ADDENDA

1. Section 3.5, page 57.

The finite difference equations given are valid when

the conductor is non-magnetic or of infinite

permeability. The contour integration technique

described by Sto1118

should be used to obtain the

general result.

2. Fig 5.8, page 134.

The figure is incorrect since it was drawn by assuming

that the primary and secondary currents were in both

space and time antiphase.

NUM=CAL FORDTLATIOIT OP LOW FREQUENCY EDDY CURRENT PROBLEMS

USING MAGNETIC SCALAR POTENTIAL

by

Edward Andrew Hyatt, B.Sc. (Eng.),

July , 1977

A thesis submitted for the degree of

Doctor of Philosophy of the University

of London and for the Diploma of

Imperial College.

Electrical Engineering Department,

Imperial College,

London SW7 2AZ.

2.

ABSTRACT

The thesis is concerned with the calculation of low frequency

electromagnetic fields in the regions of large electrical machines

and transformers where the current density vector is not

unidirectional. The approach employs a magnetic scalar potential

which completely specifies the magnetic field outside current

carrying regions. Within these an electric vector potential is

used to describe the current density and a combination of the two

potentials then yields the magnetic field.

Numerical algorithms were written to determine electro-

magnetic fields using both triangular and rectangular meshes. The

solution technique was applied to two engineering problems of

interest. First to be considered was a linear induction machine

transverse edge effect problem involving the calculation of flux

densities and forces in a situation where the solution domain was

infinite. Two methods of dealing with the infinite domain

difficulty were developed and tested. The second problem was to

determine the magnetic fields in a turbo-generator end region with

a view to estimating stator core end and flux screen power losses.

In both applications experimental observations were made in order

to substantiate the theoretical results.

The problem of organising computer programs for general use

was analysed and a suite of computer programs was developed which

made possible the solution of a wide range of low frequency

electromagnetic field problems not involving magnetic non-

linearity. The possibility of extending the scalar potential

method to non-linear problems was considered and some simple

computations were performed. An attempt was made to allow for the

@ffects of magnetic saturation by modelling saturated solid iron

3. with complex surface impcaances whose values were calculated

using a one dimensional analysis of conventional type.

it.

D7CLARATION

The work described in this thesis was carried out by myself

with the guidance and help of my supervisor, Mr. C.J. Carpenter.

Certain experimental results were provided by colleagues and

acknowledgement of source is given wherever these results are

used.

The Appendix contains two papers which are submitted in

support of the thesis. i.y contribution to each is as follows:

1) 13fficiency of numerical techniques for comouting eddy

currents in two and three dimensions.'

Finite difference formulation, numerical algorithms, computer

programs, computed results, experimental measurements.

2) 'Computation of eddy current losses in solid iron under

various surface conditions'.

The Du Fort — Frankel finite difference formulation in

cylindrical coordinates and its associated computer program.

The Frohlich magnetisation curve results.

We, the undersigned, agree that the above details are correct

and that they form a good assessment of E.A. Wyatt's

contribution to these publications.

5.

ACKUMLIMG3Y2NTS

There are many people who have made the author's work both

challenging and enjoyable. First and foremost, I thank !Ir. C. J.

Carpenter for his patience, even temper and great enthusiasm. His

leadership was most valuable and, although his perseverance

could sometimes cause mild annoyance, he commanded respect.

Thanks are also due to Mr. H. Hindmarsh of C. A. Parsons Ltd.

both for his hospitality and for his help in providing

experimental results. Not every man in such a position of

responsibility would devote as much time to a lowly research

student as did Er. Hindmarsh, and I an deeply grateful.

Several colleagues have been particularly helpful.

Messrs. N. Greatorex, T.G. Bland, D.A. Lowther and R. Ashen

immediately spring to mind. Discussions with these gentlemen

have invariably been both stimulating and useful.

Finally, I thank 0. Szpiro for his proof that telephonic

communication is unnecessary and the Science Research Council for

financial support.

6.

To Elizabeth, for her patience and courage in dealing

with an obstinate would—be author.

Believe not those who say,

the upward path is smooth,

lest thou shouldst stumble in the way,

and faint before the Truth.

7•

ANNE BRONTE

8.

CONTENTS

Chapter Page

1. INTRODUCTION 13

1.1 Statement Of Problem 13

1.2 Factors Contributing To The Choice Of 14

Field Describing Quantities

1.3 Numerical Methods For Electromagnetic 16

Field Problems

2. SOLUTION OF MAGNETOSTATIC PROBLEMS - 19

INTRODUCTION OF THE ELECTRIC VICTOR

POTENTIAL

2.1 The Electric Vector Potential 20

2.2 T Vector Construction 22

2.3 ABoundary Value Construction 26

2.4 Determination Of SZ Using Finite 30

Differences

3. SOLUTION OF EDDY CURRENT PROBLEMS 37 3.1 Electric Vector Potential Eddy Current 40

Formulation

3.2 Finite Difference Solution Of The T 44

Governing Equations

3.2.1 Determination Of The Nodal Equations 45 At Points Distant From

Discontinuities In T

3.2.2 Boundary Conditions Satisfied By T 47

At Conductor Surfaces

3.2.3 Determination Of T At Nodes Lying 48 On Conductor Surfaces

9.

Chapter Page

3.2.4 Determination Of T At Nodes 52

Lying On Interfaces Between

Non-magnetic Conductors And

Infinitely Permeable Non-conducting

Iron

3.3 T Vector Travelling Wave Formulation 54

3.4 The One Component T Formulation 57

3.5 Determination Of SL At The Surfaces 57

Of Magnetic Conductors

3.6 Two Dimensional Triangular Meshes 59

3.6.1 Magnetic Scalar Potential Finite 60

Element Formulation

3.6.2 The Branch Integration Method 62

3.6.3 Implications Of The Branch Method 71

3.7 Determination Of The Force Vector 74

3.7.1 The Volume Integral Method 75

3.7.2 The Surface Integral Method 78

4. SOLUTION METHODS FOR DISCRETIZED 80

ELECTROMAGNETIC FIELD EQUATIONS

4.1 Iterative Solution Using Successive 83

Overrelaxation (SOR)

4.2 An Analysis Of The Iterative 86

Characteristics Of The Rectangular

Mesh Equations

4.3 Behaviour Of SOR In Large Node 96

Number Problems

10.

Chapter Page

4.4 Assembly And Solution Of The 99

Triangular Mesh Matrix Equations

4.4.1 Solution Of Large Numbers Of 105

Asymmetric Linear Simultaneous

Equations Using Gaussian

Elimination

4.4.2 Magnetic Field Contour Plotting 110

For Triangular Meshes

4.4.3 A Sub-optimal Node Numbering 115

System

5. THE LONGITUDINAL FLUX LINEAR INDUCTION 121

MOTOR

5.1 Machine Details And Experimental 125

Results

5.2 Computer Prediction Of Flux 132

Densities And Forces

5.2.1 Programming Details 132

5.2.2 Analysis Of Computed Results 135

5.2.3 Convergence Characteristics Of 137

The Numerical Solutions

6. NUMERICAL SOLUTION OF OPEN 141

BOUNDARY PROBLEMS

6.1 The Coordinate Transformation 142

Method

6.1.1 Derivation Of Finite Difference 146

Equations And Network Models That

Are Valid Within The Transformed

Open Boundary Region

11.

Chapter Page

6.2 Analysis Of A E-Core Plate 153

Levitator

6.2.1 Machine On Open Circuit 155-

6.2.2 Machine With Conducting 159

Secondary

6.2.3 Convergence Properties Of The 169

Numerical Solution

6.3 The Exterior Element Method 172

7. SOLUTION OF TURBO-GENERATOR END 181

FIELD PROBLEMS

7.1 Basic Numerical Formulation 183

And Assumptions

7.2 Geometric Details Of The 188

Generator On Which The Analysis

Was Tested

7.3 Stator Winding Representation 189

7.4 Rotor Winding Representation 194

7.5 Governing Equation For 4 196

Expressed In Cylindrical

Coordinates And Applicable In

Regions Containing Known Values

Of Current Density

7.6 Calculation Of Eddy Currents In 199

The Copper Flux Screen

7.7 Computer Program And Mesh 201

Description

7.8 Excitation Conditions Used For 204

The Computer Solution

12.

Chapter Page

7.9 Characteristics Of The Numerical 204

Solution

7.10 Comparison Of Measured And 207

Calculated Results

7.10.1 Symmetric Short Circuit 210

Conditions

7.10.2 Open Circuit Stator Conditions 220

7.11 Conclusions 221

7.11.1 Determination Of The Short 221

Circuit Air Gap MMF

7.11.2 Representation Of Non-Magnetic 221

Flux Screens

7.11.3 Representation Of Magnetic 222

Saturation

8. SIMPLE REPRESENTATIONS OF 224

MAGNETICALLY NON-LINEAR MATERIALS

CARRYING EDDY CURRENTS

8.1 The Constant Permeability 226

Approximation

8.2 Surface Impedance Methods 228

8.3 Application Of Surface Impedance 235

Methods To An E-Core Plate

Levitator Problem

8.4 Calculation Of Forces Acting On 244

Magnetic Parts

REFERENCES 248

APPENDIX 255

13.

1. INTRODUCTION

1.1 Statement Of Problem

For many years, manufacturers of large turbine generators

have shown interest in the calculation of the magnetic fields

produced by end windings. These fields form an important

contribution to the leakage reactance of the machine. Moreover,

the end windings produce flux that can enter the stator core

ends in the axial direction. Eddy currents are induced, and

these cause undesirable heating effects. The axial flux also

creates high inter—laminar voltages which can break down the

insulation between adjoining stator laminations. Such breakdowns

rapidly cause extensive damage to the core. Flux screens are

employed to reduce the flux densities along the core end surfaces.

These screens contain multi—directional currents and a study of

their effects was required.

Other end—region electromagnetic phenomena have caused

concern. Heating of fans and conducting gas baffles to

unacceptable temperatures has occurred, and ultimately, machine

reliability has been adversely affected.

The initial aim of the investigation was to develop a three

'dimensional eddy current calculation procedure that could be used

by turbine generator designers and that was both relatively

simple and economical. In the event, although a three

dimensional formulation was developed, only two dimensional and

two dimensional travelling wave versions were programmed.

Moreover, since experimental measurements on turbine generators

were difficult to obtain and were sometimes of uncertain accuracy,

two linear induction machines that were available in the

14. laboratory were used for testing purposes. Both a single phase

and a polyphase device was considered, ana attention was directed

to several features of linear machine problems that were not

appropriate for turbo—generator applications. In particular,

the difficulties of representing the effects of open regions

that extended to infinity were investigated. Two techniques for

solving these open boundary problems were developed, tested and

found to be particularly satisfactory.

Finally, the effects of magnetic saturation in conducting

materials were considered in relation to a linear machine having

a solid iron secondary. The conclusions drawn, however, were

equally applicable to turbo—generator problems; particularly

those in which the stator core end, core frame and end doors

were to be represented.

1.2 Factors Contributing Tc The Choice Of Field Describing

Quantities

In the applications considered in the previous section the

magnetic fields are usually three dimensional and are influenced

by the presence of eddy currents. Solutions to these problems

are particularly difficult and time consuming to obtain in terms

,of the conventional field vectors since at least two vector

quantities (E and Hor B and 3, say) are required to describe

the interactions between the electric and magnetic fields. Thus

six vector components are, in general, required within

conductors. By introducing a magnetic vector potential (A)

defined such that its curl is the magnetic flux density (B),

only three vector components need be used to describe the

electromagnetic field everywhere. In consequence, the computation

15. effort can be substantially reduced by using an A formulation

rather than one in which B and J, say, are calculated directly.

In eddy current problems of the transverse magnetic type, A has

only one non—zero component and can be regarded as a scalar.

Three dimensional calculations are more involved since A then

has three non—zero components everywhere. In addition, but

depending on the choice of gauge (divergence) for A, an

electrostatic scalar potential may have to be determined

throughout the solution domain.

It has long been recognised that in three dimensional

magnetostatic problems it is advantageous to introduce a magnetic

scalar potential (0.0 whose negative gradient is the magnetic

field strength (H). Although St cannot describe the magnetostatic

field within conductors, the restriction can easily be removed 1

by employing the 'magnetic shell' principle. It is found that St

satisfies Laplace's, or Poisson's, equation throughout the

solution domain and, since the function is a scalar, only one

variable must be determined at every point. In contrast, both

A and B must have three non—zero components to describe the

field. The major disadvantage of the magnetic• scalar potential

technique is that Si alone cannot be used to describe the

_interactions that occur between the electric and magnetic fields

present in eddy current problems. This property is a consequence

of the requirement that H should satisfy the Kagnetic Circuit Law.

That is, it should have a non—zero curl.

It was decided to investigate ways in which the magnetic

scalar potential method could be extended for use in eddy current

problems. The difficulty seemed to be that although Si could

describe the magnetic field adequately, it could not describe the

16. currents. Thus a vector whose function was to describe current

density was introduced. This vector was related to J in the same

way as the magnetic vector potential was to B, and consequently

was termed the 'electric vector potential'. The symbol used to

represent this function was T.

The T vector exhibits the useful property that its value

need only be calculated within conductors. Furthermore, only two

non—zero components of T are required to describe three of J.

Thus, when a three dimensional T — eddy current solution is

required, only one function must be computed within insulators

(including free space) and a maximum of three functions must be

computed elsewhere.

The thesis contains an exposition of the T'— f, method as

it relates to the calculation of both magnetostatic and eddy

current problems. A more detailed comparison of the relative

advantages and disadvantages of using the magnetic and electric•

vector potentials is contained in Chapter 3, whilst Chapters 5 to

8 inclusiize describe typical applications. Displacement currentr

which is negligible at power frequencies, has been neglected

throughout.

1.3 Numerical Methods For Electromagnetic Field Problems

The various electromagnetic field formulations fall into

three major categories. In the first, an integral equation is

derived and solved by summing the contributions to the field

from each elemental region. The method is ideal for open regions

. containing small amounts of material having simple shape, since

the elemental volumes are confined to conductors and magnetic

materials. It has been used very successfully in the solution of

11E A similar technique has been described in the following puolications: 1. WOLFF W.,'Ihree dinensional eddy current. ca1cu1ations',Comumar•

Proceedins,hutherforu Labcratory,1976,pp231-240. 2. limmerische Losungen der P:axwellschen Gleichungen',ISSh O043-6601

Wiss.L3er.ALEFIUKZN,Voi.li9,1976,14r.3. (In German). 3. DE:ELM:IAN K.S., ChECHURIN V.L. & SARMA M.S.,'Scalar potential concept

for calculating the steady magnetic field and eddy currents', Joint KM-Intermag Conf.,Pittsburgh,P.A.,1976.

17. 2,3

three dimensional non—linear magnetostatic problems, but the

eddy current formulation is complicated and may be uneconomic.

Only in exceptional circumstances can integral equations be

solved analytically. ilumerical techniques produce a large order

matrix equation whose coefficient matrix is full and, in general,

is not diagonally dominant. 3oth matrix inversion and iteration

have been used to solve the simultaneous equations.

The second category includes techniques that represent the

field quantities in terms of partial differential equations which

are then solved directly subject to given boundary conditions.

This approach is well suited to closed region problems. When the

boundary and conductor geometry is simple, the solution domain

may be regarded as a single region and an analytical solution,

usually in terms of a Fourier series, obtained. More complicated

problems are difficult to solve in terms of analytical functions

and recourse is usually made to numerical methods of which that

6 of 'finite differences' is best known. A set of simultaneous

equations is formed and it is found that the coefficient matrix

is of high order, sparse, and often diagonally dominant. Iteration

is thus the most popular solution method. 4,5,7,8

The third category of formulations includes the variational

and weighted residual methods. Variational techniques have

assumed great popularity in recent years but their validity has

been mathematically proved only for linear, non—dissipative,

scalar problems. In essence, the method involves the derivation

of a functional that normally is associated with stored energy

considerations. This functional is then maximised with respect to

every point in space. When numerical solution methods are

employed, a set of simultaneous equations results. These

18.

equations have no clearly defined general characteristics since

their properties depend on the way in which the field variables

are allowed to vary within the elemental regions. It is usual to

refer to these regions as finite elements, though this tern is

vague and is often used to describe discretizations associated

with other field solution procedures. In contrast to variational

techniques, the weighted residual method Sis generally applicable.

Surprisingly, however, it has rarely been used to solve

electromagnetic field problems.

In this thesis, which is primarily concerned with closed

region boundary value problems, the partial differential equation

formulations will be used throughout.

19.

2. SOLUTION OF 1UGNETOSTATIC P:ZO3LE1:3 — INTRODUCTION OF THE

ELECTRIC VECTOR POTENTIAL

Although the thesis is primarily concerned with the

calculation of eddy currents, it is necessary to consider the

representation of windings carrying fixed currents as a

preliminary; both because such windings are usually present in

eddy current problems and because it is then possible to present

the essence of the magnetic scalar potential approach not

complicated by the introduction of Faraday's Law.

Electromagnetic field problems usually involve the

calculation of vector quantities, such as flux density and force,

which have in general got three non—zero components. However, it

has long been recognised that the complexity of field computations

can be reduced by introducing (where possible) a scalar field

quantity whose gradient yields the components of the vector

ultimately required. Thus, in three dimensional electrostatic

problems requiring a knowledge of the electric field strength (E)

distribution within a given region, an electrostatic scalar

potential (0) defined by

E 7 —grad 95 2.1

may be used in order to reduce the number of functions calculated

at each point in space from three to one.

The major limitation of the scalar potential technique is

that it requires the field vector described by the potential to

have zero curl. Proof of this may be obtained using equation 2.1

for if we take the curl of both sides we obtain the relation

curl E = curl (— grad 0) 2.2

which is zero by vector identity. Now the magnetic circuit law

shows H to have curl sources (the currents) so that a scalar

20.

potential of the type defined by 2.1 cannot be used.

One way of circumventing this difficulty is to employ the

magnetic vector potential function A which is related to the flux

density B by

curl A = B 2.3

In unidirectional electric-two dimensional magnetic problems A

has one non-zero component whilst B has two and the advantages

usually ascribed to the use of scalar potentials are obtained

without the disadvantage of requiring 3 to be non-solenoidal.

However, in two and three dimensional electromagnetic problems

A has more than one non-zero component and its use becomes

complicated, particularly in non-linear media where the components

of A are interrelated at every point. In this chapter we shall

investigate the Possibility of re-formulating the scalar potential

method so that it may be used to solve magnetostatic problems

involving multi-directional currents.

2.1 The Electric Vector Potential

In regions where currents flow, a magnetic scalar potential,

(denoted by A) used alone is insufficient to describe the

electromagnetic field interactions. If, however, it is

supplemented by a vector function T, which is related to H and a,

through the equation

H = T - grad SL 2.4

is employed, this restriction is removed. Equation 2.4 defines 10,11

a magnetic shell transformation and since the magnetic circuit

law requires that

curl H = J 2.5

T is related to the current density by the equation

21.

curl T = J 2.6

Thus T and H have the same curl. In magnetostatic problems J is

specified and it remains to construct a T vector such that 2.6

is satisfied together with the condition that the currents are

contained within conductors. In vector notation this condition

is

n . J = 0 2.7

where n is a local unit vector perpendicular to the surface of

any conductor in contact with insulating material. Substitution

of 2.6 into 2.7 yields

n . curl T = 0 2.8

which may be rearranged to yield

div (T x n) = 0 2.9

This equation allows a multitude of directions for T since its

general solution is

Txn= c 2.10

where c is avector function . If this vector is set to zero, the

direction of T at a conductor surface is perpendicular to that

surface. However, such a condition places no restriction on the

direction of T within the main body of a conductor.

The governing equation for the magnetic scalar potential, 4,

is obtained by applying the continuity condition imposed on

magnetic flux density. Since

div B = 0 2.11

and B =p1I, taking the divergence of 2.4 pre—multiplied by F.

yields

div µ grad4= div µT 2.12

We now have a governing equation for ,that is valid in both

magnetically linear and non—linear problems and which has been

22.

easily obtained without recourse to physical reasoning. It is

interesting to note that when 11910, 2.12 reduces to

r7 ^ V 2 J G = div T

and this accords with the magnetic pole formulation of electro-

10 magnetic phenomena when

p = 1 div T 2.14

where py is the pole volume density. Thus T may be thought of as

a magnetic polarisation vector. However, since T is also a current

describing function defined by 2.6 and is the electric analogue

of the magnetic vector potential A, its most appropiate title is

the electric vector potential and this is the one that we shall

use throughout.

One consequence of adopting the T vector formulation is

that the condition

div J = 0 2.15

is automatically satisfied as can be seen by taking the divergence

of 2.6 and noting that the divergence of a curl is always zero.

Thus, 2.12 embodies all the magnetostatic Eaxwell equations

and this is an advantage, particularly when analytical methods are

employed to obtain the Afield.

2.2 T Vector Construction.

T may be given any orientation provided that T and J are not

codirectional. This condition is imposed by the curl relationship

which links these two quantities. In mathematical terms we require

that

TxJ* 0 2.16

2.13

One normally chooses a direction for T that results in the

minimum computational complexity when an SL solution is attempted.

23.

A del operator property of relevance here is that the curl of a

vector confined to a plane has non—zero components in all three

coordinate directions. If we confine the direction of T to such a

plane, this property allows all three components of current

density to be described. For example, consider a Cartesian

reference frame and let T = 0 . Then the curl T = J relationship

yields

J = 8Tz 2.17 ; J = 8Tx - aTz 2.18 ; J = - 6 x 2.19 . x ay Y az ax z ay

These equations indicate that T can be determined uniquely from

Jx

and similarly Tx

from Jz. Once Jx

and Jz have been described

by Tz and Tx respectively then the (implied) condition that the

divergence of J is zero yields J directly and this is reflected

in 2.18 .

Integrating 2.17 and 2.19 with respect to y yields

Tz = Jx dy fi(x,z) 2.20 and Tx = - f Jz dy Y f2(x,z) 2.21

where f1 and f2

are functions of integration. They are independent

of y and may be arbitrarAl set to zero since they alter the

absolute value of T but not its y directed gradient.

Equation 2.20 and 2.21 are particularly interesting

since they reveal that in problems exhibiting symmetry in the

y-z plane T is confined to the conductors and to any holes t

which they may contain. As an example of how the T distribution

might be calculated in a general magnetostatic problem let us

consider the asymmetric busbar geometry of Fig 21. The busbars

are assumed to have equal cross sectional area and to contain x

directed currents evenly distributed across each conductor section.

We shall restrict our attention to the determination of Tz1 since

. T may be obtained by applying the same procedure to J which is here

lt Note that by setting fiend f2to zero, H and T are made to be very different functions. Although both have the some curl, the divergence of T is position dependent whilst that of H is zero (except where p varies). T and H can be made identical by choosing fl and f2 to be functions such that divf=O in regions where p is constant.

We here define a 'hole' as a non-conducting region in which T is non-zero. Thus around each hole a circulating current must flow.

y

riMaaat ••••••■ ■••••■■•

oj

■•■•••••• ONNINIma sm•■•■• ■•■ •••■••

A

C

B

L >

.■■■=0111, ■■•=0. .11■111b ..m1..1110 .11■•■■• .01■1•111, ■ ••■• ••••■•

- X

Fig 2.1

<11•••■■•■■■• L

24.

F

assumed to be zero. Now let us apply 2.20 to the y directed

line marked by letters A and B in Fig 2.1.To the left of point A

Jx is zero and there is no contribution to T. In consequence,

this function is zero at point A. However, between points A and B

Jx

is non—zero but constant and Tz is thus required to vary

linearly with distance. From point B to +m no further non—zero

Jx is encountered and Tz remains constant. Hence Tz is not

restricted to the region near the conductors but extends to

infinity. This property can cause computational difficulties in

solution domains not surrounded by iron. Chapter 6 contains an

analysis of the problems involved when open boundaries must be

represented and the techniques described there may be easily

extended to the application of Fig 2.1 .

Along y directed

lines which out both conductors/Tz is

distributed differently. Consider, for example, the line marked

by the letters C D E and F. As before, it is found that Tz is

zero to the left of point C and isanon—zero but linear function

of distance between points C and D. Between D and E Tz is

constant but at point E it begins to decline since negative Jx

25.

is encountered. At F, Tz is zero and t remains so beyond this

point since no more non—zero x directed current density is

traversed. Fig 2.2 contains a map of the Tz distribution in which

the arrows denote non—zero values. Note that discontinuities in

Tz

occur at all the conductor surfaces which lie in the x—y plane.

Furthermore, a discontinuity occurs within the right hand

conductor as well as at its surface. At the surfaces of the left

.11

Fig 2.2

hand conductor which are in the x—z plane we find that Tz is

continuous although its gradient with respect to y is not. This

property reflects the fact that Jx is zero outside the conductor

but non—zero at its surface. The discontinuities may be

interpreted as magnetic poles 10. T then becomes a polarisation

vector whose divergence is the magnetic pole strength.

The discontinuities in T are important features which can

be difficult to represent correctly in differential terms. Let us

make the reasonable, but unnecessary, assumption that 11 is

constant within the solution domain. Then 2.12 yields

N72,11 = div T

2.13

which is the governing equation for .derived from the condition

26.

that the magnetic flux density is continuous. 2quation 2.13

is valid wherever the derivatives of both T andst are continuous

and defined. It should thus not be used at T discontinuities.

Instead, the div 3 = 0 condition should be re-formulated to

yield

a - a4) a - aJL) = -n

TE1 tl ST1 at2 -2 at2 an 2.22

where n is the direction normal to the discontinuity and n, ti

and t2 form a right handed local coordinate set. All the

derivatives in 2.22 are defined since Hn is continuous across

any T discontinuity. This procedure may seem pedantic but it has

direct relevance to the discretization techniques adopted later

in this chapter.

Note thatthe divergence of T is zero away from

discontinuities. This is a result of assuming the currents to be

evenly distributed across the conductors. If any other distribution

had been assumed, divergence sources would have existed wherever

T was non-zero. However, the discontinuities always dominate the

overall magnetic scalar potential distribution.

2.3 06-Boundary Value Construction

We have so far considered magnetostatic field problems not

involving the representation of ferro-magnetic material. Such

Problems are rare in practice because most magnetic field

applications require the maximisation of magnetic inductance

and this can only be achieved by introducing materials whose

relative permeabilities are greater than unity. The requirement

usually leads designers to place ferro-magnetic material (which

we shall call 'iron') in the vicinity of the current carrying

27.

conductors. At the surfaces of the iron certain boundary

conditions must be satisfied and the effect of these on the T

and %%distribution will here be examined.

Oven when iron is highly saturated its absolute permeability

is very much greater than that of free space. Oonsequently, an

approximation often made is that the iron is infinitely permeable.

This approximation makes it possible to neglect the magnetic field

inside the iron and to perform a field solution in low

permeability regions only. In a later chapter (8) the non—linear

aspects of introducing iron into a magnetic circuit will be

considered but here we shall assume the infinite permeability

representation.

Let us first consider the boundary conditions which must

be satisfied at the surface of an isolated iron block placed in

a region such that T is —zero at all its surfaces. Within the

iron the flux density is finite. Moreover, since the permeability

is assumed infinite it follows that the magnetic field strength

is zero. Now it is known that the tangential component of H is

continuous across any interface. Consequently, its value must be

zero at the surface of the iron. Equation 2.4 then requires •

that the tangential gradients of be zero. Hence the boundary

conditionA must satisfy is that its value along the interface be

constant. The choice of constant is arbitmrl, although zero is

normally chosen.

In many applications iron occupies regions in which T is

non—zero. The boundary conditions for JI, must then be modified

accordingly. On sections of periphery at which T is non—zero, the

tangential H interface condition requires that

aJ, T t at

2.23

28.

where t is any coordinate direction tangential to the interface.

This equation implies that h is not constant but is given by

A = f Tt dt + a function of integration, f. 2.24

As an example of how the h distribution may be determined in a

practical situation let us consider the arrangement of Fig 2.3

which consists of two parallel busbars having equal dimensions

and carrying equal but opposite total currents and separated by

an unsaturated iron core. As in the asymmetric busbar problem,

the currents are assumed to be evenly distributed over the

conductor cross sections.

F

6

P L.

H

L

0 x

iron

B Fig 2.3

A

We have noted that his constant along interfaces where there are

no tangential T components. Thus, one expects the interface

lengths D 3 F G and H A B C (taken clockwise) to be at constant

scalar potential. However, as a consequence of 2.24 the two

constants will be different. 3efore making use of 2.24 we must

ascertain the nature of the integration function f. Since A is

constant when Tt is zero it follows that f must also be a constant

and not a function of the space coordinates.

29.

As it is known that Stf, is constant along the interface

section H A B C, let us define this value to be zero. Now at any

point Q lying between points C and D we find that is given by

Tz dz + f 2.25

where f is zero since A. 0 at noint C.

Because we have established the condition that the currents

are evenly distributed across the conductors, Tz is constant

between C and D. Consequently,4Q must vary linearly with the

distance (z —c) and reach a maximum value

= Ll Tz 2.26

In terms of Jx

this becomes

D = Ll L Jx 2.27

so that D is the total mmf of one conductor. The potential of

the interface sectionDEFGis constant atilt,D but J1. is found to

vary between points G and H. For a point P lying between points

G and H, SLis given by

P = dirT

z dz + F 2.28

where F is now equal to AD because 4+= JVD at point G.X6pis

found to fall linearly with distance and to reach zero when the

points P and H are coincident. Hence, the complete variation of 0,

with peripheral distance along the air—iron interface is that of

Fig 2.4 .

Fig 2.4 Peripheral distance

30.

2.4 Determination Of AUsing Finite Differences

Thus far we have considered the derivation of the magnetic

scalar potential (a) governing equation, the construction of the

electric vector potential (T) from a specified current density

distribution, and the restrictions placed on Jt at interfaces

between infinitely permeable iron and air. It remains for us to

consider how Jt may be determined using this information.

There are many ways in which partial differential equations

may be solved. At one extreme the solution domain may be

considered as a single region. The field quantity is then

described by a continuous algebraic function of the space

coordinates and is obtained using analytic solution methods.

The single region technique has the advantage that the effects of

changing parameters can often be readily appreciated, but it

cannot be used when the boundary geometry is complicated. At the

other extreme, we may divide the solution domain into many small

elements of simple shape and assume that within each element the

field is some simple function of distance. The contribution of

each element to the overall field distribution can be calculated

31.

by several methods, of which the best known are the finite

difference and finite element techniques. When finite elements

are employed, the governing equation is used to form a functional

which is maximised with respect to each nodal field value.

Although the functional need not be determined using energy

considerations, the procedure is nevertheless equivalent to

making the stored energy a maximum. It is particularly useful in

magnetostatic applications but difficulties in formulation

arise when dissipative systems are considered. These difficulties

are particularly acute in magnetic scalar potential eddy current

problems and this point will be discussed further in a later

chapter.

The finite difference method directly approximates the

governing partial differential equations and any boundary

conditions must be explicitly applied. A commonly held view is

that the method can only be used when the nodes at which the

field values are computed are arranged so that they form

rectangular mesh patterns. In fact, any nodal arrangement may be 12

employed. Thus, finite differences provide no less a general

method of discretizing solution domains than do finite elements.

Furthermore, the finite difference nodal equations are easily

derived and it is found that there is no difficulty in using the

technique to solve the eddy current expressions for T and &For

these reasons it was decided to adopt finite differences for the

solutions of both the magnetostatic and eddy current problems

considered in this thesis.

The magnetic scalar potential and finite difference

techniques are completely general methods of respectively

formulating and solving low frequency electromagnetic problems.

32.

In this section, however, we shall consider a particularly

simple situation in which the following assumptions are made:

1) the permeability takes a constant value everywhere;

2) any pole sheets pass through mesh nodes and are not inclined

relative to the coordinate system chosen; and

3) the finite difference mesh is rectangular and has a uniform

mesh interval in each of the three coordinate directions consictereaserrateta.

All three assumptions can be removed and the implications of

doing so are explored in later chapters. Here it is intended to

present the essence of the discretisation procedure.

First, let us consider the three dimensional regular

rectangular mesh of Fig2.5. This mesh could be used in problems

such as that of Fig 2.1 although busbar calculations are usually

performed using planar meshes.

ICS

4

Fig 2.5 Points A, B, C and D of Fig2,5 are midway along the branches on

which they are placed and the node spacings in the X, Y and Z

directions are uniform and given by Nih, N2h and h respectively.

Let the equation requiring solution be the general form of 2.13

having two non—zero components of T, Tx and Tz. The one

component T equations then become a special case of the finite

33.

difference relationships derived. In order to overcome the

discontinuity problem mentioned earlier the governing equation

for hmust be expressed as the relevant form of 2.22 . Thus if

the y direction is never normal to a discontinuity

2 a — a (T - atfli) a atiL) 2.29

— x ax (T z

az ay2

ax az

Then, since Hxand Hz are continuous everywhere, the two bracketed

terms are continuous and their respective X and Z

derivatives are always defined. Let any subscript applied to J6

indicate the value of a, at the node number given by that

subscript. Similarly, let Tx and Tz be given the subscripts A,

B, C or D to indicate their values at points A, B, C or D of 13,14

Fig 2.5 • Using a conventional Taylor series expansion about node

0 for the left hand side of 2.29 yields

2 a J11 = J113 ± J116 - 2 JP

0+ 0(h2)

ay2 (112h)

2

Let us now consider the first term on the right hand side of

2.29 . At point A, a Taylor series expansion about node 0 yields

a jb) - ( - 111,3)0(T T xA x Ti

Nth

Similarly, at point B

(TXaft') TxB . — 'As 0) 717c

Nth 1

Thus

(TX tr.!: 1 ((T "Li -40)) (TxA- - xB (60 - h3))) 2.33

ax x ax Nth Nth N1h

In an identical manner the second term on the right hand side is

found to be

(T 6J16 ) 1 ( (T ( - )) - (T - (J4 - ))) 2.34 zC 0 4 — zD 2 0 az z az

2.30

2.31

2.32

a

so that the finite difference approximation to equation 2.29 is

132 4 + + N2

2 (A, + ) + N2 (A,2 + 4) - 2(1 + -2 + 112, )4, =

5 6 7 3 2 N1

N2 1

1 Nh UT - T )) -7 + (T - T )) 2.35

2 xB xA N1 iD z0

An inconvenience associated with this equation is that the value

of T is required midway between nodes. This difficulty may be

overcome by making T.(A, for example, the average of the Tx values

at nodes 0 and 3. Unfortunately this procedure has to be used with

care because we have made the assumption that any discontinuities

in Tx

and Tz

occur at the nodes. In general such discontinuities

could occur at all the nodes in the computation molecule, but

this is very unlikely. To avoid ambiguity, let Txi+ be the value

of Tx an infinitessimal distance away from node i measured in the

positive x direction. Similarly, let Txi- be the value of T

x an

infinitessimal distance away from node i measured in the

negative x direction. Then if we apply the same convention to Tz

with regard to the z direction, the. mid branch values of the

components of T are

TxA

Txo- + Tx3+ TxB 4t Tx0+ Txl- 2 2

2.36

TzC Txo- + T

z4+ TzD

Tzo+ + Tz2- 2 2

and 2.35 may be expressed in the form

2 N2

6 5 + 2(A A 1 3 2 +)+ N2 (

2 +

4) - 2(1 + N2 + N22 )S60 = N

-- N 2 N2

1 1

N2 h 1 2 ( (T

xl- -Tx3+ + T

x0+ - T

x0- ) + T

z2 Tz4+

+ Tz0+

- Tz0-) 2.37 ri -

2 ' 1

- -

35.

Where there are no discontinuities in Tx or Tz at node 0,

T = T and T xo+ xo- zo+ 2

+ 2 (41 + J166 '

N

N2 - 1

= T so that 2.37 zo-

.3) + N2 ( + JV4) N2 2

simplifies to

2 - 2(1 N 2 + 2 + N

77 2

1

0

1122 h (T - T ) 2.38

xl- x,+ z2- ,I+

2 Ii 1

In many problems the magnetic fields may be assumed to vary two

dimensionally. A planar mesh can then be used and the nodal

equations can be obtained by removing appropriate terms from

equation 2.37. In contrast to the busbar problem of Fig2.1 let

us assume that 3i+ is y invariant. Then the equation for ''0 is

obtained from 2.37 by removing the A+5 and &6 terms together

with their associatediN0 coefficients. This yields

+ 463 + (42 + 44) - 2(1 + ),111,0 =

2.39 N2 h 1 1 ( (T - T + T - T ) + Tz2-

- T + T z) - T )

2 til xl- x3+ x0+ x0- z4+ 0+ z0-

Finally, if 15%, and T are specified to have a sinusoidal distribution

in.space (the y direction, say) and time then they can be replaced

by phasors and T defined. by

(x, y, z, t) = Re CA (x, z) exp j(wt- P))

and

T (x, y, z, t) = Re ( T (x, exp j (wt- P) ) 2.40

The three dimensional problem may then be reduced to two since

") 2

a 151' =- ( ) 2 a

ay 2

where p is the half wavelength of the y directed sinusoidal

distribution. No other aspect of the finite difference

derivation is affected so that the approximation to 2.29 then

becomes

1 4- 1 N2 (it Lit ) - (2 4. 2N2 + co =

3 2 4 0

2.41

2.1+3 a. N, 1-1 7C 2 7-- where

36.

14 2 h 1 "." .rto

1 ( ( T — T + T - T ) +Tz2- - 5z1++ + 5z0+ xl- x3+ x0+ x0- 2ijl 2.1+2

and nodes 5 and 6 are redundant.

37. 3. SOLUTIOIT Or EDDY CURT-LaIT PqMLEMS

One way of solving low frequency eddy current problems is

to use a magnetic vector potential defined by

curl A = B 2.3

This is a popular technique for two dimensional transverse

magnetic problems since two non-zero components of flux density

may be described by one of A. For example, if it is assumed that

A = iAy where i is a y directed unit vector relative to a -

Cartesian reference frame then

aA aA B = - _y ; B = 0 ; B = Oz ax

3.1

and the flux density vector is confined to the x - z plane.

Furthermore, since B is planar the only non-zero component of

current density which can exist is that having the same

orientation as A.

