Transformer Winding Deformation and Insulation Characteristic Effects on Frequency Response

Transformer Winding Deformation and

Insulation Characteristic Effects on

Frequency Response Analysis

By

Mehdi BAGHERI

Supervisor: Dr.Toan PHUNG

Co-supervisor: A/Prof. Trevor BLACKBURN

A THESIS IN FULFILMENT OF THE REQUIREMENT FOR

THE DEGREE OF DOCTOR OF PHILOSOPHY

SCHOOL OF ELECTRICAL ENGINEERING AND TELECOMMUNICATIONS

FACULTY OF ENGINEERING

March 2014

THE UNIVERSITY OF NEW SOUTH WALES

Thesis/Dissertation Sheet Surname or Family name: BAGHERI First name: Mehdi

Other name/s:

Abbreviation for degree as given in the University calendar: Ph.D.

School: Electrical Engineering and Telecommunications

Faculty: Engineering

Title: Transformer Winding Deformation and Insulation Characteristic Effects on Frequency Response Analysis

Abstract 350 words maximum:

Frequency Response Analysis (FRA) is considered an accurate, fast, economical and non-destructive method for the detection of winding deformation within power transformers, providing detailed information on electrical properties of this asset. Changes in winding configuration, as well as other transformer active part structures would almost certainly cause variation in the frequency response spectrum. This can be exploited for mechanical defect recognition. On the other hand, transformer oil deterioration, temperature variation as well as water absorbed by the paper can cause transformer insulation characteristics to change over the time. In fact, capacitances, self- and mutual inductances and conductor resistances might be altered due to any changes in above mentioned factors. In turn, the frequency response of the winding will change accordingly. Thus in the interpretation of the FRA spectrum for evidence of winding deformation, the influence of insulation characteristic on the spectrum must be taken into consideration. FRA deviation due to the winding deformation or insulation characteristic changes becomes even more complicated to interpret when FRA baseline and measured spectra are taken under different temperatures and moisture contents. In such a case, existing FRA evaluation methods using statistical indicators are likely to reveal incorrect prognosis. Hence in this thesis, the aim of the research is to distinguish the insulation characteristic impacts on FRA spectrum from winding deformation. To this end, resonances and anti-resonances in FRA spectrum over different frequency bands are examined in detail and interpretations are provided. FRA deviation due to the transformer winding deformation is discussed analytically, modelled and simulated. The results are then compared to practical measurements. Insulation characteristic changes in transformer are studied through temperature and moisture variations to recognise their influences on FRA data. FRA capability in recognising moisture migration from the paper insulation of transformer winding is recommended and its potential application in transformer winding dry-out process evaluation is revealed in this research. Finally, possible offline and online solutions to distinguish the impact of moisture and temperature variations on winding deformation diagnosis are provided. Online FRA measurement and its required setup as a potential future approach in transformer condition monitoring are discussed.

Declaration relating to disposition of project thesis/dissertation

I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all property rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstracts International (this is applicable to doctoral theses only). ……………………………………………………… Signature

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……14/06/2014……….

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FOR OFFICE USE ONLY Date of completion of requirements for Award:

COPYRIGHT STATEMENT

‘I hereby grant the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstract International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation.'

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Date……………14/06/2014.…………………..............

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Signed…………………………………………..............

Date……………14/06/2014.…………………..............

ORIGINALITY STATEMENT

‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.’

Signed…………………………………………..............

Date……………14/06/2014.…………………..............

A THESIS DEDICATED TO MY FATHER AND SISTER

IN LOVING MEMORY OF MY MOTHER

Acknowledgment

I would like to express my sincere gratitude to my supervisor, Dr. Toan PHUNG, for his

continuous support throughout my PhD study. I cannot thank enough for his guidance,

insightful instruction, and ceaseless encouragements during the past years.

I would also like to give my special appreciation to my co-supervisor, A/Prof. Trevor

BLACKBURN. His inspiring minds and firm support have definitely taken my research to

another level. I have been deeply influenced by his enthusiasm for science, ambitious heart

in research, and greatest dedication to work.

In addition, I would deeply thank Dr. Mohammad SALAY NADERI my former supervisor

for his detailed instruction and fruitful discussions. His ongoing and valuable advice

allowed me to expand my knowledge broadly and gain very useful skills.

Last but not least, I am forever indebted to my father and sister, who gave me a beautiful

life and supported me for studying overseas emotionally. I also give my special thank to

my mother who could not survive to see this great time in my life. It is your love that

embraced me through the PhD and my life.

i

Abstract

Frequency Response Analysis (FRA) is considered an accurate, fast, economical and non-

destructive method for the detection of winding deformation within power transformers,

providing detailed information on electrical properties of this asset. Changes in winding

configuration, as well as other transformer active part structures would almost certainly

cause variation in the frequency response spectrum. This can be exploited for mechanical

defect recognition. On the other hand, transformer oil deterioration, temperature variation

as well as water absorbed by the paper can cause transformer insulation characteristics to

change over the time. In fact, capacitances, self- and mutual inductances and conductor

resistances might be altered due to any changes in above mentioned factors. In turn, the

frequency response of the winding will change accordingly. Thus in the interpretation of the

FRA spectrum for evidence of winding deformation, the influence of insulation characteristic

on the spectrum must be taken into consideration. FRA deviation due to the winding

deformation or insulation characteristic changes becomes even more complicated to

interpret when FRA baseline and measured spectra are taken under different temperatures

and moisture contents. In such a case, existing FRA evaluation methods using statistical

indicators are likely to reveal incorrect prognosis. Hence in this thesis, the aim of the

research is to distinguish the insulation characteristic impacts on FRA spectrum from

winding deformation. To this end, resonances and anti-resonances in FRA spectrum over

different frequency bands are examined in detail and interpretations are provided. FRA

deviation due to the transformer winding deformation is discussed analytically, modelled

and simulated. The results are then compared to practical measurements. Insulation

characteristic changes in transformer are studied through temperature and moisture

variations to recognise their influences on FRA data. FRA capability in recognising

moisture migration from the paper insulation of transformer winding is recommended

and its potential application in transformer winding dry-out process evaluation is revealed

in this research. Finally, possible off-line and on-line solutions to distinguish the impact of

moisture and temperature variations on winding deformation diagnosis are provided.

Online FRA measurement and its required circuit setup as a potential future approach in

transformer condition monitoring are discussed.

ii

Nomenclature

A Sinusoidal signal amplitude Ch.2

Ac Cross sectional area of core limb Ch.5

Arog Cross sectional area of each small loop in Rogowsky coil Ch.8

Ay Cross sectional area of core yoke Ch.5

B Magnetizing flux density Ch.2, Ch.5

c Local moisture concentration Ch.7

cs Series capacitance per unit length Ch.4

cg Shunt capacitance to ground per unit length Ch.4

C Total winding capacitance Ch.5

Cc Substance concentration Ch.7

Cd Total series capacitance of entire disks Ch.3

Ci Capacitance at ith resonance frequency Ch.7

Cs Total series capacitance Ch.3

Ct Total turns’ capacitance Ch.3

Ctt Turn-to-turn capacitance Ch.3

Cs-pair Pair-disk equivalent series capacitance Ch.3

CgHV(meas) Total measured shunt capacitance of winding Ch.3

CgHV(calc) Total calculated shunt capacitance of winding Ch.3

Csh(b) Winding shunt capacitance of phase b when phase B isolated from the ground

Ch.5

Cʹsh(b) Winding shunt capacitance of phase b when phase B is grounded Ch.5

CHT Shunt capacitance of HV winding respect to transformer tank Ch.5

CHL Shunt capacitance of HV winding respect to the LV winding Ch.5

CLC Shunt capacitance of LV winding respect to the core Ch.5

Cdeform Shunt capacitance of deformed section Ch.6

Cnorm Shunt capacitance of normal section Ch.6

C0 Absolute permittivity of vacuum Ch.7

Cʹg Shunt capacitance of buckled winding Ch.6

Cʹtt Total turn-to-turn capacitance of the deformed disk Ch.6

d Distance between circular turns Ch.3

dʺ Average length of conductor turn Ch.4

dAc Infinitesimal of the core cross section area Ch.5

dp Paper insulation thickness or pressboard Ch.7

D Diffusion coefficient Ch.7

Dave Winding average diameter Ch.2

D0 Pre-exponential factor Ch.7

Dielectric displacement Ch.7

iii

Eint Number of disks used for interleaving Ch.3

e Potential to ground (time domain) Ch.4

ē Induced voltage Ch.5

er Measurement error Ch.8

E0 Activation energy of diffusion process Ch.7

Electric field Ch.7

Ed Total disk-to-disk energy App. C

Et Total turn-to-turn energy App. C

Etot Total energy stored in a pair of disks App. C

f Operational frequency Ch.2

fi ith resonance frequency Ch.7

fr Resonance frequency Ch.4

fr-low Anti-resonance frequency Ch.5

fsweep Variable frequency Ch.2

Fradial Imposed force during transportation Ch.2

Fpredicted Predictable force Ch.2

Fstop Stop force Ch.2

Fwind Wind force Ch.2

g Shunt turn-to-turn conductance per unit length Ch.4

G Shunt conductance to ground Ch.4

h Axial dimension of conductor cross section Ch.3

Hm1 Height of LV winding Ch.2

Hm2 Height of HV winding Ch.2

Hw Winding height Ch.2, Ch.3,

and Ch.6

i Conductor current Ch.4, Ch.5

icg External capacitive current per unit conductor length Ch.4

ics Internal capacitive current per unit conductor length Ch.4

ig External conductance current per unit conductor length Ch.4

iG External conductance current per unit conductor length Ch.4

i2100 Current square harmonic at 100 Hz Ch.8

I Filament current App. B

I(jω) Conductor current (frequency domain) Ch.4

Ish Short-circuit current Ch.2

Iinsul Current traversing the medium Ch.7

j Imaginary operator Ch.4

J The current density vector App. B

ka LV winding frequency response magnitude of phase a Ch.5

kaB(sc) LV winding frequency response magnitude of phase a when phase B of HV side is short-circuited

Ch.5

kaC(sc) LV winding frequency response magnitude of phase a when phase C of HV side is short-circuited

Ch.5

kʹ Dimensionless parameter and is equal to 0.5 Ch.7

kʹa Generated frequency response magnitude of phase a Ch.5

k0 Circular inductance decrement factor Ch.3

iv

Km Mass coefficient of oil App. F

Kmag Frequency response magnitude Ch.2

KN Nagaoka’s coefficient Ch.3

KR Rogowsky coefficient Ch.2

l Inductance per unit length Ch.4

lc Mean magnetic path length of core limb Ch.5

ly Mean magnetic path length of yoke Ch.5

Lʹ The self-inductance of a single non-circular filament App. B

Leq Winding equivalent inductance Ch.3

LHV(calc) Total calculated inductance of winding Ch.3

LHV(meas) Total measured inductance of winding Ch.3

L Total winding self-inductance Ch.5

La LV winding inductance of phase a Ch.5

LaB(sc) LV winding inductance of phase a when phase B of HV side is short-circuited

Ch.5

LaC(sc) LV winding inductance of phase a when phase C of HV side is short-circuited

Ch.5

Li Inductance at ith resonance frequency Ch.7

LOC HV winding inductance at 50 Hz when LV side terminals are left open circuit

Ch.6

LSC HV winding inductance at 50 Hz when LV side terminals are short-circuited

Ch.6

L1 Self-inductance of primary winding Ch.8

L2 Self-inductance of secondary winding Ch.8

m Total transformer mass Ch.2

M Mutual-inductance of coaxial circular filaments Ch. 3, Ch.6

Mʹab The mutual-inductance between a circular and non-circular filaments

App. B

Mʺab The mutual-inductance between two non-circular filaments App. B

Mc Disk-to-disk mutual-inductance Ch.3

Meq Winding equivalent mutual-inductance Ch.3

Mʹ Mutual-inductance of the circular filaments whose axes inclined to one another

Ch. 6

Mʹʹ Mutual-inductance between the turns for asymmetrical axial deformation of a disk

Ch.6

Mʹʹʹ Mutual-inductance of circular elements with parallel axes Ch.6

n Turn number Ch.3

N Number of disk turns Ch.3

NB Number of transformer core limbs surrounded by HV and LV windings

Ch.2

Nd Number of transformer winding disks Ch.3

Nt Number of winding turns Ch.2

NI R.M.S. winding’s ampere-turns value Ch.2

Nw Number of winding turns Ch.3, Ch.4,

and Ch.5

Nsh Number of shield turns per disk Ch.3, App.C

v

Ns Number of elements (samples) App. A

Nrog Number of coil turns in Rogowsky coil Ch.8

P Total mass of paper scraps App. F

q Coupling coefficient Ch.8

r Conductor resistance per unit conductor length Ch.4

r The distance between the filament center and associated point for flux density

App. B

r' Radius of deformed section Ch.6

r1 Average radial dimension of winding Ch.3

r2 Average radial dimension of tank Ch.3

R Mean radius of disk Ch.3

Ra Mean radius of circular turn a Ch.3, Ch.6, and App.B

Rb Mean radius of circular turn b Ch.3, Ch.6, and App.B

Rd The distance between the current source and the associated point for flux density

App. B

Rn Resonance ratio number Ch.7, App. G

S Transformer apparent power Ch.2

Si Measured scatter parameters Ch.8

Sref Scatter parameter Ch.8

T0 Reference temperature Ch.7

Tk Current temperature Ch.7

u Estimated uncertainty Ch.8

u2100 Voltage square harmonic at 100 Hz Ch.8

U Voltage across a pair-disk Ch.3, App. C

Uinsul Voltage across the insulation medium Ch.7

Ux Short circuit reactance Ch.2

v The volume containing current App. B

υtank-100Hz Tank vibration frequency at 100 Hz Ch.8

V(jω) Potential to ground (frequency domain) Ch.4

Vn Nominal voltage of winding Ch.2

Vin Injected voltage at line-lead Ch.2

Vout Measured voltage at neutral-lead Ch.2

VO Output voltage Ch.8

Vphase Phase voltage Ch.8

Vtap Bushing tap voltage Ch.8

Vx Shield turn potential App. C

W Radial dimension of winding cross section Ch.3

Wspacer Spacer width Ch.2

Wref Reference paper humidity in percent Ch.7

Wll Lower limit of the paper humidity in percent Ch.7

Wul Upper limit of the paper humidity in percent Ch.7

x Substance movement position Ch.7

X Axial displacement of the outermost turn Ch.6

vi

Xi ith elements of the reference FRA spectrum App. A

Y Total mass of accumulated water App. F

Yi ith elements of the measured FRA spectrum App. A

Za Transformer LV winding impedance of phase a Ch.5

ZaB(sc) LV winding impedance of phase a when phase B of HV side is short-circuited

Ch.5

ZaC(sc) LV winding impedance of phase a when phase C of HV side is short-circuited

Ch.5

Zin Input impedance of measurement cable Ch.5

Zout Output impedance of measurement cable Ch.5

ZOC(50 Hz) Inductive reactance of HV winding at 50 Hz when LV side terminals are left open circuit

Ch.6

Zp Impedance paralleled with bushing tap Ch.8

Zsc Short circuit impedance Ch.8

ZSC(50 Hz) Inductive reactance of HV winding at 50 Hz when LV winding is short-circuited

Ch.6

Zw Transformer winding impedance Ch.5

Z11 Open circuit input impedance Ch.8

Z12 Open circuit reverse transfer impedance Ch.8

Z21 Open circuit forward transfer impedance Ch.8

Z22 Open circuit output impedance Ch.8

α Initial impulse voltage distribution coefficient Ch.4

αʹ The angle between Ra and Rd App. B

γ Self-inductance per unit conductor length Ch.4

δt Inter-turn insulation thickness Ch.3

ε Dielectric permittivity Ch.3

εoil Oil insulation permittivity Ch.7

εpaper Paper insulation permittivity Ch.7

εspacer Spacer permittivity Ch.7

εr Relative permittivity of dielectric Ch.7

εt Paper insulation permittivity Ch.3

ε0 Vacuum permittivity Ch.3

έ0 Static dielectric constant Ch.7

ε∞ Infinite-frequency dielectric constant Ch.7

ε' Real part of complex permittivity Ch.7

εʹʹ Imaginary part (loss factor) of complex permittivity Ch.7

ε* Complex permittivity Ch.7

η Ratio of entire trigonometric circular span Ch.6

θʹ The angle between the differential current vector and the vector directed from it to the point associated for flux density

App. B

θs Angle between spacers Ch.2

θto Top oil temperature Ch.8

ϑa Phase angle in the first loop for mutual-inductance calculation App. B

ϑb Phase angle in the second loop for mutual-inductance calculation App. B

λ Winding length Ch.2, Ch.4

vii

λrog Rogowsky winding length Ch.8

λ' Entire conductor length in one disk Ch.3

μ Mutual inductance per unit conductor length Ch.4

μc Core permeability Ch.5

µs Static friction constant Ch.2

μ0 Vacuum permeability Ch.2, Ch.5

σ Dielectric conductivity Ch.7

σoil Oil insulation conductivity Ch.7

σpaper Paper insulation conductivity Ch.7

τ Diffusion time constant Ch.7

ϕ Magnetizing flux Ch.5

ϕb The induced magnetizing flux on the second loop due to the current initiated by the first loop

App. B

χ Electric susceptibility Ch.7

ω Angular frequency Ch.4, Ch.8

R Magnetic core reluctance Ch.5

Ra Equivalent magnetic reluctance for phases a/A Ch.5

Rʹa Generated magnetic reluctance for phase a Ch.5

RaB(sc) Equivalent magnetic reluctance of phases a/A when phase B of HV side is short-circuited

Ch.5

RaC(sc) Equivalent magnetic reluctance of phases a/A when phase C of HV side is short-circuited

Ch.5

Rl Leakage reluctance Ch.5

Rc Transformer core limb reluctance Ch.5

Ry Transformer core yoke reluctance Ch.5

FSD FRA Spectrum Deviation App. G

MAMD Mean absolute magnitude distance Ch.8

MAPD Mean absolute phase distance Ch.8

RH Air relative humidity in percent Ch.7

WCO Water content in oil Ch.7

WCP Water content in paper Ch.7

WCP Moisture in paper in percent by weight Ch.7

viii

Contents

Abstract i

Nomenclature ii

Contents viii

List of Figures xiii

List of Tables xix

Chapter 1 Introduction 1

1.1 Problem Statement 1

1.2 Motivation 2

1.3 Current State of the Art 3

1.4 Research Objectives 6

1.5 Thesis Overview 7

1.6 Key Contributions 9

1.7 Publications 10

Chapter 2 Transformer Winding Deformation and Diagnosis Techniques 13

2.1 Introduction 13

2.2 Transformer Winding Deformation and Displacement 13

2.2.1 Short Circuit Current 14

2.2.2 Transformer Transportation Causing Active Part Displacement 18

2.3 Winding Deformation Diagnosis Methods 21

2.3.1 Short Circuit Impedance Measurement 21

2.3.2 Transfer Function Methods (FRA/LVI) 24

2.3.3 Deformation Coefficient Method 32

2.4 FRA vs. SCI 33

2.4.1 SCI Measurements 33

2.4.2 FRA Measurements 34

2.4.3 Discussion on FRA and SCI 38

2.4.4 Summary on FRA and SCI Methods 38

2.5 Conclusion 38

Chapter 3 Transformer Winding Parameters 39

3.1 Introduction 39

3.2 Self and Mutual Inductances of Transformer Winding (Analytical Approach) 40

3.2.1 Self - Inductance 43

3.2.2 Mutual – Inductance 44

3.3 Series and Shunt Capacitances of Transformer Winding 46

3.3.1 Series Capacitance 46

ix

3.3.2 Shunt Capacitance 51

3.4 Verification of Calculated Parameters Using Manufactured Model Transformer 51

3.4.1 Manufactured Model Transformer (Test Object) 51

3.4.2 Inductance Calculation of the Test Object 53

3.5 Conclusion 56

Chapter 4 Transfer Function Model of Air-Core Transformer Winding 57

4.1 Introduction 57

4.2 Modelling 57

4.3 Discussion on Resonant Frequencies 63

4.4 Verification of Mathematical Calculation Using Practical Measurement 64

4.5 Practical Study 66

4.5.1 Case Study 1 66


4.5 Conclusion 69

Chapter 5 Low Frequency Interpretation of FRA Signature 71

5.1 Introduction 71

5.2 Flux Division Theory 72

5.2.1 Technical Concept 72

5.2.2 Flux Division Measurement (FDM) 73

5.3 Mathematical and Practical Approach to Interpret Low-frequency Band 76

5.3.1 General Interpretation of FRA Trace 76

5.3.2 Practical Approach 79

5.4 Mathematical Approach 83

5.5 Effect of Core Configuration on FRA Trace 89

5.6 Shunt Capacitance Influence 91

5.6.1 Practical Approach 92

5.6.2 Physical and Mathematical Approach 93

5.7 Conclusion 95

Chapter 6 Axial and Radial Deformation of Transformer Winding 97

6.1 Introduction 97

6.2 Axial Deformation and Its Impacts on Winding Parameters 98

6.2.1 Mutual Inductance of Circular Filaments Whose Axes Are Inclined to One Another 98

6.2.2 Capacitances of Circular Filaments Whose Axes Are Inclined to One Another 103

6.3 Radial Deformation and Its Impacts on Winding Parameters 104

6.3.1 Self and Mutual Inductances in Radial Deformation 104

6.3.2 Series and Shunt Capacitances in Radial Deformation 106

6.4 Numerical Example 107

6.4.1 Axial Deformation of a Disk 107

6.4.2 Radial Deformation along the Winding 111

6.5 A Summary on Axial and Radial Deformations 112

6.6 FRA Simulation Study and Practical Measurement Results 113

6.6.1 Inductance Variation (Simulation) 113

6.6.2 Inductance Variation (Practical Study) 115

6.6.3 Shunt Capacitance Variation (Simulation) 117

x

6.6.4 Shunt Capacitance Variation (Practical Study) 119

6.6.5 Series Capacitance Variation (Simulation) 121

6.6.6 Series Capacitance Variation (Practical Study) 123

6.6.7 Resistance Variation (Simulation) 125

6.6.8 Resistance Variation (Practical Measurement) 127

6.6.9 Conductance to Ground (G) Variations (Simulation) 130

6.6.10 Turn-to-Turn Conductance (g) Variations (Simulation) 132

6.6.11 Conductance Variation (Practical Study) 133

6.7 Conclusion 133

Chapter 7 Temperature and Moisture Content Influences on FRA Signature 135

7.1 Introduction 135

7.2 Transformer Water Dynamic 135

7.2.1 Transient 136

7.2.2 Equilibrium 137

7.3 Practical Study 138

7.3.1 Test Object and Setup 138

7.3.2 Case Study 1 (‘Wet’ Model Transformer) 139

7.3.3 Discussion 1 144

7.3.4 Case Study 2 (‘Dry’ Model Transformer) 149

7.3.5 Discussion 2 151

7.3.6 Case Study 3 (Three-Phase Transformer) 153

7.4 Verification of Practical Results Using Modelling and Simulation 154

7.5 Influence of Temperature and Moisture Content on FRA Statistical Indicators 160

7.6 Practical Solution to Modify Statistical Indicators 161

7.7 Transformer Winding Dry-out Influence on Frequency Response Trace 165

7.8 Conclusion 167

Chapter 8 On-line Transformer Winding Deformation Diagnosis 169

8.1 Introduction 169

8.2 Advanced Methods in On-line Transformer Winding Deformation Diagnosis 170

8.2.1 Vibration Method 170

8.2.2 Communication Method 170

8.2.3 Current Deformation Coefficient Method 171

8.2.4 Ultrasonic Method 171

8.2.5 Online Short Circuit Impedance and Winding Stray Reactance Method 171

8.2.6 On-line Frequency Response Analysis (On-line FRA) 173

8.3 Discussion 175

8.4 Problem Statement on Online FRA Setup 179

8.5 Challenges with On-line FRA Setup 180

8.6 Case Studies 181


8.6.2 Test Procedure 182

8.6.3 Interpretation 183



xi



8.7 Discussion 191

8.8 Conclusion 192

Chapter 9 Conclusion and Future Research 193

9.1 Conclusion 193

9.2 Future Research 196

9.2.1 FRA Test Setup Development 196

9.2.2 Transformer Humidity Recognition Using FRA 196

9.2.3 Transformer Dry-out Assessment Using FRA 197

9.2.4 Oil and Paper Insulation Aging and Oil Replacement 198

9.2.5 On-line Transformer Winding Deformation Recognition 198

Appendix A Developed Software to Calculate Statistical Indicators 199

A.1 Introduction 199

A.2 Implemented Statistical Indicators 200

A.2.1 Correlation Coefficient (CC) 200

A.2.2 Maximum Absolute Difference (DABS) 200

A.2.3 Absolute Sum of Logarithmic Error (ASLE) 200

A.2.4 Min-Max (MM) 200

A.2.5 Standard Deviation (SD) 201

A.2.6 Spectrum Deviation (σ) 201

A.2.7 Cross Correlation Coefficient (CCF) 201

A.2.8 Relative Factor (R) 201

A.3 Developed Software 203

Appendix B Tables and Formulas for Inductance Calculation 204

B.1 Introduction 204

B.2 Tables and Formulas 204

B.3 Self and Mutual Inductances under Buckling 206

B.3.1 Biot-Savart Law and Inductance Calculation 206

B.3.2 Mutual-Inductance under Buckling 209

B.3.3 Self-Inductance under Buckling 215

B.3.4 Numerical Example 217

Appendix C Calculation of Series Capacitance of Intershield Winding 220

C.1 Introduction 220

C.2 Series Capacitance of Intershield Winding 220

C.2.1 Method to Calculate Total Series Capacitance 223

C.2.2 Application of the Proposed Method 225

Appendix D Glassy Model Transformer 227

D.1 Introduction 227

D.2 Test Object Overview 227

D.3 Winding Photos and Winding Schematic 228

Appendix E Dry-out Process of Model Transformer 235

xii

E.1 Introduction 235

E.2 Model Transformer Dry-out 235

E.3 Procedure 236

Appendix F Recognition of Moisture Content in Transformer 239

F.1 Introduction 239

F.2 Water Content Recognition in Oil-impregnated Paper 240

F.2.1 Karl-Fischer Titration 240

F.2.2 Capacitor Method 247

F.2.3 Paper Sample Method 247

F.2.4 Electrical Methods 249

Appendix G Study on Recommended Solution 251

G.1 Introduction 251

G.2 Case Study 1 251

G.3 Case Study 2 254

G.4 Winding Deformation vs. Temperature and Moisture Influences on FRA Spectrum 257

G.4.1 Influence of Internal Short-circuit 257

G.4.2 Influence of Tank Grounding 258

G.5 Conclusion 260

Bibliography 261

xiii

List of Figures

Figure 2.1 Winding deformation (Buckling), (a) Free buckling (top view), (b) Forced buckling (top view), (c) Free buckling (side view) 16

Figure 2.2 Winding deformation, (a) Before tilting, (b) After tilting, (c) Bending (side view), (d) Bending (close view) 17

Figure 2.3 Transformer transportation schematic 18

Figure 2.4 Transformer transportation, (a) Truck, (b) Rail road, (c) Sea carrier, (d) Air carrier 21

Figure 2.5 Schematic model of primary, secondary and leakage inductances of a transformer 22

Figure 2.6 Schematic of transformer core and windings 23

Figure 2.7 Short circuit impedance measurement setup, (a) Single phase transformer, (b) Three-phase transformer 23

Figure 2.8 Transfer function measurement techniques 24

Figure 2.9 Low, mid and high frequency bands of a typical FRA spectrum (measured on 400 MVA transformer) 26

Figure 2.10 FRA test setups, (a) End-to-end measurement, (b) Inductive inter-winding measurement, (c) Capacitive inter-winding measurement, (d) End-to-end short-circuit measurement 29

Figure 2.11 FRA test setups (detailed connections) 30

Figure 2.12 The measured and fingerprint frequency response magnitudes for phases A, B and C of the transformer HV side 36

Figure 2.13 Buckled HV winding of phase B, (a) Side view of the middle disks, (b) Front view of the upper disks 37

Figure 3.1 Air-core transformer winding model 43

Figure 3.2 Coaxial circular conductors, (a) Coaxial filaments, (b) Coaxial disks 45

Figure 3.3 The overall layout of a layer winding including equivalent capacitance network 46

Figure 3.4 Continuous disk winding schematic 47

Figure 3.5 Equivalent capacitance network of the continuous disk winding 48

Figure 3.6 Pair of disks, cross-section overview and voltage distribution along disks pair 49

Figure 3.7 The interleaved disk winding 50

Figure 3.8 The intershield disk winding 51

Figure 3.9 Manufactured glassy model transformer (a) Bird’s-eye view, (b) Side view 52

Figure 3.10 Measured value for the inductance of HV winding 53

Figure 3.11 HV winding schematic (glassy model transformer) 54

Figure 3.12 Measured value for shunt capacitance, HV winding 55

Figure 3.13 Model transformer schematic, top view 55

xiv

Figure 4.1 Equivalent detail circuit of a transformer winding (dx is the infinitesimal length of winding), (a) Entire winding schematic, (b) Close view of the conductors and modeled parameters 58

Figure 4.2 Common FRA test setup, (a) For single winding (b) For Transformer 61

Figure 4.3 Simulated and measured frequency response traces for HV winding of the manufactured glassy transformer 65

Figure 4.4 Continuous and interleaved disk windings, (a) Continuous winding, (b) Interleaved winding 67

Figure 4.5 Frequency response traces of continuous and interleaved disk windings 67

Figure 4.6 FRA traces of HV windings phases A for 45 MVA (continuous disk winding) and 66 MVA (interleaved disk winding) power transformers 69

Figure 5.1 Magnetic flux flow due to the transformer winding phase A excitation when short-circuit occurred in phase B 74

Figure 5.2 Five-limb transformer active part as well as its magnetic circuit 75

Figure 5.3 Magnetic flux division in five-limb transformer due to phase a/A excitation, short circuit is occurred phase b/B 75

Figure 5.4 Frequency response traces of phases a, b and c 77

Figure 5.5 Equivalent magnetic circuit of three-phase transformer 78

Figure 5.6 FRA measurement setup for phase a when A and C are short-circuited 80

Figure 5.7 Equivalent magnetic circuit for FRA measurement setup of phase a when the terminals A and C are short-circuited 80

Figure 5.8 Reference and measured FRA traces for phase a when limb C is surrounded by short circuit loop 80

Figure 5.9 FRA measurement setup for phase a when B and C are short-circuited 81

Figure 5.10 Equivalent magnetic circuit for FRA measurement setup of phase a when the terminals B and C are short-circuited 82

Figure 5.11 Reference and measured FRA traces for phase a when limb B surrounded by short circuit loop 82

Figure 5.12 Frequency response traces of phase a (open circuit) and deliberate short circuit on phase B and C 83

Figure 5.13 Equivalent magnetic circuit of transformer when frequency response trace is measured for phase a, (a) Normal three-phase, (b) HV winding phase B is short-circuited, (c) HV winding phase C is short-circuited 86

Figure 5.14 Original and generated frequency response traces of phase a 88

Figure 5.15 Equivalent magnetic circuits when deliberated short circuit is applied for various limbs 90

Figure 5.16 Frequency response traces for test setups of Figures 5.15 (b) and (h) 91

Figure 5.17 Frequency response traces for test setups of Figures 5.15 (c), (d), (f) and (k) 91

Figure 5.18 Frequency response traces for phase b when HV side phase B is left open circuit, short-circuited, short-circuited and grounded 92

Figure 5.19 Active part and related shunt capacitances, (a) Active part upper view schematic, (b) Shunt capacitance configuration for b where HV side phase B is just short-circuited and isolated from the ground, (c) Shunt capacitance configuration for b where HV side phase B is grounded 94

Figure 6.1 Symmetrical axial deformation of a disk, (a) Axial deformation pattern, (b) Deformed disk 100

Figure 6.2 Asymmetrical axial deformation of a disk, (a), Axial deformation pattern, (b) Deformed disk 101

xv

Figure 6.3 Cross-section overview and voltage distribution along deformed winding 103

Figure 6.4 Radial deformation schematic (free-buckling) 104

Figure 6.5 Radial displacement pattern of a disk 105

Figure 6.6 Axially deformed interleaved winding 110

Figure 6.7 Reference and measured frequency response spectra for interleaved windings 110

Figure 6.8 The modelled winding through finite element (r1= 280 mm, r2 = 400 mm, r'= ±50 mm, φ= π/2) 112

Figure 6.9 FRA simulation results of winding due to the inductance reduction for 20 and 40 %, (a) Entire FRA spectrum, (b) Expanded view of dash-line rectangle in Fig. 6.9(a) 114

Figure 6.10 FRA test setup on HV side, (a) Open circuit on LV side, (b) Short circuit on LV side 116

Figure 6.11 Frequency response of HV winding for open circuit and short-circuited LV winding 116

Figure 6.12 FRA simulation results of winding due to the shunt capacitance reduction for 20 and 40 %, (a) Entire FRA spectrum, (b) Expanded view of dash-line rectangle in Fig.6.12 (a) 118

Figure 6.13 FRA test setup for HV winding, where LV side is short-circuited, and test object tank is isolated from the ground 119

Figure 6.14 Active part and related shunt capacitances, (a) Shunt capacitance configuration for HV, where LV side is short-circuited and transformer tank is grounded, (b) Shunt capacitance configuration for HV, where LV side is short-circuited and transformer tank is isolated 120

Figure 6.15 Frequency response of HV winding of single phase transformer, LV winding is short-circuited and also LV winding is short-circuited, transformer tank is grounded and isolated 120

Figure 6.16 FRA simulation results of winding due to the series capacitance increase, (a) Entire FRA spectrum, (b) Close-up view of dash-line rectangle in Fig.6.16 (a), (c) Close-up view including some resonance frequencies 123

Figure 6.17 FRA measurement results of faulty transformer for phase b and c, (a) Entire FRA spectrum, (b) Close-up view of dash-line circle in Fig.6.17 (a) 124

Figure 6.18 FRA simulation results of winding due to winding resistance increment, (a) Entire FRA spectrum, (b) Close-up view of region ‘b’ , (c) Close-up view of region ‘c’ , (d) Close-up view of region ‘d’ 127

Figure 6.19 FRA measurement results of winding due to the resistance increment, (a) Entire FRA spectrum, (b) Close-up view of region ‘b’ at very low frequencies, (c) Close-up view of region ‘c’ at high frequency resonant, (d) Close-up view of region ‘d’ at very high frequencies 130

Figure 6.20 FRA Simulation results of winding due to the conductance to ground (G) increment, (a) Entire FRA spectrum, (b) Close view of the first resonance point 131

Figure 6.21 FRA Simulation results of winding due to the turn-to-turn conductance (g) increment, (a) Entire FRA spectrum, (b) Close view of the first resonance point 133

Figure 7.1 Water dynamic in paper and oil insulation for different temperatures T1 and T2 (T1 < T2), WCO and WCP; t denotes the time 136

Figure 7.2 Manufactured glassy air-core transformer (setup preparation to study temperature and moisture impact) 138

Figure 7.3 FRA test setup to examine temperature and moisture variation 139

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Figure 7.4 FRA spectra for ‘wet’ model transformer, HV side (a) Entire trace for 30 and 90 °C, (b) Close-up view of region 1 shown by dash-line in Fig. 7.4(a), frequency band 800 kHz-3 MHz, (c) Close-up view of region 2 shown by dash-line in Fig. 7.4(a), frequency band 3 MHz – 10 MHz 142

Figure 7.5 FRA spectra for ‘wet’ model transformer, LV side (a) Entire trace for 30 and 90 °C, (b) Close-up view of region 1 shown by dash-line in Fig. 7.5(a), frequency band 500 kHz - 3.5 MHz, (c) Close-up view of region 2 shown by dash-line in Fig. 7.5(a), frequency band 3.5 MHz – 10 MHz 144

Figure 7.6 Deviation of total capacitance for HV and LV windings (average deviation at 90 °C is 6.22 %) 147

Figure 7.7 Moisture content of oil and paper (wet model transformer) 148

Figure 7.8 Vacuum process of the model transformer 149

Figure 7.9 FRA spectra for ‘dry’ model transformer, HV side 150

Figure 7.10 FRA spectra for ‘dry’ model transformer, LV side 150

Figure 7.11 Moisture content of oil and paper (dry model transformer) 151

Figure 7.12 HV winding spectra at 10 and 60 °C (1.6 MVA transformer) 154

Figure 7.13 Schematic of the paper content and the spacer coverage in the insulation duct 158

Figure 7.14 Simulated FRA spectra for model transformer (shunt capacitance deviation) 160

Figure 7.15 The chart on preliminary calculation on FRA traces to distinguish insulation deviation from winding deformation; Xi and Yi were defined in Chapter 2 164

Figure 7.16 Glassy model transformer 165

Figure 7.17 FRA traces for glassy test object (a) HV winding spectra before and after dry-out (frequency band 3 kHz-20 MHz, 0 dB < 3 kHz), (b) LV winding spectra before and after dry-out (frequency band 6 kHz- 20 MHz, 0 dB < 6 kHz). The measurements have been performed for oil-filled model transformer 166

Figure 8.1 Two-port network 172

Figure 8.2 Side cut-off of a capacitive bushing 174

Figure 8.3 Paralleled impedance with bushing tap (test tap) on phase U 175

Figure 8.4 Wye circuits for on-line frequency response measurement 176

Figure 8.5 Delta circuits for on-line frequency response measurement 176

Figure 8.6 On-line FRA setup for a transformer interleaved winding 181

Figure 8.7 Close view of test setup 182

Figure 8.8 Frequency response traces of normal and defected winding, off-line setup 183

Figure 8.9 Frequency response traces of normal and defected winding, on-line setup 183

Figure 8.10 FRA test setup behaviour with and without bushing connection, (a) Common FRA test setup, (b) FRA test setup with bushing connected, (c) FRA test setup with bushing connected (low-frequency behaviour) (d) FRA test setup with bushing connected (mid-frequency behaviour), (e) FRA test setup with bushing connected (high frequency behaviour) 185

Figure 8.11 Frequency response traces of normal and defected winding, off line and on-line setup 186

Figure 8.12 Frequency response traces of interleaved (off-line setup and on-line setup through one coupling capacitor) 187

xvii

Figure 8.13 Paralleled coupling capacitors on conventional bushing 188

Figure 8.14 Frequency response traces of interleaved (off-line setup and on-line setup through coupling capacitor) 189

Figure 8.15 Frequency response traces of continuous disk winding (off-line setup and on-line setup through coupling capacitor) 189

Figure 8.16 Frequency response traces of single phase transformer (off-line setup and on-line setup through coupling capacitor) 190

Figure A.1 Developed software to calculate statistical indicators, snapshot 203

Figure B.1 Biot-Savart law 206

Figure B.2 Magnetic flux determination for circular filament 207

Figure B.3 Concentric circular filaments 208

Figure B.4 Concentric circular filaments, inward buckling demonstration for the second loop 210

Figure B.5 Concentric circular filaments, demonstration of inward buckling for both loops 212

Figure B.6 Magnetic flux determination for non-circular filament 215

Figure C.1 Equivalent RLC network for intershield disk winding 221

Figure C.2 Manufactured windings from top to bottom: Interleaved, Continuous, Intershield with one shield turn, and Intershield with six shield turns in each disk 222

Figure C.3 Overall scheme and the sample point of voltage for the simulated winding, (S.P=Sample Point) 224

Figure D.1 Manufactured glassy test object, (a) Side view of winding, (b) Side view without oil, (c) Entire view, (d) Side view with oil 228

Figure D.2 HV technical drawing, (a) HV technical schematic, (b) Specifications, (c) Backward step drawing 231

Figure D.3 LV technical drawing, (a) LV technical schematic, (b) Specifications, (c) Backward step drawing 234

Figure E.1 Dry-out equipment, Dry-out equipment, (a) Wet silica gel, (b) Half dried silica gel, (c) Silica gel dry-out process (oven view), (d) Dried silica gel (oven view), (e) Dried silica gel, (f) Silica gel container to break the vacuum after oil dry-out process, (g) Oil circulator, (h) Internal piping of oil circulator, (i) Oil circulator vacuum gauge, (j) Transformer oil, (k) Vacuum pump, (l) Internal motor pump of circulator, (m) Glassy anti-vacuum container of oil circulator, (n) Karl-Fischer equipment, (o) Metal stopcock, (p) Vacuum gauge for test object dry-out, (q) Model transformer dry-out process (oven view) 238

Figure F.1 Oil sample containers, (a) Glass laboratory bottle, (b) Glass syringe 241

Figure F.2 Oil sample containers, (a) Side view, (b) Front view 242

Figure F.3 KFT equipment, (a) Digital micro scale, (b) Glassy container, main unit and keyboard 243

Figure F.4 KFT unit main compartments, (a) Double platinum wire electrode (0.8 x 4 mm), (b)Generator electrode for Karl Fischer titrations, with diaphragm, (c) KF absorber tube for coulometer cell, (d) SGJ stopper, (e) Stopper, (f) Plastic tube, (g) Titration vessel holder for coulometric cells, (h) KF titration vessel, (i) Keyboard for 756 KF Coulometer 244

Figure F.5 Oommen equilibrium curves for oil-paper system 246

Figure F.6 MIT-developed curves for water equilibrium in cellulose/mineral oil systems 246

xviii

Figure F.7 Capacitance sensor layers 247

Figure F.8 Dean-Stark apparatus, (a) Condenser, (b) Receiver, (c) Flask, (d) Assembled parts 248

Figure F.9 Different views of measuring water content of the sampled papers using Dean-Stark method 248

Figure F.10 Variation of loss factor of oil-immersed paper with frequency and dominant influences 250

Figure G.1 Reference and derivative spectra for HV winding of glassy model transformer, (a) at 30 °C, (b) 90 °C 252

Figure G.2 Frequency response spectra for HV winding when the LV winding was left open and short-circuited (test object without oil), (a) Entire frequency band (20 Hz – 20 MHz), (b) The area enclosed by dash-line rectangle in Fig. G.2(a), (100 kHz – 20 MHz) 255

Figure G.3 HV winding spectra (a) Reference spectrum and its derivative (100 kHz – 20 MHz), (b) Measured spectrum and its derivative (100 kHz – 20 MHz) 256

Figure G.4 (a) HV winding frequency response spectrum when LV winding is open-circuited (original spectrum) and short-circuited (affected frequency-band 20 Hz-3 MHz), (b) FRA spectra of HV winding for isolated and grounded tank (affected frequency-band 300 kHz-20 MHz), (c) FRA spectra for HV winding due to moisture migration from paper into the oil insulation at 30°C and 90°C, re-plotted from Fig. 7.4 for comparison (affected frequency-band 800 kHz-20 MHz) 259

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List of Tables

Table 2.1 FRA measurement connections (phases A, B and C) 30

Table 2.2 CC and SD criteria 32

Table 2.3 Level of deformation based on Rxy 32

Table 2.4 400 MVA Step-up transformer parameters 34

Table 2.5 Measured SCI Values 34

Table 2.6 CC, SD values 36

Table 2.7 Rxy values 36

Table 4.1 Transformer specifications 68

Table 5.1 Transformer specifications 78

Table 5.2 FRA setup connections for the configuration in Fig. 5.15 90

Table 6.1 Calculated capacitance between the winding and metal container (tank) 112

Table 6.2 Technical specifications of single phase test object 115

Table 7.1 Ambient air relative humidity and moisture in paper 140

Table 7.2 HV winding electrical parameters for 30 and 90 °C 146

Table 7.3 LV winding electrical parameters for 30 and 90 °C 146

Table 7.4 HV winding capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig.7.4(b) and Fig. 7.4(c ) 147

Table 7.5 LV winding capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig.7.5(b) and Fig. 7.5(c) 148

Table 7.6 HV winding capacitance ratio, anti-resonance and resonance frequencies 151

Table 7.7 LV winding capacitance ratio, anti-resonance and resonance frequencies 151

Table 7.8 FRA deviation and total capacitance variation for 0.5 % WCP change 152

Table 7.9 Measured DDF at 5 kV before and after dry-out process 153

Table 7.10 HV winding capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.12 154

Table 7.11 Calculated statistical indices 160

Table 7.12 Capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.16(a) 167

Table 7.13 Capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.16(b) 167

Table 8.1 Typical values for bushing capacitances 174

Table B.1 Values of KN for single layer coil 204

Table B.2 Values of R0 for inclined circles 205

Table B.3 Values of F for parallel circles, rd/2R=Λ 205

Table C.1 Initial voltage distribution (%) 225

Table G.1 Calculated values for Zî n and Zî m 253

Table G.2 Calculated values for reference spectrum Zî n, and measured spectrum Zî m 257

Chapter 1. Introduction

1

Chapter 1

Introduction

1.1 Problem Statement

Power transformers are put in service under different environmental, electrical and

mechanical conditions, and may be subject to enormous hazards during the course of

operation. They are commonly considered as the heart of the transmission and distribution

sectors of electric power systems. Thus, monitoring their condition and diagnosing faults

are important parts of the maintenance function. Utility engineers strive to keep power

transformers in service, and to prevent even short-term outages. Failure of a transformer

can cause extensive damage to connected equipment owned by consumers and/or the

utility.

Mechanical defects are probably the most common cause of problems in transformers

which can put this asset out of service for a long time. They may be due to short circuit

currents, earthquakes, careless transportation between sites, explosion of combustible

gases accumulating in the transformer oil, etc. In fact, transformer internal short circuits

will adversely affect transformer insulation system which includes paper and pressboard,

while external short circuits in power networks will subject the transformer winding to

substantial electro-dynamic forces. Direction of forces is perpendicular to the magnetic

field vector according to electromagnetic theorem. They cause winding deformation in the

axial and/or radial directions, hoop buckling, tilting, spiraling, displacements between high

and low voltage windings, shorted or open-circuited turns, partial winding collapse,

loosened clamping structures, core movement, faulty grounding of core or screens, broken

clamping structures, and intermittent internal connections. Repair of such defects may

require taking the transformer out of service, which could prove costly to the utility. To

recognize transformer mechanical defect without opening the transformer tank and also

avoid undesirable costly maintenance, the method of Frequency Response Analysis (FRA)

has been introduced and widely employed in the power industry. FRA is considered a

highly accurate, fast, economical and non-destructive method of detecting winding defects

and damage in the transformer core. This method has been studied for many years, while


2

FRA data (trace) interpretation is still under development. On the other hand, there has

been a concern that the changes in transformer insulation characteristics, in particular the

temperature and moisture content, can influence FRA results.

Hence, not only interpretation of FRA data due to the winding deformation has still

remained as a challenge, but also FRA trace alteration due to the temperature and moisture

content variation should be clarified.

Furthermore, another challenge that is worthwhile to tackle is to be able to distinguish

whether the measured FRA deviations are indicative of transformer winding deformations

or due to insulation characteristic changes.

1.2 Motivation

The IEC Standard 60076-18, Ed.1 [1], on FRA test technique points out that not only FRA

traces is affected by winding deformation, but also it could be affected by the temperature

changes. A figure showing deviated FRA trace from the reference value is provided in this

standard. However, deviation amounts in resonance peaks as well as their causes and

effects are not discussed.

In addition, IEEE Standard C57.149 [2] on application of frequency response analysis for

oil-immersed transformers mentions that “large temperature difference, typical much more

than 10 °C, between two measurements will slightly influence the response at higher

frequencies”, though, further discussion is not provided in this standard either.

The research undertaken in this thesis targets the IEC and IEEE Standard comments by

developing more comprehensive and flexible transformer models that will facilitate a

profound study on this issue.


3

1.3 Current State of the Art

The AC impedance or admittance of any RLC network is a function of frequency. Dick and

Erven have utilized this idea [3] in transformer diagnosis and introduced Frequency

Response Analysis (FRA). Nowadays FRA measurement is widely accepted as an effective

diagnosis approach in transformer mechanical integrity investigation [4]- [7].

The initial FRA measurement during transformer factory tests is known as the transformer

winding fingerprint (baseline or signature) [8]- [9]. The distributed resistance, capacitance

and inductance of the transformer winding in the frequency range 20 Hz–2 MHz determine

the shape of the reference frequency response signature. These parameters are influenced

by not only the material electrical characteristics but also the structural geometry. Thus,

transformer geometric structure changes will lead to frequency response trace deviation

and winding deformation recognition, accordingly [10].

Since the last decade, there has been an interest in progressing the diagnostic benefits

associated with FRA. Also different approaches have been tried to improve FRA

interpretation ability. Based on literatures, studies in this area can be categorized into five

major groups: FRA classification, FRA development, FRA assessment, FRA interpretation

and on-line FRA application.

In the case of FRA classification, the literatures are trying to explore transformer winding

deformation type through measured FRA spectrum [11]- [14]. For instance, Rahimpour et al

[12], have determined and classified mathematical coefficients to explore winding axial

deformation through deviated FRA trace.

In the case of FRA development, researchers have tried to extend the diagnostic potential of

the FRA method [15]- [19]. They believe not only FRA can be utilized as a method to explore

transformer mechanical integrity but also it can provide significant information about

transformer internal condition. Hence, FRA might eventually replace some conventional

methods such as turn ratio, flux division, vector group and short circuit impedance

measurements.

Regarding FRA assessment, the literatures have concentrated on transformer winding

deformation diagnosis through measured FRA spectrum [20]- [26]. In fact, the main

concern of these studies has focused on using statistical indicators to recognize winding

deformation or active part displacement apart from deformation type. Researchers believe

that such an approach can lead to development of an intelligent system for FRA

investigation and diagnosis of transformer mechanical condition.

The next category, namely FRA interpretation, is concerned with the interpretation of FRA

fingerprint [27]- [29]. The correlation between FRA spectrum oscillations and transformer


4

physical configuration has been discussed. It has been emphasized that physical

interpretation of FRA spectrum will lead to a methodical and hence better understanding of

transformer physical condition.

The last category, namely on-line FRA application, is focused on real-time FRA

measurement under normal operating conditions. The literatures in this area [8], [10],

[30]- [37] believe that since on-line high voltage monitoring systems are under

development now, all of off-line diagnosis methods need to be replaced with on-line

application. Therefore, the potential problems of adapting off-line FRA to on-line situations

need to be identified and addressed.

Apart from discussed categories, a number of researchers have made significant

contributions in transformer modeling to support FRA interpretation [38]- [43]. These

literatures contain valuable mathematical calculations. Transformer active part modeling

through different approaches has been elaborated to study frequency response. Other

studies in this field show interest on FRA circuit setup, connections as well as

environmental impacts on frequency response trace deviation [44]- [50].

Despite all attempts there are many aspects in the FRA concept that need to be addressed

and overcome. One of those aspects is the interpretation of resonances and anti-resonances

within FRA trace for each and every transformer. Previous efforts should be further

developed with supporting theory to help interpret FRA fluctuations.

In addition, as discussed earlier, changes in winding configuration would almost certainly

cause changes in the frequency response trace. On the other hand, transformer oil

deterioration as well as water absorbed by the paper insulation cause transformer

insulation characteristics to change over the time [51]- [52]. In fact, insulation characteristic

can change due to a number of factors. Temperature, humidity, oil acidity, oil interfacial

tension, oil contamination, oil viscosity, oil breakdown voltage and degree of

polymerization in paper are major factors to be considered, to name a few. Any change in

one or more of these factors will be reflected in some change in winding electrical and

magnetic behaviour. Among the factors, water content and temperature variation would be

more significant.

