Finite Element Modeling of the F'requency Characteristics of Tkansformers

Y e Liu

X thesis submittd in conformity with the requirernents for the Degree of Master of Applied Science in the

Department of Electrical and Computer Engineering, University of Toronto

@Copyright by Ye Liu 1996

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Finite Element Modeling of the

Requency Characteristics of Transformers

by

Ye Liu

M.A.Sc Thesis, 1997

Dept. o f Eiectncal and Cornputer Engineenng

lIniverstty of Toronto

Abstract

The transformer is one of the most difficult power devices to rnodel accurately, especially in

the high frequency rangc. A transformer model which is suitable for simulating high frequency

behavior in the Electromagnetic Transients Program (EMTP) is available. This model is based on

the frequency characteristics of the transformer terminai admittance matrix over a given range of

frequencies. Normally, the terminal admittance matrix is obtained from memurements. However,

this procedure is very costly and, in addition. it is not useful to predict the current/voltage profile

inside the transformer windings. In order to avoid these disadvantages. in this thesis a finite element

method to mode1 the transformer is proposed. This model can be used to obtain the electromagnetic

field inside the windings and the circuit parameters such as R, L, C to derive the equivalent circuit

network. The terminal admittance rnatrix is then obtained which can be used by ESITP.

Acknowledgment s

1 am eternally grateful to my supervisor, Professor Gdalbert Konrad, for his guidance, patience.

and financial support during the period 1 am working on the thesis. His scientific thinking, his

enthusiasm and dedication make me felt great to work with him.

1 sincerely thank Professor A. Morched for his valuable suggestions and technical discussions.

My special thanks go to the members of my M.A.Sc committee.

1 am also thankful to Dr. K.V. Narnjoshi who spent much of his spare time reading the draft of

my thesis and gave me a lot of good advice.

Last but not the least, 1 would like to thank my parents for their understanding. encouragement

and mental support throughout the last two years.

Contents

1 Introduction 1

1.1 Staternent of t h e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Literature Review 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 State of the Art 5

1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 O u t h e of Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 A Simplistic Transformer Mode1 9

2.1 Basic Equivalent Circuit Mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 ReaIization of the Slodel 1 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Measurernent Method 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Finite Element Mode1 13

2.5 DetailedEquivalentCircuitStodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Capacit ance Calculat ions 17

3.1 Principle of Capacitance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Self- and Mutual- Capacitances 18

. . . . . . . . . . . . . . . . . . . . 3.1.2 Capacitance of a Multi-Conductor System 18

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Floating Potential Approach 22

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Validation 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Cornputer Program 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Example 25

. . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Cornparison with Analytical Results '16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Rmults 27

. . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Capacitance of .4i r-Core Windings 29

. . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Capacitance of Iron-Core Windings 31

4 Inductance Calculation 32

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Principle of Inductance Calculation 33

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Self- and Mutual- Inductances 33

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 General Formulation 34

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Formulation for Transformer 37

. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Approach to Inductance Calculation 39

. . . . . . . . . . . . . . . . . . 4.2.1 Calculation of the Magnetic Vector Potential 39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Calculation of L and M 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Validation 4 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Validation of the Program -42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Validation of the Approach 44

4.4 Resuits . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . 46

. . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Inductances of Air Clore Windings 47

4.4.2 Inductances of Iron Core Windings . . . . . . . . . . . . . . . . . . . . . . . . -28

5 Loss Estimation 57

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Winding Loss 57

. . . . . . . . . . . . . . . . . . . . . 5.1. 1 Formulation of the Skin Effect Problern 58

5.1.2 Brief Description of the Cornputer Program . . . . . . . . . . . . . . . . . . . 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Results of Resistances 59

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . 2 Corc Loss Evaluation 63

5.2.1 Calculation of Core Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Discussion of Core Loss 65

6 Equivalent Admittance Matrix 66

. . . . . . . . . . . . . . . . . . . . . . . . 6.1 Air-Core Transformer Admittance Matrix 67

. . . . . . . . . . . . . . . . . . . . . . . . 6.2 Iron-Core Transformer Admittance Matrix i l

. . . . . . . . . . . . . . . . . . . . . . . . 6.3 Traditional and New Equivalent Circuit 72

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Summary 78

7 A More Realistic Transformer Mode1 79

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Ceometry Description 79

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Capacitance Calculatioa 81

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Inductance CaIculation 81

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 LossEstimation $2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Adrnittance Matrix 82

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Discussion 84

8 Conclusions 91

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Surnrnary and Contributions 91

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Future Work 92

A Appendix

B Bibliography

vii

List of Tables

3.1 Se!f and Mutual Capacitances of Two Spheres . . . . . . . . . . . . . . . . . . . . . . 27

3 .2 Self- and Mutual- Capacitances of Air Core Transformer (pf) . . . . . . . . . . . . . 30

3.3 Self- and Mutual- Capacitances of Iron Core Transformer(pf) . . . . . . . . . . . . . 3 1

4.1 Results of Self and Mutual Inductances ( x I O - ~ H ) . . . . . . . . . . . . . . . . . . . 46

4.2 Self and Mutual Inductances of Air Core Transformer( x 10-' H) . . . . . . . . . . . 49

4.3 Self and Mutual Inductances of Air Core Transformer ( x 10-'H) . . . . . . . . . . . 5û

4.1 Seifand M u t u a l I r ~ J u c t a n c e s o f A i r C o r e T r a n s f o r m e r ( x l ~ ' ~ ~ ) . . . . . . . . . . . 51

4.5 Self and Mutual Inductances of Iron Core Transformer(x l0-'H) . . . . . . . . . . . 52

4.6 Self and Mutual Inductances of Iron Core Transformer(x 1 0 " ~ ) . . . . . . . . . . . 53

. . . . . . . . . . . 4.7 Self and Mutual Inductances of Iron Core Transformer( x 1 0 " ~ ) 54

. . . . . . . . . . . . . . . . . . . . . . . . . 5 . 1 DC Resistances of Conductors ( x 10-4ir) 60

. . . . . . . . . . . . . . . . . . . . . . 5.2 Relative Resistances RAC/RDC vs . Frequency 61

. . . . . . . . . . . . . . . . . . . . . . 3 Relative Resistances RACI RDC vs . Frequency 62

. . . . . . . . . . . . .5 .4 Grade Designat ions and Maximum Core-Loss Limits for 36F 14.5 64

7.1 Self- and Mutual- Capacitances of Transforrner(pf) . .

7.2 Self and Mutual Inductances of Transformer( x IO-~H)

A. 1 Self and Mutual Inductances of Transformer( x 10-5 H )

A.2 Self and Mutual Inductances of Transformer(x 1 0 ' ~ H )

A.3 Self and Mutual Inductances of Transformer( x I O - ~ H )

A.4 Self and Mutual Indlictances of Transformer( x 10" H )

A.5 Self and Mutua1 Inductances of Transformer(x 10-'H)

A.6 Self and Mutual Inductances of Transformer(x H )

A.7 Self and Mutual Inductances of Transformer(x 10'5H)

A.8 Self and 11 utual Inductances of Transformer( x 1O"H)

A.9 Self and Mutual Inductances of Transformer( x 10" H )

A. 10 Self and bfutual Inductances of Transformer(x IO-' H)

A . 1 1 Self and Mutual Indiictances of Transformer( x 1 0 ' ~ H)

A. 12 Self and Mutual Inductances of Transformer( x 10" H)

A. 13 Self and Mutual Inductances of Transformeri x 10" H)

A.14 Self and Slutual inductances of Transformer( x I O - ~ ~ I ) . . . . . . . . . . . . . . . . . 10'7

A.15 Self and Siutual Inductances of Transformer(x 1O"H) . . . . . . . . . . . . . . . . . 108

A.16 Self and Mutual Inductances of Transformer(x 10" H) . . . . . . . . . . . . . . . . . 109

A . 17 Self and Mutuai Inductances of Transformer( x IO-~H) . . . . . . . . . . . . . . . . . 110

A. 18 Self and Mutual Inductances of Transformer( x IO-' H) . . . . . . . . . . . . . . . . . 1 1 1

A.19 Self and Mutual Inductances of Transforrner(x 10' '~) . . . . . . . . . . . . . . . . . 112

List of Figures

. . . . . . . . . . . . . . . . . . . . . 2.1 Basic Equivalent Circuit Model of Transformer 10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Transformer Configurations 1 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Axisymmet tic Transformer 12

. . . . . . . . . . . . . . . . . . . . . . . 2.4 Measurement of Transformer Characteristics 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Sketch of the Finite Elernent %iode1 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Detailed Equivalent Circuit Mode1 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.L A System of 'N' Conducting Bodies 19

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Two Concentric Spheres 2.5

. . . . . . . . . . . . 3.3 Relationship between Electric Potential and Boundary Position 26

. . . . . . . . . . . . . . . . . . . . . . . . 3.4 Equipotential Lines of the Solution Region 27

. . . . . . . . . . . . . 3.5 Detail of Finite Element Mesh for the two Concentric Spheres 28

. . . . . . . . . . . . . . . . . . . . . . . 3.6 Geometry of the Axisymmetric Transformer 29

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Anaiysis of Mutual Inductance 3.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Analysis of Self Inductance 36

4.3 Basic Geometry of Transformer Winding Cross-section . . . . . . . . . . . . . . . . . 38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Integration in Polar Coordinates 39

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Configuration of Four Conductors 40

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Configuration of Solenoid 42

. . . . . . . . . . . . . . . . 4.7 Magnetic Vector Potential vs . Distance of Boundary ' c ' 43

. . . . . . . . . . . . . . . . . . . . . 4.8 Relationship between Grid Size and Inductance 43

. . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Two Parallel Coaxial Conducting Rings 44

. . . . . . . . . . . . . . . . . . . . . 4.10 Solution Region for the Two Conducting Rings 44

. . . . . . . . . . . . . . . . . . . . . . . . . 4.1 1 Magnetic Vector Potential vs Distance 'c' 45

. . . . . . . . . . . 4.12 Equipotential Lines of Magnetic Field for Two Conducting Rings 46

