8/19/2019 A Unified Wavelet-based Approach to Electrical Machine Modeling

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A unified wavelet-based approach to electrical machine modeling

S. Fedrigoc), . Gandellic),A. Monti < ),

F. Ponci (')

7

ipartimento di Elettrotecnica

Politecnico di Milano

Piazza Leonard0 da Vinci 32

20133 Milano (Italy)

epartment of Electrical Engineering

University of South Carolina

Columbia, SC 29208 USA)

Abstract-

The paper presents an original approach to

electrical machine modeling based on wavelet transformation.

The main purpose of the approach is to give a more detailed

description

of

the field in the airgap removing the hypothesis of

sinusoidal field distribution typical of traditional space-phasor

approach.

As

result, more detailed closed-form analysis is

possible including torque ripple evaluation.

1.

INTRODUCTION

The modeling of air-gap electromotive force for motor

design or motor control has followed different ways. In

particular both numerical, through finite elements

computation and analytical ways have been followed.

All

the

proposed methods are in general time consuming [11-[2] and

are structured to supply high level

of detail. On the other

hand the traditional space phasor approach implies a rough

approximation neglecting the presence of the slots and

assuming ideal materials and ideal conductors distribution.

The method here proposed allows the synthesis of a more

realistic model of the machine still requiring a low

computational effort.

The traditional approach for describing magnetic

interactions within electrical machines air-gap is based on a

simplified space representation of the magnetic field. In

particular, under ideal material conditions, the magneto-

motive force along the air gap is square-shaped as reported in

Figure 1. For calculation purpose only the first harmonic in

space, corresponding to a sinusoidal winding distribution, is

considered.

Figure

1:

electro-magnetic force along the air gap

and

its

irst

space

harmonic within a rotating electrical machine

In this paper we propose a new simplified method to

analyze the magneto-motive force without these assumptions.

This method will lead to satisfactory results compared with

traditional approach both in term of torque and torque ripple

analysis.

Starting from this new approach, we will achieve an

original and unified way to study rotating electrical machines

without the limits related to the electric currents waveform

and physical structures.

To

achieve this result the Haar wavelet approach is used.

Haar wavelets proved to be particularly efficient in

synthetically represent a piecewise constant function with

compact support such as the magnetic field along the air gap

in electrical rotating machines.

11. HAAR RANSFORM

BASICS

The basics of Haar wavelet approach and applications are

widely presented in [l-71. In particular we recall here that the

orthogonal set of Haar functions, har(k,t), is defined from the

Haar mother wavelet

1 i f O l t < 1 / 2

har (l ,t )= -1 if 1 / 2 1 t < l

0 for any other interval

by translation of the non-negative integer

n =

k - and

by contraction of

2'

where j is the scale parameter. The

function har(O,t), equal to unity in the interval [0,1], closes

the orthogonal set.

In our studies we deal with finite sequences of

N

samples

of the analysed signal,

{ x ( n N)}

,where

N =

2'

.

A band-

limited Haar-transformable function is gven whose

maximum wavelet frequency equals to N/2.

Uniform

sampling in the interval [0,1] is carried out, yielding a

sequence of N samples univocally related to the Haar-

sequence of

N

wavelet-coeficients. By ordering these

sequences respectively in the column vectors of samples

xNand coefficients X,

,

he matrix relation follows:

where we have introduced the square matrix [H,]

,

whose

N rows are made up of samples of ordered Haar-functions:

0 7803 70910/01/$10°2001 IEEE

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Due to the orthogonal properties of the Haar basis,

[H 1

[H

]=

[IN

the following synthesis relation can be

inferred

x N = [ H N k N 3

111

THE ELECTROMAGNETIC COUPLING JOINT: A SIMPLE CASE

Let us analyze a simple case of coupling joint (see Figure

2).

We suppose to operate with an ideal iron material with

infinity permeability.

As result of that, we have:

1.

2.

The superposition heorem holds.

The wave-shape of the magneto-motive force is

reported in Figure 1.

..&et us suppose to analyze the magneto-motive force using

Haar wavelets. Simply by inspection, we can conclude that

only one component is present in the transformed domain. If

we put the space reference on the half-way between the two

terminals only the mother wavelet will be present.