Although a large number of eddy current problems can be

expressed in transverse magnetic terms, there are important

problems which cannot. For example, the effects of transverse

edges in a linear induction motor can only be accurately

determined if the two non-zero components of current density

in the secondary are represented. Moreover, turbo-generator end

field problems require the inclusion of all three components of

current density if the stator end windings are to be correctly

modelled.

Multi-dimensional eddy currents are difficult to

compute using magnetic vector potential and little has been

published concerning the techniques involved. Equation 2.3

implies that only two components of A are required to describe

38.

three of 3. Ue shall, however, show that when A is restricted

in this way it is incapable of interrelating the electric and

magnetic fields correctly. In this section we shall briefly

investigate a few of the many possible three dimensional A

formulations and point to the major difficulties invol7ed in

obtaining their respe-:.tive field solutions. Now Amperes Law

requires that

curl H = J 2.5

and eliminating B and H between 2.3 and 2.5 using

3 =pH. yields

curl - curl A = J 3.2

which may be expanded using the vector identity

curl p n= grad p xg+ P curl a 3.3

to give

1 1 grad x curl A + (grad div A - r72A) = 3.4 v Now it is usual (but not necessary) to choose the Coulomb

gauge for A which is

div A = 0 3.5

so that 3.4 becomes

v2A - p grad 1 - x curl A = -11 J 3.6

Equation 3.6 provides the important result that J and A must

have the same number of components. Furthermore, the Faraday Law

condition

curl 13 = -B 3.7 where the dot indicates differentiation in time, taken in

conjunction with 2.3 shows E and A to be related by

E = -A - grad V 3.8

where the grad V term is an integration function. When 3.8

is substituted into 3.6 we obtain a governing equation for

39.

A which incorporates all Kaxwellts equation except the condition

that the current density be solenoidal. This condition is

satisfied by taking the divergence of 3.8 premultiplied by

the conductivity, a . Since div A is zero, the resulting equation

is

div o- grad V = -A .grad o 3.9

Equations 3.6 and 3.9 are a coupled set and must be solved

simultaneously. Thus, at every point in space four functiOns

(V and the three components of A) must be determined.

It is possible to avoid calculating V by choosing the gauge

div ciA = o 3.10

since the div J = 0 condition applied to 3.8 then requires

V to be zero everywhere. Hence

= -A 3.11 and A is time dependent even when E and J are not. By vector

identity

div 0- A= (T div A+ A • grad 0-

so that

div A = 1 - ( div aA - A . grad cr)

3.12

and the governing equation for A in terms of J becomes

1 v2A + grad( cr ( A . grad a)) - u( grad x curl A ) = -u J 3.13

This expression is only slightly more complicated than equation

3.6 to which it reduces when the conductivity is uniform.

It is therefore probable that the gauge defined by 3.10 is

the more useful of the two considered.

In addition to the difficulties associated with the

solution of equations 3.6

or 3.13 within conductors,

interface conditions must be met. These are considered in detail

15 by Carpenter and it is sufficient to mention here that p and a

40.

are discontinuous at magnetic and electric interfaces

respectively. Their gradients are therefore undefined and in

consequence the governing equation(s) . for A must be

re—formulated.

3.1 1lectric Vector Potential i]ldy ;:urrent Formulation

An alternative to formulating eddy current problems in

magnetic vector potential terms is to employ the electric vector

potential, T, introduced in the previous chapter. Whereas in

magnetostatic problems T is specified by the known current

density distribution, in the presence of eddy currents T must be

computed as a function of flux density. The interlinkages between

flux density and electric field may be expressed using the

relation

curl 2 = —B 3.7

Since curl T = J it follows that T is linked to B by the

expression

curl — curl T = —B cr 3.14

which is analogous to the magnetic vector potential governing

equation

curl — curl A = J 3.15

It is important to note that here the similarity between A and

T ends since the properties of p and a- are not analogous. The

conductivity is not electric field dependent in general; but the

permeability often is a function of the magnetic field strength.

However, the major difference between t and 0- is that a is

invariably zero within a large part of the domain of interest

whilst LI is never less than the permeability of free space..

Thus T need only be computed at the surfaces of, and inside,

conductors; unlike A which must be computed at every point

within the solution domain. It could be argued that this property

of T is a disadvantage since one reason why J is not usually

calculated directly from its governing equation is that it is

zero outside conductors, and a magnetic field function must be

introduced if interface conditions are to be satisfied. However,

we know that T is related to the magnetic field strength by

H = T grad JL

2.4

Consequently, the interactions bet::een the electric and magnetic

quantities T and H respectively are expressible in terms of a

scalar quantity,A, from which II may be calculated directly in

regions where T is zero.

The above results may be summarized by stating that when the

magnetic vector potential is used to solve three dimensional eddy

current problems, at least three functions must be computed at all

points in space, these functions being the components of the

vector A. In contrast, the electric vector potential is usually

confined to conductors and the spaces between them so that A

alone can describe the magnetic field over much of the solution

domain.

The governing equation for T is obtained by eliminating B

and H between 2.4

and 3.14 using the constitutive relation

B =p. H. This yields

1 curl — cr curl T = — µ (T — grad4)

which, when solved in conjunction with the relation

div p grad dl, = divtiT 2.12

obtained from the condition that div 3 = 0, completely satisfies

Maxwell's equations. If T is given three non—zero components,

3.16 and 2.12 completely define the T —afield if adequate

3.16

42.

boundary conditions are provided. However, we noted in Chapter 2

that only two non-zero components of T were strictly necessary

to describe a three dimensional current distribution. If T is

restricted in this way one obtains four equations containing

three unknowns. Thus, one equation must be redundant and since

2.12 defines and not the divergence of T, the redundancy

must be in 3.16 .

In order to achieve economy of expression, it was decided

to use the two component T formulation whenever three components

of current density were to be described. A Cartesian reference

frame in which T had non-zero x and z components was usually

adopted. Equation 3.16 then gave the following interrelations

a ( 1 8T ( "'

( OT aT )) s p( _ ) 3.17 Ty o- ay ' az 0- az -87 x ax

Tx` Z77- -87 az ay ay aT ) ( ,P.Tz ) 1.3 = 3.18

a ( air a 1 aT aT ay( ayz ) jr

(Fcz - az )) = Bz = V( Tz 8z - 3.19

One of these equations can be discarded and the remaining two

used to find Tx

and Tz. Since 3.17 and 3.19 formed a

symmetrical set it was decided to use these as governing

relations for T±

and Tz

respectively. In theory it should be

possible to derive equation 3.18 from a combination of 2.12,

3.17 and 3.19 . Attempts to do this failed and the nearest

expression to 3.18 that could be obtained was

ax( (

a- ay aT ) L( ) f(x,z)

cr 3.20

where f(x,z) was a function of x and z only. Clearly the

difficulty is a mathematical one and it needs to be resolved.

43. An interesting special case of the T vector eddy current

description occurs when J is assumed to he confined within a

plane perpendicular to a coordinate direction. This situation is

16 fully dealt with by Carpenter and we shall simply consider the

prominent features. Consider the arrangement of Fig 3.1 in which

J is confined to the x - y plane.

Conducting material

z = 0

Fig 3.1

If the flux density is three dimensional in and around the

conductor, then restricting J to the x - y plane implies that

the conductivity in the z direction, a, is zero. Thus 3z = / a Z z

is indeterminate but non-zero as can be established experimentally

using a stack of laminations insulated from one another. Ez may

be found after a field solution has been completed by applying.

the equation

dE Ez = f ( —x - B ) dx dz

or • aE + B ) dy

EL = ( (-7 x

These are derived from the y and x components of 3.7

respectively. The z component of 3.7 is

aE aE — Tpr Bz

3.21

3.22

3.23

44. and since T

z alone is sufficient to describe gm,y)

E =x = 1 2, _ a a--lz

x y =6 = 0--x ay ra ax

aX

Eliminating Ex and E between 3.23 and 3.24 yields the

governing e(Tuation for Tz which is

a ( I Ez ax ax '

) ( I . 1 824 = B 3.25 "gY aY

where crx and "Y 0" are the material conductivities in the x and y

directions respectively. 3.25 has a -lirect physical

interpretation in that it shows the currents in a lamination to

be solely determined by the component of flux density normal to

the plane of that lamination. Flux can travel in the tangential

directions unimpeded and this property is examined in detail by 17

Carpenter and Lowther.

3.2 Finite Difference Solution of the T Governing Equations

Having derived equations 3,17 and 3.19 which yield the

x and z components of the vector T respectively, it is necessary

to obtain a discrete formulation capable of numerical solution.

Rectangular mesh finite differences will be employed for this

purpose since they are relatively easy to apply in both two and

three dimensions, particularly when the boundaries consist of

flat planes meeting at right angles. Two dimensional solutions

using the popular triangular mesh technique will be considered in

section 3.6 .

We shall assume that all the electromagnetic field vectors

vary sinusoidally in time; thus implying that there are no

magnetically non—linear conducting regions within the domain of

interest. The field vectors may then be described by phasors

3.24

8 7

10 Ni

Fig 3.2

45. consisting of real and imaginary parts. The superscript Al was

previously used to indicate phasor quantities but we shall

dispense with it here and assume phasors throughout. Equations

3.17 and 3.19 then become

---x + ---x - a2Tz = jwax =

T —a4) a2T a2T 2 d x sax

ay az2 axaz

and

02T 82T a2T

2 2 z + ---z ay ax

az8x

respectively, where

= JwaBz = j- ( Tz az - :LIP) 3.27

• de

d = 1/( >)3 3.28

is the effective depth of penetration of the conductor.

3.2.1 Determination Of The Nodal Eauations At Points Distant

From Discontinuities In T

The simplest finite difference mesh we can use to discretize

3.26 and 3.27 contains eleven nodes arranged as shown in

Fig 3.2 „It will be noted that the computation molecule used to

obtain a discretization of the magnetic scalar potential

governing equation did not contain nodes 7, 8, 9 and 10. They

are required here since 3.26 and 3.27 contain double

derivatives with respect to two different coordinate directions.

At nodes distant from discontinuities in T we may use the Taylor

13,14 series method to obtain the required finite difference equations.

3.26

N Tx6 + Tx5 2 + N2 ( Tx2

+ Tx4 ) - 4 -- N1 z ( T 7 - Tz10 - Tz8 + T z9 )

N2 ' 2 + j 2 p ( ..11., 1 - A 3 ) - 2( 1 + N2 ( 1 + jp )) Tx0 = 0 3.37

N h 1

the following finite difference equations for Tx and Tz 2

46. Thus the double derivatives of Tx, Tz

and.% with respect to the

single coordinates may be approximated as follows:

T 2 a2T . T + T - 2T 2

8-Ix ..t. x6

+ Tx5 - 2T -- .n.x0 3.29 -x -- x2 x4 x0 3.30

ay az 2

(N2h)2 (h)

2

82T jL T

z6 + T

z5 - 2T 3.31 a2T A. + T

z3 - 2Tzo 3.32 ---z -- z0 2z -- zl

ay2

(N2h)2 dx (N

1h) 2

aJI, ..A. j1,2 '' J1/4 O" .n. J1,1 - A3 3.33 ax az 3.34 2N1h

2h

The double derivatives with respect to different coordinates

require careful treatment. About nodes 3,0 and 1 the Taylor

8T series method yields the following expression for -0

Tz8 Tz9 Tz2 - T )4. 2 Tzi

- Tz10

2h 2h 2h

aT respectively. Thus, expanding Tiz in the x direction we obtain

the relation

a` aT (

Tz7

- Tz10 ) - ( Tz8 - T

z9 ) 2h 2h 3.35

aT Similarly, --xis given by 8x

2N1h

T T x10 x9

2N1h

Txl - Tx3

2N,h

Tx7

- Tx8

2N1h

at nodes 4, 0 and 2 respectively. Thus

a -

(dam ) _A_ ( Tx7 - Tx8 ) - T T x2_) az Ox 2N 1h i 2N1h

2 h Substitution of 3.29 to 3.36 into 3.26 and 3.27 yields

3.36

47.

where P = 2d2 3.39

T +T+ ( - 2 (T-T T +T)

+ j 2 p ( J1:2 - J1,4) - 2( 1+ 1N1 ( 12 + JR )) Tzo = 0 3.38

z6 z5 z1 z3 x7 x8 - x10 x9

N2

N22

1

N2 N2

1

h2 1

4N

N

These expressions seem complicated but if we consider equation

3.3 8 say, and transfer the Tx and cit, terms to the right hand

side so that -they act as sources, then the linkages between the

Tz values are found to be simple. Furthermore, the

equation becomes diagonally dominant and this is an advantage

when iterative methods of solution are employed.

3.2.2 Boundary Conditions Satisfied By T At Conductor Surfaces

At any conductor surface across which no current flows

J . n = 0 3.40

where n is a unit vector normal to the surface. In the previous

chapter we showed that this condition imposed a restriction on

T which could be expressed mathematically as

div (T x n) = div c = 0

3.41

where c, in general, was a function of x, y and z. Equation

3.41 implies that if c is zero, then T is confined to the

conductor. We have already shown that Tx

and Tz

are related to

Jz and Jx

respectively by

Tx =

z dy 2.21 T

z = J

x dy 2.20

Thus, although we may set Tx and Tz

to zero at one surface

coincident with the x - z plane, at all other such surfaces these

functions will not be zero. The only exception to this rule

occurs when the conductor is symmetrical and is placed in a

symmetrical solution domain. T is then zero at all surfaces

48.

lying in the x z plane.

Let us illustrate the application of these boundary

conditions by considering the isolated, conducting rectangular

block of Fig3.3a. If we 'let T be zero at face ABCD then the

boundary conditions of Fig343b satisfy eauation 3.41 .

Surface Boundary Condition

ABCD T =T =0 x z

EFGH Txn= c

AHD Tz= 0

BFGC Tz= 0

AEFB Tx= 0

DHGC Tx=

(a)

Fig 3.3 (b)

Note that if the block is placed in an asymmetric domain then

T is non—zero within the rectangular region bounded by EFGH in

the x — z plane, and stretching to-t-Coin the y direction measured

from plane ABCD. Moreover, it is important to remember that T is

non—zero within all holes surrounded by conducting material.

3.2.3 Determination Of T At Nodes Lying On Conductor Surfaces

To avoid undue confusion, we shall consider the rectangular

block of Fig 3.3 to be a typical conductor and shall derive

finite difference equations for T which are valid at nodes lying

on its surfaces. Clearly surface ABM) presents no problem since

T is zero and no calculations are necessary. Let us leave surface

EFGH for the moment and concentrate attention on the remaining

four faces of the block. If we define a local right handed

coordinate set (ti,t2,n) such that the origin is on a surface

49.

and n is the direction perpendicular to that surface at any

point, then equations 3.26 and 3.27 may be expressed in the

single relationship

a2T 2 ---n + a --- T n = j ( T - a ) 2 ot2

1 at2 n an 3.1+2

Ttl

or Tt

do not appear in this equation since their tangential 2

gradients are zero at the surfaces considered. Thus the finite

difference computation molecule need only contain the seven nodes

given in Fig 3./i. Nodes 0,1,3,5 and 6 are assumed to lie on the

= cr

= o

N h 3 1 fi = p1

cr 0

conductor side of the surface. The finite difference

approximation to the left hand side of 3.42 is

Tnl + Tn3 - 2Tn0

(N1 h) 2 Tn6 + Tn5 - 2Tn0

(N2h) 2 3.43

Now the right hand side of 3.42 is more difficult to

approximate since the condition that the normal component of

flux density ( Bn ) be continuous across the interface reauires

that

aJta 11A ( Tn ) IA = /133 ( Tn 67.1 ) 1 B 3.44

where the letters A and B refer to the materials separated by

the interface. Equation 3.44 reveals that since Tn is

discontinuous, so must be the normal gradient of J. Hence,

because a discontinuity occurs within the expansion interval,

the Taylor series method cannot be used to obtain this gradient

1 Fig 3.4

50.

at node 0. Instead we note that

j aJt, d2 n an - ( T - ) = jwo- B

n

and as Bn

is continuous across the interface we may assume that

Bn

at node 0 is the average of its values obtained midway along

mesh branches 0 - 2 and 0 - 4. Thus til" tri 40 ) = tt 1 • ( ° + ( Tn0 + Tn4 - ̀b0

- )) n

2

3•46 where Tn midway between nodes 0 and 4 is given by the average of

the Tn values at these nodes. Substitution of 3.46 and 3.43

into 3.42 yields

0 4 II 2 O

2 4 - A A A - T + T + N2 (T + T ) + jN2 p ( + p l ) n5 n6 ni n3 2 h h

N1

2

111

- Tn4 ) - ( 2 + N2

2 - + jp/2 )) TnO = 0 3.47

2 2

2 2 N2 1

p = w pi a h2/2

Let us now turn our attention to surface EFGH of Fig 3.3 341.At this

face the normal component of T is zero and the tangential

components are related through the equation

aT aT = --z az ax

-which is obtained from 3.41 • Using 3.49 the Tz and Tx terms

in equation 3.26 and 3.27 respectively may be eliminated to

yield relationships of the form

a2

a 2 = 841'

d2 t - at )n

Tt1

(T

3.50

where tI can be coincident with either the x or z directions.

As before, the n direction is normal to the surface. Now the

tangential components of T are continuous ructions of n measured

3.45

where

3.49

Nh 4

Fig 3.5

111 0

conductor

;mai ••■•■ •••• &OM WWI. 4■11. 0.11•

51. across the interface, but their normal gradients are not. Thus

the Taylor series method cannot be used to obtain a finite

difference approximation to 3.50 . Uote that this equation

contains derivative terms for only one tangential direction. In

consequence, the planar mesh of Fig3.5 may be used as a

computation molecule. Points P,Q,R and S bisect their respective

= t10 cr = 0

mesh ce11-61 — insulator I I

mesh branches and define the mid points of the sides of a

rectangle which we shall call the 'mesh cell'. Instead of

employing the Taylor series discretisation method we shall adopt

18 that described by stall . First we integrate 3.50 over the

surface of the mesh cell to yield

If an2 d2 dt do = ( Tt at - dt do 3.51

where the tI subscript has been changed to t for manipulative

convenience. We then use Stoke's theorm to change the left hand

side into the line integral

OT ( an

t) i . dl 3.52

where i is a t directed unit vector and c forms the periphery

of the mesh cell. The value of aTt/ an at point S may be found

using a Taylor series expansion about node 0, at which no

52. discontinuities occur. At point 0,a Tt/an is zero since Jt1 =

Jt2

= 0. Thus the finite difference appro-cimation to 3.52 is

Tto Ttl ) Nh 3.53

Now within the insulator the effective depth of penetration is

infinite so that there is no contribution to the right hand side

of 3.51 . This expression may be approximated by multiplying the

integrand by half. the mesh cell area. Since the tangential

components of both H and T are continuous across the interface,

so are the tangential gradients ofh, and nkat may be determined

using a Taylor series expansion about node O. Thus

atft, (Jt, 1 - it, 3 ) at z -21411 3-4

Substitution of 3.54 , 3.53 and 3.52 into 3.51 yields

T + (JL 1 - 3 ) - jr3 ) Tt0 = 0 3.55 apth

This equation is valid for both tangential directions and forms a

surprisingly simple result.

3.2.4 Determination Of T At Nodes Lying On Interfaces Between

Non-YaEnetic Conductors and Infinitely Permeable

Non- Conducting Iron

In many engineering problems it is found that conductors

lie. with-at least one surface against some ferro-magnetic

material. For example, a linear induction motor secondary is

invariably placed along the surface of laMinated or solid

backing iron. It is usual to assume that the iron has infinite

permeability and resistivity so that it becomes of fixed magnetic

scalar potential (often zero). We shall here determine finite

53. difference relationships that yield the normal component of T

(Tn) at interfaces between conductors and any iron which has been

represented in this way.

As before, we shall consider the rectangular block of Fig 3.3

to be a typical conductor. If face ABCD is placed in contact with

iron, no special equations are necessary since T is zero.

Similarly, although the finite difference relationships for the

4 governing equation must be modified at surface EFGH, the

equation for Tt is still valid since its computation molecule

does not include nodes lying outside the conductor. When iron is

placed at any of the other faces of the block, partial

differential equation 3.42 applies. This equation includes a

normal derivative of A which must be evaluated in finite

difference terms by incorporating the magnetic boundary conditions

applicable at the interface.

It is convenient both not to have nodes within the iron and

to avoid deriving new finite difference expressions. Consequently

we shall develop an image system that mimics the known interface

conditions. Now the iron is infinitely permeable and resistive

so that the tangential components of magnetic field strength are

zero at its surface. In an equivalent image system the tangential

-components of current density must therefore be continuous across

the line along which the image is taken. This condition requires

that T be symmetric. Furthermore, since the normal component of

flux density is continuous

( T -a51+ , T -an, n , I = an image n an conductor 3.56

and the requirement of symmetry for T makes Aanti-symmetric

about the image line. Thusr the image system is that of Fig 3.6 .

iron

= CC) = 0

z tot 30

5

Fig 3.6

0 Nlh1 0

(image node)

0 ■..-

conductor St = 2 LO.0 - J12

T T = —4 -2

2 Nth to 6

54.

Bn

at node 0 is given by

Bn = po ( Tna- ( 2 - ) 2( ai 2 - 111 ,

3.57

2h ( TnO

- 2h

0) )

and substitution of 3.57 , 3.45 and 3.43 into 3.42 yields the following expression for Tn

at node 0:

_2 01,72 a Tn5

+ Tn6

+ N2 ( T

nl + Tn3 ) + j —2 r (A

2 - A 0 ) 3.58 h N2 2 2

1 - ( 2 + N ( 2 + j2p )) Tno = 0

3.3 T Vector Travelling Wave Formulation N1

any low frequency electromagnetic devices are designed to

produce sinusoidal travelling waves of fixed amplitude moving in

a given direction. Analysis of such devices is particularly easy

if the geometry can be assumed invariant with distance measured

in the direction of the travelling wave. Under such circumstances

T and slio may be described by phasor equationsof the type

T (x, y, t) = Re 02 (x,z) exp j( wt - 211 )) 3.59

tit (x, y, z, t) = Re (:(x, z) exp j( wt - P)) 3.60

Here the travelling wave is y directed and of half wavelength p.

Double differentiation of 3.59 with respect to y produces the

result

( ) 2 1,1' P 3.61

55.

Substitution of 3.61 into 3.26 and 3.27 yields the following

governing equation for the x and z components of T respectively:

2" 24 a T a T , x ,2 ---x - ---z - t - ) -.1.-x = juxr II'x

3.62 axaz P az

2

2^' 2" 8 T a T x 2 F.' N --7z ---x - ( - ) Tz

= jwo7Bz

ax axaz

Note that the travelling wave assumption is valid only when the Ona conclut

permeabilityAis invariant with time and travelling wave direction.

Kagnetic non-linearity generates harmonics which can only be

accurately calculated by employing three dimensional meshes and

by working explicitly in time.

Since 3.62 and 3.63 contain no y derivatives, their

finite difference formulation is in two dimensions. The travelling

wave finite difference equations are obtained by applying 3.35

3.30 , 3.33 , 3.32 , 3.36 and 3.34 to yield the

following equations

1X2 '1X4 1 Ni(5z7 7z10 7z8 ) jA- ( 1 — 3 ) 3.64

"

— 2( 1 +

2

jp + ) acco = 0 .

2N1

NP 7E1 — N1 ( 5x7 — 5)(8 73c10 4 X9 ) + j ( 2 - ) 3.65

4 h 2 - 2( 1 + jNl(3 + q/2 ) Tzo = 0

where a = ( N1nh/P )

2 3.66

and the node numbering is that of Fig 3.2 . These equations are

valid at points distant from discontinuities in T.

Let us now determine the boundary conditions that T must

satisfy at the surfaces of a rectangular conductor. Consider the

arrangement of Fig 3.7a .

3.63

Tz =

= PO

o = air

Tx

= 0

P. =

= 0

(a) Fig 3.7 (b)

2

X

Tx

= 0

\

56.

T and J are related by the equations

Tx

= - )( Jz

dy 2.21 Tz = jeJx

dy 2.20

Therefore, since Jz

is zero at interfaces in the x - y plane,

so must Tx

be. Similarly, the condition that Jx

is zero at y - z

plane interfaces requires T to be zero. At every surface, one

component of T is non-zero and its value may be calculated using

the appropriate three dimensional mesh result by omitting the

terms associated with nodes 5 and 6 of Fig 3.4 and inserting a

travelling wave term. The computation molecule is then that of

Fig3.7b and equation 3.47 yields the following relationship

for Tn at node 0:

j N2 p° tert .c) (

Ae

— '340 ) T33.4. ) 3.67hn1 n3 1 h • 2 N2

ev - ( 2 + j 1

If there are interfaces between the conductor and unsaturated

non-conducting iron, equation 3.58 may be used in its two

dimensional form which is

2112 Tn1 + T115

+ j 113 21 :11' 2 0 ) 2+ i2N1P +a) Tn0 =0

Al A,

E3/2+a ) Tno = 0

h 3.68

57.

3.4 The One Component T Formulation

In many practical problems the conductors are rectangular

in cross section and have one major dimension that is much smaller

than the other two. Under such circumstances the current density

vector is almost completely confined to a plane and its smallest

component may be neglected. Only one non—zero component of T is

then required to describe the current flow paths. Contours drawn

between points of constant T become current flow lines in much the

same way as lines of constant magnetic vector potential are flux

lines in Cartesian geometry transverse magnetic problems.

Furthermore, restricting T to one non—zero component (To) reduces

the computation effort required to obtain an electromagnetic field

solution, since only two functions, T and A, must be computed

within conductors. Tc

is determined from the two component T

finite difference equations derived previously by setting an

appropriate component to zero. In this context note that To can

only describe the currents flowing in the plane perpendicular to

its coordinate direction.

3.5 Determination Of At The Surfaces Of magnetic Conductors

When the finite difference equations for diiin terms of T

were derived in Chanter 2 it was assumed that the conductors

were non—magnetic. This assumption was valid for the magnetostatic

problems considered there. However, eddy current problems often

involve the calculation of magnetic fields and currents within .

ferro—magnetic materials so that it is necessary to consider the

effect of magnetic interfaces on the magnetic scalar potential

finite difference formulation. For convenience we shall assume the

magnetic conductor to have a constant permeability, pl. This

58 restriction can be removed at the cost of increased nodal

equation complexity.

We shall use the local coordinate system and computation

molecule of Fig3.4.Kodes 0,1,3,5 and 6 are assumed to lie on the

conductor side of the interface. The value of Jt at any point is

determined by applying the condition that the divergence of B is

zero. It is a simple matter to evaluate the tangential derivatives

of 3 since 11 is constant along the tangential mesh branches.

However, 11 is discontinuous in the normal direction and care must

be exercised when a finite difference approximation to the normal

derivative of B is sought. Let us consider a general T vector

having three non-zero components. Furthermore, let T I. . be the

value of T midway between nodes i and j. Then, using the usual

Taylor series method we find that

aB

1

ae 2 at 2

aB --n an

=

=

=

4 -1

( p 0

( Tt1101

- -

co ) ( - T t1

1I 03 3.69

3.70

3.71

N1h 0N

1 h )

Tt2

1 06 ( 4'6 Nlh

) - Tt2105

J , o 5) ) -

N2h N

2h

( T

N2h

( T n 04

(A° - A4) - I _ _ p 1

h

Note that this last approximation is valid because the normal

gradient of Bn (unlike that of Hn) is continuous across the

interface and no discontinuities occur within the series expansion

interval. Addition of 3.69 3.70 and 3.71 yields the

following finite difference equation for it at node 0:

2 p1N2h (.-T. t21

I

06 4 Tt 105 ) + P1 h (-Tt 101 + Tt 103 ) 2 N

1 1 1

+ N22 h (-

P1 TnIO2 4. Po TnI04 ) Pi (j115 4. 46) 4' Vi N2(Al + J113)

N2 1

2 + N2 ( tioA2 + pit5L4) - ( 2pi + 2µ1N2 2 + N ( 110 + pi ) )1110 = 0

N2 1

3.72

59. When the permeability of the magnetic conductor is infinite, the

',2 termin this equation disappears andho only depends on the

values of St at nodes within the conductor. This is a well known

result which shows the magnetic field distribution in unsaturated

iron to be completely independent of that in surrounding non-

magnetic materials.

If we assume a travelling wave to exist in the t2 direction,

say, and if the geometry under consideration is t2 invariant,

A/ then T anda may be represented by travelling wave phasors T and

St, such that

.%1

aBt2 = _

(

P )2 irb

at2 within the conductor. Thusons at node 0 satisfies the equation

("Tt 101 4 lilt 103) 11111 (/11%102 110TnI04)

111(31 + 1'3) + 1.121( c rb2 +1-1 131,4 ) (2Pi 1123.( c) + h_ )

41m),510 = 0 3.74

where a. is defined by 3.66 .

3.6 Two Dimensional Triangular Ileshes

So far we have only considered rectangular mesh finite

difference discretizations for the T and &partial differential

governing equations.These discretizations produce algebraically

simple equations which are easily applied to most field problems

provided that each boundary line lies in one of the coordinate •

directions. Curved boundaries are difficult to model except by a

'staircase' approximation which is not always satisfactory. In

order to increase boundary matching accuracy, the finite element

method is often employed. This technique involves approximation of

3.73

60 . the field within a given set of sub-domains by simple algebraic

functionsof distance. Nodes are associated with the sub-domains

and the value of the field quantity in question at these nodes

is obtained by finding the stationary point of a functional which

can often be derived from stored energy considerations. The sub-

domains can be two or three dimensional and have any shape, but

usually they are chosen to be triangular (in two dimensions) or

have triangular faces (in three dimensions) since any closed

figure having flat sides can be divided into triangles or

triangular prisms. Three dimensional finite element techniques

are at an early stage of development because of the concentual

and data handling problems involved. For the moment, let us

briefly consider the fundamentals of the finite element method as

they relate to the calculation of electromagnetic fields using

magnetic scalar potential.

3.6.1 Kagnetic Scalar Potential Finite Element Formulation

A large class of electromagnetic field governing equations

take the form

zx ( k ax ) + ( k a PI ay ay

) -Sz( k az ) = P(xa'z) 3.75

where 0 is a scalar. In general, p and k are functions of all

three coordinate directions. The solution of this equation,

obtained with appropriate boundary conditions (usually given in

terms of 0 or its normal gradient), gives the answer to the

physical problem. It can be shown (by applying the Euler 19

conditions of the calculus of variations) that the problem

defined above is identical to that of finding a function 0 which

minimises the functional f defined by

f = )(k -2 )2 + ( ) 2 + ( ) 2idv - fpg5 dv 3.76 ay

61.

and on which are im7osed the same boundary conditions. The

integration is assumed to extend over the whole solution domain.

In the two component T magnetic scalar potential formulation

we have three equations of type 3.75 , and thus three

corresponding functionals. By using the Principle of Restricted

Variations we may minimise each functional by holding constant

the variables described by the other two and combine the results _

to yield a complete field solution. In regions where the permea-

bility and conductivity are functions of position, the functionals

associated with equations 2.12, 3.17 and 3.19 are

respectively

2 2 2 aa. )

1h = -a- /11 ( ax (---"" ) + ( ay ) + =I") + ( — ) 3 dv -

fT . ail ( Tx) 2+ i L.r .)2 dv _pp ei, 4 0:3- a y

( d'A. r x x- x

f .Ez) 2+ Ez, 2 ) dv -/J.T (T - Tz 21 0- ay ax z z

jai div p.2." dv 3.77

axot

ax) dv -iT x.z (c-• Ez) dv

3.78 art OT dv a 0_ 737) dv

3.79

where the dot denotes differentiation in time. wince divIIT is

a discontinuous function largely confined to the surfaces of

conductors, it is convernient to transform the last term of 3.?7

into the difference between a surface and a volume integral. By

vector identity

f J div p.T dv 5(it( p.T . n) da ( pT . dv

3.80 where S is a surface enclosing volume R and n is a unit vector

everywhere normal to that surface. Thus 3.77 becomes

f = a)0( 2b)2 ( .a.L')21_( ilb)2 idv (11TV ).51,dv - st, 2 1. fax ay az

'V V

.1(4.n) da S

3.81

62. In this thesis we have always tried to avoid solving for the

field quantities explicitly in terms of time. Instead, we have

adopted phasor methods and assumed there to be no magnetically

non—linear conducting materials within the solution domain. If

these methods are employed here, the functionals become phasors.

Unfortunately, the validity of the finite element method when

its associated functionals are complex numbers is not

mathematically proven. Nevertheless, if it is assumed that one

can treat phasor functionals in the same way as the real number

20 types, then correct results can be obtained. To avoid this

mathematical difficulty it was decided to investigate other

discretization methods which would allow nodes to be randomly

placed within a solution domain and which, unlike the finite

element method described above, had a firm theoretical base.

3.6.2 The Branch Integration Kethod 21

It is possible, as Denegri, Kolinari and Viviani have shown,

to use a generalised Taylor series to obtain finite difference

equations for irregular node arrays in two and three dimensions.

If the series is stopped at the Nth order terms, then in two

dimensions each computation molecule must contain (N + 1)(N + 2)/2

nodes. After the nodal values have been computed in a field

solution, those at intermediate points may be easily found using

the original Taylor series. A disadvantage of the method is that

the expansion interval must not contain discontinuities. If it

does then alternative procedures that can be computationally

inconvenient must be employed.

Another possibility is to use the two dimensional nodal

22 method described by Hanalla and Macdonald. This technique

63.

involves integration of field variables along contours which

enclose nodes. It is applicable to both linear and non—linear

problems and may be extended to include high order field

variations within the elemental areas. In view of the flexibility

of the nodal method, it was decided to apply the contour

integration approach to the magnetic scalar potential eddy

current equations. In order to make the formulation more

innovative) contours that enclosed mesh branches rather than

mesh nodes were chosen. For simplicity, a planar triangular

mesh was used and magnetic non—linearity was neglected.

Furthermore, a travelling wave situation was assumed. Thus, at

each time instant the field quantities were allowed to vary

sinusoidally with distance measured in the direction perpendicular

to the plane of the mesh. As in previous work, Cartesian geometry

was used and T was assumed to have non—zero x and z components.

Consider a typical triangular element having a node at each

vertex as shown in Fig3.8. Let us assume thath and T vary

1 2

x2'z2 ) x4.z.11

Fig 3.8

linearly across the area enclosed by the triangle. This is a

first order approximation and a function satisfying the

linearity condition is

64. 3

;57 (x,z) (or T(x,z)) = ( a. + b x + c.z (or Ti)

i=1

where the index i refers to the node numbering system which,

incidentally; we shall assume to have an anticlockwise

orientation for every mesh triangle. It is interesting to note

the implications of 3.82 . First, since curl T = J, the x and z

components of current density are both linear functions of

distance within each triangular element. J , however, is constant

and given by

3 3 biTzi

J = i

E cT - 22 Y xi

i=1 i=1

3.83

Second, we find that grad J+ is a constant and

3.82

3 3 b T + c

iT

i xi zi i=1 i=1

3.84 divH

Thus, div H is not automatically zero. Nevertheless, since

the governing equation forA is derived from the condition that

B has no divergence, a simultaneous solution of the T

equations must require 3.84 to be zero.

We require JL and T to have the values SI, and T at any node i.

Thus, 3.82 implies that

al + b

lx1 + clzl = 1

al + b

1x2 + c

1z2 = 0

al +blx3 +c!3 = 0

Solving for al' b1 and c1 using Cramer's rule or its

equivalent yields

al = (x2z3 - z2x3) / 2A; bl = (z2 - z3) / 2A

3.86 e l = (x3 - x2 ) / 20

3.85

65.

where

= area of triangle = z1(x3 - x2) z2(x1 - x3) +

z3(x2 - xl)

3.87

Using an identical procedure with regard to nodes 2 and 3 we

obtain the values of a2, b2, c2 and a3, b3, 03 which are

a2 = (x3z1 z3x1) / 2A ; b2 = (z3 z1) / 2A

3.88 02 = (x1 - x3) / 2A; a3 = (xlz2 - zix2) /2A

b3 = (zi - z2) / 2A; 03 = (x2 - xl) / 2A

Thus far we have followed the procedure which would be adopted

for both finite element and nodal method discretizations.Here,

the similarity with the finite element method ends for instead

of deriving an energy functional we consider the governing

partial differential equations for and T which are

v2 Sti = div T 2 2T v „Tx - as z = j/d x 2 (Tx - g---atrti) 6-1.67

V 2 T xy z - a2Tr = i/ 2 ( 82!) a z T a

and integrate them with respect to x and z to yield

fiV2`51' dz =/div T dx dz 3.89

T - a2T dx dz Tx z = I/ 2(Tx ox

— PA) dx dz

yz aXez 3.90

3.91

We have assumed quasi-static conditions to exist so that A

and T are phasors defined by 3.59 and 3.60 . The y

derivatives of A and T are given by

2.13

3.26

3.27 axaz

2 r, Tz - o2T

x dx dz = fj 2(Tz 86) dx dz

axaz az

a2T' =

ay?

7c 2 3.61 and a2111) ay2

f )2f, 3.92 p 3"

66.

respectively where P(known as the 'pole pitch') is the half

wavelength of the y directed travelling wave. If no such wave

exists, P is infinite and the problem becomes two dimensional

magnetic / one dimensional electric and can be solved at less

expense by using the magnetic vector potential, A.

Now Green's Theorm in a plane states that if P and Q are

functions of x and z only

7( dz ax ) dx dz = O P dx + Q dz 3.93 R C

where c is the contour bounding region R over which the double

integral is to be taken. By applying this theorm to the left

hand sides of 3.89 3.90 and 3.91 after first taking the

y derivatives of T and Ato the right hand side of these

equations and including results 3.61 and 3.92 yields

.95

at% ( dz -

a.11, dx ) = "( div T + (p) 2,1, ) dx dz 3.94ax

aT aT x dx + --z dz ) = (T - 81) + Tx ) dx dz az az d2 x ax

C

R 3.95

aT .57 aT dz + dx) = ( 12(T - .2.111:) + (L)2 T ) dx dz d z az P z

C R 3.96

Consider two adjacent triangles having vertex noaes numbered

.123 and 134 as detailed in Fig3.9 . Furthermore, define a contour

c passing through the centroids of each triangle and nodes 1 and

3. Let the centroids of triangles 123 and 134 be given the

numbers 6 and 5 respectively. Then the areas of triangles 163

and 135 are

A163 = 1(21(x3 - x6) z6(x1 - x3) 4- z3(x6 - x1)) 3.97

and A135 = 2(z1(x5 - x3) + z3(xl - x5) + z5(x3 - xi)) 3.98

3 3

R is bounded by c

Fig 3.9

67.

respectively where

and

(x5,z5) =

(x6' b z-) =

[Ix1 + x3 + x4 , z1 + z3 + 2

4 3.99

3.100

3 x1 + x2 + x3

3 Z1

+ Z2

+ Z3

A

ye---.x

We shall now perform the line integrations associated with 3.94

3.95 and 3.96 around contour c. Since this contour encloses

a mesh branch, the procedure to be described may be succinctly

termed the 'branch method'', in contrast to the nodal method

which involves a line integration path enclosing a mesh node. The

positions of nodes 5 and 6 within their respective triangles

is arbitrora but the centroid position chosen gives the

algebraically simplest result.

