ac winding analysis using winding function approach

7/26/2019 AC Winding Analysis using Winding Function Approach

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AC Winding Analysis using Winding Function Approach

Gojko Joksimovi

Department of Electrical Engineering

University of Montenegro, 20000 Podgorica

Montenegro

[email protected]

Abstract One of the crucial parts of every electrical machines course is the introduction to the basic

elements of alternating current (AC) windings design. After introductory part regarding definition

of the full-pitch coil magnetomotive force (MMF), usually continues analysis based on fundamental

MMF harmonic. This paper presents winding function approach to the AC winding analysis.

Although this approach is well known and intensively used in the scientific circles, especially last

few years, authors impression is that this approach is not sufficiently present in the education

process. This approach has at least two principal advantages over the conventional one. Firstly, it

takes into account exact MMF waveform i.e. all of the MMF space harmonics simultaneously.

Secondly, this approach is mainly computer oriented and on such a manner it promotes very

important aspect of electrical engineering education - an interdisciplinary approach. All of the basic

terms in the field as distribution factor, pitch factor, synchronous speed could be and are derived in

this paper, using winding function definition. Additionally, basic elements of Fouriers analysis

using built-in MATLAB

functions are described and its applications are illustrated.

Keywords Turns function, Winding function, MMF, distribution factor, pitch factor, Fourier

analysis

INTRODUCTION

With the advent of low-cost personal computers and various easily accessible software packages,

computer-aided teaching tools have become an essential part of both classroom lectures and

laboratory experiments in electrical machinery education [1]-[5]. These tools have particular


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importance in those cases when it is not convenient to organize laboratory exercises as in the case

of AC winding design and analysis.

The fundamentals of Electric machines are taught at the University of Montenegro in two main

courses. The first course is an introductory course which covers the fundamental principles of

electromechanical energy conversion and transformers. The second one deals with rotational

electrical machines in depth. In the first course, one of the crucial parts is AC winding design and

analysis, one of the most difficult topics in electric machines for students to understand. This is why

a special attention must be given to this topic with extensive use of graphics and animation.

Presented approach to this complex matter at the Department of Electrical Engineering at

University of Montenegro was carried out last two years. After learning the materials in the

traditional manner, by presentation of slides and animation of rotating MMF wave, the students take

active participation in the modeling of the concrete winding in the MATLAB

environment. During

the classroom lectures students have a chance to explore and look at real winding machine, too, Fig

1. The quality of this way of active participation in the educational process describes a high

percentage of success at the mid-term test: almost 86%.

Fig 1. Stator AC winding in a real induction machine

The described approach facilitates students understanding of a very important concept in electric

machinery: spatial harmonics of MMF. It makes a good basis for distinguishing these from the

others, later introduced time harmonics of voltage or current. The described approach has the


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following additional good side: it provides an animation of the rotating MMF wave. Spectral

analysis of rotating MMF wave at different instants of time assures students of a constant harmonic

content of the wave regardless of its different waveforms in different instants of time.

WINDING FUNCTION DEFINITION

AC windings in electrical machines basically have a twofold purpose. In electrical generators AC

windings are place where electromotive force (emf) should be induced. In electrical motors,

primarily AC windings goal is to produce rotating MMF wave. In any case, AC windings have to be

designed on such a manner that induced EMF in the AC windings or produced rotating MMF wave

consists predominantly of the space-fundamental sinusoidal component.

Keeping in mind that AC windings are a complex subject having a books dedicated to it, [6], [7],

and that are even today interesting from different point of views, [8]-[11], we will treat here first its

basics. Lets look, for the beginning, the simplest form of the stator phase winding. It is a single

concentrated coil withNnturns. Positions of coils sides along stator circumference are described by

mechanical angles 1and 2, Fig 2.

1

2

A

1

2

23

4

C

A

Fig 2. Turns function definition


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Assuming uniform air-gap length i.e. neglecting existence of stator and rotor slots, as well as

assuming infinitely permeable iron, Amperes circuital law, in common shape,

=SC

dd SJlH (1)

get much simpler form,

( ) ( ) ( )= nigHgH 0 (2)

where H() is radial component of magnetic field intensity in the air gap at position described by

angle , gis air gap length, iis coil current and n() is turns function, [12].

From Fig 2 it is obvious that by changing the position of the closed loop side 2 (this closed loop

takes part in the Amperes circuital law), turns function could be defined on following manner:

( )


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From (2) and (8) follows,

( ) ( ) 002

12

0

=

+

dng

iH (9)

( ) ( ) 02

0

2

0

=

+

dng

iH (10)

( ) ( )= ng

iH 0 (11)

where ( )n is the mean value of the turns function. From (11) and (2) follows:

( ) ( ) ( )[ ]inngH = (12)

AsH()gis MMF per definition, last expression could be written as,

( ) ( )[ ] ( )iNinnMMF == (13)

where,

( ) ( ) ( )= nnN (14)

is winding function, [13], [14]. Obviously, winding function is nothing else but MMF per unit

current. In further text terms winding function and MMF per unit current will be used

interchangeable. Winding function could be alternatively written as,

( )


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( )

ac winding analysis using winding function approach

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