Chapter 4 permanent magnet ac machines

121

4.1. INTRODUCTION

The permanent-magnet ac machine supplied from a controlled voltage or current source

inverter is becoming widely used. This is attributed to a relatively high torque density

(torque/mass or torque/volume) and ease of control relative to alternative machine

architectures. Depending upon the control strategies, the performance of this inverter–

machine combination can be made, for example, to (1) emulate the performance of a

permanent-magnet dc motor, (2) operate in a maximum torque per ampere mode, (3)

provide a “fi eld weakening” technique to increase the speed range for constant power

operation, and (4) shift the phase of the stator applied voltages to obtain the maximum

possible torque at any given rotor speed. Fortunately, we are able to become quite

familiar with the basic operating features of the permanent-magnet ac machine without

getting too involved with the actual inverter or the control strategies. In particular, if

we assume that the stator variables (voltages and currents) are sinusoidal and balanced

with the same angular velocity as the rotor speed, we are able to predict the predominant

operating features of all of the above mentioned modes of operation without becoming

involved with the actual switching or control of the inverter. Therefore, in this chapter,

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.

PERMANENT-MAGNET

ACMACHINES

4

122 PERMANENT-MAGNET AC MACHINES

we will focus on the performance of the inverter–machine combination assuming that

the inverter is designed and controlled appropriately and leave how this is done to

Chapter 14 .

4.2. VOLTAGE AND TORQUE EQUATIONS IN MACHINE VARIABLES

A two-pole, permanent-magnet ac machine, which is also called a permanent-magnet

synchronous machine, is depicted in Figure 4.2-1 . It has three-phase, wye-connected

stator windings and a permanent-magnet rotor. The stator windings are identical wind￾ings displaced at 120°, each with N s equivalent turns and resistance r s . For our analysis,

we will assume that the stator windings are sinusoidally distributed. The three sensors

shown in Figure 4.2-1 are Hall effect devices. When the north pole is under a sensor,

its output is nonzero; with a south pole under the sensor, its output is zero. During

steady-state operation, the stator windings are supplied from an inverter that is switched

at a frequency corresponding to the rotor speed. The states of the three sensors are used

to determine the switching logic for the inverter. In the actual machine, the sensors are

not positioned over the rotor, as shown in Figure 4.2-1 . Instead, they are often placed

over a ring that is mounted on the shaft external to the stator windings and magnetized

in the same direction as the rotor magnets. We will return to these sensors and the role

they play later.

The voltage equations in machine variables are

v ri abcs s abcs abcs = + pl (4.2-1)

where

()[ ] fabcs T as bs cs = fff (4.2-2)

Us= diag[ ] rrr sss (4.2-3)

The fl ux linkages may be written as

l l abcs s abcs m = + L i ′ (4.2-4)

where, neglecting mutual leakage terms and assuming that due to the permanent

magnet the d -axis reluctance of the rotor is larger than the q -axis reluctance, Ls may

be written as

/s

LLL LL LL ls A B r A B r A B r

=

++ − + − ⎛

⎝

⎜ ⎞

⎠ cos cos cos 2 ⎟ −+ +

1

2

2

3

1

2

2

3

θ θ π θ⎛ π

⎝

⎜ ⎞

⎠

⎟

−+ − ⎛

⎝

⎜ ⎞

⎠

⎟ ++ − ⎛

⎝

⎜ ⎞

⎠

⎟ − 1

2

2

3

2 2

3

1

2

LL LLL A B r ls A B r cos cos θ π θ π L L

LL LL L

AB r

AB r AB r

+ + ( )

−+ + ⎛

⎝

⎜ ⎞

⎠

⎟ −+ +

cos

cos cos ( )

2

1

2

2

3

1

2

2

θ π

θ π θ π ls A B r ++ + L L ⎛

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥ cos2 2

3

θ π

(4.2-5)