The branch method proceeds as follows. The variation of T

or within the triangles is given by 3.82 so that the

,various derivatives of T and hrequired for substitution into

3.94 , 3.95 and 3.96 are

3

a7c I ") 5 123 ii 3.101

.8151' I 1 ax 134 = 7(b.%Sb.) ; . „ (c.A.)

i=1,3,4 134 i=1,3,4

123 i=1

i=1

c.T ) 1 zi

i=1

i=1 Using an identical procedure )07 dx may be obtained. The

C az left hand side of 3.94 is then given by

f 35a" dz = [ Eb i -

6 ( z3 z1 ) 3.103

i=1

68.

c1.T ) xi 123

c1.T. ) xi

i=1,3,4 3

:571(7 1 b.T ) , xi 123 i=1

= (b,Txj)

134

aTz az 123

aTz = 71(ciTzi az

134 i=1,3,/+ 3

aTz 71(b.Tz.1) ax

1123 i=1

aTz = >-110.T zi) ax

134 i=1,3,4

aTx az

aT

Z7-

aTx ax

aT

ax

3.102

134

Note that within the triangles 123 and 134 the derivatives in

3.101 and 3.102 are constants with respect to position.

dz ah ax

integral associated with the line joining nodes 1, 6 and 3 is

given by 3

dz =:111 ax 1123 ( z3 z1 ) = [ bijbij ( z3 ) 163 i=1

Similarly

tu7x dz = ax 1134 ( z1 z3 ) = ( z3 ) a,s13 adt,

so that 3

Consider now the evaluation of The part of this

i=1,3,1+

( dz - az dx ) = ax a41) ash _ biJbi 2 z3 - z, )

i=1,3,4 C

i=1

The left hand sides of 3.95 and 3.96 may be similarly

3.101+

obtained and it remains to consider the discrete representation

of the right hand sides of equations 3.94 , 3.95 and 3.96 .

69. Clearly, the derivatives under the double integral sign may be

replaced by the values given by 3.101 and 3.102 and since these

are constant within each triangle

T ax dz = rdiv Tfax dz1 [idly TAx dz 163 135

3

=!(biTxi

+ ciTzi) 163

1=1

1:(b1Txi + c.Tzi ) 1 LA 135 3.105

R

dx dz =

aq% az dx dz =

3

i=1,3,4

biai A + Tbist,i A 3.106 163 i 135

i=1,3,4

c. A. A 1 1 LA 163 +

ciai A135 1 i.1 i=1,3,4

i=1

3.107

The pole pitch terms which occur under the double integral signs

of equations 3.94 — 3.96 pose more of a problem since within

triangles both T and A, are functions of position. A suitable

Gaussian quadrature formula may be employed to evaluate the

required double integrals accurately but the algebra is tedious.

Here we shall make the approximation that the value of T ora,

at nodes 6 and 5 are representative of the values within triangles

163 and 135 respectively. Thus

T dx dz T- Q . A 6 163 5 "135

and Pr dx dz a A A +a A 3.108

6 163 5 "135

Substitution of 3.104 and its T'equivalent for the left hand

sides of 3.94 , 3.95 and 3.96 followed by substitution of

3.106 , 3.107 , 3.105 and 3.108 for the right hand sides

yields three equations, each of the form

70•

dl Al d2 A2 + d33A3

+ d4 JL4 + e1Txl

+ e2Tx2

+ e3Tx3

+ e4Txl

+ f1T +

f2Tz2 + f3Tz3

+ f4T = 0 3.109

The values of the d, e and f coefficients are contained in

Appendix 1. The 3.109 equations are determined for each pair of

adjacent triangles having the common vertex 1 and are then

added together to yield the complete expressions for Al and T1.

Thus, the total area over which integrations are performed to

determine the equations for a single vertex field value has the

form shown in Fig 3.10.

integration area

Fig 3.10 A difficulty occurs when it is required to determine T at an

unsaturated iron surface since one does not normally want to

introduce triangles lying within the iron, and yet the

discretisation procedure requires a closed contour to be defined.

One way of overcoming the difficulty is to introduce an image

triangle reflected about the iron surface as per Fig 3.11. An

alternative is to define a triangular contour (having vertices at

points 1, 5 and 3, say) that encloses the iron surface and along

which the line integrations are performed. However, special iron

surface equations must then be written into the computer program.

In general, the image method is the most convenient of the two.

Thus, although the branch method is capable of implicitly

Physical System

them e:cplicitly.

Fig 311

T2

= Ti+ 61,2

= 2 a - 3t 4

Image

71.

incorporating the boundary conditions, it is usual to impose

3.6.3 Implications Of The Branch I:ethod

Having considered the construction of general nodal

equations using the branch method, it is instructive to apply

the technique to a rectangular node array so that the equations

obtained may be compared with those derived using finite

differences. There are many triangular meshes which will fit a

given rectangular node array but two of the simplest are given in

Fig 3.12 .

1 2

3 1 2

3

& uniform

(a) Fig 3.12 (b)

Part (a) of this figure contains a mesh which is completely

regular but for which it is relatively difficult to obtain the

required nodal equations since the computation molecule of a

typical node, 6 say, is symmetrical only about its main diagonal

which is the line between nodes 1 and 11. Consequently, a minimum

72.

of 144 coefficients (36 each for branches 6-11, 6-7, 6-2 and 6-1)

must be computed. On the other hand Fig3abcontains a mesh for

which there are two types of computation molecule. Since the

molecule ceLtred on node 7 has two degrees of symmetry and no

diagonal mesh branches, only 36 coefficientS need be computed in

order to obtain the complete nodal equations for T and a . In the interests of simplicitly, this molecule was chosen. It should

be mentioned at this stage that although the prospect of

computing 36 equation coefficients per mesh branch may seem

daunting, the process can be automated with relative ease when a

computer is available.

To obtain the equation for T and .14 at node 7 one first

calculates the values of the constants associated with the

describing functions(3.82) of these quantities for triangles 7,

11, 8 and 7, 8, 3. The coefficients given in Appendix I may then

be evaluated for mesh branch 7-8. If we assume that the node array

is square then the coefficients applying to mash branches 7-3,

7-6 and 7-11 will have identical form. Summing the contribution

for each mesh branch yields the following equations for T and Sio

(Tx8 + Tx6) (49 [i/d2 + a] )+ (Tx3 + Tx11)(149[3/d2 + m] )

+ j/3d2 (A8 — 416) — Tx7(2 + 2/9 (j/d2 + ) = 0 3.110

(Tz8 Tz6) (1-)9[ /d2 al ) (Tz3 Tz11)(9C/d2 4- a] )

j/3d2 ( - Tz7(2 + 2/9 Li/d2 + a.!) = 0 3.111 01,3 — ail)

(J1,3 + J1,6 + ,51,8 + (1 - i) ( 11 .1.

- 7 9 (4 12) =0

— T + T — Tz. ) x8 z11 3

3.112

73.

It has here been assumed that the node spacing is unity. The

finite difference equations corresponding to 3.110 , 3.111

and 3.112 are

- Tx3 + Txii *.(Tz4 Tz12 Tz2 Tz10)

3/d2(418- J16)

- (2 + j/d2 + a )Tx7 = 0 3.64

Tz6 Tz8 41(Tx4 Tx2 Tx12 Tx10) j/d2"/3 -411)

- (2 + j/d2 + m)Tz7 = 0 3.65

and J/6 48 43 4. ji'11-(4+a)47÷ -12'(Tx6 -

Tx8 + Tzli - Tz3) = 0

2.42 respectively. The first point of difference between the two secs

of equations is that the branch method relationships contain

travelling wave terms in nearly all the nodal value coefficients

whilst the finite difference relationships do not. This is also

true with regard to the eddy current terms associated with the

equations for T. Most of the other differences between the two

sets of equations may be explained by noting that the finite

difference eauations are associated with an area 1 unit whilst

those of the branch method are associated with an area 4/3.

Furthermore, since the branch method computation molecule has

only 5 nodes (compared with 9) the double derivatives of T with

respect to mixed coordinate variables are found to have rather

different discrete representations.

It is difficult to assess the comparative accuracies of the

branch and finite difference methods. The branch method allows

both Jx and Jz to be linear functions of distance within each

triangle and thus is probably the more accurate since the finite

74.

difference method involves the assumption of constant current

density within each mesh element. However, the approzimations

made to the integrals forming the right hand sides of 3.94

3.95 and 3.96 are such as to put this conclusion in doubt.

An important advantage of the branch method is that interface

conditions between magnetic (and non—magnetic) conductors and

insulators are implicit to the method which assumes that

discontinuities in T and the gradient of J, occur only at

triangle edges. Its major disadvantage is that a great deal of

computational work is required to derive each nodal equation.

Consequently, use of the branch method is best reserved for

problems in which the solution domain boundaries are curved or

highly irregular.

3.7 Determination Of The Force Vector

The purpose of a large majority of electro—magnetic field

calculations is to determine the force acting on conducting

structures lying within a region of interest. Since it is

experimentally observable that in the absence of displacement

current the force acting at any point is equal to the vector

cross product of the current and flux densities at the point,

the total force acting on an enclosed volume V of conductor

is given by

F = sirJ x B dv 3.3.13

There are several ways of evaluating 3.113 The most direct

is to substitute curl T for J and tk(T gradA) for B.

This results in a volume integral expressed in terms of T andil

only and the integral may be evaluated with ease numerically for

75. simple conductor shapes. When the conductor is of complex shape

it is often advantageous to use surface integral expressions to

determine F. A simple surface which encloses the conductor may be

defined, and integration over this surface then yields the

23 reauired value of force. The technique is due to Maxwell who

developed it from a consideration of the electro-mechanical

stresses within an element of conductor. Consequently, the

technique is often refer-red to as that of Maxwell Stresses.

It is found that when the field vectors are given phasor

form

F = fRe (J*x B) dV 3.114 V

where the asterisk denotes complex conjugate. In the calculations

to follow we shall assume 3.114 in preference to 3.113 since

the phasor equations immediately reduce to the non-phasor ones

when the conjugate quantities are replaced by instantaneous

values. Having derived the non-phacor force expressions, however,

it is not immediately obvious how to obtain the phasor force

expressions from them.

3.7.1 The Volume Integral Method

Substitution of 2.4 and 2.6 into 3.114 yields F in

terms of T and SL thus

= fRe(curl T*x (T - grad (510) dV 3.115

This expression is valid in all coordinate, systems. If we choose

the Cartesian reference frame and assume Tx and T

z to be the

only non-zero components of T then in component form F is given

by

n (6T dr, ) (T as) aa 8Tx dV 3.116 F = 1.1.11e [k x - ) z -- A X

az a ay V az ax

76.

Fy

rz

= 11; Re[

V

:/(11 ReQdTx

V

a T _ x

6Tz

atm, ax

) (rx

aTz

+ _

6J1,)] do az

a z dV

3.117

3.118

a Y a y

qiu) L

To determine the force vector in rectangular mesh finite

difference terms it is necessary to associate some elemental

volume with the points on which the derivatives of T and.% are

based. A suitable volume, shown in relation to a three

dimensional grid, is contained in Fig 3.13. This volume is a

rectangular block for which each side is coplanar with two

coordinate directions and each face bisects a single mesh branch.

mesh intervals

hx , by , h

z

The force calculation proceeds by finding the derivatives and

values of T and A centred on node 0. The square bracketed terms of equations 3.116 to 3.118 are then evaluated and their real

part found. Kultiplication of the result by the rectangular

block volume and ti (which is assumed constant within the block)

yields the elemental contribution to the force vector, F. The

sum of these elemental contributions taken over the limits of

any conductor yields the total force acting on that conductor.

The above procedure needs to be modified when a surface node is

77.

to be consi:lered since the rect.angular block is then not bod.y

centred. There are several ways of dealing Hith this situation

but it is suggested that the tanGential derivatives be ce~tred

on the surface node itself whilst the normal derivatives be cen~re1

half a noie spacing on the inside of the conductor ~easured nor~al

to the s-'.lrface. This rule has been founel to yield Good numerical

results (sea Cha~ter 5 ) but may nevertheless not do justice to

the accuracy of the finite difference method used to obtain the

field solution.

A further difficulty arises at conductor edges and it may be

expressed \-lith reference to a plane slice of mesh taken through

a conductor as in Fig 3.14. Although it is possible to obtain the

force contributions from elemental volumes associated \'lith nod3s

2, 6, 8, 4 and 5 using the procedures outlined above, the force

contributions from volumes A, B, C and D cannot be so obtained.

The Simplest Hay of overcomine- the problem is to determine, say

for volume A, the x derivatives half way betHeen nodes 1 and 2,

the z derivatives half way between nodes 1 and 4, and the y

conductor surfaces Fig 3.14

derivatives and absolute values at node 1. The accuracy of this

technique is difficult to estimate and since the largest current

and flux densi ties often occur at or ~ear conductor edges the net-

78.

force Calculated could possibly be substantially in error.

When the'two—dimensional branch method is used to determine

the magnetic fields the elemental forces are easily obtained

since the derivatives of T and A are constant within each

triangle and may thus be taken outside the integral signs of

3.116 to 3.118 The non—gradient St. and T terms may also be taken outsile the integral signs if one assumes that the values

of these terms at any triangle's centroid are typical of those

elsewhere within the said triangle. The difficulties associated

with the corners and edges of conductors in the finite difference

volume integral determination do not arise when the branch method

is used. In this respect the branch method is superior to that of

finite differences.

3.7.2 The Surface Integral Eethod

The surface integral method has been little used by the

author and the interested reader is referred to two papers by 24,25

Carpenter The first of these is a general review of force

calculation methods whilst in the second, surface integral finite

difference evaluations are considered.

Let S be a closed surface in air that totally encloses some

material on which electromagnetic forces act. If n denotes a

direction everywhere perpendicular to S and (ti, t2, n) form a

right handed local coordinate set whose origin is on SI then the

material experiences a force

F = E 'Ft Ft2'12). i 1

PF1111+ Pt 1

Ft 1;2).1'111 2

((Fnn F, t, /1 I

Pt t2)41k 2

da

3.119

79. o2 , * ,

where Fn = Re ( n Hn

Ht Ht — H

t2Ht2

)) 1 1

3.120

Ft = Re ( H H )

1 Po n t

1 3.121

= Re (0 HnHt2

) 3.122 Ft2

are known as the Maxwell Stress components and H is a phasor.

j and k are unit vectors in the x, y and z Cartesian directions

respectively. Similarly n, t1 and t2 are unit vectors in the

n, t1 and t2 directions respectively.

Equation 3.119 is important since it yields the force due

to the currents togther with that due to reluctance effects. In

contrast, the volume integral technique based an equation 3.113

only yields the force due to the currents.

An application of the surface integral approach to force

calculation is given in Chapter 8 where an E—core plate

levitator fitted with a magnetic conducting secondary is

considered.

80 .

4. SOLUTICU METHODS FOR DIS..'1RETIZE1)

ELECTRO:JAUETIC FIELD 2..---qATIOir3

All the partial differential equation approximations so far

discussed have produced a set of equationshaving no less an order

than the number of nodes. This set must be solved subject to the

boundary conditions given and it my be expressed in the form

L1 x1 (A, T) = fl 4.1

where L1 is a sparse coefficient matrix, is the vector of tit

and T component values, and f1 is a constant vector that

incorporates the boundary conditions. 4.1 may be solved using

direct methods, such as Gaussian elimination, or by iteration.

Combinationsof these two methods are possible and are sometimes

useful. For example, 4.1 may be separated into three sets

of equationsthus

L2 x.2 (A) = Q1L3 (Tx) + Q22s4 (%) + 4.2

L3 x3 (Tx) — (Tz) Q312 (A) 4.3

L4 x4

(T z) = x3 (Tx) + Q412 (A) 4. 14 4.4

Qi, (12, Q3 and Q4 are rectangular matrices9 f2, f3, f4 contain

tho boundary conditional and x2, x3, x4 contain only SI', TX and

Tz values respectively. Since L3 and L

4 are usually of much

lower order than L2, a useful solution procedure might be to

solve 4.3 and 4.4 by elimination whilst iterating to find

the values using 4.2 .

It.is possible to partition equation 4.1 such that T may

be eliminated by manipulating the sub—matrices. For example,

consider the following two equation splitting of 4.1

* An alternative to employing the complex matrix Ll is to split 4.1 into real and imaginary parts and then solve the pair of coupled equations as per MAMAR R.S. & LAITHAVAITE E.R.,'F.umerical evaluation of inductance and a.c. resistance',Froc.IEE,Vol.108C,1961,p252.

81.

L5 L5(St) = Q5 3.6(S1,) 16 4.5

L6 2s7(A0 = Q6 x6 (A)

7 4.6

This is valid when there is only one conducting region. 'Tote that

Q5 and .(16 are rectangular matrices, and z5 contains the values

of A at nodes outside the region within which T is non-zero. x

contains the I% values at the nodes within the zero T region the non-zero T region

that are connected to X by single mesh branches. x7contains the

remaining J and the T component values. The elements of x5 and

x6 are mutually exclusive. x7 is an ordered vector arranged so

thatforanyx.contained within x7 1 x. =A for O<Xia,

xi = Txi for n<g2n, and xi = Tzi for 2n<iOn . n is the

number of nodes within and on the surface of the region where T

is non-zero. As before, the f vectors incorporate the boundary

conditions and it should be noted that f7 is invariably zero

since it only incorporates the reference T values. Consider now

a partitioning of L6 such that 4.6 becomes

L11 •=18(A) 4- L12 x'..9(Tx) + L13 x10(Tz) = 4.7

L21 4(4) L22 -..9(Tx) + L23 lcao(Tz) = L2(4‘,) 4.8

L31 .2061,) + L32 x9(Tx) + L33 ic.10(Tz) = .3(61) 4.9

where f is assumed zero, L 6 L for l‘ign, 7

Lid 6 x7 = (=8' 39' E10)

and x, = (12_,. L3). Elimination of Tx and Tz between these -o

equations yields

L7 x8A) = diag[-P 0 -R,Q] 2s6(A) 4.10

where

L7 = QL

31 L32

L22 -1L21 - L11 + L12 L22

-1L 21

4.11 P = unit matrix of order n

82. -1 L

12L22-1

14 —1 —1L

—1 4.11 L23) (L33 — Ll,L

= ( 13 Ll2L22 22 23)

= QL32L22 "'

Equation 4.10 has a significance which is easily clouded by the

algebra involved in its derivation. For 4.10 and 4.5 may be

combined to yield a single matrix equation in A only. Thus .a, arreo-rs to 6s

alone A capable of describing three dimensional electromagnetic

fields. However, the nature of the linkages between nodes lying

in the conducting region is complicated since L7 is a full matrix

expressing the fact that within conductors each node is coupled

to every other. A further result of interest is obtained by

combining 4.10 with 4.5 and performing an elimination of

values at nodes within the conductor. If a network model of the

new set of equations is made, it is found that the conductor is

modelled by surface impedances connecting each member of x6 (A)

with each and every conductor surface node.

Although elimination of T from the linear simultaneous

equations yields another physically recognisable set, the

technique is unlikely to be useful because of the considerable

number of matrix inversions and manipulations required.

It was mentioned previously that either direct or iterative

methods could be used to solve equations of type 4.1 . It is

conventional to use iterative methods for sets of equations

appertaining to rectangular meshes since these sets have the

regular structure necessary for efficient iterative solution. In

contrast, direct methods are usually employed to solve irregular

mesh eqUations. The sectiomwhich follow contain details of the

solution techniques used for the discrete scalar potential

equations derived in the previous two chapters.

83.

4.1 Iterative Solution Using Successive Overrelaxation (SCR)

One of the simplest and most effective iterative methods is

point successive overrelaxation. The method is defined in the

following way. Let the set of ec:uationsto be solved have the

form •

L1 Y = fa 4.1

and express the matrix L1 as the mc,.trix sum

L1 = D—E— F

4.12

where D 1 and E and F are = (nag [L11, L22' • • • L

respectively lower and upper triangular m x m matrices whose

entries are the negatives of the entries of L1 respectively below

and above the main diagonal of Li. Then the SOR iterative scheme

is defined by

(K + 1) =

Lw

(K) + w( I —-1D-1fi Lc4 4.13

where

Lw . (1 - wL1)-1 ( (1 - w) I + wD-1F) 4.11+

and w is an acceleration factor lying between zero and two. K is

the iteration number and L is termed the point successive w

overrelaxation matrix. Convergence of the SOR scheme occurs only 26

when the spectral radius of Lw, p(L ), is less than unity.

It is known that any L1 matrix which has a consistently ordered

Jacobi matrix and satisfies Young's 'Property A' has an L matrix

27 which satisfies this condition. 'Property A' and consistent

ordering are difficult to establish directly for any given matrix

so thatequivalent definitions in terms of p — cyclic matrices 28

and directed graphs are often used. The directed graph of a

matrix may be defined as follows. Let A = (aid) be any m x m

81f.

complex matrix, and consider any m distinct points Pi, P2...Pm

(called 'nodes') that are confined to a plane. For every non—zero

entrya_ofthenatrixweoanneotthenodeP.to the node P. by 13 1

meallsofapaulp.p.l airectecIfromP.toP..In this way, with

every m x m matrix A can be associated a finite directed graph

G (A). For example, consider the matrix

1 0 1 A = [0 1 1

1 0 1

To construct the finite directed graph we first set down on

paper a number of nodes equal to the order of the matrix (3).

These are denoted by P1,

P2 and P3. 7:Te consider one element of the

. . matrix — that in the (i,j)th position, say. If the element is

non—zero we join P. to P with a branch on which is placed an 1

arrmthatisdirectedfromnodeP..If the element is zero, no 1

action is taken. When all the elements of A have undergone this

treatment, we obtain the following finite directed graph:

At this stage it is useful to define what is meant by a

p — cyclic matrix. Let A ..0 be an irreducible m x m matrix,

and let p be the number of eigenvalues of A of modulus p(A).

If p = 1, then A is primitive. If p >1, then A is cyclic of

index?. Now the value of p for any given matrix may be

determined using a directed graph. The technique is quoted by

29 Varga and is as follows.

Let A = (aii);> 0 be an irreducible m x

G (A) as its directed graph. For each node P. 1

allclosedpathsoonnectingP.to itself. Let

m matrix, with

of G (A), consider 0.

the number of IrroAches A

85.

path be m.. If S. is the set of all the m. values and p. is the 1 1 1 1

greatest common denominator (g.c.d.) of these values then

p.1 = g.c.d. (' mim .E S . 1 1

Defining p as the g.c.d.of the set of pi values we state that

when p = 1, the matrix A is primitive and when p> 1 A is

cyclic of index p. As an example consider the matrix A examined

previously and for which

pl = g.c.d. (3,2,1) p2 = g.c.d.(1) ; p3 = 0

so that p = 1 and A is primitive. These directed graph results

have been presented as a pre—amble to an important theorm30 which

states that if the point Jacobi matrix of L1 defined by

B = I — D1

(where D is defined by 4.12) 4.15

is weakly cyclic of index 2 then L1 satisfies Young's 'Property

AI. Now B is weakly cyclic when it is irreducible and non- 31

primitive. Irreducibility is assured when A is an electromagnetic

field nodal equation coefficient matrix since the value of a

field quantity at one node is dependent on its values at all

other nodes. This property is reflected in the result that the

inverse of A is always a full matrix.

Finally, it is possible to determine whether or not a given

matrix is consistently ordered by employing a modified finite A 28

directed graph. Let G(B) be the directed graph of type 2 for

thernatrix B.(b.)/ so constructed th i

atifb.j q20, then

ij

the path from the node P. to the node P. is denoted by a double

arrowed path only if j> i; otherwise a single—arrowed path is

used as in our previous graphs. The former paths can be called

major paths; the other paths can be called minor paths. Then.

the matrix B is consistently ordered only if every closed path

A Nodes

0 (r— 0 For clarity, links between T &

0 Tz omitteCi. 0 0

Tz Nodes

— 7---0 0

Tx Nodes

0 0

86. A

of its directed graph G(3) has an equal number of major and

minor paths. For example, consider -the matrix

[ 0 1 0] B = 1 0 1

0 1 0

A whose G(B) is F)

1

so that B is consistently ordered (as well as of index 2).

4.2 An Analysis Of The Iterative Characteristics Of The

32

The rectangular mesh magnetostatic scalar potential

equations are discrete forms of Poisson's equation and do not

contain T as a variable. Using directed graph techniques, the

33 equations are shown by Varga to have L

1 matrices that are

consistently ordered and satisfy Young's 'Property A'. In order

to ascertain whether the eddy current equations also had these

properties the following procedure was adopted. Consider the

nine point x — z plane mesh of Fig 3.2 . This mesh was

employed to obtain a two dimensional solution of 4 and T within,

but not at the surface of, conductors. In our analysis, the

values of a and T at a point in space will each be associated

with a node. Now form the part of G(B) which shows the

linkages between T and S' at a single computation molecule,

using equations 2.42 3,64 and 3.65 . The result is

Rectanr,ular esh T]guations

Examination of the loops formed by this process shows that four

87. branches are always used in each loop. Thus B must he no greater

than 4 - cyclic. Consider now the part of G(B) appertaining to

the top surface of a conductor. Equations 2.42 and 3.67 are

used for this purpose. The result is

SL Nodes Tz Nodes

and it will be noted loop a,b,c contains three branches. The

Previous graph contained a loop of four branches and since the

highest common denominator of four and three is unity, B is

primitive. Thus L1

does not satisfy Young's 'Property A' and

convergence of the SOR iterative scheme is not guaranteed. These

examples show how powerful the directed graph technique is; for

it enables us to deduce some of the important properties of a

large matrix by considering the structure of only a few rows

and columns.

It should be mentioned here that one further sufficient, but

not necessary, condition for convergence of SOR is that the 34

,matrix L1

should be strictly diagonally dominant. In

magnetostatic scalar potential applications this condition is

met. However, examination of equations 2.42 3.64 and

3.65 reveals that when eddy currents are modelled the L1

matrix is no diagonally dominant, though it may approach this

condition when the simultaneous equations are suitably

manipulated.

In view of these results it was of no surprise to discover

88. that iter,Aive convergence was slow when the magnetic scalar

potential technique was first used to solve a model eddy current

problem. The geometry employed was similar to that of Fig 5.1

except that the air gap was 1.4cm, and there were 90 and 120

nodes in the air gap and end regions respectively. The conducting

secondary was placed lcm above the stator (primary) and was

0.39m (26 horizontal mesh intervals) in width and 1.5cm (3

vertical mesh intervals) thick. Furthermore, its physical

properties were such that the p constant was equal to 0.005. In

order to obtain a direct comparison with a magnetic vector

potential solution of the problem, the pole pitch was made

infinite. When the scalar potential distribution was calculated

using unity acceleration factor, 850 iterations were required to

reduce the maximum error to 2 x 10-5 of the maximum potential.

In contrast, only 137 iterations were required when the

conducting secondary was removed.

Attempts at overrelaxation were made but(nly a small increase

in convergence rate was achieved. Use of the Carre — st01135,36

technique for finding the best acceleration factor4E caused

numerical instability. This result was to be expected since the

technique relies on there being two eigenvalues of Lw having

modulus f) (Lw) i.e. that the point Jacobi matrix, B, is

2 — cyclic. Since we have shown that B is primitive in eddy

current calculations it follows that use of the Carr — Stoll

method has no theoretical justification.

A solution of the model problem was attempted using

magnetic vector potential but the dominating effect of the iron

surfaces made the convergence rate too slow to be practicable

without the use of various acceleration techniques (including

irC A more systemmatic analysis of methods that yield the best acceleration factor for complex successive overrelaxation is given by H.E. Kulsrud in: Comm.Assoc.Comput.mach., 1961, pp184-10.

14-.17 0

--1 BA 2 2

0

N = I - D-lA =

89.

specifying the flux linkage instead of the excitation current).

Thus quantitative comparison between the magnetic vector and

magnetic scalar potential solutions was difficult and perhaps not

very meaningful because it was so problem dependent.

The discovery of poor convergence rates for the magnetic

scalar potential solution prompted an investigation into the

possibility of using block SOR as a means of increasing these

rates. Consider, for example, the matrix Li partitioned into

block tridiagonal form such that the matrices on the main

diagonal are square and non-singular. L1 thus has the structure

4.16

The block Jacobi matrix, LI , of L1 is given by

where D = diag t Bi, B2...BM] . The block directed graph of A,

type 2 of M I GM, is

N-1

9n. so that M is both 2 — cyclic and consistently ordered. Thus,

convergence of the block iterative scheme

A Kia) (E B. = A. x.

+1) + C x.

(K) + f1 . 16:i.;ll 4.18 1 —1

( 1 —1-1 i1 —+1 —

(K+1) ,A (K+1) — (K) (K)

x. x = w k. x. ) + x. 1.i<IT 4.19 B —3. 1

is guaranteed. Here the x.'s and f.'s are the respective vector —s —a

partitions of x1 and f

a corresponding to the partitioning given

A

d

(K4.1)

in equation 4.16 . x is simply the non—accelerated form

of x.a after the (I\+1) `'h iteration.

Now the scalar potential eddy current equations may be

expressed in the form

LII 2;8(A) = Q9 2s9(A)

4.20

L9 9(S t) = Q10 4(St) Q11 3S10(I) 19 4.21

L10 1:10(2) = Q12 19(A) 4. 110 4.22

where x8 contains the A values associated with nodes at which T 0

x9 contains the St values associated with nodes

within and on the surfaces of regiorrswhere T 4:0

x10 contains the values of the non—zero components of

T

L8,L9 are square. 2 — cyclic non—singular irreducible

matrices

Q9'Q10'Q11 and Q12 are rectangular matrices

f8' f9 and f

ao incorporate the boundary conditions. The

—— — and

• n (K1-1) = • ±8

A (K+1) L 19 =

, A (K+1) L'10 110

(K) f8 -8

(K+1) (K+1) Q10 18

(K) Q x f 11 -10 -9

(K+1) = 0 x + f '12 -9 . -10

4.24

4.25

4.26

91.

matrix form of 4.20 - 4.22 is

--(1 -

Le, 9

0

-Q10 L -011 -11 0 . -Q12

L10_

f8

±9

-10.

4.23

The block coefficient matrix is seen to be tridiagonal and of the

form 4.16 with N = 3. The equations of type 4.18 associated

with the partitioning 4.23 are

Each of these equations may be solved in many different ways and

it is not obvious which are likely to give high rates of

convergence. As a first guess it was decided to solve 4.26 for

10 using ten Gauss-Seidel iterations and assume a known x „.

Then x9 was determined using the computed xio values and an •

arbitary x8. Finally, x10 was computed using the calculated x,

values. These last two steps were accomplished using one point

SOR scan each. The process was repeated using the latest x

estimates and an wB

value of unity until convergence was

achieved. The Gauss-Seidel method was used for 4.26 since L10

was primitive and consequently there was no satisfactory way

of determining the best acceleration factor for SOR. Ten

iterations were used because it was thought that 4.26 would

converge more slowly than 4.24 and 4.25 since L10, unlike

92.

L8 and L99

did not satisfy Young's 'Property Al..

Using the above algorithm in the model problem improved

convergence by a factor of 5 (to 173 cycles) and there was a

reduction in computing time by a factor of 3.5. Furthermore, when

the Carr.6- — atoll method wasepolied to 4.24 and 4.25 the

number of iterations was further reduced (to 115 cycles). It will

be noted that, assuming consistent ordering of L8 and L9, the only

block iteration constituent equation that could diverge is 4.26 .

Now, convergence of the block method is guaranteed only when

each constituent block has a numerically stable solution. Hence,

in some applications the equation for T (4.26) may have to be

evaluated by direct rather than iterative methods. Such a solution

may be accomplished in the following way. Consider a one

component (Tz) T formulation for which, under travelling wave

conditions and within conductors, the governing equation is

Tzl 2

+ Tz3 + j Ni p (A2 — A4 ) — (2 + j2N2 a)Tzo =

4.27

(obtained from 3.65 by setting T'x to zero)

where both T and SL are phasors and the node numbering is that

of Fig 3.2 . 4.27 may be re—arranged to yield

Tz0 — (T

z1 + Tz3 )Q = SN21 p Q

(J12 — JL4 ) 4..28

h

where Q = (2 + j2N5 + a )-1

4.29

and thus take the form of 4.26 with the SL values acting as

sources. Now, at top and bottom x — y plane conductor surfaces

the appropriate Tz euuations are

2 Tz0

*(Tzl + T' z3 1 .) V + j N1p v Tz4

= j Nip v ('22 - SI% ) 2

4.30

93.

and

j 1T12 pv Tz2 = j Ill {3V ( La2 s2,4) Tz0 (Tzl Tz3) V + 2

4.31

respectively where

V = (2 j Ni2 p a )-1

4.32 2

Assembly of 4.30 , 4.31 and 4.28 for each node yields a

matrix equation of order equal to the number of nodes at which

Tz

is a non—zero variable. Unfortunately the assembled equations

are not tridiagonal so that very efficient direct solution

39 procedures like the Thomas algorithm may not be employed. Note

that the lack of tridiagonality is due to the surface node

equations 4.30 and 4.31 which involve Tz terms appertaining

to rows of nodes having different z values. Before leaving this

aspect of the numerical technique it should be mentioned that

line iterative methods may be employed to solve 4.26 . Consider,

for example, the line of nodes on the top surface of a conductor.

Then the line method assumes that the Tz4

values are known and

puts them on the right hand sides of the equations so that they

act as source terms. In practice the Tz4 values from the

previous iteration are employed. When assembled, the equations

for the top line take the form

am,

Tz0

2 = 1p v

j /T2 pv

Tz4

2

AI=

4.33

in which the matrix pre—multiplying the unknown Tz values is

94. tridiagonal. Thus the Thomas algorithm may be employed to

determine these values. Within conducting regions distant from

surfaces, equations of type 4.28 apply and a tridiagonal set

for each line may be assembled. The matrix form of these lines

is similar to that of 4.33 except that V is replaced by Q and

the Tz4

term is omitted. At a bottom surface another set of

equations similar to 4.33 is obtained; the only difference

being that thez4 vector is replaced by one containing the

z2 values.

The line method proceeds by systematically solving the T

equation line by line from the top of the region to the bottom.

Such a set of calculations forms one iteration. After each line

of values has been determined, overrelaxation can be employed

and when this is used the iteration technique is known as

successive line overrelaxation. When the bottom line of T

values is determined, the Tz2 values calculated at the line

above, rather than those appertaining to the previous iteration,

are employed. This ensures that the most recent values of Tz

are used at all times. 37,38

In a model problem Varga shows that line iteration can

'involve no more arithmetic operations per node than point SOR and

at the same time it can improve the asymptotic rate of iterative

convergence by a factor ofiV7. When the unaccelerated line method

described was used to determine Tz in our model problem, the

number of overall iterations required to solve 4.24 , 4.25

and 4.26 was unaffected. The value of p was increased by an

order of magnitude to 0.052092 and the number of cycles required

using either line or point iteration for 4.26 was found to be

160 (as compared with 173 for p = 0.005 ). Thus, not only was

4.34 0

0

L13

L12

L11

95. the rate of convergence high, but it was almost independent of

the effective depth of penetration of the conducting secondary.

This behaviour was attributed to the fact that the model problem

contained boundaries which were iteratively well conditioned (i.e.

of the Dirichlet type).

Finally, let us consider the convergence properties of the

line method. First we note thl,_+, the method corresponds to a

partitioning of L10 given by

L10 0

11 L12

L13

0

so that the block Jacobi matrix associated with this partitioning

is

= 4.35

The block directed graph of id is given by

1 2 rem-1 mm

where mm is the block order of L10

and is equal to the number

of lines. Examination of G(M) shows +hat M is an unconnected

96. non—cyclic matrix and consequently graphical methods cannot be

used to determine its iterative characteristics. In short, no

guarantee of convergence of the line method can be made. The

argument preLented here is restricted to single eddy—current

region one component T formulations but is equally applicable to

the general case.

4.3 Behaviour Of SOP, In Large ode Number Problems

In many low frequency electromagnetic field problems the

level of accuracy required necessitates the use of large numbers

of nodes (greater than 1000, say). Irregular meshes utilise the

available nodes more efficiently than do rectangular ones, since

large numbers can be placed in regions where the field is

expected to vary rapidly and only a few nodes may be used

elsewhere. However, iterative solution is often impractical when

irregular meshes are used since the spectral radius of the

corresponding iteration matrix is invariably close to unity;

thus making convergence slow. Consequently, more complicated 40

methods of solution such as sparse matrix Gaussian elimination

must be employed. Solution of the final set of linear

simultaneous equations is much easier when high order mesh

,elements are used since the number of nodes is then relatively

small. However, the equations are much more time consuming to

generate., and high order elements are difficult to use

effectively when the conductors and boundaries are not

rectangular. In this section we shall consider the effect on

iterative convergence of increasing the number of nodes in a

model problem. The aim of the investigation is to suggest methods

by which iterative computation times may be reduced.

97.

