.
It is interesting and instructive to take a moment to return to Chapter 1 and to begin
the derivation of torque using arbitrary reference-frame variables. From (1.3-21)

dW dW dW
efm
=−
(6.6-5)
where dW
e
is the change of energy entering the coupling fi eld via the electric inputs,
dW
f
is the change of energy stored in the coupling fi eld, and dW
m
is the change of
energy entering the coupling fi eld via the mechanical input. We can turn the energy
balance equation given by (6.6-5) into a power balance equation by dividing (6.6-5)
by dt . Thus,

pW pW pW

efm
=−
(6.6-6)
where p is the operator d / dt . The power input to the coupling fi eld may be expressed
from the voltage equations (6.5-10) – (6.5-15) by multiplying each voltage equation by
the appropriate current and removing the i
2
r terms. Thus, with e
qs
= ν
qs
− i
qs
r
s
, and so
on, we can write from (6.5-10) – (6.5-15) and (3.3-8)

2
3
22
00 0
⎛
⎝
⎜
⎞
⎠
⎟
=++ +
′′

+
′′
+
′′
pW e i e i e i e i e i e
eqsqsdsds ssqrqrdrdr r
ii
r0
(6.6-7)
Removing the ir voltage drop is not necessary; we have done this to be consistent with
the work in Chapter 1 . Appropriate substitution of (6.5-10) – (6.5-15) yields

2
3
00
⎛
⎝
⎜
⎞
⎠
⎟
=+++
′′
+
′′
+
′
pW i p i p i p i p i p i
eqsqsdsds s sqrqrdrdr
λλλλλ

000rr
ds qs qs ds dr qr qr dr dr qr qr
p
iiii i
′
+−+
′′
−
′′
−
′′
−
′
λ
ωλ λ λ λ λ λ
()(
′′
ip
dr r
)
θ
(6.6-8)
Comparing (6.6-8) with (6.6-6) , it is clear that the right-hand side of (6.6-4) is
pW
f
− pW
m
. Now
TORQUE EQUATION IN ARBITRARY REFERENCE-FRAME VARIABLES 231

pW T p
merm
=−
θ
(6.6-9)
where θ
rm
is the actual angular displacement of the rotor and

θθ
rrm
P
=
⎛
⎝
⎜
⎞
⎠
⎟
2
(6.6-10)
where θ
r
is the electrical angular displacement and P is the number of poles. Therefore,
(6.6-9) can be expressed in terms of the electrical angular velocity of the rotor as

pW T
P
p
me r

=−
⎛
⎝
⎜
⎞
⎠
⎟
2
θ
(6.6-11)
Therefore, (6.6-6) can be written as

2
3
2
⎛
⎝
⎜
⎞
⎠
⎟
=+
⎛
⎝
⎜
⎞
⎠
⎟
pW pW T
P

p
efe r
θ
(6.6-12)
If we compare (6.6-12) with (6.6-8) , and if we equate coeffi cients of p θ
r
, we can express
torque, positive for motor action, as

T
P
ii
eqrdrdrqr
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
′′
−
′′
3

22
()
λλ
(6.6-13)
It is important to note that this expression for torque is valid for linear or nonlinear
magnetic systems and that it was arrived at without evaluating fi eld or coenergy. It
would appear that since we are working in the arbitrary reference frame that (6.6-13)
would be valid in all reference frames. This is not the case; there is one exception. In
the rotor reference frame where p θ = p θ
r
, the qr ′ and dr ′ variables disappear as coef-
fi cients of p θ
r
and the expression for torque becomes

T
P
ii
eds
r
qs
r
qs
r
ds
r
=
⎛
⎝
⎜

⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
−
3
22
()
λλ

(6.6-14)
where the raised r is used to denote variables in the rotor reference frame. It is interest-
ing that this approach for deriving an expression for torque was used by Park in his
1929 paper [2] ; however, his expression was the negative of (6.6-14) since he was
dealing with a synchronous generator and considered power output as positive where
we are considering power input as positive. There is another slight difference; Park did
not consider a “complete” power balance since he did not include the fi eld or damper
circuits in his derivation. However, since, in the rotor reference frame, the voltage
equations associated with these circuits do not contain a p θ
r
term; therefore, the expres-
sion for torque given by Park is consistent with the power balance approach.
232 SYMMETRICAL INDUCTION MACHINES
It would seem that if in any reference frame where terms other than
′′