. . . . . . . . . . . . . . . . . 4.13 Self Inductance of Air Core Transformer vs . Frequcncy 47

. . . . . . . . . . . . . . . . . . 4.14 Equipotential Lines of Air Core Transformer at 60Hz 48

. . . . . . . . . . . . . . . . 4.1.5 Equipotential Lines of Air Core Transformer at 200 kHz 55

. . . . . . . . . . . . . . . . . 4 .16 Self Inductance of Iron Core Transformer vs Frequency 5.5

. . . . . . . . . . . . . . . . . 4.17 Equipotential Lines of Iron Core Transformer at 60 Hz 56

. . . . . . . . . . . . . . . 4.18 Equipotential Lines of Iron Core Transformer at 200 kHz 56

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Winding Configuration 60

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Relative Resistance vs Frequency 63

. . . . . . . . . . . . . . . . 6.1 Detailed Equivalent Circuit of the Air-Core Transformer 67

. . . . . . . . . . . . . . . . . . . . . . . 6.2 Y 1 1 of Air-Core Transformer vs . Frequency 68

. . . . . . . . . . . . . . . . . . . . . . . 6.3 Y2 1 of Xir-Core Transformer vs . Frequency 69

6.4 Y22 of Air-Core 'hansformer vs . Frequency . . . . . . . . . . . . . . . . . . . . . . . 69

. . . . . . . . . . . . . . . . . . . . . . . 6.5 Y 1 1 of .Ai &ore Transformer vs . Frequency 70

. . . . . . . . . . . . . . . . . . . . . . . 6.6 Y2 L of Air-Core Transformer vs . Frequency 70

. . . . . . . . . . . . . . . . . . . . . . . 6.7 Y22 of Air-Core Transformer vs . Frequency 71

6.8 Detailed Equivalent Circuit of the Iron-Core Transformer . . . . . . . . . . . . . . . 72

. . . . . . . . . . . . . . . . . . . . . . . 6.9 Y 11 of Iron-Core Transformer vs Frequency 73

. . . . . . . . . . . . . . . . . . . . . . . 6.10 Y21 of 1 ron-Core Transformer vs Frequency 74

. . . . . . . . . . . . . . . . . . . . . . . 6.11 Y22 of Iron-Core Transformer vs Frequency 14

. . . . . . . . . . . . . . . . . . . . . . . 6.12 Y 1 1 of Iron-Core Transformer vs Frequcncy 75

-- . . . . . . . . . . . . . . . . . . . . . . . 6.13 Y21 of Iron-Core Transformer vs Frcquency 1 3

. . . . . . . . . . . . . . . . . . . . . . 6.14 Y22 of Iron-Core Transformer vs . Frequency 76

-- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.15 Simplified Equivalent Circuit r r

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.16 Traditional Equivalent Circuit 78

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration of the Transformer $0

Equipotential Lines of the Electric Field of the More Realistic Mode1 . . . . . . . . . 8 1

Magnetic Field of the More Realistic Mode1 (60 Hz) . . . . . . . . . . . . . . . . . . 84

Magnetic Field of the More Redistic Model (200 kHz) . . . . . . . . . . . . . . . . . 85

. . . . . . . . . . . . . Self Inductance of More Realistic Transformer vs . Frequency 86

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Y1 1 of More Realistic Transformer 86

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Y21 of More Realistic Transformer $7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Y22 of More Realistic Transformer 87

. . . . . . . . . . . . . . . . . . . . . . . . . . . . k'l 1 of More Realistic Transformer 88

7.10 Y21 of More balistic Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.11 Y22 of More Realistic Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.12 Y 1 1 of More balistic Transformer (Inductance at 60Hz) . . . . . . . . . . . . . . . . 89

7.13 Experimental Results for Y1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

... X l l l

Chapter 1

Introduction

in this chapter, a brief review of transformer modeling and investigation of frequency characteristics

is given. The problem is stated and various methods are dixussed to solve it. Finally, outline of the

approach and organization of thesis are given.

1.1 Statement of the Problem

The transformer is one of the rnost familiar devices in electrical engineering. .4lso, it is one of the

most difficult power devices to mode1 accurately. Failure of a large power transformer is costly.

According to[l], in the fa11 of 1990, one unit of a 990 MV.4 EHV generator s t e p u p transformer bank

f a i l d in one of Ontario Hydro's major generating stations. This was the second unit experiencing

such failure in the sarne station. However, a physical examination could not determine the cause of

the failure. The investigation showed that the unit failed in relative isolation of switching activities.

faults, or system conditions that could clearly establish a causal link to failure. Thcrefore, it was

necessary to look for the possibility that may impose high stresses on insulation. The stresses cause

insulation fatigue and then lead to a seemingly unrelated failure.

EMTP (Electromagnetic Transient Program) simulations suggested that the fiiilure of an SF6

bus induced higher and faster transients than the regular switching operations do. To this point. one

needs a frequency dependent model of the transformer to investigate the behaviour of the unit during

fast transients. Researchers at Ontario Hydro have already developed such a frequency dependent

model. This model is based on the known frequency characteristic of the transformer. In order to

obtain the frequency characteristic of the transformer, researchers have developed met hods w hich

can be categorized as: (a) Measurernent and (b) Sumerical Calculation. Measurernent niethods

takc into account the frequency dependent characteristics such as skin effect. stray mpaci tanc .~ and

proximity effmts automatically. However, this approach has some disadvantages:

(1) It is impossible to carry out any measurements when the transformer is in service.

(2) It is impossible to obtain the voltage/current profile inside the transformer windings.

(3) The rneasurement approach cannot be used to design a new transformer or to predict the

performance of an unavailable transformer.

Because of these disadvantages and limitations, we need to develop a frequency dependent ter-

minal admittance matrix which can be used in EMTP. In this thesis work, the elements of the

equivalent circuit such as capacitance (C), inductance (L . M) and resistance ( R ) are obtained by

finite etement calculations. The terminal admittance matrix is then obtained by using these circuit

element values as input to an admittance rnatrix calculation proqram.

Lit erat ure Review

Numerous papers have been pubtished on the behaviour of transformer windings subjected to impulse

voltages. Some of thern contributed towards a better understanding of the phenornena encountered.

Abetti's papers [2] [3] give an extensive survey of the major contributions in this field up to the

1950's. Since that tirne, the complexity of calculations prevented investigators from producing

forrnulae which could be adopted in practical transformer design. The elect romagnetic model that

Abetti developed is used in [4]. However, this physical scaled model is expensive and time-consuming

to build, and whenever a new transformer is designed, one must build a new model.

Later, Dent and Okuyama used numerical methods to investigate the transformer transients

in the windings. The early works only consider some simple cases. In [:il and [6], the authors

only considered one winding and losses were neglected. Also, the nonlinearity of the iron core was

neglected.

In 1972, Fergestad [î] investigated the transient oscillations in transformer windings based on

an equivaIent circuit model. In this thesis, the author established an equivalent network for the

transformer winding, and calculated the elements (C, L. hi, R) from geornetric information. The

influence of iron core is included by choosing a typical permeability (p , = 60) for the high frequency

range (0.06-200kHz).

After Fergestad's work, there appeared also many research works aimed towards transformer

modeling. The main streams in the computer modeling for analysis and design of transiormers can

be surnmarized as follows:

The first analytical atternpt following an approach based on self- and mutual- inductances was

presented by Rabins (81 followed by many others çuch as Fergestad and Henriksen [9] (101 and

continued recently by Wilcox et al. [Il].

An approach based on leakage inductances was initiated by Blume Il21 and improved by many

others [la] [14]. A three-phase rnulti-winding generalization was presented by Brandwajn et al. [lj].

Dugaii [16] u d the same technique for the modeling of multi-section transformers. These rnodels

represent adequately the leakage inductance of the transformer.

Another approach based on the principle of duality was introduced by Cherry Ili] and generalized

by Slernon [18]. With this approach the iron core can be modeled accurateiy. However, rnodels

based only on this approach have the inconvenience that the leakage inductances are not correctly

represented.

There exists a large number of high frequency transformer rnodels derived from measurement

[19]. Tests are made for the determination of the model paramctms in t h i frequ~ncy domain or time

dornain. Models obtained from measurements have the drawback that their performance can only

be guaranteed for the tested transformers. Although some general trends can be inferred frorn the

tests, according to design, size. manufacturer, etc., accurate predictions for untested transformers

cannot be assared.

Designers of large transformers use electrornagnetic field approaches for the calculation of the

design parameters. The finite element rnethod is the most accepted numerical solution method for

fie!d problems [20] today.

It is impossible to list al1 of the related work here. The rnodels can be roughly categorized a s

two broad trends. One is detailcd interna1 winding models [IO] [19]. This kind of model consists of

large networks of capacitances and inductances. The size of the matrices rnakes this kind of mode1

impractical for EXITP system studies. The other is terminal models [21]. This type of mode1 is

based on the simulation of the frequency and/or time domain characteristics a t the terminals of the

transformer.

1.3 State ofthe Art

As stated in section 1.1, a model that can simulate the behaviour of power transformers in a wide

frequency range is presented in [21]. This model can predict the behaviour of transformers efficiently

and accurately. The model is based on the frequency characteristics of the transformer admittance

rnatrix. The admittancc rnatrix can be obtained from measurements or from calculation.

Due to the disadvantages of the measurernent approach we mentioned previously, we will obtain

the admittance matrices by finite etement calculations.

1.4 Objectives

Based on the above discussion, the objectives of the research work are as follow:

( 1 ) Use the finite element method to obtain the electric field and calculate the self- and mutual-

capacitances for the windings.

(2) Use the finite element rnethod to obtain the magnetic field and calculate the self- and mutual-

inductances for the windings.

(3) Estimate the loss of the windings and iron core based on the geometry and empirical formulae

to calculate the equivalent resistance.

(4) Gse these calculated circuit elements to establish an equivalent circuit for the transformer

windings.

(5) Obtain the frquency dependent terminal admittance matrix.