First, let us consider the effect of increasing the number

of nodes on the amount of computation required to obtain a

solution. In the point successive overrelaxation (SOR) iterative

scheme it is clear that the computation time per iteration is

directly nroportional to the number of nodes. Now the asymptotic

rate of convergence of point SOR as the number of iterations 41

approaches infinity may be defined as

R = 1 logo p(Lw) 1 4.36

If we consider a hypothetical problem involving 499 nodes (say)

arranged to form a square mesh, and if we let the moduli of the

eigenvalues of the point SOR iteration matrix, Lw, be evenly

spaced between the limits of zero and unity, then

R = in 422 I = 8.69 x 10-4

Increasing the number of nodes to 999 and 1999 yields R values

of 4.34 x 10-4 and 2.17 x 10-4 respectively so that the

asymptotic rate of convergence is halved when the number of

nodes is doubled. Thus, the overall computation effort required

to obtain a solution of fixed accuracy seems to be proportional

to the square of the number of nodes. Obviously, the conditions

imposed on the model problem considered are unlikely to occur

in practice. Furthermore, only the asymptotic rate of

convergence, which is usually reached after a considerable

number of iterations, has been examined. :Tevertheless, this

brief analysis gives an indication of the relationship between

node number and computation effort.

By choosing a good acceleration factor for SOR it is

possible to reduce considerably the number of iterations

required to obtain a solution. However, the best acceleration

500

98.

factor (s gib) is a difficult parameter to estimate but if the

coefficient matrix of the full set of simultaneous equations (Lw)

is consistently ordered and satisfies Young's 'Property A' then

the method of Carr-6 and Stoll is often useful. The details of

the method may be found in the literature and it is sufficient

to state here that a good estimate of wb can only be obtained

when the dominant (W(1) and sub—dominant ( Xsd

) eigenvalues

of L are not close in value. ::ow we know that if Lw is

convergent then all the eigenvalues of this matrix must have

moduli between the limits of zero and unity. Thus, as the number

of nodes is increased, the probability of X d and X sd having

similar values must also increase. This helps to explain why

the Carre. — Stoll method works well for small numbers of nodes

(less than 1000, say) but may produce acceleration factors which

cause oscillation of the nodal values or even divergence when the

number of nodes is large.

One way of overcoming many of the problems so far described

is to use a block method, such as line iteration, throughout the

solution domain. Each block then acts, as far as the block

iteration matrix (LB ,;f

)is concerned, as if it were a single

node. When line iteration is employed, the number of eigenvalues

,of LBW is equal to the number of lines. Thus, given the

assumption that the block Jacobi matrix is consistently ordered

and weakly cyclic of index 2, the Carrg — Stoll method should

be very effective since the number of eigenvalues of L3,4 is

much less than that of L. The success of successive line

overrelaxation (SLOR) depends very much on the ease with which

the nodal values along each line may be determined. When the

problem exclusively involves a two dimensional approximation to

99- Laplace's (or Pogssonts) equation, the simultaneous equations for

each and every line are found to be tridiagonal and thus easily

solved. Lost other elliptic partial differential equations have

finite difference forms that do not produce tridiagonal line

equatiens.

In this section we have shown that the amount of computation

effort required for the point SOR solution of large numbers of.

equations is approximately proportional to the square of the

number of nodes. Furthermore, it has been suggested that line

methods may be considerably more effective than point methods,

particularly in problems where Laplace's or Poisson's equation

apply over most of the solution domain. However, if rapid

convergence of the line method is to be expected, the line Jacobi

matrix must be weakly cyclic of index 2 and this condition does

not hold true for the magnetic scalar potential eddy current

formulations examined in this thesis.

4-4 Assembly And Solution Of The Triangular lesh Latrix Equations

Each item of triangular mesh data is usually placed in one

of two categories. The first contains the node numbers and their

position coordinates whilst the second contains the number of

each triangle, the node numbers of its vertices and a flag which

indicates the properties of the material covered by the triangle.

This data may be produced manually, or automatically using a

mesh generation computer program. The former method has the

advantage that nodes may be placed where experience indicates

them to be necessary; but the latter has the advantage of speed

(in human, though not necessarily in computer, terms) and is

often preferred.

100.

Given the mesh data, we must create a matrix equation akin

to 4.1 . Let L be the coefficient (or 'stiffness') matrix.

Then the required equation has the form

L x = f 4.37

where x is the vector of unknowns and f is a constant. Since

L has few non—zero terms per row and is of high order, it is best

generated as a list (P) that contains only the non—zero elements.

An addressing list (a) is associated with P and the Kth

row of

a contains the number (i 1000j) where row K of P contains

the (i7j)th element of L. Before constructing P it is

useful to form an array giving the number of each node together

with the numbers of the nodes to which it is connected by a

single mesh branch. This array is particularly valuable when

the branch formulation is employed since the integrations

involved are taken along contours that enclose the mesh branches.

The T and A eddy current equations form an asymmetric set.

In consequence, every non—zero element of L must be calculated

separately. If the set of equations had been symmetric then only

the main diagonal and the upper (or lower) triangular coefficients

need have been generated, the rest being filled in by symmetry.

When P is under construction it is found convenient to

partition 4.37 as follows

L21

L31

L12

L22

L32

L13

L23

L33 —z

1

2

3

4.38

This partitioning gives L a very large bandwidth. Since the

101.

time taken for solution by Gaussian elimination, or related

direct methods, is proportional to the square of the bandwidth,

a reordering of 4.38 is necessary after it has been constructed

(see section 4.4.3 ). A high degree of computational efficiency

can then be achieved. When successive overrelaxation is used to

solve 4.38 , no re—ordering is necessary because the technique

is bandwidth independent.

In the branch method, the elements of L are obtained by

systematically considering given nodes and the branches which

connect them to surrounding nodes. The line integral

contribution of each branch is calculated and entries are placed

in P and a accordingly. In this connection it should be

mentioned that the direction of the line integral is important

and must be anticlockwise for each and every closed contour

considered. Thus, it is necessary to decide which mesh element

is to the right of a directed line joining the given node to a

surrounding one, and which is to the left. Consider the directed

Fig 4.1 line of Fig 4.1 which is drawn relative to the coordinate origin

of x and z. The problem is to determine which of elements

(i,j,111) and (i,n,j) is to the right of the directed line ij.

Let the coordinates of points i and j be. (x.,z.) and (x.,z )

,A r

aA

t .41.

01 7 . no

de t J v

- ? a

Fraitt lki

r et 1 r , drfr

4i;A

l

1 $1

.I O

il I

/111

44 F

et4

740* F

iletb

‘ .

lk1/4

ir"

ILA

i.$4.1

0 ■

/!/b

6

446

41

1141r 141 IW

O° krw

4lj

w ,A

4 9

a V

I a

ti bli

gir

ak

. PA 1 ke

ttA

fr 1

i r

iff

A TA

IP W

P 1

frl .W•

..;1

air ■))w

103. respectively. Then we proceed by shifting the coordinate origin

to the point i using the transformation

7 1 x = x — x.

1 1

Z = Z - Z. 1 4.39

The coordinate directions are then rotated so that one is

orientated along the line ij. The required transformations are

x11

xlcos = x. )cosh z. )sine

4.40

z11

z clcos — xlsin 6 = (z — z.) os 6 — — xi) sin

4.41

Now cos 0 = (x. — x.) / length ij 4.42 J

and sin 6 = (z. — z.) / length ij

4.43

so that as long as points i and j are distinct we may define a

quantity W given by

W = (z — z. j ) (x —x .) — (x x . j) (z —z .) 4.44

such that W>0 when thepoint (x,z) is to the left of ij

and W<0 when the point (x,z) is to the right of ij.

As can be seen, the concepts associated with the generation

of the elements of the stiffness matrix are fairly simple.

However, the computer programming effort required is considerable,

particularly when program efficiency is given a high priority.

The author's version had 769 card images and required 68.6

execution seconds of CDC 6400 time to assemble the stiffness

matrix of order 355 appertaining to the mesh of Fig 4.2 .

Magnetostatic field conditions were assumed. The computer time

quoted does not include that required for calculation of the

shape function constants. These were determined using a separate

program which required 8.84 execution seconds.

104. MAGNETIC SCALAR POTENTIAL

FORMULATION }OR THE

CARTESIAN COORDINATE X-Z PLANE.

ONE 0) COMPONENT MAGNETIC VECTOR

POTENTIAL FORMULATION }DH THE

CYLINDRICAL COORDINATE R-Z PLANE.

Generate the mesh (node and element )data.

Re-number the :lodes to reduce the

bandwidth of the stiffness matrix.

Determine the shape function constants.

Form the stiffness matrix and

determine the vector of

excitation values.

Form the stiffness matrix and

determine the vector of

excitation values.

Re-number the nodes. Re-order

the stiffnees matrix and

excitation vector accordingly.

Solve the linear simultaneous equations

using sparse matrix Gauenian elimination

and print out the solution.

Plot out lines of constant flux

linkage.

Plot out lines of constant

magnetic scalar potential.

STOP

Fig 4.3 Two Dimensional Irregular Memh Co=7..uter

Program - Flow Chart

105.

In the following sections various aspects of the equation

solution and result presentation procedures will be considered.

These aspects were investigated with the intention of developing

a series of independent tackages which could be combined as the

application under study demanded. When all these packages were

included, the computer program had the structure of Fig 4.3 .

4.4.1 Solution Of Larr;e :lumbers Of Asymmetric Linear

Simultaneous Eauations Usinr; Gaussian Elimination

It is generally accepted that one of the most efficient

direct solution methods for linear simultaneous equations is

Gaussian elimination, details of which may be found in the

references 42 This procedure is easy to program for a computer

and one elimination can provide solutions for several sets of

excitation values. In this section the application of Gaussian

elimination to the magnetic scalar potential eddy current

equations will be considered; particular attention being given

to the limitations imposed by the type of computer available.

A typical magnetic scalar potential irregular mesh contains

in excess of 300 T and J, nodes so that the associated stiffness

matrix has over 90,000 entries. Furthermore, in eddy current

problems these entries are complex numbers. Hence, over 180,000

words of computer core are required if the full stiffness matrix

is to be stored. The computer available to the author contained

only 50,000 words of core and consequently direct elimination

using the full matrix was not possible except by using disk or

tape backing store. Since transfers of numbers to and from core

are expensive in terms of computer time and peripheral device

costing, it was decided to investigate ways of reducing the

106.

storage requirement for Gaussian elimination so that all

operations could be done using coefficients stored in core.

hIxamination of the elimination process shows that in the reduction

of a banded matrix the (zero) terms outside the bandwidth are not

employed. Thus, one way of reducing the storage requirement is

to keep only those matrix coefficients lying within the region

one bandwidth to each side of the main diagonal. If N is the

order of the stiffness matrix and B its bandwidth then the

number of complex storage locations needed is given by

S = B( f N — l) 4.45

As an example, consider a matrix for which B = 75 and N = 300.

S is then equal to approximately 32,000 and the storage

requirement is reduced by a factor of three. If the matrix is

symmetric, S may be reduced further since only the main diagonal

and upper (or lower) triangular terms need be stored. However,

the magnetic scalar potential eddy current equations are not

symmetric so that this procedure cannot be adopted.

The band method described suffers from the disadvantage

that many zeros are unnecessarily stored within the band. When

the stiffness matrix is symmetric a better technique, due to

40 Jennings, is available. In this technique all the elements below

the leading diagonal are stored in sequence by rows, but with all

elements preceding the first non—zero element in each row left

out. For example, the matrix

[ 1

6 0

6 1 0

0 0 1

would'be stored in the computer as a 'main sequence' given by

107.

In addition, an 'address sequence' would be used to locate

the positions of the leading diagonal elements within the main

secruence. For the example this would be

[1 3 4]

so that a total of seven storage locations would be required

in place of the original nine. TheJennings method may be

extended to asymmetric matrices but (at least) two indexing lists

are then needed. One can be the address sequence already described

whilst the other can mark the beginning of each new row of

coefficients in the main sequence. In the above example let the

coefficient at position (2,1) of the matrix be set to zero so

that asymmetry is introduced. Then the sequences stored could be

E 1 6 1 1] (main)

[ 1 3 4 J (diagonals)

and

C 1 3 4 J (row marker)

In typical applications the asymmetric matrix Jennings method

requires far fewer storage locations than does the band

technique. Both have the important feature that when the

equations are solved by elimination and backsubstitution without

row or column interchan.ze, all the build up of non—zero elements

occurs within the main sequence and no new storage locations

need be found as the elimination:progresses.

It was found that even using the Jennings method the storage

requirements of many stiffness matrices encountered were too

great for the computer available. Now in principle, the most

108.

economic storage arranement for any matrix is to store only its

non—zero elements in a single list. The most important

disadvantage of the technique is that new storage locations must

be generated as the elimination progresses and the indexing lists

accordingly reordered. These steps are time consuming and are

difficult to program. However, since corn-outer core store was

strictly limited and computer time (within reason) was not, the

non—zero element storage system was that adopted by the author.

Several types of indexing arearailable for the system and

most involve the same computation effort per location sought. The

type chosen took the following form. Let a3.3 . . be a non—zero

element of the stiffness matrix and let there be an indexing list

q which contains the ordered column (j) locations of each a. .

stored in a main sequence of the Jennings type. Then, for the 3 x 3

matrix previously considered

El 6 6 1 3.]

would be the main seauence stored row by row and q would be

given by

El 2 1 2 3]

Now we define a second indexing list r such that the last

non—zero element of row i is stored at location

q (i 1) )

in the main sequence. In the example r is thus

Co 2 4 5]

where r (1) is arbitrary and has been set to zero. The storage

109.

technique described was employed for a stiffness matrix of order

355 containing approximately 2400 non—zero entries and at the

end of the Gaussian elimination the triangulated matrix contained

5877 further entries. The stiffness matrix bandwidth was found

to be 72 and 1153 seconds of CDC 6400 execution time were required

for the elimination.

In general, the discrete forms of elliptic partial

differential ecruations are stable with respect to the growth of

numerical rounding errors produced during the course of the

solution. If, however, mesh elements of severe aspect ratio are

employed then rounding errors can be significant. In order to

combat their growth, partial pivoting with row scaling may be

employed. Partial pivoting proceeds as follows. When the ith

row

of the coefficient matrix is to be eliminated, a search is made

th down the i column (starting from row i) for the element of

largest modulus. Having found this element its row is interchanged

with the ith. In this way the pivotal value is not allowed to

approach zero and rounding errors are severely controlled when,

at each stage of the elimination, the row norms are the same

order of magnitude. The norm condition can be satisfied by

scaling the matrix after each row has been eliminated. In

practice, scaling is found to be necessary for only the first

few eliminations. The choice of norm for scaling is not critical43

and in the author's computer program that given by n

> 'laiji

2

;1- = 0.75 4.46

j=1

was used. Strictly speaking, the columns should also be scaled

but this is time consuming and unnecessary for the vast majority

of applications.

4.4.2 Magnetic Field Contour Plotting For Triangular Meshes

110.

The nodal values of a calculated field quantity are often

used to determine the 'global' properties of a solution domain;

force, power loss and current density being e::amples. Nevertheless

it is often useful to have a two—dimensional plot describing the

field, particularly when the positions of point phenomena are to

be identified, Where successive overrelaxation is employed the

nodal values are usually stored in a matrix whose elements are so

ordered as to be directly related to the spatial nositions of

the nodes. Thus, when convergence has been reached the matrix

may be printed out and immediate assessment of the field

distribution made. As a subsidiary matter, contour lines may be

printed out using computer software and there are many packages

available which perform this function for regular rectangular

meshes. Where irregular meshes are employed the list of nodal

values produced by the solution program usually has little

direct meaning and the provision of field contouring computer

packages is then important. Some irregular mesh packages are

already available but those examined by the author required

cumbersome data input and used interpolation based on nodes

rather than elements. The only irregular meshes considered by

-the author were first order and triangular, and it was decided

to write a contouring program which assumed this mesh form.

Since it is known that within each first order element the

field quantity (A, say) varies linearly with position, the entry

and exit points of a given contour through every triangle serve

to define the pen movements of the plotter. Let us consider

how the positions of these entry and exit points may be

determined. Fig 4.4 contains a typical triangular element and it

(x2, z2)

1 2 i=1

then 3.82 may be expressed as

3 3 a = Ea .2 St .1

; 'o = bi St.1 ;

1=1 i=1

and c = ]): a.

will be assumed that 3

.51(.,z) = (a. + b.x + ciz) LSI • 1 2 1 i

i=1 within the element where St is real and a_,b.andc.a.re

2. 2. 2.

3.82

defined by 3.86 and 3.88.

(x3,z3)

(x1, z1)

Fig 4.4 If we let

(a — St(x,z) ) bx + cz = 0

Assuming that SL(x,z) takes the required contour value of Ac

(which is bounded by the vertex St values), 4.48 becomes the

equation of the contour line crossing triangle 123. The problem

is now one of finding the positions at which the line defined

by 4.48 crosses the sides of the triangle. The equation of

each of these sides takes the form

z. —z. x [ 3 1-]

z.x — z .x. z =

where tiw. subscripts denote vertex number. Substitution of

4.1+9 x — x. x. x.

3 1

4.47

4.48

I Determine the values of a,b and c.

Find where the contour enters the triangle]

ISet the pen of the plotter at this point.

Find where the contour leaven the triangle.'

Draw a line between the entry and exit points.

112. READ IN

1. Node position data

2. Element construction data

3. Shape function constants (ai,bi,ci)

4. Number of contours required

5. Plot scaling factor

DETEREINE

1. The maximum and minim m contour values

2. The contour increment

and print out the values of these quantities.

Set the contour level to its minimum value.

Has each triangle been checked to determine whether or

NO

not the contour level under consideration crosses it?

Find a triangle not yet considered at this

contour level.

Check the vertex potentials to check if the

contour crosses the triangle.

Does the contour cross it ? E-N0-4

Increment the contour level

—TES

Are these any more contours to be drawn ?

NO V

Write out the plot title and computation time.

I STOP

Fig 4.5

First order triangular memh contour plotting computer program

- Flow Chart

113.

4.49 into 4.48 yields the x and z values at which the

contour crosses a given triangle side. The relations obtained

are

- (a + cv) xc = z = um

c + v

b + cu

where z31. z.

1

z.x. — z.x. and v = • 1 a 1

—x. 1 4.51

The flow chart of the program incorporating this technique is

given in Fla' 445 •

Thus far we have considered contour plotting for Cartesian

coordinate problems. When the r — z plane of the cylindrical

coordinate system is under consideration, exactly the same

techniques apply. However, when this system is used it is often

required to plot a simple function of the field values computed

rather than the field values themselves. For emample, an r — z

plane plot of flux linkage may be required using calculated

magnetic vector potential values. Alternatively, a plot of the

peripheral component of magnetic field strength may be required

given values of magnetic scalar potential. These requirements

have attendant difficulties. Let us consider the former

application. If the magnetic vector potential, A, is peripherally

directed so that 'A = A then at radius r the flux linkage is 0

given by

• 0 = rA

0 = r(a + br + cz)

where A0 is assumed to be invariant with 6 and a, b, c are

constants defined by 4.47 with Ae replacing J1, . Thus 0 is

a quadratic function of radius within each triangular element.

Rearranging 4.52 yields

4.50

4.52

114. b r

2 + r(a — Lle ) + c r z = 0 4.53

and elimination of z using 4.49 with x replaced by r

yields the position of the point at which the Ib c contour crosses

each triangle edge. This position is given by

r — (a + cv) ± V(a + cv)

2 + 4 0 c (b + cu)

2(b + cu) 4.54

and zc = UT

c -I- V

4.55

The correct sign in 2.54 must be determined by trial and

error. It is found that the wrong choice of sign places the

point (rc z

c) outside the triangle under consideration. In

certain circumstances rounding error puts the value of the

expression under the square root sign slightly less than zero

and the computer program must be able to cope with this

eventuality.

In the second application mentioned, the H distribution 6

is required given, for example, that A is sinusoidally

distributed in the peripheral direction. The peaks of A and H e

m and H

em respectively) at a given point in the (r,z) plane

are 90 electrical degrees apart measured peripherally and their

magnitudes are related by

It m

= 1 r m 4.56

Thus, if the contour value N. mc cfosses a given triangle it O

does so at the points

re . (a + cv) / (b + cu — H emc)

4.57

and

zc

= urc + V 4.58

Numerical rounding error can affect 4.57 significantly,

115- particularly when the denominator is much less in magnitude

than the numerator and when both numerator and denominator are

small in magnitude. Indeed, identification and rectification of

rounding error problems is an important facet of program operation

when either of the applications mentioned here are considered._

The contour plotting program written took 14.3 seconds of

CDC 6400 execution time to plot 32 flux linkage contours in a

355 node mesh of structure given in Fig 4.2 representing the

(r z) plane of a turbine generator end region which was assumed

to be operating under open circuit conditions.

4.4.3 A Sub—Optimal ':ode Yumberin7 System

It has already been noted that the computation time

required for sparse matrix Gaussian elimination is proportional

to the square of the bandwidth. Thus, any procedure which lowers

this bandwidth can drastically reduce the overall cost per

solution. This is particularly true when a series of solutions

utilising the same mesh but with a different set of excitation

values is required since the stiffness matrix then remains

unaltered and the bandwidth reduction process need be applied

only once.

The bandwidth of a matrix has a topological interpretation

for it is given by the sum of unity and the maximum difference

between-the node numbers of pairs of nodes linked by a single

edge in the computation mesh. Thus, the sequence in which the

nodes are numbered sets the value of the bandwidth. Consider, for

example, the mesh of Fig 4.6 . In this particular example the

nodes have been numbered to give the lowest bandwidth possible.

It illustrates that when a regular rectangular node pattern is

116. employed the lowest bandwidth is obtained by considering the

Fig 4.6 Bandwidth = 5

side having fewest nodes and by numbering along that side

consecutively from unity, starting at a corner node. When the

side has been numbered attention should be transferred to the

row of nodes to which it is connected and an identical procedure

adopted; the node first numbered being that connected to node 1

by the shortest edge. When the node array has a random pattern

it is usually difficult to establish a simple yet effective

numbering scheme. Furthermore, node numbering by hand is

impracticable except for small numbers of nodes. A computer may

be employed to number .nodes and Silvester et al44favour the

Cuthill Mc' Ree algorithm for this purpose. Briefly, the

algorithm is as follows. Suppose R (i) is the ordered set of

all nodes which have already been numbered. Let Q (i) denote

,the set of yet unnumbered nodes, which are connected to at least

one node in R (i) by at least one edge. Order the nodes in Q (i)

so as to take those first which are connected to the Fewest

unnumbered nodes not in Q (i), those last which are connected

to the most unnumbered nodes not in Q (i). Let R(i + 1) be

the ordered set composed of R(i) and Q(i), taken in that

order. Then the algorithm is described by

R (i 1) = (11 Q i 4.59

117. The choice of starting node is important. In practice several

are tried and the one which yields the lowest bandwidth is

chosen. In the example previously citea, the Cuthill — McKee

algorithm yields the numbering of Fig 4.7 when the bottom left

Bandwidth = 8 Fig 4.7

2 6

hand node is used as a starting point. Note that the numbering

given is not unique with respect to the bandwidth achieved. For

example, when i = 1 there are two nodes connected to node 1

which are connected to the same number of unnumbered nodes. Thus,

both the sequence shown in the figure and that in which nodes 2

and 3 have interchanged numbers are valid.

Let us now consider an algorithm developed by the author for

which the next node numbered is that which is connected to the

most numbered nodes. This procedure yields the numbering of

Fig 4.8 . Again the sequence shown is not unique. From a

Bandwidth = 7

2 6 computation point of view, this ambiguity is inconvenient. To

some extent it may be overcome by applying the condition that if

two or more nodes satisfy given criteria then the next node

Fig 4.8

5 1 2

Fig 4.10 3

8

7 Bandwidth = 6

118. numbered is that for which the average difference between the

next number in the sequence and the numbers of nodes to which it

is connected is the largest. Nevertheless, there still is an

ambiguity of choice when the sixth node is being numbered. If

the average difference condition is replaced by one based on the

mean of the squares of the differences then the unambiguous

result of Fig 4.9 is obtained.

7

3 Fig 4.9 4

5 Bandwidth

9

6

1 2 8

Note that both the node numbering methods described

produce a poor result in the example given. This illustrates

that the methods are not always very useful. Furthermore, the

performance of each is highly problem dependent and it is not

possible to state with certainty which is the better of the two.

The example also illustrates the importance of choosing a good

starting node or nodes. For instance, starting with nodes 1 and 2

as per Fig 4.10, the author's algorithm using the mean scruare.

condition yielded the minimum (5) bandwidth attainable. In

irregular node array problems it is found that the best starting

119.

node is close to the region(s) of highest node density. Whole

rows of nodes may be numbered by hand before the chosen

numbering algorithm is applied but if this is done then the

sequence should be reversed before Gaussian elimination is

attempted. This is necessary since the matrix rows associated

with the initially numbered nodes will have a large band spread

and consequently should be eliminated first.

The author's algorithm was applied to a 355 node mesh which

covered a turbine generator end region and was found to yield

bandwidths of 79 and 72 when the average difference and mean

square conditions were employed respectively. The mesh used is

contained in Fig 4.2 which includes the element, but not the

node, numbers. Several different starting nodes were tried and

the one shown in the figure gave the best results. The node

renumbering then took 53 seconds of CDC 6400 time. A one

component magnetic vector potential finite element formulation

which included the effect of eddy currents in the copper flux

screen was employed and the time taken for Gaussian elimination

of the nodal equations was in excess of 1000 seconds; thus

showing that the renumbering time forms only a small proportion.

of overall computation effort involved in obtaining a field

solution.

In conclusion, it should be mentioned that in this

treatment we have assumed each node to be spatially distinct.

When magnetic scalar potential eddy current problems are

considered this is not so since T and A are associated with

the same points in space.. The problem is overcome by giving each

node as many numbers as there are functions to be evaluated at

that node: In other words, the finite directed graph of the

120.

stiffness matrix, rather than the mesh of elements, is that to

which the node numbering strategy should be applied.

121.

5. TH LONG:I 'UDI"PL PLIR IIIDUCTION roTm

In recent years there has been renewed interest in the use

of linear induction motors for high speed ground transport

applications. For reasons of safety these motors must work with

large air gaps. Thus the analytical field solution methods often

used for the analysis of linear induction machines yield poor

results since they are incanable of modelling the transverse edge

effect accurately. Dption effects also cause difficulties and

the presence of longitudinal machine edges has a considerable

influence on the machine's performance, particularly at low

slip values.

Numerical methods may be used to advantage for linear

induction motor analysis. In the interests of. simplicity it is

usual to assume that longitudinal edge effects are negligible

and that the machine primary is moving at constant speed relative

to the secondary. A two dimensional travelling wave solution

may then be obtained. Since the machine's secondary usually

consists of a conducting plate whose thickness is much less than

its other dimensions, only two components of current density

need be considered. This implies that in magnetic. vector

potential formulations employing the Coulomb gauge, two non—zero

components of A must be employed together with an electrostatic

scalar potential. The two components of A are however,

interdependent and only one need be directly computed in a

magnetic field solution. The magnetic scalar potential

formulation involves the determination of SI, and the component

of T normal to the major dimension plane of the secondary.

Backing iron is often used to increase the air gap flux density.

122.

If this iron is solid rather than laminated, saturation effects

(which are imcompatible with the travelling wave formulation)

must be considered.

There are many types of linear induction motor, but all fall

either into the longitudinal or transverse flux categories.

Polyphase transverse flux machines, of which the U—core magnetic

45 river is an example, have two major flux paths; one in the

direction of motion and one in the transverse direction. The

windings of these machines lie in slots aligned both with, and

perpendicular to, the direction of motion so that there are a

large number of magnetic- discontinuities within the primary

member. To obtain accurate nredictions of machine performance

it is necessary to use a three dimensional mesh, and this has

been accomplished by Nahendra46 in terms of magnetic scalar

potential. He analysed a circular transverse flax motor and used

a single current sheet representation of the plate secondary as

47 did Carpenter and DuroviC in an earlier work concerned with

the prediction of transformer end fields using three dimensional

meshes. In Chapters 6 and 8 we shall consider the analysis of

a transverse flux single phase E—core plate levitator. The mesh.

required for this application is two dimensional since the

machine has longitudinal, but no transverse, slots.

Longitudinal flux linear induction motors are geometrically

much simpler than those of the transverse flux type and have

been used for many years in sliding door and conveyor belt

applicationsBy neglecting longitudinal edges, such machines

may be analysed accurately using a two dimensional mesh provided

that they are longer than approximately four pole pitches.

A check on the accuracy and numerical behaviour of the

123.

Fig 5.1 Thickness of secondary = 6.35 mm

Longitudinal Conductivity = 3.3x107S/m

Flux Motor Iat 20C]

7.3 cm

Nh 5cm

Boundary For

Y Numerical Solution

Laminated Backing Iron

Nh=4.67mm

Lx h mm h, =4.5mm i

Pole Pitch= 9.54 cm

31 hi ° Nh 2.8crn

4

2

—hi

4- Nh

6 cm

6.35 cm Conductor

Laminated Primary

7.63 cm

V

[-7̀,

co End

3 Winding

—3.5 cm V < >

1.5 cm

Line Of Symmetry

' I 7

•• • t - • -----

• --- --7.7 . 1--- • : • 1 - - --1 . 1 • : . -- --'- `---- i • ; : : : ;I

.! ' 1 : r: 1 . 1- 1 Slot/pale/phase ; '. ' 2/3 Chorded '. - ; • ;E. , - ; ::

, : ; , • • • : : • : : : 1 ; ; : : : . i i i : : :

; • . I .- i ; • . ' ,

: ,

, 1 : : : • . BO Turns • /: C oil ' ' ' • . :

: ; I -; . 1 . I : ; : FT 7- I : TT T : i ' rr 1 -1 •-•i- .-i

124.

Series' Connected , , . ' r

i . .

F.

--B R

• ;

B 4 1 '

.1 • I .. •••• ! • . • ; • '11 : I • 'I ::

• • • I l• • • I • .

i • :

I • . i

-- - - I.

• i

- ---- - • • ! • - - -- „ : • •

. 4. • t 1- • • ' .•

•-•.:. • '

! •

•

•• - : •

• : ;

-R

,

. .. •

Slot depth = 3.B1 cm Open slots

Slot width =1.905 cm

L Tooth width =1.27cm -

:.Lbrigitudinal Flux Linear Induction Motor Winding Data

7 • • • 1 • I • • •

: ; . • • .

Phase Current Used In Tests = 5 A . ms r : „ .

, • , : • Fig 5.2 ; ; ; ;• :

I L ;

•• I I • .

• i

i • • • • ; I . • _: ,

-J •

magnetic scalar potential eddy current formulation was required.

A longitudinal flux linear induction motor was available in the

laboratory and it was decided to use this machine in an initial

investigation of the T St method's accuracy in predicting

both magnetic flux density distributions and thrust forces. A

two dimensional mesh in the plane perpendicular to the direction

of motion was used.

5.1 Lachine Details And qxperimental Results

The major dimensions of the machine are given in Fig 5.1 ,

which is a transverse section taken through a tooth centre.

. Backing iron was employed in order to increase the flux density

in the air gap. To keep the end ring resistance low, a practical

machine would have a conducting secondary that was wider than the

primary. For the purposes of this investigation, it was convenient

to contain the secondary within the air gap since a crude model

of the open boundary could justifiably then be made.

A large air gap was chosen for ease of access, even though

this resulted in low values of normal flux density in the gap

at rated phase current. The values of flux density were measured

using three calibrated search coils wound concentrically so that

.each coil axis was orthogonal to the other two. The search coil

set formed a cylinder of diameter and length equal to 0.4cm

which was mounted on a beam. This beam could be moved in the

x and y coordinate directions by means of a hand or motor drive.

The primary winding data is contained in Fig 5.2 . Note

that the slots are fully open and are wider than the teeth.

Consequently it was expected that the slot ripple would be

considerable. To reduce the effects of such ripple on

26, T- r , . Tr- t

I ;

I I ; 1 . I

;

A mT

-22

20-

. . • - ... s .

. i .

0

5 Cm

2 0.

127.

measurements, the conductor was placed not on the backing iron

surface where it would be under normal operating conditions, but

on the surface of the primary. It was thought that by taking

measurements between the conductor and backing iron, slot edge

effects would not dominate the flux density readings.

To exclude longitudinal edge effects from the measurements,

and thus make them comparable with the computed values, all

readings were taken at the machine centre plane. Furthermore,

in order to make some allowance for slot effects, two readings

were taken for each x value; one at a slot and one at a tooth

centre. The two readings (which were up to 15% different 2mm

below the backing iron) were averaged to obtain the nominal

experimental result. A phase current sufficiently low to avoid

overheating when the machine was continuously operated for

twenty minutes or so was chosen. reasurements could then be

taken over a relatively long period of time without the fear that

the material properties of the machine were changing significantly.

The choice of low nhase current also ensured that magnetic

saturation occurred only at iron edges.

A meter that measured average volts and then applied a

form factor of 1.11 was connected to the search coils. The flux

)density waveforms were found to be sinusoidal in time at every

point and, in consequence, conversion of the search coil voltages

to flux density values produced true RMS results.

The measured and calculated flux density results for the

machine without a conducting secondary are given in Fig 5.3 •

There was good agreement between the two sets of values. Note

that the normal component of flux density, Bz, is predicted

particularly well. The x and y components of flux density are

Ei is measured value

129.

small since the search coils were placed close to the backing

iron surface at which, in the absence of magnetic saturation,

there are no tangential magnetic field components. It is not

surprising that the Bx values shown in Fig 5.3 do not compare

favourably since the backing iron surface was not smooth but had

some laminations protruding more than others.

The conducting secondary was introduoed and placed in a

symmetrical position on the surface of the primary. The effect of

conductor asymmetry was not investigated since this is not of

importance in longitudinal flux machines which are designed to

produce thrust only. Pig 5.4 contains the results analogous to

those of Fig 5.3 but with the conducting elate present. The

computed values of Bx

and By were more accurate than they were

when the conductor was absent, but the B7 predictions were

comparatively poor near the air gap entry corner. Initially this

was thought to he a magnetic saturation effect, but the results

of Fig 5.3 do not seem to substantiate this view. The maximum

direct measurement error was 7% (search coil calibration

voltmeter 2%), yet that in Bz was approximately 15X. Further

investigations showed that Bz

varied rapidly with z near the air

gap entry corner. The discrepancies could thus be accounted for in

terms of search coil positioning error.

The thrust .of the machine was measured by placing the

conducting secondary in a two wire sling such that the wires

were vertical. A load cell, which had previously been calibrated

using known weights, was then attached. Although all the flux

density measurements were made at a phase current of 5A, it

was decided to measure the thrust at different currents so that

a check on the magnetic linearity of the machine could be made.

0 Magnitude Of B 4 mm Above 6.35 mm Plate

26

24

22

20

18

8 cm

0 is measured value

....... I , I

.......

130.

131.

The results are contained in Table5.1.These show that the thrust

Phase Current (A) Thrust (11) Thrust At Iph = 4A Obtained

By Quadratic Scaling -__

2.5 2.54 6.51

3.02 3.66 6.43

3.52 4.97 I 6.41

Rating Of Load Cell = 511

TABL7, 5.1

is proportional to the square of the phase current. The machine

is thus magnetically linear over the range of excitation values

employed. When a one component T formulation was used to calculate

the thrust, a value of 6.25 N was obtained. The two component T

formulation predicted a value of 6.5 N. The respective errors to

a base of average measured force were thus 3.5 and 0.4. Both

these values were less than the expected experimental error.

Further measurements of flux density were made, this time

4mm above the conductor. These results are contained in Fig 5.5

which shows very poor agreement between computed and measured

values. The fact that the measured value of B (the longitudinal

component of B) is much greater than that computed is evidence

that slot effects dominate. In the light of these results it is

somewhat surprising that the thrust was predicted to such

accuracy. Had the normal force been measured it would probably

not have compared favourably with that predicted since

experience shows that the accuracy of normal force computations

132.

is heavily dependent on a correct prediction of the tangential

flux density comnonents.

5.2 Comnuter Prediction Of Flux 7)ensities And. Forces

5.2.1 Programming Details

The domain used in the computer solution is shown in Fig 5.1.

The laminated iron regions were assumed to have infinite

permeability and resistivity. In consequence, they were of

pre—determined scalar potential value. Within the air gap a

uniform rectangular mesh was employed but for maximum node

economy a near—square mesh was used in the end region. The open

boundary problem present at the transverse ends of the machine

was overcome by setting the outermost S/ values to zero, thus

simulating an unsaturated iron surface. This approximation was

thought to be reasonable since the conductor did not protrude

outside the air gap.

In order to determine the effective winding MF, a cross

section of the end winding was considered at a centre of slot y

value. If it is assumed that the conductors of the two phases

cut are evenly intermingled and that, say, the number of turns

per phase is IT , then the mnF will be a linear function of x

and have a maximum value in time of 1.5 NwIph

. A detailed

account of end winding representation methods is given in

Chapter 7 which deals with turbine generator end field

calculations.

Inherent in the computer solutions are the assumptions that

there is no vertical (z) component of winding current, and that

there are no MMF space harmonics in the longitudinal direction

at any instant in time. The first assumption is reasonable

133 .

whilst the second is dubious because the machine has only 1

slot /pole /phase. The effects of space harmonics can be

assessed by performing a Fourier analysis on the longitudinal

distribution of MEF, Separate T — A solutions are then obtained

for the fundamental and the required number of harmonics. Finally,

the solutions are combined to yield the transverse T —

distribution as a function of longitudinal (y) distance at a

given time instant. Due to the presence of the harmonics, the

longitudinal distribution will not be sinusoidal. The technique

does not include any magnetic effects of slots. Since these

were thought to outweigh the discrete current effects, it was

decided not to include space harmonics in the numerical solution.

Two dimensionless quantities, a and p , must be computed

before the numerical solution commences. The first is given by

3.66 and defines the ease with which flux can travel

longitudinally. For the machine considered its value was 0.0241.

The second, p , incorporates the electric and magnetic constants

of the conductor and is given by h2/2d2 where h is the z

mesh interval within the conductor and d is its effective depth

of penetration defined by 3.28 . Since the h / d ratio should

be less than unity for reasonable eddy current modelling, it

follows that p should always be less than 0.5. In the present

application h was set to 2mm and the secondary was spanned in

the z direction by three mesh intervals. The conductor was

therefore assumed to be 6mm, rather than the actual 6.35mm, thick.

To compensate for this, it was decided to use a conductivity

value in the computer Program that was higher than that of the

actual secondary by a factor of 0.35 / 6. p was then equal to

0.0276.

134.