This result makes clear

that

wavelets are as comfortable in

this application field as Fourier spectrum used to be in the

traditional space-phasor approach.

The whole waveform is synthesized without any

approximation, with just one coefficient.

Let us now suppose to superimpose the action of a second

coil

so

to create a joint structure where one coil is located on

the external structure and one

on

the internal structure (see

Figure 2 .

Every coil will contribute to the field in the airgap creating

a magneto-motive force with the same waveform.

In

this case we have to consider that the space allocation is

different and then the space reference should be changed in

order to obtain the same result in terms of spectrum

de fition .

Thanks to the linearity of the transformation, the same

result can be obtained simply rotating the columns in the

Haar transformation matrix. In a few words, we can say that

to multiply the Haar matrix by a rotated vector corresponds to

multiply the matrix, with translated columns, by the fix

vector. This is the equivalent of Kennelly-Steinmetz operator

in the phasor theory (e ?.

Combining this result with the previous one, we can

conclude that we are able to superimpose whatever number of

coils in whatever position to define the total magneto-motive

force in the airgap.

Once the magneto-motive force is calculated, it is easy to

develop an equivalent expression to define the magnetic

induction in the airgap thanks to the following relationship:

4

Figure

2:

The coupling joint, the wire current density and

the

generated

magneto-motive force

where:

1.

2.

pais the air permeability

3.

4

b

is the magnetic induction as function of the position

m

is

the magneto-motive force

as

function of the position

t

the size of the airgap

Every coefficient involved is constant so that the spectrum

for the magneto-motive force can be immediately converted

in the spectrum for the magnetic induction.

The main question is now: how many wavelets do we need

to achieve a good signal reconstruction? The answer is

related to the structure of the machine, in particular there is

virtually no approximation if we suppose to use as many

wavelet as the slots in the airgap

1v.

ENERGY

ND TORQUE EVALUATION

Following the typical procedure applied for the space-

phasor approach, we can now define the energy and the

torque in the new domain.

First of all, the Haar transformation is an orthogonal one

and then energy does not change changing the domain. This

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means that we can calculate the energy stored in the airgap

directly in the new domain. The calculation is rather simple

thanks to the linear algebra properties of the Haar domain.

With reference to the simple magnetic joint structure, the

energy stored can be defined as:

jb2d29=

00

T O V C

034

a 6

39

W =

2 0

2

N,,

where

p]

is the vector containing the sampling of the

Let

us

now suppose to apply the Haar transform matrix

magnetic induction, and

Nwv

is the number of samples that

will be equal to the number of wavelets.

4

a42

[HI, we have: 44

046

048

2

Nwllv

2tP Nwm

Q 5

[by[HI'

[HIbl=

n

[bY[b]=

=

n

where I is the length of the magnetic structure.

The main problem now is the calculation of the integral

defined over the airgap.

Working with a discrete-space approach, and considering

the step-wise wave-shape of the magnetic induction, the

following relationship holds:

.

.

.

.

0 1 O P r 4 J 9 1 Q x

In this case we suppose a two-phase winding on the stator

and a single coil on the rotor. For sake of simplicity we

perform the calculation having 4 slots on the stator and

2

on

the rotor. We want to study the torque as a function of the

rotor position in steady state condition, when the rotor is

rotating synchronous with the field created by the stator coils.

The current distribution at the machine air gap surface

shows four pulses (the stator current sources) with variable

amplitude, and two pulses with fixed amplitude but moving

location along the periphery. Applying the same procedure

described in what above, the total field can be calculated

superimposing the action of the three windings.

As result the field will be represented with a restricted

number of coefficients in the Haar domain. After that the

torque can be estimated for every rotor position.

The results of this analysis are reported in Figure 3.

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, .

I :

, , ,

. , . .

. , . .

, .

, I

, , ,

, , ,

Figure 4: the new sampling basis

for

the analysis

of

three-phase synchronous

machine

The torque ripple obtained with respect to the displacement

angle between stator and rotor field is shown in Figure 5.