Specifically, we shatl use the finite element method to analyze the electromagnetic field in the

transformer windings and obtain the admittance rnatrix under different frequencies ranging from

60Hz to 200 kHz. The numerical information obtained will be the rnagnetic vector potential A and

electric scalar potential V. We can derive the admittance matrix of the transformer at a specified

frequency.

1.5 Outline of Approach

In a very fast transient, high frequency components are p r e n t . .4t frequencies above a few kilohertz,

the transformer characteristics are strongly depcndent on the frequency. Eddy currents induccd in

the iron core and the transformer windings are more significant than those at the power frequency.

Also the non-linearity of the iron core will distort the electric and magnetic fields in the windings.

There are other factors whicfi rnay affect the distribution of the electric and magnetic fields in the

windings. However, it is irnpractical to take al1 of the factors into consideration. Therefore, we need

to simplify our mode1 by rnaking the following assumptions based on the study and knowledge of

the particular transformer:

(1 ) T h e fieId in the transformer windings is assurnecl to be axisymmetric.

(2) The current in the conductor is assumed to be directed along the conductor.

(3) T h e magnetic vector potential .< has only a 0 component.

(4 ) Displacement currents are neglected in the frequency range from 60Hz to 200kHz.

( 5 ) Temperature effects on the parameters are neglected.

We consider the transformer under three-phase balanced condition. Hence we can only calculate

one phase. However, the calculation can be extended to cover three phases.

Real transformers have complicated 3D geometries and exhibit resonances of certain frequencies

in the range of interest (O - 200 kHz). These resonances show up as peaks in the plots of the

admittance matrix elements as a function of frequeacy. Since we intend to establish 3 new and

rather lengthy numerical procedure based on the finite element method. we will unciertake this first

atternpt on a very simple axisymmetric transformer geometry. Furtherrnore. the dimensions of our

first mode1 will be such a s to obtain a nurnerically stable, rather smooth variation of the adrni ttance

matrix parameters with frequency. Consequently, our first model will be a simplistic and quite

unrealistic transformer with relatively few winding sections.

Once experience has been gained with the sirnplistic transformer rnodel and satisfactory. well-

behaved dmi t tance matrices are obtained, we will turn our attention to a more realistic, though

geometrically still manageable transformer model.

Only when we gain confidence in our new finite elerncnt modeling procedure, can we think of

applying the approach to a geometrically more complicated real 3D transformer. However. such an

arnbitious undertaking is outside the scope of this master'i thesis.

The electric and magnetic fields can be calculated by solving the following equations for the

electric scalar potential V and the magnetic vector potential .<:

Equation (1.1) is a cornplex Poisson's equation and (1.2) is the frequency domain diffusion

equation. Equation (1.1) can be derived from the time harmonic current continuity equation:

di" J = - j u p

Equation (1.2) can be derived from the tirne harmonic !vlaxwelt's equation:

From the electric field Ë, we can obtain the values for self- and rnutual- capacitances: Froni the

magnetic vector potential .x, we can obtain the values for self- and rnutual- inductances.

1.6 Organization of the Thesis

In chapter 2, a simplistic transformer mode! is given; then in chapters 3 . -1 and .5 we calculated the

capacitances, inductances and resistances based on this unrealistic transformer model. The terminal

admittance matrix is derived in chapter 6. h more realistic transformer model is considered in

chapter 7 and the corresponding capacitances. inductances and losses are calculated. The terminal

admittance matrix is obtained and cornpared with rncasurement results. Finally. in chapter 8.

conclusions are given.

Chapter 2

A Simplist ic Transformer Model

Whenever we need to analyze the performance or calculate the parameters of a physicai element.

first we should establish a mathematical model. In this chapter, we start with the well-known basic

circuit equivalent transformer model. Then we discuss the realization of this model. Based on the

real configuration of the transformer, we propose the finite elernent model. Finally, the detailed

equivalent circuit mode1 is presented.

2.1 Basic Equivalent Circuit Model

As we stated in Chapter I , the transformer is one of the rnost familiar equiprnent in electrical

engineering, yet it is one of the most difficult power device for rnodeling accurately. During the last

40 years, many researchers were interested in transformer modeling.

From the most basic electric machine books [22]. we can find the equivalent circuit model for

transformers (Fig. 2.1).

Figure 2.1: Basic Equivalent Circuit Mode1 of Transformer

Ri and R? are the resistances of the primary and secondary windings. .Yll, .Yr2 are the leakage

inductances of the primary and secondary windings. RCi and Srni correspond to the core loss in

the magnetic material and the magnetizing inductance, respectively. Since a practical rnagnetic core

has finite permeability, a magnetizing current I,,, is required to establish flux in the core.

From this model we can see that only inductances and resistances are shown. It doesn't include

the capacitances between difkrent parts of the same winding, bctween two different windings. and

between the windings and ground which rnay play a very important sole in the high frequency

analysis. In this model, the transformer is represented by lumped components. The whole winding

(high voltage winding and low voltage winding) is represented by a single inductor. So this model

is valid only for low frequency steady-state analysis. For transient analysis in which the excitation

may include high frequency components. we can consider the transformer as an assernbly of discrete

components and include the effects of capacitances.

2.2 Realization of the Mode1

Corresponding to the basic equivalent circuit model shown in Fig. 2.1, wc have two basic transformer

configurations shown in Fig. 2.2(a) and Fig. 2.2(b)

(a) Cors 'Type (b) SheU Type

Figure 2.2: Transformer Configurations

Consider these two configurations and the specific transformer we are interestcd in. N'P find that

(a ) is very difficult to analyze using a 2 - 0 finite element rnodel. However, in (b) if the iron core is

round as shown in Fig. 2.3, we can use an axisymmetric finite element model to analyze i t . This

configuration is similar to the transformer model we created. Once we prove that the apprmch we

propose works, we can calculate the parameters of the real transformer accurately by using a 3-D

finite element model.

2.3 Measurement Method

The high frequency transformer model (HFT) described in Chapter 1 requircs the nodal admittance

matrix as a function of frequency. By measurernent a frequency scan is made with a variable

11

- Cylindrical Core

+Tank W a l l

Figure 2.3: Axisymmetric Transformer

frequency generator connected to one of the terminals, while al1 other terminals are grounried. 'The

currents (magnitude and phase angle) measured at each of the terrninals are directly proportional

t.o one column of the nodal admittance matrix. Repeating the procedure with each of the terminals

results in the nodal admittance matrix [ Y ( J ) ] as a function of angular frequency J. . h u m e the

sirnplest case in which there are only two windings and four terminals a s shown in Fig. 2.4. The

admittance matrix will be a 2x2 matrix:

First we connect the voltage source V ( f ) to terminal 1. the other terminals are grounded. iCé

can rneasure the currents at terminals I and 2. The result will be

- - -

Figure 2.4: Measurernent of Transformer Characteristics.

then the value of the first column of the admittance rnatrix which is the value of current provided

that the magnitude of 1: is I . By this rnethod we can obtain each column of )',,(&)

2.4 Finite Element Mode1

Based on the transformer shown in Fig. 2.3. we derive the finite element model in Fig. 2.3.

Since it is very difficult to model each turn as a separate conductor. we bunch several turns

as a section. In a real transformer, the voltage drop across adjacent turns is small cornpared with

the total voItage drop. In electric field calculations, we consider each section as an equal potential

conductor. -4lthough at high frequency, the current distribution in the conductor is non-uniform due

13

Figure 2.5: Sketch of the Finite Elernent Model

to skin and proximity effects, the cross-section of each conductor compared to the cross-section of

the whole winding is small and thus we still could assume that the current distribution in the whole

winding cross-section is uniform when we calculate the inductance. Our magnetic field calculation

is based on this assumption. However, when we calculate for the los, we will consider the non-

uniform distribution of the current density across the conductor cross-section. In Fig. 2.5, each of

the small quadrilaterals represents one section. This is an axisyrnmetric problern. At this point, we

could calculate the electric field, obtain the self-capaci tances of each section and mutual-capaci tances

between any two sections. Then we coutd calculate the magnetic field, obtain the self-inductances

of each section and mutual-inductances between any two sections. As for the loss of the windings.

we will calculate it according to the length and cross-section of each turn . And we will estimate

the iron core loss by empirical forrnulae. Here, we will ignore the loss of insulating materials. since

compared with the Ioss in the windings, insulating loss is very small. After we obtain the value of

each circuit element, (C, L , hl , R) we can derive a det~i led equivalent circuit network.

2.5 Detailed Equivalent Circuit Model

Baed on the above, we propose the detailed equivalent circuit mode1 which is shown in Fig. 2.6.

Figure 2.6: Detailed Equivalent Circuit 5Iodel

In Fig. 2.6, the shunt capacitances which are connected to ground represent self-capaci tances

of each winding section. The capacitances between each section are rnutual capacitances. The

inductances are in series with the winding loss. This part is different frorn the equivalent circuit

provided in [7] which put the winding loss in parallel with the inductances. Al1 the inductances are

coupled with each other. The core loss is in parallel with the inductances and winding loss. A more

detailed explanation is given in chapter 6.

Then we can simulate the measurernent procedure. Apply a frequency dependent voltage Ci(&)

a t terminal 1, ground the other terminals and calculate the current at each terminal.

Chapter 3

Capacitance Calculations

There exist several methods to calculate capacitances. For a simple geometry, capacitances can

be calculated from the gmrnetry or it can be calculated from the charges on the conductor and

the potential difference between the two conductors. However, for a multiconductor system such

as the transformer, the above mcntioned two methods are not appropriate. Since we can not find

an analytical solution easily. In this chapter. another approach for calculating self- and mutual-

capacitances is described. The main idea for this approach is to find the potentiat coefficient matrix

from which we can obtain the capacitance matrix with Little difficulty. The irnplementation of this

procedure is described in detail. .i\ simple example is used to validate the approach and numerical

results for the zelf- and rnutual- capacitances of transformer coils are presented.

3.1 Principle of Capacitance Calculation

There are two different kinds of capacitances in the equivalent network. One is a shunt capacitance

which is between windings or between windings and ground, the other is a series capacitance which

is between different sections of one winding.