In fact, capacitances, self and mutual inductances and conductor resistances might be

altered due to any changes in above mentioned factors. In turn, the frequency response of

the winding will change accordingly. The concern on the influence of insulation

characteristic on the frequency response trace becomes significant when this trace is

required to be interpreted for diagnostic purposes. FRA deviation due to the winding

deformation or insulation characteristic changes turns into a more substantial challenge


5

when FRA baseline and measured traces are taken at different temperatures and moisture

contents of the transformer winding. In this regard, investigation of the effect of changes in

complex dielectric permittivity of transformer insulation on its high frequency response is

the primary aim of a study by Abeywickrama et al [16]. In this study, a single-phase

transformer without a core was first modeled and then developed to three-phase by means

of a lumped parameter circuit to illustrate the insulation impacts on frequency domain

response [16]. This study concludes that FRA technique could be implemented for

diagnostics of the insulation quality in power transformers but more work is necessary for

establishing reliable interpretation.

The impact of oil and temperature on the initial voltage distribution for air and oil

immersed transformer layered windings has been studied by Florkowski et al [53]. This

investigation was subsequently extended for different disk windings in [54] by the same

authors. The oil and temperatures influences on the frequency characteristic have been

experimentally investigated and discussed in both works.

Reykherdt et al [19] studied the sensitivity of the FRA signatures to winding temperature

and moisture content. However, the reason for changes in the FRA spectrum was not

rigorously discussed in detail.

The influence of moisture ingress on transformer winding distributed parameters was

mentioned in a study by Abu-Siada et al [55], while the FRA method as a technique for

moisture diagnostics of power transformers was elaborated by Yadav et al [56]. However,

the abstract of the latter was too promising in relation to what was actually reported in the

article.

A study by Bjerkan [57] claims that the transformer insulation aging will impact the FRA

signature, but this impact is not significant. The IEC Standard 60076-18, Ed.1 [1] provides

information of deviated FRA spectra due to just temperature changes. This standard also

clearly states that “temperature affects the frequency response” but supporting discussion

is not provided.

Hence, a comprehensive study on FRA trace alteration due to the temperature and

moisture content changes is necessary to accurately quantify the FRA signatures variation

under such circumstances. Statistical indicators as available FRA evaluation methods

should be also examined once temperature and moisture content are changed. Correlation

between the moisture content in paper insulation and FRA trace variation can also be

considered as another important research area.


6

1.4 Research Objectives

Based on aforementioned discussion, it appears that a number of key areas are in need of

further research.

At first, the main reasons of resonances and anti-resonances of FRA signature should be

clarified through the mathematical modeling of transformer winding. Next, it would be a

novel idea to fabricate a model transformer as an experimental test object to study

mechanical integrity as well as temperature and moisture variation on its FRA signature,

synchronously. This manufactured model transformer can also be utilized for mathematical

model verification. The verified model can then be used for studying the axial and radial

deformation of transformer windings of FRA signature. In addition, it could be utilized to

investigate the insulation characteristics influences on FRA data. Based on this, the

objectives of this thesis are:

Transformer winding modeling to interpret the resonances and anti-resonances in

FRA trace,

Manufacturing a model transformer as a unique test object to experimentally verify

the developed model and studying the mechanical integrity,

Study on axial and radial deformation of transformer winding as the main roots of

any transformer winding deformation category,

Explore the influence of temperature and moisture contents on FRA signature and

compare the results to those coming through mechanical defects,

Estimation of moisture variation through FRA trace,

Development of a comprehensive software including all available FRA evaluation

indicators to investigate the impact of FRA trace deviation due to the temperature

and moisture variation on indicators,

Investigate and suggest possible solution to distinguish the winding deformation

from insulation characteristic impacts on FRA spectrum.


7

1.5 Thesis Overview

This study is focused on the influence of different origins on the transformer winding FRA

signature. Accurate recognition and interpretation of these origins are crucial for

developing a reliable transformer condition monitoring system. The different reasons

causing FRA trace deviations are going to be examined theoretically and through practical

experiments. To satisfy this objective, the thesis is structured in the following manner.

Chapter 2 provides a review on transformer winding deformation. It also discusses

available off-line methods for transformer mechanical defects recognition. The

fundamentals of frequency response analysis as the main subject of this study are

introduced and through a practical approach compared to the conventional method of

Short Circuit Impedance (SCI) measurement. FRA evaluation methods based on statistical

indicators are then discussed in detail to support the work in later Chapters.

Chapter 3 derives the self and mutual-inductance of an air-core transformer winding in

detail. To do this, literatures including some over a hundred years old have been taken into

account. In addition, calculations of the series and shunt capacitance of the transformer

winding are discussed. At the end, to verify the presented analytical approach on winding

parameters, a numerical example is provided and the calculated parameters based on the

analytical approach are compared to the measured parameters in manufactured test object.

Chapter 4 develops a mathematical model for air-core transformer winding to account

for oscillations in the FRA trace. This model is verified through FRA measurement on a

model transformer manufactured. This in turn enables interpretation of the mid-frequency

oscillations once the transformer core is not taken into consideration.

Chapter 5 examines the FRA low-frequency band. Through practical experiments, the

main resonances and anti-resonances in FRA trace for star connection of the transformer

windings are analyzed and discussed. The technique developed in this Chapter can be

utilized for transformer core defects recognition.

Chapter 6 discusses specifically on axial and radial deformations in transformer

winding. The mathematical foundation developed in this Chapter is believed to be essential

for future researches on winding deformation. A small transformer winding is considered

as a simple model and winding parameters are derived in detail for each and every

deformation category. The results are then used for simulation of different winding

deformation category. In the case of radial deformation, through a simple winding

deformation, the results achieved by the analytical approach are compared to Finite

Element Method (FEM) results.


8

Chapter 7 experimentally discusses the effect of temperature and moisture content

variation on FRA trace. Accurate measurements are performed on the manufactured test

object and the influence of moisture migration on FRA data is derived. Practical results are

then validated in this Chapter through the modeling and simulation and the main reason of

FRA deviation due to insulation characteristics changes are derived. FRA evaluation

methods are examined under such circumstances. This Chapter also opens up discussion on

recognition of moisture content variation in paper insulation using FRA. In addition, a

solution to distinguish the transformer winding deformation from insulation characteristic

impacts on FRA spectrum is also discussed in this Chapter.

Chapter 8 introduces the online FRA measurement as a possible solution to distinguish

winding deformation from insulation characteristic effects on the FRA trace. This Chapter

also reviews all recommended online transformer winding deformation recognition

methods and practically compares them with online FRA application. As the last part of the

thesis, this Chapter opens the doors for future research in this regard.

The conclusion to this research is given in Chapter 9 including a summary of the results

achieved and a discussion on future research.

The thesis also includes a number of appendices. Appendix A provides detailed

information of the software developed to calculate the various statistical indicators to for

evaluating FRA. This software is able to analyze the FRA spectra different frequency sub-

bands.

Appendix B gives the tables and formulas for inductance calculation used in Chapters 3

and 6.

Appendix C provides a detailed calculation on series capacitance of intershield winding.

It examines how the difference in the number of shields can produce a significantly

different series capacitance.

Appendix D provides detailed information on the glassy model transformer including

technical drawing. This transformer was specifically designed and fabricated for this

research work.

Appendix E describes transformer dry-out process and specifically discusses the process

used in this thesis for the work required in Chapter 7.

Appendix F gives an overview on moisture content measuring techniques in transformer

and specifically focuses on the Karl-Fischer Titration (KFT) used in Chapter 7.

Appendix G provides a practical example on the solution recommended in this thesis to

distinguish the insulation characteristic changes from the mechanical deformation in their

influence on the FRA trace.


9

1.6 Key Contributions

Demonstrated analytically how axial and radial deformation in transformer

winding will influence winding parameters,

Developed a comprehensive model of transformer winding to interpret mid-

frequency resonances and anti-resonances in FRA trace,

Demonstrated experimentally that low-frequency oscillations in FRA trace are

influenced by the middle and lateral transformer core limbs,

Demonstrated experimentally that FRA trace can be influenced through

temperature and moisture content variation,

Examined FRA evaluation indicators when moisture content of transformer paper

insulation is changed,

Developed estimates for the moisture content variation in transformer paper

insulation through FRA trace study. This work may impact significantly the future

of moisture content recognition in transformer,

Demonstrated that the effectiveness of the transformer drying process could be

verified by examining the FRA trace. This can serve as a fast, reliable and non-

destructive technique for transformer dry-out evaluation,

Developed a comprehensive software package to calculate all evaluation indicators

of FRA trace,

Recommended a technique to distinguish insulation characteristic variation from

winding deformation in FRA trace, also recommended on-line FRA measurement as

another possible solution to distinguish the impact of these phenomena on the FRA

spectrum,

Discussed the practical challenges in on-line FRA measurement and a technique to

extract maximum information through the online FRA setup recommended.


10

1.7 Publications

Journal Papers

1. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, “Advanced Transformer Winding Deformation Diagnosis: Moving from Off-line to On-line”, IEEE Transaction on Dielectric and Electrical Insulation, Vol. 19, Issue. 6, pp. 1860-1870, 2012.

2. Mehdi Bagheri, Toan Phung, Trevor Blackburn, “Transformer Frequency Response Analysis: Mathematical and Practical Approach to Interpret Mid-frequency Oscillations”, IEEE Transaction on Dielectric and Electrical Insulation, Vol. 20, Issue 6, pp. 1962-1970, 2013.

3. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, Toan Phung “Frequency Response Analysis and Short Circuit Impedance Measurement in Detection of Winding Deformation in Power Transformer”, IEEE Electrical Insulation Magazine, May/June, Vol. 29, no. 3, pp. 33-40, 2013.

4. Mehdi Bagheri, Toan Phung, Trevor Blackburn, “Influence of Temperature and Moisture Content on Frequency Response Analysis of Transformer Winding”, IEEE Transaction on Dielectric and Electrical Insulation, Vol. 21, Issue 3, pp. 1393-1404, 2014.

5. Mehdi Bagheri, Toan Phung, Trevor Blackburn, “Axial and Radial Transformer Winding Deformation: An Analytical Step towards Frequency Response Understanding”, IEEE Transaction on Dielectric and Electrical Insulation, (submitted) 2014.

Conference Papers

6. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung, “Practical Challenges in Online Transformer Winding Deformation Diagnostics”, IEEE International Conference on Electric Power and Energy Conversion Systems (EPECS’11), Sharjah, UAE, Nov. 15-17, pp. 1-6, December 2011.

7. Mehdi Bagheri, Mohammad S. Naderi, “Moisture Diagnostics of Power Transformers Using Dielectric Response Methods and Paper Samples Method”, IEEE Electrical Insulation Conference (EIC’11), Maryland, U.S.A., pp. 36-40, June 2011.

8. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung, “LED vs. CFL Effects on Distribution Transformer Efficiency”, 4th International Engineering Conference (EnCon’11), Kuching, Sarawak, Malaysia, pp. 1-5, December 2011.

9. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung, “Online Transformer Winding Deformation Diagnosis: A Profound Insight to Methods”, 26th International Power System Conference (PSC’2011), Tehran, Iran, pp. 1-9, Oct and Nov 2011.

10. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung “Bushing Characteristic Impacts on On-line Frequency Response Analysis of


11

Transformer Winding”, IEEE International Conference on Power and Energy (PECON’12), Kota Kinabalu, Malaysia, pp. 956-961, December 2012.

11. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, Damin Zhang, “Transformer Frequency Response Analysis: A Mathematical Approach to Interpret Mid-frequency Oscillations”, IEEE International Conference on Power and Energy (PECON’12), Kota Kinabalu, Malaysia, pp. 962–966, December 2012.

12. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung “Dean-Stark vs FDS and KFT Methods in Moisture Content Recognision of Transformers”, IEEE International Conference on Power and Energy (PECON’12), Kota Kinabalu, Malaysia, pp. 712-717, December 2012.

13. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung, “Frequency Response Analysis to Recognize Inductance Variation in Transformer Due to Internal Short Circuit”, IEEE International Power and Energy Conference (IPEC’12), Ho Chi Minh, Vietnam, pp. 1-5, December 2012.

14. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung, “Case Study on FRA Capability in Detection of Mechanical Defects within a 400MVA Transformer,” CIGRE, 21, rue d’Artois, F-75008 Paris, France, pp.1-9, August 2012.

15. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung , “FRA vs. Short Circuit Impedance Measurement in Detection of Mechanical Defects within Large Power Transformer”, IEEE International Symposium on Electrical Insulation (ISEI’12), San Juan, Puerto Rico, June 10-13, pp. 301-305, June 2012.

16. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung “Frequency Response Analysis vs. Flux Division Measurement in Detection of Transformer Winding Internal Short Circuit”, IEEE International Conference on Power System Technology (POWERCON’12), Auckland, New Zealand, pp. 1-5, November 2012.

17. Mehdi Bagheri, Mohammad S. Naderi, Trevor Blackburn, B.T. Phung “Transformer Efficiency and De-rating Evaluation with Non-Sinusoidal Loads”, IEEE International Conference on Power System Technology (POWERCON’12), Auckland, New Zealand, pp. 1-6, November 2012.

18. Mehdi Bagheri, Toan Phung, Trevor Blackburn, “On-line Transformer Frequency Response Analysis: Moisture and Temperature Influences on Statistical Indicators”, IEEE International Conference on Smart Instrumentation, Measurement and Applications (ICSIMA’13), Kuala Lumpur, Malaysia, pp. 119-124, November 2013 (Best Paper Award).

19. Mehdi Bagheri, Toan Phung, Trevor Blackburn “Transformer Core Influence on Frequency Response Spectrum Oscillations”, International Symposium on High Voltage (ISH’13), Seoul, South Korea, pp. 1772-1777, August 2013.

20. Mehdi Bagheri, B. T. Phung, Trevor Blackburn “Impact of Transformer Winding Dry-Out on Frequency Response Analysis”, IEEE International Conference on Electrical Insulation and Dielectric Phenomena (CEIDP’13), Shenzhen, China, pp. 1113-1116, October 2013.


12

21. Mehdi Bagheri, B. T. Phung, Trevor Blackburn, “Impacts of Transformer Winding Series Capacitance on FRA Spectrum Oscillations”, CIGR ’13, Int’l Study Committee Meeting and Colloquium, Brisbane, Australia, pp. 1-9, September 2013.

22. Mehdi Bagheri, Toan Phung, Trevor Blackburn, Ali Naderian “Shunt Capacitance Influences on Transformer FRA Spectrum”, IEEE International Electrical Insulation Conference (EIC’13), Ottawa, Canada, pp. 225-229, June 2013.

23. Mehdi Bagheri, Toan Phung, Trevor Blackburn, Ali Naderian “Influence of Temperature on Frequency Response Analysis of Transformer Winding”, IEEE International Electrical Insulation Conference (EIC’13), Ottawa, Canada, pp. 40-44, June 2013.

24. Mehdi Bagheri, Toan Phung, Trevor Blackburn, “A Practical Solution to Distinguish Winding Deformation from Insulation Characteristic Impacts on FRA Spectrum”, International Conference on Condition Monitoring and Diagnosis (CMD’14), Jeju, South Korea, (submitted) 2014.

25. Mehdi Bagheri, B. T. Phung, Trevor Blackburn, “The Influence of Dielectric Dissipation Factors on Transformer Frequency Response Analysis”, IEEE International Conference on Electrical Insulation and Dielectric Phenomena (CEIDP’14), Des Moines, IA, U.S.A., (submitted) 2014.

26. Mehdi Bagheri, Toan Phung, Trevor Blackburn, “Influence of Moisture Content Variation on Frequency Response Analysis of Transformer Winding”, IEEE International Electrical Insulation Conference (EIC’14), Philadelphia, U.S.A., (approved for publication) June 2014.

27. Mehdi Bagheri, Toan Phung, Trevor Blackburn, “Paper Moisture Variation vs. Mechanical Deformation Impacts on Transformer Frequency Response Spectrum”, 7th International Symposium on Electrical Insulating Materials (ISEIM’14), Toki Messe, Niigata City, Japan, (approved for publication) June 2014.

Chapter 2. Transformer Winding Deformation and Diagnosis Techniques

13

Chapter 2

Transformer Winding Deformation and Diagnosis Techniques

2.1 Introduction

This Chapter discusses transformer winding deformation and available diagnosis

techniques. The discussion begins with investigation on the main causes that put

transformers out of service. It then proceeds to highlight the impacts of short circuit on

transformer winding. It specifically debates on radial and axial deformations and then

introduces existing diagnosis methods. Frequency Response Analysis (FRA) as the main

tool of this study and its evaluation methods are elaborated and discussed in detail. The

advantages FRA as compared to other available methods for winding deformation

diagnosis are demonstrated through carefully-designed practical work.

This Chapter is structured in the following manner. Winding deformation is discussed in

Section 2.2 and its diagnosis techniques are described in Section 2.3. Comparison between

FRA and short circuit impedance (SCI) measurement is presented in Section 2.4 with

concluding remarks in Section 2.5.

2.2 Transformer Winding Deformation and Displacement

Different types of transformer windings are designed and manufactured based on voltage

level and electromagnetic relations. Spacers, barriers, dense woods and other materials

are used to provide mechanical support for the winding. Spacers are employed to separate

one disk from another disk in order to provide easy heat dissipation as well as mechanical

support. They can also help as a part of the cooling system by providing appropriate

distance between two adjacent disks for oil flow. Larger space between the disks might

result in oil flow speed reduction and inefficient cooling whereas small space can impede

oil flow and make cooling function useless. Hence, there is an optimum disk space based

on transformer winding design and its physical configuration and also oil viscosity.


14

On one hand, the spacers can be placed in layers to provide suitable disk-to-disk distance;

on the other hand they can be arranged horizontally in a disk to provide mechanical

support. In this regard, the number of spacers per disk (horizontally) is given by:

,ave

spacers spacer

DN

W

(2.1)

where, Dave is the average diameter of winding, Wspacer is the spacer width, and θs

represents the desirable angle between spacers based on mechanical support concepts.

Transformer mechanical defects can occur due to the many disturbances like as short

circuit currents, severe explosion of combustible gas in transformer oil, earth quake, or

even improper transportation. Winding deformations are reportedly happening between

spacers and barriers. Statistical evidence has shown that well placed spacers can prevent

winding deformation during short circuit, especially for inner winding subjected to the

radial inward forces. In fact, winding sections between spacers are considered the weak

points of a winding during short circuit events.

2.2.1 Short Circuit Current

Of all the possible causes of transformer failure, mechanical deformation of windings as a

result of high short-circuit currents is probably the most common. Such currents may

generate radial, axial or combined forces acting on a transformer winding. The result

could be radial, axial or angular deformation of the windings, or conductor rupture.

Transformer winding deformation may be categorized as follows [18]:

Radial forces

Forced buckling

Free buckling (hoop buckling)

Hoop tension (stretching)

Relaxation buckling

Axial forces

Tilting (cable-wise tilting, strand-wise tilting)

Conductor bending between radial spacers

Combined forces

Spiralling

Telescoping

Twisting


15

In all cases the force F acting on the transformer winding is given by:

.shF I d B

(2.2)

where, Ish is the short-circuit current, B is the vector of magnetic induction, and λ is the

winding length.

2.2.1.1 Radial Forces

Radial forces produced by the axial leakage field act outwards on the outer winding

tending to stretch the winding conductors and it can cause hoop stress. On the other hand,

the radial forces cause the inner winding to experience radial compressive stress. The

radial force due to the axial leakage flux in the gap between the two windings is calculated

as follow [10], [58]- [60]:

02

( 2 ),2averadial

w

NIF D NI

H

(2.3)

where, Hw is the winding height, Dave is the average winding diameter, NI denotes R.M.S.

winding’s ampere-turns value, and µo represents vacuum permeability.

Various types of deformations due to the radial forces were discussed earlier. Amongst all

of them, the buckling type of deformation is the most often reported in transformer

windings [10]. Radial forces in concentric windings lead to free buckling and forced

buckling as illustrated in Fig. 2.1.

2.2.1.2 Axial Forces

Axial forces due to short circuit current are produced by the radial magnetic flux. Axial

forces lead to tilting or bending of conductors and will be more dangerous when the

windings are not placed symmetrically. Any ampere-turn mismatch between LV and HV

windings will strengthen axial forces. Titling and bending of conductors between spacers

due to axial forces are shown in Fig. 2.2. Any small displacement during transformer

transportation or due to earth quake would result in intensified axial forces when short

circuit occurs. As the most common reported deformation type, symmetrical and

asymmetrical axial deformation in transformer winding will be discussed in detail in the

next Chapters.


16

(a) (b)

(c)

Figure 2.1. Winding deformation (buckling), (a) Free buckling (top view), (b) Forced buckling (top view), (c)

Free buckling (side view).

(a)


17

(b)

(c)

(d)

Figure 2.2. Winding deformation, (a) Before tilting, (b) After tilting, (c) Bending (side view), (d)Bending (close view).


18

2.2.2 Transformer Transportation Causing Active Part Displacement

Transformer transportation is one of the major causes in transformer active part

displacement and winding deformation. Transformer must be carefully transported,

especially in the case of large power transformers. In this regard, there are four different

means to transport a transformer:

Truck

Railroad

Sea carrier

Air carrier

In each region, based on available transportation infrastructures one of the mentioned

carriers will be employed. From the safety point of view, the best approach should

produce the lowest vibration and bring high reliability even if it is slow and time

consuming. While a transformer is transported, the main tank is filled with dry air, dry

nitrogen or any other gas which does not have any chemical impact on transformer major

and minor insulation system and at the same time completely prevents the core and

windings from absorbing moisture until final on-site oil injection. Therefore, the pressure

of the gas inside the transformer tank must be higher than the outside. Also, bushings are

disassembled for easier carrying. A typical arrangement for transformer transportation is

shown in Fig. 2.3.

Figure 2.3. Transformer transportation schematic.


19

All mechanical forces which can have serious effect on the transformer tank during

transportation need to be considered when the transformer is settled on the portable

surface. Hence, in mechanical forces calculation, all of ramps, winding routes, wind force,

stop shock as well as bumps should to be taken into account. Indeed, all effective acting

forces must be less than the static friction force between the transformer tank and

portable surface plus stop forces. In this regard, the overall acting force can be calculated

as:

s stopradial wind predicted mgF F F F (2.4)

where, Fradial is the imposed forces during transportation, Fwind is the wind force, Fpredicted

denotes other predictable forces, Fstop is the security force like bracing or stop forces, m is

the total transformer mass, and µs represents the static friction constant between the

transformer tank and portable surfaces and wooden bars if available. Figure 2.4 illustrates

different transformer transportation classes.

(a)


20

(b)

(c)


21

(d)

Figure 2.4. Transformer transportation, (a) Truck [61], (b) Rail road [62], (c) Sea carrier [63], (d) Air carrier

[64].

2.3 Winding Deformation Diagnosis Methods

Mechanical diagnostic methods have been developed to recognize transformer active part

displacement as well as winding deformation. Hence, various methods such as Low

Voltage Impulse (LVI), Frequency Response Analysis (FRA) and Short Circuit Impedance

(SCI) have been employed for off-line mechanical defects recognition in transformers [18].

In fact, off-line methods have been employed and their advantages from recognition

perspective have been debated quite extensively in the literature [18].

2.3.1 Short Circuit Impedance Measurement

The Short Circuit Impedance (SCI) may be considered as a parameter which highlights

imperfect magnetic coupling between primary and secondary windings. It contains

resistive and inductive terms, the latter being much more dominant than the former. The

short circuit impedance (or leakage inductance) can be represented as an additional

inductance in series with the transformer primary inductance, as shown in Fig. 2.5. A high

SCI value leads to a high voltage drop across the transformer terminals and thus affects

network voltage regulation, while a low value influences the network short circuit current.


22

The distances between the HV winding, the LV winding and the core of the transformer

have considerable influence on the SCI value (Fig. 2.6). The SCI is given by:

2,0.248

502( )n

x xx R

m Bw

SC D fU K

V N H N

(2.5)

where Ux is the short circuit impedance, NB is the number of transformer core limbs

surrounded by HV and LV windings, KR is the Rogowsky coefficient taken as 1 for most HV

and LV winding arrangements [18] but which can be calculated if necessary, f is the

operational frequency, Hm=(Hm1+Hm2)/2, where Hm1 is the height of the LV winding and Hm2

is the height of the HV winding, Vn is the nominal voltage of the winding, Nw is the number

of winding turns and S is the apparent power quoted on the transformer nameplate. In

addition, Dx=DC2 + (BOW2-BOW1)/3 and Cx=C2 + (BOW1+BOW2)/3, where the distances C2, DC2,

BOW1 and BOW2 are shown in Fig. 2.6. Clearly, winding deformation (changes in the

geometrical factors) will result in a change of Ux.

Figure 2.5. Schematic model of primary, secondary and leakage inductances of a transformer.

The measured SCI for a transformer should be compared to the value printed on the

nameplate or quoted in factory test results. Winding displacement that may have

occurred since the factory tests were performed may then be detected. According to [65],

changes of more than ±3% should be considered as indicating winding deformation or

core displacement. IEC standard 60076-5 states that, changes should not exceed ±1% for

transformers with power rating capacities above 100 MVA [66].

SCI measurements can be performed on single or three-phase transformers, usually on the

HV winding, with the LV winding short-circuited. The cross-sectional area of the cable

used to short-circuit the LV winding must be at least 30% greater than that of the winding

conductor ‎[67], and it must be as short as possible. The resistance of the connection

between the LV terminals and the shorting cable must also be as small as possible. The SCI

measurement test setup for single and three-phase transformers is shown in Figures

2.7(a) and 2.7(b) respectively.


23

Figure 2.6. Schematic of transformer core and windings.

(a)

(b)

Figure 2.7. Short circuit impedance measurement setup, (a) Single phase transformer, (b) Three-phase transformer. The test is made by short-circuiting the line-leads of the low-voltage windings and applying a single-phase voltage at rated frequency to terminals of the other winding. Three successive readings are taken on the three pairs of leads [65]. If the neutral terminal is available, the measurement can be conducted through the line-lead and the neutral-lead.


24

2.3.2 Transfer Function Methods (FRA/LVI)

Transfer function is basically a way of describing a system behaviour. It is an established

convention for quantifying the system response as well as other means such as differential

equations. Transfer functions encapsulate information about frequency response and time

response given certain input signals, and system characteristics can be picked out through

the transfer function as well. There are two popular methods that can be used for transfer

function measurement (Fig. 2.8). The first one applies in the time domain and named as

Low Voltage Impulse (LVI), while the second one operates in the frequency domain and

called Frequency Response Analysis (FRA).

Figure 2.8. Transfer function measurement techniques.

Frequency domain measurement is performed by injecting a swept sinusoidal waveform

within a predetermined frequency band as it is described in (2.6).

maxmin .

( ) sin( ) sin(2 ),sweep

sweep

U t A t A f t

f f f

(2.6)

where A is the sinusoidal signal amplitude and fsweep denotes the variable frequency. It is

worth noting that for commercially-available FRA equipment, A varies from 5.66 Vpp to 25

Vpp for different manufacturers.

FRA data are commonly presented as magnitude Bode diagrams, with the x-axis for

frequency and the y-axis for the response magnitude. In some cases, more information is

also provided in the form of the FRA phase diagram, although this does not seem to

provide useful diagnosis on the mechanical integrity of transformer. In fact, the magnitude

diagram conveys more detail information than the phase diagram and that is the reason

most of literatures are just concentrated on the magnitude. Likewise, this study is focused

only on the magnitude response.

In general, FRA measurement is performed in the frequency band 20Hz – 1MHz for

transformers with highest voltage of > 72.5 kV, and in the range of 20Hz – 2MHz for

transformers with highest voltage of ≤ 72.5 kV ‎[1]. To be on the safe side, FRA measurement

can be performed over the range 20Hz – 2MHz for all transformers irrespective of their

voltage rating. However, in the case of special transformers or reactors, the upper limit may


25

be shifted to even higher frequencies. For instance with air-core reactors, this limit could be

increased up to 20 MHz. It should be noted that on-site FRA measurement beyond 2 MHz is

likely to experience undesirable oscillations or additional fluctuations.

Some researchers believe that reliable results are obtained only within the range 10 Hz - 1

MHz ‎[1] for routine transformers, while others have recommended an upper bound of 10

MHz ‎[73].

To facilitate classification, the frequency response data may be divided into three bands,

namely low-, medium-, and high-frequency bands (see Fig. 2.9). In [1] it is stated that the

data are dominated by the transformer core at low frequencies, by the winding structure at

medium frequencies, and by the connection leads at high frequencies. However, the

boundaries between the bands are not widely agreed.

Apart from the high-frequency band which is influenced by connections leads as well as

other setup contacts, the mid- and low- frequency bands of FRA spectrum should be

interpreted, precisely. To reach to this level, this study will argue that although the

transformer core affects mainly the low-frequency band, elimination of the transformer

core would totally removes its impacts on the FRA trace, and enable a more accurate

interpretation of the mid-frequency oscillations are become discussable. Thus, to conduct

precise modeling of transformer winding for mid-frequency interpretation, an air-core

model transformer was manufactured and studied in detail in Chapter 4. Chapter 3 also

provides the calculation of air-core transformer winding parameters to utilize in Chapter 4.

To conduct interpretation of low-frequency band oscillations, a full-size 100 kVA

transformer was used as the test object and low-frequency oscillations are discussed in

detail in Chapter 5.


26

102

103

104

105

106

-90

-80

-70

-60

-50

-40

-30

-20

-10

Frequency [Hz]

Magnitude [

dB

]

Low Mid High

Figure 2.9. Low, mid and high frequency bands of a typical FRA spectrum (measured on 400 MVA

transformer).

In the time domain method, the impulse waveform is injected into the test object input and

the time domain response is measured from the test object output. Once the time domain

measurement data is obtained, the transfer function in frequency domain can be

determined by using FFT technique.

Generally, the purpose of both methods is to excite the natural frequencies of the test

object. Technically speaking, the transfer function of a certain test object determined by

using frequency domain method is not identical to that calculated by employing time

domain measurement and then utilizing FFT technique. For consistency, researchers have

recommended applying only one method in the reiteration of the measurements on a

particular item of equipment over a certain period of time. LVI is generally faster than

FRA, while FRA is more accurate. The trend in industry shows more interest in the latter.

In transformer winding studies, the FRA could be measured for HV windings or LV

windings. In three phase transformers windings are inter-connected (eg. as wye, delta or

zigzag); the FRA data can be extracted collectively or independently for the windings.

In fact, the FRA can be measured for any proposed circuit when the transformer is out of

service. Definitely, if we are setting up a proposed circuit for transformer frequency

response measurement now, the same proposed circuit needs to be applied in future

studies of transformer frequency response. Otherwise the results would not be

comparable.


27

2.3.2.1 FRA Measurement Setups

FRA has been widely used as a comparative diagnosis method for some years now. The

initial FRA measurements during factory testing serve as the winding fingerprint

(reference, baseline, or original trace). Changes in winding configuration would almost

certainly cause changes in the frequency response trace ‎[17]. The initial measurements of

the distributed resistance, capacitance and inductance of a winding may also be usefully

compared with the same measurements following transformer maintenance, repair or

transport ‎[15], ‎[72].

Independent measurement of the frequency response for each individual winding is more

convenient for future comparison. Hence it is recommended for transformer frequency

response measurement the best simple and independent circuit to be considered.

Appropriate circuit for Yn winding connection will be phase bushing lead as input and

neutral bushing lead as output. When the transformer includes isolated wye or delta

connection, two phase leads need to be regarded as input and output. Based on this,

different test setups of FRA are given as follow ‎[1].

2.3.2.1.1 End-to-end Measurement

In this setup, FRA measurements can be made on a winding by injecting a preset signal Vin

at the line-lead, and detecting the response Vout at the neutral-lead of the transformer, as

shown in Fig. 2.10(a). The frequency response magnitude Kmag (the voltage attenuation in

dB) is given by:

1020log outmag

in

VK

V

(2.7)

2.3.2.1.2 Inductive Inter-winding Measurements

In this setup, FRA measurements can be made on a winding by injecting a preset signal Vin

at the line-lead (for instance HV winding, phase U), and detecting the response Vout at the

line-lead of the corresponding concentric winding (for instance LV winding, phase u),

where the neutral-lead of both windings are grounded as shown in Fig. 2.10(b).

2.3.2.1.3 Capacitive Inter-winding Measurements

In this case, FRA measurements can be made on a winding by injecting a preset signal Vin at

the line-lead (for instance HV winding, phase U), and detecting the response Vout at the line-


28

lead of the corresponding concentric winding (for instance LV winding, phase u), where the

neutral-lead of both windings are left open circuit as shown in Fig. 2.10(c).

2.3.2.1.4 End-to-end Short-circuit Measurements

In this case, FRA measurements can be made on a winding by injecting a preset signal Vin at

the line-lead, and detecting the response Vout at the neutral-lead of the transformer, where

all terminals of the other side are short-circuited as shown in Fig. 2.10(d).

Transformer

FRA

Vin

Vout

Zin

ZoutV

sourc

e

(a)

Transformer

FRA

Vin

Vout

Zin

Zout

Vso

urc

e

(b)


29

Transformer

FRA

Vin

Vout

Zin

Zout

Vso

urc

e

(c)

Transformer

FRA

Vin

Vo

ut

Zin

Zout

Vso

urc

e

(d)

Figure 2.10. FRA test setups, (a) End-to-end measurement, (b) Inductive inter-winding measurement, (c)

Capacitive inter-winding measurement, (d) End-to-end short-circuit measurement.

Figure 2.11 and Table 2.1 provide detailed FRA setups for both star and delta connections

(star and delta connections do not influence illustrated terminals in Table 2.1).


30

A B C

cba

A1

B1

C1

C2

B2

A2

a1

b1

c1

a2b2

c2

HV

Win

din

gsLV

Win

din

gs

Figure 2.11. FRA test setups (detailed connections).

Table 2.1. FRA measurement connections (phases A, B and C).

Setup Source (Vin) Response (Vout) Terminals

earthed

Terminals connected

together

End-to-end A1 A2 none none

Inductive inter-winding A1 a1 A2 and a2 none

Capacitive inter-winding A1 a1 none none

End-to-end short-circuit A1 A2 none a1-a2-b1-b2-c1-c2

End-to-end B1 B2 none none

Inductive inter-winding B1 b1 B2 and b2 none

Capacitive inter-winding B1 b1 none none

End-to-end short-circuit B1 B2 none a1-a2-b1-b2-c1-c2

End-to-end C1 C2 none none

Inductive inter-winding C1 c1 C2 and c2 none

Capacitive inter-winding C1 c1 none none

End-to-end short-circuit C1 C2 none a1-a2-b1-b2-c1-c2


31

2.3.2.2 Evaluation Indicators

A common method to interpret frequency responses of transformer windings is by using

evaluation indicators (statistical indices or statistical indicators). Such indicators have

been introduced by various researchers over the years. In order to exploit FRA test results

for extracting more information, a software package was developed to calculate statistical

indicators (see Appendix A). Statistical indicators with mathematical expressions as below

are implemented in the software to quantify the gap between reference and deviated

traces:

Correlation Coefficient (CC) [20]- [21], [23], [25]- [26]:

1( , )

2 2

1 1

s

s s

i i

N

i iiX Y

N N

i iX Y

X YCC

(2.8)

Maximum Absolute Difference (DABS) [20]:

1

( , )

| |s

s

N

i iiX Y

Y XDABS

N

(2.9)

Minimum Maximum (MM) [20]:

1

( , )

1

min( , )

max( , )

s

s

N

i iiX Y N

i ii

X YMM

X Y

(2.10)

Standard Deviation (SD) [21]:

2

1( , )

1

s

i i

s

N

iX Y

Y XSD

N

(2.11)

Spectrum Deviation (σ) [22], [68]:

2 2

( , ) 1

1 2 2

2 2

s

i iN

X Y i

i i i i

i i i is

X Y

N

X Y X Y

X Y X Y

(2.12)


32

where Xi and Yi are the ith elements of the fingerprint and measured FRA traces

respectively, and Ns is the number of elements (or samples). CC is thus a number whose

absolute value lies between 0 and 1 [21]. Maximum coefficient is achieved by complete

overlapping between the reference and measured traces and minimum coefficient is when

there is no overlapping at all.

SD values less than or equal to 1 indicate normal condition for windings, while values

greater than 1 reveal a problem associated with winding deformation [21]. The highest

possible similarity level of FRA traces can be determined by DABS, MM and σ parameters

once they take values as 0, 1 and 0, respectively but there are no predetermined limits for

maximum deviation for them. The Relative factor (Rxy) has been introduced and discussed

in [25] and [74]- [75] (see Appendix A). Tables 2.2 and 2.3 provide the criteria for CC, SD

and Rxy. The equation for Rxy is provided in Appendix A.

Table 2.2. CC and SD criteria [21].

Statistical indices Boundary values of parameters for deformation indication

CC (Magnitude spectrum) <0.9998

CC (Phase spectrum) <0.95

SD (Magnitude spectrum) >1.00

SD (Phase spectrum) >10.0

Table 2.3. Level of deformation based on Rxy [25], [74]- [75].

Deformation category Rxy

Severe RLF < 0.6

Obvious 1.0> RLF >= 0.6 or RMF < 0.6

Slight 2.0> RLF >=1.0 or 0.6 =< RMF < 1

Normal RLF >= 2.0, RMF >= 1.0 and RHF >= 0.6

LF: 1 kHz - 100 kHz , MF:100 kHz- 600 kHz, HF: 600kHz - 1 MHz

2.3.3 Deformation Coefficient Method

In this method which is recommended by [76], instead of sweep frequency, only three

measurements at selected high frequency and at selected low frequency are considered.

The main approach of this method is capacitance measurement at both ends of the

winding. Joshi et al [76] have introduced Deformation Coefficient (DC) as a judicable

parameter in transformer winding deformation recognition.

Transformer windings are generally represented by a lumped parameter equivalent

circuit which includes series capacitances and shunt capacitances. The Deformation

Coefficient is a function of changes in series capacitance as well as shunt capacitance.


33

Deformation Coefficient parameter which can be calculated through off-line measurement

is given by [76]:

1 1

102 2

,log H H

H H

C CDC

C C

(2.13)

where, C1H and C2H are the fingerprint values of measured terminal capacitances at the

selected high frequency and 1H and 2H are the terminal capacitance (after deformation)

values at terminals 1 and 2, respectively. These terminals are discussed in ‎[76] in detail.

As discussed in ‎[76], the proposed capacitances in (2.13) can be measured through off-line

process and DC deviation indicates transformer winding deformation. This method has

not been industrialized, but registered as a patent ‎[77].

2.4 FRA vs. SCI

Frequency response analysis and short circuit impedance measurement as two popular

methods for transformer winding deformation diagnosis will be employed to get insight

into transformer active part condition, but which one is more sensitive for checking

winding deformation? The answer to this question via a practical case study below

indicates FRA is a better technique.

A failed 400 MVA step-up transformer was used in order to compare the effectiveness of

FRA and SCI measurements. The principal parameters of the transformer are given in

Table 2.4. The transformer was suspected to have winding deformation due to the short

circuit fault. Therefore, both SCI and FRA measurement methods were performed to

assess the mechanical integrity of transformer and the results were compared.

2.4.1 SCI Measurements

The LV winding terminals were short-circuited, and SCI measurements were made

between the HV windings and the neutral terminal. The measured impedances are

presented in Table 2.5.


34

Table 2.4. 400 MVA step-up transformer parameters.

Parameter Value Parameter Value

Manufacture date May 1993 No-Load current (%) 0.37

Rated Voltage [kV] 242/20 Number of phases 3

Rated power [MVA] 400 Number of limbs 5

Rated current [A] 954/11550 Frequency [Hz] 50

No. of coolers 12 Cooling system OFAF

Table 2.5. Measured SCI values.

Measurement Scheme Measurement Zmeasured

[ohm]

Zfactory

[ohm]

Change (%)

HV-LV A-Neutral 17.028 17.290 1.515

HV-LV B-Neutral 17.798 17.290 2.938

HV-LV C-Neutral 17.088 17.290 1.168

Accuracy of multimeter: 0.5 mV

The percentage change, defined as 100 Zfactory – Zmeasured/ Zfactory, is smaller than the

maximum permitted change (3%) suggested in [65], for each of the three phases. Thus one

standard [65] indicates that winding deformation or displacement had not occurred.

(However, the change of 2.93% for phase B is close to the limit and thus might raise

concern to an expert). On the other hand, each of the three changes exceeds the 1% figure

as suggested in [66] for transformers with capacities above 100 MVA. Thus the latter

standard [66] indicates that deformation or displacement had occurred in each of the

three windings. Clearly the two standards yield conflicting indications.

2.4.2 FRA Measurements

FRA measurements were made by applying a 5.66 V signal across each winding, at 801

discrete frequencies in the range 20 Hz - 2 MHz. The characteristic impedance of the

measurement cables was 50 . The measured FRA traces were compared with fingerprint

traces obtained during transformer overhaul (before failure).

The results are shown in Fig. 2.12. In order to avoid compression of the higher frequency

data, the plots of the frequency response magnitudes are presented on a logarithmic

frequency scale. It can be seen that the measured and fingerprint spectra for phases A and

C are very nearly identical. However, this is not the case for phase B.


35

(a)

(b)

102

103

104

105

106

-90

-80

-70

-60

-50

-40

-30

-20

-10

Frequency [Hz]

Magnitude [

dB

]

Phase A (Measured trace)

Phase A (Fingerprint)

Frequency Band: 20 Hz - 2 MHz

102

103

104

105

106

-90

-80

-70

-60

-50

-40

-30

-20

-10

Frequency [Hz]

Ma

gn

itu

de

[d

B]

Phase B (Fingerprint)

Phase B (Measured trace)



36

(c)

Figure 2.12. (a), (b) and (c) show the measured and fingerprint frequency response magnitudes for phases A, B and C respectively of the transformer HV side.

Low-, medium- and high-frequency bands for RXY, and deformation levels related to the RXY

values, have been defined in the Chinese standard [74] and by some other workers [25],

[75] (see Table 2.3). These definitions have been widely used for FRA trace evaluation, and

have been adopted in the present work. The calculated values of CC and SD for each of the

three phases are given in Table 2.6, and the corresponding values of RXY in Table 2.7. It can

be seen that, for phase B, CC and SD indicate deformation, and RXY indicates slight

deformation at low frequency and no deformation at medium and high frequencies. There

is no indication of deformation of phase A or phase C.

Table 2.6. CC, SD values

Frequency Band CC SD

Phase A Phase B Phase C Phase A Phase B Phase C

20 Hz - 2 MHz 0.9999 0.9977 1 0.5859 3.6367 0.5004

Table 2.7. Rxy values

Frequency Band RXY

Phase A Phase B Phase C

Low 10 1.2674 10

Medium 10 1.0724 10

High 10 2.3831 10

Low: 1 kHz - 100 kHz, Medium:100 kHz - 600 kHz, High: 600 kHz - 1 MHz.

102

103

104

105

106

-90

-80

-70

-60

-50

-40

-30

-20

-10

Frequency [Hz]

Magnitude [

dB

]

Phase C (Measured trace)

Phase C (Fingerprint)



37

These findings were checked and validated by internal inspection after de-tanking the

transformer. Figure 2.13 shows the side and front views of phase B of the HV winding.

Clearly the winding had suffered slight outward deformation (buckling) in two different

parts.

(a)

(b)

Figure 2.13. Buckled HV winding of phase B, (a) Side view of the middle disks, (b) Front view of the upper

disks.


38

2.4.3 Discussion on FRA and SCI

SCI measurements have been used for many years to detect and locate transformer

winding deformation and displacement. Given (2.5), the SCI values would be expected to

change, due to changes of Bow1 and Bow2, if significant radial movement occurred within a

winding. Changes in Hm would also be expected to influence the SCI value. However,

internal axial movement between transformer winding disks, e.g., bending and tilting of

conductors, would not be expected to influence the SCI value, since they would not change

any of the factors in (2.5). The results presented in this part suggest that FRA is capable of

providing reliable information on the level of deformation of transformer windings, based

on single values of the statistical quantities CC, SD and Rxy. However, it does not at present

provide information on the type of deformation. A more precise interpretation of FRA data

needs to be developed. This interpretation is provided in Chapters 4, 5 and 6.

2.4.4 Summary on FRA and SCI Methods

Off-line SCI and FRA measurements were conducted on a transformer which had failed

because of deformation of the B phase of the HV winding. Interpreted according to IEEE

Standard 62-1995 [65], the SCI values indicated that winding deformation had not

occurred, but when interpreted according to IEC Standard 60076-5, Ed. 3.0, 2006 [66],

they indicated that deformation had occurred in each of the three windings. Two indices

derived from comparison of fingerprint and measured FRA data indicated deformation of

phase B winding, and the third index indicated slight deformation of that winding. Visual

inspection of the failed transformer showed that the B phase winding had indeed been

deformed.

Since SCI measurements are made only at the operating frequency, it is expected that they

would provide less detailed information on the state of the windings than the more

comprehensive FRA measurements. Nevertheless, it is recommended that both types of

measurement be performed on power transformers (along with other conventional test

measurements) as part of transformer routine maintenance.

2.5 Conclusion

This Chapter provided a review of the main reasons causing winding deformation and

specifically discussed the impact of short circuit current. The various methods to

recognize winding deformation and active part displacement were introduced. It was

highlighted that FRA is superior to SCI in terms of winding deformation recognition.

Statistical indicators as the evaluation methods of frequency response were discussed.

Chapter 3. Transformer Winding Parameters

39

Chapter 3

Transformer Winding Parameters

3.1 Introduction

In order to gain new knowledge on FRA trace interpretation through analytical approach,

the focus should be specifically on the transformer winding electrical parameters. To this

end, the present Chapter investigates inductance and capacitance calculation in

transformer windings. This in turn will help in verification of the developed model in the

next Chapter. It is also quite crucial to study axial and radial deformation in transformer

winding in Chapter 6.

To conduct the calculations, the test object is a transformer winding. This is just a model to

clarify how to calculate inductance and capacitance. The model winding is enclosed in a

cylindrical metal container (tank). At first, the self- and mutual-inductances as well as

series and shunt capacitances are calculated for the winding in normal (not deformed)

condition. Afterwards, the analytical approach is validated through practical measurement

on the manufactured small model transformer with air-core continuous disk type HV and

LV windings. This circumstance verifies that the calculated results are accurate.

It should be highlighted that the problem of inductance calculation has been quite

thoroughly treated in a number of significant studies. Formulas for circular filaments were

first given by Maxwell ‎[78]. Subsequently, Rayleigh ‎[79], Lyle ‎[80], Butterworth ‎[81], Snow

‎[82], Rosa ‎[83], Curtis and Sparks ‎[84], Grover ‎[85], Babic ‎[86], Conway ‎[87], etc., have

developed and derived others formulas and tables to calculate self and mutual

inductances.

In this study, the Grover’s formulas ‎[85] are used for the self-inductance calculation of

circular coils of rectangular cross section as well as mutual-inductance calculation for

coaxial circular filaments, circular filaments whose axes are inclined and circular coils

with parallel axes. Grover’s formulas are then developed in Chapter 6 for the calculation of

transformer winding inductance matrix for the case of axial deformation.


40

It should be noted that to simplify inductance calculations, the thickness of paper

insulation wrapped over the conductors is ignored in this study, but for capacitance

calculation this thickness is taken into consideration.

3.2 Self and Mutual Inductances of Transformer Winding (Analytical

Approach)

In the absence of magnetic materials, self and mutual inductances are parameters

dependent only on the system configuration and independent of the current ‎[85].

Formulas for the self-inductance calculation of the disk winding have been discussed

widely in the literatures ‎[88]-‎[89]. However, the calculation of mutual-inductance between

disks has not been formulated ‎[88]. The inductance matrix of the winding (Leq) illustrated

in Fig. 3.1 is given by (3.1) in general form and (3.2) in detailed form:

A AB AC AD

B BDBA BC

CA CB C CD

DB DDA DC

L M M M

M L M MLeq

M M L M

M M M L

(3.1)

1 12 13 14 15 16 17 18 19 110 111 112 113 114 115 116

2 23 24 25 26 27 28 29 210 211 212 213 214 215 216

3 34 35 36 37 38 39 310 311 312 313 314 315 316

4 45 46 47 48 49 410 411 412 413 414 41

.

.

.

eq

L M M M M M M M M M M M M M M M

L M M M M M M M M M M M M M M

L M M M M M M M M M M M M M

L M M M M M M M M M M M

L

5 416

5 56 57 58 59 510 511 512 513 514 515 516

6 67 68 69 610 611 612 613 614 615 616

7 78 79 710 711 712 713 714 715 716

8 89 810 811 812 813 814 815 816

9 910 911 912 913 914 915 916

1

.

. .

. .

. .

. . .

. . .

M

L M M M M M M M M M M M

L M M M M M M M M M M

L M M M M M M M M M

L M M M M M M M M

L M M M M M M M

L 0 1011 1012 1013 1014 1015 1016

11 1112 1113 1114 1115 1116

12 1213 1214 1215 1216

13 1314 1315 1316

14 1415 1416

15 1516

16

. . .

. . . .

. . . .

. . . .

.

. . . . . . . . . . . . . . .

M M M M M M

L M M M M M

L M M M M

L M M M

L M M

L M

L

(3.2)

where:

1 12 13 14

21 2 23 24

31 32 3 34

41 42 43 4

A

L M M M

M L M ML

M M L M

M M M L

(3.3)


41

5 56 57 58

65 6 67 68

75 76 7 78

85 86 87 8

B

L M M M

M L M ML

M M L M

M M M L

(3.4)

9 910 911 912

109 10 1011 1012

119 1110 11 1112

129 1210 1211 12

C

L M M M

M L M ML

M M L M

M M M L

(3.5)

13 1314 1315 1316

1413 14 1415 1416

1513 1514 15 1516

1613 1614 1615 16

D

L M M M

M L M ML

M M L M

M M M L

(3.6)

15 16 17 18

25 26 27 28

35 36 37 38

45 46 47 48

AB

M M M M

M M M MM

M M M M

M M M M

(3.7)

19 110 111 112

29 210 211 212

39 310 311 312

49 410 411 412

AC

M M M M

M M M MM

M M M M

M M M M

(3.8)


42

113 114 115 116

213 214 215 216

313 314 315 316

413 414 415 416

AD

M M M M

M M M MM

M M M M

M M M M

(3.9)

59 510 511 512

69 610 611 612

79 710 711 712

89 810 811 812

BC

M M M M

M M M MM

M M M M

M M M M

(3.10)

513 514 515 516

613 614 615 616

713 714 715 716

813 814 815 816

BD

M M M M

M M M MM

M M M M

M M M M

(3.11)

913 914 915 916

1013 1014 1015 1016

1113 1114 1115 1116

1213 1214 1215 1216

CD

M M M M

M M M MM

M M M M

M M M M

(3.12)

In the above, LA is the inductance matrix of disk A, MAB is the mutual-inductance matrix of

disks A and B, L1 denotes the turn inductance, and M12 is the turn-to-turn inductance

between the turns 1 and 2. Kirchhoff ‎[90] has shown that the equivalent self-inductance of

a disk is equal to the summation of the self and the mutual-inductance of each turn with

respect to all the other turns in that disk. Hence, for the proposed winding, the self-

inductance of the first disk (LA) is given by:

1 2 3 4 12 13 23 14 24 342AL L L L L M M M M M M (3.13)

while the mutual-inductance of the same disk is given by:


43

12 13 23 14 24 342AM M M M M M M (3.14)

Equations (3.13) and (3.14) can be extended for any number of disks and not restricted to

a single disk; hence, the self-inductance for the winding of Fig. 3.1 is given by:

2B D BDA C AB AC BC AD CDeqL L L L L M M M M M M

(3.15)

The mutual-inductance of the winding is then defined as:

2 BDAB AC BC AD CDeqM M M M M M M (3.16)

1 2 3 4 4 3 2 1

8 7 6 5

9 10 11 12

5 6 7 8

12 11 10 9

MAB

A

C

B

16 15 14 13 13 14 15D

MBC

MCD

MAC

MBD

MAD

M13 M24 M34 M23 M12

R

B w

W

δd

1 2 3 4

8 7 6 5

9 10 11 12

16 15 14 13

1234

5 6 7 8

9101112

13 14 15

δt

εt

h

d

Air-Core

Figure 3.1. Air-core transformer winding model.