λ
dr qr
i
and
′′
λ
qr dr
i

products are coeffi cients of p θ
r
, then (6.6-13) would not be valid. We found that this
was true when p θ = p θ
r
. Now, the rotor circuits of a synchronous machine are unsym-
metrical since the fi eld winding and the damper windings do not form a symmetrical
two- or three-phase set. Let us backup for a moment, (6.6-8) is valid for a machine
with symmetrical stator windings and symmetrical rotor windings and can be used to
obtain an expression for torque in any reference frame regardless if the magnetic system
is linear or nonlinear. If either the stator or rotor windings are unsymmetrical, then
(6.6-8) can still be used to obtain an expression for torque (linear or nonlinear magnet
system) if we select the reference frame to be where the asymmetry exists; p θ = p θ
r
,
for rotor asymmetry and p θ = 0 for stator asymmetry.
It is also interesting that in the case of a linear magnetic system, the coeffi cient of
p θ in (6.6-8) becomes zero. Whereupon, (6.6-13) becomes valid in all reference frames.
This would seem to fl y in the face of (6.6-14) . However, for a linear magnetic system,
(6.6-13) and (6.6-14) are equal. Moreover,

T
P
Lii ii
e M qs dr ds qr
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
′
−
′
3
22
()
(6.6-15)
is also a valid expression for torque if the magnetic system is linear. Therefore, in the
case of a linear magnetic system we have identifi ed three expressions for torque in
terms of arbitrary reference-frame variables that are identical; (6.6-13) , (6.6-14) with
the raised r removed, and (6.6-15) . The reader may wish to verify these statements.
Also, the above expressions for torque are often written in terms of fl ux linkages
per second and currents. For example, (6.6-13) can be written as

T
P
ii
e
b
qr dr dr qr
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
′′
−
′′
3

22
1
ω
ψψ
()
(6.6-16)
It is left to the reader to show that in terms of fl ux linkages per second the electromag-
netic torque may be expressed as

T
PX
D
e
M
b
qs dr qr ds
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎛

⎝
⎜
⎞
⎠
⎟
′
−
′
3
22
ω
ψψ ψψ
()
(6.6-17)
where D is defi ned by (6.5-39) .
6.7. COMMONLY USED REFERENCE FRAMES
Although the behavior of a symmetrical induction machine may be described in any
frame of reference, there are three that are commonly employed. Namely, the station-
ary reference frame fi rst employed by H.C. Stanley [3] , the rotor reference frame that
is Park ’ s transformation [2] applied to induction machines (Brereton et al. [4] ), and
PER UNIT SYSTEM 233
the synchronously rotating reference frame [5] . The voltage equations for each of
these reference frames may be obtained from the voltage equations in the arbitrary
reference frame by assigning the appropriate speed to ω . That is, ω = 0 for the sta-
tionary, ω = ω
r
for the rotor, and ω = ω
e
for the synchronously rotating reference
frame.

Generally, the conditions of operation will determine the most convenient reference
frame for analysis and/or simulation purposes [1] . If, for example, the stator voltages
are unbalanced or discontinuous and the rotor applied voltages are balanced or zero;
the stationary reference frame should be used to simulate the performance of the induc-
tion machine. If, on the other hand, the external rotor circuits are unbalanced but the
applied stator voltages are balanced, then the reference frame fi xed in the rotor is most
convenient. Either the stationary or synchronously rotating reference frame is generally
used to analyze balanced or symmetrical conditions. Linearized machine equations that
are used to determine eigenvalues and to express linearized transfer functions for use
in control system analysis are obtained from the voltage equations expressed in the
synchronously rotating reference frame. The synchronously rotating reference frame is
also particularly convenient when incorporating the dynamic characteristics of an
induction machine into a digital computer program used to study the transient and
dynamic stability of large power systems. The synchronously rotating reference frame
may also be useful in variable frequency applications if it is permissible to assume that
the stator voltages are a balanced sinusoidal set. In this case, variable frequency opera-
tion may be analyzed by varying the speed of the arbitrary reference frame to coincide
with the electrical angular velocity of the applied stator voltages. A word of caution is
perhaps appropriate. Regardless of the reference frame being used, the stator and rotor
voltages and currents must be properly transformed to and from this reference frame.
In most cases, these transformations are straightforward and can be accomplished
implicitly. However, it may be necessary to actually include or implement a transforma-
tion in the analysis or computer simulation of an induction machine whereupon special
care must be taken.
6.8. PER UNIT SYSTEM
It is often convenient to express machine parameters and variables as per unit quanti-
ties. Base power and base voltage are selected, and all parameters and variables are
normalized using these base quantities. When the machine is being considered sepa-
rately, the base power is generally selected as the horsepower rating of the machine in
volt-amperes (i.e., horsepower times 746). If, on the other hand, the machine is a part

of a power system and if it is desirable to convert the entire system to per unit quanti-
ties, then only one power base (VA base) is selected that would most likely be different
from the rating of any machine in the system. Here we will consider the machine sepa-
rately with the rating of the machine taken as base power.
Although we will violate this convention from time to time when dealing with
instantaneous quantities, the rms value of the rated phase voltage is generally selected
as base voltage for the abc variables while the peak value is generally selected as base
234 SYMMETRICAL INDUCTION MACHINES
voltage for the qd 0 variables. That is, if V
B