3.1.1 Self- and Mutual- Capacitances

By definition, capacitance is the ratio of charges on the conductor and the potential of the conductor

relative to ground potential. However, capacitance is determined only by the rnaterial properties

and the geometry. It does not depend on the potential or charge. The capacitance between the

conductor and the ground is referred to as self-capcitance.

Lisually when we talk about capacitance, we talk about the capacitance between two conducting

bodies which are oppositely charged. The capacitance of such a two-conductor system is the ratio of

charge on one conductor to the potential difference between the two conductors. This capacitance

is referred to as mutual-capacitance.

3.1.2 Capacitance of a Multi-Conductor System

For a rnulti-conductor system. calculation of the self- and mutual- capacitances is not as straight-

forward as for a two-conductor system. However, the results obtained for a twcxonductor system

can be extendeû to the case of the multi-conductor system [23].

Assume there are N conducting bodies in empty space as shown in Fig. 3.1.

From the uniqueness theorem for solutions to Laplace's equation, we have the following results:

The distribution of electric charges over the outer conducting surfaces S,, is fully specified if one

Figure 3.1: A System of 'Y ' Conducting Bodies

knows either (1 ) the potential of each conductor or (2) the total charge on each conductor.

It is well known that Laplace's equation for electrostatics is linear. Linear equations have the

property of superposition.

Based on the above theory, now suppose that a positive unit charge is placed on the first conductor

while al1 other conductors are uncharged. This produces the potentials pli. p z l , ...,Ps, on the S

conductors respectively. Then if a charge QI is placed on the first conductor. with al1 others left

uncharged, the potentials of each conductor will be:

Similarly, if placing a positive unit charge on the nth conductor Sn and leaving the others

uncharged produces potentials:

Then placing Q, on Sn and maintainhg the other bodies uncharged will yield potentials

p i n Q n - p l n Q n ...- p.\*nQn

The effect of charges Q I , Q?, ..., Q.v on the !Y' bodies is described by the following 1V equations

The above equations give the potentials in terms of the charges. The factors p,, are known

as potential coef ic ients; They are purely geornetrical quantities which dcpend on the size, shape.

orientation, and position of the various conductors.

The following :V equations give the charges in terms of the potentials:

The coefficient ci j represents ratios of two determinants involving the p , j s . Thus the c,,s are

also purely geometrical quantities, depending on the size, shape, orientation, and position of each

conducting body. The q,s are called coeficients of capucrtonce, and the c,,s(i f j ) are called

cwflcients of electrostatic tnductton. I t is ciear t hat the rnductron coepcient m a t n r can be obtained

by inverting the potential coeflcient m a l r i r . Due to symmetry we have

From equations (3.1) and (3 .2 ) . we have

[cl = b1- l

Equations (3 .2) may be rewritten in a more revealing form by making the substitutions

which leads to the following !V equations

From (3.7) and the definiton of self- and mutual- capacitances, we can easily identify the quan-

tities Ci, and Ci, as the self-capacitance of the it h body and the mutual capacitance between the

ith and j t h bodies, respectively. Thus the entire set of equations may be interpreted in terms of a

capacitance Cii between the ith body and ground, plus capacitances C,J between the ith and j th

bodies of the systern. These capacitances are purely geometrical quantities and cas be deterrnined

by finite element calculat ions.

3.2 Floating Potential Approach

Csually, in the finite elernent method, the capacitance will be calculated by computing the energy

stored in the electric field surrounding ;V conductors [23]. This procedure requires .t'(.Y + t ) / 2

separate FE solutions [24].

Xccording to the principle we described in the above section, for a multi-conductor system. if

we cari calculate the potential coefficient matrix. then by inverting the potential coefficient matris.

we can obtain the induction coefficient matrix. From the induction cofficient matrix, it is easy to

calculate the self- and mutual- capacitances shown in (3.5).

From (3.1), if we first impose charge QI on conductor Si and leave al1 the other conductors

uncharged. and then calculate the Aoating potential of each conductor by solving Poisson's equation,

we have

From this equation, the first colurnn of the potential coefficient rnatrix can be obtained. By

following the sarne procedure described above but by imposing the charges on conductors 2. 3. ....

S. respectively, we can calculate the potential coefficient matris column by column. This recently

published approach is called the f h t t n g potential appmch (FPA) [24].

23

The FPA requires Poisson's equation to be solved 1%' times. However, because of syrnmetry

p;, = p j i , and the fact that the conductor pcrssessing the charge must De at the most positive

potential 1231, it suffices to solve Poisson's equation oniy once and to solve Laplace's equation N - 1

times. When Laplace's equation is solved, the j-th conductor's potential is fixed by previous results.

while the other N - 1 conductors have floating potentials (FP) .

After we obtain the potential coefficient matrix, we can simply invert it to find the induction

coefficient matrix.

Validation

Before we use the FPX to solve for the mutual- and self- capacitanccs. we give a n esaniple to prove

that our approach is correct and our program works properly.

3.3.1 Cornputer Program

The cornputer program we used is called RLAP2D. This program can solve the equation

for the steady-state tirne-harmonic electric scalar potential V. Here LY. is the angular frequency of

the source, is the real part and i' is the irnaginary part of cornplex charge density. and u is

the conductivity. E' and E" are the real and irnaginary parts of comples perrnittivity. In al1 our

calculations, 5" and are set to zero.

3.3.2 Example

The case we use to validate the approach is to calculate the self- and rnutual- capacitances of two

concentric spheres as shown in Fig. 3.2. Since the geometry is symmetric, we only show a quarter

of the spheres.

Figure 3.2: Two Concentric Spheres

First we move the outer boundary so that the influence of the boundary on the electric potcntiaI

in the region of interest can be ignored. Fig. 3.3 shows the relationship between outer boundary

position and the value of electric potential as a function of r.

In Fig. 3.3, we can see that when the circular outer boundary moves from 2685 mm to 7270 mm.

there is no big difference between the electric potentials. So, we use '268.5 mm s the outer boundary

when we remesh (make the grid finer) the area of interest.

Fig. 3.4 shows the equipotential lines of the solution region when we put the charges on the inner

sp here.

Figure 3.3: Relationship betwecn Electric Potcntial and Borindary Position

3.3.3 Cornparison with Analytical Results

The 2 x 2 [Pl-niatris cornputed by the FP FE11 compares with the analytical [Pl. The finite element

rnesh is shown in Fig. 3.5.

Analytical [pl-rnatrix Finite elernent [pl-matrix

2.96575 2.2 1598 [3.00000 2.?1000] [ ] 2.25000 2.25000 2.2 1598 2.2 1598

Table 3.1 shows the results of self- and rnutual- capacitances of the two concentric conducting

spheres. LVe can see that when the grids become small. the value of self- and mutual- capacitances

converge to the analytical results. This means that the prograrn and approach we used to calculate

the capacitances are accurate and converge to the analytical solution.

SA V N Conductor Sphere

250 - Equipotential Lines

200 -

Figure 3.4: Equipotential Lines of the Solution Region

3.4 ResuIts

We will use the approach described above to calculate the capacitances of a simple transformer. The

configuration of the transformer is shown in Fig. 3.6. Several bunches of winding coils are considered

as one conductor. We make the following assumptions about the transformer:

The electric field of the transformer winding is axisymrnetric.

Table 3.1: Self and Mutual Capacitances of TWO Spheres

sphere

sphere

Figure 3.5: Detail of Finite Element hfesh for the two Concentric Spheres

Displacement currents are neglected.

Temperature effects on the permittivity and conductivity are neglected.

O Frequency effects on the capacitances are neglected.

In this section, we will first treat the transformer as an air-core transformer then treat it as an

iron-core transformer to see if and how the capacitances will change. In [il_ the author calculates

the capacitance frorn the geometric information. By this approach ic is very difficult to consider the

effect of the core. Actually. the author of [il didn't take the influence of the core into consideration.

However, i f we use the finite element method, we can consider the effect of the core with l e s dificulty,

CORE Grounded

Tank Wall

Figure 3.6: Geornetry of the .Axisymrnetric Transformer

3.4.1 Capacitance of Air-Core Windings

In this section, we disregard the iron-core. The transformer is treated as an air-cote transformer.

The calculation frequency is 60 Hz. The induction matrix is shown below.

Table 3 2: Self- and Mutual- capacitances of Air Core Transformer (pf)

The self- and mutual- capacitances are calcutated. The results of self- and niutual- capacitarices

are shown in Table 3.2.

The other self- and mutual- capacitances can be deduced by the symmetric propetty of the

transformer.

Table 3.3: Self- and Mutual- Capacitances of Iron Cote Transformer(pf)

3.4.2 Capacitance of Iron-Core Windings

Here we consider the iron-core. The iron is an electric conductor compared with the insulation

rnaterials. The presence of the iron core will influence the capacitance calculation of the coi1 segments.

The c ipacitance matrix when iron core is considered is shown below.

7.8049 - 1.6072 -0.0055 -0.000 1 -3.256 1 -0.38 18 -0.0090

- 1.6071 8.4208 - 1.6052 -0.0055 -0.3628 -2.8007 -0.36115

-0.00.55 -l.60*52 8.4208 -1.8Oïl -0.00 13 -0.36 15 -2.8007

-0.0001 -0.0055 - 1.6072 7.8049 -0.005 1 -0.0090 -0.38 18

-3.256 1 -0.3627 -0.00 14 -0.005 1 8.2042 -2.ai3 12 -0.1849

-0.3819 -2.8008 -0.361.5 -0.0089 -2.5311 9.4526 -2.453.5

-0.0089 -0.3615 -2.8008 -0.3819 -0.1850 -2.4535 9.4526

1 -0.0051 -0.0014 -0.3627 -3.2561 -0.0858 -0.1849 -2.5312

The self- and mutual- capacitances are calculated by ( 3 5 ) . Table 3 .3 shows the results of self-

and mutual- capacitances of the transformer when we consider the presence of iron core.