It should be noted that (3.2) is a symmetrical matrix for a winding with normal structural

condition. It will be shown latter that this matrix will turn into an asymmetrical matrix

once axial deformation occurred.

3.2.1 Self - Inductance

The self-inductance of a circular winding with rectangular cross sections is a function of

winding shape ‎[85]. The significant parameters defining the winding shape are the mean

radius of the turns per disks, and axial and radial dimensions of the conductor cross


44

sections. These parameters are illustrated in Fig. 3.1. Based on this, the self-inductance (L)

of an air-core circular disk for h=W is given by ‎[85]:

0

20.001L RN P H (3.17)

where, N is the number of disk turns, R is the mean radius of disk turns, h is the axial

dimension of the conductor cross section, W is the radial dimension of the winding cross

section, and P0 is a function of W/2R. For relatively small cross section category such that

(W/2R < 0.2), P0 is given by ‎[85]:

0

22 21 1 2

4 ln 8 0.84834 0.20412 12 2 2

W R WP

R W R

(3.18)

For a thin circular disk with rectangular cross section of any desired proportions, the self-

inductance is given by:

0

220.019739 ( )

N

RL N R K k H

W

(3.19)

where KN comes from Nagaoka’s formula ‎[91] and can be derived through Table B.1 in the

[Appendix B], and k0 is a factor that specifies circular inductance decrement due to the

separation of turns in radial direction. For a single turn with significant mean radius

dimension, k0 can be zero (see Table B.2).

3.2.2 Mutual – Inductance

The mutual inductance of a transformer winding (Mc) can be calculated for one disk with

respect to another disk as given by (3.20), or it can be achieved through summation of the

turn-to-turn mutual inductances in the inductance matrix of a winding as obtained by

(3.21).

1 2 0M N N M Hc (3.20)

N1 and N2 denote the numbers of turns of different disks, and M0 is calculated using Lyle’s

method ‎[80].

In this study, as the main goal is specifically focused on the axial and radial deformation in

a disk; thus, instead of winding mutual-inductance in (3.20), the turn-to-turn mutual-

inductance calculation is further discussed in detail. Indeed, winding mutual-inductance is


45

simply achievable through turn-to-turn mutual-inductance summation. The mutual-

inductance of coaxial circular filaments (turn-to-turn) is given by ‎[85]:

abM f R R H

m (3.21)

where Ra and Rb are the mean radius of the turns a and b, respectively (see Fig. 3.2(a)). fm

is a function of parameter k which is given by:

2 2

2 2(1 )

, ,(1 )

a

b b

R dk

R R

(3.22)

d is the distance between circular turns as illustrated in Figures 3.2(a) and 3.2(b).

Rb

d

Ra

(a)

Ra

Rb

d

(b)

Figure 3.2. Coaxial circular conductors, (a) Coaxial filaments, (b) Coaxial disks.

For usual transformer windings, k ≤ 0.1; thus, fm can be obtained by ‎[85]:

10

10.014468 log 0.53307f

m k

(3.23)

For the mutual-inductance between two turns in a common disk β=0, and α=1 is

considered for the inter-disk mutual inductances of the conductors with equal mean

radius. Therefore, all mutual inductances (Mxy) in (3.2) can be obtained through the use of

(3.21).


46

3.3 Series and Shunt Capacitances of Transformer Winding

3.3.1 Series Capacitance

3.3.1.1 Layer Winding

Figure 3.3 is a schematic of layer winding which shows its series and shunt capacitances.

According to Fig. 3.3, there is a capacitance between each conductor and its adjacent

conductors. If Nw is the number of turns in layer winding, the number of series

capacitances would be Nw-1, and the series capacitances can be determined through ‎[92]:

1w

CttCsN

(3.24)

where Ctt is the turn-to-turn capacitance and Cs is the total series capacitance in a layer

winding. It should be noted that the value of series capacitances will be reduced when the

number of series conductors is increased.

U

Ctt

Air-CoreCg

3

4

1

2

Ctt

Ctt

Next turn

1

2

3

4

Tank

Cg

Cg

Cg

Figure 3.3. The overall layout of a layer winding including equivalent capacitance network.

3.3.1.2 Disk Winding

Various types of disk winding are used in power and distribution transformers.

Continuous disk winding (also called conventional disk winding) and interleaved winding


47

are the most popular. In some distribution and power transformers continuous disk

winding is used, while interleaved winding is employed in large power transformers at

voltage level of 230 kV and above.

3.3.1.2.1 Continuous Disk Winding and Related Series Capacitances

In Fig. 3.4, a continuous disk winding containing two disks is shown. Also, Fig. 3.5 shows

the equivalent capacitive network of continuous disk winding. The series capacitance is

divided into two parts:

1) Total series capacitance between the turns, Ct.

2) Total series capacitance between the disks, Cd.

To calculate the equivalent series capacitance in continuous disk winding, the energy

summation method is used. According to this method, the summation of energies in the

capacitances along a pair of disks is equal to the total energy which exists in the winding

with those two disks. It is assumed that the number of conductor turns in each disk is N.

The number of series capacitors between turns, as shown in Fig. 3.5, will be 2N-2 for a pair

of disks.

Therefore the total equivalent capacitance between the conductors, Ct, is given by ‎[92]:

21 1 1 12 (2 2)

22 2 2 2tt ttU N

C U N C C Ct tN N

(3.25)

4 3 2 1

5 6 7 8

Air-Core

1234

5 6 7 8

Next disc

U

Line-lead Tank

Figure 3.4 Continuous disk winding schematic, taken and modified ‎[92].


48

4 3 2 1

5 6 7 8

Air-Core

1234

5 6 7 8

Next disc

CttCttCtt

Cdd Cdd Cdd U

Line-leadTank

Figure 3.5. Equivalent capacitance network of the continuous disk winding.

where U is the voltage drop across the pair of disks (see Fig. 3.5), and Ctt is given by:

0 .2

2t

tt tt

hC R

(3.26)

In the above formula, δt is the thickness of inter-turn insulation, εt is the relative

permittivity of paper insulation, and ε0 is the vacuum permittivity.

Calculation of the equivalent capacitance between disks (Cd) is based on the voltage

distribution demonstrated in Fig. 3.6.

According to Fig. 3.6, when moving from end points starting from conductor number 1 or

number 8 towards middle of the winding (conductor number 4 and number 5) the voltage

on corresponding conductors will change linearly and continually. Hence, the steady state

voltage distributions for conductor in upper and lower disks are as follows:

2( ) , ( )

2 2up down

x xU n U U n U

(3.27)

where n is the turn number, λ' is the total length of conductor in one disk, and U denotes

the voltage across the disk pair. The equivalent inter-disk capacitance between two disks

is given by ‎[92]:

2 2

2 2

0

1 1( ( ) ( ))

2 21

(1 )2

upd d dd down

dd

E C U C U n U n

xC U dx

(3.28)

.

3dd

d

CC (3.29)


49

1 2 3 4

8 7 6 5

U/2N

l

x

U Cdd

Cttl

UU/2N

0

Figure 3.6. Pair of disks, cross-section overview and voltage distribution along disks pair (paper insulation has

been ignored).

The summation of Ct and Cd gives the equivalent series capacitance Cs-pair for a pair of disks

in a continuous disk winding:

.ts pair dC C C (3.30)

The first part in (3.30) corresponds to the capacitance between the disks, which is

obtained with stored energy and the second part corresponds to the capacitance between

the conductors. The equivalent capacitance for the entire winding (Cs) is then obtained by:

2

111

4

w

ddttd

d dw

d

N

NNC C Cs

N NN

N

(3.31)

where Nd is the number of transformer winding disks, and Nw is the number of winding

turns.

3.3.1.2.2 Interleaved Winding and Related Series Capacitances

In an interleaved winding, Cs increases considerably as compared to continuous disk

winding, therefore it is utilized to improve the electric stress distribution. Various

methods to interleave the disk windings are available. One of the simplest techniques is

shown in Fig. 3.7.


50

4 3 2 1

5 6 7 8

Air-Core

1526

3 7 4 8

Next disc

U

Line-lead Tank

Figure 3.7. The interleaved disk winding.

The series capacitance of this interleaved winding is given by [92]:

int

( 1)

4

C NttC Es

(3.32)

where Eint is the number of disks used for interleaving and equal 2 for Fig. 3.7. Increasing

Eint will increase the manual welding work required within the interleaved winding

considerably. The magnitude of series capacitance is increased in interleaved windings

and this leads to a more uniform voltage distribution throughout the winding. However,

the turn-to-turn potential difference is increased considerably for steady state operation

and thus the turn-to-turn insulation should be made properly to withstand against this

potential.

3.3.1.2.3 Intershield Winding

In Fig. 3.8, the configuration of a disk winding with electrostatic shields in each disk is

shown. In this winding, the shield turns, which can be made from copper or aluminum

conductor, are placed between the winding main conductors at predetermined places,

while the shield or shield turns of each disk are insulated from the conductors [Appendix

C]. Electrostatic shield conductors from the upper disk (in a pair of disks) are connected to

the electrostatic shield conductors of the lower disk at outermost shield turn as shown in

Fig. 3.8. For instance, the shield conductor between main conductors 1 & 2 of the upper

disk is connected to the shield conductor between conductors 7 & 8 from the lower disk

and both are isolated from the main conductors.


51

4 3 2 1

Air-Core

1234

Next disc

U

Line-lead

321

4 3 2 18765 456

Tank

Figure 3.8. The intershield disk winding.

The series capacitance of the intershield winding is given by [Appendix C]:

2

2 ( 1)1 1

2 3sh

s ttddN NN

C CN N

C

(3.33)

where Nsh is the number of shield turns per disk.

3.3.2 Shunt Capacitance

The shunt capacitance between the winding and the cylindrical metal container (tank) is

given by:

2 1

2

ln

wg

HC

r r

(3.34)

where, Hw is the height of the winding, ε is the dielectric permittivity, r1 is the radial

dimension of the winding, r2 denotes the radial dimension of the tank, and δt as compared

to r2 is small and thus ignored.

3.4 Verification of Calculated Parameters Using Manufactured Model

Transformer

3.4.1 Manufactured Model Transformer (Test Object)

In order to conduct this thesis, a specific test object was designed and fabricated by the

author. This test object is a model transformer with air-core concentric continuous disk

type HV and LV windings. The HV winding consists of 8 disks with 8 conductor turns per


52

disk. The LV winding has 10 disks with 6 conductor turns per disk. The cylindrical tank

housing the windings was made from plexiglass material. The line and neutral leads of the

windings were brought out from the tank through appropriate HV and LV bushings. A

drain valve was installed on the top plate to enable oil injection and also taking oil sample.

The plexiglass tank was air-tight sealed, proper for vacuuming and winding dry-out. Also

when required, an aluminum foil was wrapped over the glass tank to simulate the metal

tank. Detailed information of the manufactured test object is provided in Appendix D. The

test object is shown in Fig.3.9.

(a)

(b)

Figure 3.9. Manufactured glassy model transformer (a) Bird’s-eye view, (b) Side view.


53

3.4.2 Inductance Calculation of the Test Object

The inductance of the HV winding of the test object was measured using a precision LCR

bridge (AIM-TTI LCR400) and compared with the value calculated based on the analytical

approach developed in this Chapter. The measurement value obtained is:

( ) 710.7000HV measL H (3.35)

as shown on the LCR Bridge in Fig. 3.10.

Figure 3.10. Measured value for the inductance of HV winding.

For the inductance calculation using analytical approach, the HV winding schematic of the

manufactured test object as shown in Fig 3.11 was considered. Based on this and ignoring

the paper insulation thickness of the conductors, the calculated value for inductance is

given as (3.36):

21.1170 17.0926 11.8346 8.7993 6.6873 5.0901 3.8249 2.7928

17.0926 21.1170 17.0926 11.8346 8.7993 6.6873 5.0901 3.8249

11.8346 17.0926 21.1170 17.0926 11.8346 8.7993 6.6876 5.0901

8.7993 11.9346 17.0926 21.1170 17.0926 11.834HVL

6 8.7993 6.6873

6.6873 8.7993 11.8346 17.0926 21.1170 17.0926 11.8346 8.7993

5.0901 6.6873 8.7993 11.8346 17.0926 21.1170 17.0926 11.8346

3.8249 5.0901 6.6873 8.7993 11.8346 17.0926 21.1170 17.0926

2.7928 3.8249 5.0901 6.6873 8.7993 11.8346 17.0926 21.117

(3.36)

Hence, the total value of the HV winding inductance (LHV) is given by:


54

( ) 743.2651HV calcL H (3.37)

Equation (3.37) in turn verifies the analytical approach. The discrepancy between

measured and calculated values is less than 10 percent and could be due to the paper

insulation thickness being ignored.

1 2 3 4

Air-Core

MAB

MBC

MCD

M13 M24 M34 M23 M12

R

B

w

W

δd

1 2 3 4

δt

εt

d

1 2 3 45 6 7 8

1 2 3 416 15 14 13 1 2 3 412 11 10 9

1 2 3 417 18 19 20 1 2 3 421 22 23 24

1 2 3 432 31 30 29 1 2 3 428 27 26 25

1 2 3 433 34 35 36 1 2 3 437 38 39 40

1 2 3 448 47 46 45 1 2 3 444 43 42 41

1 2 3 449 50 51 52 1 2 3 453 54 55 56

1 2 3 464 63 62 61 1 2 3 460 59 58 57

MDE

MEF

MCG

MGH

MBH

MCE

MEG

Figure 3.11. HV winding schematic (glassy model transformer), R= 95 mm, w= 3mm, W= 24 mm d= 11mm, δd=

6 mm and conductor height h= 7 mm.


55

To verify the capacitance calculations, the HV winding shunt capacitance of the test object

was measured using Omicron Mtronix MI600 DDF measurement system and the result

was compared to analytical approach (see Fig. 3.12). This capacitance contains the

capacitance of HV winding with respect to LV winding including pressboard and the

capacitance between HV winding and the aluminum tank. The measured and calculated

results are obtained as (3.38) and (3.39), respectively. Figure 3.13 shows the top view of

the model transformer schematic.

( )

98.2700HV measg

C pF (3.38)

( )

87.2861HV calcg

C pF (3.39)

Figure 3.12. Measured value for shunt capacitance, HV winding.

55 mm

95 mm

116 mm

18 mm

2 mm

24 mm

8 mm

LV

HV

Plexiglass Tank

Pressboard

Figure 3.13. Model transformer schematic, top view.


56

The measured value for the HV shunt capacitance shows a larger value than the calculated

one. This difference is not quite significant particularly when the measurement is

performed in the range of (pF). Nevertheless, the reason should be taken into

consideration. In fact, this discrepancy comes through the impact of the capacitance from

the bushing and the aluminum tank. Physically speaking, the bushing capacitance is in

parallel with the other calculated capacitances and in turn increases the total shunt

capacitance of the HV winding in practice.

3.5 Conclusion

This Chapter has derived the self and mutual inductances between the transformer

winding turns. The turn-to-turn inductance relationships were used to develop the

inductance matrix and derive the final self and mutual inductances of the winding. The

series and shunt capacitances of transformer winding were then derived through a similar

approach. A small air-core glassy model transformer was fabricated for experiment and

analytical calculations were verified against practical measurement results on this test

object. The inductance matrix and capacitance values are essential components in the case

of winding model verification as well as axial and radial deformation studies in Chapters 4

and 6.

Chapter 4. Transfer Function Model of Air-Core Transformer Winding

57

Chapter 4

Transfer Function Model of Air-Core Transformer Winding

4.1 Introduction

FRA data are typically reported as Bode diagram over a determined frequency band. It was

discussed earlier that FRA data in the mid-frequency band are influenced by the winding

structure, while data in the low-frequency band are affected mainly by the transformer core.

To study just the impact from winding structure on the FRA spectrum, this Chapter has

concentrated on the mid-frequency oscillations. Mathematical approach using travelling

wave theory is employed to explore frequency response trace behaviour. Practical studies

on a small air-core model transformer as well as two 66 kV, 25 MVA continuous and

interleaved disk windings have been performed to validate the mathematical calculations. In

addition, two 245 kV, 45 MVA and 66 MVA power transformers are tested to obtain mid-

frequency oscillations and compare these results with mathematical evaluation.

4.2 Modelling

Significant studies have concentrated on wave propagation in transformer windings [93]-

[98]. Loss-less charging and discharging currents and also wave propagation in transformer

winding using Maxwell’s equations for nth turn have been calculated in [94]. In addition,

travelling wave and multi-conductor transmission line theories have been utilized in [95] to

calculate voltage and current propagation for transient studies. In [95], it was emphasized

that the oscillation characteristics in the transformer primary winding are essentially the

same as that obtained when the secondary winding is ignored. Hence, it is appropriate to

perform the analysis for a single independent winding. The equations then become greatly

simplified and easy to visualize [95].

In this Chapter, the above mentioned fundamental studies as well as calculations are utilized

and extended in detail to interpret FRA mid-frequency band. To this end, the equivalent


58

detailed circuit of transformer winding as shown in Fig. 4.1 is considered. According to this

model, the charging currents for an infinitesimal length of transformer winding are given by:

2

,sc gs

e ec gi i

x t x

(4.1)

(a)

(b)

Figure 4.1. Equivalent detail circuit of a transformer winding (dx denotes an infinitesimal length of winding), (a)

Entire winding schematic [95], (b) Close view of the conductors and modeled parameters.

where, ics is the internal capacitive current per unit conductor length, cs denotes series

capacitance, e is the potential to ground (time domain), ig represents external conductance

current per unit conductor length and g is the shunt turn-to-turn conductance.

Also, discharging currents through shunt capacitance as well as conductance for

infinitesimal length of transformer winding flow to ground are calculated as equation (4.2):

,

g g Gci ce

i eGt

(4.2)


59

where, icg is the external capacitive current per unit conductor length, cg denotes the shunt

capacitance to ground, iG is the external conductance current per unit conductor length, and

G represents the shunt conductance to ground. Equation (4.3) is valid if the transformer

winding includes numerous number of turns and also if the detailed distribution of the wave

propagation along a single turn length is not of interest [94]. Therefore, the space derivative

of the current is calculated as equation (4.3) [95]:

( ) 0,

s gc g c Gi i i i ix

(4.3)

where, i is the conductor current. Substituting of equations (4.1) and (4.2) into equation

(4.3) will lead to equation (4.4):

2

( ) 0gse e

c g i cx

eeG

x t x t

(4.4)

According to (4.4), the space derivative of the winding current for an infinitesimal length of

transformer winding is given by:

3 2

2 2 gsi e e

c g cx

eeG

tx t x

(4.5)

On the other hand, the self-inductance of an infinitesimal winding turn contributes an

induced voltage. In addition, two adjacent turns coupled by the mutual-inductance cause

additional induced voltage in the turn.

Therefore, the space derivative of winding voltage including consideration of conductor loss

for an infinitesimal length of transformer winding can be calculated as equation (4.6) while

the partial mutual inductances due to other conductors’ turns are neglected:

e i

rix t

(4.6)

where:

l (4.7)


60

γ and μ are the self- and mutual-inductance per unit conductor length of the turn and

adjacent turn, respectively, r is the conductor resistance, and l is the inductance coefficient.

Subsequently, the voltage difference along the coil can be calculated as follow:

l

e iri

x t

(4.8)

In fact, the voltage as well as the current in a long coil varies from turn-to-turn as a function

of time and also space.

In [42], current and voltage studies have been converted from time domain to frequency

domain using Laplace transform while initial conditions have been supposed to be zero.

Therefore, transferring equations (4.5) and (4.8) from time domain into frequency domain

yield:

2

2

0( , ) ( , )

0( , ) ( , )

0 0 ( , )

0 ( ) ( , )

j l rV j x I j x

j c Gx I j x V j xg

I j x

j c g V j xxs

(4.9)

where, V represents potential to ground (frequency domain), I is the conductor current

(frequency domain), ω is the angular frequency, and j denotes the imaginary operator.

According to equation (4.9), the differential equation for voltage propagation along

transformer winding is as follows:

2 2

2

( , ) ( , )( ) ( , ) ( )

( )g s

V j x V j xj c G V j x j c g

j l r x x

(4.10)

2

2

( )( )( , )( , ) 0.

(1 ( )( )

g

s

j l r j c GV j xV j x

j l r j c gx

(4.11)

Therefore, the solution for the voltage at any point x along the transformer winding is given

by:

1 2( , ) x xV j x Ae A e (4.12)


61

( )( )

1 ( )( )

g

s

j l r j c G

j l r j c g

(4.13)

β denotes the propagation coefficient and A1 and A2 are constants that depend upon the

boundary conditions. It is worth noting that x=0 and x=λ correspond to the neutral- and line-

terminal points of the transformer winding, respectively where the winding length is λ.

According to equation (4.9), the current at any point of the transformer winding is

calculated as equation (4.14):

1 2( , ) .x xI j x A e A e

j l r

(4.14)

The frequency response magnitude for a transformer winding is calculated as equation

(4.15). Fig. 4.2 demonstrates common FRA test setups for a single winding and also

transformer including the source voltage as well as Vin and Vout. Based on Fig. 4.2, the output

voltage with respect to input voltage should be calculated to reach frequency response. This

calculation was given in (2.7).

(a) (b)

Figure 4.2. Common FRA test setup, (a) For single winding (b) For transformer.

According to equation (2.7) and the FRA test setup shown in Fig. 4.2; Vin is measured at the

line-terminal of the winding while Vout is measured at the neutral-terminal. Also, Zin and Zout

are the input and output measurement cable impedances and commonly represented by 50

Ω resistors. Since, Zout has small value compared to equivalent impedance of the winding, it is

approximated that Vout << Vin. Therefore in equation (4.12), A1 = - A2 and the frequency

response trace is obtained as:


62

0 0( ,0)

( , )

out out

in

outZ I j

V j

V Z e e

V j l r e e

(4.15)

It should be emphasized that according to equation (4.15) and above mentioned

assumption, the transformer frequency response magnitude exhibits small value especially

in very high frequencies. Although, according to equation (2.7) when equation (4.15) is

subject to logarithm operation, reported frequency response magnitude in Bode diagram

would be significant and completely traceable. Simplification of (4.15) using complex

algebra is given by:

1

sin( )

out

in

V

V j j

(4.16)

1( ) , woutZ j l r N d (4.17)

where, Nw is the number of winding turns, and dʺ is the average length of a conductor turn.

Transfer function resonances occur when the denominator in equation (4.16) experiences

minimum absolute value. It can be yielded once β2λ2=-(krπ) 2. Therefore:

2 2 2( )( )

1 ( )( )

gr

s

r

j l r j c Gk

j l r j c g

k 1,2,3,.......

(4.18)

then:

2 2 2 2 1

2 2 2 1

2 2 2 1 2 2 2 1

( ( ) )

(( ) ( ) ( ) )

( ) ( )

g r s

g r s

r r

lc k lc

rc lG j k rc lg j

rG k rg k

(4.19)

Moving from low frequencies to mid and ultimately high frequencies will lead to ω2 >> ω,

therefore:

2 2 2 2 1 2 2 2 1( ( ) ) ( )g r s rlc k lc k (4.20)

Hence, the resonance frequency (fr) will be calculated as (4.21):


63

2 2 2 1

1

2 ( ( ) )

rr

g s

kf

c c k l

(4.21)

4.3 Discussion on Resonant Frequencies

Technical specifications and physical characteristics for each and every transformer are

major factors in determining the limits of the low-, mid- and high-frequency bands. The most

appropriate limits for each are still not widely agreed. Nevertheless, the flux division theory

and short circuit principles ‎[17] can be used to determine the low-frequency band. High-

frequency band can be recognized through test setup variation and then the remaining gap

constitutes the mid-frequency band.

Regarding the low-frequency band, as the inductive reactance of the winding is greater than

the capacitive reactance; low frequency behaviour is affected mainly by transformer

inductive reactance and the Bode diagram experiences a falling trend. Furthermore, the self-

inductance of transformer winding is greater than mutual inductances. Hence, low

frequency behaviour is affected mainly by transformer inductive reactance due to the

winding self-inductance.

Mainly comprising series and shunt capacitances, the capacitive reactance in the case of

high-frequency band could be neglected as its reactance is reaching sufficiently low value.

Hence, the measurement cables as well as connection resistances of the test setup would

become dominant at the high-frequency band and determine oscillations.

Therefore, equation (4.21) just represents mid-frequency oscillations of the frequency

response trace. Based on equation (4.21), mid-frequency resonances of transformer winding

frequency response trace are completely dependent on winding inductance as well as series

and shunt capacitances. Any changes in the values of inductance or capacitances as well as

winding length may lead to changes in resonance frequencies.

On the other hand, different types of transformer windings are designed and manufactured

based on voltage level and electromagnetic relations. When a unit-function voltage is

applied to transformer, the initial distribution of voltage is determined entirely by the

capacitive network.

According to the literature ‎[99], the initial impulse voltage distribution intensity is

determined through α parameter. A low α value will lead to a more uniform initial impulse

voltage distribution, whereas a high α value will create severe stress on the upper disks of

the transformer winding. It is worth noting that α parameter can be calculated from


64

equation (4.13) and is equal to equation (4.22) when the capacitive network is considered as

the equivalent circuit of transformer winding:

g

s

c

c (4.22)

Since the initial distribution is controlled entirely by the ratio of capacitances, numerous

studies have been conducted in the past aiming to explore methods to decrease the value of

α in various types of transformer windings ‎[99]. This may be done either by reducing the

shunt capacitance cg or alternatively by increasing the series capacitance cs. In fact, each and

every transformer winding type has its own α value. It is mathematically proven that

interleaved winding shows lower α compared to continuous disk winding ‎[93],‎[99]-‎[100].

According to equation (4.21), significant value for shunt capacitance compared to series

capacitance (large α) will lead to maximum number of resonance frequencies. On the other

hand, large series capacitance with respect to shunt capacitance (small α) results in

minimum number of resonance points. This behaviour was also discussed in ‎[27]. Hence, the

frequency response trace for continuous disk winding has more oscillations in mid

frequencies while interleaved winding has fewer changes over the same frequency band.

This hypothesis will be explored through practical measurements in the next subsections in

order to validate the modelling and calculation carried out in the last Section.

4.4 Verification of Mathematical Calculation Using Practical Measurement

In order to verify mathematical calculation for mid-frequency oscillations of FRA trace, the

glassy model transformer (Fig. 3.9) in the absence of transformer oil was used as the first

test object.

An aluminium foil has been wrapped over the glass casing and grounded to simulate the

metal transformer tank. To carry out the frequency response simulation of manufactured

test object, electrical parameters of the glassy transformer should be calculated and utilized

in mathematical result. This work has been done using the formulas provided in the last

Chapter. Calculated electrical parameters for the test object were validated through practical

measurement using RLC meter and dielectric dissipation factor (DDF) measurement devices.

The maximum applied voltage for DDF measurement was 5 kV to avoid any kind of

undesirable flash-over in the test object.

Afterwards, the frequency response was simulated for the HV winding of the test object

based on mathematical calculation results and also achieved parameters.


65

Note that the significant value for air-core magnetic reluctance of the test object will lead to

small self-inductances for the windings of the glassy transformer. Therefore, the resonance

frequencies in FRA trace for HV and LV windings will be shifted to higher frequencies.

Hence, the upper band limit for FRA simulation was extended from 2 to 20 MHz to monitor

FRA trace oscillations. Practical measurement of FRA trace was performed for the test object

to compare with simulation result. FRA measurements were made by applying a 20 Vpp

swept sinusoidal signal from 20 Hz to 20 MHz. The reference signal was injected through

input-lead of the HV winding and the response signal was recorded through its neutral-lead

(end-to-end measurement). The characteristic impedance of the measurement cables was

50 , the cable braids were grounded through the aluminium tank. The ambient

temperature surrounding the test object was 27 °C. The simulation and measurement

results for the HV winding of the test object are shown in Fig. 4.3.

Figure 4.3. Simulated and measured frequency response traces for HV winding of the manufactured glassy

transformer.

In the case of FRA trace oscillations, it can be observed that the resonance and anti-

resonance points in the simulated result are ‘synchronized’ with the measured trace.

However, the discrepancy between simulated and measurement results could be due to the

existence of additional loss mechanism, loss in the aluminium tank or maybe it comes

through boundary condition approximation (A1 and A2) in the calculations.

According to Fig. 4.3, moving from 20 Hz to 10 kHz both traces display almost constant value

due to small value of inductive reactance. Moving to higher frequencies, the reactance due to

102

103

104

105

106

107

-100

-80

-60

-40

-20

0

20

Frequency [Hz]

Ma

gn

itu

de

[d

B]

Simulated

Measured


66

the self-inductance become significant and Vout in equation (2.7) becomes smaller; therefore,

the frequency response magnitude reported in Bode diagram follows a falling trend.

Interaction between self-inductance and capacitances starts in the first anti-resonance point.

Oscillation behaviour for both spectra in the rest of frequency band comes through equation

(4.21).

4.5 Practical Study

4.5.1 Case Study 1

In order to have precise investigation on mathematical calculation and evaluate continuous

and interleaved disk windings frequency responses, experiments were carried out on two

66 kV, 25 MVA transformer windings. The continuous disk winding contains 72 disks with 8

turns per disk while the interleaved winding has 32 disks with 24 turns per disk.

For this study, FRA measurements were made by applying a 20 Vpp sinusoidal signal across

each winding, at various frequencies in the range 20 Hz - 2 MHz. The measurements were

performed for 801 points (frequencies). Input and output impedances of measurement

cables were 50 Ω. The ambient temperature was 29 °C. The test circuit was connected as in

Fig. 4.2(a). Voltage was injected through the winding input end and output voltage was

measured at the winding output end. Tested windings are shown in Figures 4.4(a) and

4.4(b). Note that an aluminium cylinder was fitted inside each winding and grounded to

simulate the earthed core.

Figure 4.5 shows the FRA measurement results for interleaved and continuous disk

windings; the mid-frequency band in frequency response traces has been highlighted by a

dashed-line rectangle. It can be seen that the frequency response of the continuous disk

winding experiences a large number of oscillations within the mid-frequency range

including resonances (maxima) and anti-resonances (minima). These oscillations occur due

to the kr parameter substitution in equation (4.21) where cg >> cs..


67

(a)

(b)

Figure 4.4. Continuous and interleaved disk windings, (a) Continuous winding, (b) Interleaved winding.

Figure 4.5. Frequency response traces of continuous and interleaved disk windings.

102

103

104

105

106

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Magnitude [

dB

]

Interleaved Disc Winding

Continuous Disc Winding


68

In contrast, the interleaved winding displays a monotonously uniform trend in mid

frequencies as it involves significant series capacitance value. On the other hand, in moving

from low-frequencies to mid-frequencies, the first resonance point relies on the interaction

between the winding self-inductance and winding capacitances. Frequency responses of

both continuous and interleaved windings demonstrate a rising trend in high frequencies as

they are more influenced by the shunt capacitance and connections’ resistances in this

frequency range. As the outer metal tank is absent in this particular case, the shunt

capacitance is from only between winding conductors and the earthed aluminium core.

4.5.2 Case Study 2

Practical studies were also carried out on two 45 MVA and 66 MVA transformers to examine

mathematical result for power-rated transformers. Frequency response spectra were

measured for HV windings of specified transformers in Table 4.1. The 45 MVA transformer

has continuous disk winding while the 66 MVA contains interleaved winding. For both

transformers, input signal was injected through phase A and output signal was taken

through the neutral bushing while other transformer terminals were left open circuit

(floating). The measurements were performed for 801 points (frequencies) and FRA traces

were recorded from 2 Hz-2 MHz. Input and output impedances of measurement cables were

50 Ω. The ambient temperature at the time of measurements was 21°C. The FRA traces are

shown in Fig. 4.6.

Table 4.1. Transformer specifications.

Specifications

Transformer no. Capacity Voltage [kV] Vector Group HV winding type

T1 45 MVA 245/10.5 YnD11 Continuous

T2 66 MVA 245/11.5 YnD11 Interleaved

According to Fig. 4.6, two anti-resonances (minima) are encountered when moving from the

left-hand side to the right-hand side on FRA traces. The first anti-resonance is due to

influence of the transformer middle limb (B) while the second one is caused by the

transformer lateral limb (C). This will be discussed later in Chapter 5. Moving from low

frequencies to mid frequencies shows discrepancy between the two spectra. The continuous

disk winding experiences oscillatory behaviour while the interleaved winding follows a

rising trend in the mid-frequency band as it was discussed earlier.


69

Figure 4.6. FRA traces of HV windings phases A for 45 MVA (continuous disk winding) and 66 MVA (interleaved

disk winding) power transformers.

4.5 Conclusion

Travelling wave and transmission line theories were utilized to analyse the frequency

response trace oscillations in the case of mid frequencies. The mid-frequency resonance

points in FRA trace were verified through mathematical calculations. FRA mid-frequency

oscillation dependency on inductance as well as series and shunt capacitances was explored.

The glassy model transformer was utilized to compare the simulation and measurement

results. Mathematical calculations were validated through practical measurements.

For experiment, a continuous disk winding, an interleaved disk winding as well as two

power transformers were used to find series and shunt capacitance influences on FRA

fluctuations. It was revealed that a winding with lower α will lead to more oscillations in the

mid frequencies of FRA trace while greater α will result in a steady trend in the mid-

frequency band.

To eliminate the impact of the transformer core on FRA spectrum, the modelling was

conducted on an air-core transformer winding and validation was achieved through

measurements on a model winding and also two full-size windings – all with air-core. FRA

measurement on two power transformers also validates the mathematical model,

specifically in the mid-frequency band. It was demonstrated that mid-frequency oscillations

in FRA spectra of power transformer are affected by series and shunt capacitances,

102

103

104

105

106

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Magnitude [

dB

]

Continuous Disc Winding

Interleaved Disc Winding


70

significantly. The transformer metal core influence is negligible in terms of mid-frequency

interpretation.

Chapter 5. Low Frequency Interpretation of FRA Signature

71

Chapter 5

Low Frequency Interpretation of FRA Signature

5.1 Introduction

Mid-frequency oscillations in FRA spectrum was discussed in Chapter 4. It was shown that

the transformer winding structure is the main reason of the resonances and anti-

resonances in this frequency band.

This Chapter aims to interpret the oscillations in the low-frequency band of FRA spectrum.

Based on literatures [23], [102], the low-frequency band oscillations in FRA trace is

affected mainly by the transformer magnetic core. Actually, it refers to the impedance of

the transformer magnetic core. This parameter is affected through the winding

inductance, and the winding inductance itself comes through the transformer core

reluctance.

Therefore, on one hand to interpret the low-frequency band of FRA trace the core

reluctance should be taken into consideration; on the other hand, this parameter is not

simple to measure or calculate as the frequency is swept. Also, different transformers have

different magnetic core laminates, and their characteristics can be quite different.

Specifically when the frequency is swept from 20 Hz to 2 MHz, their behaviour becomes

non-linear. Hence, working on this case requires a technique to derive the frequency

variable magnetic core reluctance. The technique introduced in this Chapter uses reverse

calculation on the FRA signature. Indeed, the information of magnetic core reluctance is

supposed to be a kind of hidden data available in the winding FRA signature.

In addition, each and every winding experiences its own core reluctance. Therefore,

another method is required to practically distinguish the winding magnetic reluctance of a

phase from that of another phase. This method is also discussed in this Chapter.

This Chapter intends to show that the oscillations occurring in the low-frequency band of


72

FRA trace are mainly affected by the middle and lateral core limbs of the other windings.

This means that the first anti-resonance in FRA trace comes through the middle limb

influence, while the second one is affected through the lateral limb. If one of those limbs is

excluded from the FRA setup or blocked, its corresponding anti-resonance will be

eliminated.

At first, this idea is validated by experiment in this Chapter, and then mathematical

approach is used to verify the practical findings. To this end, the frequency response of a

winding in a power transformer is generated mathematically through the reluctance

spectrum of the other limbs and the result is compared with practical measurements. It

reveals that the generated trace is quite similar to the measured spectrum, and its

resonances and anti-resonances are completely matched.

The interaction between the winding shunt capacitance and inductance to produce anti-

resonance in the low-frequency band is another topic for discussion in this Chapter.

In this Chapter, a 10/0.4 kV, 100 kVA three-phase core type transformer is taken as a test

object. Note that to differentiate between the two winding sides; lower-case symbols are

used to denote low voltage parts (a, b, c, n). Similarly, upper-case symbols refer to high

voltage parts (A, B, C).

5.2 Flux Division Theory

5.2.1 Technical Concept

Essentially, in a transformer the alternating current in the primary winding creates a

magnetic flux in the transformer core. The magnetic flux density is given by:

1( , )c w

B r t dA N i (5.1)

where ϕ represents the flux, B is the flux density, dAc is an infinitesimal area of the core

cross section, R is the magnetic core reluctance, i denotes the current, and Nw is the total

number of winding turns.

According to Faraday's Law, when a conductor surrounds a time varying magnetic field -

in the core - it can induce a voltage onto the conductor. This induced voltage consequently

will result in an alternating current through the conductor. The induced voltage is given

by:


73

w

de N

dt

(5.2)

Presumably there is a load connected to the secondary side of the transformer, meaning

that the induced current will be carried by the load as well.

On the other hand, Lenz’s law is a simple rule to find the direction of induced current.

Lenz’s law implies that the induced current will flow in such a direction so as to oppose

the cause that produces it. Therefore, if a turn-to-turn or disk-to-disk short circuit occurs

in a transformer winding there will be a circulating current in that short-circuited loop

which tends to oppose the flow of magnetic flux in the transformer core – the cause that

produces it. Eventually, the magnetic field initiated by the circulating current in the short-

circuited loop will not allow the transformer main magnetic flux pass through the

corresponding limb. Hence, theoretically the induced voltage across the defected winding

is equal to zero which is approximately verified in practical measurements. Based on this

fact, transformer Flux Division Theory (FDM) has been employed for decades as an on-site

diagnosis test to recognize transformer winding turn-to-turn and disk-to-disk short

circuit.

FDM can be conducted on single-phase or three-phase core type transformers. This

method will yield adequate information about turn-to-turn or disk-to-disk short circuit in

transformer winding on a certain limb.

5.2.2 Flux Division Measurement (FDM)

FDM can be performed on a transformer on HV or LV side. The test voltage in FDM method

must be applied onto phase A, B or C while the other phases are left open circuit. Under

such a circumstance, only one single winding is excited and the main magnetic flux is

generated in the corresponding limb as per Faraday’s law. The generated magnetic flux

will pass through other limbs to complete flux loop.

Magnetic flux passing through other limbs will create an electromotive force on the

windings of other phases and therefore a voltage will be induced. If an excitation voltage is

applied to a HV winding, the algebraic sum of the induced voltages on the other two HV

windings will be equal to the excitation voltage, theoretically. This can be approximately

validated in practical measurements for wye and delta winding connections.

For instance, in transformers with wye connection, if 230 V is injected to HV side phase A,

the sum of induced voltages in HV side winding phases B and C will be around 230 V as

calculated in (5.3):


74

( ) ( ) ( ).BA CV t V t V t (5.3)

Practical experience shows that discrepancy between the injected voltage (230 V) and

induced voltages can be up to a few volts.

According to Lenz’s law, the induced voltage in other windings would be around zero while

turn-to-turn or disk-to-disk short circuit occurs. It is assumed that short circuit has

happened in a winding that belongs to a phase other than the associated phase of the

excited winding. When a short circuit occurs in a winding then there will not be any flux

flow in the associated limb for that winding and the generated flux will flow through the

other limbs of the core. Hence, there will not be any voltages induced on the windings with

short-circuited disks or turns. This effect is utilized to distinguish the faulty winding from

the others and that is considered to be the main aim of the FDM method.

The magnetic flux flow due to the excited winding of phase A when disk-to-disk short

circuit has occurred in HV side of phase B is shown in Fig. 5.1. For shell type and five-limb

core type transformers, there are some non-wound auxiliary limbs that always carry some

magnetic flux. This is not an obstacle to distinguish the windings with short-circuited

turns or disks. A transformer with five limbs as well as its associated magnetic circuit is

illustrated in Fig. 5.2.

Figure 5.1. Magnetic flux flow due to phase A excitation when short-circuit occurred in phase B.

Magnetic flux division in a five-limb transformer is illustrated in Fig. 5.3 while short circuit

has occurred in HV or LV side of phase b/B. According to Fig. 5.3, measured voltage on the

side phase will be floating around a certain value which depends on the transformer

construction and technical specifications.

In addition, turn-to-turn or disk-to-disk short circuit in transformer winding can change

winding DC resistance values as well as the no-load current. Although altered DC


75

resistance or no-load current cannot be interpreted as internal short circuit in transformer

winding directly. FDM was introduced in this Section in preparation for interpretation of

the FRA low-frequency band using the short circuit technique.

Figure 5.2. Five-limb transformer active part as well as its magnetic circuit.

Figure 5.3. Magnetic flux division in five-limb transformer due to phase a/A excitation and short circuit

occurred phase b/B. Limbs 2, 3 and 4 are assumed as phase A, B and C respectively.


76

5.3 Mathematical and Practical Approach to Interpret Low-frequency

Band

In order to interpret FRA low-frequency oscillations, general interpretation of FRA trace is

discussed first. Then, practical approach and mathematical calculation are utilized to

analyse oscillations.

5.3.1 General Interpretation of FRA Trace

A 10/0.415 kV, 100 kVA three-phase core type transformer was taken as a test object to

examine FRA low frequency oscillations. The HV windings were left open circuit.

Transformer winding frequency response traces of phases a, b and c were measured and

shown in Fig. 5.4 over the frequency range 20 Hz to 2 MHz. The equivalent magnetic

schematic of three-phase transformer is shown in Fig. 5.5. Test object specifications are

provided in Table 5.1.

Ch

apter 5

. Lo

w F

requ

ency

Interp

retation

of F

RA

Signatu

re

7

7

Figu

re 5.4

. Freq

uen

cy resp

on

se traces of p

hases a

, b an

d c.

ot··----- -.............

-5

jjj' :!2. -10

.g "" c ~ -15 :::!:

-20

-25

. , -. '• '•

Low Freq.

• . , ......... , .,

' \

\ \

\ ,, \

\ .. \ • \

• • • \ \

• • • • • ' I •• • • ..

-30 102 10 l ...............

104

Frequency (Hz)

..

Mid Freq. High Freq. « ,

•·•·• LV side phase a ----· LV side phase b -LV side phase c

I

/ V/ ! lr! I •

105 10'

'


78

Figure 5.5. Equivalent magnetic circuit of three-phase transformer.

Table 5.1. Transformer specifications.

Rated Voltage [V] 10000/415 Impedance Volts (%) 4.17

Rated Power [kVA] 100 Number of Phases 3

Rated Current [A] 5.77/139 Number of Limbs 3

Cooling System ONAN Vector Group Dyn1

According to Fig. 5.4, in the low frequency range, the inductive reactance of the

transformer winding is considerably greater than the capacitive reactance and plays a

pivotal role in trace formation. Hence, the frequency response magnitude as given by (2.7)

follows a falling trend. In fact, the low frequency behaviour of transformer winding

impedance (Zw) is completely dominated by its inductive reactance. Therefore, Zw can be

expressed as 2πfL where f denotes frequency and L is the total winding self-inductance.

Mutual inductances between HV and LV windings do not contribute to this part of

spectrum as the HV winding is left open circuit without carrying any alternating current

during the test on LV winding, However, the turn-to-turn mutual-inductance of LV winding

has significant impact on total self-inductance formation.

Moving from left to right in Fig. 5.4, frequency response traces of lateral windings (a and c)

exhibit two minimal peaks (anti-resonances) whereas the middle winding spectrum

shows one minimal peak (This behaviour is a common pattern of FRA trace for all

transformers having star connections, see Figures 2.12 and 4.6). For frequencies above 10

kHz there is no significant discrepancy between recorded traces. In the case of delta

connection, these two minimal peaks are replaced by a minimal peak, though, it depends

on terminal configurations. In this study, attempt has been made to interpret double anti-

resonance which is quite a common occurrence in all routine transformers.


79

Low frequency anti-resonances can be generally interpreted through interaction of the

winding self-inductance and shunt capacitances.

Precise investigation on magnetic flux flow reveals that the flux generated by the lateral

windings should pass through different limbs. For instance, in terms of FRA measurement

on phase a some part of the flux generated in limb a due to the FRA test setup on phase a

will flow through the transformer middle limb b while the other goes through the outer

limb c. In terms of middle winding FRA test, the generated flux runs through similar

electromagnetic paths (limbs a and c).

Based on this fact, FRA lateral traces exhibit two minimal peaks while the middle

spectrum shows one minimal peak in low frequencies.

In addition, the first minimal peak in lateral traces is associated with the transformer

middle limb while the second one is influenced by the outer limb. In the next subsection,

this hypothesis is validated through practical and mathematical approach.

5.3.2 Practical Approach

According to the flux division theory, if an internal short circuit occurs in a transformer

winding the magnetic field initiated by the circulating current in the short-circuit loop will

not allow the transformer main magnetic flux pass through the corresponding limb (the

limb with the short-circuited winding). Based on this, to prevent the magnetic flux flows

through a specific transformer core limb, a deliberate short circuit could be made between

the input and neutral leads of the transformer winding associated with that limb. This in

turn can model the short circuit loop within the winding, block the flux flow through that

limb and divert the magnetic flux into other limbs. It is thus a convenient technique to

eliminate a predetermined limb from the core magnetic path of winding. It is then possible

to distinguish the transformer core limb influences on the frequency response trace.

To clarify the introduced technique and interpret the low-frequency oscillations in FRA

spectrum, the line and neutral leads of phase C of the test object were short-circuited to

simulate short-circuit loop around the transformer core limb c/C. This created short-

circuit will cause the transformer main magnetic flux unable to flow through the core limb

c/C. Other terminals were left open circuit and the frequency response trace was recorded

for phase a over the frequency band from 20 Hz to 2 MHz. The test setup and the magnetic

core schematic are shown in Fig. 5.6 and Fig. 5.7, respectively. The result in Fig. 5.8 shows

the reference and measured FRA traces for phase a.


80

Figure 5.6. FRA measurement setup for phase a when A and C are short-circuited.

Figure 5.7. Equivalent magnetic circuit for FRA measurement setup of phase a when terminals A and C are

short-circuited (this will block the flux flow in limb c/C).

Figure 5.8. Reference and measured FRA traces for phase a when limb C is surrounded by short circuit loop.

When scanning across Fig. 5.8 from low frequencies towards high frequencies, it can be

seen that the anti-resonances in the reference trace at 822.2 Hz and 1.101 kHz have been

102

103

104

105

106

-25

-20

-15

-10

-5

0

5

Frequency [Hz]

Ma

gn

itu

de

[d

B]

LV side phase a (Short circuit in HV side phase C)

LV side phase a (Reference)

1.101 Hz825.5 Hz

822.2 Hz


81

replaced by one anti-resonance at 825.5 Hz in the measured trace. When moving to higher

frequencies, both traces experience similar oscillations. Note that the magnetic flux flows

through transformer core legs a and b when the line and neutral leads of phase C are

short-circuited.

Therefore, the single anti-resonance in the low-frequency band of the measured trace

represents the interaction between self-inductance of winding a due to core magnetic path

of phases a and b, and the total shunt capacitance of phase a. It should be noted that the

discrepancy between 822.2 Hz and 825.5 Hz is negligible and comes through

measurement accuracy.

Afterwards, the line and neutral leads of the winding on phase B were short-circuited to

model the short-circuit within the transformer centre leg. Other terminals were left open

circuit and the frequency response trace for phase a was re-recorded. The test setup and

the magnetic core schematic are shown in Fig. 5.9 and Fig. 5.10, respectively. The result in

Fig. 5.11 shows the reference and measured FRA traces for phase a.

According to Fig. 5.11 moving from left hand side to right hand side on FRA traces the anti-

resonances in the reference trace at 822.2 Hz and 1.101 kHz have been replaced by a

single anti-resonance at 1.101 kHz in the measured trace. Moving to higher frequencies

reveals similar oscillations for both spectra. In this case with phase B short-circuited, the

magnetic flux flows through transformer core legs a and c. Therefore, the single anti-

resonance in the low-frequency band of the measured trace at 1.101 kHz represents the

interaction between the self-inductance of winding a due to the magnetic core path of

phases a and c, and the total shunt capacitance of phase a.

The frequency response traces of phase a due to a deliberate short circuit on B and C are

shown in Fig. 5.12.

Figure 5.9. FRA measurement setup for phase a when B and C are short-circuited.


82

Figure 5.10. Equivalent magnetic circuit for FRA measurement setup of phase a when terminals B and C are

short-circuited (this will block the flux flow in leg b/B).

Figure 5.11. Reference and measured FRA traces for phase a when limb B is surrounded by short circuit loop.

102

103

104

105

106

-25

-20

-15

-10

-5

0

5

Frequency [Hz]

Ma

gn

itu

de

[d

B]

LV side phase a (Short circuit in HV side phase B)

LV side phase a (Reference)

1.101 Hz

822.2 Hz

1.101 Hz


83

Figure 5.12. Frequency response traces of phase a (open circuit) and deliberate short circuit on phase B and C.

According to Fig. 5.12, the frequency response trace of phase a due to a deliberate short-

circuit loop in phase C shows a single anti-resonance (in the low frequency region). The

frequency of this anti-resonance coincides with the first anti-resonance of the reference

trace but has a different magnitude. This magnitude discrepancy is initiated by changes in

the equivalent core reluctance.

Similarity, the frequency response trace of phase a due to a deliberate short-circuit loop

on phase B displays another single anti-resonance (in the low frequency region). The

frequency of this anti-resonance corresponds to the second anti-resonance of the

reference trace.

Therefore, it has been practically demonstrated that the centre and lateral legs of the

transformer core have different influence on the FRA spectrum. In the next subsection,

this practical achievement is going to be validated through mathematical calculations.

5.4 Mathematical Approach

Frequency response trace of phase a when the other leads are left open circuit (end-to-end

measurement) is given by (5.4), see Fig. 2.10(a) and equation 2.7.