(abc)
is the rms voltage selected as base voltage
for the abc variables, then
VV
B qd B abc() ()0
2=
. The base power may be expressed

PVI
B B abc B abc
= 3
()()
(6.8-1)
or

PVI
B B qd B qd
=
⎛

⎝
⎜
⎞
⎠
⎟
3
2
00()()
(6.8-2)
Therefore, since base voltage and base power are selected, base current can be
calculated from either (6.8-1) or (6.8-2) . It follows that the base impedance may
be expressed

Z
V
I
V
P
B
B abc
B abc
B abc
B
=
=
()
()
()
3
2

(6.8-3)
or

Z
V
I
V
P
B
Bqd
Bqd
Bqd
B
=
=
⎛
⎝
⎜
⎞
⎠
⎟
()
()
()
0
0
0
2
3
2

(6.8-4)
The qd 0 equations written in terms of reactances, (6.5-22) – (6.5-33) can be readily
converted to per unit by dividing the voltages by V
B

(

qd

0)
, the currents by I
B

(

qd

0)
, and the
resistances and reactances by Z
B
. Note that since a fl ux linkage per second is a volt, it
is per unitized by dividing by the base voltage.
Although the voltage and fl ux linkage per second equations do not change form
when per unitized, the torque equation is modifi ed by the per unitizing process. For
this purpose, the base torque may be expressed as

T
P
P

B
B
b
=
(/ )2
ω
(6.8-5)
where ω
b
corresponds to rated or base frequency of the machine. A word of caution is
appropriate. If, in (6.8-5) , P
B
is the rated power of the machine, then base torque T
B

will not be rated torque. We will fi nd that in the case of an induction machine, rated
power generally occurs at rated speed that is less than synchronous. Hence, T
B
will be
less than rated torque by the ratio of rated speed to synchronous speed.
ANALYSIS OF STEADY-STATE OPERATION 235
If the torque expression given by (6.6-16) is divided by (6.8-5) , with (6.8-2) sub-
stituted for P
B
, the multiplier
3
2
21
()
(/)(/ )P

b
ω
is eliminated, and with all quantities
expressed in per unit, the per unit torque becomes

Ti i
eqrdrdrqr
=
′′
−
′′
ψψ
(6.8-6)
If the electrical variables are expressed in volts, amperes, and watts, then the inertia of
the rotor is expressed in mks units. If, however, the per unit system is used the inertia
is expressed in seconds. This can be shown by fi rst recalling from (6.3-8) that the
inertial torque T
IT
for a P -pole machine may be expressed

TJ
P
p
IT r
=
⎛
⎝
⎜
⎞
⎠

⎟
2
ω
(6.8-7)
where ω
r
is the electrical angular velocity of the rotor and J is the inertia of the rotor
and connected mechanical load expressed in kg·m
2
. In order to express (6.8-7) in per
unit, it is divided by base torque, and the rotor speed is normalized to base speed. Thus

T
JP
T
p
IT
b
B
r
b
=
(/ )2
ωω
ω
(6.8-8)
By defi nition, the inertia constant expressed in seconds is

H
P

J
T
P
J
P
b
B
b
B
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛

⎝
⎜
⎞
⎠
⎟
1
2
2
1
2
2
2
2
ω
ω
(6.8-9)
Thus, in per unit (6.3-8) becomes

THp T
e
r
b
L
=+2
ω
ω
(6.8-10)
It is important to become familiar with both systems of units and to be able to convert
readily from one to the other. We will use both systems interchangeably throughout
the text.