Chapter 4

Inductance Calculat ion

Inductance is a very important parameter of the transformer. Actiiaily, the opcration principle of

the transformer can be traced to the nature of self- and mutual- inductances. Accurate calculation

of inductance values is crucial for the modefing of the transformer. T hus. inductance calculation is

an important subject of the transformer investigation and this thesis work.

There are several methods for calculating self- and mutual- inductances. They can be obtained

from (i) geometry and material property; (ii) the energy stored in the system: (iii) the vector

potential A. In this thesis, we calculate the self- and rnutual- inductances from the magnetic vector

potential.

4.1 Principle of Inductance Calculation

This section starts with a brief definition of self- and mutual- inductances; followed by a general

calculating formula; finally, a specific formulation for obtaining self- and mutual- inductances of

transformer windings is presented.

4.1.1 Self- and Mutual- Inductances

Inductance can be classified as self-inductance and mu tual-inductance. in an isotropic. linear

medium, the rnagnetic flux density B is proportiona! to the rnagnetic field fi. if the rnagnetic

field is induced by a closed current loop, then the magnetic flux through the area defined by the

circuit is proportional to the current. In other words, the magnetic flux linkage is proportional to

the current in this circuit.

where h is self magnetic flux linkage. L is self-inductance and I is the current.

Assume there are two circuits: circuit 1 and circuit 2. Magnetic flux linkage .IZ1 is induced by

current I l and it is linked with circuit 2. In a linear medium, .Izl is proportional to I I , i.e.

Here ,Cf2i is the mutual-inductance of circuit 1 with respect to circuit 2. It is easy to show that

.\Il? (mutual-inductance of circuit 2 with respect to circuit 1) and have the same value.

It is worthy to point out that, in a linear medium, self- and mutual- inductances are determined

only by the geometry and material property.

In order to calculate the inductance, first we need to calculate the flux. We can obtain the flux

O either from the surface integral of the msgnetic flux density 8 or from the line integral of the

rnagnetic vector potential À:

tn this thesis, we use the line integral to calculate the flux.

4.1.2 General Formulation

We derive a general formula to obtain self- and mutual- inductances from a known rnagnetic vector

potentiai .i.

In order to be simple and clear, we first investigate the rnutual inductance between two filamen-

tary circuits shown in Fig. 4.1. Assume that the permeability of the conductor and surrounding

m d i a is po. Current I I flows in circuit 1. The magnetic vector potential .-I produced by I r at dl?

is given by

So. the mutual flux linkage produced by II and linked with circuit 2 is given by

Figure 4.1 : Analysis of kf utua l Inductance

where 1, a n d II a r e t h e lengths of circuits 1 and 2, respectively; r represents the dis tance between

d l l and d l 2 .

The magnet ic vector potcntial .x produced by i2 a t dl 1 is given by

T h e m u t u a l flux linkage produced by I2 and linked with circuit is givcn by

The mutual- inductance between these two circuits is ob ta ineâ as follows:

If circuit 1 and circuit 2 have N1 and 1V2 turns respectively, the mutual-inductances can be

described as

where 1 1 and I:! represent the length of one turn of each circuit.

Figure 4.2: Analysis of Self Inductance

If the geometry and shape of circuit 1 and 2 are the same, overlapping as shown in Fig. 4.2, the

self inductance cannot be evaluated by (4.9) . Since here 1 , and 12 overlap. r in the above equation is

q u a 1 to zero, and the integral value becornes infinitely large. However, this difficulty can be tackled

by the following method:

The self-inductance of a filarnentary circuit can be categorized as outer-self-inductance and inner-

self-inductance [25]. The nurnber of times the flux (corresponding to outer-self inductance) linked

by the current is always an integer. So when we calculate the outer flux. we should use l2 as the path

of integration. However, the effect of current on the outside was shown in Fig. 4.2 as dashed line

1 1 . Thus, the calculation of outer-self-inductance is the same as that of mutual-inductance between

two circuits, I l and 1 2 .

For a circuit which contains :V turns, the outer-self-inductance is

Ci'sually, the value of the inner-self-inductance is much srnaller than that of the outer-self-

inductance. Therefore, the self-inductance can be approximated as follows:

4.1.3 Formulation for Transformer

In the previous section, we discussed the general formulation for finding inductances. However.

in practical cases, usually the generai formulation can be greatly simplified duc to sonie specific

properties of the problem. A s for the transformer. in order to make the calcuIation practical and

more economical, very often we treat it as an aaxisymmetric problem as described in chapter 2.

We describe below briefly the method used to calculate the flux linkage frorn a known magnetic

vector potential .i.

In Fig. 4.3 the small rectangular region represents one section of the transformer winding rnodel.

One section contains several turns of the winding. In order to obtain the flux linkage. first we need

to calculate the magnetic vector potential ï. By solving the equation

Figure 4.3: Basic Ceometry of Transformer Winding Cross-section

we obtain the numerical result for -4. Here represents current density. The differential amount of

flux is given by

Therefore, the differential flux linkage di1 is given by

where I is the total current, I' is the portion of current which linked with the flux. f i s the current

density. By integrating d.1 we obtain the flux linkage:

Figure 4.4: Integration in Polar Coordinates

Obviously, once we obtain the flux linkage. it is not difficult to calculate the self- and mutual-

inductances frorn ( 4 . L) and (4 .2 ) , respectively.

4.2 Approach to Inductance Calculation

4.2.1 Calculation of the Magnetic Vector Potential

For the purpose of analysis. the transformer is assumed rotationally symmetric about a core leg. as

indicated in Fig. 2.3. The space occupied by the insulation and windings is assumed te be rectangular

in the r-z plane. Furthermore, we will consider the skin effect. since skin effect has influence on the

inductances.

In the insulation and winding space, the magnetic vector potential .4 must satisfy (4.13). To solve

this equation, we use the finite element program S L A P 2 D . This program needs a data input file

containing information on nodes. elements. boundary conditions, rnaterial properties and applied

frequency. The output da ta file of SLAP2D gives the numerical value of the magnetic vector

potentia] at each node with r-z coordinates. Also, the magnetiç vector potential .x will Vary with

the frequency W .

4.2.2 Calculation of L and M

Once we obtain the value for the magnetic vector potential .4 at each node, we can integrate it to

get the flux linkage.

For instance, consider four conductors as shown in Fig. 4.5:

Figure 4.5: Configuration of Four Conductors

We assign the ctirrent I l to conductor 1. In matrix form the flux linkages can be represented as

follows:

Then we have

where L,] is self inductance if i = j; Lij is rnutual inductance if i # j . ;\,, is the flux linkage.

By solving the diffusion equation (4.13) once, we can get one colurnn of the inductance matrix

corresponding to a specified frequency. Using the sarne methodology, by assigning the current at

conductors 2.3 and 4 , we can calculate the inductances in columns 2, 3 and 1.

4.3 Validation

For any numerical method we need to verify its correctness and accuracy. Norrnally. there are two

methods to do this. One is to compare the numerical results with the experimental results: the

other is to compare the numerical result with the analytical results. The former method is direct

yet usually expensive. The latter method can be readily applied by finding a simple geometry which

has an analytical solution.

In this thesis, we use the latter rnethod to verify the approach. There are two steps: First, we

will show that the program we use is correct; Then we show that the approach used to calculate the

mutual inductance and self inductance is correct.

4.3.1 Validation of the Program

First, we calculate the inductance of an infinitely long solenoid. Fig. 4.6 shows the configuration.

This is an axisymmetric problem. The boundary at 'a' is a Dirichlet boundary. The boundaries at

'b' and 'd' are Neumann boundaries. As for boundary 'c', we move it outward along the r-mis until

its influence on the field solution can be ignored. Fig. 4.7 shows the results for shifting the position

of boundary 'c'

Figure 4.6: Configuration of Solenoid

From Fig. 4.7 , we can see that when the boundary rnoves from 5Omm to 100mm. the resu1t

change is more significant than that when the boundary moves from lOOrnm to 200rnrn. The results

for the boundary at LOOmm and 200mm are sirnilar, so we set out boundary at lOOmm to do the

rest of the calculations.

Fig. 4.8 shows the relationship between grid size and the value of inductance. From this Figure

we can see that when the grid becornes finer, the numerical results become closer to the analytical

results. This means that the numerical results of the program SL.4P'LD converge and are correct.

Figure 4.7: Magnetic i'ector Potential vs. Distance of Boundary ' c '

Figure 4.8: Relationship between Grid Size and Inductance

4.3.2 Validation of the Approach

In this example, we calculate the self and mutual inductances of two parallel conducting rings with

the same geometry pararneter as shown in Fig. 4.9. Fig. 4.10 is an illustration of the solution region.

Figure 4.9; Two Parallel Coaxial Conducting Rings

Figure 4.10: Solution Region for the Two Conducting Rings

Fig. 4.11 shows the relationship between boundary position and the rnagnetic vector potential.

Frorn the Fig. 4.11. we can see that for the rnagoetic vector potential .x. the boundary at 200mm

and 300mm, there is only a slight difference. So we use the boundary at 200mm as our boundary

when we calculate the magnetic vector potential.

Figure 4.11: Slagnetic Vector Potential vs* Distance 'c '

Fig. 4.12 shows the equipotential lines of the two parallel rings. The current is imposed on

conductor 1.

Table 4.1 shows the results of self- and mutuzl- inductances we obtained frorn the magnetic

vector potential.

From Table 4.1, we can see that when the grid becomes srnall, the numerical results are getting

closer to the analytical results. The error is acceptable. This means the approach can be used to

obtain the self- and mutual- inductances of a multi-conductor system.

- z

Figure 4.12: Equipotential Lines of Magnetic Field for TWO Conducting Rings

4.4 Results

In this section, we give the results calculated based on the approach described above. The frequency

range is from 60 Hz to 200 kHz. Cornpared with the capacitance, the inductances are strongly

dependent on frequency.

First, the transformer is treated as an air core transformer. Tben it is treated a s an iron core

Table 4.1: Results of Self and Mutual Inductances ( x 10-9H)

*e%: percent error.

transformer.

