Chapter 2 distributed windings in ac machinery

53

2.1.  INTRODUCTION

Many ac machines are designed based on the concept of a distributed winding. In these

machines, the goal is to establish a continuously rotating set of north and south poles

on the stator (the stationary part of the machine), which interact with an equal number

of north and south poles on the rotor (the rotating part of the machine), to produce

uniform torque. There are several concepts that are needed to study this type of electric

machinery. These concepts include distributed windings, winding functions, rotating

MMF waves, and inductances and resistances of distributed windings. These principles

are presented in this chapter and used to develop the voltage and flux-linkage equations

of synchronous and induction machines. The voltage and flux linkage equations for

permanent magnet ac machines, which are also considered in this text, will be set forth

in Chapter 4 and derived in Chapter 15. In each case, it will be shown that the flux￾linkage equations of these machines are rather complicated because they contain rotor

position-dependent terms. Recall from Chapter 1 that rotor position dependence is

necessary if energy conversion is to take place. In Chapter 3, we will see that the com￾plexity of the flux-linkage equations can be greatly reduced by introducing a change

of variables that eliminates the rotor position-dependent terms.

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.

DISTRIBUTED WINDINGS IN

AC MACHINERY

2

54 Distributed Windings in ac Machinery

2.2.  DESCRIBING DISTRIBUTED WINDINGS

A photograph of a stator of a 3.7-kW 1800-rpm induction motor is shown in Figure

2.2-1, where the stator core can be seen inside the stator housing. The core includes

the stator slots in between the stator teeth. The slots are filled with slot conductors

which, along with the end turns, form complete coils. The windings of the machine are

termed distributed because they are not wound as simple coils, but are rather wound in

a spatially distributed fashion.

To begin our development, consider Figure 2.2-2, which depicts a generic electrical

machine. The stationary stator and rotating rotor are labeled, but details such as the

stator slots, windings, and rotor construction are omitted. The stator reference axis may

be considered to be mechanically attached to the stator, and the rotor reference axis to

Figure 2.2-1. Distributed winding stator.

Figure 2.2-2. Definition of position measurements.

fsm

qrm

Stator

Reference

Axis

Arbitrary

Position

Stator

Rotor

Rotor

Reference

Axis

frm

Describing Distributed Windings 55

the rotor. Angles defined in Figure 2.2-2 include position measured relative to the stator,

denoted by ϕsm, position measured relative to the rotor, denoted by ϕrm, and the position

of the rotor relative to the stator, denoted by θrm. The mechanical rotor speed is the time

derivative of θrm and is denoted by ωrm.

The position of a given feature can be described using either ϕsm or ϕrm; however,

if we are describing the same feature using both of these quantities, then these two

measures of angular position are related by

θ φ rm + = rm φsm (2.2-1)

Much of our analysis may be expressed either in terms of ϕsm or ϕrm. As such, we

will use ϕm as a generic symbol to stand for either quantity, as appropriate.

The goal of a distributed winding is to create a set of uniformly rotating poles on

the stator that interact with an equal number of poles on the rotor. The number of poles

on the stator will be designated P, and must be an even number. The number of poles

largely determines the relationship between the rotor speed and the ac electrical fre￾quency. Figure 2.2-3 illustrates the operation of 2-, 4-, and 6-pole machines. Therein

Ns, Ss, Nr, and Sr denote north stator, south stator, north rotor, and south rotor poles,

respectively. A north pole is where positive flux leaves a magnetic material and a south

pole is where flux enters a magnetic material. Electromagnetic torque production results

from the interaction between the stator and rotor poles.

When analyzing machines with more than two poles, it is convenient to define

equivalent “electrical” angles of the positions and speed. In particular, define

φ φ s s  P m / 2 (2.2-2)

φ φ r r  P m / 2 (2.2-3)

θ θ r r  P m / 2 (2.2-4)

ω ω r r  P m / 2 (2.2-5)

In terms of electrical position, (2.2-1) becomes

Figure 2.2-3. P-pole machines.

P = 2 P = 4

Ns

Ns

Ns

Ss

Ss

Ss

P = 6

Sr

Sr

Nr

Nr Nr

Sr

Ss

Ss

Sr

Sr

Ns Ns

Nr

Nr

Nr

Ss Ns

Sr