1020log out

aa out

Zk

Z Z

(5.4)

102

103

104

105

106

-25

-20

-15

-10

-5

0

5

Frequency [Hz]

Ma

gn

itu

de

[d

B]

LV side phase a (Short circuit in HV side phase C)

LV side phase a (Short circuit in HV side phase B)

825.5 Hz

1.101 Hz


84

where, ka and Za represent the magnitude of frequency response trace and the transformer

LV winding impedance of phase a, respectively, and Zout is the output impedance depicted

in Fig 2.10. The output impedance is typically 50 Ω in the FRA test setup, thus, if Za is

obtained then the frequency response spectrum can be determined using (5.4). On the

other hand, the frequency response trace of phase a when phase B is short-circuited and

other leads are left open circuit can be calculated using (5.5). Also, (5.5) can be replaced by

(5.6) when phase C is short-circuited.

10( )

( )

20log outaB a outsc B sc

Zk

Z Z

(5.5)

10( )

( )

20log outaC a outsc C sc

Zk

Z Z

(5.6)

where, kaB(sc) and kaC(sc) represent the trace magnitude of phase a when phase B and C are

short-circuited respectively and other leads are left open circuit. ZaB(sc) and ZaC(sc) denote

transformer LV winding impedance when phases B and C are short-circuited respectively

and other leads are left open circuit.

The discrepancy between (5.4), (5.5) and (5.6) comes through Za, ZaB(sc) and ZaC(sc). In fact,

various winding impedances will lead to discrepancy in low-frequency band between the

traces. Za, ZaB(sc) and ZaC(sc) are given by:

( ) ( )

( )( )

2

a a

a aB Bsc sc

aa CC scsc

Z L

Z f L

LZ

(5.7)

where, La is the transformer winding self-inductance of phase a. LaB(sc) and LaC(sc) denote

transformer winding self-inductances of phase a when phase B and C are short-circuited

respectively and other leads are left open circuit. La, LaB(sc) and LaC(sc) are then given by:


85

2

1

1

( ) ( )

1

( ) ( )

aa

a w aB Bsc sc

a aC Csc sc

L

L N

L

(5.8)

Substitution of (5.8) into (5.7) becomes:

1

2 1

( ) ( )

1

( ) ( )

2

aa

a w aB Bsc sc

a aC Csc sc

Z

Z fN

Z

(5.9)

Ra and Nw denote equivalent magnetic reluctance and number of winding turns in phase a.

RaB(sc) and RaC(sc) are the equivalent magnetic reluctances of phase a when phase B and C

are short-circuited respectively and other leads are left open circuit (see Fig. 5.13).

According to Fig. 5.5, equivalent magnetic reluctances for (5.9) can be calculated through

(5.10), and (5.11) where Rc ||Rl ≈Rc, and Rl denotes the leakage reluctance.

(a)


86

(b)

(c)

Figure 5.13. Equivalent magnetic circuit of transformer when frequency response trace is measured for phase

a, (a) Normal three-phase, (b) HV winding phase B is short-circuited, (c) HV winding phase C is short-circuited.

2 2(3 2) 4 2c c y ya

c y

(5.10)

4 2( )

2 2

( )

aB ysc

a cC

sc

(5.11)

Rc and Ry represent the transformer limb and yoke reluctances, respectively. The limb and

yoke reluctances are given as (5.12), independently:


87

0

1c

y c

lc

Ac

ly

Ay

(5.12)

lc and ly denote the mean magnetic path length of the core limb and yoke respectively, Ac is

the limb cross-sectional area, Ay is the yoke cross-sectional area, μ0 is the vacuum

permeability and μc represents the core permeability.

In a reverse procedure, Rc and Ry are calculated based on RaB(sc) and RaC(sc) as (5.13) and

(5.14) and substituted into (5.15) to calculate R’a. R’a is the generated magnetic reluctance

for phase a through deviated (or measured) magnetic reluctances.

( )( )

2

2

a aBC scsc

c

(5.13)

( ) ( )

2

a aB Csc sc

y

(5.14)

Then, R’a is given by:

2

( )

( )( )

14

aB sc

a aB asc C sc

(5.15)

Therefore, we obtained the magnetic reluctance seen by phase a (R’a) based on RaB(sc) and

RaC(sc). However, RaB(sc) and RaC(sc) are not calculable. Thus, one approach is to use the result

of practical measurement in Fig. 5.12. RaB(sc) and RaC(sc) are given as (5.16) using reverse

calculation on measured traces depicted in Fig. 5.12.

1

1

( ) 2

( )

2

aB sc

aaC sc

fN

(5.16)

Γ and Λ can be derived through kaB(sc) and kaC(sc) based on measurement results in the last

subsection as given in (5.17) and (5.18).


88

120

( )10

outout

aB sc

ZZ

k

(5.17)

120

( )10

outout

aC sc

ZZ

k

(5.18)

Substitution of (5.16) into (5.4) will lead to the generation of frequency response trace of

phase a as it is given by:

2

10

2

12

20log1

2

a outa

out

a a

fN Zk

Z

fN

(5.19)

The original and generated frequency response traces of phase a are shown in Fig. 5.14.

Figure 5.14. Original and generated frequency response traces of phase a.

According to Fig. 5.14, the frequency response trace of phase a is generated through

deviated traces. The frequency of resonances and anti-resonances in the generated trace

are completely synchronized with the reference spectrum.


89

Different magnetic reluctances for traces in Fig. 5.12 led to different anti-resonance

magnitudes between generated and reference spectra. Hence, the generated trace in Fig.

5.14 shows slightly greater magnitude for anti-resonances compared to reference trace as

it is created through the deviated spectra in Fig. 5.12.

5.5 Effect of Core Configuration on FRA Trace

Practical and mathematical approaches showed that the transformer core reluctance

obtained for each winding can initiate the low-frequency anti-resonance formation in the

FRA trace.

It is also shown that the core reluctance of a specific winding certainly depends on the

magnetic path associated with that winding. Thus, a hypothesis arises that for windings

with similar magnetic core reluctance in primary or secondary side, the frequency

response would be identical specifically in the low-frequency band. This can occur when

the numbers of winding turns are quite equal in different windings.

To examine this hypothesis, the transformer equivalent magnetic circuits in Fig. 5.15 were

considered. Table 5.2 shows the frequency response measurement setup for various

configurations in Fig. 5.15.

Different FRA measurements were conducted based on the setups recommended in Table

5.2. Measured frequency responses show that similar core configurations in test setup for

the transformer under test will lead to identical frequency response traces. In fact, the

frequency response trace of the test setup in Fig. 5.15(a) is completely matched with the

test setup in Fig. 5.15(k) due to identical transformer core reluctances as well as

configurations. Similar frequency response traces are achieved for test setups in Figures

5.15(b) and (h). Also, FRA spectra for Figures 5.15(c), (d), (f) and (g) are completely

matched.

It should be noted that a, b and c have similar number of winding turns. Frequency

response traces for test setup in Figures 5.15(b) and (h) are shown in Fig. 5.17. Figure 5.18

shows frequency response spectra of Figures 5.15(c), (d), (f) and (g) while the traces of

other setups are shown in Fig. 5.4.


90

(a) (b) (c)

(d) (e) (f)

(g) (h) (k)

Figure 5.15. Equivalent magnetic circuits when deliberate short circuit is applied for various limbs.

Table 5.2. FRA setup connections for the configurations in Fig. 5.15.

Phase

State a b c n A B C

Figure 5.15 (a) Vinput O.C. O.C. Voutput O.C. O.C. O.C.

Figure 5.15 (b) Vinput O.C. O.C. Voutput B A O.C.

Figure 5.15 (c) Vinput O.C. O.C. Voutput O.C. C B

Figure 5.15 (d) O.C. Vinput O.C. Voutput C O.C. A

Figure 5.15 (e) O.C. Vinput O.C. Voutput O.C. O.C. O.C.

Figure 5.15 (f) O.C. Vinput O.C. Voutput O.C. C B

Figure 5.15 (g) O.C. O.C. Vinput Voutput C. O.C. A

Figure 5.15 (h) O.C. O.C. Vinput Voutput B A O.C.

Figure 5.15 (k) O.C. O.C. Vinput Voutput O.C. O.C. O.C.

O.C.=Open Circuit


91

Figure 5.16. Frequency response traces for test setups of Figures 5.15 (b) and (h).

Figure 5.17. Frequency response traces for test setups of Figures 5.15 (c), (d), (f) and (k).

This in turn can be used as a technique for core deformation localization in transformer.

5.6 Shunt Capacitance Influence

Transformer winding capacitance contains series and shunt capacitances. Series

capacitances consist of turn-to-turn and disk-to-disk capacitances whereas shunt

capacitance is the capacitance of winding with respect to the core and to the electrostatic


92

screens/tank. It is well-known that shunt capacitance effects can be explored in the FRA

low frequencies but there is no practical evidence and physical validation on this

hypothesis. This hypothesis is going to be validated in the next subsection.

5.6.1 Practical Approach

To validate shunt capacitance effects on FRA low frequency spectrum, at first the

frequency response of phase b was recorded when input and output leads of phase B were

short-circuited together. Afterwards, this short-circuit was further connected to the

ground and FRA spectrum was measured once more. This experiment should cause

changes to the shunt capacitance of phase b. This in turn results in frequency response

deviation of this phase as well. Figure 5.18 shows FRA measurement results. The reference

FRA spectrum of phase b is also provided for comparison with the measured traces.

Figure 5.18. Frequency response traces for phase b when HV side phase B is left open circuit, short-circuited, short-circuited and grounded.

According to Fig. 5.18, frequency response of phase b has altered from the reference trace

for the short-circuited and isolated B as well as the short-circuited and grounded B.

For isolated B (Trace (2) in Fig. 5.18), the frequency response trace shows smaller

magnitude for the first minimal peak due to the greater magnetic reluctance as compared

to reference trace. In addition, the minimal peak is slightly shifted to higher frequency due

to the lower self-inductance. For the grounded B (Trace (3) in Fig. 5.18), the first minimal

peak occurred in lower frequency. The entire low-frequency band of FRA spectrum has

also shifted to lower frequencies. Since the self-inductance of winding b has experienced


93

similar values for both isolated and grounded traces (trace (2) and trace (3)), the

discrepancy between the spectra comes from shunt capacitance alteration. This practical

result is going to be validated through physical and mathematical approaches in the next

subsection.

5.6.2 Physical and Mathematical Approach

From a physical point of view, comparing grounded with isolated traces of phase b reveals

that the winding shunt capacitance has changed considerably. This alteration has led the

entire low frequency band of FRA trace to shift to lower frequencies.

On the other hand, the frequency of the first minimal peak in FRA trace is given by:

1

2r lowf

LC (5.20)

where, fr-low is the anti-resonance frequency, L denotes the total winding self-inductance

and C denotes the winding capacitance. The winding capacitance (C) contains series and

shunt capacitances.

In this circumstance the transformer winding structure was not changed; therefore, the

transformer series capacitance has remained unchanged. Thus, according to Fig. 5.18 and

equation (5.20), the transformer shunt capacitance has increased. This increment impacts

FRA low frequencies, specifically the anti-resonance frequency.

Shunt capacitance topography for the LV winding of phase b is shown in Fig. 5.19. Figure

5.19 (a) demonstrates active part configuration. Figures 5.19(b) and 5.19(c) illustrate the

shunt capacitance configuration for isolated and grounded B.

To validate the measurement result, the shunt capacitance of the phase b is calculated and

compared to isolated and grounded B in (5.21) and (5.22) respectively.

( )

HL HTLCsh

b HL HT

C CC C

C C

(5.21)

( )

sh HLLCbC C C

(5.22)

where, Csh(b) and Cʹsh(b) are the shunt capacitances of HV winding phase b while the HV side of

phase B is short-circuited and isolated, and grounded respectively (see Fig. 5.19). CLC, CHL and


94

CHT denote the shunt capacitances of LV with respect to the core, the shunt capacitance of HV

with respect to the LV and, the shunt capacitance of HV with respect to the transformer tank.

(a)

CLC

CHL

LV

Tank

HVCore

CLC

CHL

LV

Tank

HVCore

CLC CHL

LV

CHT

HVCsh(b)

CLC CHL

LV

C’sh(b)

(b) (c)

Figure 5.19. Active part and related shunt capacitances, (a) Active part upper view schematic, (b) Shunt capacitance configuration for b where HV side phase B is just short-circuited and isolated from the ground, (c) Shunt capacitance configuration for b where HV side phase B is grounded.

From a physical point of view:

, 0HL HTLCC C and C (5.23)

Therefore:


95

( )( )sh

HL HTHLLC LCsh bb HL HT

C CC C C C C

C C

(5.24)

Validating (5.24) is based on the following expressions:

HL HT

HLLC LCHL HT

C CC C C

C C

(5.25)

2HL HT HL HT HL HT HL HTLC LC LC LCHL

C C C C C C C C C C C C C

(5.26)

20HL

C (5.27)

Equation (5.27) is always true. Therefore, the assumption in (5.24) is true and:

( ) ( )

sh shb b

C C (5.28)

According to (5.28), the shunt capacitance of grounded connection is greater than that of the

isolated case, mathematically. Therefore, based on (5.20) the frequency response trace of

phase b experienced an anti-resonance in lower frequency as compared to the isolated one. In

addition, it is clarified that the first minimal peak can be interacted through self-inductance

and shunt capacitance. Therefore, the shunt capacitance influences the low frequency band of

the FRA trace.

5.7 Conclusion

Interpretation of the low frequency band of FRA trace was discussed. Mathematical calculation

and practical measurements showed that the first minimal peak in the FRA spectrum comes

through the transformer middle limb while the second one is influenced by the lateral limb. To

validate the mathematical and practical approach, one of the winding frequency response

trace for the test object was generated through deviated traces.

According to the data obtained in this Chapter, similar equivalent magnetic circuits for FRA

setup will lead to identical FRA traces. This concern was investigated through practical

measurement. This in turn suggests a new prognosis technique for transformer core defect

localization. The reason lies in the fact that any deformation or defect in the transformer core


96

will cause the entire transformer FRA trace to change, accordingly. This should be taken into

account in FRA result interpretation.

In addition, the shunt capacitance influence was studied and discussed through mathematical

and practical approach. It was found that the shunt capacitance effects can change the low

frequency minimal peak position. It also has considerable effect on the frequency response

trend and its approach in low frequencies.

In summary, it is worth noting that the transformer winding inductance is not expected to

increase. The winding inductance is strictly related to the transformer magnetic reluctance.

Transformer design normally aims to achieve the optimum (low value) magnetic reluctance.

Thus each and every deformation or undesirable lamination in the transformer core sheets

will cause the magnetic reluctance to increase. Accordingly, the winding inductance will

experience lower value.

Reduction in winding inductance will in turn cause the first anti-resonance in the low-

frequency band to move to higher frequencies. Therefore, the shifting toward the right hand

side in the low frequency band can be interpreted through inductance reduction or shunt

capacitance increment. If this deviation is limited to the low-frequency band until the first anti-

resonance, the winding inductance alteration can be concluded. Otherwise, deviation in FRA

spectrum from the first minimal peak towards higher frequencies indicates changes in the

shunt capacitance.

Chapter 6. Axial and Radial Deformation of Transformer Winding

97

Chapter 6

Axial and Radial Deformation of Transformer Winding

6.1 Introduction

Mechanical defects in transformer winding such as axial or radial deformation may be due

to short circuit currents, earthquakes, careless transportation between sites, explosion of

combustible gases accumulating in the transformer oil, etc. They can cause changes to the

mutual-inductance, series and shunt capacitances and eventually FRA trace deviation. This

Chapter is specifically concerned with symmetrical and asymmetrical axial deformation of

a disk in transformer winding. An analytical approach is provided to calculate self- and

mutual-inductance variation under such circumstances. Also, the radial deformation in

transformer winding is modelled through the “buckling” and discussed analytically.

Afterwards, a numerical example is provided on a model winding.

In the case of capacitance calculation in radial deformation, the 2D Finite Element Method

(FEM) is also used to evaluate capacitance variation and the results are compared to the

analytical approach.

To explore the influence of winding parameters variation on FRA spectrum deviation, the

inductance, series and shunt capacitances, resistance, conductance to ground and turn-to-

turn conductance in transformer winding are changed and the frequency response is

simulated for the glassy model transformer presented in Chapter 4. The reason of FRA

spectrum deviation in each case is then discussed physically. To verify the simulation

result for each and every case, practical measurements are performed and deviated parts

in the measured FRA trace are compared to simulation results. The practical experiments

conducted verify the simulation result as well as the modelling.


98

6.2 Axial Deformation and Its Impacts on Winding Parameters

It is possible for axial deformation to occur symmetrically or asymmetrically for one or

several disks of a transformer winding.

Symmetrical axial deformation means that when one side of a disk has moved up closer to

the upper disk, the other side is moved down closer to the lower disk as illustrated in

Fig.6.1. In this condition, the axes of the deformed disk will be inclined to other disks.

Asymmetrical axial deformation of a disk refers to the situation when one side of the disk

moves closer to the upper disk whereas the outermost conductor on the opposite side

remains unchanged in original state as depicted in Fig. 6.2. Figures 6.1(a) and 6.2(a) show

the deformation patterns, and Figures 6.1(b) and 6.2(b) illustrate the implementation of

the windings.

Indeed the distance between the transformer winding disks (inter-disk distance) can

change due to the axial deformation. This causes the turn-to-turn mutual-inductance to

change accordingly, and further to that the total self-inductance is altered. An analytical

approach to investigate the mutual-inductance variation under axial deformation is

discussed hereinafter.

6.2.1 Mutual Inductance of Circular Filaments Whose Axes Are Inclined to One

Another

6.2.1.1 Symmetrical Axial Deformation of a Disk

The mutual-inductance of circular filaments whose axes are inclined to one another is

given by ‎[85]:

0 0cos cosabM R M R f R R H

m (6.1)

where M is calculated as (3.21), θ is illustrated in Fig.6.1, and R0 is a function of θ and given

by:

2 4 63 5 7

5 73

0 2 4 65 73

2 22 2 1

2 2

( ) ( ) ( )1 1 51 ( ) ( ) ( ) ...

4 8 641 1 5

1 ( ) ( ) ( ) ...4 8 64

where: , , sin , cos1 1 a

P P PP P P

RP P P

XR

(6.2)


99

Note that X denotes the axial displacement of the outermost turn. The formulas for Pm(μ)

and P'm(ν) are provided through (B.1) and (B.2) in Appendix B ‎[78]. In the case of

transformer winding, the parameter ψ generally takes a value between 0.6 and 0.9. Hence

for convenience, in the range of 0.6 < ψ < 0.9 some of pre-calculated values of R0 are

provided in Table B.2 ‎[85] in Appendix B.

Based on Table B.2, R0 is a value less than 1 for routine transformer windings. In addition,

cosθ ≤ 1; hence, M' ≤ M (M was discussed in Chapter 3). Therefore, the mutual inductances

between the winding turns for the case of symmetrical deformation of a disk take smaller

values as compared to the normal winding. This in turn influences the total self-inductance

of the winding and makes it smaller.


100

Ra

d

θ

Rb

(a)

Air-core

1 2 3 4 4 3 2 1

56

78

θ

W

d

9 10 11 12 12 11 10 9

16 15 14 13 13 14 15 16

d

87

65

(b)

Figure 6.1. Symmetrical axial deformation of a disk, (a) Axial deformation pattern, (b) Deformed disk.


101

Rb

D

θ

d

Ra

(a)

Air-core

1 2 3 4 4 3 2 1

5 6 7 8

θ'

W

d

9 10 11 12 12 11 10 9

16 15 14 13 13 14 15 16

d8 7 6 5d

'

d'

D'

D

(b)

Figure 6.2. Asymmetrical axial deformation of a disk, (a) Axial deformation pattern, (b) Deformed disk.


102

6.2.1.2 Asymmetrical Axial Deformation of a Disk

The mutual-inductance between the turns for asymmetrical axial deformation of a disk is

given by:

0 0cos cosa abM R M R f R R H

m (6.3)

According to Fig. 6.2, the distance between the second and third disks has been changed

from d to D, while D > d. Hence, χ in (6.2) should be replaced by χdown for mutual inductance

calculation between the turns in the second and third disks, and given as follows:

sinadown

b b

d RDR R

(6.4)

For the mutual-inductance between disk two and disk one, χup is then defined as (6.5) and

represents the ratio of the distance between the first and second disks and Rb:

sinaup

b b

d RDR R

(6.5)

Other parameters in (6.2) should be also replaced by:

222 2 1

2 2, , sin , cos

21 1x

ax x

XR

(6.6)

χx can take χdown or χup in (6.6) for the mutual inductance calculation of the second disk with

respect to the third or first disk, respectively. In addition, μ can be replaced by μ', κ2 by κ'2

and ν2 by ν'2 in (6.6) and k by kdown or kup in (3.22) and (6.3), accordingly.

Since χup < χ < χdown , therefore, kup < k < kdown and fm(up) > f > fm(down). Hence, we define A1=

fm(up) – fm > 0, and A2 = fm(down) – fm < 0. Based on (6.3), if fm(down) < fm(up) then |A1| >|A2| and

eventually the total value of fm'' for the asymmetrical deformation would be less than the

mutual inductance coefficient, fm, in the normal winding. This factor causes M'' to

experience larger value than M (see equation (3.23) to recall fm calculation).

On the other hand, as discussed earlier R0 is a value less than 1 for typical transformer

winding. In addition, cosθ' ≤ 1. Hence, M'' can take a value less than M.

Therefore, two factors (cosθ and R0) will reduce M'' whereas one factor (fm'') would

increase it. Thus, it is difficult to argue analytically on the magnitude of M'' as compared to

M. In order to determine which factors are more effective in changing their relative

magnitude, a numerical example is provided in Section 6.4 and the results are discussed.


103

6.2.2 Capacitances of Circular Filaments Whose Axes Are Inclined to One Another

6.2.2.1 Series Capacitance

Series capacitance comprises total turn-to-turn (Ctt) and total inter-disk capacitances (Cd).

Turn-to-turn transformer winding capacitance can be influenced slightly by the axial

symmetrical and asymmetrical deformation of a winding’s disk.

For the deformation illustrated in Fig. 6.1, equation (3.26) for the turn-to-turn capacitance

of the second disk is obtained as:

0

tan 2 (1 tan ).

2t

tt tt

h wC D

(6.7)

1 2 3 4

87

65

U Cdd

Ctt

9 10 11 12

16 15 14 13

U

CttCtt

tt

tt

tt

CttCttCtt

U 2Ul

9

1

8

7

5

14

12

15

16

1011

U

4

13

x

2N

U

2N

U

2N

U

2N

0

23

6

Figure 6.3. Cross-section overview and voltage distribution along deformed winding.

The voltage distribution in along transformer winding length (voltage per turn) is mostly

influenced by number of winding turns as well as winding height. Therefore in the case of

inter-disk capacitance, if assuming the voltage distribution along the pair of disks still

remained uniform, in axial deformation the total inter-disk capacitance (Cd) will not

change (see Fig. 6.3).

Base on this, the total series capacitance in Fig. 6.1 will slightly reduce. This reduction

could be negligible as the transformer winding disks are practically close together and

tanθ experiences small value. Similar calculations can be performed for Fig. 6.2.


104

6.2.2.2 Shunt Capacitance

The shunt capacitance between the winding and the container (metal tank) can experience

similar value as given by equation (3.34). This is because the outward structural

configuration of winding is still similar to the normal winding.

6.3 Radial Deformation and Its Impacts on Winding Parameters

In this study, radial deformation of transformer winding is modelled through free-buckling

as shown in Fig. 6.4, where the winding exterior has faced buckling towards metal tank.

Inward buckling occurs when the circumference of the winding disk is bent towards the

disk centre. Based on this, the winding inductance as well as series and shunt capacitances

are going to be discussed in this Section.

60

240

120

300

180

0

60

240

120

300

180

0

r0

r1

r2

Metal container

Winding

Figure 6.4. Radial deformation schematic (free-buckling).

6.3.1 Self and Mutual Inductances in Radial Deformation

Analytical calculation of the self and mutual inductances for the transformer winding

which has experienced inward or outward buckling is quite a challenging task to address.

Most of the literatures have referred to the work and equations presented by Maxwell in


105

‎[78]. Work by Chattopadhyay in ‎[103] on super conductors has tried to address this issue

through analytical approach.

In this thesis, it has been tried to improve this issue analytically to move one step forward

and find the loop inductance value under deformation. To this end, the provided equations

by Maxwell ‎[78] and Paul ‎[104] for the calculation of self and mutual inductances of the

circular loops are used and extended to determine the self and mutual inductances of the

non-circular filaments. However, the resultant integrals obtained in some cases are quite

non-linear and should be solved numerically. Thus, numerical calculation is conducted to

complete the inductance values calculations in radial deformation. In detail, this task is

provided separately in Appendix B due to significant volume of calculations.

Furthermore, radial displacement is a different concept as compared to radial

deformation, but in some literatures ‎[102] it has been considered and discussed as radial

deformation (see Fig. 6.5). In this case, the mutual inductance of circular elements with

parallel axes is given by ‎[85]:

pM M F (6.8)

R

d

R

rd

ρ

θ"drd

θ"

Figure 6.5. Radial displacement pattern of a disk.

where, Mp is calculated as M with a distance rd between the disks and F is given as Table

B.3 in Appendix B.

Since rd > d, therefore Mp < M. According to Table B.3, F can take a value larger or smaller

than 1. Based on this and also data in Table B.3, M''' will not change significantly for the

usual buckling of transformer winding.


106

6.3.2 Series and Shunt Capacitances in Radial Deformation

6.3.2.1 Series Capacitance

According to equations (3.26) through (3.30), it can be concluded that the series

capacitance including turn-to-turn and inter-disk capacitances of the buckled winding

shown in Fig. 6.4 is not changed significantly because the dimensions still remained

unchanged for these parameters.

6.3.2.2 Shunt Capacitance

Shunt capacitance calculation for the deformed winding is discussed in ‎[13]; however, the

provided formula requires to be modified. Therefore, a more accurate approach is

suggested here.

The electric field generated by the transformer winding is not uniform across the

deformed turns. Therefore, calculation of the shunt capacitance requires the electric field

determined by finite element method. This approach is demonstrated in the next

subsection through a numerical example, and the result is then compared with the

analytical approach.

In the case of analytical approach, if assuming a uniform electric field across the deformed

section (highlighted in Fig 6.4), the equivalent shunt capacitance can be calculated through

the summation of the shunt capacitance of the normal section, Cnorm, as given by (6.9)

paralleled with the shunt capacitance of the deformed section, Cdeform, as given by (6.10):

20

2(2 )

2

1

ln

r wnormal

HC d

r

r

(6.9)

2

0

02

1

ln0.5 (cos( ) 1)

r wnormal

HC d

r

r r

(6.10)

where, η is the ratio of entire trigonometric circular span (2π) over the deformation span

(rad) as illustrated in Fig. 6.4, Hw is the winding height, and r' represents the deformation

radius. The total shunt capacitance of the winding that had buckling is then obtained as:

g normal deformC C C (6.11)


107

6.4 Numerical Example

In this Section, a numerical example is provided to examine the inductance and

capacitance variation in winding deformation.

It is assumed that the model winding illustrated in Fig. 3.1 has single strand conductor.

Radial dimension of conductor is w = 7 mm, axial dimension is h = 11 mm, inter-disk

distance is δd = 6 mm, thickness of paper insulation is δt = 0.5 mm, relative permittivity of

paper insulation is εt = 3.2, vacuum permittivity is ε0, mean radius of winding disk is R=

280 mm, mean radius of the tank is R= 400 mm, and winding as well as tank height are 62

mm and 100 mm, respectively. The winding has four disks and four conductors per disk.

Note that centre to centre distance for pair of disks is d = 5.5 + 5.5 + 6 = 17 mm.

6.4.1 Axial Deformation of a Disk

Axial deformation of model winding is assumed to occur for the second disk such as to

what happened in Figures 6.1 and 6.2. Hence, the maximum axial displacement in Fig. 6.1

and Fig. 6.2 for the conductor numbered 8 would be X = 6 mm. Therefore, the inductance

and capacitance of the model are calculated as follow using the mentioned formulas.

6.4.1.1 Inductance Calculation

Equations (3.19) and (3.21) were used to calculate the self and mutual inductances. Thus,

the detailed inductance matrix (μH) of the normal winding obtained is:

1.8221 1.3679 1.0997 0.9387 0.8500 0.9385 1.0190 1.0653 0.8128 0.7910 0.7564 0.7131 0.6006 0.6277 0.6498 0.6655

1.3679 1.7354 1.3250 1.0630 0.9066 0.9863 1.0310 1.019 0.791 0.7845 0.7631 0.7291 0.6036 0.6253 0.6409 0.6498

1.0997

eqL

1.3250 1.6507 1.284 0.9529 0.9969 0.9863 0.9385 0.7564 0.7631 0.7565 0.7354 0.6010 0.6164 0.6253 0.6277

0.9387 1.0630 1.2840 1.5682 0.9629 0.9529 0.9066 0.8500 0.7131 0.7291 0.7354 0.7287 0.5922 0.6010 0.6036 0.6006

0.8500 0.9066 0.9529 0.9629 1.8221 1.3679 1.0997 0.9387 0.8500 0.9066 0.9529 0.9629 0.8128 0.7910 0.7564 0.7131

0.9385 0.9863 0.9969 0.9529 1.3679 1.7354 1.3250 1.0630 0.9385 0.9863 0.9969 0.9529 0.7910 0.7845 0.7631 0.7291

1.019 1.0310 0.9863 0.9066 1.0997 1.3250 1.6507 1.2840 1.0190 1.0310 0.9863 0.9066 0.7564 0.7631 0.7565 0.7354

1.0653 1.019 0.9385 0.8500 0.9387 1.0630 1.2840 1.5682 1.0653 1.0190 0.9385 0.8500 0.7131 0.7291 0.7354 0.7287

0.8128 0.791 0.7564 0.7131 0.8500 0.9385 1.019 1.0653 1.8221 1.3679 1.0997 0.9387 0.8500 0.9385 1.019 1.0653

0.791 0.7845 0.7631 0.7291 0.9066 0.9863 1.0310 1.0190 1.3679 1.7354 1.3250 1.0630 0.9066 0.9863 1.0310 1.019

0.7564 0.7631 0.7565 0.7354 0.9529 0.9969 0.9863 0.9385 1.0997 1.3250 1.6507 1.284 0.9529 0.9969 0.9863 0.9385

0.7131 0.7291 0.7354 0.7287 0.9629 0.9529 0.9066 0.8500 0.9387 1.0630 1.284 1.5682 0.9629 0.9529 0.9066 0.8500

0.6006 0.6036 0.6010 0.5922 0.8128 0.791 0.7564 0.7131 0.8500 0.9066 0.9529 0.9629 1.8221 1.3679 1.0997 0.9387

0.6277 0.6253 0.6164 0.6010 0.791 0.7845 0.7631 0.7291 0.9385 0.9863 0.9969 0.9529 1.3679 1.7354 1.3250 1.0630

0.6498 0.6409 0.6253 0.6036 0.7564 0.7631 0.7565 0.7354 1.019 1.0310 0.9863 0.9066 1.0997 1.3250 1.6507 1.2840

0.6655 0.6498 0.6277 0.6006 0.7131 0.7291 0.7354 0.7287 1.0653 1.0190 0.9385 0.8500 0.9387 1.0630 1.284 1.5682

(6.12)

For the symmetrical axial deformation occurred in the second disk (Fig. 6.1), the

inductance matrix is calculated as (6.13) using equation (6.1), while this matrix for the

asymmetrical deformation (Fig. 6.2) is obtained as (6.14) using equation (6.3):


108

1.8221 1.3679 1.0997 0.9387 0.8483 0.9365 1.0168 1.0629 0.8128 0.791 0.7564 0.7131 0.6006 0.6277 0.6498 0.6655

1.3679 1.7354 1.3250 1.0630 0.9047 0.9842 1.0287 1.0168 0.791 0.7845 0.7631 0.7291 0.6036 0.6253 0.6409 0.6498

1.099

eqL

7 1.3250 1.6507 1.284 0.9508 0.9947 0.9842 0.9365 0.7564 0.7631 0.7565 0.7354 0.6010 0.6164 0.6253 0.6277

0.9387 1.0630 1.284 1.5682 0.9607 0.9508 0.9047 0.8483 0.7131 0.7291 0.7354 0.7287 0.5922 0.6010 0.6036 0.6006

0.8483 0.9047 0.9508 0.9607 1.8221 1.3679 1.0997 0.9387 0.8483 0.9047 0.9508 0.9607 0.8110 0.7893 0.7548 0.7117

0.9365 0.9842 0.9947 0.9508 1.3679 1.7354 1.3250 1.0630 0.9365 0.9842 0.9947 0.9508 0.7893 0.7827 0.7614 0.7276

1.0168 1.0287 0.9842 0.9047 1.0997 1.3250 1.6507 1.284 1.0168 1.0287 0.9842 0.9047 0.7548 0.7614 0.7548 0.7338

1.0629 1.0168 0.9365 0.8483 0.9387 1.0630 1.284 1.5682 1.0629 1.0168 0.9365 0.8483 0.7117 0.7276 0.7339 0.7271

0.8128 0.791 0.7564 0.7131 0.8483 0.9365 1.0168 1.0629 1.8221 1.3679 1.0997 0.9387 0.8500 0.9385 1.019 1.0653

0.791 0.7845 0.7631 0.7291 0.9047 0.9842 1.0287 1.0168 1.3679 1.7354 1.3250 1.0630 0.9066 0.9863 1.0310 1.0190

0.7564 0.7631 0.7565 0.7354 0.9508 0.9947 0.9842 0.9365 1.0997 1.3250 1.6507 1.284 0.9529 0.9969 0.9863 0.9385

0.7131 0.7291 0.7354 0.7287 0.9607 0.9508 0.9047 0.8483 0.9387 1.0630 1.284 1.5682 0.9629 0.9529 0.9066 0.8500

0.6006 0.6036 0.6010 0.5922 0.8110 0.7893 0.7548 0.7117 0.8500 0.9066 0.9529 0.9629 1.8221 1.3679 1.0997 0.9387

0.6277 0.6253 0.6164 0.6010 0.7893 0.7827 0.7614 0.7276 0.9385 0.9863 0.9969 0.9529 1.3679 1.7354 1.3250 1.0630

0.6498 0.6409 0.6253 0.6036 0.7548 0.7614 0.7548 0.7338 1.019 1.0310 0.9863 0.9066 1.0997 1.3250 1.6507 1.2840

0.6655 0.6498 0.6277 0.6006 0.7117 0.7276 0.7338 0.7271 1.0653 1.019 0.9385 0.8500 0.9387 1.0630 1.2840 1.5682

(6.13)

1.8221 1.3679 1.0997 0.9387 0.8495 0.9380 1.0184 1.0647 0.8128 0.7910 0.7564 0.7131 0.6006 0.6277 0.6498 0.6655

1.3679 1.7354 1.3250 1.0630 0.9061 0.9857 1.0304 1.0184 0.791 0.7845 0.7631 0.7291 0.6036 0.6253 0.6409 0.6498

1.0

eqL

997 1.3250 1.6507 1.2840 0.9523 0.9963 0.9857 0.9380 0.7564 0.7631 0.7565 0.7354 0.6010 0.6164 0.6253 0.6277

0.9387 1.0630 1.284 1.5682 0.9623 0.9523 0.9061 0.8495 0.7131 0.7291 0.7354 0.7287 0.5922 0.6010 0.6036 0.6006

0.8495 0.9061 0.9523 0.9623 1.8221 1.3679 1.0997 0.9387 0.8495 0.9061 0.9523 0.9623 0.8123 0.7905 0.7560 0.7127

0.9380 0.9857 0.9963 0.9523 1.3679 1.7354 1.3250 1.0630 0.9380 0.9857 0.9963 0.9523 0.7905 0.7840 0.7626 0.7287

1.0184 1.0304 0.9857 0.9061 1.0997 1.3250 1.6507 1.284 1.0184 1.0304 0.9857 0.9061 0.7560 0.7627 0.7560 0.7350

1.0647 1.0184 0.9380 0.8495 0.9387 1.0630 1.284 1.5682 1.0647 1.0184 0.9380 0.8495 0.7127 0.7287 0.7350 0.7283

0.8128 0.791 0.7564 0.7131 0.8495 0.9380 1.0184 1.0647 1.8221 1.3679 1.0997 0.9387 0.8500 0.9385 1.019 1.0653

0.791 0.7845 0.7631 0.7291 0.9061 0.9857 1.0304 1.0184 1.3679 1.7354 1.3250 1.0630 0.9066 0.9863 1.0310 1.0190

0.7564 0.7631 0.7565 0.7354 0.9523 0.9963 0.9857 0.9380 1.0997 1.3250 1.6507 1.284 0.9529 0.9969 0.9863 0.9385

0.7131 0.7291 0.7354 0.7287 0.9623 0.9523 0.9061 0.8495 0.9387 1.0630 1.284 1.5682 0.9629 0.9529 0.9066 0.8500

0.6006 0.6036 0.6010 0.5922 0.8123 0.7905 0.7560 0.7127 0.8500 0.9066 0.9529 0.9629 1.8221 1.3679 1.0997 0.9387

0.6277 0.6253 0.6164 0.6010 0.7905 0.7840 0.7626 0.7287 0.9385 0.9863 0.9969 0.9529 1.3679 1.7354 1.3250 1.0630

0.6498 0.6409 0.6253 0.6036 0.7560 0.7626 0.7560 0.7350 1.019 1.0310 0.9863 0.9066 1.0997 1.3250 1.6507 1.2840

0.6655 0.6498 0.6277 0.6006 0.7127 0.7287 0.7350 0.7283 1.0653 1.0190 0.9385 0.8500 0.9387 1.0630 1.2840 1.5682

(6.14)

As compared to (6.12), those elements which have changed due to the winding axial

deformation are highlighted by enclosing rectangles in (6.13) and (6.14).

The data of (6.15), (6.16) and (6.17) are obtained as the overall inductance matrix through

the summation of elements in (6.12), (6.13) and (6.14), respectively. These results

represent the inductance matrix in (3.1).

20.9330 15.3627 12.0587 9.9310

15.3627 20.9330 15.3627 12.0587

12.0587 15.3627 20.9330 15.3627

9.9310 12.0587 15.3627 20.9330

eqL

(6.15)

20.9330 15.3296 12.0587 9.9310

15.3296 20.9330 15.3296 12.0329

12.0587 15.3296 20.9330 15.3627

9.9310 12.0328 15.3627 20.9330

eqL

(6.16)


109

20.9330 15.3545 12.0587 9.9310

15.3545 20.9330 15.3546 12.0517

12.0587 15.3546 20.9330 15.3627

9.9310 12.0517 15.3627 20.9330

eqL

(6.17)

According to (6.13) and (6.14), the elements in Mab, Mba, Mbc, Mcb, Mbd and Mdb are changed

due to the axial deformation of the second disk. In fact, all related mutual inductances to

the second disk have changed, while other elements have remained unchanged.

Furthermore, the self-inductance of the second disk (LB) has not changed, while the total

self-inductance of the model winding (Leq) has altered.

In addition, the variation of mutual inductances as well as total self-inductance for the

symmetrical axial deformation is more than asymmetrical deformation. This in turn means

that the deformation angle, θ, is quite a significant parameter in inductance variation.

6.4.1.2 Capacitance Calculation

The series capacitance of the normal winding of (Fig. 3.1) is:

67.950C pFs (6.18)

This value was obtained as (6.19) and (6.20) for the configurations in Fig. 6.1 and Fig. 6.2,

respectively.

67.827C pFs (6.19)

67.889C pFs (6.20)

The shunt capacitance will remain constant as the outward configuration has not changed

much.

6.4.1.3 Practical Experiment on Axial Deformation

The inductance reduction was discussed in the last subsections for the winding having an

axially deformed disk. To examine this result, a 66 kV, 25 MVA air-core interleaved

winding was used to conduct an experiment on axial deformation. This winding contains

32 disks with 24 turns per disk (examined test object in Chapter 4). At first, frequency

response end-to-end open circuit measurement was performed on the original

(undeformed) winding. The FRA spectrum was recorded over the frequency range of 20

Hz - 2 MHz. Afterwards, the fourth disk of the winding was deformed axially with its


110

outermost turn shifted toward the upper disk, as shown in Fig. 6.6. This was achieved by

inserting a plastic wedge to simulate the asymmetrical axial deformation. The winding

frequency response was re-recorded and the two spectra (before and after deformation)

are shown in Fig. 6.7.

Figure 6.6. Axially deformed interleaved winding.

102

103

104

105

106

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Magnitude [

dB

]

After Axial Deformation

Fingerprint

Frequency Bankd: 20 Hz - 2 MHz104.2

104.3

104.4

104.5

-60

-55

-50

-45

-40

-35

Frequency [Hz]

Magnitude [

dB

]

After Axial Deformation

Fingerprint

Frequency Bankd: 20 Hz - 2 MHz

105

-45

-40

-35

-30

-25

Frequency [Hz]

Magnitude [

dB

]

Figure 6.7. Reference and measured frequency response spectra for deformed interleaved windings.

As achieved in Chapter 5, it was shown that within the FRA spectrum of a transformer

winding, the first minimal peak (anti-resonance) in the low frequencies is initiated

through the interaction between the winding inductive and capacitive reactances. Thus,

study on this anti-resonance will help to estimate which winding’s parameter has altered.

Study on other frequency bands of the FRA spectrum can also narrow down the

investigations, help to analyse the first anti-resonance better, and improve the diagnosis.

According to Fig. 6.7, moving from lower towards higher frequencies, it can be seen that

the spectra in the very low-frequency region are almost matched. After around 1 kHz, the

discrepancy becomes noticeable in the low-frequency band while the first minimal peak


111

shows FRA spectrum deviation due to the axial deformation. This in turn could be due to

the inductance or shunt capacitance reduction of the winding. In addition, mid-frequency

oscillations have altered and resonance and anti-resonances are randomly shifted. Since

the mid-frequency oscillations are initiated through the mutual-inductance within the

total winding inductance as well as series and shunt capacitances, this alteration tends to

support a hypothesis that the mutual-inductance or total capacitance may have been

changed. Furthermore, close matching of the FRA spectra in the high and very high

frequency regions indicates that deviation in spectra comes through inductance rather

than capacitance variation. Available discrepancy in FRA spectra in low frequencies can

also validate this hypothesis. In fact, the low-frequency band of the FRA spectrum is

mainly affected by winding inductance and deviation in this region comes through self- or

even mutual-inductance alteration. This is the fact that the first anti-resonance is shifted to

higher frequencies due to the inductance reduction in this winding, see equation (5.20).

Practical measurement results of this study on a single air-core interleaved winding verify

the analytical approach and numerical calculation on axial deformation.

6.4.2 Radial Deformation along the Winding

Here, it is assumed that radial deformation caused part of the model winding to be

stretched towards the metal tank as shown in Fig. 6.4. The mean radius of deformation is

r'= ±50 mm, and the deformation angle is φ= π/2.

6.4.2.1 Inductance Calculation

Self- and mutual-inductance variation due to the radial deformation in transformer

winding is discussed in details in Appendix B.

6.4.2.2 Capacitance Calculation

In this particular case, the shunt capacitance of the winding can change significantly due to

the radial deformation, while the series capacitance will remain almost constant.

The shunt capacitance is simulated and calculated for a buckled winding similar to Fig. 6.4

using finite element method as well as the formula presented in this Chapter. The

parameters of the simulated winding are same as those presented in the numerical

example shown in Fig. 6.8. Simulated and analytical results are presented in Table 6.1. The

data obtained in Table 6.1 reveal that the analytical approach to calculate the shunt

capacitance is reasonably accurate. However, this accuracy might not be adequate for

complicated radial deformations. For both finite element and analytical approaches, the


112

winding which is bent inward experiences less shunt capacitance than that winding bent

outward.

Figure 6.8. The modelled winding through finite element (r1= 280 mm, r2 = 400 mm, r'= ±50 mm, φ= π/2).

Table 6.1. Calculated capacitance between the winding and metal container (tank).

Condition Finite Element Analytical Calculation

Normal Winding 9.6893 pF 9.6658 pF

Bent outward 10.6976 pF 10.5604 pF

Bent inward 9.2218 pF 9.1997 pF

6.5 A Summary on Axial and Radial Deformations

Modelling of transformer windings due to radial and axial deformations is considered a

significant challenge for those researchers studying transformer winding through the

detailed model. This Chapter is focused on inductance and capacitance variations due to

winding deformation. A winding model was proposed, self- and mutual-inductances as

well as series and shunt capacitances were studied in detail. Inductance and capacitance

variation due to axial and radial deformations was discussed analytically. A numerical

example was presented and it showed that the inductance value will be changed due the

axial deformation; this alteration for capacitance was insignificant. Study on radial

deformation revealed the shunt capacitance variation in transformer winding. This in turn

almost verifies experimental findings in ‎[102]. It should be emphasized that different

winding deformations can lead to different inductance and capacitance variation and in a

complex deformation; these might not even be distinguishable. Therefore, having obtained

the frequency response spectrum, it is suggested to carry out calculations using the

analytical approach prior to FRA interpretation. However, deformations with similar

parameter changes lead to similar deviation pattern in the FRA spectrum, viz., inductance

reduction either in a single winding or a single phase transformer results in identical

patterns (not identical deviation) in FRA trace movement.


113

6.6 FRA Simulation Study and Practical Measurement Results

Transformer winding modelling to interpret oscillations was discussed in detail in Chapter

4. In addition, axial and radial deformations of the transformer winding and their

influences on winding parameters were investigated in this Chapter. The current Section

tries to explore the FRA trace deviation due to each and every parameter variation in

transformer winding. To gain this knowledge, the mathematical model presented in

Chapter 4 was used. Transformer winding inductance, series and shunt capacitances, turn-

to-turn and turn-to-ground conductance are changed and their impacts on the FRA trace

are discussed. Afterwards, the simulation results are compared to practical measurements

to verify our approach in the interpretation of FRA spectrum due to winding deformation.

It should be noted that due to the high costs associated with a transformer or perhaps

winding, the experiments associated with this research had to be non-destructive.

Therefore in order to study all parameters to obtain the FRA data associated with different

levels of parameter changes, it was decided to utilise different methods which could

emulate inductance, capacitance, resistance and conductance variations. In addition,

different test objects are used to demonstrate that fundamental parameters variation in

transformer winding will lead to similar deviation pattern in the FRA spectrum. Hence, the

simulations are carried out on the unique model transformer and verification is performed

through different test objects.

6.6.1 Inductance Variation (Simulation)

One of those crucial parameters in transformer winding is the winding inductance.

Winding inductance contains self-and mutual-inductance as discussed in (3.1). It is mostly

influenced by winding configuration, magnetic core reluctance or even active part

condition. Any changes of these factors can alter the self- or mutual-inductance and

ultimately the total winding inductance. In practice, the transformer winding self-

inductance cannot be increased easily. In fact, it is tightly related to the transformer

magnetic reluctance, see equation (5.8). It was discussed in Chapter 5 that typical designs

by transformer manufacturers aim to minimize transformer magnetic reluctance. Hence,

any deformation or undesirable lamination of transformer core sheets will lead to increase

in the magnetic reluctance and accordingly, the winding self-inductance will have lower

value. The other parameter which influences inductance is the number of winding turns. If

the number of winding turns is increased, the winding inductance would be increased as

well, but this obviously is not possible practically.


114

On the other hand, this Chapter revealed analytically that the axial deformation of winding

will lead to a decrease in winding inductance. This was further demonstrated by a

numerical example. Hence, decreasing of the winding inductance is more feasible in

practice as compared to increasing Based on this, it was supposed that the total

inductance of the winding has decreased for 20% and 40% from the reference value, and

the simulation was carried out on the mathematical model of winding presented in

Chapter 4. Figure 6.9 shows the simulation result.

(a)

(b)

Figure 6.9. FRA simulation results of winding due to the inductance reduction for 20 and 40 %, (a) Entire FRA

spectrum, (b) Expanded view of dash-line rectangle region in Fig. 6.9(a).

104

105

106

107

-100

-80

-60

-40

-20

0

20

Frequency [Hz]

Magnitude [

dB

]

L

0.8 x L

0.6 x L

Frequency Band : 1 kHz - 20 MHz

106

107

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

Frequency [Hz]

Magnitude [

dB

]

L

0.8 x L

0.6 x L

Frequency Band : 1 MHz - 10 MHz


115

According to Fig. 6.9, self-inductance reduction will lead to shifting of the first anti-

resonance in transformer FRA trace to higher frequencies. Therefore, deviation of FRA

spectrum toward the right hand side in the low-frequency band can be interpreted

through self-inductance reduction. However, this movement in FRA trace could be affected

through the shunt capacitance reduction as well. The shunt capacitance will be discussed

in following subsections. In addition, the anti-resonances in Fig. 6.9(b) have reduced in

magnitude. This can be interpreted through smaller inductive impedance in (5.9) and so

less magnitude in (2.7) and/or (5.4). Therefore, inductance reduction in transformer

winding will cause anti-resonances shifting to higher frequencies and having smaller

magnitude. This deviation is more significant for the first anti-resonance in the FRA

spectrum. This is verified through practical measurement in the next subsection.

6.6.2 Inductance Variation (Practical Study)

In order to validate the simulation result, an 11/0.25 kV, 25 kVA transformer was used as a

test object and inductance reduction of the HV winding was studied. Technical

specifications of this transformer are given in Table 6.2.

Table 6.2. Technical specifications of single phase transformer.

Manufacture year 2010 Frequency [Hz] 50

Rated voltage [kV] 11/0.25 Number of phases 1

Rated power [kVA] 25 Cooling ONAN

At first, the frequency response trace was recorded from the HV side while the LV side was

left open circuit. This measurement configuration is considered a common setup for FRA

measurement (end-to-end measurement, see Fig. 2.10(a)). To study the inductance

reduction, the LV winding is deliberately short-circuited and the frequency response of HV

winding is recorded.

According to Section 5.2.2 and also Lenz’s law, making short circuit across the LV winding

will cause the magnetic flux initiated in HV winding unable to flow through the

transformer core. This in turn leads to significant increase in magnetic core reluctance and

the winding inductance is decreased. The FRA test setups for normal and reduced winding

inductances are illustrated in Figures 6.10(a) and 6.10(b). The measured result is shown

in Fig. 6.11.

According to Fig. 6.11, when the LV winding is short-circuited, the frequency response of

the HV winding changes in the range from 20 Hz to 40 kHz. For frequencies above 40 kHz,


116

there are no significant discrepancies between the recorded traces. The reason lies in the

fact that the self-inductance of the winding has changed due to the deliberately created

short circuit.

(a) (b)

Figure 6.10. FRA test setup on HV side with (a) Open circuit on LV side, (b) Short circuit on LV side.

In addition according to Fig. 6.11, the frequency response magnitude when the LV winding

is left open circuit starts from -60.78 dB and following its descending trend reaches the

minimal peak at 605.7 Hz. Also, the frequency response magnitude when the LV winding is

short-circuited starts from -8.99 dB at 20 Hz and following its descending trend reaches

minimal peak at 7.971 kHz, meaning that the frequency response trace has shifted to the

right considerably.

Figure 6.11. Frequency response of HV winding for open circuit and short-circuited LV winding.