4.4.1 Inductances of Air Core Windings

Fig. 4.13 shows the relationship between self-inductances and freqiiency of the air core transformer

windings.

O 0 L3

O r 0 r r x ~f

x

+ I -

I

Figure 4.13: Self Inductance of Air Core Transformer vs. Frequency

From Fig. 4.13, we can see that when the frequency increases, the inductances decrease. This

result is consistent with the phenornenon of skin effect. When the frequency increases. the &in

effect becomes more and more important. The current starts to be distributed more and more on

the surface of the conductor. Thus, the performance of the soiid conductor is like a hollow conductor,

and the inductance of a hollow conductor is smaller than that of a solid conductor.

Fig. 4.14 and Fig. 4.15 show the equipotential lines of the air core transformer at frequencies

6OHz and 2OOkHz. respectively.

Air Core 1-00 Hz

Figure -1.14: Equipotential Lines of Air Core Transformer at 6OHz

The results for self- and rnutual- inductances are shown in Tables 4.2. 1.3 and 4.4

4.4.2 Inductances of Iron Core Windings

Fig. 4.16 shows the relationship between self-inductances and frequency for the iron core transformer

windings.

The relationship between mutual inductances and frequency is sirnilar to that of self inductances.

Fig. 4.17 and Fig. 4.18 show the equipotentiaf lines for the iron core transformer at 60Hz and

200 kHz, respectively.

The results of self- and rnutual- inductances are shown in Tables 4.5. 4.6 4.7'.

Table 4.2: Self and Mutual Inductances of Air Core Transformer( x IO" H )

Table 4.3: Self and Mutuai Inductances of Air Core Transformer ( x IO-' H )

Table 4.4: Self and Mutual tnductances of Air Core Transformer( x ~ o - ' H )

Table 4.5: Seif and Mutual Inductances of Iron Core Transformer(x 1 0 " ~ )

Table 4.6: Self and Mutual Inductances of Iron Core Transformer( x IO'? H)

Table 4.7: Self and Mutual Inductances of Iron Core Transformer(x 10-'1

7.0 1.2130 1.0690 0.8507 1.1090 0.8328 0.6460 1 .1260 8.0 1.2070 1.0630 0.8464 1.1030 0.8286 0.6426 1.1210 9.0 1.2000 1.0570 0.8416 1.0980 0.8239 0.6389 1.1 150 10 1.1930 1.0510 0.8363 1.0910 0.8187 0.6348 1.1080 20 1.0950 0.9615 0.7640 1.0060 0.7484 0.5786 1.0160 30 0.9770 0.8533 0.6765 0.9035 0.6631 0.5104 0.90-57 40 0.8638 0.7494 0.5922 0.8056 0.5809 0.4446 0.7995 50 0.7652 0.6584 0.5183 0.7208 0.3088 0.3866 0.7069 60 0.6824 0.5815 0.4559 0.6499 0.4478 0.3374 0.6290 70 0.6135 f 0.5173 0.4036 0.5912 0.3967 0.2961 0.5643 80 0.5560 0.4635 0.3597 0.5425 0.3539 0.2612 0.5105 90 0.5079 0.4182 0.3227 0..5018 0.3178 0.2317 0.4655 100 0.4671 0.3798 0.2912 0.4674 0.2871 0.2066 0.4277 110 0.4322 0.3469 0.2642 0.4380 0.2608 0.1850 0.3955 120 0.4021 0.3185 0.2409 0.4125 0.2381 0.1663 0.3681 130 0.3759 0.2938 0.2206 10.3902 0.2184 0.1000 0.3444 140 0.3529 0.2721 0.2028 0.3706 0.2011 0.1337 0.3238 150 0.3325 0.2530 0.1870 0.3530 0.1859 0.1231 0.3059 160 0.3143 0.2360 0.1731 0.3373 0.1725 0.1120 0.2900 170 0.2980 0.2208 0.1606 0.3230 0.1605 0.1021 0.2635 180 0.2832 0.2072 0.1494 0.3100 0.1498 0.0933 0.2523 190 0.2698 0.1950 0.1394 0.2981 0.L402 0.0855 0.2423 200 0.2576 0.1838 0.1303 0.28'71 0.1315 0.0784 0.2283

"

Air Core f=200 kHz

Figure 4.15: Equipotentiai Lines of Air Core Transformer at 200 kHz

Figure 4.16: Self Inductance of Iron Core Transformer vs. Frequency

bon Core

Figure 4.17: Equipotential Lines of Iron Core Transformer at 60 Hz

Iron Core 1=200 kHz

I Core

Figure 4.18: Equipotential Lines of Iron Core Transformer at 200 kHz

Chapter 5

Loss Estimation

The loss in a transformer is also an important parameter in design. manufacture and operation. .r\lso

it is the most difficult part to calculate accurately. Most of the references estimate it by empirical

data or experiments.

The loss in a transformer can be roughly categorized a s winding loss, insulation material l o s ,

rore loss and stray loss. The l o s is dependent on the operating temperature. source frequency. It

is very difficult to predict. Compareci with the winding Ioss and core l o s , insulation loss and stray

loss are rather srnall, hence we neglect them in this thesis.

Loss

The winding loss is due to the existence of the conductor. N'hen source frequency changes, because

of the skin and proximity effects, the winding loss changes. Skin effect causes uneven current

distribution in the conductor. In general, the resistance of the conductor increases with the frequency.

The winding loss applies to both the air core and iron core transformer we investigated. LVe

assume that the winding loss is the same for both cases. The geometry of the winding in both cases

is the same. Cornpared with the windings, the conductivity of the iron core is much lower than

that of the copper. Therefore, the influence due to the proximity of the iron core on the equivalent

winding l o s is negligible.

5.1.1 Formulation of the Skin Effect Problem

Skin effect is due to the uneven current distribution in the conductor when a time varying source

is applied. The relationship between the time vsrying source current density and the resulting

magnetic field can be derived frorn Maxwell's equations [26] and is given by:

where ,u is permeability, J is the angular frequency and u is conductivity. Displacement currents

are neglected. The total measurable current can be expressed as:

From (5.2), can be expressed in terms of the known current I and the unknown rnagnetic

vector potential -4 which then yields the following equation:

Here a is the cross-sectional area of the conductor carrying the current I . The total current densi ty

Jt can be calculated as:

where J, can be obtained from (5.2). Once we obtain the current distribution, the loss per unit

length of conductor is given by

Thus, by solving (5.3), we can obtain the frequency dependent resistance of the conductor

5.1.2 Brief Description of the Computer Prograrn

In the thesis, we consider the skin effect and proximity eff't, and use the finite element method

to calculate the frequency dependent resistances to represent the frequency depcndent losses of the

windings. The program [27] we used is skin61.m. It is a rnatlab program developed by Sarui [27]. We

modified the storage rnethod of the program frorn a full matrix storage to a sparse matrix storage.

Hence, now it can solve relatively large practical problerns.

The program needs three input data files: ( i ) skinls.dat contains the node information. ( i i )

skin2s.dat contains the element information, and ( i i i ) skin3s.dat contains the boundary information.

Since this program has already been verified by rnany examples in [27], we use it directly to

calculate the winding resistances.

5.1.3 Results of Resistances

Assume that each section of the winding consists of two conductors a shown in Fig. 5.1.

where there is an insulating layer between the two conductors. In this specific example. we have 16

conductors in total. The properties of the copper are a = 5.8 x 10CQ-' m- l . p , = 1. LVe assume

that the conductivity of air and insulating materials is zero.

Figure 5.1: Winding Configuration

Table -5.1: DC Resistances of Conductors ( x 10-'R)

- - 1 1

Rg Rio Ri1 R i z , Ria Rr4 Ris R i s b 3.298 9.298 9.298 9.298 9.960 9.960 9.960 9.960 L.

The DC resistances of each conductor are listed in Table 5.1. These DC resistances are calculated

from the geometry of the transformer.

Fig. 5.2 shows the relationship between the ratios of AC to DC rmistances and frequency. In this

figure, only the curves for conductors 1, 2, 3 and 4 are shown. The tendencies of the other curves

are similar.

Table 5.2 and Table 5.3 contain the ratios of AC to DC resistances a t different frequencies. In

order to be consistent with the inductance calculations, the frequencies we calculated are from 60

Hz to 200k Hz.

From Fig. 5.2 and Tables 5 . 2 and .5.3 we can see that when the frequency increases. the equivalent

AC resistances increase. Above a certain frequency, the curves for the AC resistances becorne flat.

Table 5.2: Relative Resistances RAc/ RDC VS- Frequency

Table 5.3: Relative Raistances RA=/ RDc vs. Frequency

Figure 5.2: Relative Resistance vs. Frequency

These results are consistent with our expectations

5.2 Core Loss Evaluation

The core loss of a transformer has two parts: ( i ) eddy current l o s ; and [ii) hysteresis loss. In modern

transformers, the value of hysteresis 1 0 s is about one third of the eddy current loss of the core. In

this thesis, we only consider the eddy current loss while neglecting the hysteresis 10s.

In order to estimate the core loss, we select the core rnaterial according to XSTM Standard A677

[28] as 36F145. This is a kind of nonoriented electrical steel. The properties of this rnaterial are

Table 5.4: Grade Designations and Maximum Core-Loss Limits for 36F 145

listed in Table 5.4.

The laminated iron core carries the magnetic flux which links with the transformer windings. [ts

-

size and magnetic stress determine the capacity and no load characteristics of the transformer. The

AIS1 type M-15

ASTM *, type

36F145

no load losses consist of the eddy current loss and hysteresis loss of the iron cote.

The eddy current loss is caused by the eddy currents induced in the laminations of the core. Its

Nominal t mm 0.36

value is estimated by [29]

where P, is the eddy current loss in W. f is the frequency in H z , Bm is the peak value of the

flux density in T, t is the thickness of the individual steel lamination sheets (minimum dimension

perpendicular to the magnetic flux lines) in m, p is the specific resistance of the sheet material in

I lm, d is its density in kg/m3 and m is the mas of the iron core in kg.