6.9. ANALYSIS OF STEADY-STATE OPERATION
The voltage equations that describe the balanced steady-state operation of an induction
machine may be obtained in several ways. For balanced conditions, the zero quantities
of the stator and rotor are zero, and from our work in Chapter 3 , we know that for
balanced steady-state conditions, the q and d variables are sinusoidal in all reference
236 SYMMETRICAL INDUCTION MACHINES
frames except the synchronously rotating reference frame wherein they are constant.
Hence, one method of obtaining the steady-state voltage equations for balanced condi-
tions is to fi rst recall that in an asynchronously rotating reference frame the steady-state
voltages are related by


FjF
ds qs
=
(6.9-1)
and with θ (0) = 0


FF
qs as
=
(6.9-2)
Since the induction machine is a symmetrical device, (6.9-1) and (6.9-2) also apply to
the stator currents and fl ux linkages. Likewise, the steady-state rotor variables are
related by


′
=

′
FjF
dr qr
(6.9-3)
and with θ (0) and θ
r
(0) both selected equal to zero


′
=
′
FF
qr ar
(6.9-4)
Appropriate substitution of these equations into either (6.5-22) and (6.5-25) or (6.5-23)
and (6.5-26) yields the standard steady-state voltage equations in phasor form. A second
method is to express (6.5-22) , (6.5-23) , (6.5-25) , and (6.5-26) in the synchronously
rotating reference frame. Since for balanced steady-state operation the variables are
constants, the time rate of change of all fl ux linkages is zero. Therefore, the phasor
voltage equations may be derived by employing the relationships

2

FFjF
as qs
e
ds
e
=−

(6.9-5)

2

′
=
′
−
′
FF jF
ar qr
e
dr
e
(6.9-6)
We will proceed using the fi rst approach and leave the latter as an exercise for the
reader.
If, in (6.5-22) and (6.5-25) , p is replaced by j ( ω
e
− ω ), the equations may be written
in phasor form as



VrI j
qs s qs
b
ds
e
b

qs
=+ +
−
⎛
⎝
⎜
⎞
⎠
⎟
ω
ω
ψ
ωω
ω
ψ
(6.9-7)



′
=
′′
+
−
⎛
⎝
⎜
⎞
⎠
⎟

′
+
−
⎛
⎝
⎜
⎞
⎠
⎟
′
VrI j
qr r qr
r
b
dr
e
b
qr
ωω
ω
ψ
ωω
ω
ψ
(6.9-8)
Substituting (6.9-1) and (6.9-3) into the above equations yields
ANALYSIS OF STEADY-STATE OPERATION 237




VrI j
qs s qs
e
b
qs
=+
ω
ω
ψ
(6.9-9)



′
=
′′
+
−
⎛
⎝
⎜
⎞
⎠
⎟
′
VrIj
qr r qr
er
b
qr

ωω
ω
ψ
(6.9-10)
The well-known steady-state phasor voltage equations, which are actually valid in all
asynchronously rotating reference frames, are obtained by substituting the phasor form
of (6.5-28) and (6.5-31) for

ψ
qs
and

′
ψ
qr
, respectively, into (6.9-9) and (6.9-10) , and
employing (6.9-2) and (6.9-4) to replace qs and qr variables with as and ar variables,
respectively.


VrjXIjXII
as s
e
b
ls as
e
b
Mas ar
=+
⎛

⎝
⎜
⎞
⎠
⎟
++
′
ω
ω
ω
ω
()
(6.9-11)



′
=
′
+
′
⎛
⎝
⎜
⎞
⎠
⎟
′
++
′

V
s
r
s
jXI jXII
ar r e
b
lr ar
e
b
Mas ar
ω
ω
ω
ω
()
(6.9-12)
where the slip s is defi ned as

s
er
e
=
−
ωω
ω
(6.9-13)
Equations (6.9-11) and (6.9-12) suggest the equivalent circuit shown in Figure 6.9-1 .
All voltage equations are valid regardless of the frequency of operation in that ω
e

and ω
b
are both given explicitly. Since ω
b
corresponds to rated frequency, it is generally
used to calculate and per unitize the reactances. Although the ratio of ω
e
to ω
b
is gener-
ally not included in the steady-state voltage equations, this ratio makes (6.9-11) and
(6.9-12) valid for applied voltages of any constant frequency. It should be clear that all
products of the ratio of ω
e
to ω
b
times a reactance may be replaced by ω
e
times the
appropriate inductance.
The steady-state electromagnetic torque can be expressed in terms of currents in
phasor form by fi rst writing the torque in terms of currents in the synchronously rotating
Figure 6.9-1. Equivalent circuit for steady-state operation of a symmetrical induction
machine.
+
–
+
–
r

r
r
s
¢¢
¢
s
V
as
¢V
a
r
I
as
I
ar
s
w
e
w
b
jX
ls
w
e
w
b
jX
M
w
e

w
b
jX
lr
~
~
~
~
238 SYMMETRICAL INDUCTION MACHINES
reference frame and then utilizing (6.9-5) and (6.9-6) to relate synchronously rotating
reference frame and phasor quantities. The torque expression becomes