Fig 5.6

Real Part

Fig 5.7

Imaginary Part Scalar Potential Solution [3 = 0.0276

Fig 5.8

; Variation In Typical Layer Of Plate

135. The computation matrix was partitioned into three, one part

being for the A values and the remaining two for the T vector.

The SL and T values at a given node had the same column, but a

different row, reference number. Since the number of nodes at

which T is non-zero is small in comparison with the number of

nodes at which is computed, considerable wastage of computer

store can take place if the two T partitions are given the same

number of nodes as the St one. In the present application it was

possible to store the T values at matrix locations within the

stator iron region since the magnetic fields within the iron were

not determined. A was calculated at 661 nodes and 56 nodes were

required for each component of T. The iteration algorithm is

described in detail in section 14.2 . After every point successive

overrelaxation scan of the St meshes within and outside conducting

regions, there were ten Gauss -Seidel scans for each non-zero

component of T.

5.2.2 Analysis Of Comnuted Results

First, let us consider the one component T numerical

solutions. A contour plot of the real part of A is given in

Fig 5.6 . This part yields the magnetic field distribution when

,the primary IMF is at its peak value. The field in the regions

where the z mesh interval is 4.5mm is compressed by treating all

nodes, for plotting purposes, as having the same spacing as in

the air gap region. The imaginary part of J1 (corresponding to

the instant when the primary MMF is zero) in the air gap and

plate is drawn to an enlarged scale in Fig 5.7 . The diagrams

show the discontinuity in the normal gradient of Si' at the plate

surface which is caused by the discontinuity in T. As is

136.; . •• • r -1 • 7.1- • -1 .1-- • •

1 T

J x105 A / m2

X CM

• I machine .1 centre tine

Plane of peak primary MMF

1_ Line of nodes 21-thrl below top surface of I secondary.

I

Plane of zero primary MMF

J -

6.35 mm Secondary — Variation Of J With Transverse Distance. MMF Of Primary =100 Arms:.

_L -6 • L •

137.

typical with these proportions, the variation of T with z is

comparatively small, even though the effective depth of penetration

is only 20,-S greater than the plate thickness. The variation of

Tz with x ana y over the entire plate is shown in Fig.5.8 for

one instant of time (the calculation assumes no end effects in

the y direction). Since T varies sinusoidally with y, as well

as in time, the real and imaginary parts of Tz are obtained by

choosing appropriate sections of the diagram. It is noteworthy

that Tz(x) is approximately sinusoidal.

A second non—zero component of electric vector potential (Tx)

was introduced so that the z directed component of current

density could be represented. A numerical solution was performed

and it was found that the ft and TZ fields were almost

unaffected. The predicted thrust was approximately 3;,, greater

than that obtained using the one component T formulation. Fig 5.9

contains a plot of the calculated current densities along a line

2mm below the top surface of the secondary. Tote that although

Jz

forms about a 20(/'; contribution to the modulus of J at points

near the left hand conductor edge, it falls off rapidly with x

and is zero at the machine centre line. Between the conductor

edge and the centre line, jz reverses direction twice. As

-expected, the transverse component of current density (Jx) was

dominant at the machine centre line, and the longitudinal

component (J ) was dominant at the left and right hand edges

Of the secondary.

5.2.3 Converence Characteristics Of The Numerical Solutions

Successive averrelaxation was used to obtain the numerical

results. The initial accelerating factor was unity and a new

value was calculated after every twenty iterations using the

35,36 Carre — Stoll technique. Table 52 contains the relevant iterative

138.

Program Details

',?umber Of

Iterations

Final Accelerating Factor

As Calculated By The

Carr — Stoll Kethod

ITo secondary 86 1.745

P = 0.0

6.35mm secondary

p = 0.0276 91 1.711 + J0.00874

T . k T — — z

6.35mm secondary

p = 0.0276 104 1.7539 + J0.06562

T . iTx + 22z _.

The convergence criterion that the modulus of the

maximum residual (potential error) be less than

2 x 10-6 of the maximum real magnetic scalar potential

was used throughout.

TABLE5.2

convergence results. The linear iduction motor solution domain

boundary consisted mainly of laminated iron and in consequence

was iteratively well conditioned with respect to the magnetic

scalar potential. Hence convergence was always rapid. Note,

however, that introduction of the second non—zero component of

electric vector potential had an adverse effect on the number of

iterations required.

139.

rig 5.100

Real Part

Fig 510b

Imaginary Part Scalar Potential Solution 13 = 0.0248

Fig 5.11

Power Loss Density Contours

140.

p was increased an order of magnitude (to 0.276) so that

the sensitivity of the convergence rate with respect to this

quantity could be ascertained. The change in p was equivalent

to raising the operating frequency of the machine from 50 to

500 H. It was found that the number of iterations required

remained the sae as in the 50 Hz calculations. Thus the rate of

convergence would seem to be independent of p over a large

range of frequencies.

The conductor thickness was increased to 19.05mm which was

made to correspond with ten intervals in the finite difference

mesh. The value of p used was thus 0.0248 and the number of

iterations required was found to be 165 with the acceleration

factor converging to 1.7955 + j0.059036. The real and imaginary

parts of the St solution are contained in Fig 5.10..- and Fig 5.10fr

respectively where, due to the large number of nodes within the

conductor, the nature of the SI, field within and around the

plate is more readily appreciated than in the plots for the

6.35mm plate. A contour plot of the power loss densities within

the conductor was obtained and is shown in Fig 5J1 . Each

contour is a line of constant rate of temperature rise. Note

that the points of maximum heat generation are at the conductor

corners in contact with the stator. Furthermore, there is a

rapid variation of power loss density with distance, particularly

near the end region edges of the conductor.

141.

6. I.:U=10AL SOLUTIC' OF OP= 3OU7DARY FZ073LEMS

An open boundary problem is one in which some boundary

conditions associated with the solution of a given partial

differential equation are specified at an infinite distance from

the region of interest. In the analysis of the longitudinal flux

linear induction motor considered in the previous chapter, SL

was set to zero. at a short distance from the air gap entry points.

This was the most computationally convenient form of open

boundary approximation but was valid only when the end region

flux densities were low in comparison to those in the air gap.

If the conducting secondary had been of greater width than the

primary; inaccurate results would have been obtained since the

a= 0 boundary acts as an unsaturated iron surface and attracts

too much flux from the air gap region. An alternative to this

Dirichlet (J. = 0) condition is the Neumann relation

0 6.1 a n

where n is a local direction everywhere perpendicular to the

1 end regions periphery. The periphery then becomes a flux line

and the end region is found to draw too little flux from the

air gap. Furthermore, the iterative convergence rate is impaired.

Accurate representation of open boundary effects is vitally

important in the analysis of linear induction motors having no

backing iron. These machines are employed whenever large forces

of repulsion are required between the primary and secondary

members. In this chapter, two alternative methods of dealing with

open boundaries are presented together with some computational

and experimental results.

1142.

6.1 The Coordinate Transformation lethod

It is common practice when faced with an open boundary

problem to grade the computation mesh (whether it be of finite

element or finite difference type) and extend it to a distance,

dictated by experience, at which the field quantity being

determined is set to zero. This procedure requires many nodes

and often leads to inaccurate results caused by the poor mesh

termination conditions.

By defining a circular contour (or a sphere, in three

dimensions) of radius r1' which surrounds a region of interest

such that all sources are enclosed, an open boundary can be

represented easily by considering Laplace's equation in

cylindrical (or polar) coordinates and applying the transformation

r' r r1 / r. Its effect is to transform an infinite region outside

a contour of radius r1 into a finite region bounded by an

identical contour. The mapping is shown diagrammatically in

Pig 6.1 . The field problem is solved by having two meshes, one

within contour c and the other within contour c'. In conducting

paper analogue terms this is equivalent to having two circular

sheets of different 'square resistance' connected together only

at their peripheries.

The circular contour method is particularly attractive

when irregular meshes are to be used since nodes may then be

X43 .

placed at convenient points on the chosen contour. When regular

rectangular meshes are employed however, matching of the interior

and exterior regions becomes difficult. Since the interior region

usually has a rectangular periphery, it is advantageous to use

a transformation that maps infinite space into a finite-region

having rectangular inner and outer boundaries. There are many

transformations that will achieve this, though to be useful they

must satisfy several further conditions. Let u, v and w form a

Cartesian right handed local coordinate set with origin on the

external boundary of the interior (finite) region. Thus the

boundary conditions are specified at points where u, v and w

are infinite. Now let x, y and z form another local coordinate

set which is coincident everywhere with that of u, v and w. Then

we need to find a transformation linking the two coordinate sets

which has the following properties.

1) x, y and z are finite when u, v and w respectively are infinite.

2) The external boundary of the interior region is coincident

with the internal boundary of the external region. i.e. u = x,

y = v and z = w at the external boundary of the interior

region.

3) In source—free regions the field describing function to be

calculated should have continuous spatial derivatives. This

implies that x, y and z must be continuous functions of, and

have continuous derivatives with respect to, the coordinates

u, v and w respectively.

4) The inverse transformation should be of closed form. This

requirement is not strictly necessary but it reduces the

algebraic complexity of the formulation.

All four properties are satisfied if transformations (due to

A z

f

,zy x d

144.

Mr. C. J. Carpenter of Imperial College) typified by •

x = u d d

y = v e e v

z = w f f w

6.2

are employed. Expressed in terms of x, y and z the values of

u, v and w are given by

u =xd;v= e ; w= z f d— x e— y f— z

6.3

respectively. Note that these are coordinate, and not conformal,

transformations. The two dimensional diagrammatic form of 6.2

in terms of x and z only is given in Fig 6.2 . We shall now

Region undergoing

transformation

Equation 6.2

Fig 6.2 Transformed region

assume that within the exterior region the field describing

quantity, which we shall denote by the scalar V, is Laplacian

with respect to the (u,v,w) coordinate system and is zero when_

u, v or w are infinite. In the remaining part of this section

we shall derive the (x,y,z) plane equivalent to Laplace's

equation. The next section deals with the finite difference

approximation to this equivalent.

Within the non—transformed exterior region V satisfies

Oa V + o 2 v + .2 v = o dug 817.2 aw2

6.4

Let us consider the transformation of the first term of 6.4

into (x,y,z) coordinates. By identity

8Vax+ avay. ate d2

a u a x a u ay au a x (a u)2

6.5 Taking the u derivatives of 6.5 yields

13 2 v a ( =I a ( a v)) a x

au2 a u. au a x a u a u

(3.2 a r av d2 I (from 6.5)

We shall now determine the x derivative of the square

bracketed term in 6.6 . expanding this derivative using the

product formula yields

a C av d2 1 = 82 V d2

a x ax (a + u)2 ax2 (d u)-2

av a [ d2 •

6.7 ax ax (a. u)

2

From 6.3 d2 1 (d x)2

(d u)2 d2

so that

a [ d2

I = (d x) a x (d u)2 d2

and 6.6 becomes (after some reorganisation)

a2 V = (d — x)

2

8112 d2 a [ (d x)2 a v

6.10

ax d2 ax

Similar expressions may be derived for the second and third

11+5.

(d + u)2 x ax (a + u)2 6.6

6.8

6.9

terms of 6.4 so that the transformed Laplace equation becomes

z

05

Fig 6.3

o 01

04 1111

4

2

146.

P a (p a + q a (q aV ) t r a (r a V)= 0

a x x a y a y a z a z

6.11 where

p = fd — x12 ;

d 1 te

e y1

2 and r If — zl2 f

6.12

6.11 is an equation of the non—linear Laplacian type and has

position dependent coefficients. rote that wherever x, y and z

are zero, 6.11 reduces to 6.4 •

6.1.1 ' Derivation Of Finite Difference .7:cTuations And retwork

nodels That Are Valid Within The Transformed. Open

Boundary Region

Let us now derive a set of finite difference equations that

is valid at nodes in the (x, y, z) plane. The approximations

employed will be those described, or referred to, in Chapters 2

and 3. Consider the (x,y,z) plane rectangular finite difference

mesh of Fig 6.3 .

The finite difference approximation to the first term on the

left hand side of 6.11 is

p a fp aV PO r (V1 — V ) p01 — (V0

— V3) p03

a x 111 h L N1 h N1 h

6.13

Corresponding expressions for the y and z dbrivative terms may be

derived. Thus 6.11 takes the finite difference form

11+7.

P0 ( (V

1 — v0) poi — ) (v0 - v3) p03)

N1 h Nt h N

1 h

(V - v0) (v0 - V -0 f 6 0 q06 — 0 v5) q05 )

112 h N2 h N2

h

r0 / (V2 SIO) r02 — (V0 — V4) r04

0 6.14 h 1

The subscripts applied to p, q and r require some explanation.

pi say, p i, whilst sa is the value of at node ihilst . . is the value - D13

at a point midway between nodes i and j measured along the

branch joining i to j. An identical convention is applied to

variables q and r.

By rearranging the terms in 6.14 we obtain the equation

N2 q0 (v5 + V 0 5 q 05 V6 q06) + 72 po (Vi P01 4. V3 1303)

2 N2 r0

(V2 r02

+ V4 ro4) — (q0 (q05 c106)

2

N2 P (P P ) 2 (r + r )) = O 01 O3 112 1'0 02 04 0

N1

0 6.15

6.15 has a corresponding resistive equivalent circuit given

in Fig 6.4 .

z

4

Fig 6.4 The resistance values are obtained by applying Kirchhoff's

II1

148.

Current Law at node 0. This procedure yields the equation

6

IT1 . v0 0 6.16

0 i

i=1 Comparing coefficients of 6.16 and 6.14 we find that

R02 = h

2 ; R03 = (N1 h)

2 RO1 (1T1 h)

; = 2 ;

r0r02 P POP01 0 03

6.17 R04

h2 ; R05

= (N2 h)2 ; Rob

(11

2 h)2

r0r04 g0-a 05 goclo6

and since the p, q and r coefficients are dimensionless, the R

values have dimensions of length squared. To give the R values

dimensions of reluctance, the right hand sides of equations

6.17 must be multiplied by µ / (N1N2h3).

If it is known that at any instant of time V is distributed

sinusoidally in space with respect to, say, the v direction then

we may define a space phasor (V) by

V (u,v,w,t) = Re (V (u,w,t)e—v/p ) 6.18

A solution in two, instead of three, space dimensions may then

be used to determine V and thus V. Since

"*" 2 a2 v = — tai V

1 a v2 Pi 6.19

and it is unnecessary to transform v into y because 6.18 is

valid for all v from -cm to +co, the space phasor form of 6.14

is

PO r(vi vo) Poi — ( Vo v 3) P03 Nlh L N1h N

1h

Fig 6.5

w wf z = few x = ud

diu y x

149.

r02 f

r0 r t 2 V0) 02 - 71 r04

h L

0

6.20 which may be re—arranged to yield

p0 (p011 + p

033 ) + N

1 2 r0 (r022

+ r04 54

) —

(p0 (p01 P03) + Ni r0 (r02 + rod + m ) VO = 0

6.21

where

a =r 211x h 2 6.22

p

The resistance equivalent circuit of 6.21 is that of Fig 6.4

with R05

and 2306

removed and a resistance of value (p )2

connected between node 0 and a point where V -is zero.

Let us now consider how the Cartesian coordinate

transformation which we have developed can be employed in a

problem solving situation. Pig 6.5 shows how an infinite region

surrounding a rectangular finite region may be transformed into

a 'picture franc' shape using 6.2 . For simplicity, only the

u and w directions are here assumed to undergo transformation.

Let the contour c and its interior be identical in the (u,v,w)

150 .

151. and (x,y,z) planes. Then within regions such as R

1, trans-

formation in only one coordinate direction is required; but

within the corner regions (R2 for example) both u and w must be

transformed into x and z respectively using 6.2.

For computational convenience it was decided to make the

coordinate origin in the (x,y,z) plane position dependent. This

origin was allowed to take up one of four positions — points

Al, B

l, C

l and D1 of Fig 6.6 . In this Figure the positive x

and z directions have also been made position dependent since

this ensures that computations need only be made within the

positive quarter plane of the (x,y,z) coordinate system.

Thus far no mention has been made of the way in which the

numerical values of the space independent constants d, e and f

are determined. Reference to the transformation relationships

(6.2) indicates that when x = d, y = e and z = f the point

(x,y,z) is at infinity in the (u,v,w) plane. Consequently, if we

choose d to be the width of regions R3'

R7 and f to be the z

directed width of regions R1,

R5

then contour c1 of Fig 6.6 is

at infinity in the (u,v,w) plane. In this connection, it is

instructive to plot w against z in order to appreciate the

nature of the transformation defined by 6.2 . Fig 6.7 contains

F ig 6.6

lh Nti

h= 2 mm Nh= 6 mm

Boundary at infinity

Picture frame

E-core Plate Levitator — Geometric Data

6.35 mm thick plate 41 u Resistivity = 2.902x10 Jim

- 8.9cm -

c4.45cmjc

2r4-1 cm

4cm

4.45cm

26.7 cm

152.

Jw O. Machine lehgth = 30.5 cm Winding has 80 turns

Square mesh of side 6mm used in slots

Fig 6.8

153.

such a plot which shows w to vary rapidly with z near the point

z = f. If a regular finite difference mesh is used in the

(x,y,z) plane, a considerable number of nodes (usually between

eight and twelve) should separate contours c and c1

of Fig 6.6 .

This ensures that the Taylor series truncation error is

acceptably low for nodes in the (x,y,z) plane which are close to

contour c in the (u,v,w) plane. It is difficult to estimate

algebraically the effect of truncation errors generated in the

-picture frame on the potentials at nodes within the region

bounded by c, and this was one reason for performing the numerical

experiments of the section following.

6.2 Analysis Of An 2—Core Plate Levitator

An 2—core plate levitator is a transverse flux machine

designed to repel a conducting secondary. It contains no backing

iron and produces no thrust. Eoreover, the secondary is laterally

unstable and, when marginally displaced from a symmetrical

position above the primary, is quickly ejected.

An analysis of a given 2—core levitator was required and

this provided an excellent opportunity to test the Cartesian

coordinate transformation method. The finite difference mesh

employed was two dimensional and, since the machine was a single

phase device, the pole pitch was assumed to be infinite. In

consequence, current density was restricted to the direction

perpendicular to the plane of the mesh, and a one component

electric vector potential formulation was used to calculate the

current distribution within the secondary.

The dimensionsof the machine are given in Fig 6.8 which also

contains the boundaries used in the numerical solution. Henceforth,

154. and for simplicity, we shall assume that u, v and w no longer

form a local coordinate set but have the fixed orientation given

in this Figure. 3ince large lateral displacements of the

secondary from the symmetrical position were not anticipated,

the mesh was not extended to enclose the machine. Furthermore,

the slot leakage flux paths were not thought to be important in

relation to the machine's levitation properties, and this

explains why a coarse square mesh was employed within the slot

regions. Outside the slots a rectangular mesh of aspect ratio

3 : 1 was used since it was anticipated that the magnetic scalar

Potential would vary more rapidly with the w coordinate direction

than with u.

The primary of the levitator was symmetrical about a w

directed line placed at its centre, and although the tests were

performed using a symmetrically placed secondary of rectangular

cross section, no advantage of the overall machine symmetry was

taken. Thus, although the number of nodes was double that

strictly necessary, the possibility of using asymmetric

secondaries was not precluded.

When a mesh of the size indicated in Fig 6.8 was used to

fit the machine's geometry, the slot to tooth width aspect ratio

mas represented correctly but the computer model assumed the

width of the primary to be 25.2 (instead of 26.7) centimetres.

An irregular finite difference mesh could have been used to fit

the dimensions of the primary exactly, but to avoid undue

programming complexity the width discrepancy was allowed for

by scaling up the coordinate lengths, and scaling down all three

components of computed flux density, by the ratio 25.2 : 26.7

before comparisons were made with experimental measurements.

155. These measurements were performed by Dr. T. G. Bland of

Imperial College. Flux densities were measured using a Hall

probe connected to a 'time average' meter whose readings were

converted to R.II.S by assuming there to be no waveform distortion.

Thus, a form factor of 1.11 was employed.

6.2.1 Lachine On Open Circuit

since secondaries tend to contain the magnetic flux within

the air gap, magnetic field calculations on an open circuited

E—core machine provide a strict test of the open boundary

transformation method. Within the picture frame it was found

convenient to use a regular rectangular mesh which necessarily

had the same aspect ratio (measured in the (x,y,z) plane) as

that used within the interior region.

The primary aim of the investigation was to determine the

minimum number of nodes that must be placed in the picture frame

to achieve a given accuracy within the interior region. The

entire computation mesh contained 44 w directed, and 76 u

directed, nodes and it was decided to keep the total number of

nodes constant throughout the tests. Consequently the inner

picture frame boundary was moved closer to the machine when the

number of nodes in the picture frame was increased.

Numerical solutions were performed for several different

numbers of picture frame nodes. After each magnetic scalar

potential solution had been completed, the flux density values

at the nodes were calculated using central differences. Plots of

magnetic flux density as a function of u were then obtained along

a line 6mm above the primary. These were compared with the

measured flux densities. The comparisons are summarised in

156.

Table&lvihich suggests that the picture frame should be at

least nine nodes wide if the w (normal) component of flux density

at the machine's centre is to be predicted to within 5% of the

measured value. The Table also reveals that doubling the width

(in node terms) of the picture frame approximately doubles the

B:!..t The i:achine w

Centre (mT)

Width Of Picture

Frame In Nodes

Number Of

Iterations

21.47 0 69

14.6 3 96

10.17 6 203

9.93 9 271

9.907 12 401

leasured Value

= 9.44

Table 6.1

iteration time required to obtain a solution of fixed accuracy.

In all the numerical solutions the magnetic scalar potential was

assumed to be zero at an infinite distance from the machine. An

acceleration factor of value 1.8 was chosen. This decision was

based on experience with similar magnetostatic problems. The

convergence criterion was that both the maximum real and maximum

imaginary residuals be less in magnitude than 2 x 10-5 of the

maximum real potential.

157.

Magnitude Of B 6mm Above Primary

Current/Turn =10A rm s 0 is measured value

9 nodes in picture frame No secondary

Fig 6.9

Magnetic Scalar Potential Plot - E-core Machine With Secondary Absent.

159.

Fig 6.9 contains a comparison of measured and computed

flux densities for a picture frame nine nodes in width. Overall

agreement to within 15';.: will be noted. !.long much of the air gap

region, however, the differences are far smaller. The largest

discrepancies occur in the vicinity of the inner slot corners

which are probably subject to considerable magnetic saturation.

Both the normal (w) and transverse (u) components of flux

density are found to vary rapidly with u. This result indicates

that a choice of square, rather than rectangular, finite difference

mesh would have been more appropriate. lIevertheless, the

rectangular mesh results were reasonably accurate and good node

economy was achieved.

A plot of the magnetic scalar potential distribution for

open circuit conditions is contained in Fig 8.5 . The plot is

drawn to a scale such that outside the slot regions the w

coordinate direction is scaled up by a factor of 3 relative to

the u coordinate. This procedure, which was also applied to

Figs 6.11 and 6.12, was adapted to facilitate assessment of the

field distribution's characteristics.

6.2.2 Machine With Conducting? Secondary

A 6.35mm thick aluminium plate•of equal width and length to

the primary was placed symmetrically above the machine so that

there was a constant air gap of !cru. The plate spanned three

mesh intervals in the w direction and contained a total of 164

nodes. Reasonably accurate eddy current modelling was achieved

since the ratio of the largest mesh interval in the plate (6mm)

to the effective depth of penetration was 0.64. Good results were

obtained when a similar ratio was employed for analysis of the

o is measured value 6 nodes in picture frame

Current/Turn =10 Arms

Fig 6.10

160.

CM

Magnitude Of B 4 mm Below 6.35 mm Plate

3w

Cane Pal •.ze Devi rum

Magnetic Scalar Potential Plot - E-core Machine With Secondary. Time Instant When The

Current In The Winding Is A Maximum. Fig 6.11

162.

longitudinal flux linear motor described in Chapter 5.

Numerical solutions were obtained using a picture frame 9,

and then 6, nodes wide. There was no significant difference

between the results obtained. In consequence a 6 node picture

frame was adopted as standard since a saving in cost per solution

could thereby be achieved.

Once again, the flux density was plotted as a function of

transverse distance and compared with measurements. Fig 6.10

Contains the relevant plots, which agree well. Comparison of

the open circuit flux densities (Fig 6.9) with those occurring

when the conducting secondary was present (Fig 6.10) shows the

transverse (B ) components to be almost identical in magnitude

and spatial distribution. However, the normal component of flux

density is greatly suppressed when the secondary is introduced;

particularly at the machine centre plane. Across the centre pole

of the machine, the computed open circuit and conducting

secondary flux density values were higher than those measured.

This was to be expected since, in section 6.2.1, we showed that

the picture frame errors were always such as to increase the

computed flux densities at the machine centre line.

Fig 6.11contains the magnetic scalar potential plot at the

instant of time when the winding current is a maximum. Note how

the field is contained between primary and secondary and that

virtually no flux emanates from the centre pole of the machine.

Furthermore, the two 'blips' on the top of the secondary indicate

the dominance of the inner slot corners which seem to punch flux

through. the plate. Very high magnetic field gradients occur near

these corners and it is surprising that the measured and computed

flux density values agree so well. It is found that the measured

Magnetic Scalar Potential Plot - E-core Machine With Secondary. Time Instant When The

Current In The Winding Is Zero. Fig 6.12.

ecoste razmxtFn 0 .Z3 1101 raw

164.

A/m2 x10-4

20

Real parts 16

Bottom surface of plate 12

Imaginary parts

1

Real part — time instant when coil current is maximum Imaginary part-- -- zero

_Winding current = 1.25 Arms / Turn

Predicted Current Densities In The Top And Bottom Surfaces

Of The Conducting Secondary

Fig 6.13

165.

values are more dependent on the accuracy of lateral, rather than

vertical, Hall probe positioning.

At the instant of time when the winding current is zero,

the field distribution is that of Fig 6.12. Here, the normal

component of flux density at the machine centre plane is

relatively high (2.5mT), but the flux densities in the vicinity

of the inner slot corners are small compared with those that

existed a quarter cycle earlier in time. A large proportion of

the total flux lies outside the air gap and in consequence it is

at this time instant that the picture frame technique is most

severely tested. The magnetic scalar potentials shown have a

complicated shape and this result is reflected by the secondary

current density distribution. The current density distribution in

the top and bottom surfaces of the conductor are given in Fig 6.13.

At the time instant when the slot current is maximum, one current

loop is present. A quarter cycle later there are three, though

one of these (that nearest the machine's centre) is of negligible

proportions in comnarison with the other two. The current density

peaks at each time instant and each level in the conductor occur

at the mid—slot positions. The modulus of the current density

at these positions decreased by as the conductor was

;traversed from bottom to top surface. Considering that the

conductor's thickness was 25,E greater than its effective depth

of penetration, this difference was less than that expected. The

phase shift across the conductor at the mid slot point was lin

with the current density in the top surface leading that in the

bottom. Thus an upward travelling wave, which exerted a lift force

on the secondary, was produced. The normal (lift) force was

measured and found to be 200gms when a current of 9.1 A was rms

166.,

peak value= 21.85—of

- - -

no secondary

• 1 ins.L. / / i(7. • ,

Primary , w.i

6mm 1 , , I- i • . .

Current /turn. =i 10 Arrns -

• 1 • I • Fig 614 1---

. . _, i I , . , 1 , • 1 . , . ,

Measured B As A Function Of Machine

167.

12t rnT

8

0 Bw

0 "TrEl 10 12 worn

- 0 1:1

__ Magnitude Of B 4 mm Below 6.35 mm Plate

J J

\ \ \ w

\V

N

o is measured value 9 nodes in picture frame

Fig 6.15

168. passed through the winding. This force was predicted numerically

and the result was found to be 17.4 high. An error of this

magnitude was expected since the machine length (30.5cm)

was only marginally greater than its width (26.7cm) and hence

the longitudinal edge effect was significant. In contrast, the

analysis assumed there to be no longitudinal edges or transversely

directed current density.

The machine was traversed longitudinally at its transverse

centre and the normal component of flux density was measured

6mm above the primary, both with and without the conducting

secondary. The results obtained are included in Fig 6.14 and they

show the normal flux density to be approximately constant over

only one half of the machine length. In order to reduce the

effects of the longitudinal edges, a secondary of identical

thickness,conductivity and width to that used previously but of

length 41.9cm was placed lcm above the primary in a symmetrical

position. 7.2 Arms

per turn of the primary winding was then

required to give a normal force of 200 gms and the computed value

was found to be 4.4°/, low.

It was thought that a possible source of error in the

computed results might be the scaling used to compensate for the

fact that the machine width used in. the computer model was

slightly less than that of the actual machine. To check this,

the computer program was altered to make the machine analysed of

width 26.8cm (compared with the actual width of 26.7cm). The flux

densities calculated are contained in Fig 6.15. The measured and

computed values are shifted in space relative to one another

since the penalty for getting the overall width correct was that

the slot to tooth width ratio became incorrect. Nevertheless, the

169.

value of normal force remained unaltered from that previously

computed.

6.2.3 Convercence Properties Of The Hunerical Solution

When the conducting secondary was introduced above the

Primary, iterative convergence was found to be slow in comparison

to that achieved for magnetostatic conditions. Originally each

iteration consisted of one successive overrelaxation a mesh scan

followed by ten Gauss — Seidel scans of the T mesh. Let r be

a complex number defined so that Re (r) is the maximum nodal

change in Re (a), and Im (r) is the maximum nodal change in

Im (St), between two successive iterations. If Rmax is the

modulus of r, then after 300 iterations and with an acceleration

—3 factor of unity applied to ,

the maximum magnetic scalar potential. Surprisingly, increasing

the acceleration factor above unity produced no significant

change in the rate of convergence. This behaviour was attributed

to the fact that the numerical algorithm employed was a weak

block method (see Chapter 4 ). In order to strengthen the

division between the blocks, the following procedure was adopted.

At the start of a given iteration all the St nodes outside the

conducting secondary were scanned once. Then the a nodes within the secondary were scanned three times. Finally, the T nodes

were scanned three times using Gauss — Seidel line iteration for

each transversely directed row of nodes.

When the new algorithm was employed, Rmax was1(0.55 +

• j2.57)I10-3 after 300 iterations using an acceleration factor of

unity. The rate of convergence was thus no better than that

achieved previously. However, the new algorithm responded to

Rmax was1(1.66 + j2.55)110 of

170. acceleration of the St values and when an acceleration factor

of 1.4 was employed, the value of Rmax after 300 iterations was

1(5.4 + j6.0)110-4. Nevertheless, a further 600 iterations reduced

by a factor of only three. :xamination of the results max

obtained after 300, 600 and 900 iterations revealed that there

was little point in using more than 300. It is interesting to

note that after 50 iterations, the computed normal force was

within of its final value even though the magnetic scalar

potentials had not converged.

The numerical behaviour of the solution is reminiscent of

that which occurs in many magnetically non-linear problems.

Convergence of such problems has been successfully accelerated

67 using block change techniques. Ahamed, for example, notes that

in two dimensional magnetic vector potential problems

f H dl 6.23

is not equal to the current enclosed by contour c except when

convergence is reached. He accelerates the iteration process by

performing line integral 6.23 after each iteration and dividing

the known enclosed current by the value calculated. If the

finite difference mesh is square, each nodal value within, and

-on, contour c is then multiplied by this ratio. The 1Tagnetic

Circuit Law is thereby satisfied throughout the solution process.

Equation 6.23 cannot be used when magnetic scalar potential

solutions are considered since the houndary conditions applied

to T ensure that the Magnetic Circuit Law is always satisfied.

Instead we use the condition that

fdiv B dV = 0 6.21+

for any volume R. By the Divergence Theorm 6.24may be transformed

171.

to the surface integral

513 . n da = 0

6 .25

where n is a unit vector that is everywhere perpendicular to,

and outward from, surface S. In two dimensional problems 6.25

becomes the line integral

„0-B . al = o = a 6.26

At intermediate stages in the magnetic scalar potential

calculation, 6.26 is non—zero and equal to the net flux crossing

contour c. This net flux must be zero at convergence. The block

change technique is best exemplified by reference to the

magnetic equivalent circuit of Fig 8.2b. Consider a flux 0 to

be injected at node 0. Application of Kirchhoff's Current Law

reveals that

1+ a3 + N2(312 + 94 ) +

— N0 2 2 N

2 4 (2 2112)

6.27 After each iteration, line integra16.26 is performed to yield

the net outward flux from the region bounded by a contour that

we shall assume to enclose n nodes placed in a regular

'rectangular array. 0 is then given the value X / n and each

nodal potential has the last term of equation 6.27added to it.

Thereby 6.24 is satisfied at every stage in the iteration process.

Note that although we have considered only the Laplacian finite

difference equation here, the same 0 term appears in the

Poissonian expression for R.

Due to lack of time the block change technique was

unfortunately not tested numerically.

50,51 172.

6.3 The ?-4teriorMoment Method

It has been mentioned that a common method of approximating

boundary conditions which are specified at infinity is to extend

the computation mesh to a distance from the region of interest

at which a given field quantity is considered to be constant.

This mesh extension is performed with much greater ease when

irregular, rather than regular rectangular, meshes are used.

However, since their associated point SOR matrices are slowly

convergent, irregular mesh problems are usually solved by direct

methods. Consequently only a small number of nodes (4:100, say)

can be employed if the use of computer backing store (which is

expensive in terms of peripheral device transfer time) is to be

avoided. This limitation implies that long 'thin' mesh elements

must be used to extend the mesh away from the region of interest.

The accuracy of such elements is limited by numerical rounding

error considerations. An investigation into ways of best utilising

the available nodes was made, and an exterior element technique

(first proposed by Professor Silvester of McGill University,

Montreal, Canada) that modelled open boundaries using few nodes

was developed.

The essence of the exterior element method is as follows.

-Consider a given region containing all the sources and having

a closed boundary, and let each mesh branch be associated with an

electric circuit element. Then the exterior element method yields

a set of impedances which represent infinite space and which

connect each boundary node to every other. Thus, a numerical

solution need be obtained only within, and on the surface of,

the enclosed region.

Consider the bounded region (a) of Fig 6.16. Now construct

CB

\ ptz 1,\x \\N\ \ (a)

C2

0 x •

6.28 0

6E4

0 I

173.

a contour CB

which is a mapping of the boundary C1 such that for

each node pl lying at (xilzi) on C1 there is a corresponding

node p2 at (Rxi,Rzl) on CB. R is assumed to be greater than unity

and is known as the 'mapping ratio'. The mapping is possible

only when C1 is 'star shaped' (i.e. non—reentrant) with respect

to a point P at which the coordinate origin is defined. In the

Figure C1 is a rectangle so that the point P is at its centre;

but in general it may be difficult to determine the position of

P. Consequently, an inner contour exhibiting at least two planes

of symmetry should be chosen since P is then at their intersection.

Row let there be a discretization within the region bounded by Cl

Fig 6.16 and C

B such that no mesh elements straddle the boundaries of

these contours. If the boundary nodes are numbered first and the

interior nodes last, then a matrix eauation of the form

S11

S12

521 522

is obtained where 0 contains the boundary node, and 0 I the

interior node, field values respectively. Sll and S22 are found

to be square non—singular matrices whilst S21 is the transpose of

S12. 0 I may be eliminated from 6.28 to yield

S11 s12 11

Si Si 21

0 B

0 2 22

•••••

0

1 S12

Si 22

0 1

0 B

OOP

0

0

0

6.32

yields the matrix equation

*PO

1 Si11

S12

S21 Si 4. S22 22 11

21

0 S2 1

174.

(S11 — s12 S22-1 S21) ° x 0 6.29

which provides a relationship between the nodal values of C1

and those of CB only. 6.29 may be partitioned into the forms

1 Si °11 12 6.30

Si b22 21

where 01 contains the field values at nodes on Cl' and

0

contains the corresponding values at nodes on CB. The sub—matrices

of 6.30 are such that those on the main diagonal are square and

non—singular. Let us consider one more contour, C2, which

surrounds CB

and is star shaped with respect to P. If the nodes

on C2

are related to those on CB

by the same mapping as the nodes

on CB are related to those on C1

(i.e. the mapping ratio, R, is

kept constant), then the coefficient matrix of equation 6.30

also describes the links between the nodes on CB and C2

. Thus 0 2'

which contains the field values at nodes on C2' and 0B

satisfy

the equation

1 0

0 B 0

obtained by analogy with 6.30

00

6.31

. Addition of 6.30 and 6.31

°2

WON. ••••• PM/

S(2) s(2) 11 12

6.33 (2) s(2) 621 22

s1 - s1 (s1 s1 )-1 51 -s1 (S1 + S1 )-1 31 11 12 11 22 21 12 11 22 12

0 1

0 2

1 1 1 -1 _1 -621 ' (S11 + 52 2) S21

sl sl isl s1 71,1 22 21‘ 11

4. 22 612

[

,(k) ,(),(k),() 611 - 612 11 621

(k) (k) (k) - S12 A S12

3(k) ,(k)A(k)S(k) 622 - 621 12

(k) (k) (k) - 021 A S21

931

MI%

175.

Elimination of 0B between the first two equations and then the

last two yielas

Ow. ■■•=1 OM.

[o0

which is symmetric. It is now possible to envisage an infinity of

contours; each Ck being related to each Ch - 1 as C2 is to C1.

Thus R is kept constant and for each of the contour pairs there

exists a relation of type 6.33 given by

s(h+1) 5(k+1) (h+1) s(k+1) (k+1) 11 12 0 0

q(k+1) 5(k+1)

[ -21 22 •=0111

6.34

0

0 4.•

,,1k1 () (k, where A(k) = kb + S22))-1• It will be noted that the

176.

shortest distance between Ck and C1 is given by

d = g k2(k-1) k

where g is a constant. If C1 is a square, then when R=2, g is half the

length of one side. Equation 6.35 tells us that very few steps

of the recurrence relation 6.34 are required before Ck is a

great distance from Cl. In the limit as k approaches infinity

it is often assumed that d (k+1) approaches zero (there being no -

sources in the exterior region) so that

(s(c° (s(o) ) + ,(ap ) \-1 (co X

12 11 '22 1 321 /0 1 = 6.36

The matrix premultiplying 0 1 contains coefficients which link

each element of 0 to every other and, furthermore, these

coefficients form a surface impedance model of the region

exterior to that bounded by Cl. Thus, addition of 6.36 to the

set of equations obtained from a discretization of the region

enclosed by C1 yields a further set which then includes the

open boundary effect.