Hence in low frequency range, the trace has shifted upward and toward the right

considerably. In order to interpret the discrepancy between traces, equation (2.7) was

102

103

104

105

106

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

Frequency [Hz]

Ma

gn

itu

de

[d

B]

HV winding - LV (open circuit)

HV winding - LV (short-circuited)



117

taken into consideration. If assuming that inductance and capacitance are frequency

invariant, to calculate the self-inductance variation in open and short circuit situations,

(6.21) and (6.22) are employed:

(50 )

10

(50 )

20log 11.9out Hz

in Hz

VdB

V

(6.21)

(50 )

10

(50 )

20log 62.33out Hz

in Hz

VdB

V

(6.22)

According to Fig. 6.11, at the fundamental frequency (50 Hz), the frequency response

magnitude when the LV winding is short-circuited reaches -11.9 dB, while for open circuit

LV winding, it reaches –62.33 dB. According to (6.21) and (6.22), the impedance ratio for

open and short circuit situations is calculated as (6.23):

(50 )

4(50 )

0.2540

7.647 10

outOC Hz

outSC Hz

Z Z

Z Z

(6.23)

where, ZOC(50 Hz) is the inductive reactance of the HV winding in open circuit state at 50 Hz

and ZSC(50 Hz) is the inductive reactance of the HV winding at the same frequency when the

LV winding is short-circuited. Therefore, the ratio of open circuit and short circuit self-

inductance at the fundamental frequency for the tested transformer is given by:

(50 )

(50 )

445OC Hz

SC Hz

L

L (6.24)

where, LOC is the open circuit self-inductance and LSC represents short circuit self-

inductance. Thus, the open circuit self-inductance of this single phase transformer is 445

times greater than self-inductance of the HV winding when the LV winding is short-

circuited. Since the number of turns of the HV winding has not changed, the calculated

ratio for inductances can be applied to reluctances accordingly. This experimental result

verifies the simulation study in the previous subsection.

6.6.3 Shunt Capacitance Variation (Simulation)

In order to explore the shunt capacitance variation on FRA spectrum, the total shunt

capacitance of the winding was decreased for 20% and 40% from the reference value, and


118

the simulation was carried out on the mathematical winding model presented in Chapter

4. Figure 6.12 shows the simulation results due to the shunt capacitance variation.

(a)

(b)

Figure 6.12. FRA simulation results of winding due to the shunt capacitance reduction of 20 and 40 %, (a)

Entire FRA spectrum, (b) Expanded view of dashed-line rectangle in Fig.6.12 (a).

104

105

106

107

-100

-80

-60

-40

-20

0

20

Frequency [Hz]

Magnitude [

dB

]

C (shunt)

0.8 x C (shunt)

0.6 x C (shunt)


106

107

-100

-80

-60

-40

-20

0

20

Frequency [Hz]

Magnitude [

dB

]

C (shunt)

0.8 x C (shunt)

0.6 x C (shunt)


I I

L " ·I I' h l!l li

~~ I I' ' 1: I

li ! I !

H~=~~.---------------~LI ----~ ---+-. ~-----~-==·-==,_---. .....,!/!~-'0)j~\-\ _p·=~jl)'

1=·==·=·---------------; I ! i -----· I l::=:=~---------------1


119

According to Fig. 6.12, resonances and anti-resonances in FRA trace have moved to higher

frequencies and anti-resonance magnitudes have increased.

Resonant movements to the right hand side in the Bode diagram can be interpreted

through equation (5.20), while magnitude increase in FRA trace comes through interaction

between capacitive and inductive reactances. Since the shunt capacitance has decreased,

the resonant frequency has shifted to higher frequencies, and in the meantime, the

inductive reactance experiences higher amplitude due to the higher frequency (see

equation 5.20). Therefore, greater capacitive reactance is required to interact with

inductive reactance and creating an anti-resonance point in FRA trace.

6.6.4 Shunt Capacitance Variation (Practical Study)

In order to study the shunt capacitance variation, the single phase transformer introduced

in Table 6.2 was again taken as the test object. To change the shunt capacitance, the test

setup was configured as in Fig. 6.13 where the LV winding was short-circuited and

transformer tank was isolated from the ground. Note that the transformer tank was

originally grounded (see Fig. 6.10(b)).

Figure 6.13. FRA test setup for HV winding, where LV side is short-circuited, and test object tank is isolated

from the ground.

Obviously from a physical point of view:

, 0HL HTLCC C and C (6.25)

where CLC, CHL and CHT denote shunt capacitances of LV to the core, HV the LV and HV to the

transformer tank, respectively. For the shunt capacitance configuration in Fig. 6.14:

( )( )sh

HVHV

HL LC HL LCHTsh

HL LC HL LC

C C C CC C C

C C C C

(6.26)

where, Csh(HV) and Cʹsh(HV) are the HV winding shunt capacitances when the tank is grounded

and isolated, respectively. Equation (6.26) is always true. Therefore, we expect to get a lower


120

value of the shunt capacitance for HV winding through isolating the transformer tank. This

hypothesis was verified by FRA measurement on the HV side of single phase transformer for

the test setups in Figures 6.10(b) and 6.13. The results are shown in Fig.6.15.

HV

Tank

CHL

LV

CLC

CHT

CHT CHL

HV

TankLV

Core

CLC

CHT CHL

HV

CLC

LVCsh(HV)

CHL

HV

C’sh(HV)

CHT

(a) (b)

Figure 6.14. Active part and related shunt capacitances, (a) Shunt capacitance configuration for HV, where LV side is short-circuited and transformer tank is grounded, (b) Shunt capacitance configuration for HV, where LV side is short-circuited and transformer tank is isolated.

Figure 6.15. Frequency response of HV winding of single phase transformer, LV winding is short-circuited,

transformer tank is grounded and isolated.

102

103

104

105

106

-80

-70

-60

-50

-40

-30

-20

-10

Frequency [Hz]

Ma

gn

itu

de

[d

B]

Shortened LV side

Shortened LV side - Tank Isolated

Frequency Band: 20 Hz - 2MHz

9.003 kHz-69.16 dB

7.883 kHz-68.7 dB


121

According to Fig. 6.15, the frequency response trace for grounded tank does not deviate

from the isolated one over the very low frequency range. In fact, the self-inductance of the

HV winding does not change for grounded and isolated tank.

Moving from very low to low frequencies reveals that the minimal peak has shifted to a

higher frequency (from 7.883 kHz to 9.003 kHz) and its absolute magnitude has increased

from 68.7 dB to 69.16 dB. In addition, in mid and high frequencies the trace of isolated

tank displays greater absolute magnitude compared to grounded tank, while oscillation

trends in both traces remain roughly the same.

To interpret mid and high frequency discrepancies in Fig. 6.15, it is worth noting that the

first minimal peak in FRA trace can be calculated as (5.20). Since HV winding configuration

has not changed and just the tank has been isolated, the series capacitance of HV winding

has not varied. Based on this fact, minimal peak movement in Fig. 6.15 from 7.883 kHz to

9.003 kHz can be interpreted as due to winding shunt capacitance variations. In fact, anti-

resonance movement in the trace of isolated tank to higher frequency implies a slight

reduction in shunt capacitance of the winding. This showed in (6.26) and in turn verifies

the simulation results for the case of shunt capacitance reduction in subsection 6.6.3.

On the other hand, capacitive reactance increases when capacitance of the winding

decreases. This in turn results in greater absolute frequency response magnitude in mid

and high frequencies.

Note that, the self-inductance does not change during isolated and grounded tank while

the shunt capacitance has decreased.

6.6.5 Series Capacitance Variation (Simulation)

The series capacitance can be changed due to changes of turn-to-turn or disk-to-disk

distances of transformer winding. Any change or movement of the conductor or slanting of

the vertical form can also influence the winding series capacitance. Tilting, bending and

conductor deformation can change this parameter as well. In order to examine the impact

of series capacitance of the winding on FRA trace, this parameter was increased by 20%

and 40% and simulation was performed to obtain the corresponding FRA traces. The

result showed that such changes in series capacitance did not influence the FRA trace

significantly. Hence, changes of series capacitance were examined for larger values. It was

supposed that the series capacitance of the winding experiences 100 times and 200 times

greater than the original value. This could be happened when the transformer winding


122

turns are tilted. Figure 6.16 shows the FRA simulation result for the series capacitance

variation.

(a)

(b)

104

105

106

107

-100

-80

-60

-40

-20

0

20

Frequency [Hz]

Magnitude [

dB

]

C (series)

100 x C (series)

200 x C (series)


106

107

-100

-80

-60

-40

-20

0

20

Frequency [Hz]

Magnitude [

dB

]

C (series)

100 x C (series)

200 x C (series)


t

~ ~

I I !

~ I

' I I I

I I

I I

" ~

i

J II ll

\ 1/1 I II'

r---------------------1 ~ v \j ~ ~ 1··-· I I I I ----- I l-----

-T-- 1 _l

i·-- -~--- - - -~--~1

I ! ----· ! L-=--=.-=--=------------------J

I I I I I I I

I ' i I

I I

! ~ .. I I'

I 1\i ' Ill

iit ! 1 il\ i

L i I' '' 1,1 r-----__ : I( I· I

1 ~ ~\ 11 \

i '/ l /i I

------ / ""-/ v ~ I \ ·y \ j _l_


123

(c)

Figure 6.16. FRA simulation results of winding due to the series capacitance increase, (a) Entire FRA spectrum,

(b) Close-up view of dash-line rectangle in Fig. 6.16 (a), (c) Close-up view including some resonance

frequencies.

According to Fig. 6.16, series capacitance variation will influence the mid frequencies

towards the high-frequency band in the FRA spectrum. In fact, it has insignificant impact

on the first anti-resonance in FRA trace. In addition, as can be seen in Fig. 6.16, series

capacitance alteration of the transformer winding does not influence the low-frequency

band. Also, it tends to change the higher frequencies more than the lower frequencies.

Indeed, series capacitance variation influences the FRA spectrum less than shunt

capacitance. In other words, the shunt capacitance has more impact on the FRA spectrum

than the series capacitance.

6.6.6 Series Capacitance Variation (Practical Study)

A transformer had been de-energized by emergency disconnection system actuated by

gassing and differential protection systems, as well as two safety valves. Thus, in order to a

detailed investigation of the transformer prior to re-energizing, various diagnostic tests

were conducted. One of those tests was FRA measurement on different phases of the faulty

transformer. Fig. 6.17 shows the FRA result for phases b and c of the LV side. According to

this, the transformer was suspected to have winding deformation. Hence, the transformer

oil was drained and transformer tank was removed for internal inspection. Visual

inspection showed that the winding conductors of phase c in LV side have tilted. It was

107

-100

-80

-60

-40

-20

0

20

Frequency [Hz]

Magnitude [

dB

]

C (series)

100 x C (series)

200 x C (series)

Frequency Band : 4 MHz - 20 MHz

13.990 MHz13.770 MHz

13.560 MHz


124

highlighted earlier that conductor titling can in turn cause the series capacitance to change

significantly. In practice, the FRA trace has been deviated in mid frequencies for phase c

due to the titling, while the low and high frequencies are following normal oscillations (see

Fig. 6.17).

(a)

(b)

Figure 6.17. FRA measurement results of faulty transformer for phase b and c, (a) Entire FRA spectrum, (b)

Close-up view of dash-line rectangle in Fig. 6.17 (a).

102

103

104

105

106

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

Phase b (LV side)

Phase c (LV side)


105

106

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

Phase b (LV side)

Phase c (LV side)

Frequency Band: 15 kHz - 1.1 MHz

~ ---· I

L-:_-:_-:_· ______ _

~ ---· I

:::::::::::::::::::::J:::::c_ __ :::: __ _L::::L_ ______________ c_ ______ _. ______ c_ __ _. __ _. __ _. ____ _

l-:_-:_-:_·______ l


125

Figure 6.17(b) provides the close view of the FRA traces. According to simulation results,

changes in series capacitance will affect the mid-frequency band of FRA trace; practical

results illustrated in Fig. 6.17 verify the simulation results.

6.6.7 Resistance Variation (Simulation)

In order to study the impact of winding resistance (r) variation on the FRA spectrum, this

parameter was changed for 5 and 10 %, respectively, and the frequency response was

simulated for the model winding. Figure 6.18 shows the simulation results. According to

Fig. 6.18(a), variation in winding resistance can influence the very low and also very high

frequencies. Figures 6.18(b), (c) and (d) show the close-up views of those areas enclosed

by dash-line in 6.18(a). Other parts of the spectrum remain intact.

103

104

105

106

107

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Frequency [HZ]

Ma

gn

itu

de

[d

B]

r

1.05 x r

1.1 x r

Frequency Band : 20 Hz - 20 MHz

b

c

d

(a)


126

(b)

(c)

103

104

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency [HZ]

Ma

gn

itu

de

[d

B]

r

1.05 x r

1.1 x r

Frequency Band : 20 Hz - 45 kHz

107.27

107.271

-25

-20

-15

-10

-5

0

5

Frequency [Hz]

Magnitude [

dB

]

r

1.05 x r

1.1 x r

Frequency Band : 18.61 MHz - 18.68 MHz


127

(d)

Figure 6.18. FRA simulation results of winding due to winding resistance increment, (a) Entire FRA spectrum,

(b) Close-up view of region ‘b’ , (c) Close-up view of region ‘c’ , (d) Close-up view of region ‘d’.

At very low frequencies, the capacitive reactance due to the total capacitance of

transformer winding shows value high enough to be considered as open circuit. Therefore,

the winding impedance in this frequency range is more influenced by the inductive

reactance and winding resistance.

As the frequency increases, the magnitude of the inductive reactance becomes

considerable and thus winding resistance impact becomes negligible. Hence, winding

behaviour follows the inductive reactance trend. This occurred at 10 kHz in Fig. 6.18. In

fact, before 10 kHz the winding resistance value and its changes are significant as

compared to the winding inductive reactance. However, this is not so after 10 kHz.

At very high frequencies, the inductive reactance due to the total inductance of winding is

large enough to be considered as open circuit. Consequently, the capacitive reactance due

to total winding capacitance and the winding resistance are more dominant in FRA trace.

However, the capacitive reactance experiences small value as the frequency is significant

in this region. Therefore, the winding resistance and its variation become more significant

as depicted in Fig. 6.18.

6.6.8 Resistance Variation (Practical Measurement)

In order examine the winding resistance variation practically; the glassy transformer

model was taken as a test object. The AC resistance of the HV winding was measured and a

107.30101

107.30101

-85.3304

-85.3302

-85.33

-85.3298

-85.3296

-85.3294

-85.3292

-85.329

Frequency [Hz]

Magnitude [

dB

]

r

1.05 x r

1.1 x r

Frequency Band : 19.99 MHz - 20 MHz


128

resistor having 5 % value of the total HV winding resistance was connected to the winding

(series combination). This setup can model the winding resistance increment. Afterwards,

the frequency response of HV winding was re-measured. Similar experiment was

performed for the 10 % increment in the total winding resistance value using the same

approach. The results of measured frequency response are shown in Fig. 6.19.

According to Fig.6.19, practical measurements show a trend similar to what happened for

simulation results. In fact, very low and also very high frequencies have been influenced

and deviated due to the winding resistance variation. Comparing simulation and

measurement results, it is clear that measurement result verifies simulation result.

104

105

106

107

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Magnitude [

dB

]

r

~1.05 x r

~1.1 x r

Frequency Band: 2 kHz - 20 MHz

b

d

c

(a)


129

(b)

(c)

104

-7

-6

-5

-4

-3

-2

-1

Frequency [Hz]

Ma

gn

itu

de

[d

B]

r

~1.05 x r

~1.1 x r

Frequency Band: 2 kHz - 4.5 kHz

107.17

107.18

107.19

107.2

107.21

107.22

-23.5

-23

-22.5

-22

-21.5

-21

-20.5

-20

-19.5

-19

-18.5

Frequency [Hz]

Magnitude [

dB

]

r

~1.05 x r

~1.1 x r

Frequency Band: 14.60 MHz - 16.82 MHz


130

(d)

Figure 6.19. FRA measurement results of winding due to the resistance increment, (a) Entire FRA spectrum,

(b) Close-up view of region ‘b’ at very low frequencies, (c) Close-up view of region ‘c’ at high frequency

resonance, (d) Close-up view of region ‘d’ at very high frequencies.

6.6.9 Conductance to Ground (G) Variations (Simulation)

The insulation conductivity can be changed due changes of insulation parameters such as

the loss factor – also called the Dielectric Dissipation Factor (DDF). DDF is affected mainly

by temperature and moisture content of the insulation medium. Also, DDF in paper

insulation could be affected through the degree of polymerization (DP). In the case of oil

insulation, DDF might be influenced by oil contamination, viscosity, acidity, interfacial

tension, etc. All in all, increase in loss factor deteriorates the insulation quality and

changes the insulation conductance in the insulation model. The insulation system

becomes vulnerable due to significant conductance increment. As a rule of thumb, a

temperature increase of 10°C will halve the insulation resistivity and make insulation

conductivity two times greater ‎[105].

Based on this, the conductance to ground (G) of the transformer winding was changed and

the frequency response was simulated to realize its impact on FRA spectrum. Figure 6.20

shows the FRA simulation result, where the conductivity to ground for transformer

winding was changed to 5 and 10 times greater than the original value.

107.25

107.26

107.27

107.28

107.29

107.3

107.31

-32

-30

-28

-26

-24

-22

-20

Frequency [Hz]

Magnitude [

dB

]

r

~1.05 x r

~1.1 x r

Frequency Band: 17.5 MHz - 20 MHz


131

(a)

(b)

Figure 6.20. FRA Simulation results of winding due to the conductance to ground (G) increment, (a) Entire FRA

spectrum, (b) Close-up view of the first resonance point.

104

105

106

107

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Frequency [Hz]

Ma

gn

itu

de

[d

B]

G

5 x G

10 x G


106.61

106.63

106.65

106.67

106.69

106.71

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Frequency [Hz]

Ma

gnitu

de [

dB

]

G

5 x G

10 x G


4.664 MHz, 8.367 dB

4.664 MHz, -11.97 dB

~

"" "" " " " I

""' I I I " I ' I I -I - -

! ~ H ------- I / \ J J lJ L--------r------- - l

I J . _l l

rl ==~~~=~~~: I I I

I ". I -·-·-·-·•-·-·-·-··

~

L _____

// ~ \

I!'\ -~

I '\ / \.

/ "' ~ ~ :::::;::: ~ --


132

According to the simulation, changes of conductance (G) in the winding model can

influence the frequency response since it modifies the dielectric permittivity and the

losses. This causes resonance damping in the FRA trace. For instance in the current case,

the magnitude of the resonant peak in Fig. 6.20 (b) was changed from 8.367 dB to -11.970

dB while the resonance frequency remained unchanged.

6.6.10 Turn-to-Turn Conductance (g) Variations (Simulation)

Similar to the last subsection, the turn-to-turn conductance (g) was also examined for

values 5 and 10 times larger than the original conductance and frequency responses were

simulated. Fig. 6.21 illustrates the simulation results.

(a)

104

105

106

107

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Frequency [Hz]

Ma

gn

itu

de

[d

B]

g

5 x g

10 x g



133

(b)

Figure 6.21. FRA Simulation results of winding due to the turn-to-turn conductance (g) increment, (a) Entire

FRA spectrum, (b) Close-up view of the first resonance point.

According to Fig. 6.21, similar to (G), the change of turn-to-turn conductance (g) has

influenced the frequency response, since it modifies dielectric permittivity and the losses.

This in turn causes a slight magnitude reduction at the resonance points in the FRA trace.

The magnitude deviation in Fig.6.20 due to the conductance (G) variation was 20.337 dB,

while similar experiment for (g) led to 4.624 dB deviation in FRA spectrum magnitude.

The former is almost 5 times greater than the latter. This in turn reveals that FRA trace is

more influenced by the variation of conductance to ground (G) rather than turn-to-turn

conductance (g) alteration.

6.6.11 Conductance Variation (Practical Study)

Practical study on turn-to-turn (g) and turn-to-ground (G) conductance will be discussed

in detail in Chapter 7, where the influence of moisture and temperature changes is going

to be studied.

6.7 Conclusion

Study on axial and radial deformation was conducted analytically in this Chapter. It was

addressed that the transformer winding inductance will be changed due to the axial

deformation to winding disk. In addition, radial deformation will change the shunt

capacitance of the winding significantly. The analytical approach was completed through a

106.66

106.67

106.68

-20

-15

-10

-5

0

5

Frequency [Hz]

Mag

nitude [

dB

]

g

5 x g

10 x g


4.664 MHz, 3.737 dB

4.664 MHz, 8.362 dB


134

numerical example as well as simulation study using finite element method. A practical

study was also conducted to verify the analytical approach.

This Chapter also paid attention to the FRA trace deviation due to the winding parameters

changes. Simulation study was performed on the winding model presented in Chapter 4

and practical results were achieved on different test objects. Practical results have verified

the simulation studies. In summary, the influences of winding inductance, series and shunt

capacitances, resistance, conductance to ground and turn-to-turn conductance on FRA

trace were clarified in this Chapter.

Chapter 7. Temperature and Moisture Content Influences on FRA Signature

135

Chapter 7

Temperature and Moisture Content Influences on FRA Signature

7.1 Introduction

This Chapter is focused on the influence of temperature and moisture migration on the

FRA trace of transformer winding. It also discusses the feasibility of FRA capability in

moisture diffusion recognition in transformer paper insulation. To conduct this

investigation, the manufactured glassy model transformer and a 20/0.4 kV, 1.6 MVA three-

phase two windings transformer are used as test objects in the experiment.

At first, transformer water dynamic is discussed. After that frequency response

measurements on HV and LV windings of the test object at different moisture contents for

various temperatures are performed and deviation in resonance frequencies discussed.

Total capacitance deviation of transformer windings due to the moisture and temperature

changes is calculated using mathematical approach. In the meantime, Karl-Fischer

Titration (KFT) is utilized as a method to measure the Water Content of the Oil insulation

(WCO), and the Water Content of the Paper insulation (WCP) is then derived using MIT

equilibrium curve. In addition, Dielectric Dissipation Factor (DDF) is measured for each

and every stage. The result achieved through the model transformer is then examined on a

real (power-rated) transformer. Based on practical results, a hypothesis is proposed for

“the main reason of FRA trace deviation due to the moisture and temperature variation”.

This hypothesis is verified through simulation result. Finally, FRA statistical indicators are

calculated and discussed in detail, and possible solutions to distinguish insulation

characteristic impacts on FRA trace from winding deformation is provided.

7.2 Transformer Water Dynamic

Residual moisture in a transformer due to the water ingress through atmosphere,

insulation aging, cellulose decomposition or even after dry-out process will transfer from


136

the oil into the paper insulation and from the paper towards oil insulation in low and high

temperatures, respectively. Indeed moisture migration from one phase (liquid/solid) to

the other phase (solid/ liquid) could be due to the moisture concentration, temperature

and pressure gradients ‎[106]. Nevertheless, water dynamics in transformer can be

classified into transient and steady state periods ‎[107]. The transient period involves

moisture diffusion activity whereas water equilibrium between paper and oil insulations

is attained in the steady state. Fig. 7.1 shows the water dynamic in paper and oil

insulations for different temperatures.

Figure 7.1. Water dynamic in paper and oil insulation for different temperatures T1 and T2 (T1 < T2), WCO and WCP; t denotes the time.

7.2.1 Transient

In the case of moisture diffusion, Fick’s second law as one of the basis equations can be

expressed as:

2

2c c

C CD

t x

(7.1)

where D is the diffusion coefficient [m2/s], Cc is the concentration of substance (moisture)

[mol/m3] and x denotes substance movement position [m].

Temperature changes can lead to different activation energy for the molecules in adjacent

regions and ultimately moisture migration. Guidi and Fullerton have employed a diffusion

model to estimate moisture migration from the transformer paper insulation ‎[52], ‎[106]

and ‎[108]. They have specifically focused on power transformer drying time, temperature

and moisture adsorption rates for insulation when exposed to the atmosphere ‎[106]. Their

estimation is expressed by:


137

0

00

1 1

k

k c ET T

D D e

(7.2)

where c is the local moisture concentration [H2O/kg], Tk is the current temperature [°K], T0

is the reference temperature (298 °K), kʹ is a dimensionless parameter and is equal to 0.5

‎[109]-‎[110], D0 is a pre-exponential factor [m2/s] and E0 is the activation energy of the

diffusion process [kJ.mol-1]. D0 and E0 have been estimated for oil-free as 2.62×10-11 and

8140 and for oil-impregnated paper as 1.34×10-13 and 8074 by ‎[109]-‎[110], respectively.

Substitution of (7.2) into (7.1) will lead to (7.3) which can explain moisture migration as a

function of temperature. The solution of (7.3) is detailed in ‎[111]:

02

21 10.5 8140112.62 10 k

cT Tc c

C Ce

t x

(7.3)

7.2.2 Equilibrium

Water content equilibrium between paper and oil insulation has been widely discussed in

‎[110], ‎[112]-‎[115]. Equilibrium curves showing paper water content versus oil water

content for different temperatures have been achieved; hence, it is possible to determine

the value of one of them once knowing the other ‎[112]. Most of the above-mentioned

literatures have provided the equilibrium curves up to at most 100 ppm, a wider range of

equilibrium curve from 0 °C to 100 °C and moisture in oil up to 800 ppm was presented in

a work by Du et al at MIT university and named MIT oil-paper equilibrium curve ‎[110].

Therefore, due to the wide range the MIT equilibrium curves were employed in the

current study to derive the moisture content.

The diffusion time constant for moisture diffusing from one side of insulation is also

calculated as ‎[110]:

2

2,4 p

d

D

(7.4)

While, this equation for double-side diffusion through paper insulation is given by ‎[110]:

2

2.

pd

D

(7.5)

where τ is the diffusion time constant, and d represents the thickness of paper insulation

or pressboard.


138

7.3 Practical Study

7.3.1 Test Object and Setup

To conduct an accurate practical study on temperature and moisture influences on FRA

spectrum of winding, the glassy model transformer was used in the experiment. Detailed

information on this test object is provided in Appendix D as well as Chapter 4.

The test object is shown in Fig. 7.2. Transformer oil is required to be injected into the

glassy container. Hence, a drain valve was fitted onto the top plate to enable oil injection

and also taking oil sample.

Figure 7.2. Manufactured glassy air-core transformer (setup preparation to study temperature and moisture

impact).

Three different methods can be used for heating the test object: Low Frequency Heating

(LFH), oil circulation through an oil circulator, and using an electric oven. Among these,

the last approach was chosen for this study as it would be more accurate for controlling

the temperature. The test object was placed inside the oven and wiring connections for

FRA measurement were brought out through a bushing mounted on a small opening on

the oven top. The oven was equipped with a sensitive thermostat and a digitally controlled

heater to govern the internal temperature. Two thermocouples were used to monitor the


139

internal temperature of the oven as well as the test object. The FRA test setup then

remained unchanged for the entire experiment. FRA measurement was performed on a

winding by injecting a swept sinusoidal signal Vin at the line-lead, and detecting the

response Vout at the neutral-lead as shown in Fig. 7.3.

Figure 7.3. FRA test setup to examine temperature and moisture variation.

As discussed in Chapter 2, FRA measurement is performed in the frequency band 20Hz –

1MHz for transformers with highest voltage of > 72.5 kV, and in the range of 20Hz – 2MHz

for transformers with highest voltage of ≤ 72.5 kV ‎[1]. To be on the safe side, FRA

measurement is performed at least over the range 20Hz – 2MHz for all transformers

irrespective of their voltage rating. However, in the case of special transformers or

reactors, the upper limit may be shifted to higher frequencies. For instance with air-core

reactors, this limit could be even increased up to 20 MHz. In this study, the significant

value of the air-core magnetic reluctance of the test object results in a small self-

inductance for the windings. Small self-inductances will lead to small inductive reactances.

Therefore, the resonance frequencies in FRA traces for HV and LV windings would be

shifted considerably to higher frequencies. Hence, the upper band limits for FRA

measurements were extended from 2 MHz to 20 MHz to display entire oscillations similar

to what conducted for FRA measurement in Chapter 4.

7.3.2 Case Study 1 (‘Wet’ Model Transformer)

To study the temperature and moisture variation on FRA trace, at first the ‘wet’ test object

was examined. Initially, the transformer drain valve was opened and the test object was

deliberately left exposed to the laboratory ambient for two weeks so that the paper

insulation was saturated with moisture. The average readings of ambient temperature and


140

relative humidity were 23 °C and 26 %, respectively. Hence, the initial moisture content

for paper insulations was 4.1 %, calculated using the following data in Table 7.1 and also

equation (7.6) provided by Du et al ‎[109] on air relative humidity method.

Table 7.1. Ambient air relative humidity and moisture in paper (polynomial fitting parameters for various temperatures), taken and modified ‎[109].

30 °C 40 °C 50 °C 60 °C 70 °C

a0 (x 10-1) 2.4131270 1.6954583 1.0483257 1.3978572 0.7441865

a1 (x 10-1) 3.2828657 2.9079147 2.4316118 2.1359436 1.7762623

a2 (x 10-3) -14.929696 -11.950117 -7.2850779 -6.2300223 -2.7797731

a3 (x 10-4) 4.3831525 3.2448905 1.2731316 1.1731076 -0.27101029

a4 (x 10-6) -6.3395879 -4.2926236 -0.37397578 -0.57129397 2.2473555

a5 (x 10-8) 4.2446633 2.5228351 -1.1755019 -0.75286519 -3.3218692

a6 (x 10-11) -9.3468655 -3.8729882 9.5319144 7.4380470 16.252499

6 5 4 3 2

56 4 3 2 1 0( ) ( ) ( ) ( ) ( ) ( )WCP a RH a RH a RH a RH a RH a RH a (7.6)

where WCP is the moisture in paper in percent by weight and RH is the air relative

humidity in percent.

Afterwards, the transformer oil was dried-out (see appendix E), and the glassy tank was

filled with dry transformer oil (< 5ppm, at 70 °C) then left until the paper is fully

impregnated by oil, and eventually both of them reached equilibrium. The equilibrium

time was calculated as 244 hours using (7.2) and (7.4) at 23 °C. Thus, the oil sample was

taken from the container using a glassy syringe after 11 days (>244 hours). This is long

enough to ascertain that oil and paper insulation are in equilibrium.

The moisture content of the oil was measured using KFT method at 23 °C and moisture

content of the paper insulation was derived through MIT oil-paper equilibrium curves

‎[110], accordingly. These values were 11 ppm and 4 %, respectively.

Derived moisture contents of the paper insulation through air relative humidity and KFT

methods led to similar results (4.1 % and 4 %).

Next, the test object was heated up to 30 °C to be ready for the initial stage of FRA

measurement. FRA measurements were performed over the range of 30 to 90 °C in 10 °C

increments but the spectra were recorded for HV and LV windings at 30, 50, 70 and 90 °C.

The frequency response trace for HV winding was recorded over the range 20 Hz – 20

MHz when the LV winding terminals were left open circuit (end-to-end measurement).

Likewise, FRA measurement was also performed for the LV winding. On completion, the


141

oven temperature was increased gradually to reach 50 °C. The test object was left standing

for 25 hours (equilibrium time at 50 °C) to complete heat exchange between the test

object insulations and oven environment and also to pass the equilibrium time.

(a)

(b)

104

105

106

107

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

HV winding spectrum (at 30 °C )



Region 1 Region 2 Region 3

106

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]




HV winding spectrum (at 90 °C )1.223 MHz (at 30 °C)

1.209 MHz (at 50 °C)

1.190 MHz (at 70 °C)

1.184 MHz (at 90 °C)

1.705 MHz (at 90 °C)

1.802 MHz (at 90 °C)

2.540 MHz (at 90 °C)

1.733 MHz (at 70 °C)

1.742 MHz (at 50 °C)

1.761 MHz (at 30 °C)

2.552 MHz (at 70 °C)

2.594 MHz (at 50 °C)

2.623 MHz (at 30 °C)

1.861 MHz (at 30 °C)

1.841 MHz (at 50 °C)

1.832 MHz (at 70 °C)

__ . .,....- · -- -- ---fll-----


142

(c)

Figure 7.4. FRA spectra for ‘wet’ model transformer, HV side, (a) Entire trace for 30 and 90 °C, (b) Close-up view of region 1 shown by dash-line in Fig. 7.4(a), frequency band 800 kHz-3 MHz, (c) Close-up view of region 2 shown by dash-line in Fig. 7.4(a), frequency band 3 MHz – 10 MHz.

Then, frequency response traces for HV and LV windings were re-measured. Similar

experiments were done for 70 °C and 90 °C. FRA traces for HV and LV windings for

different temperatures are shown in Figures 7.4 and 7.5 over the frequency range of 5 kHz

– 20 MHz. The frequency response magnitude for the frequency range 20 Hz – 5 kHz was 0

dB and thus is not shown in Fig. 7.4 and Fig. 7.5. Figures 7.4(b), 7.4(c), 7.5(b) and 7.5(c)

show the close-up view of the measurement results.

Synchronous to FRA measurement, the moisture content of the oil insulation was

measured for all temperatures using KFT method. Moisture content recognition

techniques including KFT are described in Appendix F.

107

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]





4.178 MHz (at 90 °C)

4.199 MHz (at 70 °C)

4.267 MHz (at 50 °C)4.314 MHz (at 30 °C)

4.464 MHz (at 90 °C)

4.537 MHz (at 70 °C)

4.560 MHz (at 50 °C)

4.610 MHz (at 30 °C)

7.344 MHz (at 90 °C)

7.463 MHz (at 70 °C)

7.501 MHz (at 50 °C)

7.668 MHz (at 30 °C)

8.863 MHz (at 90 °C)

8.908 MHz (at 70 °C)

9.052 MHz (at 50 °C)

9.052 MHz (at 30 °C)


143

(a)

(b)

104

105

106

107

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

LV winding spectrum (at 30 °C)


Frequency Band: 5 KHz - 20 MHz

Region 2 Region 3Region 1

106

-60

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Frequency [Hz]

Ma

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0.951 MHz (at 30 °C)

0.946 MHz (at 50 °C)

0.941 MHz (at 70 °C)

0.928 MHz (at 90 °C)

1.122 MHz (at 30 °C)

1.117 MHz (at 50 °C)

1.099 MHz (at 70 °C)

1.096 MHz (at 90 °C)

~' I 1

,------

I.

~ i

J\ ~ ! .

""' -( i

" \ l ~ I ! i

' \ j l !

~ I

'! I I

++ + + + + + + +

!-·-·· .. . . -- ---- ---- 1

! !-· .. ··

I

L--·-·-· - ·r - ·-·-·- · + + + · - · - · - ·

! · - · · -·-· - · .J

c____


144

(c)

Figure 7.5. FRA spectra for ‘wet’ model transformer, LV side, (a) Entire trace for 30 and 90 °C, (b) Close-up view of region 1 shown by dash-line in Fig. 7.5(a), frequency band 500 kHz - 3.5 MHz, (c) Close-up view of region 2 shown by dash-line in Fig. 7.5(a), frequency band 3.5 MHz – 10 MHz.

Water content of the paper insulation was then derived through MIT equilibrium curves

for each and every temperature, independently.

The majority of the insulation system for manufactured test object was between HV and

LV windings including Kraft paper, oil canal, spacers and pressboard. Hence parallel to the

other measurements, the DDF values between HV and LV windings were measured at the

power frequency (50 Hz) for different temperatures to quantify the quality of insulation.

To avoid any kind of flash-over in the test object, the maximum applied voltage for DDF

measurement was kept low at 5 kV.

7.3.3 Discussion 1

According to Figures 7.4 and 7.5, when examining from low frequencies to 800 kHz all of

the traces are perfectly matched. Moving from 800 kHz to higher frequencies, the

discrepancy between the traces becomes obvious.

As the test object temperature changed from 30 to 90 °C, the FRA traces have slightly

shifted to lower frequencies. In the case of 90 °C, this movement seems to be more

significant. Some of the resonances and anti-resonances have been highlighted in Figures

7.4(b), 7.4(c), 7.5(b) and 7.5(c).

107

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Frequency [Hz]

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4.464 MHz (at 90 °C)

4.528 MHz (at 70 °C) 4.553 MHz (at 50 °C)

4.624 MHz (at 30 °C)

6.817 MHz (at 70 °C) 6.855 MHz (at 50 °C)

6.963 MHz (at 30 °C)

7.676 MHz (at 90 °C)

7.785 MHz (at 70 °C)

7.915 MHz (at 50 °C)

8.039 MHz (at 30 °C)

4.453 MHz (at 30 °C)

4.404 MHz (at 50 °C)

4.380 MHz (at 70 °C)6.722 MHz (at 90 °C)

4.367 MHz (at 90 °C)


145

Detailed investigation of Figures 7.4 and 7.5 shows that the oscillations trend of the

spectra seems to be similar, while the resonance frequencies have shifted to lower

frequencies as the temperature increased and equilibrium time passed.

In the meantime, some of the resonant magnitudes are reduced. This deviation comes

certainly through windings’ inductance, total capacitance, resistance or insulation

conductance changes. From a mathematical point of view, resonances and anti-resonances

in FRA trace can be generated due to the interaction between inductive and capacitive

reactances. As frequency increases, each resonance point indicates the changing from

capacitive towards inductive behaviour, while the anti-resonance point shows the turning

from inductive to capacitive behaviour of the winding impedance. Hence, every resonance

or anti-resonance in FRA trace can be explained through (7.7) if it is considered

independently:

1

2if

L Ci i

(7.7)

where fi is the ith resonance frequency, Li and Ci are the inductance and capacitance

(involving series and shunt capacitances) at ith resonance frequency. According to (7.7),

resonance frequencies in FRA trace would be changed if the inductance or total

capacitance is changed. Thus to clarify whether inductance variation due to the

temperature changes can influence FRA trace, the windings’ inductances were measured

for some frequencies and given in Tables 7.2 and 7.3 at 30 and 90 °C for HV and LV

windings, respectively. These tables also provide the measured values for windings’

resistances, inductive reactances and impedances.

According to Table 7.2 and Table 7.3, the maximum inductance deviation due to the

temperatures changes is less than 1.1 % for the entire measurement. Hence, it has

insignificant impact on FRA resonant peaks; therefore, deviation of resonant points in FRA

trace seems to be coming through changes in the total capacitance.

Total capacitance variation could be influenced by series and/or shunt capacitances

alteration.


146

Table 7.2. HV winding electrical parameters for 30 and 90 °C.

Parameter R X Z θ∡ L

100 Hz (30 °C) 0.087 Ω 0.444 Ω 0.453 Ω 78.91 707.90 μH

100 Hz (90 °C) 0.104 Ω 0.449 Ω 0.461 Ω 76.95 715.30 μH

Change 19.54 (%) 1.12 (%) 1.76 (%) 2.48 1.04 (%)

1 kHz (30 °C) 0.333 Ω 4.13 Ω 4.14 Ω 85.39 657.43 μH

1 kHz (90 °C) 0.347 Ω 4.17 Ω 4.19 Ω 85.24 664.57 μH

Change 4.20 (%) 0.96 (%) 1.20 (%) 0.17 1.08 (%)

100 kHz (30 °C) 11.65 Ω 337.76 Ω 337.95 Ω 88.02 537.55 μH

100 kHz (90 °C) 13.12 Ω 338.57 Ω 338.82 Ω 87.78 538.85 μH

Change 13.49 (%) 0.23 (%) 0.25 (%) 0.27 0.24 (%)

Table 7.3. LV winding electrical parameters for 30 and 90 °C.

Parameter R X Z θ∡ L

100 Hz (30 °C) 0.056 Ω 0.174 Ω 0.183 Ω 72.15 278.3 μH

100 Hz (90 °C) 0.067 Ω 0.176 Ω 0.188 Ω 69.15 280.3 μH

Change 19.64 (%) 1.14 (%) 2.73 (%) 4.15 0.71 (%)

1 kHz (30 °C) 0.114 Ω 1.67 Ω 1.68 Ω 86.09 267.25 μH

1 kHz (90 °C) 0.123 Ω 1.69 Ω 1.69 Ω 85.84 268.97 μH

Change 7.89 (%) 1.19 (%) 0.59 (%) 0.29 0.64 (%)

100 kHz (30 °C) 2.94 Ω 148.55 Ω 148.57 Ω 88.86 236.41 μH

100 kHz (90 °C) 3.33 Ω 148.93 Ω 148.97 Ω 88.72 237.03 μH

Change 13.26 (%) 0.25 (%) 0.27 (%) 0.15 0.26 (%)

Technically speaking, series and shunt capacitance can be altered due to the moisture

migration. As a hypothesis, the shunt capacitance is more influenced by transformer oil

insulation and thus it could be more significant in changing the total capacitance when

temperature and moisture change. This hypothesis is verified through simulation result in

Section 7.4.

Tables 7.4 and 7. 5 provide detailed information of deviated frequencies for FRA traces at

30 and 90 °C where f1 and f2 represent the identical frequencies of the anti-resonances and

resonances for HV and LV windings at 30 and 90 °C respectively. In addition, total

capacitance deviation of transformer windings for 30 and 90 °C have been calculated and

shown in the last column using (7.7).

According to Tables 7.4 and 7.5, average capacitance deviations for HV and LV windings

were calculated as 6.24 % and 6.20 %, respectively. Figure 7.6 shows the total capacitance

deviation of HV and LV windings for each 20 °C temperature increment. In Fig. 7.6, when


147

the temperature increases, the rising trend in capacitance deviation for both windings is

obvious due to the moisture migration from the paper insulation towards oil insulation.

Figure 7.6. Deviation of total capacitance for HV and LV windings (average deviation at 90 °C is 6.22 %).

Table 7.4. HV winding capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.4(b) and Fig. 7.4(c).

Frequency f1(30°C) f2(90°C) C2/C1=( f1/f2 )2 ΔC%

1st minimum 1.223 MHz 1.184 MHz 1.0669 6.7 %

1st maximum 1.761 MHz 1.705 MHz 1.0667 6.7 %

2nd minimum 1.861 MHz 1.802 MHz 1.0665 6.7 %

2nd maximum 2.623 MHz 2.540 MHz 1.0664 6.6 %

3 rd maximum 4.314 MHz 4.178 MHz 1.0661 6.6 %

3rd minimum 4.610 MHz 4.464 MHz 1.0664 6.6 %

4 th minimum 7.668 MHz 7.344 MHz 1.0901 9.0 %


6th minimum 10.810 MHz 10.580 MHz 1.0439 4.4 %

7th minimum 13.630 MHz 13.200 MHz 1.0662 6.6 %

4 th maximum 16.090 MHz 15.750 MHz 1.0436 4.4 %

The moisture content of the oil and paper insulations for each and every temperature was

recorded and shown in Fig. 7.7. Moisture migration from paper into the oil is clearly

obvious in this figure. The water content of the oil has increased from 18 ppm at 30 °C to

95 ppm at 90 °C.

20 30 40 50 60 70 80 90

0

1

2

3

4

5

6

Temperature ( °C )

De

via

tio

n (

%)

LV winding

HV winding

HV 2.43 %

LV 2.55 %

HV 6.24 %

LV 4.59 %

HV 4.12 %

LV 6.20 %


148

Table 7.5. LV winding capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.5(b) and Fig. 7.5(c).






3 rd maximum 6.963 MHz 6.722 MHz 1.0729 7.2 %



Figure 7.7. Moisture content of oil and paper (wet model transformer).

In addition, the water content in paper insulation has decreased from 3.6 % at 30 °C to 1.7

% at 90 °C. In fact, changes of 1.9 % moisture content in paper insulation results in almost

6.22 % alteration in total capacitance of the windings at mid and high frequencies. This in

turn ultimately caused FRA spectrum deviation for HV and LV windings, respectively.

Additionally, based on Tables 7.2 and 7.3, the maximum deviation for the resistance of the

windings is 19.54 %, while this value is negligible for inductive reactance alteration. The

winding resistance variation can certainly influence the FRA trace magnitude in very low

frequencies, even though, when it comes to impedance and combine to the inductive

reactance its influence would be insignificant. Increasing the temperature will certainly

cause the dielectric loss in paper and oil insulations to increase. Dielectric loss variations

20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

100

Temperature ( °C )

WC

O (

pp

m)

Water content in oil (ppm)

Water content in paper (%) 4.5

3.5

2.5

1.5

0.5

1.7 %1.8 %

2.0 %

52.5 ppm

63 ppm

95 ppm

35 ppm

25 ppm21 ppm

18 ppm

13 ppm

15 ppm

WCP (%)

4%

3.8%

3.6%

3.0%

2.6 %

2.2 %


149

will marginally influence the magnitude of resonant peaks. Hence, some of the resonant

peaks in Figures 7.4 and 7.5 are only slightly damped due to the temperature increase.

Considerable dielectric loss will increase attenuation of the peaks. This in turn verifies the

simulation results on conductance variations in Chapter 6 (see subsections 6.6.9, 6.6.10

and 6.6.11).

Study on frequency response spectra when the temperature decreases from 90 to 30 °C

led to similar results. Although moisture absorption for paper insulation is different with

desorption, the equilibria should be the same and hence similar spectra were observed in

the reverse procedure.

The temperature and moisture influences on FRA trace for the ‘wet’ model transformer

(WCO 4%) were discussed. In order to realize the influence of initial moisture content

value on FRA spectrum deviation, the ‘dry’ model transformer is studied in the next

subsection and their results are compared.

7.3.4 Case Study 2 (‘Dry’ Model Transformer)

To study the impact of the temperature and moisture variation on FRA trace for dry

transformer winding, the oil was drained and the model transformer was heated up to 90

°C in the electric oven and then vacuumed (less than 750 mTorr) to remove the moisture

content from the paper insulation.

Figure 7.8 shows the vacuum process. After that, the vacuum was broken through a three-

way valve and then dry transformer oil was injected into the container once more. The

moisture content of the oil was measured after 11 days through KFT method and gave a

result of 3 ppm at 23 °C. The test object was heated up to 30 °C same as before to prepare

for frequency response measurement.

Figure 7.8. Vacuum process of the model transformer.


150

The FRA spectra for HV and LV windings were then measured at 30 and 90 °C over the

frequency range of 20 Hz – 20 MHz (end-to-end measurement), similar to the last

experiment. Also WCO, WCP and DDF at power frequency were measured as in the last

experiment. Figures 7.9 and 7.10 show the frequency response spectra just for 30 and 90

°C.

Figure 7.9. FRA spectra for ‘dry’ model transformer, HV side.

Figure 7.10. FRA spectra for ‘dry’ model transformer, LV side.

106

107

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Frequency [Hz]

Ma

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HV winding spectrum (at 30 ºC), After dry-out

HV winding spectrum (at 90 ºC), After dry-out1.289 MHz (at 90 ºC)

1.857 MHz (at 90 ºC)

1.963 MHz (at 90 ºC)

4.549 MHz (at 90 ºC)

9.651 MHz (at 90 ºC)

1.879 MHz (at 30 ºC)

4.604 MHz (at 30 ºC)

9.768 MHz (at 30 ºC)

1.305 MHz (at 30 ºC)

Frequency Band: 900 kHz- 20 MHz

1.986 MHz (at 30 ºC)

106

107

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Frequency [Hz]

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LV winding spectrum (at 30 ºC), After dry-out

LV winding spectrum (at 90 ºC), After dry-outFrequency Band: 600 kHz - 20 MHz

1.011 MHz (at 30 ºC)

1.194 MHz (at 30 ºC)

1.182 MHz (at 90 ºC)

4.865 MHz (at 30 ºC)

4.816 MHz (at 90 ºC)

7.407 MHz (at 30 ºC)

1.001 MHz (at 90 ºC)

7.332 MHz (at 90 ºC)


151

Figure 7.11. Moisture content of oil and paper (dry model transformer).

Moisture content migration in different temperatures in oil and paper insulations is

illustrated in Fig. 7.11. Measurements of electrical parameters for dry type study of HV and

LV windings resulted in values similar to those in Tables 7.2 and 7.3.

Table 7.6. HV winding capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.9.






Table 7.7. LV winding capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.10.






7.3.5 Discussion 2

It is noteworthy to mention that as a result of the dry-out process, all resonance

frequencies in Fig. 7.4 and 7.5 (‘wet’ model transformer) are shifted to higher frequencies

20 30 40 50 60 70 80 900

5

10

15

20

25

Temperature (°C)

WC

O (

pp

m)

Water content in oil (ppm), After dry-out

Water content in paper (%), After dry-out

1.5

1

0.50.4 %

0.5 %0.6 %

10.3 ppm

16 ppm

21 ppmWCP (%)

0.8 %

0.9 %

1.5 %

1.3 %

1.1 %

1.0 %

3.0 ppm3.0 ppm

3.0 ppm 3.0 ppm

5.0 ppm

8.0 ppm


152

in Fig. 7.9 and Fig. 7.10 (‘dry’ model transformer), and the test object moisture content has

changed from 3.6 % to 1.1 % at 30 °C. For instance, the first resonance frequency in Fig.

7.4(b) has moved from 1.223 MHz to 1.305 MHz in Fig. 7.9. Such a shift can be interpreted

through total capacitance changes of the test object due to the dry-out process which will

be discussed in the last Section of this Chapter.

Further examination of the FRA traces in Fig. 7.9 and Fig. 7.10 reveal that the minimum

and maximum resonance frequencies have moved to lower frequencies as the

temperature increases. This movement is less than the movement of the ‘wet’ test object

spectra for 30 and 90 °C. Tables 7.6 and 7.7 provide detailed information of deviated

frequencies for FRA traces at 30 and 90 °C and also total capacitance variation for some of

the identical frequencies of HV and LV winding traces, respectively.

Based on the values reported in Tables 7.6 and 7.7, average capacitance deviations for HV

and LV windings were calculated as 2.37 % and 2.0 %, respectively. In addition, Fig. 7.11

shows that as the temperature increases the moisture content of the paper insulation is

changed from 1.1 % at 30 °C toward 0.4 % at 90 °C. Also, the oil moisture content has

increased from 3 to 21 ppm at 30 and 90 °C, respectively.

Technically speaking, from 30 to 90 °C the reduction of 0.7 % moisture content in paper

insulation caused around 2.18 % deviation in the total capacitance value for the ‘dry’ test

object, while these values were 1.9 % (WCP) and 6.22 % for the ‘wet’ test object.

Based on the result achieved from study of the test object under different moisture

contents, it is possible to calculate the total capacitance variation through FRA spectrum

deviation for 0.5 % moisture diffusion in paper insulation using linear interpolation (see

Table 7.8).

Table 7.8. FRA deviation and total capacitance variation for 0.5 % WCP change.

Model Transformer WCP Change FRA Spectrum Deviation (FSD) ΔC%

Wet 0.5 % 0.81 % 1.63 %

Dry 0.5 % 0.77 % 1.55 %

According to Table 7.8, each 0.5 % moisture migration from the paper insulation can cause

0.79 % displacement in FRA spectrum (as an average value for FSD) towards higher or

lower frequencies. In fact, 0.5 % moisture migration from the paper into the oil insulation

will lead to FRA spectrum moving to lower frequencies, while the same value of moisture

migration from the oil to paper insulation results in 0.79 % FRA movement to higher

frequencies.