T
PX
jI I
e
M
b
as ar
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞

⎠
⎟
′
3
2
ω
Re[ ]
*

(6.9-14)
where

I
as
*
is the conjugate of

I
as
.
The balanced steady-state torque–speed or torque–slip characteristics of a singly
excited (singly fed) induction machine warrants discussion. The vast majority of induc-
tion machines in use today are singly excited, wherein electric power is transferred to
or from the induction machine via the stator circuits with the rotor windings short-
circuited. Moreover, a vast majority of the singly excited induction machines are of the
squirrel-cage rotor type. For singly fed machines,

′
V
ar

is zero, whereupon


′
=−
′
+
′
I
jX
rsj X
I
ar
ebM
rebrr
as
(/)
/(/)
ωω
ωω
(6.9-15)
where
′
X
rr
is defi ned by (6.5-36) . Substituting (6.9-15) into (6.9-14) yields

T
PXr
s

I
r
s
e
e
b
M
b
r
as
r
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟

′
⎛
⎝
⎜
⎞
⎠
⎟
′
⎛
⎝
⎜
⎞
⎠
⎟
3
2
2
2
ω
ωω

22
2
2
+
⎛
⎝
⎜
⎞
⎠

⎟
′
ω
ω
e
b
rr
X
(6.9-16)
Now, the input impedance of the equivalent circuit shown in Figure 6.9-1 , with

′
V
ar

equal to zero, is

Z
rr
s
XXX j
r
s
XrX
sr e
b
Mssrr
e
b
r

ss s rr
=
′
+
⎛
⎝
⎜
⎞
⎠
⎟
−
′
()
+
′
+
′
⎛
⎝
⎜
⎞
⎠
ω
ω
ω
ω
2
2
⎟⎟
′

+
′
r
s
jX
re
b
rr
ω
ω
(6.9-17)
where X
ss
is defi ned by (6.5-35) . Since



I
V
Z
as
as
=
(6.9-18)
the torque for a singly fed induction machine may be expressed as

T
PX
rsV
rr s X

e
e
b
M
b
ras
sr
e
b
M
=
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
′
′
+
⎛
⎝
⎜
⎞

⎠
⎟
3
2
2
2
2
2
ω
ωω
ω
ω

(
−−
′
⎡
⎣
⎢
⎤
⎦
⎥
+
⎛
⎝
⎜
⎞
⎠
⎟
′

+
′
XX rX srX
ss rr
e
b
rss srr
)( )
2
2
2
ω
ω
(6.9-19)
ANALYSIS OF STEADY-STATE OPERATION 239
Figure 6.9-2. Steady-state torque–speed characteristics of a singly excited induction machine.
MotorGenerator
T
e
–
1.0
2.0
–0.6
1.6
–0.2
1.2
1.2
–0.2
1.6
–0.6

2.0
–1.0
0.2
0.8
0.6
0.4
w
r
/w
e
slip
In per unit, (6.9-19) becomes

T
X rsV
rr s X X X
e
e
b
Mr as
sr
e
b
Mssrr
=
′
′
+
⎛
⎝

⎜
⎞
⎠
⎟
−
′
⎡
⎣
⎢
⎤
⎦
⎥
+
ω
ω
ω
ω
2
2
2
2
2

()
ωω
ω
e
b
rss srr
rX srX

⎛
⎝
⎜
⎞
⎠
⎟
′
+
′
2
2
()
(6.9-20)
The steady-state torque–speed or torque–slip characteristics typical of many singly
excited multiphase induction machines are shown in Figure 6.9-2 . Therein, ω
e
= ω
b
.
Generally, the parameters of the machine are selected so that maximum torque occurs
near synchronous speed, and the maximum torque output (motor action) is two or three
times the rated torque of the machine.
An expression for the slip at maximum torque may be obtained by taking the
derivative of (6.9-19) with respect to slip and setting the result equal to zero. Thus

srG
mr
=
′
(6.9-21)

where

G
rX
XXX rX
ebs ss
Mssrr
e
b
srr
=±
+
−
′
⎛
⎝
⎜
⎞
⎠
⎟
+
′
⎡
⎣
⎢
−
(/)
()
ωω
ω

ω
22 2
22
2
22
⎢⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1
2
(6.9-22)
Two values of slip at maximum torque are obtained, one for motor action and one for
generator action. It is important to note that G is not a function of
′
r
r
; thus the slip at
maximum torque is directly proportional to
′
r
r
. Consequently, with all other machine
parameters constant, the speed at which maximum steady-state torque occurs may be
varied by inserting external rotor resistance. This feature is often used when starting