- ., :

Displacement:

I

. ,. I .

,. ,.

I . I . I O .. 0 I..

.I

.,

*

,

I I , e

I , 1 0 I ,

I . 4 1

3 0

. I

. ~ . ' ' ' ' ' '

Figure

5:

Torque ripple

vs

displacement

for

three phase synchronous

machine

In

case

of

more slots for each phase, the magneto-motive

force presents a more irregular form. Its analysis within the

Haar domain simply needs a larger matrix size. As already

stated, the minimum size of the sampling matrix is related to

the number of slots in the machine.

Once the magneto-motive force is synthesized for each

phase, the torque ripple is obtained. In Figure 6 the torque

ripple is reported for a three phase synchronous machine

with

slots per pole per phase ratio q=3.

VII. AIRGAP

NISOTROPY

One of the limits of space-phasor approach

is

the

hypothesis of constant airgap. The anisotropy condition can

be analysed anyway, but it requires more complex analytical

elaboration like a rotating reference.

The problem can be easily managed in the Haar domain.

The concept of modularity of the calculation, proper of the

proposed approach, is introduced and also reproduced in the

study

of

the constructive characteristics of the machme.

As

shown in what follows, linear algebra allows us to

consider anisotropy in Haar domain as a variable. The

expression in Haar domain is obtained simply by multiplying

the magneto-motive force in the Haar domain by a suitable

matrix

T.

1 6 5

I

I

2 0

3 0

4 0 5 6 70 8 0 9 10

2 . 0 5 I

Figure

6:

Torque ripple

of

a three phase synchronous machine having q=3

Being:

within the Haar domain the following holds:

[Hl.[bl= Po [H l . [ ~x ) ] ,+ l

where

1 0 /g

and follows:

and eventually:

bwnvelet = Po .

TI-

wavelet

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VIII.

APPLICATIONS

A .

Unified approach to torque ripple modeling

The possibility to study machines with a typical current

waveform is one of the main aspects of the proposed

approach. No assumption is needed concerning -current. The

Haar matrix can synthesize the magneto-motive force and

calculate the torque in a closed form.

For

the same reason we

can approach the analysis of DC and AC brushless machine

without any a priori Qfference. Moreover a certain

computational speed makes the application of wavelet

approach possible even in real time.

B. Minimization

o

torque ripple

Coming to on-line application, the method can be easily

applied to determine the optimum current waveform for any

kind of rotating machine in order to limit the torque ripple.

The approach here introduced allows the definition of a

unified approach to the solution of this problem also for non-

classical machines.

In

particular, the approach can be applied to modular

machines where every single winding

is

controlled by a

single inverter, producing a non-typical induction waveform.

IX.

CONCLUSION

A new wavelet based approach in the study of electrical

rotating machines was introduced. The Haar domain

approach allows the analysis of different kind of machines;

the torque and torque ripple calculation without any

assumption related to the current waveform. Anisotropy

of

the machine

can

be handled as well. With this new approach

the differences between machines vanish and a unified way to

study modeling and control is opened.

REFERENCES

[l] K.F.Rasmussen, J;H.Davies, T.J.E.Miller, M.I.McGilp,

M.Olaru Analytical and Numerical Computation of

Air

Gap Magnetic Fields in Brushless Motors with Surface

Permanent Magnets , IEEE Trans. On Industry

Applications,

Vol. 36, No. 6, NovDec 2000

[2] J.De La Ree, N. Boules, Torque Production in

Permanet-Magnet Synchronous Motors ,

IEEE Trans.

On IA, Vo1.25,No. 1, Jan/Feb 1989, pp. 107-112

[3] S. Santoso, E. J. Powers, W. M. Grady - P. Hofmann,

Power Quality Assessment via Wavelet Transform

Analysis , IEEE Trans. Pow er Delivery, Vol. 11 (1996)

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Network Analysis using Wavelet-based Tools , 40th

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A.

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Proc. Pp 96 1-964

[6] A. Monti, M. Eva, A. Gandelli, Wavelet-Based

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[8] A. Gandelli, A. Monti, F. Ponci, State Equations in the

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