At each step in the recurrence relation, one matrix

inversion and six matrix multiplications are required. The

orders of S11 and S22

are equal to,the number of nodes on contour

C1 and this number must be small if computation of the open

boundary coefficients is to, be computationally cheap. Lines of

symmetry may be exploited in order to reduce the number of nodes

but, in general, high order elements must be used in the

exterior region since they require relatively few nodes. A

mathematical difficulty can arise here, for there must be no

mismatch between elements having a common edge along contour C1

if the finite element formulation is to be strictly valid. Thus,

th if" n order elements are used in the exterior region they ought

6.35

177.

_.<

c B

c B

Positive Quarter Plane Of Busbar

Arrangement

)

c1

A=0

c1

CI A

c1 A

Second order elements used. apping ratio (R)= 2.

A = 0

CB^

z

+I

P

-I

Cl

Busbar Arrangement

X

0.5 unit

V

1 unit > Finite Element Mesh For Model Open Boundary

Problem Fig 617

178. also to be used in the interior regior.

The exterior element technique was tested on a rectangular

busbar arrangement consisting of two rectangular non—magnetic

conductors of dimensions 1 by 0.5 unit placed so that their

larger surfaces were in contact. The conductors were assumed to

be of infinite length and contain time independent evenly

distributed currents of equal magnitude but opposite sign. Second

order triangular finite elements were employed and the function

calculated was a magnetic vector potential directed at right

angles to the plane of solution. This function was employed

since the required computer program was available in working

form. The program could generate a stiffness matrix and solve a

given matrix equation using Gaussian elimination. It was based

on the assumption that all the sources were evenly distributed

over the elements that covered the conductors. The finite element

mesh used is contained in Fig 6.17. Since the elements were

second order, each triangle was associated with six interpolation

nodes; one at each vertex and one at the mid point of each side.

Altogether there were eight 'free' nodes on the section of Cl

that was in the positive quarter plane.

The problem described was studied mainly because an

analytical solution was available and consequently the accuracy

of the model could be easily assessed. The analytical solution

49 used in comparisons is given byBinns and Lawrenson. Although

this gives the flux densities correctly, a —12ab term must be

added to the expression. within the major brackets if comparisons

are to be made with the magnetic vector potential solution

obtained here. Another aspect of the busbar problem is that the

C1 contour lies along the surface of the conductors, so making

179.

the solution a severe test of the exterior element approach..

The computer solution required 4.2 execution seconds of

CDC6400 time for the stiffness matrix generation and final

solution stages whilst generation of the exterior element

coefficients took 11.8 seconds for a k value of four. This value

of k was chosen since the C4 contour is 32 units away from the

coordinate origin and thus may be regarded as being at a

distance eauivalent to infinity. Increasing k to five and more

revealed that with a k value of four the majority of the exterior

element coefficients were accurate to three significant figures.

It was also noted that as k increased, the off—diagonal

sub—matrix elements of 6.34 decreased in magnitude and reached

numerical rounding error proportions for a k value of

approximately ten.

The magnetic vector potentials computed at nodes on C1

were compared with those determined using the exact analytical

solution and the maximum error was found to be less than 1c/S. Thus,

although determination of the exterior element coefficients is

expensive compared with the cost of forming the interior region

stiffness matrix and obtaining a field solution, high accuracy

may be achieved using few nodes. Indeed, had the mesh implicit

in the exterior element relaxation been duplicated explicitly,

a solution involving 52 (as compared with 12) 'free' nodes would

have been required. Since the computation time is roughly

proportional to the square of the number of nodes, the solution

time would then have been approximately 80 (as compared with

4.2 11.8) seconds.

It was originally intended to perform detailed tests on the

exterior element method but lack of computing facilities made

180.

this impossible. One intention was to compute the stored energy

within the conductors and compare it with that in the exterior

element. This is most easily accomplished by obtaining the former

energy using the relation

Ec = 2 A . J dV 6.37

2 conductors

and the latter energy by boundary integration. The reason why

stored energy is useful as a basis for comparison is that the

quantity includes the effects of all the errors, not just those

at a particular point in. space. The energy stored within the

conductors may be obtained using the exact analytical expression

for the magnetic vector potential together with equation 6.37 .

Since the current density within each conductor is not a function

of position 6.37 reduces to

E = J ./ A cix dz 2

2 conductors

where Ec

is here the stored energy per unit length of

conductor. The integration was carried out numerically using the

trapezium rule. Various numbers of equally spaced integration

nodes forming a square mesh were used and it was found that

20,000 nodes per quarter conductor were required if accuracy to

better than 0.05;' was to be obtained.

Although riot all the tests originally intended were

performed, it would certainly seem that the exterior element

method is well worth employing for irregular mesh open boundary

problem's since it requires few nodes, and the exterior element

may be placed on the surfaces of conductors with no attendant

loss of accuracy.

6.38

181. 7. SOLUTIOU OF TUR30—=_MATOR ED FIELD P,OU-EI is

In.recent years turbine generator unit ratings have

increased rapidly to satisfy the capital cost conditions imposed

by power system operators and designers. Unit reliability is of

increasing importance as ratings rise, since generator outages

can cause severe power system transient stability problems and

require the run up of a large amount of low merit order plant.

Furthermore, since present turbine generator production capacity

exceeds demand, manufacturers are under pressure to keep prices

(and profit margins) as low as possible in order to remain

competitive. These factors explain the continued quest by

machine designers to reduce production costs whilst maintaining

(or improving) unit reliability.

It is suspected that a number of generator failures are

due to the effects of axial flux produced by the end windings.

In this connection it has been noted that severe heating of the

stator core end can occur when the machine is underexcited;

though not when it is overexcited and operating at the same

loading level. The effect is found to be less noticable in four

pole machines than in the two pole machines used in conjunction

with high speed steam turbines. Attempts to reduce core end

heating have usually been made on a trial and error basis. Since

the excessive heating is caused by axial flux, it is accepted

practice to place a flux screen between the stator end windings

and the stator core end plate of large machines. Eddy current

power losses are associated with this screen, which is also

expensive to manufacture. Experience suggests that conducting

screens can have the undesirable effect of increasing the axial

182.

flux densities impinging on the stator teeth. These flux

densities induce increased tooth losses and in order to combat

the problem two important techniques are often employed. In the

first, eddy currents flowing in the plane of the laminations are

reduced by placing a radial slit down the centre of the stator

teeth. This slit, known as a Pistoye slot, is a source of

mechanical weakness and is placed only in the lamination packets

near the core end. The second technique increases the cooling of

the core end without substantially reducing the eddy current losses

induced by axial flux. This is achieved by reducing the stator

lamination packet thickness as the end region is approached. The

technique has the disadvantage that the stator core stacking

factor is impaired. Core end lamination packet stepping can also

be used to increase cooling and inhibit axial flux, but it suffers

from the same disadvantage and must be employed with care.

As well as stator core end heating, other end region

phenomena require explanation. The overheating of fans and

aluminium gas baffles has caused some concern but of greater

importance has been the interest shown in the possibility of using

magnetic retaining rings for the rotor end windings. The reason

for this interest is that the rating of a turbine generator is 52

-approximately proportional to the volume of its rotor. Lachine

ratings may thus be increased by either lengthening the rotors or

increasing. their diameters, or both. At present there is a limit

to rotor length imposed by vibration mode considerations. The

rotor diameter is limited by the centrifugal forces acting on

the traditional non—magnetic retaining rings. These are

mechanically weak in comparison with those made of magnetic

steel whose electromagnetic effects still require detailed

183.

investigation.

Bearing all these points in mind, an ideal analysis would

incorporate the effects of

1) eddy currents in the major conducting regions

2) stator core steps

3) saturated magnetic materials such as the rotor end winding

retaining ring, rotor balance ring, and the stator core end

and teeth

4) electrical transients

5) the electromagnetic fields at any given load condition.

It was decided to develop an end region electromagnetic field

formulation capable of including all these aspects. However,

since manufacturers expressed particular interest in the effects

of eddy currents induced in non—magnetic flux screens under

steady—state conditions, 1) and 5) were given prominence. Eddy

current phenomena in magnetic materials were investigated as a

separate exercise (see Chapter 8).

7.1 Basic Numerical Formulation And Assumptions

The turbine generator end region geometry is both

magnetically and electrically three dimensional. Some early

,attempts at end region field calculation involved three

57 dimensional integrations of the Biot—Savart type. Although the

winding .forces were predicted correctly, the methods could not

represent the effects of eddy currents and they did not include

a good geometric model of magnetic parts.

Most recent attempts at solving end field problems have been

53-56 made using magnetic scalar potential formulations. Both finite

6 difference and finite element procedures have been employed,

l84.

though the former seem to have found greater favour. come

56,58,54 formulations have included the effects of eddy currents, either

56 by using surface impedance models or by assuming conductors

to have zero resistance and imposing the condition that the

normal component of flux density is zero at conductor / air

58,54 interfaces. Calculations of the electric and magnetic fields

within conductors carrying eddy currents has usually been

attempted in magnetic vector potential terms. For the purposes

59 of these calculations either Cartesian geometry has been assumed

or the peripheral component of flux density has been neglected.

Some numerical formulations use three dimensional meshes and

59 60,61 examine either a whole pole pitch or a single tooth pitch. To

the author's knowledge, only Ewa attempt to include the effects 59,61%61

of both the radial and peripheral components of current density.

Bost of the methods use two dimensional meshes and assume that

the machine's geometry is peripherally invariant. Furthermore,

it is often assumed that any magnetic saturation causes negligible

waveform distortion. A travelling wave solution in phasor terms

may then be obtained and a drastic saving in computation cost

achieved.

The author was particularly concerned with the calculation

of magnetic fields in the air spaces and non—magnetic conductors

of turbo—generator end regions rather than with the determination

of fluxes and eddy currents in the stator core itself. With this

in mind, it was decided to proceed with a magnetic scalar

potential analysis based on the following assumptions

l) Transient phenomena were not to be investigated.

2) The permeability and resistivity of all magnetic materials

was infinite. Under steady state conditions, this assumption

185.

allows us to use phasor quantities for the field variables.

Thus, time need not be included explicitly in the governing

equations.

3) The geometry of the machine was invariant with peripheral

direction. This was a reasonable assumption since the

important non—magnetic conductors all exhibited peripheral

symmetry. However, it implied that both the rotor and stator

had smooth air gap surfaces with no slots. The validity of

computed results obtained in the vicinity of the air gap

is thus doubtful. Furthermore, the effects of three

dimensional features such as bolt holes and brackets must be

considered when comparisons are made with measurements:

The assumptions allow us to employ a two dimensional mesh

at whose nodes may be defined two Phasors,S/m and T such that

j(wt —m0) (r,e, z, t) = Re (Di (r, z) e 7.1

m = 1,3,5,7— Zin (r, 6,z, t) = Re ( T;1(r, z) ei(wt —") ) 7.2

The subscript m indicates the space harmonic number of the

harmonics generated by the stator and rotor windings. Due to the

symmetry of the windings, m does not take even values. In view.

of the approximations inherent in the numerical formulation, it

was decided that calculation of harmonics greater in frequency

than the fundamental ( m = 1 ) would create only an illusory

improvement in accuracy except, say, near the rotor end windings.

In addition to the major assumptions given above, the

following minor ones were made:

4) The stator core end plate was non—conducting. This plate is

usually non—magnetic and is approximately one effective

depth of penetration thick.

at any instant of time

186.

DUNGENESS .B' TURFO-GENERATOR DATA

1. Winding data

Rotor 32 'lots 15 teeth/pole 56 turnA/pole

Pitch angle = Angle between two adjacent slot centres = 31 radians. 47

Stator 42 slots with conductors wound fro= slot 1 to slot 18. 81% chording.

2 conductors/slot 2 parallel paths / phase

60° phase belts

Peak of the fundamental component of stator winding mmf

= (/ .I) (bg/2) (3k/2)

where I = RMS current / conductor

b = number of conductors / slot

g = number of slots / pole / phase

k = winding spread factor = k k P d

k = winding pitch factor = sin ( 1) 2

kd

m winding distribution factor = sin ( ( ) 2q 2nq

p = winding pitch (i.e. the ratio of coil span to pole pitch)

q = number of phase belts / pole

n = number of clots / phase belt

2. Miscellaneous data

Rated power = 660 Ku

Rated volt-amps = 776 MVA

Rated stator volts . 23.5 KV

Rated stator phase current = 19.076 KA Leakage reactance =

Armature resistance = 0.2;:

3. Tent Condition

Open circuit stator. 1.6 }A rotor current required to give rated stator terminal volts.

Short circuit stator. 3.08 KA rotor current required to give 19.15 KA stator current.

Fig 7.1

air gap

rotor winding balance ring

fan

gas baffle

core end plate

flux screen 0

.1.8x10 8Stm.

stator winding

187.

Dungeness '13'

End Region Fig 7.2

188. 5) The gas baffles near the end doors were constructed of

laminated iron. Often these are made of aluminium but since

the end doors are a considerable distance from the main region

of interest, a laminated iron boundary was thought to be a

reasonable substitute.

6) Stator core steps were neglected.

7) The stator windings were comprised of conductor bundles of

equal shape and spacing.

These assumptions were minor in the sense that their restrictions

could easily have been removed had the ad'Ational computation cost

been considered worthwhile.

7.2 Geometric Details Of The Generator On Which The

Analysis Was Tested

A 660 I.IN 2—pole turbo—generator manufactured by C.A.Parsons

and installed at Dungeness 'B' power station was chosen. for

analysis. This machine had a non—magnetic core end plate and

rotor end winding retaining ring. Its end door gas baffles were

constructed of 1" thick aluminium and the machine was fitted with

a copper flux screen. Data concerning the windings of the

machine is contained in Fig 7.1. The geometry of the r — a plane

used in the numerical solutions is given in Fig 7.2. In order

to simplify the computer programming of the finite difference

equations, sloping gas baffles were approximated by the series of

steps shown in this figure. Since the leakage fields in the

vicinity of the stator core frame were known to be small, it was

decided to model this frame with a cylindrical laminated iron

boundary 9" smaller in radius. The end windings were represented

by current sheets whose positions are shown in Fig 7.2 .

189.

The main reason why the Dungeness '3' machine was chosen

for analysis was that it had a large number of search coils

installed in the end region. Extensive tests had been performed

with the machine on open and short circuit and the results were

judged to be reliable. Due to power station construction delays,

no lcad tests have yet been performed but it is hoped that the

machine will be brought into service in the near future.

7.3 Stator Winding Renresentation

The stator winding was represented using the techniques

53 described by Carpenter and Locke. The formulation ignores slots

and assumes that the slot currents act at their respective rotor

and stator air gap surfaces. At these surfaces all three components

of flux density exist, even though both the rotor and stator

ironwork is assumed to be infinitely resistive and permeable.

Fig 7.1 contains the analytical expression63and associated

winding factor definitions6 used to determine the fundamental

of the stator magneto-motive force (MMF). Since the slot

conductors were represented by current sheets, it was consistent

to represent the end windings in an identical manner. Each layer

of the stator winding was associated with a current sheet which

passed along the geometric centres of its conductors as shown in

Fig 7.2 . The nose of the winding (that furthest from the stator

core end) presented a problem in this respect, since the inter-

layer connections were made at two different radii depending on

peripheral position. As an approximation, it was assumed that the

nose of the winding was always at the radius given by the mean of

these two values.

If the conductor bundles in the stator end windings are

• 9 717- 7-1 •

- -

Fundamental Of The Stator MMF - Distribution Atong The End Winding _

Fig 7.3

Stu or core end

1 p.u. --Y

T cos L. rx

Not to scale MMF-

/

MMF •

Stator end. winding current sheet

Si Axis

•

190.

191. equally spaced and of identical shape then the distribution of

conductors in the r — z plane is a reflection of that in the r —6

plane. Consequently, if the air gap current sheet currents are

assumed to vary sinusoiclally with peripheral direction at any

instant of time, then the currents in the sections of end winding

current sheet representing conductors turning into the r — z plane

must vary in an identical manner. If we define the Si axis as

the r — z plane in which the total slot current is zero and the

S2 axis as the r — z plane 90 electrical degrees away from Si,

then the I= distribution along the stator end winding current

sheets will be those of Fig 7.3 . Under short circuit conditions

and assuming the armature resistance to be negligible, the S1

and S2 axes become the d and q axes of the machine respectively.

Since the MF is distributed along the stator end windings in

a relatively simple way, the sources for a magnetic scalar

potential solution may be determined systematically using a

computer program for which the following data must be prepared:

1) the coordinates of the two ends of each straight line segment

of current sheet;

2) the type of 1I4IP variation along each current sheet segment

(constant, sinusoidal or cosinusoidal);

.3) the maximum IMF value on each current sheet segment.

rather than electric vector potential (T), distributions

have been employed in this end winding analysis since T is more

suited to the description of distributed currents rather than to

those confined to current sheets. Let us consider an example.

Because T satisfies the magnetic circuit law, we may define a

quantity F such that

F =/T.d1

C . 7.3

line representing conductor centre

rotor end. winding

0.0642

Quadrature axis

rotor pole

192.

Quadrature axis

,

Direct axis

Dungeness `B' Machine. Z-9 Diagram Of Rotor Winding

193.

mmf

1.0

Per Unit Direct Axis MMF Distribution

fundamental of 9 directed mrnf Fourier. analysis

1---

-11

—0.1 current sheet representing rotor.

• winding

_ rotor pole

Fig 7.5

. i .

Dungeness V Machine - Rotor End Winding

194.

is the MF generated between the limits of contour c. By

introducing a contour that links two opposing surfaces of a

rectangular conductor across which J is uniform, T is seen to be

the I:12? per unit thickness of winding. It is thus indeterminate

when the LMF remains constant and the conductor thickness is

allowed to ap-.roach zero. Equation 7.3 provides a direct link

between and T which will be utilised in section 7.5 when

calculation of the magnetic scalar potential sources is

Considered.

7.4 Rotor Winding Representation

The rotor winding of the machine is concentric and consists

of eight conductor groups, each of seven turns. Fig 7.4 contains

a z— 6 plane diagram of the end winding in which each conductor

group is represented by a straight line placed at its geometric

centre. For simplicity, bends in the conductors are neglected.

Since it has no slot conductors on the direct axis, the rotor

winding creates no quadrature axis LEI% Furthermore, the flux

pattern produced appears as a travelling wave relative to a

stationary reference system.

It was decided to represent the rotor end winding by a single

,current sheet joining the centres of the conductor groups in the

r — z plane. The position of the sheet is shown in Figs 7.2 and 64

7.5 . The peripheral variation of MF was harmonically analysed

at points midway between each z position where the contours of

Fig 7.4 crossed the direct axis. The variation of the first seven

MP harmonics with axial distance at the direct axis position is

shown in Fig 7.5. Note the very small harmonic content of the

air gap rotor MF and the large harmonic content of the MF

195.

acting near the balance ring side of the rotor end winding. In

the computer programs, only the fundamental component of EEF was

employed.

It is interesting to compare Fig 7.5 with Fig 3 of Carpenter

and LockPwhich contains the rotor harmonic analysis for a

machine with a slightly different concentric winding arrangement

(seven conductor groups, each of eight turns as compared with

eight conductor groups, each of seven turns). The two plots are

substantially the same except that the fundamental of the air gap

1,111F for the Dungeness '3' machine has magnitude less than that of

the actual EPF obtained by summing the currents in the individual

rotor conductors. In the analysis of Carpenter and Locke the

opposite property was noted.

The stator end windings were represented by two distinct

types of current sheet segment. Along the first the I•LiF was

constant, whilst along the second type the EFT varied sinusoidally

with distance. These functions are analytically simple and a

computer program of simple structure may be employed to calculate

the magnetic scalar potential sources associated with each mesh

branch. In contrast, the fundamental of the rotor winding MED' was

known only at discrete points along the end winding. Consequently

it was decided to model the variation of MI5' (F) with distance

measured along the rotor end winding using the expression

F = a + bz+ cz2 7.4

where a, b, c were constants and the z origin was at the outer

edge of the rotor end winding conductor group furthest from the

air gap. This expression was chosen since Fig 7.5 shows the

fundamental of F to be an approximately linear function of z

between the axial limits of the rotor end winding. Assuming F is

= (F2 - F3) - (F1 - F2) ( z.2 — z3) (z1 z2) 7.5

196.

known at three values of z, then c, b and a are given by

(z22 - - z2) (z2 z3)- z3)

(z1 - z2)

b = (F1 - F2) c(z -

(z1 z2)

a = F1 -bIz1 -cI z2

respectively. Points 1 and 2 are normally chosen to be at the

extremities of the rotor current sheet (points A and B of Fig 7.5)

whilst point 3 should be at a location approximately midway

between these.

7.5 Governing Txruation For S6 :expressed In Cylindrical

Coordinates And Annlicable In Regions Containing Known

Values Of Current Density

If we assume that the magnetic permeability is constant

throughout the solution domain, then continuity of magnetic, flux

density is assured when

div (T - grad ,!L) = 0 7.8

If the vector T is confined to the r - zplane, the cylindrical

coordinate form of 7.8 is

1a ( r( - T )) a ( 1511, - T ) _ —

r ar ar r az az z r

7.9

Now let us reformulate 7.9 in terms of F and St only in order

to obtain expressions compatible with the end winding

representations developed in the previous two sections. First

7.6

7.7

197. we note that because T has no 6 component, equation 7.3

may be written in the form

rdr + z

dz = F 7.10

where contour c must be confined to the r — z plane if the

integrals are to be path independent. Next we define the two

quantities Fr and F

z such that

Fr = rdr and Fz

Tzdz

From these definitions

T a Fr and T = a Fz r a r z a z

Elimination of T from 7.9 using 7.12 yields

11, ( r — ))+ a .(3 Ci ` F )) JL r 2

Or r z a z z

Let us express this equation in finite difference terms using

the regular rectangular mesh of Fig 7.6 . 2

z

t.r

Now

4

Fig 7.6 a [r22 - Fr )lez.- 1 [ r01 ((al - Fri ) - (n0 - Fr0 )1 Or Or Nh Nh

- r03 ((J20 — Pr0) — ('513 — F1'3 )1 ] 7.13 Nh

and 2 az2

(St — 2z)=1 [

31-1. (J/2 — Pz2) — (3/0 — Pz0)

(61/0 Fz0 ) - "14 - Pz4 ) ] h

7.11

7.12

7.12

7.14

198.

where r.. indicates the radius of tilt: mid point of the mesh

branch terminated by nodes i and j. Substitution of 7.13 and

7.14 into 7.12 yields the following finite difference

equation for a at node 0:

/12(512 + St4) + (r01 SL1 + 103 S/3) — ro 0

(2112 + 1 (I._ + r03) , u2h2) SL ul 031) ' 0

= r0 r

2

+ Fz4) + (r01 F

rl + 103 Fri) — 21i

2F z0

0 r0

1 (rol + r03) FrO 7.15

r0

The source term has here been placed on the right hand side of

the equation. In the computer program, the source term

appertaining to each node is pre—calculated and stored for use

during the iteration procedure. It is of note that when eauation

7. 15 is divided by the factor , a cylindrical coordinate

equivalent circuit model may be derived. Applying this procedure

and rearranging terms yields the equation

(j12 Fz2) (ILO Fz0) "14 Fz4 ) (J1 z0 F ) 0 1/µN 1/p14

rol ' )[ —

r Fr1) (j110 FrO) 7.16

0

)(51.3 — Fr3) — (S1,0 - FrO) r0

S1'0 = 0

ro2/(Nh2 ti

11/p.

in which the denominator of each term has the dimensions of

199.

reluctance. The magnetic equivalent circuit is contained in

Pig 7.7 .

•r

Fig 7.7

If T is zero along each mesh branch then the generators are of

zero value. Otherwise their values are found using equation 7.11

It will be noted that if the integration contour is defined as

linking two nodes along a single mesh branch then the value of

the LEP generator ( F) is given by

P = dl 7.17 i

where the positive direction is from node i to node j outwards

from node 0. In current sheet terms, 6F is simply the ELF drop

across any current sheet at the point where it cuts a given mesh

branch. It is found that the function F is like T in the sense

that only when its value changes are magnetic scalar potential

sources created.

7.6 Calculation Of 3ddy Currents In The Conner Flux Screen

The copper flux screen was the only component within the end

region in which eddy currents were calculated. Since the radial

dimension of the screen (12") was considerably greater than the

axial dimension (1"), it was initially assumed that T was axially

200.

directed within it. This assumption restricted the current density

vector to the radial and peripheral directions only.

To be strictly correct, the governing equation for T should

have been expressed in cylindrical coordinates and the

appropriate cylindrical coordinate finite difference expressions

found. However, to save time and effort it was decided to use the

Cartesian equations derived in Chanter 3 and neglect the effects

of flux screen curvature. It was suspected that this step might

cause iterative convergence problems and produce peculiar results

due to the mismatch between the cylindrical and Cartesian meshes

at the surfaces of the screen. In the event, no problems of this

nature were encountered.

The travelling wave term for the Cartesian mesh is defined by

a = (IT 7ch/p )2 7.18

Since a cylindrical machine was under consideration, the pole

pitch (p ) was a function of radius (r) and given by the product

% r. Thus a was also a function of radius and given by

a =Nh )2 ` r'

The value of a chosen for the Cartesian mesh was the average of

the two values corresponding to the maximum and minimum radii of

the flux screen. This implied that the effective radius of the

screen was

Air2 + r2. max min

In practice, rm is approximately equal to the average of rmax

and r mm . n. The dimensionsof the Dungeness 'B' flux screen were

such that the ratio of rmax to ',min was 1.335.

7.19

2r r . rm

= max mmn 7.20

201.

7.7 Commuter Program And Eesh Description

For simplicity, the finite difference method was employed

for all the solutions obtained. A superficial investigation into

the possibility of using triangular meshes was made but it was

concluded that as no computer library routine was available for

solving large sets of sparse, asymmetric, complex matrix equations

the cost per solution would be excessive. In order to achieve

reasonable finite difference node economy, two different mesh

aspect ratios were employed. Within the air gap and up to the

radial line along which the core end surface lay, a square mesh

of side 1.5" was employed. This was a convenient size since the

air gap of the machine was 4.5". A mesh of radial dimension 1.5"

and axial dimension 0.3" was used between the stator core end

and a radial line 0.6 " to the end door side of the flux screen.

Thus the effective depth of penetration of the flux screen was

approximately 30,'Q less than the axial mesh interval. It was hoped

that variations of T and St would be greatest in the axial

direction so that the choice of large radial mesh interval would

not cause too great an inaccuracy.

Within the major part of the end region, a square mesh of

side 1.5" was employed. Iven with such a relatively coarse mesh

was possible to locate the boundary features with considerable

accuracy. The air gap region was terminated 6" away from the

stator core end by a radial line. It was assumed that no axial

flux crossed this line and in consequence was made to satisfy

a zero gradient condition with respect to the axial coordinate

direction. In an early version of the computer program, it was

assumed that the flux density had zero radial and peripheral

components at points on the radial air gap boundary line. This

* This large hid ratio was not expected to result in tower less daterminations of high accuracy. Nevertheless, it wa hoped that the external ragnetic field would be predicted well. Another, though minor, consideration was that by choosing a coarse finite difference mesh, the arount of core store taken up by the computer'progrsm could be kept at a manageable level.

202.

Fig 7.8

DATA PREPARATICS PRELIMINARIES: 1. Prepare a scale drawing of the end region on graph

paper such that the graph equaree bear sore simple

relation to the proposed meal: size.

2. Write on the graph (a) the matrix position

coordinates (I,J) of all corners; and

(b) the actual r-z coordinates

of all the current sheet segment's.

3. Obtain a rotor end winding ha....:nic analysis.

START THE COI:FUER PROGRAM

Read in the following flux :screen data:

1. the conductivity

2. the effective depth of penetration

3. the valuer of the axial and radial reeh intervals

in the screen

4. the maximum and minimum radii of the ecreen.

Read in the coordinates of the bottom left hand corner of the magnetic scalar potential

mesh relative to the reference frame used for the current sheet position data.

Read in three valuer from the curve of )HF against axial distance measured along the

rotor end winding.

Read in the descriptive data for one current :sheet segment. This data includes:

1. the coordirates of each end of the segment

2. a flag varidsle indicating the type of I-IF variation occurring along the segment

(constant, :sinusoidal, cosinunoidal, quadratic)

3. the KMF phase shift between the segment ends and the maximum MMF value.

Determine the magnetic ecalar potential sources associated with each node lying outside

the flux screen and place them in the lower half of the iterative computation matrix.

Has all the current sheet data been

read in ? 110

YES

Read in the fixed non-zero magnetic scalar potential boundary valuer together with their

location coordinates. Place thece values in their correct positions within the upper

half of the iterative computation matrix.

'Iterate using aucceeseive overrelaxation and obtain a solution.

Determine the current end power lore density distributions in the flux screen. Find

the total power loss and tho electromagnetic forcer.

I

I STOP

BASIC FLOW CHART FUR THE SOLUTION OF TURBINE CENERADDR DID FIELD PROBLEMS

203.

was a poor approximation which nevertheless was shown to introduce

negligible error on the magnetic scalar potentials calculated

outside the air gap.

Successive overrelaxation was used to solve the set of

finite difference equations. The iterative computation matrix

had 44 columns and 140 rows and was orientated so that the rows

formed radial lines. The upper half of the matrix contained the

magnetic scalar potential values whilst the lower half contained

the values of the sources due to the windings together with the

T values appertaining to the flux screen. The two halves of the

matrix were geometrically identical.

The basic flow chart of the end field calculation procedure

is contained in Fig 7.8 . All the winding and boundary value data

was read in from data cards, as were the values of the material

constants. In this respect the computer program was a general

one. However, the iteration area was defined by a series of

Fortran subroutine calls. Each of these subroutines iterated

within a specified rectangular area of which there were 27 in

total. If a different machine were to be analysed, a total of 40

Fortran statements and 25 data cards would have to be changed.

In the radial direction the flux screen was spanned by eight

mesh intervals whilst in the axial direction the corresponding

number was four. Since the flux screen was thereby 1.2" wide

instead of the actual 1", it was decided that when the program

data was prepared a conductivity value lower than that of the

actual screen by a factor of 0.2 would be used. After a magnetic

scalar potential solution had been obtained, the power loss

density distribution was determined and a figure for the total

power loss per metre was printed out. To obtain the total power

204.

loss in the screen, this figure must be multiplied by 2,7trm

where rm is the effective radius defined Ly 7.20 .

7.8 xcitation Conditions Used For The Computer Solution

Two magnetic scalar potential solutions are required if the

magnetic field distribution within the end region is required at

any arbitrary load condition. Since magnetic linearity is

assumed, any such load condition can be simulated by forming a

linear combination of the solutions whose magnitudes and angular

displacements must be suitably scaled.

The only sets of experimental results available corresponded

to the open and short circuit tests (for details see Fig 7.1 )

performed at the manufacturer's premises. It was therefore decided

to obtain magnetic scalar potential solutions for these two

conditions directly.

7.9 Characteristics Of The rumerical Solution

In the original version of the computer program, point

iteration was employed for both T and Si.' The T mesh in the flux

screen was scanned ten times per St mesh scan and poor iterative

convergence characteristics were noted. The scan procedure was

then altered so that for each scan of the nodes lying outside the

flux screen there were five SL mesh and fifty T mesh scans

within it. Convergence was then found to be more rapid and

occurred in 566 iterations using an acceleration factor of 1.6

which was employed throughout the tests. The convergence

criterion was that the modulus of the maximum A residual be less

than 1.7 x 10-6 of the maximum St value. Despite the improved

convergence rate, the overall solution time (475 CDC 6400

205.

execution and compilation seconds using an FTIT Fortran compiler)

was almost unchanged from that of the earlier solution since the

computation work per iteration had been increased.

Experience obtained with the 3—core plate levitator problem

of Chapter 6 suggested that it might be advantageous to determine

T using line iteration. The algorithm was therefore adjusted so

that for each A mesh scan outside the flux screen, fifteen

point iteration scans and fifteen T line iteration scans were

applied inside. 544 iterations were then reauired to satisfy the

same convergence criterion and the total computer time was

drastically reduced from 475 to 270 seconds. In an attempt to

reduce the computer time even further, it was decided to reduce

the number of A and T mesh scans within the flux screen to three- -

a figure arbitrarily chosen. Convergence was then achieved in

549 iterations and the computation time fell to 209 seconds. A

disadvantage of using line iteration to determine the T values

is that there is a greater storage requirement. 64100 octal words

of computer memory were required by the program when line

iteration was used. This compares with the 61700 octal words

required when point iteration was used throughout.

The convergence characteristics quoted apply to the

solutions obtained assuming the machine to be subject to a three

phase short circuit. Given these conditions, the final version

of the program (three point iteration A scans and three line iteration T scans in the flux screen per iteration) cost

approximately Zl0 to execute. When the armature was assumed to

be open circuited and the excitation was adjusted to give rated

stator volts, the same program required 314 seconds to compile

and execute. 520 iterations were necessary to achieve the same

206.

, 12

I

+4 5

+3

+2.

i

I I I Si * 11

.... 4.41

colt group positions

LOCATION OF COIL GROUPS IN END REGION

Fig 7.9 •

207.

accuracy as the short circuit results and the total solution

cost was approximately 5113.

7.10 Comparison Of ?:ensured And Calculated Results

The numerical solutions were obtained in complex number form

and the direct and quadrature axis values were stored in the

real and imaginary parts of the iterative computation matrix

respectively. In regions where the electric vector potential is

zero

Ii = — grad 7.21

and the three components of flux density are given by

Br

; ar

II Si. Be = j and B

z = — P. ata 7.22

az

These components are phasors and must be expressed in terms of

magnitude and phase if comparisons with measured results are to

be made. A computer program was written which took the computed

St and T values and calculated the magnitude and phase of each

vector component of flux density throughout the finite difference

mesh. The Bz

and Br

values were determined at the centres of

axially and radially directed mesh branches respectively. Be

was determined at each mesh node. The flux densities were printed

out and placed on magnetic tape so that automatic field plotting

could be attempted if desired. Since there were no search coils

on the flux screen or between the individual end winding

conductors, no correction was made for the non—zero T values.

In other words, T was assumed to be zero everywhere.

The approximate locations of the search coil groups

(relative to the geometry of the machine) are contained in Fig 7.9.

Each cross represents the centre of a wooden block of square

cross section on which three circular search coils are mounted.

208.

The dimensions of a typical coil group are given in Fig 7.10 .

Fig 7.10 10=1, 111.

Each row of coil groups is fixed into a wooden frame so that the

position of each group relative to the others in its row is

known accurately. However, the manufacturer's data gives the

coordinates of only the centres of the wooden blocks and not those

of the individual search coils. In consequence, the axial and

, radial search coil positions are known only to within ..3/8". If

one allows a tolerance of —1/8" when the coil groups are

installed, then the overall positioning error could be in the

range 1: IA' relative to the nominal coordinates. Apparently it is

the usual practice to orientate blocks so that one search coil has

a peripherally directed axis and one of the remaining two coils

is as close as possible, and parallel to, the nearest winding or

conducting member of interest. This convention can be ambiguous.

Nevertheless, it was used wherever possible since no alternative

was available.

The measured flux densities were expressed in terms of

magnitude and phase of the fundamental components. The phase

reference was the fundamental of the flux density measured by

a search coil lying on a stator tooth 1.5 slot pitches from the

r — z plane in which all the end region coil groups were

located. This search coil was flat and measured radial flux

density. It was a poor choice of phase reference since any

installation or electrical connection errors could have

A

B mT 209.

250-

r . 150-

assumed search coil . orientation..

50

0.2 0.4 0.6 7

Origin 4.76cm from stator core end.. 9 is measured value. 1.565cm

balance ring • • t

Phase reference is the search coil 1.5 slot pitches from the d-7: axis.. . . ! 1 .

! ! Ii Magnitude Of B As A Function Of Axial Distance Along Line A-A., Dungeness '13" Under Short Circuit Conditions. Fig 7.11

210.

invalidated the phase readings obtained at every coil group

position. Furthermore, any such errors might have been difficult

to rectify. A better choice would have been to use one of the

armature currents as a phase reference for the short circuit

tests, and to use a line voltage for the open circuit tests. In

the event, no comparisons between predicted and measured phase

were made.

The measured flux density component magnitudes were

punched on da,ta cards together with their respective coil group

coordinates, coil group numbers, and individual search coil

numbers. These cards were read into the flux density calculating

program mentioned earlier. In view of the relatively large coil

group coordinate tolerances, it was decided that for each spatial

component of B the four calculated values located nearest to each

nominal coil group position would be printed out. The calculated

value of flux density at each position was then assumed to be

that of the four which was closest in magnitude to the measured

result. The program required 19 seconds of CDC 6400 computation

and execution time for its completion.

7.10.1 Symmetric Short Circuit Conditions

Let us consider operation of the machine under symmetric

short circuit conditions. A comparison of the measured and

calculated flux density components along line A—A of Fig 7.9 is

given in Fig 7.11 . Reasonable agreement will be noted for the

axial and peripheral components of flux density. However, the

radial component values do not agree along the top of the rotor

end cap. It was thought that this might be due to coil group

positionin a error but examination of the calculated flux densities

211.

• . Z

41- I Stator Core Side / / 7 /

Flux screen 7 . • •

• 1 1 . • •

-. •

• • • .: • ■ • • ■ • • . • ■ - • 1 •

0 : 10

radius = 98.7 cm

12.865 cm cm • 20 30

o is measured value

Dungeness `B' Under Short Circuit Conditions. •

• I

, Fig 7.12'i , i .

. ! , 1,'', 1 , H, •.i.','

., : , 1 1 H .', '' i '''. .1 ..! .', I •-.1_

Magnitude Of B As A Function Of Radius Along Line B-B.

212.

one radial mesh interval to each side of line A—A revealed that

the gradient of Br with respect to radius was small and that the

error could not be accounted for so simply. The important point

is, however, that over the range in which the measured and

calculated 3r values disagree by a large margin, Bz

and B are the

dominant components of B.Note that the measured flux densities

are almost always less than those calculated. This is probably the

effect of rotor magnetic saturation.

