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

v ri abcs s abcs abcs = + pl (6.2-1)

v ri abcr r abcr abcr = + pl (6.2-2)

where

fabcs

T

as bs cs ( ) = [ ] fff (6.2-3)

fabcr

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 rs and rr , 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

s sr

sr T r

abcs

abcr

⎡

⎣

⎢ ⎤

⎦

⎥ = ⎡

⎣

⎢ ⎤

⎦

⎥

⎡

⎣

⎢ ⎤

⎦

⎥

L L

L L

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

Ls

ls ms ms ms

ms ls ms ms

ms ms ls

LL L L

L LL L

L L LL

=

+− −

− +−

−− +

1

2

1

2

1

2

1

2

1

2

1

2 ms

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

(6.2-6)

Figure 6.2-1. Two-pole, three-phase, wye-connected symmetrical induction machine.

fs qr

fr ar-axis

as-axis

cs-axis

cr-axis

bs-axis

br-axis

bs

bs¢

br¢

cr¢

cs¢

ar¢

as¢

ar

as

br

cr

cs

wr

vas

vbr vcr

var

vbs

vcs

ias

ibs

iar

ics

icr

ibr

rs

rs

rs

rr

rr

rr

Ns

Ns

Ns Nr

Nr

Nr

+

+

+

+

+

+

218 SYMMETRICAL INDUCTION MACHINES

Lr

lr mr mr mr

mr lr mr mr

mr mr lr

LL L L

L LL L

L L LL

=

+− −

− +−

−− +

1

2

1

2

1

2

1

2

1

2

1

2 mr

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

(6.2-7)

Lsr sr

rr r

= L r

+ ⎛

⎝

⎜ ⎞

⎠

⎟ − ⎛

⎝

⎜ ⎞

⎠

⎟

− ⎛

⎝

⎜ ⎞

⎠

⎟

cos cos cos

cos c

θ θ π θ π

θ π

2

3

2

3

2

3

os cos

cos cos cos

θ θ π

θ π θ π θ

r r

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.

i i abcr ′ = r

s

abcr

N

N (6.2-9)

v v ′

abcr = s

r

abcr

N

N (6.2-10)

l l ′

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 ms Ls

r

= sr (6.2-12)

Thus, we will defi ne

′ =

=

+ ⎛

⎝

⎜ ⎞

⎠

⎟ − ⎛

⎝

⎜ ⎞

⎠

⎟

−

L L sr

s

r

sr

ms

rr r

r

N

N

L

cos cos cos

cos

θ θ π θ π

θ

2

3

2

3

2π θ θ π

θ π θ π

3

2

3

2

3

2

3

⎛

⎝

⎜ ⎞

⎠

⎟ + ⎛

⎝

⎜ ⎞

⎠

⎟

+ ⎛

⎝

⎜ ⎞

⎠

⎟ − ⎛

⎝

⎜ ⎞

cos cos

cos cos

r r

r r ⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥ cosθr

(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 mr L r

s

= ms

⎛

⎝

⎜ ⎞

⎠

⎟

2

(6.2-14)

and if we let

′ = ⎛

⎝

⎜ ⎞

⎠ L L r ⎟

s

r

r

N

N

2

(6.2-15)

then, from (6.2-7)