153

DDF for wet and dry test objects as a factor showing the insulation quality was measured

and compared to the other results (see Table 7.9). When examining this table, it is obvious

that the DDF values for wet and dry test objects are completely different. In addition, as

the DDF increases the FRA trace is shifting to lower frequencies, and vice versa.

Table 7.9. Measured DDF at 5 kV before and after dry-out process.

Temperature 30 °C 40 °C 50 °C 60 °C 70 °C 80 °C 90 °C

‘Wet’ test

object 0.0203 0.0319 0.0484 0.0721 0.1119 0.1858 0.2849

‘Dry’ test

object 0.0028 0.0029 0.0044 0.0069 0.0088 0.0147 0.0210

7.3.6 Case Study 3 (Three-Phase Transformer)

To examine the above-mentioned result, a three-phase two-winding core type 20/0.4 kV,

1.6 MVA transformer was taken as another test object. The windings of this transformer

were homogenous conventional disk type. It was a spare transformer which was never in

service and had been kept in stock as reserve for a long time. At first, the oil sample was

taken from the transformer sampling valve and WCO and WCP were derived using KFT

method and MIT equilibrium curves, respectively (the oil temperature was 10 °C). These

values were 3 ppm and 3 %, respectively. Then, frequency response traces for HV

windings were recorded. To heat up the transformer in order to study the temperature

and moisture impacts, the secondary side of the test object was short-circuited and the

voltage was increased through the primary side to reach the nominal current on the

secondary side. It took 8 hours to reach to 60 °C for the test object. After that, the test

object was left for 48 hours under this condition to reach moisture equilibrium between

oil and paper insulations. The oil sample was then taken and FRA spectra were re-

recorded. WCO achieved was 26 ppm and WCP was derived as 1.9 %. Figure 7.12 shows

the FRA spectra for phase U on HV side at 10 and 60 °C. In addition, Table 7.10 provides

detailed information of the deviated frequencies in Fig. 7.12.

Based on the results, the total moisture variation for the paper insulation of the

transformer is calculated as 1.1 % from 10 to 60 °C. This variation has caused almost 1.5

% alteration in FRA trace and 3.22 % capacitance changes. Based on this ratio, it seems not

far from the fact that 1.49 % deviation in total capacitance is estimated for 0.5 % moisture

migration from the paper insulation in this transformer. This in turn almost verifies the

data achieved in Table 7.8.


154

102

103

104

105

106

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-60

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Frequency [Hz]

Ma

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HV winding spectrum (at 60 °C)



642.80 Hz at (at 10 °C)635.10 Hz at (at 60 °C)

46.78 kHz at (at 60 °C)

47.55 kHz at (at 10 °C)

37.09 kHz at (at 60 °C)

37.66 kHz at (at 10 °C)

15.98 kHz at (at 60 °C)

16.22 kHz at (at 10 °C)

52.83 kHz at (at 60 °C)

53.67 kHz at (at 10 °C)

103

-84

-82

-80

-78

-76

-74

-72

-70

-68

-66

-64

Frequency [Hz]

Ma

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[d

B]

Figure 7.12. HV winding spectra at 10 and 60 °C (1.6 MVA transformer).

Table 7.10. HV winding capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig.7.12.

Frequency f1(30°C) f2(90°C) f1/f2 C2/C1=( f1/f2 )2 ΔC%

1st minimum 642.80 Hz 635.10 Hz 1.0121 1.0243 2.4 %

1st maximum 37.66 kHz 37.09 kHz 1.0153 1.0427 4.2 %

2nd minimum 47.55 kHz 46.78 kHz 1.0164 1.0330 3.3 %

3rd maximum 16.22 kHz 15.98 kHz 1.0150 1.0302 3.0 %

2nd maximum 53.67 kHz 52.83 kHz 1.0159 1.0320 3.2 %

7.4 Verification of Practical Results Using Modelling and Simulation

A mathematical model using travelling wave theory had been developed in Chapter 4. This

model was verified through practical measurement in the same Chapter, and the transfer

function was given by (4.16).

This model is utilized in the current Chapter to examine the main reason of the FRA

spectrum deviation due to changes in the insulation characteristics.

On one hand, to verify the practical measurement through the theoretical approach and

estimate the main reason for a change in FRA response due to moisture and temperature

variations, each and every fundamental quantity controlling the FRA spectrum should be

studied. On the other hand, how to distinguish the winding inductance and capacitance and

resistance from one impedance measurement at specific frequency and ultimately through

the transfer function in frequency domain would be a challenge.


155

In fact, changes of FRA spectrum due to temperature and moisture variations are influenced

through changes of the inductance, capacitance and resistance in the winding impedance. In

this work, a simulation on frequency response of the model winding showed that the

winding resistance variation will lead to insignificant deviation in very low and high

frequencies in the FRA spectrum, while the resonant peak as well as spectral shape

remained unchanged (see Fig. 6.18). This work was verified through a practical study on

changes of FRA setup connections. Indeed, a bad, loose or oxidized connection in clamping

leads of the FRA test setup will change the resistivity of the setup, and this could be a model

of winding resistance variation for the winding under test. Frequency response

measurement on this case indicated similar results as to what happened in simulation study.

Very low and high frequencies were changed, while the resonant peaks remained

unchanged (see Fig. 6.19).

In addition, the results provided in Tables 7.2 and 7.3 on resistance variation showed that

the winding resistance alteration in various temperatures is insignificant in relation to the

impedance (combined with reactance). Hence, this hypothesis that FRA spectra are

influenced by winding resistance lost ground. Therefore, the remained reactance within

measured impedance in frequency response has almost certainly affected through the

inductance and capacitance.

The inductance alteration through the temperature variation was insignificant in Tables 7.2

and 7.3. On the other hand, it was shown that the low-frequency band of FRA spectrum is

influenced by the winding inductance (self- and mutual-inductance). Careful examination of

Figures 7.4(a) and 7.5(a) reveals that despite temperature and moisture variation, the low-

frequency band of spectra which is quite relevant to inductance has not deviated from the

origin. Also, shifting of local resonances has occurred in the mid- and high-frequency bands

in both graphs. This suggests that inductance deviation in this circumstance is negligible.

Investigation on resistance and inductance suggest that FRA spectrum deviation due to the

temperature and moisture variation can be initiated by capacitance variation.

In fact, in a capacitance model of insulation system, the current densities of bounded ions Jε

and free ions Jσ traversing a medium are obtained as:

0 ,r

DJ j E J E

t

(7.8)

where is the electric displacement, εr denotes relative permittivity of dielectric, ε0 is the

absolute permittivity of vacuum, is the electric field and σ designates dielectric

conductivity. The total current density traversing medium is then given by:


156

0 0

0( ) ( )r rE j E j j E

(7.9)

χ is the electric susceptibility. Then, the complex permittivity is obtained as:

*

0 0 0

( )r r

Cj j

C

(7.10)

and current traversing the medium is given by:

0( )insul insul

I j C j U (7.11)

where Uinsul is the voltage across the insulator, εʹ is the real part of the complex permittivity

related to the stored energy in the medium, εʺ is the imaginary part of the complex

permittivity related to the dissipation (or loss) of energy within the medium, C(ω) is the

complex capacitance with the dielectric present, and C0 denotes the capacitance without the

dielectric.

Therefore capacitance behaviour comes directly through dielectric behaviour, and

specifically dependent on dielectric permittivity and conductivity. Any change in these

factors will be reflected as a kind of change in capacitance. In addition, dielectric behaviour

usually depends on the variation of εʹ and εʺ with frequency, composition, temperature, and

voltage. In this study, the voltage remained unchanged for entire experiments; thus, just the

changes in frequency, temperature and composition need to be taken into consideration.

In the case of frequency and its variation, the real part of the dielectric permittivity is given

by ‎[116]:

0

2 21

(7.12)

and the imaginary part (loss factor) is given by ‎[116]:

0

2 2

( )

1

(7.13)

where ε∞ and έ0 are the infinite-frequency dielectric constant and static dielectric constant

respectively, and τ is the dielectric time constant.

The real part of complex permittivity of dielectric to which (7.12) applies should always

decrease with frequency to reach ε∞, while εʺ tends to zero when ω approaches infinity. In


157

fact, εʹ approaches limiting values and εʺ experiences maximum value at ωτ=1 and negligible

values at ω=0 or ω=∞.

Hence, effective frequency band on εʺ is completely related to the dielectric time constant τ.

For an equal thickness of oil-paper composite insulations in transformer, τ is given by

(7.14):

paper oil

paper oil

(7.14)

where, εpaper and εoil are the paper and oil insulation permittivity, and σpaper and σoil are the

paper and oil insulation conductivity, respectively.

For study at high frequencies (1 MHz- 20 MHz) similar to what happened in this study and

shown in Figures 7.4(a) and 7.5(a); the changes in the direction of the applied field are so

fast that the ions do not have to time to move an appreciable distance from their equilibrium

position before the direction of applied field is reversed ‎[116]. This could significantly

influence the contribution of ions (τ) and subsequently εʺ at high frequencies in the

dielectric structure. Thus, from frequency point of view, the impact of εʺ seems to be

insignificant in practice and εʹ tends to a determined value (ε∞).

Apart from frequency variation, temperature has its own impact on the real and imaginary

parts of the complex permittivity, whereas moisture content variation could be classified

into the composition effect.

In this regard, the study by Abeywichrama et al ‎[16] on transformer pressboard shows that

at a specific frequency if the temperature increases, then εʹ and εʺ will be increased.

Furthermore, if the moisture content decreases, then εʹ and εʺ will be decreased.

Another study by the same authors ‎[117] explains that the moisture variation mostly

influences the real part of permittivity εʹ, and the temperature affects mostly imaginary part

εʺ. In ‎[117], it is also discussed that the changes of εʺ due to the moisture variation is

considerable for the frequency below 100 Hz, while this is insignificant at frequencies higher

than 10 kHz. In addition as far as moisture increases, εʹ will increase significantly below 1

Hz, whereas weak increase can be observed above 100 Hz for this parameter ‎[117]. A

careful study on moisture variation in pressboard has been also reported in ‎[118] from an

original work by ‎[119]. It has been shown that moisture increment in the pressboard will

increase εʹ and εʺ specifically at low frequencies.

It is worth noting that the entire frequency band for study conducted in ‎[16] and ‎[117] is

100 Hz to 1 MHz, and in ‎[118] is 0.1 mHz to 1 kHz.


158

In our study, since temperature is changed from a lower value (30 °C) to reach a higher

value (90 °C), the moisture migrates from the paper into the oil insulation. In fact, when the

water in paper insulation decreases, the oil water content increases and it is not practically

possible to separate these reactions for interpretation. Furthermore, the separation of εʹ and

εʺ would be another challenge. Therefore, the introduced model in ‎[118] for composite

dielectric permittivity in transformer is used to address the issue.

In transformer design, the major part of the oil/paper insulation system is mostly

concentrated between the concentric HV and LV windings. Hence, significant amount of

water is available in this area. To study the transformer water content, this area is crucial

and should be investigated precisely.

Coil Cspacer

Cpaper

Cpaper

Oil

Paper

Paper

a1-a

b/2

b/2

1-b

Spacer

Figure 7.13. Schematic of the paper content and the spacer coverage in the insulation duct, taken and modified

[118].

In [118], it has been stated that for modelling purposes of this part it is sufficient to

represent the insulation structure by the relative amount of spacers and barriers in the duct

as it is depicted in Fig. 7.13. The major water content is absorbed by insulation, namely

pressboard barriers. If the thickness of pressboard barriers is thin enough to be close to

paper insulation thickness; then, it could be argued that instead of pressboard barriers the

paper permittivity plays major role in the model (see Fig. 7.13). Otherwise, paper

permittivity should be replaced by barrier permittivity in (7.15) and Cpaper should be

changed by Cbarrier in Fig. 7.13. With this assumption, the composite dielectric permittivity is

given by [118]:

1( , )

1 1

spacer paper paperoil

a aT

b b b b

(7.15)

where εspacer denotes spacer permittivity and it is slightly higher than paper permittivity

εpaper, and a and b are shown in Fig. 7.13.


159

Since accurate measurement of the real and imaginary parts of the complex permittivity at

very high frequencies (f > 1 MHz) is not available, it has been assumed that εʹ and εʺ

behave as in low frequencies (f < 1 MHz). Knowing that when temperature and moisture

are changed, deviation of εʹ and εʺ from the origin under such experiment (f > 1 MHz) are

considerably less than that experienced at low frequencies (f < 1 MHz).

Based on this, if assuming the temperature of the test object increases, the moisture content

in paper insulation decreases and the oil water content increases accordingly. Hence, εoil will

take a greater value and εpaper and εspacer will experience less values. On the other hand εoil <

εpaper; therefore under such circumstance, ε in (7.15) becomes larger. This increment could

be even intensified when the oil insulation thickness is greater than paper insulation within

the composite insulation medium.

All in all, increment of ε could almost certainly influence the capacitance and conductance.

Therefore, the conductance and capacitance were changed in the model and the frequency

responses were simulated.

In the case of conductance variation, simulation results showed that the conductance

increment for both turn-to-turn and shunt-to-ground conductance causes slight damping in

resonance peak magnitudes, while resonant frequencies and FRA spectrum trend remained

unchanged. This impact was more significant in shunt conductance variation (G) rather than

turn-to-turn conductance (g).

In the case of capacitance variation, at first the reference FRA spectrum was simulated for

the HV winding. Then the relative permittivity in shunt and series capacitances was changed

gradually and different spectra were simulated and compared to the reference spectrum.

Simulation results showed that a slight change (around 6.2 %) in series capacitance does not

cause considerable deviation in FRA spectrum, while significant change (around 200 %) will

slightly move the FRA trace in the mid and high frequencies. This effect became more

significant when the shunt capacitance was changed. The increment of just 6.2 % of the

relative permittivity of the shunt capacitance caused the FRA resonant peaks moved to

lower frequencies for 3 % (see Fig. 7.14).

These results in turn suggest that FRA spectrum deviation due to the temperature and

moisture variation comes primarily through the shunt capacitance rather than series

capacitance. Therefore, the right hand side columns in Tables 7.4, 7.5, 7.6 and 7.7 are

focused on the shunt capacitance variation rather than series capacitance. More precise

study on this case could be conducted through εʹ and εʺ measurement at high frequencies (f

> 1 MHz).


160

Figure 7.14. Simulated FRA spectra for model transformer (shunt capacitance deviation).

7.5 Influence of Temperature and Moisture Content on FRA Statistical

Indicators

It was discussed in Chapter 2 that a common method of interpreting FRA data is to use

statistical indicators (indices), particularly the correlation coefficient (CC) and standard

deviation (SD). In order to evaluate the capability of these indicators in FRA spectrum

interpretation due to temperature and moisture variation, all measured results in the

above case studies were examined by CC and SD. Calculated values of CC and SD for the

FRA spectra of the model transformer under different moisture contents (‘wet’ and ‘dry’)

at 30 and 90 °C as well as the power-rated transformer at 10 and 60 °C are given in Table

7.11. It can be seen that CC and SD indicate deformation for the model transformer (for

both ‘wet’ and ‘dry’ cases) but in reality, its windings have not deformed (see Table 2.6 for

criteria). On the other hand, results for the power-rated transformer indicate that its

condition is ‘normal’.

Table 7.11. Calculated statistical indices.

Test Object CC SD

‘Wet’ Model Transformer, LV Spectra (at 30 °C and 90 °C) 0.9967 1.4989

‘Wet’ Model Transformer, HV Spectra (at 30 °C and 90 °C) 0.9974 1.5258

‘Dry’ Model Transformer, LV Spectra (at 30 °C and 90 °C) 0.9967 1.4989

‘Dry’ Model Transformer, HV Spectra (at 30 °C and 90 °C) 0.9991 0.8841

1.6 MVA Real Transformer, HV Spectra (at 10 °C and 60 °C) 0.9999 0.6000

106

107

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

Cg

1.062 x Cg

3 % deviation in resonant peak

4.376 MHz 4.510 MHz


161

This study reveals that statistical indices (CC and SD) do not have enough accuracy in FRA

spectrum interpretation when FRA traces have been taken at different temperature and

moisture contents. Hence, the existing available statistical indices need to be modified.

7.6 Practical Solution to Modify Statistical Indicators

According to results in the last Section, to obtain the maximum accuracy in transformer

frequency response evaluation, the statistical indicators need to be modified and the

moisture and temperature variations are brought into consideration. In fact how to

distinguish the temperature and humidity impacts on the FRA spectrum from the winding

deformation is the challenge.

A study by Ryder in ‎[28] did not focus on the moisture migration and its impact on the

FRA spectrum; however, as a caution he advised using inter-phase comparisons for those

cases suspected of experiencing relative humidity changes. It means that to recognize the

relative humidity changes in transformer FRA signature, the FRA spectra recorded for

different phases on HV side can be compared together and their discrepancy is taken as

the influence of humidity changes. Similar procedure can be performed for LV side.

However, it was emphasized that this suggestion is feasible only for three-phase

transformers ‎[28].

The present study believes the recommended solution by ‎[28] has fundamental

limitations. Even for the lateral windings in a three-phase transformer (phases A and C),

having similar frequency responses is often not achievable in practice. There are many

cases in which lateral FRA fingerprints are quite dissimilar to compare. Also, the

assessment of the middle winding frequency response will remain a challenge. Its

spectrum is not comparable to the others.

Ryder’s suggestion could be extended into inter-winding instead of inter-phase

comparison. Inter-winding comparison can be performed between the HV and LV

windings for each and every phase independently. In fact, a differential spectrum (Xi(HV)-

Xi(LV)) can be calculated using HV and LV winding frequency response spectra and called

DFRA. DFRA spectrum which comes through the difference between HV and LV winding

spectra should remain unchanged and also not altered due to moisture and temperature

changes. Any changes in paper moisture or temperature in HV winding would occur for

the LV winding as well. As far as DFRA spectrum is not changed for the transformer entire

life, it means that transformer humidity has remained unchanged and the measured FRA

spectra can be examined through statistical indicators to explore winding deformation.


162

This approach seems to be more feasible than what has been recommended in ‎[28], but it

has still a drawback. The volume of crepe paper insulation for HV and LV conductors as

well as paper insulation thickness for each and every winding might be different. In

addition, a transformer might have an enamelled winding in LV side and a paper insulated

winding in HV side. Thus the water absorbed through the HV and LV windings in a phase

would be different and result in an incorrect prognosis. Therefore, this approach also

appears to be not quite accurate. Hence, a comprehensive practical solution which

provides an independent result for each and every winding would be preferable. This

solution is discussed hereinafter.

Indeed before the evaluation of FRA spectra through the statistical indicators, a

preliminary calculation should be performed to distinguish insulation characteristic

influence from winding deformation. Figure 7.15 presents the flowchart of a new

technique to address this issue. This procedure distinguishes the insulation impact from

the mechanical impact on the frequency response of transformer winding. The reference

and measured spectra are numerically processed to determine whether further evaluation

by statistical indicators can proceed, or other actions are required (including possible

outcome of ‘no action’).

In this chart, α and β are considered as the lower and upper limits of FRA trace deviation

and will be determined only through the FRA signature (reference trace). At first, the

moisture content of the paper insulation should be measured during the FRA signature

measurement. Next, the measured value for the paper moisture is taken as the reference

value (WCPref) in percent for FRA trace signature. Afterwards, deviation in the measured

FRA spectrum in Bode diagram due to the moisture migration from paper insulation is

calculated (FSD), and the moisture diffusion (WCP change) is derived (see Table 7.8).

Then, α and β can be determined through (7.16) and (7.17) considering 0.5 % (Wll) and 4

% (Wul) as the lower and upper criteria for moisture content of the transformer paper

insulation (different standards or guidelines may recommend different criteria for

maximum and minimum moisture content in transformer paper insulation).

Comparing Rn ≜ RXi, Yin in Fig. 7.15, α and β , appropriate decision can then be made. In

fact, Rn should remain between α and β . In order to clarify the procedure, Appendix G

provides practical examples on the case studied in Chapter 7.

1

100

ll refW WCPFSD

WCP Change

(7.16)


163

1

100

ul refW WCPFSD

WCP Change

(7.17)

where Wll and Wul are the lower and upper limits of the paper moisture content in percent,

WCPref is the reference paper humidity in percent and should be measured during FRA

fingerprint measurement, FSD is the average FRA Spectrum Deviation for 0.5% moisture

variation in paper insulation, WCP Change is the per unit water content change in paper

insulation and equal to 0.5 in Table 7.8.


164

Zero Crossing Recognition

Derivative Curve Calculation

Start

ˆ ˆ,i ii i

i i

X YX Y

f f

1 1ˆ ˆ ˆ ˆ ˆ ˆ 0 , 0i n i i i m i iZ X S X S X Z Y S Y S Y

ˆ ˆ ˆ ˆ ( ) , ( )i i i iS X Sign X S Y Sign Y

Ratio Calculation

ˆ ˆ ˆ , ˆ

i mi i n

i n

Z YR X Y

Z X

n=m

Reference

Spectrum

Xi

Measured

Spectrum

Yi

FRA Statistical

Indicators

ˆ ˆ , i i nR X Y

No Action is

Required

Trip, Humidity

Investigation is

Required

End

No

Yes

Yes

No

Figure 7.15. The chart on preliminary calculation on FRA traces to distinguish insulation deviation from winding deformation; Xi and Yi were defined in Chapter 2.


165

7.7 Transformer Winding Dry-out Influence on Frequency Response Trace

Figure 7.16. Glassy model transformer.

The achievement in the last Sections revealed that if the paper water content is changed,

then the frequency response of transformer winding will shift horizontally towards lower

or higher frequencies. The results also shown that each 0.5 % moisture migration from the

paper insulation will lead to almost 0.79 % deviation in FRA trace. This deviation was

observable and calculable for resonant frequencies. In practice, as a common transformer

active part treatment, the transformer dry-out process removes significant amount of

moisture from the paper insulation and causes the paper moisture content to change.

Therefore, it could have significant impact on FRA trace due to paper moisture diffusion.

In order to address this hypothesis, the frequency response traces of wet and dry model

transformer are shown in Fig. 7.17 for HV and LV windings, respectively.

According to Fig. 7.17(a) and Fig. 7.17(b), moving from low frequencies to 800 kHz all of

the traces are perfectly matched. Moving from 800 kHz to the higher frequencies, the

discrepancy between the traces becomes obvious. Since, the test object moisture content

has changed from 4 % to 1 %, the FRA traces have slightly moved to higher frequencies for

both LV and HV windings.

Total capacitance variation of transformer windings due to the moisture diffusion could be

considered as the major factor influenced FRA trace before and after the dry-out process.

To calculate the total capacitance variation, some of the resonances and anti-resonances

have been highlighted in Figures 7.17(a) and 7.17(b).

Tables 7.12 and 7.13 show the numerical values of deviated frequencies for 4 % and 1 %

water content of the paper (WCP), where f1 and f2 represent the frequencies of anti-

resonances (minima) and resonances (maxima) at 4 % and 1% WCP, respectively.


166

(a)

(b)

Figure 7.17. FRA traces for glassy test object (a) HV winding spectra before and after dry-out (frequency band 3 kHz-20 MHz, 0 dB < 3 kHz), (b) LV winding spectra before and after dry-out (frequency band 6 kHz-20 MHz, 0 dB < 6 kHz). The measurements have been performed for oil-filled model transformer.

104

105

106

107

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

HV winding spectrum (at 23 °C) - After dry-out, WCP 1%

HV winding spectrum (at 23 °C) - Before dry-out, WCP 4%

1.222 MHz, WCP 4%, at 23 °C

1.318 MHz, WCP 1%, at 23 °C

1.861 MHz, WCP 4%, at 23 °C

2.006 MHz, WCP 1%, at 23 °C

1.899 MHz, WCP 1%, at 23 °C

1.780 MHz, WCP %, at 23 °C

4.313 MHz, WCP 4%, at 23 °C

4.651 MHz, WCP 1%, at 23 °C

104

105

106

107

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

LV winding spectrum (at 23 °C) - Before dry-out, WCP 4%

LV winding spectrum (at 23 °C) - After dry-out, WCP 1%

0.947 MHz, WCP 4%at 23 ºC









167

Table 7.12. Capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.16(a).

Frequency f1 (WCP 4%) f2 (WCP 1%) C2/C1=( f1/f2 )2

1st minimum 1.222 MHz 1.318 MHz 0.86

1st maximum 1.780 MHz 1.899 MHz 0.88

2nd minimum 1.861 MHz 2.006 MHz 0.86

2nd maximum 4.313 MHz 4.651 MHz 0.86

Table 7.13. Capacitance ratio, anti-resonance and resonance frequencies for quoted points in Fig. 7.16(b).

Frequency f1 (WCP 4%) f2 (WCP 1%) C2/C1=( f1/f2 )2

1st minimum 0.947 MHz 1.022 MHz 0.86

1st maximum 1.119 MHz 1.206 MHz 0.86

2nd minimum 2.427 MHz 2.588 MHz 0.88

2nd maximum 6.942 MHz 7.483 MHz 0.86

Calculation of capacitance ratios at 4 % and 1 % WCP show that due to the moisture

diffusion, the total capacitances of the windings have changed around 14%. These changes

in the winding capacitance can be interpreted through 3 % moisture diffusion from the

paper insulation due to the dry-out process.

7.8 Conclusion

The influence of temperature and moisture content on the frequency response trace of

transformer winding was discussed in detail. This study indicated that transformer

temperature and moisture variation can lead to FRA spectrum deviation. In fact, moisture

migration from the paper into the oil insulation will cause the FRA spectrum shifting

horizontally (frequency-axis) to lower frequencies, whilst moisture diffusion from the oil

into the paper insulation will shift the FRA spectrum to higher frequencies. Detailed study

on moisture migration for different test objects showed that changes of 0.5 % moisture in

paper insulation will move FRA spectrum for 0.79 %. Simulation results indicated that this

movement is significantly affected by changes of transformer winding shunt capacitance.

In the case of transformer dry-out evaluation, the total capacitance deviation from the

reference value for wet and dry paper was estimated. In summary, the conclusion is that

the FRA appears to be able to provide significant information on moisture migration. It

may even be an effective tool to double–check the efficiency of transformer dry-out

process. Thus, an FRA measurement could be performed before and after a transformer

dry-out process to ensure moisture diffusion. However, it is emphasized again that more

work is necessary for establishing a reliable moisture migration recognition method


168

through FRA. Therefore, it is recommended that FRA measurement be performed on

transformers (along with other methods) before and after the dry-out process as a part of

moisture diffusion assessment.

Study on the statistical indices revealed that these indicators should be modified or their

criteria must be revised to distinguish winding deformation influence from insulation

characteristic impacts on FRA trace. A method was recommended for the first time to

address this issue.

Apart from recommended method, a possible solution to distinguish winding deformation

from insulation characteristic effects on FRA spectrum is implementing on-line FRA

measurement. In fact, FRA spectrum will change gradually as the insulation characteristic

is changed. On the other hand, winding deformation will trigger a sudden shift of the FRA

graph. On-line FRA measurement monitors the winding frequency response continuously,

and thus is able to distinguish the gradual change of FRA spectrum due to the insulation

characteristic variation from the fast displacement of FRA trace due to the winding

deformation. However, more work is necessary to establish a comprehensive technique.

To open the discussion for future study, the online FRA and its setup is discussed in the

next Chapter in detail.

Chapter 8. On-line Transformer Winding Deformation Diagnosis

169

Chapter 8

On-line Transformer Winding Deformation Diagnosis

8.1 Introduction

In Chapter 7, an off-line technique was introduced to distinguish the impacts on FRA

traces caused by insulation characteristic changes as against winding deformation. It was

also discussed and explained that on-line FRA application is another possible solution for

this discrimination.

In fact, on-line FRA application not only helps to address this issue but it is also expected

to be a logical evolution in transformer winding deformation diagnosis. In order to

implement on-line FRA successfully, all aspects of the monitoring system should be

regarded. Technical issues and practical challenges should be addressed and overcome.

To date, transformer tank vibration [120]- [123], communication technique using scatter

parameters [124]- [126], current deformation coefficient [127], ultrasonic method [128],

short circuit impedance [129]- [134] and winding stray reactance [135]- [137], on-line

Transfer Function (TF) using time domain or frequency domain measurements [10], [30],

[33]- [35], [138]- [139] have been introduced as advanced on-line methods for recognition

of transformer winding deformation or displacement. Although they have been all

practically investigated, there is no industrial evidence of permanent implementation.

This Chapter gives a brief review on recommended online methods, and then extensively

discusses on-line FRA application. On-line FRA setup is introduced and practical

challenges are highlighted and discussed in detail. How to achieve the maximum

information through on-line FRA setup is considered a fundamental challenge. This

challenge is studied practically through different test objects and a possible solution is

recommended.


170

8.2 Advanced Methods in On-line Transformer Winding Deformation

Diagnosis

8.2.1 Vibration Method

Transformer vibration can be considered to be repetitive movement of transformer inner

parts that are enclosed within the transformer tank. This movement occurs around a

reference position. The reference position is where the transformer attains once it is out of

service. Vibration might be interpreted by using parameters such as winding

displacement, velocity and acceleration.

Based on this, [120]- [123] have introduced an on-line method. These studies show that

the extent of transformer tank vibration depends on voltage square and current square.

Furthermore, the winding vibration frequency is 100 Hz when the fundamental power

frequency is 50 Hz. Core vibration which is caused by magneto striction and magnetic

forces is as (8.1) for the proposed model in [120], [122].

100 1002 2

100( )

Hz to totank i u (8.1)

υtank-100Hz is the frequency tank vibration at 100 Hz, i2100 is the current square harmonic at

100 Hz, u2100 is the voltage square harmonic at 100 Hz, and θto is the oil temperature

measured at the top of the tank. α, β, γ and δ are proposed coefficient in [120].

Considering (8.1) and alteration in υtank-100Hz, the transformer tank vibration has been

recommended to be considered as an on-line transformer winding deformation diagnosis

method.

8.2.2 Communication Method

The communication method which is introduced in the literature [124]- [126] is based on

scattering parameters. The magnitude and phase of scattering parameters for normal

transformer winding are measured by several antennas as finger print. Proposed antennas

could be placed outside or inside the transformer tank. In this method, the mean absolute

magnitude distance (MAMD) and mean absolute phase distance (MAPD) are introduced as

displacement indices. MAMD and MAPD have been calculated in [126] and are as (8.2) and

(8.3), respectively:

1|| | | ||ref

n

ii

S SMAMD

n

(8.2)


171

1| |i ref

i

nS S

MAPDn

(8.3)

where Si is the measured scatter parameters, Sref is the scatter parameters. As mentioned

in [124]- [126], any kind of transformer winding deformation can cause the above-

mentioned indices altered and hence deformation detected.

8.2.3 Current Deformation Coefficient Method

This method has been introduced by [127]. A high frequency low voltage signal is applied

to live power system line along with power frequency signal when the standard practices

of connection are followed. The line-end and neutral-end high frequency currents are

continuously measured using isolated precision current probes and digital filtering

technique [127]. Associated capacitive reactance is changed due to the transformer

winding deformation and this change is reflected in deviations of high frequency terminal

currents from the fingerprint. When these deviations are measured, the ratio of deviations

at the two ends is calculated. Hence, current deviation coefficient (CDC) is introduced and

given by [127]:

1 1

2 210log H H

H H

I ICDC

I I

(8.4)

I1H and I2H are the fingerprint values of measured terminal currents at the selected high

frequencies, and I'1H and I'2H terminal current values at the winding terminals after

deformation.

8.2.4 Ultrasonic Method

Ultrasound is a sound with a frequency greater than the upper limit of human hearing, i.e.

~20 kHz. In this method introduced in [128], an ultrasonic signal is used as the reference

signal. The basis of this method concentrates on ultrasound reflection due to the non-

matching acoustic impedance between the oil and the winding.

8.2.5 Online Short Circuit Impedance and Winding Stray Reactance Method

Based on the literature [129]- [137], the measured short circuit impedance of a power

transformer can be compared to the value which has already been recorded. In fact, on-

line short circuit impedance method relies on time based comparison. Considering

http://en.wikipedia.org/wiki/Ultrasound


172

equation (2.5), short circuit impedance of a transformer is related to the windings

configuration and distance between windings.

Off-line transformer short circuit impedance measurement is performed when the

secondary side of transformer is shorted and voltage excitation is carried out at the

primary side. Since secondary short circuit setup is not possible for energized

transformer, the following equation for on-line short circuit impedance calculation is

utilized. A two-port network as a model for transformer is considered as shown in Fig. 8.1.

Two-Port

Network

I1 I2

U1 U2

Figure 8.1. Two-port network.

Demonstrated parameters in Fig. 8.1 are defined as:

1 11 12 1

2 21 22 2

U Z Z I

U Z Z I

(8.5)

Z11 is the open circuit input impedance, Z12 is the open circuit reverse transfer impedance,

Z22 is the open circuit output impedance, and Z21 is the open circuit forward transfer

impedance.

When short circuit happens, U2=0 and (8.5) becomes:

1 11 1 12 2

21 1 22 20

U Z I Z I

Z I Z I

(8.6)

Hence, the short circuit impedance (Zsc) is given by:

12 21

1122

Z ZZ Zsc Z

(8.7)

Any deviation in Z11, Z22, Z12 and Z21 will cause Zsc to change. All impedances can be

calculated using (8.8) and (8.9):

1 1 1

1 1 1

1 11 1 12 2

2 21 1 22 2

t t t

t t t

U Z I Z I

U Z I Z I

(8.8)


173

1 11 1 12 22 2 2

2 21 1 22 22 2 2

t t t

t t t

U Z I Z I

U Z I Z I

(8.9)

U1t1 and U1t2 are the primary voltages in Fig. 8.1 corresponding to t1 and t2, respectively, I1t1

and I1t2 are the primary currents corresponding to t1 and t2, U2t1 and U2t2 are the secondary

voltages corresponding to t1 and t2, I2t1 and I2t2 are the secondary currents corresponding

to t1 and t2, and t represents the time of measurement.

Based on equations (8.8) and (8.9), proposed impedances can be calculated using equation

(8.10):

1 1 2 1

1 2 2 211 12

1 2 2 121 22 2 2 2 1 1 2 1 21 2 1 1

1

.

t t t t

t t t tt t t t

U U I IZ Z

I I I IZ Z U U I I

(8.10)

When the left hand side matrix in (8.10) changes, Zsc will change. On-line short circuit

impedance measurement has been introduced as an advanced and economical method in

transformer winding deformation diagnosis. However, some researchers have argued on

the accuracy of this method.

8.2.6 On-line Frequency Response Analysis (On-line FRA)

While off-line FRA is now a common method for transformer winding deformation

diagnosis, its extension to on-line FRA measurement is still evolving. Since most power

transformers are equipped with capacitively-graded bushings, the transformer bushing

tap is suitable as an input point for low voltage signal injection during on-line FRA

measurement ‎[30].

Figure 8.2 shows the side cut-off of a capacitive bushing. The bushing tap is connected to

the last layer of capacitive grading which is brought out insulated at the bushing flange via

a small auxiliary bushing. This tap provides a much reduced terminal voltage due to the

capacitive divider. In fact, during transformer operation the transformer bushing tap is

grounded through the cover cap with internal spring. The cover has a contact socket into

which the contact pin locks when the cap is closed.

On-line FRA measurement requires opening the grounded bushing tap so to be able to

inject signal into the transformer without requiring a direct connection to the main feeder.

Hence, C1 and C2 values as a capacitive divider are important parameters during on-line


174

FRA measurement. Table 8.1, shows typical values of measured capacitors for 230 kV and

400 kV bushings.

Figure 8.2. Side cut-off of a capacitive bushing.

Table 8.1. Typical values for bushing capacitances [10].

Rating [kV] C1 [pF] C2 [pF] Ratio (C2/C1) Tap Voltage [kV]

230 608 6192 10.18 20.56

500 498 10021 20.12 23.67

Measured values for C1 and C2 by bushing manufacturers show that the division ratio

causes high voltage on the test tap if it is not grounded in operation time. Obviously, the

creepage distance and also air clearance between the bushing tap and the grounded flange

is not coordinated to withstand against estimated voltage as calculated by equation (8.11):

1

2 1tap phase

CV V

C C

(8.11)

where Vtap is the bushing tap voltage, and Vphase denotes the phase voltage. In addition, the signal

generator device cannot tolerate high voltage due to its low insulation level. Therefore, it

is imperative to have an appropriate shunt impedance in parallel with C2 when the

bushing tap is not grounded during on-line measurement. This is illustrated in Fig. 8.3. The

divided voltage on the bushing tap is given by:

2

2 1

|| 1/

|| 1/ 1/

ptap phase

p

Z j CV V

Z j C j C

(8.12)

where Zp is the impedance in parallel with the bushing tap, and ω is angular frequency.


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Furthermore, the capacitor values of the bushing remain relatively constant over a wide

frequency range. This should limit the adverse effect of the bushings’ own frequency

response masking the actual transformer signature [36]- [37]. This is discussed and

probable solution is recommended in the next subsection. Wye and delta circuits for on-

line frequency response measurement are shown in Fig. 8.4 and Fig. 8.5, respectively.

As it is demonstrated in Fig. 8.4, with wye connection, on-line FRA can be measured by

signal injection in the phase bushing tap and response can be recorded through the

neutral bushing tap. Also, it can be measured between two phases when the connection is

delta type as it is shown in Fig. 8.5.

8.3 Discussion

To reach to an acceptable level of technical satisfaction, there are number of gaps that

need to be addressed.

Obviously, each introduced on-line method has its own advantage and disadvantages and

what is considered as a challenge for a particular method arises from the disadvantages.

Figure 8.3. Paralleled impedance with bushing tap (test tap) on phase U.


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ZN

Z1 Z2 Z3

C1N

C2N

ZU ZV ZW

C1U C1V C1W

C2WC2VC2U

U V W

Figure 8.4. Wye circuits for on-line frequency response measurement.

Z1Z2 Z3

ZU ZV ZW

C1U C1V C1W

C2WC2VC2U

U V W

Figure 8.5. Delta circuits for on-line frequency response measurement.

Many questions could be raised regarding the introduced on-line methods. As for on-line

transformer tank vibration measurement, external factors in practical environments will

affect the measurement considerably. Any kind of electro-dynamic forces present can

interfere with transformer vibration tests. In particular, most of transformer accessories

will lead to some vibration in transformer tank. Therefore, interpretation of transformer

tank vibration test result will be difficult.

As for the communication method in on-line transformer winding deformation diagnosis,

it is worthy to note that the transformer tank is grounded when the transformer is in

service. Hence, signal penetration through the transformer tank is not easily possible since

the whole of transformer active part is shielded by the tank. Even if the transmitter

sensors are placed inside the transformer tank, concentric windings will act as a shield for

each other. This method seems to be more suitable for reactors than transformers and

only if the sensors can be located inside the reactor tank. Still, result interpretation would

be a challenge due to complicated electromagnetic wave propagation.

Amongst all, short circuit impedance and frequency response measurement in frequency

domain have fewer drawbacks as compared to the other recommended methods. These

two methods are less affected by external factors, more economical and reliable to


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perform on-line. They have also been industrially tested off-line and recognized as reliable

techniques, even though for on-line application, a number of challenges will be expected:

The basic question is which method is more accurate? To verify the accuracy, SCI and

FRA should be applied on a faulty power transformer to identify the accuracy of each

method. In the absence of such a test, the more precise method when performed off-line

should be a logical choice for on-line application. This issue was studied and addressed in

Section 2.4 and it was concluded that FRA is superior to SCI.

Some researchers believe that short circuit impedance is applicable for on-line

transformer deformation diagnosis because it can be readily obtained through voltage and

current measurement. However, others are concerned that the measured impedance has a

non-linear relationship with supply voltage as well as core magnetizing level. Therefore,

non-linear coefficient must be considered to achieve short circuit impedance [23]- [28]. In

addition, based on equation (2.5) it cannot recognize disk-to-disk axial movement as well

as tilting. .

On the other hand, in case of on-line FRA measurement; there are many complicated

practical challenges when this method is going to be performed on-line. In fact, on-line

FRA measurement concept is the same as off-line but problems arise as the transformer is

energized during measurement process:

The crucial one is validity of on-line FRA measurement results. How to isolate the

response of external system from the winding frequency response is a challenge that must

be addressed.

With off-line FRA measurements it has already been explored that transfer function of

the primary side of a transformer is different for open circuit and short circuit

configuration of the secondary in low frequencies [33]- [34]. With on-line FRA

measurements transformer is energized and secondary side is normally loaded. In this

case, load variation in operation time is inevitable. Variable load might be modeled as

parallel variable impedance with the secondary side of the transformer [31]. This in turn

will result in variations of frequency response measured data.

On-line FRA measurement needs to be performed separately for each winding. In fact,

injected signals for three phase transformers must not be exercised simultaneously. This is

to avoid the problem of signal overlap and superposition. Using a multiplexer for signal

injection rather than individual signal generator for each phase might be an appropriate

practice.

On-line frequency response measurement can be performed in time domain (LVI) or

frequency domain (FRA). On one hand time domain is much faster than frequency domain.


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Impulse excitation, storage of samples, signal processing and calculation should take only

a few seconds [145], while duration of measurements in frequency domain usually is

about a few minutes. On the other hand, frequency domain result is more accurate than

that in time domain. Based on literatures, sometimes two different signals in time domain

may lead to the same frequency domain characteristic [146]. Therefore, frequency domain

measurement is more reliable. Depending on the environmental and operational

conditions and other factors involved choosing the appropriate method for every single

case would be a challenge.

The effects of the input signal on the protection system while on-line FRA is conducted

should be explored in detail practically. As the voltage of the injected signal could be about

230 volts [30], its influence would be negligible.

A number of studies have suggested that high frequency CT should be applied as an

output probe on neutral point (if exists) to measure the response. With this arrangement,

the CT frequency response would cause additional error in results and increase the

uncertainty of the main winding’s frequency response. Error and uncertainty calculation

formulas for winding and also CT are as (8.13) and (8.14):

( )e TF e winding e CTr r r (8.13)

2 2( ) ( ) ( )u TF u Winding u CT (8.14)

where, er is the measurement error, and u denotes estimated uncertainty.

As discussed, on-line frequency response measurement would be applicable for HV side

of transformer through the bushing tap (test tap). On the other hand, the low voltage side

of transformer often is oil bushing. Since the oil bushing does not have bushing tap, a

Rogowsky coil can be used as the sensor [35] and the output voltage is calculated as:

0rog rog

Orog

N AdI

Vdt

(8.15)

Nrog is the number of coil turns for Rogowsky coil, λrog is the length of the winding, and Arog

is the cross-section area of each small loop in Rogowsky coil.

This formula assumes the turns are evenly spaced and that these turns are small relative

to the radius of the coil itself. In addition, since a Rogowsky coil has an air core rather than

an iron core, it has a low inductance and can respond to fast-changing currents. Also,

because it has no iron core to saturate, it is highly linear even when subjected to large

currents. This coil would be appropriate regarding immunity against electromagnetic


179

interference because of uniformly spaced winding. However, its use will result in the

frequency response of this coil interfering with the transformer winding transfer function.

This interference should be minimized as much as possible.

Since HV and LV transformer windings’ frequency responses are affected by self and

mutual inductances as well as series and shunt capacitances values, any changes in these

elements will lead to frequency response alteration, accordingly. Indeed, off-line

frequency response is mostly affected by self-inductance, series and shunt capacitances. In

fact, as in off-line measurement the test voltage is often applied to the primary winding

whilst the secondary winding is open circuit and without carrying any alternating current,

the mutual inductances between the windings do not contribute to frequency response

formation. The transformer winding mutual inductance is given by [92]:

1 2M q L L (8.16)

where, L1 and L2 represent the self-inductance of primary and secondary windings; q is

coupling coefficient and depends on closeness of coupling between the windings.

On the other hand, in on-line transformer frequency response measurement, as both

windings are carrying current, the mutual inductances between the windings could have

significant impact on the frequency response. Therefore, in the event of a deformation in

LV (or HV) winding or the channel width between HV and LV windings, physical changes

will be reflected in HV (or LV) winding transfer function. However, if tilting or bending

occurs on LV (or HV) winding, the frequency response of HV (or LV) winding probably will

not change. Any deformation or displacement that alters the volt per turn value of the

transformer will change transformer core flux. In turn, the self and mutual inductances

will be affected specifically in low frequencies. This will lead to frequency response

variation. In fact, while the transformer is energized, its active parts consisting of the core,

HV and LV windings as well as radial and vertical channels in between can be considered

as one single integrated system.

The last and probably the most important concern is how to get the maximum

information through online FRA test setup. This is as a significant challenge. The

remaining Sections in this Chapter specifically highlight and discuss this issue.

8.4 Problem Statement on Online FRA Setup

Based on the literature [20], the off-line transformer frequency response finger print is

commonly reported between 20 Hz and 2 MHz. This frequency band is capable of


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providing significant information on transformer core and windings as well as clamping

structures. In fact as discussed earlier, each sub-band has its own meaning. Thus, having

entire FRA spectrum is quite important for its interpretation.

The study conducted in this thesis shows that using the on-line FRA measurement setup

will mask out the low-frequency data in FRA spectrum. In fact, the data obtained from the

off-line FRA measurement will not be fully retrieved through the on-line setup. The low-

frequency band of the on-line FRA spectrum shows very rapid fluctuations. It also exhibits

a rising trend instead of falling trend which is quite different to what expected for a

transformer. This issue is also discussed in a work by Behjat et al [32] through a study on a

35 kV power transformer. In [32], it raised the concern that “the major limitation of the

online method comes from the noise and environmental effects at lower frequencies”.

Moreover, it warned that “owing to noise and environmental effects, online measurement

of the winding transfer function was nearly impossible in the frequency range of lower

than 1 kHz”.

The current study believes that these fluctuations are not apparently due to the noise and

could be overcome through appropriate design of on-line circuit components. Therefore,

there should be an investigation to address this issue and obtain a proper on-line FRA test

setup having minimum influence on on-line FRA results.

8.5 Challenges with On-line FRA Setup

Based on the recommended on-line FRA measurement setup, the fundamental issue is the

bushing impact on the frequency response spectrum of transformer. In fact, when FRA

signal is going to be injected through the transformer bushing tap, the bushing capacitance

will certainly affect the FRA data.

If assuming the winding’s behaviour is quite inductive in the low frequencies; then, in on-

line FRA setup the bushing capacitance has series connection with the winding inductance.

This in turn causes the total circuit impedance to change, and consequently the FRA trend

altered, specifically, in the low-frequency region.

In the case of an oil bushing, a Rogowsky coil or a coupling capacitor is required. The idea

of coupling capacitor implementation instead of using the bushing tap comes from the

literatures ‎[30], ‎[34] and ‎[35].

Using coupling capacitor in on-line FRA setup will affect the transformer frequency

response. In this case, the on-line frequency response interpretation would be even more

complicated. This effect should be eliminated or minimized as much as possible. To


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address this issue, recommended solutions to gain maximum information through on-line

FRA setup are discussed through the practical case studies, hereinafter. In this regard, the

first case study tries to highlight the issue for better understanding of the impact of

bushing capacitance on on-line FRA setup, the second and third case studies recommend a

solution, and the last two investigate its performance when applied to different test

objects.

8.6 Case Studies

In order to evaluate the on-line FRA measurement setup, two 66 kV, 25 MVA transformer

interleaved and continuous (conventional) disk windings, one single phase transformer,

three different 72 kV capacitor type bushings, and one 72 kV oil type bushing were taken

as test objects and examined through off-line and on-line FRA measurement setups.

8.6.1 Case Study 1

Since the focus of this experiment is on bushing effects, the transformer winding was not

energized. Hence, the transformer bushing tap was directly connected to the FRA

measuring instrument probe without any protection system (see Fig. 8.6). Then, the

conductor of 72 kV capacitor bushing was connected to 66 kV interleaved winding. This

was the same interleaved winding used in previous Chapters as the test object. Figure 8.7

shows a close view of test setup.

Figure 8.6. On-line FRA setup for a transformer interleaved winding.


182

Figure 8.7. Close view of test setup.

8.6.2 Test Procedure

At first, the frequency response of the transformer interleaved winding was recorded

directly through its terminals without bushing (end-to-end measurement). The winding

outer conductors for each disk were accessible for making connection. To have more data

on different conditions of the test object, disk-to-disk short circuits were deliberately

carried out to simulate two different winding defects: one by shorting the outer

conductors of disks 5 and 7 and the other by shorting the outer conductors of disks 5 and

9. The frequency response spectra for normal and faulted cases are shown in Fig. 8.8.

These results are going to be compared to similar measurements through the on-line

setup next.

The on-line FRA test setup was configured according to Fig. 8.6. The frequency response

trace for the winding with connected bushing was recorded. To compare on-line and off-

line FRA spectra, the same disk-to-disk short circuits were made and the corresponding

frequency response spectra were obtained. Measurement results for the on-line setup are

shown in Fig. 8.9.


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Figure 8.8. Frequency response traces of normal and defected winding, off-line setup.

Figure 8.9. Frequency response traces of normal and defected winding, on-line setup.

8.6.3 Interpretation

RLCM network can be used to model a transformer winding. The FRA test setup in Fig.

4.2(a) is re-drawn in Fig. 8.10(a). The transformer winding in Fig. 4.2(a), has been

replaced by its RLCM network in Fig. 8.10(a) to facilitate physical interpretation.

According to Fig. 8.8, low frequencies are affected mainly by transformer winding self-

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Discs 5 and 7 are shortend


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184

inductance (Lm). As the frequency increases from 20 Hz to 18 kHz, the inductive reactance

of the transformer winding increases accordingly. Therefore, based on equation (2.7), the

reference frequency response trace will experience a falling trend.

When disks 5 and 7 are short-circuited the self-inductance will change. The short circuit

causes the number of turns in the winding reduced, and consequently the self-inductance

is decreased, see equation (5.8). Therefore, the minimal peak at 18 kHz will be shifted to

higher frequencies according to equation (5.20).

In addition, the minimal peak amplitude takes a lower value due to lower self-inductance

value. A lower value for the self-inductance can be justified using (2.7) and (5.7).

Moving from the first minimal peak towards higher frequencies, the trace starts to

fluctuate. These fluctuations increase when the two disks are short-circuited. In fact, due

to the short circuit, some of turn-to-turn and disk-to-disk capacitances are eliminated and

the series capacitance in transformer winding is reduced significantly. This in turn will

result in winding series capacitance reduction in the interleaved winding as per equation

(3.32), and more oscillations in mid-frequency band, see equation (4.21).

50 ohms

50 ohms

Vin

Vo

ut

FR

A R-L-C-M

Winding

(a)

Cb

Bushing

R-L-C-M

Winding

50 ohms

50 ohms

Vin

Vo

ut

FR

A

Cb

Bushing

L

Self-inductance

50 ohms

50 ohms

Vin

Vo

ut

FR

A

(b) (c)


185

Cb

Bushing

50 ohms

50 ohms

Vin

Vo

ut

FR

AM-Cs-Cg

Mutual-inductance

Series and shunt

capacitances

Cb

Bushing

50 ohms

50 ohms

Vin

Vout

FR

A

Cg

Shunt capacitances

(d) (e)

Figure 8.10. FRA test setup behaviour with and without bushing connection, (a) Common FRA test setup, (b)

FRA test setup with bushing connected, (c) FRA test setup with bushing connected (low-frequency

behaviour) (d) FRA test setup with bushing connected (mid-frequency behaviour), (e) FRA test setup with

bushing connected (high-frequency behaviour).