Fig 7.12 contains the flux density comparisons for line B—B

of Fig 7.9 . This line is radial and close to the end door side of the flux screen. The measured and calculated values of both Br and

Bz agree well. However, the Be curves do not agree near the- region

midway between the ends of the flux screen. 3xamination of the

measured characteristic shows that the Be reading from coil group

7 (marked on Fig 7.12) is largely responsible for the poor fit.

An identical effect is noticed when the open circuit results are

plotted. It is therefore likely that the coil group 7 Be

measurements are incorrect or can be safely ignored. In any event,

discrepances in Be were expected because three dimensional

projections, such as the brass brackets that are bolted to the

flux screen, cannot be included in the travelling wave

formulation. The computed results show that B is not a strong

function of distance either along the sides, or near the corners

of, the flux screen. Bz is found to vary rapidly with z along the

radially directed sides of the screen, though not at the end

surfaces. It is not a strong function of radius except near

corners. In contrast, Br varies rapidly with radius near all the

sides of the flux screen. However, its variation with the z

direction is comparatively small except near the corners.

213.

assumed search coil orientation

•

40 , 60 •

Ly28.65 mm is measured value . . . . . .

Magnitude Of B As A Function Of Axial Distance Along Line C-C. Dungeness ( 13 3 Under Short Circuit i

Conditions

Fig 7.13

•

•

. . .

20

Variation Of JB With Radius For Three Radially

Directed Rows Of Nodes In The Flux Screen. Fig 7.14 Dungeness 'IS Under Short Circuit Conditions.

stator core side of screen 1

Copper flux screen

Numbers in [ ] are the J values at the left

[4.81] hand screen edge.

o is real part o is imaginary part

214.

Fig 7.15

Of Nodes In Flux Screen. 40 L.

o is real part o is imaginary part

-60

stator core side of screen

Dungeness 'B under short circuit conditions

215.

10 Hz kA/m

inches 5 3.0 7.5 9.0 1G5 2.0

-20

Hz As A Function Of Radius. Centre Row

z Key:

o top row of nodes

o middle

x bottom••

216. Re(Jr) x 10 A/m

2

■•■

stator core end side.

flux screen

Variation Of Jr With Radius Inside The Flux

Screen. Dungeness 'B' Under Short Circuit

Conditions

Fig 7.16

217.

The results for the line of coil groups marked C-C in

Fig 7.9 are given in Fig 7.13. The agreement between the

measured ani calculated flux densities is remarkably good

considering that the coil groups are positioned close to the

phase conductor terminations whose effects cannot be included

in a travelling wave numerical formulation since the magnitudes

of the currents they contain are not sinusoidal functions of

peripheral distance. In the region near the flux screen the radial

component of B predominates, but nearer the end doors the three

components are of similar magnitude and Bz becomes the largest.

The current densities in the surfaces of the flux screen

were not measured but their values at each node within it were

nevertheless calculated. Since the radius of the flux screen

was large in comparison to its radial thickness, the peripheral

component of current density (J15 ) was considerably larger than

that acting in the radial direction (Jr). A plot of J against

radius is contained in Fig 7.14 for three values of axial distance.

As expected, the maximum J values occurred at the axially 0

directed edges of the screen. It was surprising to find that away

from the corners the magnitude of J was almost invariant with

aAkal distance. The associated phase change was, however,

extremely rapid. The radial, current density plots given in Fig 7.16

were those expected. Note that the modulus of Jr is largest on

the end door surface of the flux screen, virtually zero in the

middle, and large on the side nearest to the stator core end.

The phase of Jr varies smoothly as the screen is traversed in the

axial direction and passes through 180° near its centre.

The plot of Fig 7.15was drawn in order to determine whether

or not the axial flux penetration into the. screen, measured

218.

radially from the axially directed screen edge nearest the

machine axis, was negligible. It was found that the axial flux

density at the axial centre of the flux screen was surprisingly

large. For example, at a distance equal to eight effective depths

of penetration radially in from the edge, Bz had decreased to

only 15': of its surface value. This result puts in doubt the

validity of flux screen models based on the surface impedance

concept since these models only allow B to be parallel to the

Conductor surfaces. A detailed investigation into the

characteristics of flux screens would seem to be required; though

this should be done as a separate exercise and under controlled

conditions.

The total power loss in the flux screen was calculated and

found to be 16.1 KW. This result is in poor agreement with the

measured values which are in the range 40 — 50 KW. There are

several probable reasons for the apparent inaccuracy of the

calculations. First, the power loss was determined using the

relation

P = f [112 /°-] dv 7.23

whose accuracy depends on the accurate determination of the

square of a derived quantity (J). It was known that the finite

difference accuracy of the magnetic scalar potential solution was

poor, and therefore large inaccuracies in P were to be expected.

Second, the power loss in the screen was determined

experimentally by measuring the input and output temperatures of

the cooling water used. The temperature differences were small

and difficult to measure. Furthermore, the technique is only

reliable when there is little heat exchange between the screen

and the hydrogen surrounding it. The screen must also be a

0.2 0.4

Origin 4.76 cm from stator...core end. is measured value. .

i 1.565 cm balance ring

219.

4

200 B mT

Magnitude Of B As A Function Of Axial Distance Along Line A-A. Dungeness `B ) Under Open Circuit Conditions. Fig 7.17

220.

temperature equipotential and steady state heat flow conditions

must be established before readings are taken. Finally, we note

that the flux screen is not homogenious but contains a large

number of holes used as either bolt anchorages or gas ducts.

These holes reduce the effective conductivity of the screen. The

reduction is,however, insufficient to account for more than a few

per cent of the power loss discrepancies.

Due to space restrictions, no coil groups were placed close

to the stator core end. However, five large, flat search coils

(numbered 1 to 5 in Fig 7.9 ) were placed in this region such

that they linked axial flux only. The calculated flux densities

were several times lower in value than those measured by the

search cond. This was expected since the numerical formulation

replaced slotted structures by smooth iron surfaces and search

coils 1 to 5 were on the axis side of the core end plate, not far

from the stator slots.

7.10.2 Open Circuit Stator Conditions

The results obtained when the machine was operated on open

circuit with the excitation adjusted to give rated stator line .

voltage were rather less interesting than those obtained for

symmetric short circuit conditions. The flux density values near

the screen were negligibly small as were the flux screen power

losses. Fig 7. 17 contains the measured and calculated flux

densities for the row of coil group positions marked A — A in

Fig 7.9 . The agreement between the two sets of results was the

best achieved. As might be expected, Br dominates B8 and Bz near

the air gap but Bz is predominant slightly to the air gap side

of the balance ring.

221.

7.11 Conclusions

In general, the numerical results here presented agree well

with those measured. Improvements are possible, particularly with

regard to the prediction of localised effects and to the correct

determination of flux screen power losses. The major points to

arise from the turbo—generator end field work are considered in

the following sub—sections.

7.11.1 Determination Of The Short Circuit Air Gap 1T1F

The short circuit results, and particularly Fig 7.11, show

that in regions near the air gap the radial component of flux

density (Br) is predicted incorrectly. Br yields a direct

indication of the EMF across the air gap. Thus the wrong /IMF

value seems to have been used in the numerical solution. Several

factors could account for this. First we note that since the air

gap KMF is approximately equal to 2.2% of that due to the stator

acting alone, correct measurement of the rotor and stator currents

is essential. Second, the harmonic analysis program used to

determine the fundamental of the rotor !IMF is accurate to only

+ — 2b. A better algorithm must thus be employed if a confident

prediction of the air gap MJ.F is to be made.

7.11.2 Representation Of Non—Farmetic Flux Screens

The flux densities measured in the vicinity of the flux

screen compared favourably with those calculated. It was therefore

surprising that the predicted total power loss in the screen was

low by atkctor of three. The coarse finite difference mesh used

must account for some (if not all) of this discrepancy.

Unfortunately, substantial reductions of the mesh branch lengths

222.

causes not only computer core store problems, but also increases

the number of iterations required for solution. An alternative

is to develop a sufficiently accurate surface impedance model for

the screen. This is comPutationally attractive since the accuracy

of the surface impedance model is independent of mesh size

considerations. However, if the approach is to be valuable it

must be thoroughly tested and this is best done by restricting

attention to a geometrically simple solution domain. An

accurate finite difference solution can then be obtained and

compared with the field solution corresponding to any given

surface impedance model.

Accuracy of the numerical solution can also be improved by

including the third (axial) component of current density. The

electric vector potential then has non—zero radial and axial

components. A solution of this type was attempted but numerical

instability occurred. Investigations are in progress to determine

the reasons for this behaviour.

The computer program did not include the effects of the

(non—magnetic) stator core end plate to which the flux screen

was attached. This plate was approximately one effective depth

of penetration thick and so could have had a considerable effect

-on the flux entering the core end side of the screen. The end

plate can easily be modelled in a numerical solution by assuming

it to contain only peripheral and radial components of current

density. One non—zero component of T is then required.

7.11.3 Representation Of flagnetic Saturation

The open circuit computed results are more accurate than

those obtained for short circuit conditions, even though both

223

rotor and stator are highly saturated when the machine is run

on open circuit and almost unsaturated when it is run on short

circuit. In the overall sense, we may thus justifiably neglect

the effects of magnetic non—linearity.Mithin specific sub-

regions, however, they may have to be represented. For example,

we note that poor numerical results were obtained near the

stator slots. Here the discrepancy was partly due to the crude

current sheet modelling of the stator conductors ana partly due

to the neglect of slots in the numerical formulation, but

saturation phenomena must have played an important role. Koreover,

if the total axial flux entering the stator core end is to be

predicted correctly then a good model of saturated conducting

laminations is necessary. A suitable non—linear surface impedance

model is considered in Chapter 8 with regard to an E—core plate

levitator. However, no experimental results are yet available

to check its validity.

224.

8. SIMPLE REPRE3EUTATIONS OF I:_AGNETICALLY NCO—LINEAR

NATERIALS CARRYING EMY CURRENTS

In the work thus far described it was assumed that all iron

parts were of infinite permeability and resistivity. This

assumption allowed us to obtain at low cost numerical results

of good accuracy when the effects of eddy currents induced in

magnetically linear conductors such as copper and aluminium were

investigated. However, in some machines the eddy currents

induced in iron structures cannot be ignored. For example, it

is probable that linear motors employed in traction applications

will have solid iron secondaries constructed of boiler plate,

since this material is cheap to produce in large quantities.

Here the thrust of the machine is totally dependent on the ability

of the primary to induce ed.dy currents in the magnetic secondary.

Thus, these currents cannot be ignored. In other machines the

representation of eddy current phenomena in saturated materials

is not as crIcial, but can nevertheless he important. For

example, the effects of eddy currents induced in turbo—generator

stator core ends and teeth by axial flux are undergoing detailed

60,61 investigation since the heat which these currents generate can

severely shorten machine lifetimes. Moreover, the axial core

flux creates high inter .L.laminar voltages which can cause

breakdown of the insulation between adjoining

laminations.

The infinite permeability requirement can be removed

relatively easily. If, however, the permeability is assumed

to be excitation dependent, travelling wave formulations cannot

be employed since these require the permeability to be invariant

225.

with distance measured in the direction of the wave.

tagnetostatic solutions using the static magnetisation curve

for the material are commonplace.65

It is normal practice to

neglect hysteresis effects which are probably negligible at

power frequencies and at the magnetic flux density levels

encountered in the iron parts of large electrical machines. If

the excitation is sinusoidal and the resistivity of the iron

can be assumed infinite, the eddy currents induced within a

solution domain may be computed using phasor methods and the time

variable may thus be eliminated from the governing equations.

If eddy currents are allowed to flow in magnetic parts then

waveform distortion is produced and it is necessary to obtain the

electromagnetic field distribution by solving governing

equations containing both time and space as variables. This is a

60,61 task that few workers have attempted, except by restricting

66 the number of'space dimensions to one. In two dimensional

asymmetric problems it has been found that laminated and solid

iron behave quite differently from one another when the same 17

excitation conditions are applied to each. Several different

modes of flux penetration into both these types of material have 16

been recognised and their implications explored..

The author wished to examine the effects of saturation on

the magnetic fields outside iron regions and was less interested

in the specific field interactions within them. Furthermore, the

time available for investigations was too short to allow attempts

at solving two dimensional magnetic field problems in both time

and space. Instead of attempting solutions of the time

dependent equations it was decided to examine simple representations

of solid magnetic materials carrying eddy currents which would

226,

allow phasor methods to be used for solution purposes. It was

recognised at the outset that these representations would be

inaccurate but it was hoped that a good qualitative indication

of the effects of conducting iron would be obtained. For

simplicity, attention was restricted to the behaviour of solid

iron at power frequency (50 Hz).

8.1 The Constant Permeability Approximation

Instead of using a non—linear characteristic to relate B

and H within a magnetic material it is sometimes assumed that the

relative permeability ( of the iron is both space and time

independent. A value of ti r , usually within the bounds of 10 and

100, is set and a magnetic field solution is obtained using

phasors. From a physical point of view this technique is difficult

to justify since the surface layer of the iron will be highly

saturated, and yet unsaturated material will exist only a few

millimetres below the surface. Furthermore, the choice of tir is

difficult since the best value is heavily dependent on the

magnitude of the excitation used.

Numerical computations using the constant permeability

approximation can require many nodes to achieve reasonable

finite difference accuracy because the conductivity of iron is

high and even when low values of 111, are employed, the effective

depth of penetration is very small. Iterative convergence is 67, 68

always slow and use must be made of block techniques if

solutions are to be obtained cheaply.

The required regular rectangular mesh magnetic scalar

potential finite difference equations are contained in Chapters

2 and 3, and their derivations will not be repeated here. It is

227.

found that the magnetic fielas vary rapidly within, and near the

surface of, conducting iron and consequently fine meshes must be

used. Usually one employs a coarse mesh in regions diStant from

the iron and a special finite difference equation, which we shall

here derive, is necessary to match up the two mesh types. The

matching is normally done in the insulating regions where T is

zero. Consequently, only need be calculated. Consider the

three dimensional computation molecule of Fig 8.1 and let the mesh

interval between any two nodes,i and j say, connected by a single

brench be h... Then midway between nodes i and j the Taylor ij

series method yields

a Jt j _ + 0(max hi J )2 8.1

-671. h. ij

where n is the direction j i and movement from j to i is in a

positive coordinate direction. To obtain an approximation to

the second derivative of A with respect to x, say, we expand

the first x derivative about node 0 using a Taylor series. Thus

we find that

a2dun

al_ 2 [ 1 - 9'0 - n3) ax2 h

01+ h

03 h01

h03

8.2

Similar expressions may be formed to approximate the other two

second derivatives of A. Addition of the three expressions

1 + 1

+

1 + 1

+

1 + 1

2 hOl h03

[

h05 h06 h02 h04

h01

+h03

h05+h06

h02

+h04

S.0 = 0 8.3

228.

yields the following finite difference equivalent to Laplace's

equation

[in' 1 3 2 [613 SI 6 2

hOl h03 h01+h03 h

05 h06 h051-1106

[St 2 + 614 ] 2 h02

h04

h02

+h04

IMP

Note that two dimensional forms of Laplace's equation may be

approximated by omitting appropriate terms from 8.3 .

8.2 Surface Impedance Methods

Since the effective depth of penetration of solid iron is

usually much less than its dimensions, little inaccuracy is

introduced (except near corners) by assuming that within iron

the electric and magnetic field strength vectors are both

unidirectional and parallel with respect to the surfaces.

Numerical solutions in time and one space dimension may then

be employed to determine the field distribution and power losses. -

These solutions require a knowledge of the magnitude and

wave shape of either E or H at the surface of the iron. For

simplicity, a sinusoidal waveshape is usually chosen. When this

is done the power and reactive volt—amp absorbtion may be

considered as taking place in a complex, but non—linear, surface

impedance defined by 69

Z = E / H 8.4

where E and V are the phasor representations of the time

fundamentals of E and H at the surface. This impedance may be

229.

employed as a lumped parameter which describes the effects of

the iron on media external to it. Computations have been

69 performed to determine the variation of Z with magnitude of

surface excitation. For the purposes of these computations, it

was assumed that the alternating current magnetisation

characteristic of the iron was the same as the static

magnetisation curve represented by the Frohlich equation

B = H 8.5

a ÷ bull

where a and b are material, but not excitation, dependent. It

is found that the value of Z is weakly dependent on the magnitude

of the surface excitation once saturation has occurred in the

surface of the iron. However, its value is subject to a 2 r 1

magnitude variation when the surface conditions are changed

between the two surface waveshape limits of sinusoidal E and

sinusoidal H. Unfortunately, the waveshapes of the field vectors

acting at the surfaces of iron in whose vicinity a magnetic

field solution is required are generally unknown beforehand.

Within alternating current machines having low values of armature

resistance, the total flux is sinusoidal in time. loreover, if

the air gap dominates the magnetic circuit, each flux filament

that links the iron will also be sinusoidal and it is then

appropriate to choose the surface impedance characteristic

corresponding to sinusoidal surface E. Although leakage flux

paths are usually dominated by air regions, this is not often

true of the magnetising flux since most machines are designed

to have as large a magnetising reactance as possible.

Inaccur acies introduced when surface impedances are used

to represent the effects of solid iron on the time fundamentals

230.

of the external magnetic field quantities occur for the

following reasons.

1) It has already been mentioned that the actual waveshape of

E and H at the surface of a given sample of iron is known only

when appropriate experimental measurements have been completed.

For computation purposes, a specific waveshape must be assumed

and some inaccuracy is bound to result from the choice made.

2) The surface impedance method is not valid in the vicinity of

sharp corners; yet such corners often have a considerable effect

on the overall magnetic field distribution.

3) It is usually more convenient to determine the variation of Z

with surface excitation by finite difference solution rather

than experimental observation. Each solution has an associated

70 error, but Lim and Hammond have shown this to be small when the

magnetic properties of the iron are well modelled and its

conductivity is known.

Let us assume that we require a numerical solution for the

magnetic scalar potential in the vicinity of some solid iron

which is to be modelled using surface impedances. Then it is

necessary to incorporate these impedances into the formulation so

that the distribution of St along the surfaces of the iron can

be determined. Consider the two dimensional arrangement of Fig 8.2a

which represents the lower surface of. a rectangular iron block.

In order to find A at the surface we first consider the regular

mesh finite difference model of Fig 8.2b. In this model T is

assumed zero and the branch elements have the dimensions of

reluctance. Application of Kirchhoff's Current Law to node 0

yields the standard five point finite difference approximation to

Laplace's equation. Now let node 0 be placed at the surface of th

(b) Fig 8.2 (e) (a)

solid iron s03 t h

s01

231,

conducting iron shown in Fig 8.2a. The reluctances of the air

paths between nodes 0, 1 and 0, 3 are thereby doubled to 2N40.

In order to represent the effect of iron on the magnetic scalar

potential distribution we introduce a new equivalent circuit

component, S, which has the dimensions of reluctance and is, in

general, a complex number. If we assume S.. to be the value of ij

S midway between nodes i and_ j, then the iron surface magnetic

equivalent circuit is that of Fig 8.2c.For this circuit to be

useful, a functional relationship between S and the complex

surface impedance, Z, must be found.

The values of Z and S are related in a fairly simple way.

Since Ey and 171

x are the only non—zero components of 2 and

If at the section of surface considered

, Z = Ey / Hx

At any given point on the surface

E = jw0 8.7

where w is the angular frequency of the time fundamental of S/

and 0 ^' is the total flux in the surface layer at the point

considered. Thus, midway between nodes 1 and 0

8.6

ZO1 = • IV —3110 8.8

(4.11 —1)/Nh

232. so that

';16 S110- al 8.9

(wNhijZol)

Now the denominator of 8.9 has the dimensions of reluctance

and must therefore be equal to Sol. ie

801 = -jwNh 8.10

451 1(0.5+ j Z01) + 413(0.5+ j Zo3) m2,51

wiloh coph 4

(1 + 112 + j (Z01 Z03))61 0 = 0 8.11

Woh

In a similar manner, the finite difference expressions for St

at points along the z directed surfaces of the rectangular iron

block may be obtained. When point iterative methods are used to

obtain solutions of the resulting simultaneous equations, the

numerical experiments detailed in the next section show

convergence to be relatively rapid. This result is surprising

since the formulation is non-linear in the sense that the Z

values are excitation dependent and linear scaling cannot be

.used to derive the St values at one excitation level from those

obtained at another.

If the magnetic scalar potential distribution within a

region containing solid iron is to be obtained, the computer

program must be supplied with a characteristic giving Z as a

function of surface excitation. This may take the form of an

analytic expression. Alternatively, a set of points may be

stored and interpolation procedures adopted to obtain intermediate

ZO1 Application of Kirchhoff's current law to node 0 of Fig 8.2c

yields the following equation for SI, at a typical point on the

surface:

org tZsurff

.40.0

12surf x10-4 ohms

14.0

233•

300

20.0

10.0

NN magnitude b

j I •

a = rectangular curvc b = Frohlich curve

Elsa! =1.7 85 T a- = 0.694x 107 5/m yi =2.0 x10 3 Wrn

05 1.0 fundarr.vi lel component E

V/m peak

Figure 6. Surface impedance c5ainst E

for sinucnirfal surface E

Fig 8.3

2.0

1.-- Initialisation procedure

iFind %A at nodes not on the

surfaces of any solid iron.

1

234.

N =1

Find A at the surfaces of any solid iron.

Determine the surface E values

using ecuations (8.7) & (8.9).

Find the Z values using

equations (8.12) & (8.13).

No —N=N-a—cd--- Does N=3 ?

Yes

(arbitrary criterion)

No Is the number of iterations

sufficient ?

Yes Termination procedure

STOP

Fig 8.4

235.

values. For simplicity the author decided to assume that both

the magnitude and phase of Z were linear functions of the

fundamental of electric field strength acting at the surface.

Thus

Izi = a 141 b 8.12

c 1,41 + d 8.13

Constants a and c are always negative, whilst b and d are 69

always positive. Fig 6 of the author's paper is reproduced as

Fig 8.3 here so that the nature of a typical Z —11-11 f I curve may

—

be readily appreciated. It will be noted that for lEfl 0.2V

the curve may be justifiably represented by a straight line

characteristic. At lower excitation Z is given incorrectly by a

linear approximation, but this is probably unimportant since one

usually requires good accuracy only in regions where the electric

and magnetic field strengths are high. A computer program was

written that would solve two dimensional magnetic field problems

involving solid iron represented by non—linear surface impedances.

The flow chart of the program, as it relates to calculation of

the Z values using equations 8.12 and 8.13 , is given in

Fig 8.4 .

8.3 Application Of Surface Impedance Eethods To An E—Core

Plate Levitator Problem

A problem of interest was that of calculating the electro-

magnetic fields in an E—core levitator fitted with a solid iron

secondary. The machine considered was that of Fig6.8 fitted with

a 0.6cm. thick conducting magnetic secondary of equal width and

length to the primary. The air gap was assumed constant at lcm

and a lateral displacement of 1.8cm was allowed between the

236. primary and secondary.

On the grounds of simplicity and accuracy, it was decided to use

a non—linear surface impedance representation of the solid iron

rather than the constant permeability ap-)roximation. The values

of the constants associated with equations 8.12 and 8.13

were calculated using Fig 8.3 and are given in Fig 8.5 . Note

that they are expressed in terms of the peak fundamental of E

divided by 42. Sinusoidal current was assumed to flow in the

primary winding and the MMF was set to 2000 Arms Turns, this being

the maximum that the machine could stand without seriously

overheating.

Initially a solution was sought assuming the secondary - to

be non—conducting. A difficulty was then experienced since the

value of SI. on the secondary was unknown. The value had to be constant but was not zero since this would have implied the

existence of a magnetic short circuit between the outer teeth of

the primary and the secondary. Now it is known that for every

transverse section of the machine (we neglect end effects here)

the total flux entering the secondary in the plane of the section

must be zero. In mathematical terms, for every closed contour

c surrounding, but not cutting, the secondary

J75Bn dl = 0 8.14

where Bn is the component of flux density normal to contour c

and confined to the section of interest. If c is in air and T is

zero along its whole length then 8.14 may be expressed in terms

of it as

as dl = o 8.15 c an

For this application c was chosen to be a rectangle whose sides

c>-

161 = -7.55x10-5 + 2.7x104

.ez = -7.071EFI + 33

3w

Magnetic Scalar Potential Plot - E-core Machine With Magnetic Non-conducting Secondary.

Tire Instant When The Current In The Winding Is A Maximum. Fig 8.6.

Sinusoidal current in primary winding. MMF = 2000 ArmsTurns. Six nodes in picture frame. t talt P01144151 11415 [til Malt

Lateral displacement of magnetic secondary = 1.8 cm.

238.

bisected the finite difference mesh branches joining the

surface of the secondary to the external mesh. Condition 8.15

then required that Nk

N

N2Dtk EAMI LJ 0 Mal mitt

where

1. A0 was the value of A at the surfaces of the secondary.

2. N was the ratio of horizontal to vertical mesh interval.

3. Ak was atypical value of a at a node attached to the horizontal sides of contour c by a single mesh branch.

4. SIAm was a typical value of Si' at a node attached to the

vertical sides of contour c by a single mesh branch.

When the mesh is square (N = 1) equation 8.16 simply states

that the value of A at the surface of the secondary is the average of the St values at nodes connected to it by a single

mesh branch.

A magnetic scalar potential solution was obtained in 292

iterations after which the modulus of the maximum residual was

5 x 10-5 of the maximum scalar potential. Six nodes were placed

in the picture frame and an acceleration factor of 1.6 was

employed for the point successive overelaxation. A scalar

potential plot for the time instant when the winding current is

a maximum is contained in Fig 8.6 . If the secondary is removed

the plot of Fig 8.5 is obtained at the same time instant. As

might be expected, the non—conducting magnetic secondary creates

an intense magnetic field between itself and the teeth of the

primary. There is very little flux outside the air gap and

everywhere within it the normal (w) component of flux density is

dominant.

+ See page 158

8.16 2N k + Nm )

3w

Magnetic Scalar Potential Plot - E-coro Machine With Magnetic Conducting Secondary.

Time Instant When The Current In The Winding Is A Maximum. Fig 8.7

Surface impedance model used for the air gap surface of the secondary. At other Nagnetic

Mint e sett M 111.21 m. surfaces the permeability is assumed to be infinite. OM

Magnetic Scalar Potential Plot - E-core Machine With Magnetic Conducting Secondary.

Time Instant When The Current In The Winding In Zero. Fig 8.8

Surface impedance model used for the air gap surface of the secondary. At other magnetic

row c =et roma /I-20 ticia surfaces the permeability in assumed to be infinite. =

• U

Magnetic Scalar Potential Plot - E-core Machine With Magnetic Conducting Secondary.

Time Instant When The Current In The Winding Is A Maximum. Fig 8.9

Surface impedance model used for all surfaces of the secondary.

MUM 141 MN MATZ

Magnetic Scalar Potential Plot - E-core Machine With Magnetic Conducting Secondary.

Time Instant When The Current In The Winding 18 Zero. Fig 8.10

Surface impedance model used for all surfaces of the secondary.

E. CORE MUM, 2.21 Iwo rune

21&3.

Surface impedances were introduced along the horizontal

surface of the secondary nearest the primary and once again a

solution was obtained. The rate of convergence was less than that

achieved for she magnetic non—conducting secondary. It was

reached in 379 iterations using an acceleration factor of 1.4.

Larger values caused numerical instability. The scalar potential

solutions are characterised by the plots of Figs 8.7 and 8.8

obtained at two different time instants. When the excitation is

at its maximum value the vast majority of the flux is contained

within the air gap. Yet when the excitation is zero a relatively

high proportion is found to leak outside. In Fig 8.7 the largest

nodal value of SI► on the secondary was 1190, whereas under the

conditions of Fig 8.8 the corresponding value was -359. The

quadrature field produced by the eddy currents was thus

comparatively small.

Finally, surface impedances were placed on all the surfaces

of the secondary and a numerical solution was obtained.

Convergence occurred in 343 iterations using an acceleration

factor of 1.6 and the convergence criterion given previously.

The magnetic scalar potential distributions were those of Figs 8.9

and 8.10 at the time instants when the excitation was maximum

and zero respectively. The largest values of A occurring on

the surface of the secondary at these respective instants were

1242 and.-277. Figs 8.9 and 8.10 are somewhat different from

Figs 8.7 and 8.8 and it is uncertain which pair is more nearly

correct. It was hoped that experimental measurements would be

available to clarify points such as this and to allow an

objective assessment of the accuracy of the surface impedance

model. Due to lack of time, these measurements were not made.

244

8.4 Calculation Of Forces Acting On Tlagnetic Parts

In• previous work the forces acting on a conductor were

calculated by making use of the relation

F = RefJ x B d7 8.17

in which J and B are phasors and the asterisk denotes complex

conjugate. This equation yields the forces due to the currents

but not those due to reluctance effects. One way in which the

total force acting on a magnetic secondary can be obtained is by

using the Maxwell Stress technique which will here be described.

It can be proved 24 that the force acting on a material may be

obtained by integrating a stress vector having the following

components

Fn = Re 01 (HnHn — HtlHt1

— Ht Ht* )1 2 2 2

Ft1 = Re itio HnHt*

1 8.19

Ft = Re (110 HnHt* 8.20 2 2

over any surface which totally encloses that material. The n

direction is assumed to he normal to the surface of integration

chosen and t1, t2

and n are assumed to form a right handed local

coordinate set whose origin lies on the surface.

When regular rectangular finite difference meshes are

employed to obtain the nodal magnetic scalar potential values it

is normal practice to employ an integration surface of rectangular

cross section. For a two dimensional mesh arrangement, this

surface becomes a rectangle in the plane of solution and the

forces calculated are those per metre length of machine measured

at right angles to the plane.

In the interests of accuracy, the magnetic field strength

values used to calculate the local stresses Fn'

Ft

and Ft 2 1

8.18

4-- N h C.

integration contour

Fig 8.11

245. 25

should be determined at identical points in space. This can be

difficult in finite difference terms since H is given by a

derivative of t11). The inconvenience is minimised by choosing

an integration surface that bisects mesh branches. For example,

consider the node arrangement of rig 8.11. Let us assume that

y(s3,--cfrx

there is a sinusoidal travelling wave in the positive y direction

and that the electric vector potential is zero within the given

mesh rectangle. Then at the centre of the rectangle (point 0) SZ

is given by the average of the four nearest nodal J. values and

= aa, = j 1.10 = j 1 (Ai+ iki-J13+44) ay p 17-T.

8.21

where p is the pole pitch of the y directed wave. The values of

Hx and H

z at point 0 may be obtained by considering the local

directions a and p of Fig 8.11 .Since

= 8.22

q aq

where Hq is the component of H acting in the q direction, it

follows that

Ha = A4 — A2 and It = 3 —

8.23 h 1

h N + 1

The x and z components of H are found by resolving Ha and H o

along the appropriate directions. Thus

Hx

= Ha cose H cos 6 8.24

246.

H = Ha sine — HP sine

where 0 = tan1 (1/21)

In two dimensions the surface integration reduces to a line

integral whose evaluation is easily accomplished by assuming that

H varies linearly with distance measured along the integration

contour and within each mesh rectangle. Thus, the contribution to

the total force of the contour section lying within the mesh of

Fig 8.11may be obtained by evaluating F Ft and Ft at point 0 Fn , t1 2

and multiplying by the distance Nh.

The i:axwell Stress technique was applied to the E—core

levitator problem discussed in the previous section. Initially

it was decided to use an integration contour which was lmm below

the secondary and which extended in the horizontal direction so

as to close at infinity. It was hoped that the force contribution

from the sections of contour lying in the picture frame would be

negligible, but in the event these contributions were found to

be of great importance. When the stresses were plotted as a

function of horizontal. distance along the contour it was found

that they took large positive and large negative values.

Furthermore, the net force was the result of summing two like

numbers having opposite signs. In order to assessthe effect of

choosing different contours of integration, the forces were

calculated using two,further horizontal contours, one 3 and one

5mm below the secondary. The results were surprising. It was

found that the stabilising force was independent of contour

position but that the normal force ( the larger of the two) was

not. Its value declined by 15% when the contour was moved from

the lmm to 5mm positions. These results prompted an investigation

which is still in progress.

8.25 8.26

347,

It is probable that the most accurate results were obtained when

the contour was placed 5mm below the secondary since it was then

midway between the primary and secondary members and some distance

from their corners.

248.

REFERENCES

249.

REFERENCES

1. CARPENTER C.J.,'Theory and application of magnetic shells',

Proc.I.E.E.,Vol.114,No.7,1967,pp995-1000.

2. SIMKIN J. & TRO.1BRIDGE C.W.,'Magnetostatic fields computed

using an integral equation derived from Green's Theorms',

Compumag Proceedings, Rutherford Laboratory,1976,PP5-14.

3. ARMSTRONG A.G.A.M.,COLLIE C.J.,DISFRENS N.J.,NEWMAN M.J.,

SIMKIN J. & TROWBRIDGE C.W.,'New developments in the magnet

design program GFUN3D',Fifth Int.Conf.on Magnet Technology,

Rome,1975,pp168-182.

4-6 As references 19-21 respectively.

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22. HANALLA A.Y. & MACDONALD D.C.,'Numerical analysis of transient

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25. CARPENTER C.J.,'Numerical solution of magnetic fields in the

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31. VARGA R.S.,ibid p39 Definition 2.3 & pp35-38.

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36. STOLL R.L.,'Solution of linear steady-state eddy-current

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53. CARPENTER C.J. & LOCKE D.H.,'Numerical models of three

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of turbine generators',I.E.E.E. Power Eng.Soc.Conf.Paper

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components',Compumag Proceedings, Rutherford Laboratory, 1976,

pp213-220.

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turbo-generators',Proc.I.E.E.,Vol.108 Part A,1961.

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1965,p81.

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the electromagnetic field in saturated steel plates',

Proc.I.E.E.,V01.119,1972,pp1667-1674.

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solution of linear and non-linear vector field problems',

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apparatus',I.E.E.E. Transactions on P.A.S.,Vol.93,No.3,

PP991-1002.

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losses in solid iron under various surface conditions',

Compumag Proceedings, Rutherford Laboratory, 1976,

pp269-276.

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255•

APPENDIX

256.

APPINDIX 1

Al STCTION

Eaur.tion 3.94. Discretizati on For Si,

dl dl,l + d2 ,rt,2 + d3!,3 + d4 s1,4 + eiTxi + e2Tx2 + e3Tx3 + e4Tx4

+ f1Tzl + fTz2 + f3Tz3 + f4Tz4 0

dl (bil 123 134) (Z3

Z1 ) (C li 123 C11134) (X3 x1)

- A 163 ( 7-1p )2(al blx6 clz6 )1123 -

135 ( .5.13 )2(a1 + bix5 elz5 )1 134

= b21123(z3 z1) c21123(x3 —x1) —

A163 (

) p)2 (a2 + b2x6 + c_z.4o 1 ) 123

d3 = (b31123 1'31134) (z3 — ) (°31123 C 31134) (x3 —

163 (5p) 2(a3 + -xo b3,, c3z6 )1123 -

A,35 (714) 2(a3 + b3x5 .+ c3z5 )1134

d4 —1)41134( z3 zl ) c41134(x3 xl )

46135 (7-P2) 2(a4 134x5 c4z5 )1 134

el = A163b11123 A135b11134

e — 2 A 163b21123

257.

e3 = — A163 b31123 — A135 1'31134

e4 = — A135 b41134

fl = A163 c11123 — A135 °11134

f2 = 163 c11123

f3 = — A163 031123 — A135 031134

f4 = A 135 041134

NB a31123 indicates the value of a at node 3 calculated for

the triangle having vertices 1,2 and 3.

258.

A2 SECTION

Equation 3.95. Discretization For Tx

(1141 + A2 A2 + d3 S1,3 + d444 + e1

Tx1 + e2Tx2 + e3Tx3 + e4Tx4

flTzl f2Tz2 f3Tz3 f4Tz4 = °

dl = A163j/d2 b11123 A135j/a2 b11134

d2 = A163j/a2 b2I123

d3 = A163j/d2 b3I123 L1135j/

d2 b3I134

d4 = A135j/d2 b41134

(7c ) 2\ el = (°11123 011134) (x3 — x1) A163(j/a2 /

) (al + bix6 + ozz. 0'1123 — A135(j/d2 E (p) 2)

(al b1x5 clz5)1134

(%) 2\ e2 = (021123) (x3 — x1) — 163(j/ d2 + ‘—' I P

(a2 + b2x6 + c2z6)1 123

e3 = (e31123 — 031134) (x3 — xl) A163(j/d2 )

(a3 + b3x6 + c3z6)1 123 — A135(j/d2 + (;)2 )

(a3 + b3x5 + c,z-)1 ) 134

ONO zi)

259.

e4 = c4I134(x3 xl) L135(j/d2

(a4 + b

4x5 +c 5) 1134

f1 = - (c11123

f 2 = - c 2023 ( •z3

c11134) (z3 "zi)

z]. )

f3.—

(e31123 e31134) (z3

zi )

f4 = C4I134 (z3

(E) 2 p

260.

A3 SECTION

Equation 3.96. Discretization For Tz

d +da + +da +eT + +eT -1-e 1 1 2 2 3 3 4 4 1 xl 2 x2 3 x3 4T x4

+ f1Tz1 4- f2Tz2 + f3Tz3 + f4Tz4 = 0

dl = 1 A163j/d2 011123 A135j/d2 011134

d2 = A 2 L-4163i/a2 e21123

A1633/d2 031123 A135j/ 2 031134

a4 = A135 j/d2 c41134

el = (1'11123 b• 11134) (13 x

• l

)

e2 = b21123 (x3 - x1)

e3 = (

• b

31123 b• 3I134) (x3 - x

• l

)

e4 =

• b

4I134 (x3 xl) •

f l = (b11123 b• 11134) (Z3 - z

• 1

) A163( j/a2 (7P2 )

(al blx6 clz6) 1 .123 - 6135( j/d2 (1/i)2 )

(al blx5 e1z))1134

261.

f2 ' 1321123 (z3 - z1) - A 163( j1d2 + (11i)2 )

(a2 ÷ b2x6 ÷ c

•

2z6 ) J123

f3 = (b31123 - b31134) (z3 - zi) - 0163( i/a2+ (;)2 )

(a3 + x0 b-5,,.

