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AC Winding Analysis using Winding Function Approach
Gojko Joksimovi
Department of Electrical Engineering
University of Montenegro, 20000 Podgorica
Montenegro
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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( )