The equivalent resistance corresponding to the eddy current loss is given by

P,,,,,(GO Hz) W/kg 2.53

where Er is the induced voltage in V . From [33], for a sinusoidal signal

d kg/m3

7650

P M R . m

5200

where Ni is the number of turns in the coi1 and g, is the peak value of the flux in W b .

From (5.6), (5.7) and (.5.8), we have

where S is the cross-sectional area of the iron core. For our specific case, :VI = 8, p = 52 x 108Q. m.

t = 0.36rnrn, d = 7630ky/m3. S = 1.257 x 10-~rn?, m = 0.77Cg. From these values, we obtain

R, = 4-97 x IO%?.

5.2.2 Discussion of Core Loss

The reader may be surprised by the fact that the equivalent resistance for core loss given in the

previous section is independent of source frequency. Most of us might feel intuitively that the core

loss is highly dependent on the frequency.

Actually, as shown in (5.6). the eddy current loss of the iron core is proportional to the frequency

squared. Also, the core loss is proportional to the voltage squared. From (5.8) we can see that the

induced voltage is proportional to frequency if the flux linkage remains at the same value. This

means that if the core loss is a function of the frequency squared, the resistance which represents

this los . should be a constant.

Chapter 6

Equivalent Admit t ance Matrix

The finai step is to obtain the admittance matrix frorn the known lurnped circuit elements. In

this chapter, we will calculate the admittance matrix by using ATP EMTP[34] [35]. Two cases are

considered: ( i ) air core transformer and (ii) iron core transformer. From the results of capacitances

and inductances we calculated in chapters 3 and 4, we can see that for both air-core or iron-core

transformers, we need only consider the self- and mutual- capacitances of two adjacent coils (e.g.

Cl 1 , Clz, Cis). Since the value of CI3, for instance, is alrnost & the value of Cl?, it can be ignored.

However, for inductances, we have to consider al1 the self- and mutual- inductances, since from our

finite element results, at a specific frequency, the magnitudes of al1 the inductances are of the sarne

order.

6.1 Air-Core Dansformer Admit tance Matrk

For the air core transformer, there is no core l o s , only the winding los is taken into account. The

equivalent circuit is shown in Fig. 6.1.

Figure 6.1: Detailed Equivalent Circuit of the Air-Core Transformer

Terminals 1 and 3 represent the two terminais of the low voltage winding. Terminals 2 and 4

represent the two terminals of the high voltage winding. The voltage source is connected to terminal

1. X frequency scan is made by this voltage source. The other three terminals are grounded. By

inputting the circuit parameters into EMTP. we can obtain the current at terminais 1 and 2. Then.

one column of the nodal admittance matrix can be determined. Using the sarne procedure, we can

obtain the other column of the admittance matrix.

Fig. 6.2, Fig. 6.3 and Fig. 6.4 show the magnitude and phase angle of YIL, Yll and as a

function of frequency. The frequency range is from 60Hz to 200 kHz.

Figure 6.2: Y 11 of Air-Core Transformer vs. Frequency

Frorn Fig. 6.2, Fig. 6.3 and Fig. 6.4 we can see that there is no resonance up to 200kHz. This

is because we have very small inductances (L. 11) and capacitances for the sirnplistic transforrrier

rnodel compared with a real transformer. As we stated in chapter 1, the reason for doing this is

to obtain a numerically stable, rather smooth variation of the admittance matrix parameters with

frequency. The results for phase angle are consistent with the magnitude. .\ISO. as we expected, the

resonances appear a t relatively high frequencies. In Fig. 6.5, Fig. 6.6 and Fig. 6 . 7 , we expand the

frequency range to 600hiHz. We can see that whenever there is a phase angle shift. a peak value

occurs in the magnitude.

Figure 6.3: Y2 1 of Air-Core Transformer vs. Frequency

MawmdO O( Y P m ' C a r o ) - Fr~qr#npl

Figure 6.4: Y22 of Air-Cote Transformer vs. Frequency

Figure 6.5: Y 1 1 of Air-Core Transformer vs. Frcquency

U.g"hd. 01 Y 2 t W.) - h q u q

Figure 6.6: Y21 of Air-Core Transformer vs. Frequency

Figure 6.7: Y22 of Air-Core Transformer vs. Frequency

6.2 Iron-Core Transformer Admittance Matrix

For the iron-core transformer, the problem is more complicated. In addition to the winding loss, we

also consider the core l o s . The equivalent circuit is shown in Fig. 6.8.

The other procedures are the sarne as in the case of the air-core transformer. Fig. 6.9, Fig. 6.10

and Fig. 6.11 show the magnitude and phase angle of YI 1 , \>l and Y??. respectively. The frequency

range is from 6OHz to 200 kHz.

As for the air-core transformer. in Fig. 6.12, Fig. 6.13 and Fig. 6.14, we expand the frequency

range to 600 MHz in order to see the resonances.

Figure 6.8: Detailed Equivalent Circuit of the Iron-Cote Transformer

6.3 Traditional and New Equivalent Circuit

From the new equivalent circuit we described above, at power frequency, al1 the capacitances behave

as open circuits. Therefore, the effect of capacitances can be neglected. Hence. it is readily seen that

this new mode1 can be represented as shown in Fig. 6-15. Here L i and L2 are total self-inductances

of the primary and secondary windings respectively. R i and R2 are the total winding resistances

of the prirnary and secondary windings. Xote that for the aircore transformer RcL and RC3 should

Figure 6.9: Y 11 of Iron-Core Transformer vs. Frequency

be omit.ted from the equivalent circuit shown in Fig. 6.15. In Fig. 6.15, the prirnary and secondary

windings are electrically isolatecl and rnagnetically coupled.

Consider the T equivalent circuit [30] for the transformer shown i n Fig. 6.16.

In Fig. 6.16, the right hand side is an ideal transformer. The transformation ratio is n. On the

left hand side, La, L b , R a , Rb are leakage inductances and winding losses of primary and secondary

windings, respectively. R, represents the core los . Lm is the inductance w hich governs rnagnet izing

current.

Using two port network equivalent method, we can calculate the parameters in Fig. 6.16. so that

the dashed line box in Fig. 6.15 can be represented as the dashed line box in Fig. 6.16. First. we

select the same reference a t point 1 and 2 in Fig. 6.15. The potentials of points 3 and 4 are different

with respect to the reference. However. if we rnultiply the potential a t point 4 by n. I*? = nL*i. then

the potential of point 4 is the same as that of point 3. At this point. in the electrical sense. al1 the

Figure 6.10: Y21 of Iron-Core Transformer vs. Frequency

mgrm% oi VZZ [ i rme l - Ftsquencl 1001 1

Figure 6.1 1: Y22 of Iron-Core Transformer vs. Frequency

Figure 6.12: Y 11 of Iron-Core Transformer vs. Frequency

Figure 6.13: Y21 of Iron-Core Transformer vs. Frequency

Figure 6.14: Y22 of Iron-Core Transformer vs. Frrquency

nodes which have the same potential can be connected together. Thus points 1 and 2 are connected.

points 3 and 4 are connected (2' and 4' in Fig. 6.16). Since we multiply the secondary voltage by n.

in order to maintain the sarne power, we have to divide the current of the secondary winding bu n.

So I; = 12 /n . Severtheless. a e still have problem if we only connect the corresponding nodes. the

two sides are at the same voltage level. The transformation ratio is rnissing! This problem can be

easily remedied. An ideal transformer with ratio n : 1 can be used. This is why we have the right

hand side part in Fig. 6.16.

The circuit equations for Fig. 6.1,; are given by:

Figure 6.15: Simplifieci Equivalent Circuit

In order to be clear. here ive neglected the currents which flow in Rci and R,?. since usually they

are very large compared with winding loss.

The circuit equations for Fig. 6.16 are given by:

By equating the parameters, we have the following results:

Figure 6.16: Traditional Eqiiivalent Circuit

6.4 Summary

In this chap te r , we calculated the admi t tance mat r ix from the known parameters. T h e new equivalent

circuit c a n take account of capacitances, inductances and losses. In a prcvious t ransformer circuit

network shown in [ T l , t h e winding loss is in parallei with the inductor. In o u r circuit equivalent

network, t h e winding loss is in series with the inductor. CVe think this is m o r e reasonable. T h e

results calculated are consistent wit h t h e circuit equivafent network.

However, o u r s imple t ransformer is designed without considering many factors. For instance.

there a r e only t w o windings. Each winding only has 8 turns. The distance between each t u r n is very

b i g T h i s m a y make t h e mu tual-capacitances a n d mutua l inductances between each t urn srnaller.

T h e s e factors m a y affect t h e results.

Chapter 7

A More Realist ic Transformer

In this chapter a more realistic transformer mode1 is considered. The capacitances. inductances and

resistances are caIculated. The admittance matrix is computed up to 200 k H z .

7.1 Geometry Description

The configuration and geometry parameters of the tranformer are shown in Fig. 7.1. These param-

eters are provided by Ontario Hydro. The conductors are Iabeled as in Fig. 7.1.

The windings are disc windings with the high voltage ( H V ) winding on the outside and the low

voltage (LV) winding on the inside. In order to rnake the calculation simple. the regulation winding

is neglected.

Figure 7.1 : Configuration of the Transformer

7.2 Capacitance Calculat ion

The self- and mutual- capacitances are calculated based on the geometry of the transformer shown

above. Fig. 7.2 shows the equipotential lines of the electric field when Li' disc No. 1 is excited. The

results of capacitances are shown in Table 7.1. The capacitances which are not shown in the Table

are very small cornpared with those Iisted.

Core

Figure 7.2: Equipotential Lines of the Electric Field of the More Realistic Mode1

7.3 Inductance Calculation

We used the sarne method as shown in chapter 4 to calculate the inductances. Fig. 7.3 and Fig.

7.4 are equipotential lines of the rnagnetic field when the frequencies are at 6OHz and 200 kHz,

respectively, and LV disc Yo. 1 is excited.

In Fig. 7.5 we show the curves for frequency variation of four self-inductances. The tendency

Table 7.1 : Self- and Mutual- Capacitances of Transformer(pf)

for the others are sirnilas. In this chapter, we only show one table of numerical results for the

inductances out of 20 tables. The other tables are listed i n the Appendix.