In the case of on-line FRA spectrum interpretation, the FRA test setup with bushing

connection is shown in Fig. 8.10(b). This setup is similar to what illustrated in Fig. 8.6,

where Cb represents the bushing equivalent capacitance. Figures 8.10(c), 8.10(d) and

8.10(e) show the FRA test setup for low-, mid- and high-frequency regions, respectively.

The winding behaviour in low frequencies is modelled through the self-inductance, in mid

frequencies through the turn-to-turn mutual inductance as well as series and shunt

capacitances and for high frequencies by shunt capacitances. According to the spectra in

Fig. 8.9, in very low frequencies the test setup experiences the bushing capacitance as well

as self-inductance of the transformer winding (see Fig. 8.10(c)). The inductive reactance of

the winding self-inductance is less than the capacitive reactance of the bushing

capacitance. Thus, the overall test setup reactance is quite capacitive. This means that the

measurement setup acts as a high pass filter and cannot convey the low-frequency data of

the input signal. Therefore, the low-frequency information of transformer winding is

replaced by the highly oscillating signals such as noise in the trace.

Moving from very low frequencies (less than 1 kHz) to low-frequencies band makes the

winding inductive reactance to become comparable to the bushing capacitive reactance.

The interaction between them produced a resonant point around 12 kHz. Thus, the

frequency response spectrum reaches a turning point and starts a falling trend.

Afterwards, on-line and off-line spectra are quite similar. However, the signal magnitudes

[dB] are different. The discrepancy in magnitude can be interpreted through Figures

8.10(d) and 8.10(e). Indeed, a capacitive reactance is added to the entire circuit causing

change in FRA magnitude.


186

As for the shorted winding, similar behaviours with different resonant frequencies are

observed. This means that the bushing capacitance shows similar influence for different

circumstances.

8.6.4 Case Study 2

In order to find a solution to reach maximum information in on-line setup, further studies

were conducted through replacing the 72 kV capacitor bushing with other 72 kV capacitor

bushings of different capacitive reactance. This study was again conducted on the previous

test object. Figure 8.11 illustrates the reference frequency response spectrum of the

interleaved winding through the off-line measurement setup as well as frequency

response spectra in on-line setup using different capacitor bushings. In Fig. 8.11, the first

bushing has the minimum capacitance value (lowest trace in Fig. 8.11). The capacitance

value of the second bushing is greater than the first one and much less than the third one.

Figure 8.11. Frequency response traces of normal and defected winding, off line and on-line setup.

According to Fig. 8.11, maximum deviation from the reference trace occurs when the

transformer bushing has less capacitance value (i.e. maximum capacitive reactance). The

on-line FRA trace becomes closer to the reference trace when the transformer capacitive

bushing has greater value. In this case, the on-line FRA spectrum is completely compatible

with the reference trace above 8 kHz as the transformer bushing has significant

capacitance value. In addition, transformer bushing with higher capacitance value shows

closer magnitude to the original trace. Undesirable high frequency oscillations in very low

frequency regions are eliminated considerably and more information can be extracted. On-

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Frequency [Hz]

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Measurement through thrid bushing

Measurement through second bushing

Measurement through first bushing


187

line FRA study on various capacitor type bushings showed that higher bushing capacitance

will have less influence on the on-line FRA trace especially in low frequencies.

8.6.5 Case Study 3

In order to study the oil type (conventional) bushing, the capacitor bushing in Fig. 8.6 was

replaced with a 72 kV oil type bushing. A coupling capacitor sensor was designed and

fabricated to play the role of bushing tap. The coupling capacitor was a thin layer of

aluminium strip, wrapped on the porcelain surface of the oil type bushing and isolated

from the ground. Coupling capacitor is usually employed for partial discharge (PD)

measurement as an output sensor. In this study, it was employed as the input terminal.

Figure 8.12 illustrates the reference and recorded frequency response spectra of the

previous test object through the off-line and on-line setups.

Figure 8.12. Frequency response traces of interleaved (off-line setup and on-line setup through a coupling

capacitor).

According to Fig. 8.12 (graph 2), very high frequency oscillations are experienced in the

low frequencies. These oscillations are present up to 9 kHz. Moving to higher frequencies,

a resonant point due to interaction between the coupling capacitor and the self-inductance

of the winding can be seen.

The characteristic in the remaining of the trace is quite identical to the reference trace. In

addition, the trace achieved through the on-line setup (coupling capacitor) displays

greater absolute magnitude compared to the reference trace over the entire spectrum.

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[d

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1 Reference

2 Measurement through one coupling capacitor

1

2


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Previous study (case study 2) on various types of capacitor bushings revealed that higher

bushing capacitance will have less effect on the FRA trace in on-line setup. Therefore,

coupling capacitor value should be increased to achieve maximum information and

minimum discrepancy. Hence, a new technique is introduced here to increase the coupling

capacitor value. In this technique, several coupling capacitors are mounted on the

porcelain surface of the oil bushing as illustrated in Fig. 8.13.

Figure 8.13. Paralleled coupling capacitors on conventional bushing (oil type bushing).

The bushing conductor (main rod) is considered as the common plate between all

coupling capacitors. The other sides of the coupling capacitors can be connected together

to provide parallel combination. It is worth noting that between the coupling capacitor

plates and the bushing surface on each plate, there is a rubber insulation to isolate each

capacitor plate from the bushing surface and prevent any kind of short-circuit and

decreasing the creepage distance of the porcelain insulator. This configuration will

increase the total coupling capacitance significantly. The frequency response spectra using

five and nine paralleled coupling capacitors are plotted in Fig. 8.14. For comparison, this

figure also shows the reference trace as well as the recorded spectrum for one coupling

capacitor (tested on the interleaved winding).

According to Fig. 8.14, the frequency response trace through on-line setup was

considerably improved when the coupling capacitor value became significant. Among all

traces, the second one in Fig. 8.14 shows the closest spectrum having minimum

discrepancy to the reference trace. This was achieved when nine coupling capacitors were

employed in parallel. Based on Fig. 8.14, high frequency oscillations in the low frequency

band of FRA trace were limited to lower frequencies when maximum coupling capacitors

were employed. The benefits and drawbacks of this technique will be discussed latter.


189

Figure 8.14. Frequency response traces of interleaved winding (off-line setup and on-line setup through coupling capacitor).

8.6.6 Case Study 4

In order to study other transformer winding types, a 66 kV continuous disk winding was

put to test. This particular winding has 72 disks and 8 single strand turns per disk. The

frequency response traces of continuous disk winding for off-line and on-line setup

through coupling capacitor are depicted in Fig. 8.15.

Figure 8.15. Frequency response traces of continuous disk winding (off-line setup and on-line setup through coupling capacitor).

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Magnitude [

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4 Measurement through one coupling capacitors

3 Measurement through five paralleled coupling capacitors

2 Measurement through nine paralleled coupling capacitors

High frequency oscillationsStart point

3

4

2

1

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4 Measurement through one capacitor

3 Measurement through five paralleled capacitors

2 Measurement through nine paralleled capacitors

1

2

4

3


190

According to Fig. 8.15, the same scenario as in the interleaved winding also occurred for

the continuous disk winding. Oscillation trends in the mid-frequency band of the reference

trace spectrum remained unchanged with the on-line FRA setup. It is again obvious that

the increase of coupling capacitance value brings the on-line FRA results closer to what

was recorded through the off-line setup.

8.6.7 Case Study 5

A single phase 11/0.25 kV, 25 kVA transformer was used as another test object to explore

on-line FRA setup impacts upon the frequency response trace. This transformer has oil

type bushings. Hence, an appropriate external coupling capacitor was constructed and

mounted on the porcelain HV bushing.

At first, the reference frequency response trace of HV side was measured through the off-

line FRA setup. Then, the frequency response through one coupling capacitor was

recorded. After that, paralleled combinations for two and three coupling capacitors were

configured and frequency responses were measured. Figure 8.16 shows measurement

results.

Figure 8.16. Frequency response traces of single phase transformer (off-line setup and on-line setup through coupling capacitor).

According to Fig. 8.16, coupling capacitor increment in on-line FRA test setup appears to

shift the response closer to the reference trace and improve matching. This in turn results

in maximum information achievement.

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[d

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4 Measurement through one coupling capacitor

3 Measurement through two paralleled coupling capacitor

2 Measurement through three paralleled coupling capacitor

3

1

4

2


191

On the other hand, total capacitive reactance decreases when the number of coupling

capacitors increases. Also, it has less influence on frequency response trace oscillations. In

addition, significant capacitive reactance will shift the first minimal peak to higher

frequencies according to equation (5.20).

8.7 Discussion

Off-line FRA provides significant information about transformer condition. On-line FRA

measurement is going to be employed on transformer with the aim to achieve as much

information as possible. This study examines the sensitivity and accuracy of the on-line

measurements when it is conducted through the transformer bushing tap. Since the

bushing imposes a capacitive reactance to the test setup, the trace should be interpreted

by taking the bushing effects into account. Sensitivity and accuracy have been discussed in

the first case study.

On one hand the bushing has crucial effects in very low frequency bands and some part of

the information is missing due to the bushing characteristics. On the other hand,

transformer on-line FRA setup must be designed to avoid power frequency penetration

and superposition effect on the FRA signal in the bushing tap. In fact, a built-in high pass

filter must accompany the paralleled impedance in Fig. 8.3 to avoid power frequency

superposition impacts on the transformer frequency response trace which seems to be

inevitable. In addition, high voltage disturbances in overhead line can be transferred to the

bushing tap and harm the paralleled impedance. Hence, a protection system for this

paralleled impedance needs to be developed to avoid unwanted damage.

According to case studies, on-line FRA measurement through capacitance graded type

bushings will lead to achieving more information as compared to oil type bushings.

Using paralleled coupling capacitors can improve on-line FRA trace and eliminate a part of

undesirable high frequency oscillations. In this circumstance, bushing insulation

coordination will remain as another challenge. In fact, coupling capacitors should be

designed and manufactured based on transformer bushing characteristics to satisfy

insulation coordination. It is recommended to examine the bushing capacitive reactance as

a part of test setup before on-line FRA measurement setup is employed.

It is obvious that the measurement setup for online FRA has more influenced on the very

low frequency part of the FRA trace. However, insulation characteristic changes such as

moisture and temperature do not contribute to this part of spectrum. Therefore, these

changes of transformer could be detectable through online FRA implementation.


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8.8 Conclusion

In the first part of this Chapter; advanced methods for transformer winding deformation

diagnosis were discussed. The problem in migrating off-line transformer winding

deformation recognition methods to on-line methods was highlighted. In this regard, on-

line frequency response measurement was described and discussed.

In the second part of this Chapter, recommended practical setups for on-line FRA

measurement were elaborated and technical issues associated with the measuring process

were discussed. The possibilities to collect as much information as possible about

transformer condition through the on-line FRA setup were also studied using different

case studies.

Possible solution to get maximum information via on-line setup was introduced and

discussed, and the results through this solution in on-line FRA test setup were examined.

In order to yield traces without interferences over the very low frequency band, it was

recommended to install a built-in high pass filter in the paralleled impedance of the test

setup and determine the cut-off frequency.

In summary, it was discussed that the online FRA measurement is viable solution for on-

line moisture migration recognition from the transformer paper insulation as well as

distinguishing the insulation characteristics impacts through winding deformation on FRA

trace. Further studies are recommended to investigate theoretical and practical challenges

in replacing off-line methods by on-line schemes.

Chapter 9.Conclusion and Future Research

193

Chapter 9

Conclusion and Future Research

9.1 Conclusion

One of the most crucial defects that can put transformers out of service for a long time is

the transformer winding deformation or active part displacement. Over the years, a

number of diagnosis methods have been introduced and employed to detect the mechanical

defects within this valuable asset. Among all, the FRA method is quite capable of providing

worthy information about mechanical integrity as well as electromagnetic behaviour of the

transformer windings. The idea of using FRA as a method for monitoring the condition of

power transformers is not new. It has already been established that FRA spectrum is quite

sensitive to figure out the mechanical defects and the FRA method is widely used

nowadays. However, FRA spectrum interpretation is still under development. In addition,

its sensitivity to the transformer insulation characteristic variations such as the

temperature and humidity is quite significant. This sensitivity can lead to an incorrect

prognosis and concerns on this issue had been raised in a number of Standards.

Since one of the long term goals in transformer condition monitoring is to prevent any

incorrect decisions in transformer diagnosis, the intention of the work presented in this

thesis was to study and highlight the discrepancy between the mechanical defect and the

impacts of insulation characteristic (temperature and moisture) on the FRA spectrum.

Based on this, the preliminary step was to review of the main reasons of winding

deformation and specifically discussed the effects of electro-dynamic forces caused by

short circuit currents. The FRA method and its circuit setup were introduced. It was

demonstrated that FRA is superior to other available methods such as SCI in terms of

winding deformation recognition. Statistical indicators as the available evaluation

methods for frequency response interpretation were also discussed in detail.

Afterwards, using turn-to-turn inductance calculation the self- and mutual-inductances of

the transformer winding were analytically derived and the inductance matrix was


194

achieved. Series and shunt capacitances of transformer winding were then derived

through similar approach. An air-core glassy model transformer was designed and

manufactured and used as a test object. The analytical calculation on winding parameters

was verified using practical measurement on this object. The approach developed here can

potentially help other researchers studying transformer winding through the detailed

model.

Having obtained an air-core winding model through the travelling wave and transmission

line theories, a detailed analysis and interpretation for mid-frequency oscillations in FRA

spectrum was provided. The reason of resonances and anti-resonances in the mid-

frequency band of FRA spectrum was clarified through the mathematical calculations. FRA

mid-frequency oscillations dependency on the inductance as well as series and shunt

capacitances was explored and the simulation was carried out to examine this achievement.

The glassy model transformer was utilized to compare the simulation and measurement

results. This research revealed that a winding with small impulse voltage distribution

coefficient (α) will result in more oscillations in the mid-frequency region of FRA trace while

greater α will give a steady trend in this frequency band. It was also demonstrated that the

influence of transformer metal core would be negligible in terms of FRA mid-frequency

interpretation.

The interpretation of the low-frequency region in FRA trace was also discussed. This study

showed that the first minimal peak in FRA spectrum comes through the transformer middle

limb impact while the second one is influenced by the lateral limb. In order to validate this, one

of the winding frequency response trace of a power transformer was generated through other

deviated traces. Furthermore, according to achievements in this study similar equivalent

magnetic circuits for a given FRA setup will lead to identical FRA spectra. This concern was

studied through practical measurement. This in turn results in a new procedure developed for

transformer core defect recognition. In addition, the impact of the shunt capacitance on the

first anti-resonance in FRA spectrum was studied and discussed through mathematical and

practical approaches. It was discovered that not only the shunt capacitance can shift the

minimal peak position in the low-frequency region; it also has considerable effect on the

frequency response trend in the approach to the first anti-resonance in FRA spectrum.

In terms of winding mechanical defects, the study on axial and radial deformations in

transformer winding was rigorously conducted through the analytical approach. This was

then completed through numerical examples as well as simulations. It was found that the

transformer winding inductance will be changed due to the axial deformation in a winding


195

disk. In addition, radial deformation will change the shunt capacitance of the winding

significantly.

Since the frequency response of transformer winding is completely dependent on its

parameters, the FRA spectrum deviation due to changes in the winding parameters was

also studied. In this case, simulation studies were verified by practical measurements on

different objects. Particular influences of the various parameters (winding inductance,

series and shunt capacitances, resistance, conductance to ground, and turn-to-turn

conductance) on FRA trace were discovered and clarified in this thesis.

The influence of temperature and moisture content on the frequency response spectrum

has recently been a topic of vigorous discussion among researchers. This thesis indicated

that the transformer temperature and moisture variation can lead to FRA spectrum

deviation. In fact, moisture migration from the paper into the oil insulation will cause the

FRA spectrum shifting horizontally (frequency-axis) to lower frequencies, whilst moisture

diffusion from the oil to the paper insulation will shift the FRA spectrum to higher

frequencies. Detailed study on moisture migration for different test objects suggested that

changes of 0.5 % moisture in paper insulation will move the local FRA resonance and anti-

resonance peaks by 0.79 %.

Based on the achievements in this thesis, the FRA method appears to be able to provide

significant information on moisture migration from the paper insulation. It may be utilized

as an effective tool for double–checking the efficiency of transformer dry-out process.

Thus, an FRA measurement could be performed before and after a transformer dry-out

operation to ensure satisfactory moisture diffusion.

Study on the statistical indicators for different temperatures and moisture contents in

transformer winding revealed that they are quite susceptible to these effects and thus can

lead to incorrect prognosis. Indeed, available indicators should be modified or their

existing criteria must be revised to distinguish winding deformation from insulation

characteristic impacts on FRA trace. This was discussed in detail and a method was

recommended for the first time to address this issue.

Apart from the recommended method, a possible solution to distinguish winding

deformation from insulation characteristic effects on FRA spectrum is by implementing

on-line FRA measurement. In reality, insulation deterioration occurs gradually over a long

time scale and so the FRA spectrum is expected to also change gradually. On the other

hand, winding deformation will trigger a sudden shift of the FRA graph. On-line FRA

measurement monitors the winding frequency response continuously, and thus is able to


196

distinguish the gradual change of FRA spectrum due to the insulation characteristic

variation from the fast displacement of FRA trace due to the winding deformation.

Undoubtedly, more work is necessary to establish a robust technique. To open the doors

for future study, the online FRA and its setup was discussed.

9.2 Future Research

Based on the findings in this study, there are a number of subjects that can be targeted for

future research.

9.2.1 FRA Test Setup Development

To date, different test setups have been recommended to measure the frequency response

of the transformer winding. These setups were discussed in detail in this study.

The main goal of each different FRA test setup is to recognize the winding deformation;

however, they are also capable of checking the transformer core integrity. As a result

discovered in this thesis, detailed investigation on core integrity is feasible by adding some

extra wiring into the available FRA setups. This can be achieved through additional extra

short circuit connection on predetermined open terminals of a specific winding. This

work provides the opportunity to localize the fault in the transformer core. Note that the

additional short circuit is not required to be performed on all open winding terminals,

synchronously. It should be implemented step by step for different open winding

terminals as discussed in Chapter 5. Indeed, deliberate short circuit on the terminals of a

winding will impede the flux to flow through that winding. This work helps to direct the

flux flow through a specific core leg and concentrate on the fault recognition in that area.

Swapping deliberate short circuit in the setup from one terminal-pair to the others can

narrow down the fault investigation and lead to fault localization. Hence, rather than using

the available setups for FRA measurement, a study focused on different FRA test setups

would be quite valuable.

9.2.2 Transformer Humidity Recognition Using FRA

The moisture influence on the insulation aging of the transformer is quite obvious for all

utility engineers as well as transformer researchers. Having continuous information about

the humidity variation in the paper insulation is highly desirable. Therefore as discussed

in this study, different methods including off-line and online, direct and indirect

techniques have been employed for moisture recognition in transformer for many years.

These methods are able to provide absolute value of the moisture content in the paper

insulation [Appendix F]. However, the implementation of these methods requires that the


197

moisture equilibrium is achieved between the paper and oil insulations. Otherwise, the

results will not be accurate. On the other hand, whenever the transformer load is changed,

the moisture migrates from one phase (solid/liquid) to the other phase due to the

temperature variations. Thus, new equilibrium should be achieved for moisture

measurement. To reach the new equilibrium, the equilibrium time must be reached.

Transformer equilibrium may take several hours or even much longer. It also depends on

insulations’ aging as well as their contaminations. Hence, real-time measurement of

moisture content in transformer paper insulation is not available at present.

The study in this thesis revealed that FRA signature is quite sensitive in detecting moisture

migration from the paper into the oil insulation and vice versa. Although FRA sensitivity

cannot disclose the absolute value of the moisture in paper insulation, it is able to show

the migration trend. Having obtained the moisture migration trend through FRA and also

the initial condition of FRA signature as well as the initial value of moisture content; it

could be then feasible to recognize the moisture content in paper insulation. Indeed,

perfect implementation of online FRA measurement can change this hypothesis to a real-

time application. Additional research is required to reach real-time humidity recognition

using FRA.

9.2.3 Transformer Dry-out Assessment Using FRA

Transformer dry-out is required after its manufacturing, commissioning, prolonged storage

without nitrogen, maintenance, detection of high moisture content or internal inspection.

As discussed earlier, different methods could be employed for transformer drying. To

evaluate transformer moisture content after the dry-out process, Frequency Dielectric

Spectroscopy (FDS), Karl-Fischer Titration (KFT), paper sample method (Dean-Stark) or

Dielectric Dissipation Factor (DDF) measurement could be implemented. In addition, DDF

measurement over a wide frequency range which is quite similar to FDS has been used to

provide real-time information on the insulation's water content during the actual drying

process. The research in this thesis showed that the FRA spectrum is quite sensitive to

deviation from the reference value due to the moisture variation in paper insulation. This

deviation can be exploited to evaluate the efficacy of the dry-out process. In order to reach

an acceptable level of satisfaction, additional research on this recommendation is

required. Extensive study on different test objects as well as study on insulation

parameters of wet and dry transformers may lead to an industry-accepted method to

assess transformer dry-out process. Different measurement implementations and building


198

a comprehensive database can help significantly and should be considered as the next

move.

9.2.4 Oil and Paper Insulation Aging and Oil Replacement

The influence of temperature and moisture variation on the frequency response spectrum

was studied in this thesis. These two insulation parameters seem to cause the most

impacts on FRA spectrum. However, it would be useful to investigate the influence of other

parameters such as oil and paper insulation aging as well as oil acidity, oil interfacial

tension, oil contamination, oil viscosity, oil breakdown voltage. In addition, sometimes the

oil condition is such that it requires replacing the existing oil with new oil, or maybe a

mineral oil is changed using alternative insulating liquids, these circumstances should be

studied in detail and their impacts on FRA spectrum must be explored. The results of such

studies may then lead to some more considerations in the setup of FRA signature

measurement in future.

9.2.5 On-line Transformer Winding Deformation Recognition

Since smart high voltage monitoring systems are under development now, it is required

that all of off-line diagnosis measurements are able to be upgraded to perform on-line.

Among all, FRA has unique potential to be implemented on-line and provide real-time

information on transformer active parts. Off-line FRA measurement suffers from a number

of drawbacks: costly service outage for testing, de-energising and disconnecting the

transformer from the bus bars, conducting the test with its setup in similar conditions

(temperature, moisture content, cable connection) to the previous FRA signature

measurement. All these limitations motivate researchers to study the implementation of

on-line FRA as the future work.

Appendix A. Developed Software to Calculate Statistical Indicators

199

Appendix A

Developed Software to Calculate Statistical Indicators

A.1 Introduction

A common method to interpret the frequency responses of transformer windings is using

statistical indices or indicators which have been introduced by various researchers over

the years. These indicators provide information on winding deformation of transformer.

The criteria to explore winding deformation have been determined for some of them,

while others are still under investigation. Among those methods with criteria provided,

some have been implemented in available FRA equipment in industry as a module to

assess FRA spectrum. It means that whenever the FRA test is conducted, the

technician/expert can easily import the reference and measured spectra to the developed

module and compare them together. The output would be then a colour indicating

transformer winding condition. Demonstrating red, yellow and green colours as the

outcome of the implemented method in turn means that the transformer winding is

severely, slightly deformed or may be it has been remained unchanged, respectively. Since

different manufacturers used different methods in their own developed software, this

study has tried to develop a software package to calculate most or perhaps all the

available statistical indicators. This work provides the opportunity to compare and assess

all the criteria together and make more accurate decision on transformer condition. The

idea of this work was introduced and planned by the author for the first time, and then set

as a project for other students to implement the coding. Subsequently, the developed

programs were debugged by the author, and substantial improvement was made in

reaching the current form. Detailed information on the developed software is provided in

this Appendix.


200

A.2 Implemented Statistical Indicators

A.2.1 Correlation Coefficient (CC)

The correlation coefficient method was given by (2.8) in Chapter 2. It is well-known that

this method is not sensitive to changes in the shape of the responses characterized by

variation in the magnitude. In fact, it is more sensitive to generation, elimination, and also

deviation in existing resonant frequencies [20].

Different criteria for CC have been introduced in the literatures, but implemented criteria

for developed software are based on that presented in Chapter 2.

A.2.2 Maximum Absolute Difference (DABS)

The maximum absolute difference was given by (2.9) in Chapter 2. It is well-known that

this method is sensitive to slight differences in amplitude between the FRA responses. A

table provided by Secue et al [20] discusses the characteristic of DABS.

A.2.3 Absolute Sum of Logarithmic Error (ASLE)

The formula for the ASLE is:

10 101

( , )

20log 20logs

i i

s

N

iX Y

Y X

NASLE

(A.1)

where Xi and Yi are the ith elements of the fingerprint and measured FRA traces

respectively, and Ns is the number of elements (or samples). “ASLE were proposed by the

authors in order to correct these undesirable characteristics of the CC and SD. ASLE was

presented as the most reliable parameter which was designed to make the fully log-scaled

comparison in the magnitude frequency response; its application considers a previous

process of interpolation proposed by the operator. The normal range of variation for these

parameters has not been set yet” [20].

A.2.4 Min-Max (MM)

The Min-Max method was given by (2.10) in Chapter 2. If the resonance magnitude is not

changed; MM would not be sensitive to deviation of resonant points [20]. In [20], it is also

stated that MM is quite sensitive to the changes of the magnitude variations in FRA

spectrum as a consequence of new resonant frequencies creation or the elimination of

existing resonant frequencies, or even shifting in them.


201

A.2.5 Standard Deviation (SD)

Standard deviation was discussed in Chapter 2. Its equation was provided in (2.11) and

criteria were given in Table 2.2. Standard deviation shows how much variation exists from

the average. A low value for standard deviation shows that the data is very close to the

mean or reference value, whereas high value indicates that the data points are spread out

over a large range.

A.2.6 Spectrum Deviation (σ)

Spectrum deviation was discussed in (2.12). The spectrum deviation was first proposed as

a statistical indicator for comparing FRA traces by Bak-Jensen et al [149], and then became

popular.

A.2.7 Cross Correlation Coefficient (CCF)

The Cross Correlation Coefficient is given by:

1

( , )2 2

1 1

s

s s

N

iX Y

N N

i i

i i

i i

CCFX X Y Y

X X Y Y

(A.2)

where and are the arithmetic average for Xi, Yi and n=1… Ns. In terms of CCF

interpretation, cross-correlation takes two sets of numbers and looks at how similar they

are. If assuming two series of numbers such as from two FRA spectra are perfectly or

nearly matched, the CCF obtained is close to 1.0. If they have absolutely no correlation, in

other words are completely random, the CCF would be 0. “If assuming both spectra are

related diametrical, they would have a negative CCF. In FRA analysis negative CCF are not

common, but they do occur on occasion. Regardless, negative correlation coefficients are

not considered acceptable when trying to look for deviations between traces” [26]. In [26],

it is highlighted that “inappropriately assigned region boundaries, or single resonance

shifts for example, could cause only a minor change to the CCF and in truth indicate a

substantial problem”.

A.2.8 Relative Factor (R)

This method which is basically coming from Chinese standard [74] is based on calculation

of the covariance for FRA spectra. Firstly, the standard variance for reference and

measured spectra should be calculated [25]:

211

00

1 1ss

NN

i iX is si

D X XN N

(A.3)


202

21

1

00

1 1ss

NN

i iY is si

D Y YN N

(A.4)

Then:

2 211 1

0 00

1 1 1ss s

NN N

i i i iXY i is s si

C X X Y YN N N

(A.5)

Normalization of covariance factor is given by [25]:

XYXY

X Y

CLR

D D (A.6)

Calculation of Relative factor (Rxy) is then given by [25]:

1010 , 1 10

log 1 ,

XYXY

XY

LRR

LR Otherwise

(A.7)

Next, the introduced criteria in Table (2.3) should be taken into account to interpret FRA

responses through the Relative factor.

All above mentioned methods have been used to develop a comprehensive software tool

to calculate statistical indicators. This product is believed to have strong potential to be

commercialised.

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Appendix B. Tables and Formulas for Inductance Calculation

204

Appendix B

Tables and Formulas for Inductance Calculation

B.1 Introduction

In order to calculate the winding inductance in Chapter 3 and winding inductance

variation in Chapter 6, this Appendix provides the related formulas as well as tables. It

should be emphasized that pre-calculated tables here have been taken from ‎[85], and

summarized for convenience. In addition, the analytical approach conducted in the thesis

on self- and mutual-inductance calculations under buckling is given in this Appendix.

These calculations are given for the first time, and nonlinear integrals obtained from the

calculations are solved using the MATLAB software.

B.2 Tables and Formulas

Table B.1. Values of KN for single layer coil ‎[85].

h/2R K h/2R K h/2R K h/2R K

0 0 0.10 0.203324 0.20 0.319825 0.30 0.405269

0.01 0.034960 0.11 0.217044 0.21 0.329479 0.31 0.412650

0.02 0.061098 0.12 0.230200 0.22 0.338852 0.32 0.419856

0.03 0.083907 0.13 0.242842 0.23 0.347960 0.33 0.426890

0.04 0.104562 0.14 0.255011 0.24 0.356816 0.34 0.433762

0.05 0.123615 0.15 0.266744 0.25 0.365432 0.35 0.440474

0.06 0.141395 0.16 0.278070 0.26 0.373818 0.36 0.447036

0.07 0.158119 0.17 0.289019 0.27 0.381986 0.37 0.453450

0.08 0.173942 0.18 0.299614 0.28 0.389944 0.38 0.459724

0.09 0.188980 0.19 0.309876 0.29 0.397703 0.39 0.465860

0.10 0.203324 0.20 0.319825 0.30 0.405269 0.40 0.471865

2 4

(2 1)(2 3)( )

!

( 1) ( 1)( 2)( 3). ...

2 (2 1) 2.4(2 1)(2 3)

m

m m m

m mP

m

m m m m m mm m m m

(B.1)


205

12

( ) ( )( )

1

m mm

m P mPP

(B.2)

Table B.2. Values of R0 for inclined circles ‎[85].

α=0.9

μ β=0 β =0.1 β =0.2

α=0.8

β =0 β =0.1 β =0.2

0 0.4099 0.4741 0.6078 0.5300 0.5660 0.6550

0.1 0.4114 0.4763 0.6095 0.5330 0.5680 0.6540

0.2 0.4161 0.4807 0.6150 0.5360 0.5730 0.6580

0.3 0.4229 0.4890 0.6246 0.5440 0.5780 0.6670

0.4 0.4311 0.5012 0.6389 0.5560 0.5920 0.6800

0.5 0.4472 0.5185 0.6593 0.5730 0.6110 0.7000

0.6 0.4673 0.5431 0.6886 0.5980 0.6330 0.7290

0.7 0.4969 0.5794 0.7308 0.6270 0.6680 0.7650

0.8 0.5433 0.6383 0.7951 0.6800 0.7190 0.8180

0.9 0.6278 0.7313 0.9064 0.7640 0.8070 0.9050

1.0 1 1 1 1 1 1

α=0.7

μ β=0 β =0.1 β =0.2

α=0.6

β =0 β =0.1 β =0.2

0 0.6276 0.6498 0.7104 0.7149 0.7306 0.7746

0.1 0.6291 0.6513 0.7121 0.7164 0.7321 0.7761

0.2 0.6337 0.6562 0.7172 0.7209 0.7366 0.7806

0.3 0.6420 0.6645 0.7259 0.7287 0.7444 0.7883

0.4 0.6542 0.6769 0.7387 0.7403 0.7560 0.7996

0.5 0.6714 0.6943 0.7566 0.7563 0.7718 0.8150

0.6 0.6950 0.7182 0.7807 0.7779 0.7932 0.8354

0.7 0.7279 0.7512 0.8135 0.8071 0.8218 0.8621

0.8 0.7753 0.7984 0.8585 0.8472 0.8608 0.8972

0.9 0.8503 0.8710 0.9217 0.9055 0.9160 0.9429

1.0 1 1 1 1 1 1

Table B.3. Values of F for parallel circles, rd/2R=Λ, ‎[85].

cosθ'' Λ=1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0 -4.053 -1.953 -0.467 0.525 1.145 1.485 1.622 1.621 1.531 1.382 1

0.1 -1.508 -1.023 -0.168 0.550 1.055 1.357 1.495 1.513 1.448 1.327 1

0.2 -0.724 -0.387 0.117 0.585 1.010 1.264 1.392 1.421 1.376 1.278 1

0.3 -0.237 0.013 0.348 0.696 0.989 1.195 1.308 1.341 1.310 1.233 1

0.4 0.101 0.291 0.5246 0.766 0.983 1.144 1.239 1.271 1.252 1.191 1

0.5 0.351 0.493 0.658 0.829 0.984 1.105 1.181 1.211 1.199 1.153 1

0.6 0.544 0.647 0.761 0.878 0.987 1.075 1.132 1.158 1.151 1.115 1

0.7 0.695 0.766 0.842 0.920 0.991 1.050 1.091 1.111 1.108 1.085 1

0.8 0.818 0.861 0.907 0.952 0.995 1.031 1.056 1.069 1.069 1.055 1

0.9 0.917 0.937 0.958 0.979 0.998 1.014 1.026 1.032 1.033 1.026 1

1.0 1 1 1 1 1 1 1 1 1 1 1


206

B.3 Self and Mutual Inductances under Buckling

B.3.1 Biot-Savart Law and Inductance Calculation

One of the most fundamental laws in computation of magnetic field is the Biot-Savart law

which is given by:

0

24

R

dv

J aB dv

R

(B.3)

where J is the current density vector [A/m2], v is the volume containing current, and Rd is

the proposed distance. On the other hand Jds = I; therefore, Jdv = Jdsdl = Idl. Hence,

according to (B.3) dB is given by ‎[78], ‎[104]:

0

24R

d

IdB dl a

R

(B.4)

According to Fig. B.1 and also the magnetic field generated through I, the Biot-Savart law

can be explained as:

0

2sin ,

4n

d

IdldB a

R

(B.5)

where, an is the unit vector perpendicular to the plane containing dl and ar, dl an = dl × ar.

θʹ

aR

dl

I

dB Rd

Figure B.1. Biot-Savart law.

Based on Gauss’s law for a circular filament having radius Ra and carrying the current I,

the flux density, ϕ, is given by:

2

0 0

.aR

z

s

B ds B rd dr

(B.6)

where B = Baz and can be obtained through (B.5), and rdϕʹ= dl. Thus for a circular filament

having radius Ra equation (B.5) can be obtained as:


207

0

2sin

4z a

d

IdB R d

R

(B.7)

where, Rd is the distance between the differential current segment and the point of

proposed magnetic field and obtained as (B.8) for a specific point r away from the centre.

θʹ would then be the angle between the differential current vector and the vector directed

from it to the point as shown in Fig. B.2.

2 2 2 2 cosa adR R r R r (B.8)

r

Rd

θʹ

αʹ

ϕʹ

B

Ra

I30

210

60

240

90

270

120

300

150

330

180 0

Figure B.2. Magnetic flux determination for circular filament, ‎[104] taken and modified.

Hence, equation (B.7) is modified as:

0

22 2 2

sin4 2 cos

az

d a a

IRdB d

R R r R r

(B.9)

θʹ = π/2 + αʹ; thus, sinθʹ = sin (π/2 + αʹ) = cos αʹ. Using the law of cosines ‎[104] r2=Ra2+Rd2-

2RaRd cosαʹ, and therefore ‎[104]:

2 2 2 cossin cos

2

a d a

a d d

R R r R r

R R R

(B.10)

Based on this, Bz(r) is given by:

2

0

32 2 2

0

cos( )

4 2 cos

a az

a a

IR R rB r d

R r R r

(B.11)


208

In ‎[104], it is stated that the second method for Bz determination is using the vector

magnetic potential as given by:

.

( )z

rAB A

r r

(B.12)

Therefore using (B.7) and (B.12), equation (B.6) for a filament having radius Ra is given by:

.

a a

a z

s c

B ds A dl

(B.13)

Apart from the last approach in Chapter 3 for the mutual-inductance calculation through

Grover’s formula, the mutual-inductance between two circular filaments having radii Ra

and Rb are also obtained as:

,b

ab

a

MI

(B.14)

where ϕb is the induced magnetizing flux on the second loop due to the current initiated by

the first loop. Hence, Mab is given by (see Fig. B.3):

2 20 0

10 0 2 2 2 2

12 2 2 2

cos1.

4 4 2 cos

[ Note that: 2 cos ]

b

a b

ab ab b

c

a b ab a b a

abc ca b a b a

ab a b a b a

M A dl

R Rdl dl d d

R R R d R R

R R R d R R

(B.15)

It is again obvious that the mutual-inductance between two circular filaments is only a

function of their shapes as well as orientations. This was stated in ‎[78] and ‎[104].

z

Y

X

x

y

Rab

dlb

dla

I

Ra

Rb

d

sa

sb

ϑb

ϑa

Figure B.3. Concentric circular filaments.


209

Using Maxwell’s advices in ‎[78], reference ‎[104] is discussed on the solution of (B.15). In

fact, changing the variable ϑa to 2θʺ, cos ϑa = cos2θʺ=2cos2θʺ-1, and dϑa = 2dθʺ will simplify

the equation. Thus, Mab will be calculated as ‎[104]:

0

10 2 22 2

12

0 2

1 2 20 2 2 2

2 2 2

2cos 2

24 cos

cos 2 4 ,

2 1 cos

2 2[ Note that: cos 2 2cos 1 1 cos ]

a b aab a

a b a b a

a b a a ba

a ba

a a a

R RM d

R R d R R

R R k R Rd k

R R dk

k k kk k

(B.16)

Having information about complete elliptic integrals K(k) and E(k), equation (B.15) is

given by ‎[78] and ‎[104]:

12

0

10 2 2 2

112 2 2 22

0 10 2 2 2

112 2 2 22

0 10 2 2 2

12

0

cos 2

2 1 cos

2 1 2 1 cos

1 cos

2 1 2 1 sin

1 sin

2

a b aab a

a

a b a a

a

a b a a

a

a b

R R kM d

k

R R k k dk kk

R R k k dk kk

R R kk

12 2 2 2 2

10 02 2 2

2( ) ( )

where:

1 ( ) , ( ) 1 sin

1 sina a a

a

K k E kk

K k d E k k d

k

(B.17)

Therefore, (B.17) shows the analytical approach to determine the mutual inductance for

concentric circular filaments.

B.3.2 Mutual-Inductance under Buckling

B.3.2.1 Buckling of a Single Filament

Equation (B.15) is utilized to calculate the impact of buckling on the mutual-inductance of

two concentric circular filaments as shown in Fig. (B.4).


210

30

210

60

240

90

270

120

300

150

330

1800

z

Y

X

x

y

Rab

dlb

dla

I

Ra

Rb

d

sa

sb

ϑb

ϑa

Figure B.4. Concentric circular filaments, inward buckling demonstration for the second loop.

It should be noted that in the case of inward buckling the winding radius for the span

faced buckling is defined as Rbd = Rb + 0.5rʹ(cos ηθ-1), and outward buckling is

mathematically expressed as Rbd = Rb - 0.5rʹ(cos ηθ-1). η is the ratio of entire trigonometric

circular span (2π) over the deformation span (rad) as illustrated in Fig. 6.4, and r'

represents the deformation radius. Based on this, the mutual-inductance is given by:

1 2

2 112 22 2

2 2 1

1

2 20

0 0

2 20 1

210 2 22 2 21

cos 0.5 (cos 1)

0.5 (cos 1) 2 0.5 (cos 1) cos

cos

4

cos

42

b

a b a b

aab

a b

a ab b

R rd d

R R r d R R r

d

RM

R Rd

R R d R R

(B.18)

where, the first part of (B.18) comes through the span facing deformation and second part

represents the circular part in the second filament. This integral is quite non-linear and

complex to solve analytically. Therefore it should be addressed numerically using MATLAB

software. However to continue the equation analytically, we can assume that the influence

of Rb on the total value of Rab as the denominator of the first integral is negligible as

compared to Rb influence as the numerator. This assumption is reasonable for the

filaments which are quite far away, but maybe it is not accurate for close loops. All in all,

having this assumption integral in (B.18), Mʹab is obtained as:


211

2 20 1

2 110 0 2 2 2 2

1

2 20 1 2

2 110 0 2 2 2 2

1

12

0 1110 2 2 2 2

1

( 0.5 )cos

4 2 cos

0.5 cos cos

4 2 cos

2 (1 ) cos

4 2 cos

a bab

a b a b

a

a b a b

a b

a b a b

R R rM d d

R R d R R

R rd d

R R d R R

R Rd

R R d R R

(B.19)

where the first integral in (B.18) has split in (B.19). One step forward, the calculation of

the second filament influence on the Mʹab is given by:

20 1

110 2 2 2 2

1

20 1

110 2 2 2 2

1

12

0 1110 2 2 2 2

1

( 0.5 ) cos

2 2 cos

cos

8 2 cos

(1 ) cos

2 2 cos

a bab

a b a b

a

a b a b

a b

a b a b

R R rM d

R R d R R

R rd

R R d R R

R Rd

R R d R R

(B.20)

After simplification of (B.20), Mʹab is then given by:

20 0 1

110 2 2 2 2

1

(1 2 ) cos

8 2 2 cos

a a bab

a b a b

R r R RM d

R R d R R

(B.21)

Comparing (B.21) with (B.16), it is obvious that the term having (1-2π) subtly reduces the

mutual inductance within inward buckling as it takes a negative value. Accordingly, this

coefficient will change to a positive value (1+2π) when the outward buckling occurred for

the filament (see Fig. 6.4).

B.3.2.2 Buckling of both concentric circular filaments

Compared to the previous subsection, analytical calculation of the mutual-inductance once

both filaments are facing deformation is even more challenging. In this case, having

asymmetrical magnetic flux density generated by the first filament on the second one is

considered as the first challenge. Indeed the filament circumference is not circular; thus,

considering two pairs of opposite sides, the horizontal contribution of the magnetic flux

density cannot be completely neutralized. Therefore, B is not fully aligned with az.

The second challenge would be again the calculation of non-linear integrals, analytically. In

fact in spite of analytical approach, numerical calculation is required on the final integrals


212

to reach the exact value of mutual-inductance. All in all, attempts are made in this

appendix to analytically derive near final integrals. This helps to prepare for numerical

calculations, even though such calculations show subtle changes in mutual-inductance for

usual buckling in transformer windings.

Using the same assumption on Rb in the last subsections, the mutual-inductance of the two

concentric circular filaments is modelled as Fig. B.5 and given as (B.22) using (B.15):

30

210

60

240

90

270

120

300

150

330

1800

30

210

60

240

90

270

120

300

150

330

1800

z

Y

X

x

y

Rab

dlb

dla

I

Ra

Rb

d

sa

sb

ϑb

ϑa

Figure B.5. Concentric circular filaments, demonstration of inward buckling for both loops.

1 1

12 22 2 2 20

2 110 0

2

2 21 10

2 110 2 2 2 2 2

1

0

cos 0.5 (cos( ) 1)

2 cos4

0.5 (cos( ) 1)

cos 0.5 (cos( ) 1)

4 2 cos

a

ab a b a b

b

a

b

a b a b

R r

M d dR R d R R

R r

R rR d d

R R d R R

2 21 2

2 112 0 2 2 2 2

1

2 20 1

2 112 2 2 2 2 2

1

cos 0.5 (cos( ) 1)

4 2 cos

cos

4 2 cos

a b

a b a b

a b

a b a b

R R rd d

R R d R R

R Rd d

R R d R R

(B.22)

Equation (B.22) contains four different integrals as given by (B.23):

1 2 3 4abM Int Int Int Int (B.23)

The first integral is obtained as:


213

1 1

12 22 2 2 20

2 110 0

2

2 20

2 2 10 0

20

20

cos 0.5 (cos( ) 1).

1 2 cos4

0.5 (cos( ) 1)

0.5 (cos 1)4

( ( 0.5 ) 0.5 cos )4

a

a b a b

b

b

b

R r

Int d dR R d R R

R r

A R r d d

A R r A r

2

2 10

22

02 2 1

00

20

10

1 1

12 2 2 2

1

0.5 ( 0.5 ) sin

4

(4 2 )

2 4

cos 0.5 (cos( ) 1)where:

2 cos

b

b

a

a b a b

d d

rA R r d

R r rA d

R rA

R R d R R

(B.24)

From calculation of different spans on the first and second filaments, Mʺab is obtained as:

21 10

110 2 2 2 2

1

1

12 2 2 2

10

1 1

12 2 2 2

1

cos 0.5 (cos( ) 1)(4 2 )1

2 4 2 cos

(2 )

2 cos(4 2 )

cos (cos( )2 8

2 cos

ab

a b a b

a

a b a bb

a b a b

R rR r rInt d

R R d R R

R r cos

R R d R RR r r

r

R R d R R

2

10

1

12 2 2 2

1

20 1

110 2 2 2 2

1

1

12 2 2 2

1

(2 )cos

2 cos

(4 2 ) cos( 1)

2 8 2 2 cos

cos( 1)

2 2 cos

a

a b a b

b

a b a b

a b a b

d

R r

R R d R R

R r r rd

R R d R R

r

R R d R R

(B.25)

Simplification of (B.25) leads to:


214

20 1

110 2 2 2 2

1

20 1

110 2 2 2 2

1

0

(4 2 ) (2 ) cos1

2 8 2 cos

(4 2 ) (2 ) cos( 1)

4 8 2 cos

(4 2 ) (2

4

b a

a b a b

b a

a b a b

b a

R r r R rInt d

R R d R R

R r r R rd

R R d R R

R r r R

21

110 2 2 2 2

1

) cos( 1)

8 2 cosa b a b

rd

R R d R R

(B.26)

The second integral which concentrates on the interaction of deformation in the first

filament and the circular part in the second filament is given by:

2 21 10

2 110 2 2 2 2 2

1

12

1 10110 2 2 2 2

1

12

0

0

cos 0.5 (cos( ) 1)2

4 2 cos

cos 0.5 (cos( ) 1)(1 )

2 2 cos

(1 )

2

a

b

a b a b

a

b

a b a b

R rInt R d d

R R d R R

R rR d

R R d R R

A d

1

(B.27)

where, Aʹ has been defined in (B.24) and its solution is similar to (B.26).

The third integral is obtained as:

2 21 20

2 112 0 2 2 2 2

1

20 1

112 2 2 2 2

1

cos 0.5 (cos( ) 1)3

4 2 cos

(4 2 ) cos

2 4 2 cos

ba

a b a b

b a

a b a b

R rRInt d d

R R d R R

R r r Rd

R R d R R

(B.28)

Using Maxwell’s approach, the solution of (B.28) is similar to that calculated in (B.17).

However, the asymmetrical boundary conditions are considered an issue in this case. The

fourth integral is given by:

2 20 1

2 112 2 2 2 2 2

1

cos4

4 2 cos

a b

a b a b

R RInt d d

R R d R R

(B.29)


215

As discussed earlier, the numerical solution of the above mentioned integrals will lead to

the mutual-inductance of the concentric circular filaments with inward buckling in both

filaments. In the case of outward buckling, the positive sign before rʹ should be replaced by

the negative sign.

B.3.3 Self-Inductance under Buckling

The self-inductance of the non-circular filament as shown in Fig. B.6 can be achieved

through (B.6) and (B.7). The self-inductance is given by:

a

a

LI

(B.30)

where, Ia is the current carried by filament, and ϕa is the magnetizing flux generated by Ia.

Based on this, the magnetizing flux density generated through buckling area is then

defined as:

0

20.5 cos 1 sin

4z a

d

IdB R r d

R

(B.31)

and Rd is obtained as:

22 20.5 cos 1 2 0.5 cos 1 cosd a aR R r r R r r

(B.32)

Ra+0.5r

ʹ(Cosηϑ

-1)

r

Rd

θʹ

αʹ

ϑ

B

Ra

30

210

60

240

90

270

120

300

150

330

180 0

I

Figure B.6. Magnetic flux determination for non-circular filament.

Since θ = π/2 + αʹ, therefore, sin θ = sin (π/2 + αʹ) = cos αʹ. Using the cosines law, r is

obtained as:


216

22 20.5 cos 1 2 0.5 cos 1 cosa d a dr R r R R r R (B.33)

Thus:

2 2 20.5 cos 1sin cos

2 0.5 cos 1

0.5 cos 1 cos

a d

a d

a

d

R r R r

R r R

R r r

R

(B.34)

Substitution of (B.34) into (B.31), dBz is obtained as:

0

32 22

2

0

32 22

0.5 cos 1 0.5 cos 1 cos

4 0.5 cos 1 2 0.5 cos 1 cos

0.5 cos 1 0.5 cos 1 cos

40.5 cos 1 2 0.5 cos 1 cos

a a

z

a a

a a

a a

I R r R r rdB d

R r r R r r

R r R r rI

R r r R r r

(B.35)

Therefore, Bz(r) is obtained as (B.36) including the magnetic flux density generated by the

buckling area as well as the circular part:

0

2 2

32 220

2

0

32 2 22

( )4

0.5 cos 1 0.5 cos 1 cos

0.5 cos 1 2 0.5 cos 1 cos

cos

4 2 cos

z

a a

a a

a a

a a

IB r

R r R r rd

R r r R r r

IR R rd

R r R r

(B.36)

The self-inductance through the non-circular filament as illustrated in Fig. B.6 is then

given by:


217

2 0.5 cos 10

0 0

2

32 22

22

0

32 0 2 2 2

2

0.5 cos 1 cos 0.5 cos 1 cos

0.5 cos 1 2 0.5 cos 1 cos

cos cos.

2 2 cos

a

a

R r

a a

a a

Ra a

a a

L

R r R r rdrd

R r r R r r

R R rdrd

R r R r

(B.37)

The solution of (B.37) can be achieved through numerical calculation using MATLAB

software. However, more simplification for such integrals can be evaluated analytically

using the following integrals ‎[104], ‎[150]:

3 12 22 22

3 12 22 22

4 2 ,

4

2 4

4

dx ax b

ac b ax bx cax bx c

xdx bx c

ac b ax bx cax bx c

(B.38)

Self- and mutual-inductance variations under buckling phenomenon are quite dependent

on the winding geometry. In fact, the most important parameters to influence the original

value of the inductance are the ratio η of deformation span (rad), and the deformation

radius r'. Nevertheless, according to literatures ‎[102] and also information achieved in this

study, these variations are negligible for usual buckling in routine transformer windings.

B.3.4 Numerical Example

The self- and mutual-inductance variation caused by winding radial deformation was

discussed in the last subsection. It was also stated that the final analytical formulas

developed for self- and mutual-inductance calculation should be solved and addressed

through the numerical analysis. This is done in this subsection on the example provided in

Chapter 6 (see Section 6.4 and Sub-section 6.4.2) using MATLAB software.

Based on Maxwell equations, the self-inductance of a circular turn having 280 mm

outermost radius and 7 mm width is calculated as:

1.5338MaxwellL H (B.39)


218

while this value for the same turn using Grover equations (see L33 in (6.12) with the

central turn of the disk having outermost radius 280 mm) is calculated as:

1.6507GroverL H (B.40)

If assuming the circular turn has suffered inward buckling such as depicted in Fig. 6.8 (left

side) and Fig. B.6; then, based on the calculation provided in this thesis the self-inductance

for non-circular loop is obtained as (rʹ=50 mm):

1.1954L H (B.41)

In the case of mutual-inductance, for a concentric circular turn 17 mm away from the main

loop, MMaxwell is obtained as:

0.9977MaxwellM H

(B.42)

This value was calculated as (B.43) through Grover’s approach in Chapter 6.