4- c

•

3z6 )1123 - A 135( i/a2 ÷ (;)2 )

(a3 + b3x5 + °

•

3z5 ) I 134

f - - b31i 4 - b31134 (z3 - z1) - A1350/d 2 + (VP )

(a4 ÷ b4z5 + c

•

4z5)1134

262.

PUBLICATIONS SUBMITTED IN SUPPORT OF THESIS

EFFICIENCY OF NUMERICAL TECHNIQUES FOR COMPUTING EDDY CURRENTS IN TWO AND THREE DIMENSIONS

C.J. Carpenter* and E.A. Wyatt*

ABSTRACT

The paper reports recent work on numerical techniques for solving eddy

current problems in terms of a scalar, instead of the conventional vector,

magnetic potential function. A method described previously for thin

plates has been extended to large-section conductors, and gives high

convergence rates when solved by iterative methods provided that

suitable numerical techniques are used. These techniques are described

in the paper. The method is particularly well suitedto three-

dimensional problems, but it also gives better convergence than the

conventional method of calculation in many two-dimensional problems of

practical interest, particularly in electrical machines.

PRINCIPLE SYMBOLS:

B magnetic flux density T electric vector potential(equ66)

d depth of penetration (equ.10)

x solution vector E electric field intensity x,y,z. coordinates f column vector of constants a travelling-wave term (equ.22) H magnetic field intensity

d constant given by equ.23

h mesh interval

u permeability J current density a conductivity L coefficient matrix T time N mesh element ratio (fig.l)

O magnetic scalar potential

p pole pitch

m angular frequency

* Electrical Engineering Department, Imperial College, London SW7 2AZ.

1) Introduction

Eddy currents induced by leakage fluxes are of increasing import-

ance in large electromagnetic devices such as turbo-generators, trans-

formers, and the like, where increasing sizes and ratings may lead to

severe heating problems. Eddy currents are induced in all conducting

parts in the vicinity of the windings, particularly in the end-regions;

and the end-surfaces of the core commonly need some form of screening.

The eddy current and loss densities are difficult to predict because the

problem is three-dimensional and the iron parts may be laminated and

are liable to severe saturation. The magnetic vector potential, like

the field vectors, has three components and, in general, it has to be

supplemented by an electrostatic scalar potential, so that field

calculations in these terms become formidable.

The obvious advantages of a magnetic scalar-potential formulation

in magnetostatic problems, some aspects of which are discussed in a

companion paperi, assume an even greater significance in computing

eddy currents. It is necessary to compute only one function outside

the conductors, and excellent convergence is normally obtained when

the differential equation is solved iteratively. A supplementary

current-flow function has to be used inside the conducting regions, and

this is, in general, a vector quantity, but it can usually be limited

to one component, and it has been found to be well-behaved numerically

when applied to problems in which the current flow is confined to thin

sheets2. This formulation can be applied to conductors of large cross-

section either by assuming continuous conducting properties, by

replacing them by a stack of thin plates, or using an "onion skin"

model consisting of a set of concentric layers. Some of the possibilit-

ies have been discussed elsewhere 1,3.

In the type of problem considered here the more important conduct-

ors consist of laminated iron cores, or non-magnetic plates whose

thickness is limited (although it may often be substantially greater

than the depth of penetration at the working frequency). Under these

conditions the current density component normal to the major conductor pj ON

surfaces is negligible (i.e. the z-component in fig.l), and it is \O

242

2

1. 13 0 I

4

SI

k Nh

— 73 cm —1

2.8 c m -Y.. / / / / / /

— 6.3cm —1

R2

RI

R2

convenient to assume that it is zero in all conductors, whether laminated

or not. The effect of removing this restriction will be discussed

separately.

One of the devices of interest is a linear induction motor consist-

ing of a conducting plate in an air gap between two iron surfaces, one

carrying a winding in slots. This machine has been studied partly

because the flux distributions are easily obtained experimentally. The principal parameters are given in figure 1. The winding generates a

flux wave travelling in the y direction, and gives three-dimensional

flux and current-flow patterns for which a two-dimensional description

is adequate. This simplifies experimental work and is directly

applicable to many end-field problems.1 Varying the pole-pitch in the

numerical model of the machine changes the field conditions from a rapid

variation in the y direction, at one extreme, to a simple two-dimensional

Fig. 1 Induction machine geometry.

Machine 380 mm long, pole pitch 95 mm -8 "Duralumin" plate, resistivity 3.02 x 10 Om, thickness 6.35 mm. Mesh Nh - 4.67 mm h = 2 mm in region RI

h 4.5 mm in region R2

result, at the other, when the pole-pitch is infinite. The more general

three-dimensional problem has been studied previously2 using thin plates.

Early experience3 with the scalar-potential method suggested

that it had very considerable advantages in eddy-current problems not

only in three dimensions, but also in two, because of its numerical

behaviour when solved iteratively. This has now been more closely studied

and the behaviour of different numerical formulations compared. The

object of the paper is to report some of the results of these studies,

and the preferred numerical formulation. The method is applicable to

both transient and steady-state problems, but for the present purpose

all time variations are assumed to be sinusoidal, with angular frequency ta.

2) Formulation

The problem is to solve simultaneously the magnetic field equations

curl 11 = J (1

div B = 0 (2

together with

curlE= -jw11

(3

div J = 0

(4

in which all the quantities are vectors with three space components, each

of which is complex. The usual two-dimensional method is to satisfy

equation 2 implicitly by expressing B as the curl of the magnetic vector

potential, A, giving

V2A (j/d2)(A - grad V) (5

where A is assumed to have zero divergence (Coulomb gauge). The

electrostatic potential, V, adds an electric field component which is

constant inside each conductor in a two-dimensional problem, but is not

in three dimensions, where surface charges appear at all the conductor

surfaces as a consequence of equation 4.

One alternative is to use an electric vector potential,T, defined

by

curl T = J (6

and to confine T to a simple function by placing some other constraints ri on it and allowing it to have an arbitrary divergence. As is shown in

243

reference 1, T can be restricted to one component, Tz, inside the conduct-

or, with zero value outside and a constant value in any conductor

hole, provided that J. is zero. Assuming uniform conductivity, o, equat-

ion 3 becomes

V2 ' T ■ a2Tz/3x

2 + 3

2Tzlay

2 ■ j w oBz x,y z

when expressed in terms of T in Cartesian co-ordinates. Here Tz is

operated on only in the x and y directions. From equations 1 and 6

H - T - grad

where S2 denotes a magnetic scalar potential function. It follows from

equation 2 that, in a non-magnetic conductor, 2

V SI 3Tz/az

(9

where the V2 operator refers to all three directions, in the usual way.

Outside the conductors (or in windings in which J, and therefore T, is

specified') H can be obtained by computing 0, whilst in the eddy-

current regions both equations 7a and 9 have to be solved. The sources

of S1 - i.e. the right hand side of equation 9 - are equivalent to magnetic poles distributed through the volume of the conductor, but concentrated

primarily at the upper and lower surfaces where T is discontinuous.

Substituting from equation 8 in 7a,

Vx,y T

z - (j/d

2)(T

z - 20/az)

(7b

where d is the depth of penetration defined by

d2

- 1/6810o (10

The pair of equations, 9 and 7b have to be solved simultaneously for 0

and Tz, respectively. One is Poisson's equation, and the other a form of

the Helmholtz equation, but with a two-dimensional operator and an

additional source term.

In a travelling-wave type of solution all quantities are assumed to

vary sinusoidally with y, and with time t, so that Tz, for example,

takes the form

Tz(x'z)exp j(wr -%Y/13 + 0 )

so that the second derivative terms in y can be written

a2Tz/3Y2 ■ -(x/p)2Tz (11

and likewise for 0.

In expressing these equations in discrete form the n nd Tz

functions inside the conductors are not restricted to a common node array,

and the first-order derivatives on the right-hand sides of equations

9 and 10 suggest some advantages in computing them in two staggered

arrays. The directional properties of the various terms, together

with the discontinuities in both grad A and T at the conductor surfaces,

introduce a range of possible numerical treatments, and each of the pairs

of simultaneous equations obtained can be solved b: elimination, line,

or point iteration in various combinations.

These possibilities have been somewhat restricted by adopting a

first-order (linear) interpolation in a rectangular node array,

since this is well suited to the problem under consideration. An

investigation of staggered meshes (the"split- branch" formulation4 )

showed that these have no significant advantages in either accuracy

or numerical behaviour over a single mesh in which S1 and Tz are

computed at the same nodes.

In general the conductor surface may intersect the mesh between

nodes, but it is usually convenient to place nodes on it, as shown

in fig 1. To cOmpute Tz from equation 7 at a surface node, 0, Hz is

required at the discontinuity. From equation 8, the Hz value mid-

way between nodes 0 and 2 is

(Hz)02 ° (no - 112)/h

(12

and between 0 and 4,

(Hz)4D ° (To + T4)/2 - (00 - 04)/h (13

Here the z suffix has been dropped from Tz as it is superfluous.

The two values can be averaged, or alternatively Hz at node 0 can be

(7a

(8

244

T1+T3 + jN20(02 -n4 )/h - (2+j2N28 + a)To - 0 (24

' 0 1 +03 + N2 [n2 + 04 + (T4-T2)h/2j - (2+2N

2+ a) 00° 0

(Hz)0 - 1-(Hz)02 (Hz)40/2

(17

(25

derived from the underside values according to unit area. Alternatively, from equation 8,

(Hz)0 (Hz)40 + (3Hz/3 r)0 h/2

(14 3Hz/3z aT

z/az - 120/3,2

where so that equation 9 can be written

(pHz/az)0 °-(DH

x/3x + 3H

y/3y)

0

02 n DHz/Dz

x,y (18

since div H is zero. Hence where

(3Hz/Dz)0 ° (1211/1x

2 + 3

20 /a y

2)0

(15

and this can be expressed in terms of the nodal values of 0 in the

usual way. The required Hz can likewise be derived entirely in terms of

from values on the surface and above by substituting from equations

12 and 15 in

(H,)0 (Hz)02 (3Hz/Dz)0 h/2

(16

The results obtained from equations 14 and 16 will be identical

when the solution has converged and the continuity condition (equation 2)

has been met, and likewise adding 14 and 16 together shows that averaging

the Hz values above and below the node also gives the same final result.

But at the earlier stages of the calculation the continuity condition

is not satisfied, and it has been found that the different formulations

give very large differences in numerical behaviour when iterated. In

general, the use of asymmetric expressions for Hz has been found to

produce poor convergence, and can lead to numerical instability,

depending on the sequence and method by which the 11 and T functions

are computed. No such difficulty has been encountered when using the

symmetrical expressions

(3Hz/3z)0 ° [(Hz)02 - - (H,)40]/ h

(19

Hence the nodal forms of equations 7 and 9 at the conductor

surfaces are

Ti + T3+ .51'128 \-..(0204)/h - T4/2j - (2 + jN28/2 +a)To (20

01+03 + N2[02+ 04 + (To+T4)11/23 - (2 + 2N2 + a)110 ° 0

(21

where

a ( wNh/p)2

(22

and 8 - h

2/2d

2 (23

When equation 21 is derived by the equivalent pole-sheet approach

the (To+T4)/2 term is replaced by T0, and this provides an

alternative approximation of the same accuracy. More general express-

ions for To and 00 can be derived in the same way for conductor

interfaces at which the conductivity takes different values on the

two sides, neither zero. At nodes at which the conductivity is

uniform, equations 7 and 9 become:

where the two terms are given by equations 12 and 13, and this form of

dependence has therefore been adopted.

At the conductor surface the discontinuity in the right-hand side

of equation 9 can be represented by treating it as a sheet source,

in which the equivalent pole density is numerically equal to Tz, per

245

3. Methods of solution

The method was compared with the conventional magnetic vector

potential (A) formulation in an initial study of the linear motor. To

simplify the A calculation the problem was assumed to be two-dimensional.

Consequently the pole pitch was made infinite. The scalar potential,

when calculated by simple point iteration with unity acceleration

factor, required 850 iterations to reduce the maximum error to 2 x 10 5

of the maximum potential in a rectangular mesh of 202 nodes (with

machine proportions somewhat different from those shown in fig.1).

The dominating effect of the iron surface made the convergence of the

A calculation too slow to be practicable without various acceleration

techniques (including specifying the flux linkage instead of the

excitation current), so that quantitative comparison is difficult

and is not necessarily very meaningful because it is problem-dependent.

Rut, in general terms, the well-known advantages of the scalar-potential

formulation in regions bounded by iron, because the Neumann condition

which is imposed on A is replaced by the Dirichlet condition, are

retained in eddy-current calculations. It has been found that the

ratio of the convergence rates is reduced as the frequency is raised,

but it is greatly in favour of the scalar potential formulation at the

working frequency of the machine.

Nevertheless, the preliminary results showed room for further

improvement, and experimentation with different methods of computing

n and T showed that not all of them converged well, whilst some

diverged. The numerical behaviour was therefore examined more closely.

The full set of finite difference equations for all nodal Tz and

values takes the form

LI xl(0,Tz) fl (26

where LI is a space coefficient matrix, xl is the vector of Tz and n

values, and fi is a constant vector that incorporates the boundary

conditions. Although the values of the elements of LI depend on the way

in which Hz and 311

z/az are approximated numerically, LI has some

properties that are independent of the finite difference approximation

used. The finite direction graph technique described by Varga5 shows

that LI is not consistently ordered and does not satisfy Young's

"Property A." Furthermore, LI cannot be diagonally dominant, though

it may approach this condition when the simultaneous equations are

suitably manipulated. The manipulation takes the form of elimination

of some Tz and 0 terms. The equations are derived from two different

coupling conditions, namely the induced current equation 7, and the

magnetic continuity equation 9, and numerical experimentation has shown

that best convergence rates can be achieved if these are separated.

That is, 26 is separated into two simultaneous matrix equations of the

form

L2x2(Tz) f2 +$2 (0) (27)

(28)

where 12 and f3 incorporate the boundary conditions, whilst £2 and La

are functions of a and Tz respectively. The coefficient matrices L2 and

L3 depend on the numerical approximation adopted. It is found that

L2 and L3 are consistently ordered and satisfy Young's "Property A",

although neither is diagonally dominant. Furthermore, L2, but not L3,

can be tri-diagonal. It has been found that the Tzcalculation is less

well-conditioned than the 0 one (partly because the operator diff-

erentiating Tz has one less dimension than that operating on and

improvements depend on an increase in the number of Tz iterations. This

causes relatively little increase in the computing time per complete

cycle because the Tz calculation is confined to nodes in conductors.

There are advantages in solving equation 27 by matrix inversion,

particularly when the finite-diference approximation chosen makes L2

tri-diagonal, and this gives a part line-iteration method.

In a typical calculation, the substitution of equations 27 and

28 for 26 and computing Tz by 10 Causs-Seidel iterations per step of the

main iteration cycle improved convergence by a factor of 5 (to 173

cycles). The computing time was reduced by a factor of 3.5.

One consequence of the consistent ordering of the L3 matrix is

that the Car4- Stoll method6'7 may be used to calculate the best

acceleration factor, and this has been found to work well in practice.

246

4) Results

The linear motor used for test purposes was excited by a 3-phase

winding arranged in 3 slots per pole in the laminated block on the under-

side of the air gap (fig.l). The computation method was tested for vari-

ous thicknesses and positions of the conducting plate as well as at

different frequencies, but experimental measurements were limited to a

plate having the thickness shown in fig. 1 placed on the lower iron

surface. This left an air gap above the plate in which the flux density

measurements were relatively unaffected by slot harmonics. The plate

width was reduced below the normal value to increase leakage effects.

For the field calculation a mesh of 661 interior nodes was chosen with

approximately square elements in the end-region R2, and rectangular

elements with a length-breadth ratio N of 2.25 in the air-gap region RI.

The scalar potential field sources consisted of current sheets on the

surface of the bottom laminated core, together with sheet pole-type

sources on the end-winding surfaces (the T' function of ref.1).

A contour plot of the real part of the U function is given in fig.2.

Here the field in the R2 regions is compressed by treating all nodes,

for plotting purposes, as having the same spacing as in the region R1.

The imaginary part of II in the air gap and in the plate is drawn to an

enlarged scale in fig. 3. The diagrams show the discontinuity in the

normal gradient of U at the plate surface which is caused by the

discontinuity in T.

As is typical with these proportions, the variation of T with z is

comparatively small, although the depth of penetration d is only 20Z

greater than the plate thickness. The variation of T with x and y over

the entire plate is shown in fig.4 for one instant of time (the calcul-

ation assumes no end-effects in the y direction). Since T varies sinu-

soidally with y, as well as in time, the real and imaginary parts are

obtained by choosing appropriate sections of the diagram. It is note-

worthy that T(x) is approximately sinusoidal.

The solution converged to a potential error of 2 x 10 5 of the

maximum potential in 74 iterations, and the convergence rate was found

to be virtually independent of frequency over a range of 40 to 1. The

acceleration factor, computed by the Carrg-Stoll method, settled to a

final value of 1.707 + j 0.0047, with a sufficiently high convergence

rate to make its initial value (unity) unimportant.

Computed and measured values of the magnitudes of the two large

flux-density components are plotted in fig. 5 as a function of x. The

measurements shown were made in the mid-plane of the machine, where end-

effects were expected to be least. Somewhat higher values were observedi

in other planes. Under travelling-wave conditions, equation 8 reduces

to

Hy 3(7T /p) 2

so that the y component of B provides a direct measure of Q. The

purpose of the machine is to produce forte in the y direction, and this

force was measured and compared with the computed value to obtain a

convenient criterion of solution accuracy averaged over the plate. The

calculated force was 4.4Z less than that measured.

The agreement between the calculations and measurements provided

adequate confirmation of the former, in view of the approximations made,

particularly the neglect of end-effects. Since the programme is a small

one it could be readily extended to include these2, but a more detailed

study was not considered worthwhile. The principal objective was to in-

vestigate the numerical behaviour of the n and T functions for plate

thicknesses representative of practical devices, and this behaviour

is little affected by the way in which the y variations are modelled.

5) Conclusions

When computing eddy currents numerically the formulation can be

expressed in terms of one of a range of possible quantities, all vectors,

including the four field vectors (H,B,E,J) and the two vector potentials

(A and T). The work described has confirmed the substantial advantages

in choosing an electric vector potential, T, as the current describing

function, defined so that it is constant or zero outside the conductors. nj Cn CO

247

As in magnetostatic applications, this reduces the field problem in the

non-conducting regions to that of computing a function ft which is both a

scalar and which is well-behaved numerically in regions bounded by iron.

5)

6)

Varga, R.S..matrix Iterative Analysis", Prentice Hall 1962.

Carrd, B.A. " The determination of the optimum accelerating factor

for succesive over-relaxation"Cmput.J. 1961 4 pp 73-8

Inside laminated and plate conductors the associated T function can be 7) Stoll,R.L. "Solution of linear steady-state eddy-current problems

limited to.one component. Its interaction with n can be expressed numerically in a variety of ways, and many, although by no means all, of

these possibilities have now been explored. Poor convergence, and even

divergence, has been experienced with some, but the preferred methods

give excellent convergence in a device which typifies many power-freq-

uency applications.

The method assigns SI and T values to the same nodes and is suitable

for line iteration of the T values. Accelerated point iteration gives

convergence which is very much better than that of the magnetic vector

potential A function in regions bounded by iron, and the technique is

well suited to both two- and three-dimensional calculations. The form of

the matrices is such that the Carrel-Stoll method for computing acceler-

ation factors automatically is very effective.

6) Acknowledgements

The work has been supported financially by the Science Research

Council. The authors are grateful to many colleagues for helpful

discussions, particularly Dr. C.W. Norman (Westfield College) and Dr. D.

A. Lowther (Imperial College).

7) References

1) Carpenter, C.J. and Locke, D. "Numerical models of 3-dimensional end-

winding arrays" Compumag 1976

2) Carpenter, C.J. and Djurovid, M "3-dimensional numerical solution of

eddy currents in thin plates" Proc.I.E.E. 1975 122 pp 681-

688

3) Carpenter, C.J. "Computation of magnetic fields and eddy currents"

5th Int.Conf. on Magnet Technology Rome 1975 pp 147-158

4) Carpenter, C.J. "A network approach to the numerical solution of

eddy-current problems" Trans. I.E.E.E.1975 MAC-11

pp 1517-1522

by complex successive overrelaxation" Proc.I.E.E.

1970 117 pp 1317-23

Fig. 2 Real part of n 0 - 0.0276

248

249

Fig. 5 Magnitude of B in air gap

Measured in machine centre-plane 2mm below top iron surface Fig. 4 T variation in typical layer of plate

TAX 6520

I P icas;

/ Stator /

-9

Fig. 3 Imaginary part of

0max - -29.2 0 . - min 0

Longitudinal Component

X cm

Normal Component

co" 20

6

X cm

Discussions following paper:

(Perin, CERN) I should like to point out that the vector T was used,

for the static case, about 20 years ago for the computation of the AGS

magnets at BNI. Again a long time ago it was used by R Christian in

his magnetostatic program Sybil and by myself at CERN in the MARE

program. •

(Wyatt) I thank Mr Perin for his comments. It is all too easy to

give the impression that the method presented is entirely new when

this is not the case. Maxwell. himself was responsible for the idea

of using a current flow function, though only in scalar form. Our

contribution is that we have extended the flow function concept, using

the electric vector potential T, in order to solve eddy currents in two

and three dimensions.

250

THE COMPUTATION OF EDDY CURRENT LOSSES IN SOLID IRON

UNDER VARIOUS SURFACE CONDITIONS

Dr. D.A. Lowther* And E.A. Wyatt*

Abstract

The paper compares the computation of eddy current losses by diff-

erent methods. The rectangular B-H curve approximation is used to study

the effects of sinusoidal and non-sinusoidal surface electric and mag-

netic fields.

The non sinusoidal form is convenient both for external field cal-

culations and the analysis of experimental data. The Frohlich curve

approach is examined under similar conditions of non-sinusoidal surface H.

The extension of both methods to two-dimensions is examined.

1) Introduction

The severe heating problems which can occur in large transformers and

turbo-alternators have given rise to the need for an accurate prediction

of the power losses. This requires a detailed description of the field

distribution both inside, and external to, the core.

The problem may be conveniently divided into two parts in order to

simplify the calculations. The first part involves predicting the loss

distribution inside a magnetically non-linear core for a given surface

field distribution. The second is to find a simple model which allows

the calculation of the exterior field, avoiding complex interface condit-

ions.

*Department of Electrical Engineering,

Imperial College of Science and Technology,

London. S.W. 7

At the levels of magnetisation encountered in these applications

the hysteresis loss is negligible and the eddy current loss is the major

component. The purpose of this paper is to compare different methods

of computing the losses in a non-linear medium. The resultant solutions

may be expressed in terms of surface impedance. This concept is useful

for comparing the various methods and can be used in the computation of

the external field.

Two different approaches have been used in the published work.

These may be classed according to the way in which the magnetisation

characteristic is represented. In the simpler, and historically earlier

method described by MacLean, Agarwal et al (references 1-6) a rectangular

approximation to the characteristic is used. This leads to an algebraic

solution in one-dimension and can give a useful indication of the be-

haviour of the field inside the material.

The more complex method employs a more realistic representation

of the magnetisation characteristic and uses time stepping techniques

(references 7-12). This method has become popular as large computers

have become available and it gives the field distribution in the material

accurately.

Because many electromagnetic devices operate under "current forced"

conditions, attention has been restricted to the boundary condition of sin-

usoidal surface 11 in most of the published analyses. However, this can

be considered as a limiting condition; the other limit being that of

sinusoidal surface E (or total flux). Because, in practice, the surface

conditions may vary between the two extremes and are, in general, non-

sinusoidal, the analysis in this paper deals mainly with the sinusoidal

surface E situation. In addition, the tnalysis is extended to include

surface waveforms which are non-sinusoidal in time. This is useful as

far as comparisons with experimental results are concerned.

no In regions in which the non-linear medium is subject to a high level -.a of incident normal flux, which turns along the surface after entering, • P

269

O

a two-dimensional analysis is required. Such an analysis will be con-

sidered briefly.

2) The Finite Difference Model

The finite difference model is taken to include all those models

using time-stepping techniques. (references 7-12) and in this approach

the magnetisation curve may be represented by a single function or,

alternatively, the actual curve may be stored at discreet points and

curve fitting employed. Many different finite difference schemes have

been employed. Examples include the Crank-Nicolson9 and Durbrt-Frankel

techniques]. In practice both the E and H surface waveforms are non-

sinusoidal in time.

2.1) The Sinusoidal Surface E Analysis

When the sinusoidal surface E condition is imposed a difficulty

arises which is not present if the sinusoidal surface H solution is

sought. The difficulty is that the non-linearity considered is magnetic

and thus concerns the relationship between B and H rather than E and J.

The method used here is similar to that described by Lim and Hammond7

who used a Dur'ort-Frankel time-stepping scheme in conjunction with a

Frohlich magnetisation characteristic.

If the material conductivity, o, is assumed constant and the per-

meability is a function of B as well as the spatial coordinates, the

following governing equation for E may be derived from Maxwell's

Equations used in conjuction with the constitutive relations;

a V2 E =aF ( curl pH )

which can be further modified to give

V2 E e a

(poE - H x grad P)

If (2) is applied in conjunction with

A convenient experimental model of the one-dimensional diffusion

problem is a steel rod of circular cross-section subject to sinusoidal

surface conditions. It is thus appropriate to consider the circular

cylindrical coordinate form of equation (2) which is:

1 a 7 37 (r a ) = ;1 (v0 Ea + Ho *x )

The subscripts may be dropped because there is only one component of

E and one of H. Equations (2) and (4) , however, require that the time

derivative of the permeability be known and this is an inconvenience.

A simpler equation can be obtained from equation (1) by substituting

B for pH and noting that

DB m dB Ill 3r dH 3r

The equivalent form of equation (4) is then;

1 a aE dB at , r Ir r W ) - a Zri -37 (5)

from which the following finite difference equation may be derived

using central differences:

E(i,j + 1) ° Q(E(i,j-1) (Br (A)Z7rAt) + At E(i+l,j) (5a)

(2r + Ar) + E(i-1,j) ( 2r-Ar)))

where

dB 1

w (it r(2At +0(402)

The relationship between B and H is defined by the Frohlich curve:

B H

in which H and B are in the same direction.

At each step the H distribution must be calculated from that of E

so that the magnetisation curve can be used. Because curl H J

(1)

(2)

(4)

dive 0 (3)

all of Maxwell's equations are satisfied.

E 1 D(rH)

-517—

and integration with respect to r yields

(7)

270

r

H 2- r E dr (8)

Jo

With this modification the calculation of the field distribution

follows the method of Lim and Hammond2, using the DuFort-Frankel time-

stepping scheme.

2.2) Surface Impedance.

From the solution, the fundamental components of the field vectors

at the surface can be obtained by harmonic analysis. If either E or H

is sinusoidal at the surface, the total power loss in the material may

be determined by applying the Poynting Vector to their fundamental

components. This loss can be regarded as occurring in the real part of

a complex surface impedance. The quadrature component of the impedance

may be used to describe the reactive volt-amp absorption. This surface

impedance can then be used to terminate the exterior network.

The concept of a surface impedance is useful as a basis for compar-

ison of methods and as a check against experimental results.

3.1) Sinusoidal Surface E

Using a modified form of Maxwell's equations and the co-ordinate

system of figure 2, the field at a distance x from the surface is given

by:

H G E

x

dx and E 2B Eo sinwt dt

After integration these equations give a wavefront depth of

2430 (1 - coswt) Ho

and the solution for H is

rol° sinwt sin --

GE 2 2 wt

0 2

(12)

(9)

(10)

3) The Approximate Model As in the finite difference method the resultant solutions for the

The approximate model in which the magnetisation curve is repres- surface values of E and H may be harmonically analysed. If either

ented by a rectangular characteristic has been described by several waveform is purely sinusoidal, the loss may be obtained from the fun-

authors (references 1-6). It restricts the flux penetration to a surface damental components by Fburienanalysis. The fundamental component of H

layer in which the material is saturated either one way or the other; is:

the switching point between the two magnetisation directions defining H H' (-4 coswt.+3nsinwt) (13)

a wavefront. An algebraic expression is obtained for the E and H wave-

forms. where H' is defined in equation (A.1) of the appendix. If E E, i.e. is purely real

As with the finite difference methods,much of the published

literature considers the sinusoidal surface H condition. The following H = H' (31: - 4j) (14)

analysis considers the condition of sinusoidal surface E (as in the pre-

ceeding section) and, in addition, the analysis is extended to include Again a surface impedance may be used to describe the resistive and

the non-sinusoidal surface fields. reactive components of the total volt-amps. If this impedance is

considered to consist of series components, their values are given by

271

equation (A.2). The impedance shows the power factor to be 0.9206 so

that the phase angle is 23° . This result may be compared with that for

the sinusoidal H condition given by McConne114 and Agarwal5. In the

latter case the impedance is that given in equation (A.3) and the power

factor is 0.8944 giving a phase angle of 26.6°.

These two solutions can be compared on a basis of the same peak

value of the fundamental component of E and the ratio of the magnitudes

of the two impedances is then

(15) 12s1 / 12.0 ■ 2.04745

This indicates that the power loss for the condition of sinusoidal

surface E is double that for sinusoidal surface H. This point is confir-

med by the finite difference solutions.

3.2) Non-sinusoidal Surface Fields

The above approach allows a further generalisation to include non-

sinusoidal surface excitations by expressing the surface waveforms in

Fourier series form.

The derivation described below is in terms of H although the

treatment for E follows a similar procedure.

If the surface H distribution is given by (16)

H - HI, sin wt H10 coswt H3s sin 3wt + H3c cos 3wt +...

and equation (10) is modified to become

dx = 20 Bo at x

the resultant equation for the depth of penetration any time, E, is

given by

X 471- ZOBo

where H is defined by equation (A.7) of the appendix. The solution for

E is then

E ■ H 20

Equation (18) assumes that x - 0, and hence 1. = 0, at time t ■ 0 so

that the time origin has to be displaced to the point at which H = 0.

As before, a harmonic analysis can be used to yield a surface im-

pedance, although the impedance now has harmonic components so that the

surface layer must include a series of harmonic generators.

Agarwal 5 has modified the saturation flux density by a factor of

0.75 so as to predict the loss (but not the VAr's) accurately when

the surface H is sinusoidal. This factor has been shown to be dependant

on the magnetisation leve113. In the sinusoidal surface E condition a

similar approach can be employed to model the loss accurately.

The advantage of the rectangular magnetisation characteristic lies in

the simplicity with which the surface impedance may be derived. The

approximation is satisfactory at the large values of surface mag-

netisation which occur in many problems of interest.

4) Results

The results in figures 3 to 6 show the E and H waveforms predicted

for a specific B - H curve for sinusoidal surface E and H. As can be

seen, the waveforms are similar in form for both methods and as the

magnetisation level is increased the similarity increases. The effect

is shown more clearly in the surface impedance results.

The method of section 2 can be adapted so that a finite final slope

is included on the B - H curve but the effect is small, as has been

noted elsewhere6. It would seem that the difference between the approx-

imate and finite difference methods is largely due to the fact that the

finite difference solution allows for the initial slope in the B - H

characteristic.

(17)

(18)

(19)

272

The extra loss caused by assuming a step change in flux density

requires a reduction in the saturation flux density to obtain accurate

predictions.

This result suggests that a modification of the rectangular curve

to one having a finite initial slope would improve the agreement between

the two methods considerably.

5) Extensions To Two-dimensions

Both methods may be extended to give field solutions in two dimen-

sions, as is necessary when the flux density normal to the surface is

large. This condition is commonly met in practice in the end region of

a turbo-alternator or around a transformer leg.

The finite difference approach in two-dimensions may be formulated

in terms of magnetic vector potential, A , which has only one component.

This function was chosen because it is often employed in two-dimensional

linear eddy-current problems.

The governing equation for A is

V2A r 3A ay i r DA 31/p 3A 91/111 (20) z Pa ac -37 - Lax z ax 137z ay

which may be approximated by a nodal (DuFort-Frankel) finite difference

method. Care is needed in the treatment of interface conditions which

include restrictions both on nand the normal gradient of A.

The approximate approach may also be adapted to two-dimensions

although the wavefront, which is the key to the algebraic treatment of

the one-dimensional problem, no longer becomes as clearly defined since

the angle through which the magnetisation vector switches is not

necessarily 180 degrees. The evaluation of H is complicated by the

variation of the current density within the saturated region. The

solution is still of the surface layer type and is only applicable to

high magnetisation level problems.

Two-dimensional calculations using these two approaches are being

made and it is hoped to publish the results at a later date.

6) Conclusions

The foregoing analyses have shown that the most commonly used methods

of treating non-linearity can be adapter. to allow for any specified

surface E or H time variation. A comparison between the results for

sinusoidal H and E (the two limits) indicates a region within which the

practical condition must occur. The methods have concentrated on the

fact that sinusoidal surface E may be regarded as a limiting condition

on the waveforms encountered in practice.

The surface layer concept,together with that of a characteristic

surface impedance, can simplify external field calculations considerably.

In addition,they provide an extremely useful point of reference between

different analyses and experimental measurements.

7) References

1. Rosenberg E. "Eddy Currents in Iron Masses", The Elecrican,

1923, pp 188-191

2. Haberland G. and Haberland F. "Alternating fields in Saturated Solid

Iron", Archly fur Elektrotechnic, Berlin, Germany,

Vol 30, 1936, pp 126-133.

3. Maclean W."Theory of Strong Electromagnetic Waves in Massive Iron",

Journal of Applied Physics, 25 (10) 1954, pp 1267-1270

4. McConnell M.M. "Eddy Current Phenomena in Ferromagnetic Materials",

Trans AIEE pt I (Mag) 73, 1954 pp 226-235.

5. Agarwal P."Eddy Current Losses in Solid and Laminated Iron", Trans

AIEE, ptI (Mag) 78, 1959, pp 169-181

6. Shevez W.L. "A Modified Limiting Non-Linear Theory of Eddy Current

Phenomena in Solid Iron", Trans AIEE pt (Mag) 81,

1962 pp 48-55

7. Lim K.K. and Hammond P. "Numerical Method for determining the Electro-

magnetic field in saturated Steel Plates." Proc IEE,

119,(11), 1972 pp 1667-1673. 1.‘.3)

8. Neyfem A.H.and Asfar O.R. "An analytical solution of the Nonlinear Fs

273

Eddy Current Losses in Ferromagnetic Materials"

IEEE Trans Mag - 10, 1974 pp 32- - 331

9. Poritsky H. and Butler J.M. "A-C Flux Penetration into Magnetic

Materials with Saturation" Trans AIEE,

1964, COM - 83, pp 99-111.

10. Gillott D.M. and Calvert J.F. "EddyCurrent Loss in Saturated Solid

Magnetic Plates, Rods, and Conductors". AIEE

1965, HAG - 1, pp 126-137

11. Ahamed S.V. and Erdelyi E.A "Non-Linear Theory of Salient Pole

Machines", IEEE Trans PAS - 85, 1966, pp 61-70

12. Bullingham J.M. and Bernal M.J.M. "Investigation of the Effect of

Non-Linear B-H Loops on the calculation of

Eddy Current Losses", Proc IEE 114 (8) 1967,

pp 1174-1176

13. Freeman E.M. "Universal Loss Chart for the calculation of Eddy

Current Losses in thick steel plates", Proc

IEE, 118 (1), pp278-279

Appendix

The following equations are used in the approximate model of

section 3.

H ■ oEo2 --- 12uBor

2 12mBor [ 2L214.1i 1 se aEo 9w +16 J (A.2)

for the same condition, the ratio of penetration depths is

Se 0.949

(A.6)

H,. n

eo

n 1,3,5

sing 11E4 Hn, sin niL Cosg1t) (A.7)

Acknowledgements

The authorewould like to thank Mr. C.J. Carpenter of Imperial

College for helpful discussions, and the Science Research Council

for financial support.

-671

(A.1) Figure l. The sinusoidal E model

16 roc,[ 1+ii i sh m 3a L2Hoo J J (A.3)

for the same peak fundamental component of E

zsh 16B9togb [1 + jil (A.4) 9r Eoa

X

Hi JE

propagation

12sel

I zshl

27w3 m 2.0475 (A.5) 40 (9w2 +16)1 Figure 2. The rectangular

B-H curve model

274

275

x10 ohms

L

1.0

IZsurfl x10-4 ohms

, I4.0

2.0

0

b

phase

ax rectangular curve

b. Frohlich curve

rmgnitude

Built . 1.786 T a 0.595.10 S/m Ni 3.77 x10-3 H/m

0.5 1.0 1.5 fundamental component of E

V/m peak —

Figure 5. Surface impedance against E

for sinusoidal surface H

magnitude phase •

a = rectangular curve b Frohlich curve

Bsat T 0.694 x 107 S/m

pi .2.0 x103 H/m

0.5 1.0 fundamental component of E

V/m peak

Figure 6. Surface impedance against E

for sinusoidal surface E

30.0

20.0

10.0 0.5

rectangular —curve

Figure 3. Surface E waveforms for

sinusoidal surface H

Frohlich curve

Figure 4. Surface H wavefOrms

for sinusoidal surface E

arg (Zsurfle

.40.0

300

20.0

10.0

Discussions following paper:

(Hammond, Southampton) The loss in solid iron is closely predicted by

using a rectangular B-H characteristic. Does the author think that

this loss could be obtained very simply by using an energy functional?

(Lowther, Imperial College) The rectangular B-H characteristic predicts

the loss in solid iron closely only if the saturation flux density is

reduced by a factor such as that suggested by Agarwal. In representing

the non-linear surface layer by a surface impedance (in order to simplify

the exterior field calculation) both the phase and quadrature components

of the impedance should be accurately represented. At high levels of

magnetisation the rectangular B-H curve gives a reasonable solution for

the magnitude but produces an error inphase angle. The Agarwal factor

reduces both components of the impedance whereas a better solution might

be a method in which the phase angle only is increased, improving the

accuracy of both components of the impedance.

With the above proviso the concept of an energy functional, employing a

rectangular B-H curve, to represent the loss is an interesting idea.

This would overcome some of the problems involved in an accurate repre-

sentation of the non-linearity. However, it is an idea which we have

not considered.

Following Professor Hammond's paper at the conference, which employs this

technique to obtain the relevant parameter of interest very simply; we

would be interested if he has applied the method to predict the loss in

the non-linear situation.