Loss Estimation

The l o s is estimated from the geometry. We assume there are 70 turns in each disc for the high

voltage winding and 20 turns in each disc for the low voltage winding. The properties of the copper

are a = 5.8 x 10'R-'m-',~, = 1. The resistances for each disc of high and low voltage windings

are Rh = 0.36780 and Rr = 0.043311, respectively.

Admittance Matrix

Fig. 7.6. Fig. 7.7 and Fig 7.8 show the magnitude and phase angleof YLi, and Y??. The frequency

range is from 60Hz to 1MHz. In order to see the curve more clearly in the frequency range we are

interested in (6OHz to 200 kHz), in Fig. 7.9, Fig. 7.10 and Fig. 7.11 we plot the curves up CO 200

kHz.

Since inductance is frequency dependent, by investigating the curves of inductances vs. frequenc!

Table 7.2: Self and Mutual Inductances of Transformer( x L O - ~ H )

Figure 7.3: Magnetic Field of the More Realistic Mode1 ( 6 0 Hz)

shown in Fig. 7.5, we selected the inductances and scanning frequency ranges as follows:

(a) For inductances a t 6OHz, do frequency scan from 60Hz to 1OkHz.

(b) For inductances at 20kHz, do frequency Kan from 1OkHz to 3OkHz.

(c) For inductances a t 40kHz , do frequency scan from 3OkHz to 60kHz.

(d) For inductances a t 8OkHz, do frequency scan from GokHz to 120kHz.

(e) For inductances a t 200kH2, do frequency scan from I'LOkHz to 1MHz.

7.6 Discussion

From Fig. 7 . 9 , Fig. 7 .10 and Fig. 7.11 we can see that when the phase angle shifts, there is a peak

value shown in the magnitude of the admittance matrix. Since we use a discrete inductance to

mode1 a conîinuous inductance, there are several points (IOkHz. 30kHz, GOkHz, 120kHz) a t which

the magnitude curves are not smooth. We can determine whether they are resonance points or not

Core

Figure 7.4: Wagnetic Field of the More Realistic Mode1 (200 kHz)

by investigating the curves for phase angles. Fig. 7.12 is intended to show the continuous curves for

magnitude and phase angle. In Fig. 7.12, we used the inductances at 60Hz and do the frequency

scan from 6OHz to 1MHz. In order to save space. we only plot Y 11.

Fig. 7.13 shows the experimental results of the magnitude for Yi , . We can see that the tendency

of this curve is similar to Fig. 7.9. The experirnent was done by Ontario Hydro.

Figure 7.5: SeIf Inductance of More Realistic Transformer vs. Frequency

Figure 7.6: Y 11 of More Reolistic Transformer

Figure 7.7: k'21 of More Reaiistic Transformer

U

- ; 2 5 i â ~ ! , ~ r . 0 u r r y - ~ ( n r ) i 10'

Figure 7.8: Y22 of More Realistic Transformer

Figure 7.9: Y 1 1 of More Realistic Transformer

Figure 7. IO: Y2 1 of More Realistic Transformer

Figure 7.1 1: 1'22 of More Realistic Transformer

Figure 7.12: Y 1 1 of More Realistic Transformer (Inductance at 60oz)

Frequency Hz

Figure 7.13: Experirnentat Results for Y 11

Chapter 8

Conclusions

8.1 Summary and Contributions

In this thesis, a new transformer model is developed. Based on this transformer model, a simplistic

transformer model is treated as Our fint attempt. The capacitances, inductances and losses are

calculated in the frequency range from 6OHz to 200 kHz by finite elernent method. The skin effect and

proximity effect are considered. Two cases - a simple AIR CORE reactor and a simple IRON CORE

transformer are treated in detail. The material properties and geornetry of the two transformers

are the same except for the permeability of the core. For the iron core transformer, p, = 100 is

used; For the air core transformer p, = 1. The new circuit equivalent networks for the transformers

are developed. The admittance matrix is obtained from the computed circuit parameters such as

capacitances, inductances and losses. After we gain the confidence in this new approach, a more

realistic transformer model is considered. The circuit parameters (C, L, M, R) are calculated and the

admittance matrix is obtained using the same procedure as described in the previous chapters. Also

we compared the numerical results with the experimental results, they have the same tendency. In

order to make the computation manageable, we selected less number of disc windings compared with

the teal transformer which is measured. This is the reason that the two results are not quantitatively

comparable. The whole procedure is justified step by step to show the feasibility and correctness of

the approach. The frequency responses of the terminal admittance matrix are calculated in terms

of magnitude and phase angle. The calculation results are consistent with the transformer model.

F'ut ure Work

The modeling and analysis described in this thesis could be improved. Since the mode1 is very

simple, it does not take into account many factors. For instance, instead of considering the variation

of iron core permeabitity as a iunction of frequency, we take it as a constant. As the next step,

wc could rnodel a real transformer instead of the simple model and take into account the frequency

variation of permeability. Ir! addition, we should use a 2-D axisymmetric mode1 instead of a real

3-D transformer. In the future a 3-D model should be used.

Appendix A

Numerical Result s

In this appendix, the numerical results of inductances for the more realistic transformer are iisted.

Table A. 1: Self and MutuaI Inductances of Transformer( x IO-' H)

Table A.2: Self and Mutual Inductances of Tranaformer(x ~ O - ~ H )

Table A.3: Self and Mutual Inductances of Transformer( x 1 0 - ~ H)

Table A.4: Self and Mutual Inductances of Transformer(x IO-^ H)

Table A.5: Self and Mutual Inductances of Transformer(x ~o'=H)

Table A.6: Self and Mutual Inductances of Transformer( x H )

' f(kHz) " L3.8 L3.9 L 3 . 1 0 , L3.11 L3.12 L3.13 L3.14 L3.15

0.06 2.9040 2.6660 2.4440 2.3760 2.4290 2.4670 2.4890 2.4960 "

0.30 2.9030 2.6650 2.4440 2.3760 2.4290 2.4670 2.4890 2.4960 "

0.70 2.9010 2.6640 2.4420 2.3740 2.4270 2.4650 2.4870 2.4940 '.

" 1.00 2.8990 2.6610 2.4400 2.3710 2.4240 2.4620 2.4840 2.4910

'

1.30 2.8960 2.6580 2.4370 2.3680 2.4210 2.4580 2.4810 2.4870 1.60 2.8910 2.6540 2.4330 2.3640 2.4160 2.4540 2.4760 2.4830 2.00 2.8840 2.6480 2.4280 2.3560 2.4090 2.4460 2.4680 2.4750

" 3.00 2.8610 2.6260 2.4070 2.3320 2.3840 2.4210 2.4430 2.4490 4.00 2.8290 2.5960 2.3800 2.2990 2.3500 2.3860 2.4080 2.4140

"

5.00 2.7890 2.5600 2.3460 2.2580 2.3080 2.3440 2.3650 2.3710 '

"

6.00 2.7430 i 2.5170 2.3070 2.2100 2.2590 2.2940 2.3150 2.3210 " 7.00 2.6910 2.4690 2.2630 2.1570 2.2050 2.2380 2.2590 2.2650 - 8.00 2.6350 1 2.4170 2.2150 2.1000 2.1450 2.1780 ' 2.1980 2.2040 9.00 2.5760 2.3630 2.1650 2.0390 2.0830 2.1150 2.1330 , 2.1390

'

10.00 2.5150 2.3060 2.1120 1.9750 2.0180 2.0490 2.0670 2.0720 20.00 1.9140 1.7510 1.6010 1.3600 1.3870 1.4060 1.4170 1.4210 30.00 1.4730 1.3430 1.2250 0.9298 0.9445 0.9552 0.9612 0.9625 40.00 1.1710 1.0630 0.9673 0.6634 0.6702 0.6748 0.6769 0.6764 50.00 0.9550 0.8637 0.7843 0.4974 0.4987 0.4992 0.4983 0.4961

Table A.7: Self and Mutual Inductances of Transformer( x IO-^ H )

Table A.8: Seli and Mutual Inductan

Table A.9: Seif and Mutual Inductances of Transformer(x 1 0 e S ~ )

Table A.10: Self and Mutual Inductances of Tiaasformer(x W'H)

Table A. 1 1: Self and Mutual Inductances of Transformer( x 1 0 - ~ H)

Table A. 12: Self and Mutual Inductances of Transformer( x 10e5H)

Table A.13: Self and Mutual Inductances of Transformer(x lO-=H)

- ' f(kHz) L i i , i r L i i . i s L i i , i s Li1,i . i L i i , i s Lii,io - L ~ i , l o 1 L i l , r i '

0.06 12.5800 11.4400 10.4500 9.5670 8.7800 8.0680 7.4210 1 6.8260 0.30 12.5800 1 1 .4400 10.4500 9.5660 8.7790 8.0670 7.4200 6.8250 0.70 12.5800 11.4400 10.4400 9.56 10 8.7740 8.0630 7.4160 6.8210 1 .O0 12.5700 1 1.4300 10.4400 9.5550 8.7680 8.0570 7.4 110 6.8160

'

1.30 12.5600 11.4200 10.4300 9.5460 8.7600 8.0500 7.4030 6.8090 L.

Table A.14: Self and Mutual Inductances of Transformer(x 10"H)

Table A. 15: Self and Mutual Inductances of Transformer( x 1 0 " ~ )

Table A. 16: Self and Mutual Inductances of Transformer( x 10-'H)

u' rn m

2 CV d

tn . 0

Chw - C h m =?a! 0 0

00 TP CV Y ? d

0 0 Q> u3 ? d

0 t- mua N m

z z N N ? m m

00 O O M b 54

00 % X 2

t- oa t-

2 CY Ch '9 0

a

4

rn

4

u3 b u? 4

0 rn - CV

O

g

Table A. 18: Self and Mutual Inductances of Transformer( x 10'")

Table A. 19: Self and Mutual Inductances of Transforrner(x 10-5H)

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