0.9969GroverM H

(B.43)

Under the discussed buckling, the mutual inductance is obtained as (B.44) when one of the

turns has suffered buckling (rʹ=50 mm):

0.8740abM H

(B.44)

To complete the example, it is supposed that all conductors in the second disk of the model

winding depicted in Fig. (3.1) have suffered inward buckling (rʹ=50 mm); hence the

inductance matrix for the normal and defected winding is obtained as (B.45) and (B.46),

respectively:

( )

1.6300 1.3680 1.1003 0.9400 0.8523 0.9399 1.0208 1.0662 0.8157 0.7942 0.7598 0.7171 0.6075 0.6341 0.6558 0.6714

1.3680 1.5818 1.3260 1.0644 0.9080 0.9873 1.0319 1.0208 0.7942 0.7875 0.7663 0.7326 0.6101 0.6314 0.6468 0

eq MaxwellL

.6558

1.1003 1.3260 1.5338 1.2841 0.9540 0.9977 0.9873 0.9399 0.7598 0.7663 0.7596 0.7386 0.6073 0.6225 0.6314 0.6341

0.9400 1.0644 1.2841 1.4861 0.9638 0.9540 0.9080 0.8523 0.7171 0.7326 0.7386 0.7318 0.5984 0.6073 0.6101 0.6075

0.8523 0.9080 0.9540 0.9638 1.4861 1.2841 1.0664 0.9400 0.8523 0.9080 0.9540 0.9638 0.7318 0.7386 0.7326 0.7171

0.9399 0.9873 0.9977 0.9540 1.2840 1.5338 1.3260 1.1003 0.9399 0.9873 0.9977 0.9540 0.7386 0.7596 0.7663 0.7598

1.0208 1.0319 0.9873 0.9080 1.0664 1.3260 1.5818 1.3680 1.0208 1.0319 0.9873 0.9080 0.7326 0.7663 0.7875 0.7942

1.0662 1.0208 0.9399 0.8523 0.9400 1.1003 1.3680 1.6300 1.0662 1.0208 0.9399 0.8523 0.7171 0.7598 0.7942 0.8157

0.8157 0.7942 0.7598 0.7171 0.8523 0.9399 1.0208 1.0662 1.6300 1.3680 1.1003 0.9400 0.8523 0.9399 1.0208 1.0662

0.7942 0.7875 0.7663 0.7326 0.9080 0.9873 1.0319 1.0208 1.3680 1.5818 1.3260 1.0644 0.9080 0.9873 1.0315 1.0208

0.7898 0.7663 0.7596 0.7386 0.9540 0.9977 0.9873 0.9399 1.1003 1.3260 1.5338 1.2841 0.9540 0.9977 0.9873 0.9399

0.7171 0.7326 0.7386 0.7318 0.9638 0.9540 0.9080 0.8523 0.9400 1.0644 1.2841 1.4861 0.9638 0.9540 0.9080 0.8523

0.6075 0.6101 0.6073 0.5984 0.7318 0.7386 0.7326 0.7171 0.8523 0.9080 0.9540 0.9638 1.4861 1.2841 1.0644 0.9400

0.6341 0.6314 0.6225 0.6073 0.7386 0.7596 0.7663 0.7598 0.9399 0.9873 0.9977 0.9540 1.2841 1.5338 1.3260 1.1003

0.6558 0.6468 0.6314 0.6101 0.7326 0.7663 0.7875 0.7942 1.0208 1.0315 0.9873 0.9080 1.0644 1.3260 1.5818 1.3680

0.6714 0.6558 0.6341 0.6075 0.7171 0.7598 0.7942 0.8157 1.0662 1.0208 0.9399 0.8523 0.9400 1.1003 1.3680 1.6300

(B.45)


219

( )

1.6300 1.3680 1.1003 0.9400 0.7322 0.8074 0.8816 0.9371 0.8157 0.7942 0.7598 0.7171 0.6075 0.6341 0.6558 0.6714

1.3680 1.5818 1.3260 1.0644 0.7785 0.8510 0.9054 0.9280 0.7942 0.7875 0.7663 0.7326 0.6101 0.6314 0.6468

eq MaxwellL

0.6558

1.1003 1.3260 1.5338 1.2841 0.8206 0.8740 0.8950 0.8904 0.7598 0.7663 0.7596 0.7386 0.6073 0.6225 0.6314 0.6341

0.9400 1.0644 1.2841 1.4861 0.8427 0.8635 0.8593 0.8939 0.7171 0.7326 0.7386 0.7318 0.5984 0.6073 0.6101 0.6075

0.7322 0.7785 0.8206 0.8427 1.2364 1.1954 1.0299 0.9371 0.7322 0.7785 0.8206 0.8427 0.6643 0.6613 0.6499 0.6328

0.8074 0.8510 0.8740 0.8635 1.1954 1.2781 1.2361 1.0661 0.8074 0.8510 0.8740 0.8635 0.6841 0.6907 0.6874 0.6755

0.8816 0.9054 0.8950 0.8593 1.0299 1.2361 1.3200 1.2283 0.8816 0.9054 0.8950 0.8593 0.6947 0.7109 0.7174 0.7137

0.9371 0.9280 0.8904 0.8939 0.9371 1.0661 1.2283 1.3622 0.9371 0.9280 0.8904 0.8439 0.6971 0.7215 0.7378 0.7443

0.8157 0.7942 0.7598 0.7171 0.7322 0.8074 0.8816 0.9371 1.6300 1.3680 1.1003 0.9400 0.8523 0.9399 1.0208 1.0662

0.7942 0.7875 0.7663 0.7326 0.7785 0.8510 0.9054 0.9280 1.3680 1.5818 1.3260 1.0644 0.9080 0.9873 1.0315 1.0208

0.7898 0.7663 0.7596 0.7386 0.8206 0.8740 0.8950 0.8904 1.1003 1.3260 1.5338 1.2841 0.9540 0.9977 0.9873 0.9399

0.7171 0.7326 0.7386 0.7318 0.8427 0.8635 0.8593 0.8939 0.9400 1.0644 1.2841 1.4861 0.9638 0.9540 0.9080 0.8523

0.6075 0.6101 0.6073 0.5984 0.6633 0.6841 0.6947 0.6971 0.8523 0.9080 0.9540 0.9638 1.4861 1.2841 1.0644 0.9400

0.6341 0.6314 0.6225 0.6073 0.6613 0.6907 0.7109 0.7215 0.9399 0.9873 0.9977 0.9540 1.2841 1.5338 1.3260 1.1003

0.6558 0.6468 0.6314 0.6101 0.6499 0.6874 0.7174 0.7378 1.0208 1.0315 0.9873 0.9080 1.0644 1.3260 1.5818 1.3680

0.6714 0.6558 0.6341 0.6075 0.6328 0.6755 0.7137 0.7443 1.0662 1.0208 0.9399 0.8523 0.9400 1.1003 1.3680 1.6300

(B.46)

Considering (B.45) and (B.46), the total inductance variation of the model winding due the

inward bucking on the second disk is obtained as 4.29 %.

Appendix C. Calculation of Series Capacitance of Inter-shield Winding

220

Appendix C

Calculation of Series Capacitance of Intershield Winding

C.1 Introduction

Different kinds of transformer winding have been introduced and employed since many

years ago. Among all, the continuous and interleaved windings are more common. The

Intershield Winding (IW) was introduced later by a US registered patent ‎[151] as a disk

type winding which is employed to increase the series capacitance between conductors of

the winding. This, in turn, will decrease the value of initial voltage distribution coefficient

(α) and hence provide a more uniform initial voltage distribution along transformer

windings. Uniform initial voltage distribution will lead to less stress on upper disks (line

end) of the windings. In IW, the shield turns are made of copper or aluminium and placed

between the winding main conductors at predetermined locations while the shield or

shield turns of each disk are insulated from the main conductors of the winding. Some of

major challenges at design stage and manufacturing issues have been discussed in ‎[100].

Since the determination of the series capacitance in intershield winding is not readily

available, a method is proposed in this Appendix together with the detailed calculations.

C.2 Series Capacitance of Intershield Winding

The configuration of the intershield disk winding and its connections were described in

Chapter 3. The equivalent RLC network shown in Fig. C.1 is the ladder network which is

used for transient studies. The total energy for a pair of disks is calculated from (C.1):

ttotal dE E E (C.1)


221

Figure C.1. Equivalent RLC network for the intershield disk winding.

where Ed represents the total disk-to-disk energy and is extracted from (3.28) and Et

represents the total turn-to-turn energy. The calculation of Et using total disk-to-disk

energy was presented in ‎[152]. Hence, turn-to-turn energy calculation for IW is given by:

1 2 2( 1)2 up down

t sh tt shield sh tt shield sh ttE C U N E N E N N E

s

(C.2)

where U is the voltage across the pair disks, Nsh is the number of shield turns in each disk.

Ett-shield-up and Ett-shield-down are the energies stored in the upper and lower shield turns which

can be calculated as below:

2

2122uptt shieldU

E C U V U Vtt x x N

(C.3)

2 21

2 2 12 22down

tt shield ttU U

E C U V N U V Nx xN N

(C.4)

21

22tt ttU

E CN

(C.5)

Therefore, substituting (C.3) through (C.5) into (C.2), Et becomes:


222

2 22

2 2

12 22

2 2 12 22

sht sh

sh

tt

tt

N U UE C U V U V N N C

ttx x N N

N U UC U V N U V N

x xN N

(C.6)

Appropriate relation between stored energy and the number of shield turns is extracted

using (C.6). Vx is the potential of connected shields between pair disks. In fact, it is an

unknown time-varying quantity over the initial, intermediate or final voltage distribution

on transformer winding. Since the series capacitance of IW is to be found, the initial

impulse voltage distribution of winding has to be taken into account. To figure out this

value, the only solution was to manufacture IW along with other winding types and

perform a practical test. Therefore four windings; one interleaved, one continuous, one IW

with one shield turn and one IW with six shield turns in each disk were manufactured (see

Fig. C.2). A test setup was developed to carry out desired tests on these windings, and a

method was proposed to find the total series capacitance of IW. The proposed method

will pave the way to calculate Vx and eventually calculate the total series capacitance of

transformer winding.

Figure C.2. Manufactured windings from top to bottom: Interleaved, Continuous, Intershield with one shield turn, and Intershield with six shield turns in each disk.


223

C.2.1 Method to Calculate Total Series Capacitance

The proposed method is completed in two major steps involving theoretical and practical

exercises. Firstly, the voltage between any two neighbor shields (i.e. one shield belongs to

one disk and the other shield belongs to the disk underneath) is measured by means of a 4

GS/s digital oscilloscope. Secondly, according to the winding type, a corresponding

formula is utilized to calculate the total series capacitance in which Vx is a variable (see Fig.

C.3).


224

4 3 2 1

Air-Core

1234

U

Input-lead

654

4 3 2 116151413

4 3 2567 321

4 3 2121110

48

49

4 3 2 117181920

4 3 2 132313029

4 3 2212223

4 3 2282726

424

425

4 3 2 133343536

4 3 2 148474645

4 3 2373839

4 3 2444342

440

441

4 3 2 149505152

4 3 2 164636261

4 3 2535455

4 3 2605958

456

457

4 3 2 165666768

4 3 2 180797877

4 3 2697071

4 3 2767574

472

473R

Vx

S.P =0

S.P =2

S.P =4

S.P =6

S.P =8

S.P =10

Figure C.3. Overall scheme and the sample point of voltage for the simulated winding, (S.P=Sample Point).

The measured Vx is substituted into the formula to determine the total turn-to-turn

capacitance. Calculated turn-to-turn capacitances for the intershield windings are

compared to the windings without shield. In order to validate the calculation results, a


225

standard lightning impulse voltage of reduced amplitude is applied on the manufactured

windings and then initial impulse voltage distribution is measured and compared.

C.2.2 Application of the Proposed Method

Standard reduced impulse voltage is applied on the manufactured windings and the initial

impulse voltage distribution for each winding is measured. In addition, Vx is measured

using a 4 GS/s digital oscilloscope and then (C.6) is used to determine total turn-to-turn

energy. Table C.1 gives the initial voltage distribution percentage in input-lead of the

winding as well as on even disks. According to Table C.1, amount of stress on uppermost

two disks of the continuous disk winding, intershield winding with one shield turn in each

disk, interleaved winding and intershield winding with six shield turns in each disk are

56.4%, 41.8%, 25.8% and 23.8%, respectively.

Table C.1. Initial voltage distribution (%).

Winding

Sample Point no.

Interleaved winding

Continuous winding

Intershield winding with

one shield turn in each disk

Intershield winding with six shield

turns in each disk

Input lead 100 100 100 100

S.P 2 74.2 43.6 58.2 76.2

S.P 4 52.7 21.2 34.4 55.1

S.P 6 34 10.1 19.3 35.8

S.P 8 17 4.4 9 17.7

S.P 10 ∼0 ∼0 ∼0 ∼0

Vx -- 77.1 -- 87.4

According to Table C.1, using the intershield winding with six shield turns in each disk will

lead to the most linear initial voltage distribution amongst the four manufactured

windings. Practical measurements revealed that Vx would be around U/2. The measured Vx

is substituted into (C.6) and the total turn-to-turn capacitance for the intershield disk

winding is calculated using (C.7). Equation (C.8) shows that the total turn-to-turn

capacitance of the new intershield disk winding depends on the number of shield turns.

The first part of (C.8) is the same as (3.25) whereas the second part demonstrates the

effect of shields.

2 2

2

2 ( 1)11 1

2 2

sht t tt

N NNE C U C U

N N

(C.7)

2 2

2 ( 1) 2 ( 1)1 1 1 1 1

2 2 2sh sh

t tt tt tt

N N N NN NC C C C

N N N N

(C.8)


226

When the number of shield turns increases, the total series capacitance increases and

consequently α decreases. As a result, the stress on upper disks of the intershield winding

with six shield turns in each disk is less than that of the interleaved winding. Also,

according to Table C.1, the initial impulse voltage distribution along upper disks of the

intershield winding with six shield turns in each disk is closer to the final distribution and

so its α is smaller than other windings.

The inter-disk capacitance for the intershield winding is almost obtained as (3.29).

Therefore, the total series capacitance for the pair-disk intershield winding is given by:

2

2 ( 1)1 1

2 3sh

s ttddN NN

C CN N

C

(C.9)

Appendix D. Glassy Model Transformer

227

Appendix D

Glassy Model Transformer

D.1 Introduction

Theoretical studies must be proved through practical experiment to reach an acceptable level

of technical satisfaction in research. In fact, practical studies besides theoretical concepts can

complete a scientific circle and lead to a valuable research. Hence, to study the frequency

response a model transformer was specifically designed by the author and manufactured for

this research. This model transformer was used as a test object to conduct various

investigations described in Chapters 3, 4, 5, 6 and 7. The test object specifications as well as

engineering drawing are provided in this Appendix.

D.2 Test Object Overview

This test object has been designed based on power transformer’s concepts. Since the tank of

this transformer was fabricated by plexiglass it is called glassy transformer. Industrial paper

insulations as well as pressboards have been employed in its construction. This test object can

be electrically energised, even though, changing the voltage and transferring the power would

not be simple to be achieved as it has an air-core. The HV winding consists of 8 disks including

8 conductor turns per disk. The LV winding has 10 disks with 6 conductor turns per disk. A 12

mm oil canal including one cylindrical pressboard was designed and implemented between

HV and LV windings while the thickness of the pressboard was 2 mm. The paper insulation

thickness of the HV and LV conductors was 0.5 mm from a side, and the distance between the

HV outermost conductor turns and the tank was 2 mm. The walls were manufactured with

plexiglass. Line and neutral leads of the windings were brought out from the tank through

appropriate HV and LV bushings. Bushings and leads are designed and manufactured based on

standard’s insulation withstands. Internal temperature can be raised up to 220 °C as the

plexiglass used is heat-resistant. An aluminium foil can be wrapped over the glass casing to

simulate the metal transformer tank. A drain valve was installed on the test object cap to

enable the oil injection and also taking oil sample.


228

In addition, internal insulation voltage level is 5kV. When it is filled with transformer oil, any

kind of occurrence as well as oil movement or arcing can be easily observed. Moisture

diffusion (desorption) through paper insulation is quite observable during the winding dry-

out process. The glassy transformer is a portable real transformer designed based on

standard. Its total dimension and weight are 40x40x40[cm3] and 20 kg, respectively.

D.3 Winding Photos and Winding Schematic

Different views of the manufactured test object are shown in Fig. D.1. Figure D.2 provides

the technical drawing for HV and LV windings of the test object.

(a) (b)

(c) (d)

Figure D.1. Manufactured glassy transformer, (a) Side view of winding, (b) Side view without oil, (c) Entire view, (d) Side view with oil.


229

(a)

6

7

6

en II

5

"' 0 ~

~ 4

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f ! !l---2,4,1 0 I

II Ill

111111 7

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111111

rrnn rrnn TTTTTTTT

1111111

ITITIT 2/.10 '~/0

-- - - 1U2

Topic: HV Windin8


230

(b)

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2 1 Wire ' (3x7x60) ' E- Cu 0.2

I I HV lfinding Cop~t W.tab\• U • ' Papoer 1fcol.l \ • 0.6 114 Tot.ol •~• bt - 3.1 X,

< PA I A B I c I D E I F G I H I I/J K L I H "' " c ~

" Port neme Part dimensions Standard. WatetiaJ ~f.~~

Topic: HV Winding Table

page: 04 Number of Po.eca: 01


231

(c)

Figure D.2. HV technical drawing, (a) HV technical schematic, (b) Specifications, (c) Backward step drawing.

- l Ul

* I~ 3 2 1 6 5 4 rz: ...... -- 1 - I

8 4.5 1-- I 1--

1 _,

( 7 -4.5 1-- 1--

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k(.;: 3 2 1 6 5 4 ~~

1U2 -

Topic : Backwf)rd

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232

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-------~------

202

l+ - --I:_J ~ I L+-- -f-l i l U l

I I

I I

202 2Ul

Topic: LV W'incUng

The cut. or X-Y


233

(b)

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: : : : : : ' ' '

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A (Ausf. 07) 20.5 25 4 7

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B (Ausf. 06) 131 4 ' 2 6 2 6 ' ' ' ' ' 6 .a Rolling pin DIN 40600 psP 3052 0.1 4x1S.3r131

' ' ' ' ' ' 4

2 lnsula.t ion The thickness of each side:O.~mmCrcpc popcr 0.2: '

' ' ' ' ' ' 2

1 The Wires (3x7x80) E- Cu o.s ' ' ' ' ' '

1 1 LV Winding ' ' '

L PA j A B j c i D j E i F i G i Hj l/J jK i L i H • .1> £ Part name Part dimensions I Sl..$nd$rd I llaterial r~i'~1 , :z k Q)

Topic: LV Winding Table

I


234

(c)

Figure D.3. LV technical drawing, (a) LV technical schematic, (b) Specifications, (c) Backward step drawing.

.----- 2U2

...... ~ 3 2 1 6 5 4 rz r. ... " f-- 1 ,- 10 I

4.5 - ' r--f-- 1 -' 9 ( 4.5 - r--

f-- 3 ,-'

./ 8 4.5 - ' ~

r-- 1 -' r 7 4.5 - I ~

r-- 3 ./ ' 6 ' ' 4.5 - ! ~

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f-- 1 ( _,

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r-- 3 ./ ,-'

4 4.5 - ' ~

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r 3 r--4.5 - I ~

f-- 3 ,-'

./ 2 4.5 - ' ~

1 _, 1

% 3 2 1 6 5 4 !~

2U1 -

Topie: Backward

I

Appendix E. Dry-out Process of Model Transformer

235

Appendix E

Dry-out Process of Model Transformer

E.1 Introduction

In order to conduct precise study on temperature and moisture impacts on the frequency

response spectrum, the model transformer had to be tested for high and low moisture

contents. High moisture content can be achieved simply through opening the model

transformer valve (stopcock), whereas for low moisture study the test object has to be

dried. This appendix provides information on the dry-out process.

E.2 Model Transformer Dry-out

Water ingress into the transformer tank could be due to the transformer breathing, air-bag

destruction or may be caused by conservator leakage. Cellulose decomposition as well as

insulation aging can also generate water inside the transformer. Maximum and minimum

criteria of permissible water content inside the transformer have been discussed for many

years. These criteria, substantially focused on paper insulation, are less than 4 % and more

than 0.5 %, respectively ‎[65]. Although, some researchers believe that the maximum

permissible limit for transformer water content should be taken as 3 %. The moisture

content of more than 3 or 4 % for transformer can result in bubbling, the formation of free

water and an increased risk of dielectric breakdown, while hyper dry-out and obtaining

less than 0.5 % water in paper insulation may lead to mechanical damage of cellulose

fibres. Therefore, the transformer moisture content should remain within the

predetermined criteria.

Transformer paper insulation has significant tendency to absorb the water specifically in

low temperatures. In fact, a large percentage of the total water inside the transformer is

absorbed through the paper insulation rather than oil insulation. This water should be

removed by the dry-out process to achieve desirable moisture content for transformer.


236

Hence, knowledge of transformer dry-out process is well established and different

methods have been introduced over the years. Low Frequency Heating (LFH), hot air, oil

spray, heat and vacuum with and without cold trap, oil circulation through an oil

circulator, and using an electric oven are among the available methods. Each and every

method has its own advantages. Transformer insulation conditions as well as transformer

size are considered as the major factors to make correct decision on using a specific

method for drying.

In this study, since the paper insulation of the model transformer was completely new, the

application of heat and vacuum without oil was selected as an appropriate method for the

dry-out process. This method requires that insulation oil is dried separately and then

injected into the transformer tank (glassy container). Hence, independently, transformer

oil dry-out was performed through the oil circulation method. To implement both

methods, different equipment had to be provided beforehand and ready prior to starting

the dry-out process. These include oil circulator, appropriate vacuum

pump/pipes/fittings, silica gel, oven for heating the test object, oven for drying silica gel,

three-way stopcock, etc. Figure E.1 shows photos of the various items for the setup.

E.3 Procedure

At first, the vacuum pump was switched on to suck oil from the oil drum into the glassy

anti-vacuum container of the oil circulator. Then, the tap between the oil drum and oil

circulator was closed off to stop the oil flowing towards oil circulator when the desired oil

volume in the glassy container was reached. The oil dry-out station (oil circulator) was

then used to dry the oil. The motor pump of the oil circulator was switched on to circulate

the oil in the circulator; in the meantime the vacuum pump was also running. The oil

heater was also switched on to heat the oil up to maximum 70° C. Heating the mineral oil

more than 70° or 75° C in direct heating systems may gradually burn the mineral oil. The

dry-out process performs poorly when operated at temperatures lower than 40° C ‎[153].

Therefore, during the entire oil treatment, the oil temperature inside the circulator should

be under control through a thermostat. In this study the cold trap was not used as the

mineral oil was new, but the oil circulator was operated continuously for a week. Finally,

an oil sample at 70° C indicated 3 ppm of moisture content in oil. This in turn means that

the mineral oil was dried properly.

Synchronous to oil dry-out, the laboratory oven was utilized to dry-out the test object. The

test object was heated up to 90° C and vacuumed to 750 mTorr for 48 hours to make sure

that moisture is removed from the paper insulation. After that, the metal stopcock on the


237

test object cap was closed to keep the model transformer under vacuum. Then, the output

tap of the oil circulator was connected to the stopcock of the model transformer. A three-

way valve which was connected to a silica gel container was used to break the oil

circulator vacuum. When vacuum was broken through the three-way valve, the metal

stopcock was opened to suck the oil into the model transformer. Finally, the stopcock was

closed and test object was left standing to reach equilibrium between oil and paper

insulations.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


238

(j) (k) (l)

(m) (n) (o)

(p) (q)

Figure E.1. Dry-out equipment, (a) Wet silica gel, (b) Half dried silica gel, (c) Silica gel dry-out process (oven

view), (d) Dried silica gel (oven view), (e) Dried silica gel, (f) Silica gel container to break the vacuum after oil

dry-out process, (g) Oil circulator, (h) Internal piping of oil circulator, (i) Oil circulator vacuum gauge, (j)

Transformer oil, (k) Vacuum pump, (l) Internal motor pump of circulator, (m) Glassy anti-vacuum container of

oil circulator, (n) Karl-Fischer equipment, (o) Metal stopcock, (p) Vacuum gauge for test object dry-out, (q)

Model transformer dry-out process (oven view).

Appendix F. Moisture Content Recognition in Transformer

239

Appendix F

Recognition of Moisture Content in Transformer

F.1 Introduction

Having continuous information about the trend of change of insulation system condition

and internal mechanical integrity is vitally important for the system operator and asset

owner. With uninterrupted flow of information, management of effective life span,

scheduling maintenance operations for transformers can be promoted.

Existence of humidity in the insulation systems containing oil-impregnated paper may

result in premature aging that may lead to abrupt temperature changes. This may in turn

result in the production of bubbles and internal partial discharges within transformers.

Naturally, normal aging of cellulosic material produces water. This is caused by the

molecular breakdown of the cellulose which is high in hydrogen and oxygen molecules.

The original bonds are broken by what is called de-polymerization process and loose

hydrogen and oxygen molecules reunite to form H2O, hence water ‎[154].

In addition, exposure during maintenance or repairs is another avenue for water to find its

way into the transformer. Consequently, for the evaluation of the insulation system

viability, the extent of humidity in the impregnated paper is required. In this regard, the

only direct method to determine the moisture content in the cellulose is by testing paper

samples taken from transformer. However, there are several indirect methods to

determine the moisture content as well.

A popular indirect method for determining humidity in a transformer is using oil sample.

Oil sample can be easily taken from transformer while it is service. This sample is then

analysed by what is called Karl Fisher Titration (KFT) test.

Karl-Fischer Titration as a non-destructive test with the interpolation capability from the

equilibrium curve of oil-impregnated paper is currently common in many countries and


240

transformer operators use it continuously. In view of the researches conducted, the

aforementioned method is deemed as a conventional method for the measurement of the

humidity of the insulation paper of the transformer under operational conditions ‎[155],

which will be duly discussed in this Appendix. Another indirect method is capacitance

sensor method which will be discussed in detail in Subsection F.2.2.

The method of Polarization and Depolarization Current (PDC) is regarded as a non-

destructive and simple method for measurement of dielectric response function, as well as

conductivity and humidity rate for oil-paper insulation in a transformer ‎[139]. Other

methods of significance are Recovery Voltage Measurement (RVM) and Frequency Domain

Spectroscopy (FDS). In this Appendix, these various approaches for humidity evaluation in

oil immersed transformers will be discussed.

F.2 Water Content Recognition in Oil-impregnated Paper

F.2.1 Karl-Fischer Titration

In order to measure the moisture content of the paper, oil is sampled from an energized

transformer. Then, KFT instrument is employed to provide the moisture content of the

sampled oil. The last step is to utilize equilibrium curves to determine the water content of

the oil-impregnated paper in percentage. KFT has several considerations, some of which

are mentioned hereunder:

1. Errors resulting from unsuitable and non-standard sampling;

2. Temperature changes in the oil sample, at the time of sampling until delivery of the

sample to the laboratory for the due testing;

3. Unsuitable storage;

4. Different sample preparation in laboratories. In addition, there are various

available titration systems that use different techniques;

5. Existing errors in the equilibrium curves for low humidity cases and also low

temperature analysis.

These considerations should be taken into account to reach to maximum accuracy. For

instance, sampling errors is one of the problems caused by samplers. Experience shows

that ambient temperature and humidity has direct impact on the test results. Hence,

samples for testing taken in a damp environment have significant difference with samples

taken in a dry environment. Therefore, it is always recommended to take oil samples

using special syringes ‎[156], so as to minimize the relevant error. Another problem with

oil sampling is the temperature variation of the test sample from the sampling location to


241

the lab. Also, samples shall at no time be exposed to direct sunlight. In addition, the time

between collection and analysis of the sampled oil should be not exceeding seven days

‎[156]. If the samples are left for a period more than seven days in the laboratory,

variations of the temperature will adversely impact the sample which in turn affects test

results. Obviously, when the experiment including sampling and KFT is run in the

laboratory as to what conducted in this study, there is no concern on this issue.

F.2.1.1 KFT Oil Sampling

In order to measure the water content of the transformer paper insulation through the

transformer oil, the oil sample should be taken from the transformer tank/container. The

sample can be taken using laboratory glass bottle (250 or 500 ml) or through glass

syringe. To avoid the penetration of environment humidity into the sample, it is strictly

recommended that oil sample is taken by the syringe having plastic stopcock rather than

laboratory glass bottle. Figure F.1 shows the laboratory glass bottle and glass syringe.

The plastic stopcock, snapped to the syringe, must remain firmly affixed to it at all times to

prevent leaks and tightly close the syringe for transportation to laboratory ‎[157]. The

sample should not be subjected to vacuum to avoid any moisture penetration. Then, the

sample should be shipped to laboratory and KFT is performed. Taking oil sample for KFT

is not considered to be as accurate as DGA (Dissolved Gas Analysis), but it has its own

concerns. In this study, the oil sample at different temperatures from the model

transformer was taken by a glass syringe, and then tested by KFT immediately (< 5 min) to

get maximum accuracy. Figure F.2 illustrates an oil sample taken from the test object.

(a) (b)

Figure F.1. Oil sample containers, (a) Laboratory glass bottle, (b) Glass syringe.


242

(a) (b)

Figure F.2. Oil sample containers, (a) Side view, (b) Front view.

This volume of oil sample (20 ml) is enough for KFT measurement. At least, 10

measurements on a 20 ml sample can be conducted and the average of results is taken as

the measured value.

F.2.1.2 KFT Titration

Karl Fischer (1901-1958) was a chemist working at a petrochemical company in Germany

in the 1930’s. He published a method to determine trace amount of water samples. This

method was a technique to recognize the moisture content. Nowadays, Karl Fischer

method is an analytical technique used worldwide to measure the moisture (water)

content in solids, liquids or gases.

Titration is defined as “A technique to determine the concentration of a substance in

solution by adding to it a standard reagent of known concentration in carefully measured

amounts until a reaction of definite and known proportion is completed, as shown by a

colour change or by electrical measurement, and then calculating the unknown

concentration” ‎[158].

K-F titration involves two reactions. In the first reaction, an alcohol (usually methanol or

ethanol), sulphur dioxide (SO2) and a base (RN) react to form an alkyl sulphite

intermediate ‎[158]:

3 2 3 3[ ]CH OH SO RN RNH SO CH (F.1)

In the second reaction, the alkyl sulphite reacts with iodine (I2) and the water from the

sample:


243

3 3 4 32 2[ ] 2 [ ] 2[ ]RNH SO CH I H O RN RNH SO CH RNH I (F.2)

Since water and I2 are consumed in equimolar amounts in reaction F.2, if the amount of I2

consumed is known, the amount of water that was present in the sample can be

determined.

F.2.1.3 KFT Equipment

Based on the method introduced by Karl Fischer, KFT equipment has been developed and

commercialised. Figure F.3 shows an example of KFT equipment together with accessory

(micro scale).

Two types of KFT are commercially available:

1. Volumetric, and

2. Coulometric.

In fact, both methods use bio potentiometric titration to find the amount of I2 consumed by

the water. Electrical conductivity variation of the reaction solution is the thing called bio

potentiometric titration. This technique is pretty similar to direct titration; an indicator

and reference electrodes are used and overall electric potential is calculated to reach

desirable titrant.

Indeed, coulometric which is used in this study aimed to explore the amount of matter

transformed during electrolysis reaction by measuring the produced or consumed

coulombs. Based on this, KFT equipment using coulometric contains various components

as illustrated in Fig. F.4. Detail information of the equipment is provided in ‎[159].

(a) (b)

Figure F.3. KFT equipment, (a) Digital micro scale, (b) Glassy container, main unit and keypad.


244

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure F.4. KFT unit main components, (a) Double platinum wire electrode (0.8 x 4 mm), (b)Generator electrode for Karl Fischer titrations, with diaphragm, (c) KF absorber tube for coulometer cell, (d) SGJ stopper, (e) Stopper, (f) Plastic tube, (g) Titration vessel holder for coulometric cells, (h) KF titration vessel, (i) Keypad for 756 KF Coulometer.

F.2.1.4 KFT Equilibrium Curves

In order to determine the paper humidity through oil humidity, oil-paper equilibrium

curves have been derived in 1960 for the first time as an indirect method to measure

paper humidity by Fabre and Pichon ‎[160], ‎[110]. After that Fallou worked on Fabre-

Pichon curve and summarized the moisture content of the oil and oil-impregnated

pressboard by KFT reaction method ‎[110]. In 1963, Norris ‎[162] did further work on this

and later published equilibrium curves known as “Norris curves”. Twenty years after that,

Oommen ‎[161] developed a set of moisture equilibrium curves ‎[161] as shown in Fig. F.5.

It is stated in ‎[110] that Oommen used and combined moisture in oil versus relative

humidity curves in air and moisture in paper versus relative humidity curves in air to


245

reach moisture in oil versus moisture in paper equilibrium curves. In ‎[110], it has been

also discussed that Griffin et al ‎[163] derived similar curves for mineral oil and paper in

1988 using Oommen’s method. In 1999, a careful study by Du et al ‎[110] provided

discussion on all available equilibrium curves. Since, most studies provided the oil-paper

equilibrium curves up to at most 100 ppm, they have used the Oommen’s method and

tried to extract a wider range of moisture concentration in oil insulation. This work was

conducted from 0° C to 100° C and oil-paper equilibrium curves extracted up to 800 ppm

(see Fig. F.6.). Indeed, this attempt was crucial for high moisture concentration in oil-

paper systems. Therefore, due to the wide range of humidity in oil-paper system, the MIT

equilibrium curves were employed in the current study to derive the moisture content.

This in turn provided opportunity to figure out the moisture content in oil insulation of

glassy model transformer for a wide range, and then to derive paper water content.


246

Figure F.5. Oommen equilibrium curves for oil-paper system ‎[161].

Figure F.6. MIT-developed curves for water equilibrium in cellulose/mineral oil systems ‎[110].


247

F.2.2 Capacitor Method

Using capacitive probes is another method to measure moisture in oil. Capacitive probes

measure the changes of capacitance caused by water molecules. Water molecules migrate

into the dielectric of the capacitor and change the capacitance which is measurable.

Indeed, the thin-film polymer as shown in Fig. F.7 absorbs or releases water vapour as the

relative humidity of the ambient air rises or drops ‎[164]. Electrical properties of the

polymer film depend on the amount of water contained in it. The associated instrument

has a sensor that is inserted into the oil sample and measures the capacitance of the

sensor. This capacity is converted into an oil humidity reading. Finally, equilibrium curve

is used to recognize impregnated paper humidity.

Figure F.7. Capacitance sensor layers.

F.2.3 Paper Sample Method

The Dean and Stark (Deanastarka) procedure can be employed to measure the water

content of a diverse range of samples. This method has been used in industrial

laboratories to measure water in petroleum oils and can also be performed to determine

water content in transformer paper.

As an advantage, water content percentage of transformer paper insulation can be directly

recognized through this technique. The sample of paper insulation in transformer is mixed

with a solvent (usually a toluene/xylene mix) and refluxed under a condenser using a

special receiver that is illustrated in Fig. F.8.

There are two common designs of receivers, one for solvents that are heavier than water,

and the more common one for solvents that are lighter than water. The water and solvent

are refluxed, and as they condense the two phases separate as they run into the receiver.


248

(a) (b) (c) (d)

Figure F.8. Dean-Stark apparatus, (a) Condenser, (b) Receiver, (c)Flask, (d) Assembled parts ‎[165].

Practically, a sample from the transformer paper insulation is taken. Then, this sample is

cut into maximum 10×10 cm scraps. The flask is filled up to 2/3 with solvent and scraped

papers are added to the solvent afterwards. The flask is heated up for 2 hours to condense

water vapour and finally water drops are gathered in the receiver. To calculate the

humidity in transformer paper insulation; the moisture content is given by

(%) 100

m

YMoisture

PK

(F.3)

where Y represents mass of accumulated water, P is mass of paper scraps and Km

represents mass coefficient of oil. Generally, Km=0.7 and Km=1 are taken for impregnated

and un-impregnated oil paper, respectively. Figure F.9 shows the practical setup of Dean-

Stark method.

(a) (b)

Figure F.9. Different views of measuring water content of the sampled papers using Dean-Stark method.


249

F.2.4 Electrical Methods

In order to facilitate the measuring of paper humidity without using oil samples, three

electrical methods are commonly considered worldwide. They are Polarization and

Depolarization Current (PDC), Frequency Domain Spectroscopy (FDS), and Recovery

Voltage Measurement (RVM) ‎[51].

F.2.4.1 Polarization and Depolarization Current (PDC)

This method is based on the fact that when a DC voltage is applied across a sample of

insulation material, the interaction of the electrical field with the free charges and other

types of limited charges results in the movement of electrical charges inside the insulation

material and an electric current is created, which is known as polarization. This current

over time decreases gradually and reaches a fixed value, which is related to the

conductivity of the insulation material. If the supply is removed when the polarization

current reaches to the fixed value and the two ends of the insulation material are short-

circuited, the depolarization current leaps towards a negative value and then decreases to

zero. If the polarization time is sufficient, the depolarization current shall be suitable for

the function of the dielectric reaction.

The polarization and depolarisation currents can be measured precisely by a micro-

ampere meter. Generally, function of DC charge can be employed to characterize the

influence of material properties, such as conductivity, on the dielectric response of the

insulation material. This method is used to evaluate the moisture content of cellulosic

material in power equipment.

F.2.4.2 Recovery Voltage Measurement (RVM)

RVM is based on established knowledge on the phenomenon of the polarization of oil-

impregnated paper insulation. There are different types of polarization. In the case of

moist oil/paper insulation, there is a polarization due to the water molecules contained in

the insulation material. By applying a DC voltage, these molecules, which are electrically

neutral, acquire a polarity and try to drift in the direction of the electrical field. That means

that molecules get energized. The applied voltage is then removed, the terminals are

short-circuited to allow discharging and opened afterwards. Some energy is still stored in

the molecules. A voltage due to this stored energy can be measured, which is called the

recovery voltage. Since different recovery voltages are obtained with the change of time

duration, these voltages shall be different for various insulation materials with different

humidity. By this method, the insulation condition is examined by tracing the polarization


250

spectrum. However, some research works have revealed excessive dependency of the

method on the geometry, temperature and conductivity parameters of oil. Therefore, RVM

is regarded as an outdated method nowadays ‎[166].

F.2.4.3 Frequency Domain Spectroscopy (FDS)

Instead of studying the polarization process in time domain, the study of dielectric

response could be conducted in frequency domain when an AC sinusoidal voltage U(ω) is

applied ‎[167]. The complex impedance and dielectric dissipation factor (tan δ) are

measured. In this method, the imaginary and real parts of the capacity of an insulation

system are estimated over a frequency range and humidity amount corresponding to this

capacity is determined using diagram comparison method.

If the effective relative dielectric susceptibility is defined as (7.9) and complex permittivity

is given by (7.10); then the dielectric dissipation (loss) factor shall be estimated as (F.4):

0

( )

tan ( )( )

r

(F.4)

where, all the parameters have been defined in Chapter 7. Figure F.10 shows variation of

loss factor of oil-immersed paper with frequency.

Figure F.10. Variation of loss factor of oil-immersed paper with frequency and dominant influences ‎[119],

taken and modified.

Appendix G. Study on Recommended Solution

251

Appendix G

Study on Recommended Solution

G.1 Introduction

In Chapter 7, a technique to distinguish the insulation characteristic impacts from the

mechanical deformation influence on the FRA trace was recommended. The procedure to

conduct this technique was given through a chart, and the criteria highlighted. This

Appendix provides a numerical example on this recommended solution.

Before performing evaluation through the statistical indicators, reference and measured

FRA traces should be examined through the recommended technique to determine the

next appropriate action. This Appendix is also provided the results of winding

deformation versus the temperature and moisture influences on model transformer for

comparison.

G.2 Case Study 1

In order to test the recommended method, FRA spectra of HV winding of the glassy model

transformer taken at 30 and 90 °C were examined. A program was developed, and

according to the procedure presented in Fig. 7.15, FRA spectra (Xi and Yi) were loaded in

the program. Afterwards, î and î were calculated. We need to calculate î and î to

derive the resonant frequencies. In fact, wherever î and î experience zero value a

resonant peak has happened. Figure G.1 shows the references (Xi and Yi) and derivative

spectra ( î and î) for the HV winding of the test object at 30 and 90 °C.

Z î n and Z î m should then be calculated. This in turn helps to realize number of

resonant points and ascertain whether the winding is mechanically deformed. n≠m will

require statistical indicators to be utilized as an evaluation method, while n=m reveals

normal mechanical condition for the winding. Therefore as the next step, S î and S î

were calculated and Z î n and Z î m were derived. Table G.1 shows the calculated

values for Z î n and Z î m. According to Table G.1, n=m and thus FRA statistical


252

indicators are not required to be considered. Therefore, Rn ≜ RXi, Yin were calculated and

illustrated in the last column of Table G.1.

(a)

(b)

Figure G.1. Reference and derivative spectra for HV winding of glassy model transformer, (a) Spectra at 30 °C,

(b) Spectra at 90 °C.

106

107

-100

-80

-60

-40

-20

0

20

40

60

80

Frequency [Hz]

Magnitude [

dB

]

Derivative spectrum


1.223 MHz

1.861 MHz

4.267 MHz

106

107

-100

-80

-60

-40

-20

0

20

40

60

80

Frequency [Hz]

Magnitude [

dB

]

Derivative spectrum


1.184 MHz

1.802 MHz

4.178 MHz


253

Table G.1. Calculated values for Z în at (30°C) and Z îm at (90°C).

Z î n [Hz] Z î m [Hz] Rn ≜ RXi, Yin

1223110 1184837 0.968708456

1740778 1689525 0.970557417

1861823 1802298 0.968028647

2347992 2277552 0.969999898

2481509 2407063 0.969999706

2565236 2488279 0.970000031

2771751 2688598 0.969999830

4267881 4178285 0.979006912

4559604 4422816 0.970000026

7253854 7038179 0.970267530

7510670 7275650 0.968708517

7927191 7689375 0.969999966

8952855 8684270 0.970000073

9359869 9077133 0.969792740

10686194 10365608 0.969999983

11546489 11200095 0.970000058

11804773 11550630 0.978471166

12213072 11836980 0.969205782

13480431 13176018 0.977418155

15067130 14605416 0.969356208

15393943 14932125 0.970000019

15913342 15535942 0.976284051

18182201 17627035 0.969466513

18578695 18221334 0.980765011

To obtain α and β , the moisture content of the reference trace was used, equal to 4 %, as it

was measured earlier in Chapter 7. Upper limits (Wul) and lower limits (Wll) for moisture

content were taken as 4 % and 0.5 % respectively. Also the WCP change for this case was

taken as 0.5 (see Table 7.8). According to (7.16) and (7.17), α and β are obtained as:

(0.5 4)1 0.79 0.9447

0.5 100

(G.1)

(4 4)1 0.79 1

0.5 100

(G.2)

Considering calculated values for Rn ≜ RXi, Yin in Table G.1, all values satisfy the criteria in

(G.3). It means that the winding has “Normal” condition and “No Action Required”. Based

on our knowledge about the test object in Chapter 7, the calculated result in (G.3) is quite

reasonable for this case.

0.9447 1n

R (G.3)


254

G.3 Case Study 2

In order to study the winding deformation recognition through the recommended

solution, the glassy model test object was again examined.

At first, the frequency response trace of the model transformer was recorded from the HV

side while the LV side was left open circuit and test object tank did not have oil. Next, to

model a mechanical defect in the test object, the modification involved the short circuit on

LV terminals was performed. This modification can in turn block the flux flow in the

transformer air core and model winding internal short-circuit on LV side. Figure G.2

shows the frequency response spectra for HV winding when the LV winding was left open-

and short-circuited, respectively.

(a)

104

105

106

107

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Magnitude [

dB

]

Measured Trace (HV winding)

Original Trace (HV winding)


255

(b)

Figure G.2. Frequency response spectra for HV winding when the LV winding was left open and short-circuited

(test object without oil), (a) Entire frequency band (20 Hz – 20 MHz), (b) The area enclosed by dash-line

rectangle in Fig. G.2(a), (100 kHz – 20 MHz).

According to Fig. G.2, when the LV winding is short-circuited, changes to the frequency

response of the HV winding occurred in the range from 20 Hz to 3 MHz. For frequencies

above 3 MHz, there is no significant discrepancy between recorded spectra. The reason

lies in the fact that the winding self-inductance has changed due to the short circuit

deliberately created. In addition, in the case of short-circuit, the low frequency band of

FRA spectrum (to the first anti-resonance) shows a different trend as compared to the

original spectrum. In fact, its falling trend is slightly moderate. This means that

transformer HV winding experiences less inductance when having internal short-circuit

than ‘normal’ state in LV winding. The first minimum peak has shifted to higher frequency

and its magnitude is also reduced. Similar result for inductance reduction in transformer

winding was obtained on a different test object in Chapter 6, subsection 6.6.2.

To examine the recommended solution S î and S î were calculated and Z î n and

Z î m were derived. Table G.2 shows the calculated values for Z î n and Z î m.

According to Table G.2, n≠m and the transformer is suspected to have mechanical defect.

Therefore, FRA statistical indicators are required to be taken into consideration. The

reference and measured FRA spectra as well as their derivative curves for HV winding are

illustrated in Fig. G.3.

105

106

107

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Magnitude [

dB

]

Measured Trace (HV winding)

Original Trace (HV winding)


256

(a)

(b)

Figure G.3. HV winding spectra (a) Reference spectrum and its derivative (100 kHz – 20 MHz), (b) Measured

spectrum and its derivative (100 kHz – 20 MHz).

105

106

107

-40

-30

-20

-10

0

10

Frequency [Hz]

Magnitude [

dB

]

Derivative spectrum

Refrence spectrum (HV winding)

1.840 MHz

105

106

107

-40

-30

-20

-10

0

10

Frequency [Hz]

Magnitude [

dB

]

Derivative spectrum

Measured spectrum (HV winding, short circuit in LV)

2.102 MHz


257

Table G.2. Calculated values for reference spectrum Z în, and measured spectrum Z îm.

Z î n [Hz] Z î m [Hz]

1840822.911 2102123.015

2961946.861 3382387.641

3028202.732 3496511.071

3420008.789 3654687.782

3535401.565 3777998.756

3654687.782 4765873.542

3777998.756 4872481.525

4765873.542 5626005.61

4872481.525 6284034.927

5688581.755 8952855.389

6284034.927 9673607.699

9153122.082 9781203.931

10805053.01 10805053.01

11674917.05 11546489.36

12203072.06 12203072.06

12896990.82 12896990.82

14891496.72 14891496.72

15913342.45 15913342.45

18578695.3 17972300.97

18785339.78 18172200.75

---- 18578695.3

---- 18785339.78

G.4 Winding Deformation vs. Temperature and Moisture Influences on FRA

Spectrum

To compare the frequency bands influenced by winding deformation and

Temperature/Moisture variation in FRA spectrum, some mechanical changes were

emulated on model transformer and the results are compared with insulation parameters

changes.

G.4.1 Influence of Internal Short-circuit

At first, the frequency response trace of the model transformer is recorded from the HV

side while the LV side is left open circuit. In order to study the internal short circuit, the

modification involved the short circuit on LV terminals is performed on the test object.

This modification can in turn block the flux flow in the transformer air core and model

winding internal short-circuit. Figure G.4(a) shows the frequency response spectra when

the LV winding is left open- and short-circuited, respectively. According to Fig. G.4(a),

when the LV winding is short-circuited, changes to the frequency response of the HV

winding occurred in the range from 20 Hz to 3 MHz. For frequencies above 3 MHz, there is

no significant discrepancy between recorded spectra. The reason lies in the fact that the


258

winding self-inductance has changed due to the short circuit deliberately created. In

addition, in the case of short-circuit, the low frequency band of FRA spectrum shows

different trend to reach to the first anti-resonance as compared to the original spectrum.

In fact, its falling trend is slightly moderated. This means that transformer HV winding

experiences less inductance when having internal short-circuit than ‘normal’ state in LV

winding.

G.4.2 Influence of Tank Grounding

In order to conduct this experiment, the aluminium tank of the test object was grounded

through laboratory earth. FRA trace for the HV winding was measured while LV terminals

were left open circuit. HV winding FRA spectra for isolated and grounded tank are shown

in Fig. G.4(b). According to Fig. G.4(b), the number of resonance frequencies for the case of

grounded tank has increased as compared to the un-grounded case, in particular the mid-

frequency band. It can be explained through shunt capacitance increment. In addition, the

magnitude of the first anti-resonance in HV trace has varied considerably due to changes

in conductance between HV and LV windings with respect to the tank. This can be

interpreted by winding loss factor (conductivity). In fact, any changes in winding loss

factors can result in FRA magnitude of resonance/anti-resonance points to change

accordingly. The frequency band 300 kHz – 20 MHz is affected under such a circumstance.

These results are already compared with moisture and temperature influences on FRA

spectrum on similar winding discussed in Chapter 7 for the model test object to highlight

the affected frequency bands due to different incidents.

104

105

106

107

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

HV winding (orginal spectrum)

HV winding (short circuit in LV winding)

Frequency band: 4 kHz - 20 MHz

Deviated Frequency Band

(a)


259

104

105

106

107

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]

HV winding spectrum (Grounded Tank)

HV winding spectrum (Isolated Tank)



(b)

104

105

106

107

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

Ma

gn

itu

de

[d

B]





(c)

Figure G.4. (a) HV winding frequency response spectrum when LV winding is open-circuited (original spectrum) and short-circuited (affected frequency-band 20 Hz-3 MHz),(b) FRA spectra of HV winding for isolated and grounded tank (affected frequency-band 300 kHz-20 MHz), (c) FRA spectra for HV winding due to moisture migration from paper into the oil insulation at 30°C and 90°C, replotted from Fig. 7.4 for comparison (affected frequency-band 800 kHz-20 MHz).

Based on comparison, it can be summarized that the influence of moisture migration

appears in the mid- and high-frequency band of the FRA spectrum, the indicated frequency

values are estimated in Fig. G.4 caption. In contrast, the mechanical defects can influence


260

each and every part of the entire frequency band randomly. Indeed, FRA trace deviation

due to mechanical defects is certainly influenced by the particular type of deformation.

Self- and mutual inductances, series and shunt capacitances or even winding resistance

variation due to the winding deformation has its own impacts on the FRA trace. As it was

discussed in previous section, for accurate interpretation of the FRA results, these impacts

should be distinguished from the moisture influence using recommended thechnique.

G.5 Conclusion

The recommended solution to distinguish the insulation characteristic impacts from the

mechanical deformation influence on the FRA trace was examined on deviated FRA

spectra. It led to a correct decision on the test object. This solution is recommended to be

implemented as preliminary stage in FRA interpretation.

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261

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Transformer Winding Deformation and Insulation Characteristic Effects on Frequency Response

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