fra transformer model with application on time domain reflectometry

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A FRA Transformer Model with Application on Time Domain Reflectometry

Hanif Tavakoli

Doctoral Thesis in Electromagnetic Engineering

Stockholm, Sweden 2011

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Royal Institute of Technology (KTH) School of Electrical Engineering Division of Electromagnetic Engineering Teknikringen 33 SE– 100 44 Stockholm, Sweden TRITA-EE 2011:071 ISSN 1653-5146 ISBN 978-91-7501-134-9 Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan framläggs till offentlig granskning för avläggande av teknologie doktorsexamen torsdagen den 15 december 2011 klockan 10.00 i sal F3, Lindstedtsvägen 26, Kungliga Tekniska Högskolan, Stockholm. © Hanif Tavakoli, oktober 2011 Tryck: Universitetsservice US AB

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To my mother

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Abstract Frequency response analysis (FRA) is a frequency-domain method which is used to detect mechanical faults in transformers. The frequency response of a transformer is determined by its geometry and material properties, and it can be considered as the transformer’s fingerprint. If there are any mechanical changes in the transformer, for example if the windings are moved or distorted, its fingerprint will also be changed so, theoretically, mechanical changes in the transformer can be detected with FRA. A problem with FRA is the fact that there is no general agreement about how to interpret the measurement results for detection of winding damages. For instance, the interpretation of measurement results has still not been standardized.

The overall goal of this thesis is to try to enhance the understanding of the information contained in FRA measurements. This has been done in two ways: (1) by examining the FRA method for (much) higher frequencies than what is usual, and (2) by developing a new method in which FRA is combined with the ideas of Time Domain Reflectometry (TDR). As tools for carrying out the above mentioned steps, models for the magnetic core and the winding have been developed and verified by comparison to measurements.

The usual upper frequency limit for FRA is around 2 MHz, which in this thesis has been extended by an order of magnitude in order to detect and interpret new phenomena that emerge at high frequencies and to investigate the potential of this high-frequency region for detection of winding deteriorations.

Further, in the above-mentioned new method developed in this thesis, FRA and TDR are combined as a step towards an easier and more intuitive detection and localization of faults in transformer windings, where frequency response measurements are visualized in the time domain in order to facilitate their interpretation.

Index terms: complex permeability, lumped circuit model, frequency response analysis, time domain reflectometry, high frequency modeling, transformer diagnosis, reluctance network method.

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Acknowledgements This doctoral thesis is based on results within the research group of Electrotechnical Modeling, at the Department of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology (KTH). The project which resulted in this thesis was funded by the Swedish Center of Excellence in Electric Power Engineering (EKC2). The financial support is gratefully acknowledged.

I would like to thank my supervisor Professor Göran Engdahl for his guidance during this project, interesting and useful discussions about transformers and transformer modeling.

Also, I would like to thank Dr. Dierk Bormann for his ideas and support, his fruitful collaborations with me, and his reviewing and correcting of my papers and models during this project. Without his help, most of the achievements in this project would not have been possible.

I also thank Professor Roland Eriksson and Professor Rajeev Thottappillil, former and present head of the department, respectively, for trusting me enough to employ me for this PhD project.

Furthermore, I acknowledge Associate Professor Martin Norgren for “kvalitetsgranskning” (quality review), and Peter Lönn for technical support with computer hardware and software. Also thanks to Carin Norberg for administration support.

I also thank Dr. David Ribbenfjärd, Johanna Rosenlind and Assistant Professor Patrik Hilber for our cooperation, and the rest of the people at the Electromagnetic Engineering Lab for friendship, discussions, lunches and refreshing coffee breaks.

And last but not least I would like to thank my mother for her endless love to me and for her patience during life’s trials and tribulations. Hanif Tavakoli Stockholm, Sweden, October 2011

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List of Publications

I. H. Tavakoli, D. Bormann, G. Engdahl, D. Ribbenfjärd, “Comparison of a Simple and a Detailed Model of Magnetic Hysteresis with Measurements on Electrical Steel”, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 28 No. 3, pp. 700−710, 2009.

II. H. Tavakoli, D. Bormann, G. Engdahl, “High Frequency Oscillation Modes in a

Transformer Winding Disc”, Studies in Applied Electromagnetics and Mechanics, Vol. 34, pp. 329−337.

III. J. Rosenlind, H. Tavakoli, P. Hilber, “Frequency Response Analysis (FRA) in the

Service of Reliability Analysis of Power Transformer”, Proceedings of International Conference on Condition Monitoring, Diagnosis and Maintenance - CMDM 2011.

IV. D. Bormann, H. Tavakoli, “Reluctance Network Treatment of Skin and Proximity

Effects in Multi-Conductor Transmission Lines”, submitted to IEEE Transactions on Magnetics.

V. H. Tavakoli, D. Bormann, G. Engdahl, “Time Domain Reflectometry for Fault

Localization in Transformer Windings”, submitted to International Journal of Applied Electromagnetics and Mechanics.

VI. D. Bormann, H. Tavakoli, “Reluctance Network Theory of Skin and Proximity

Effects in Strongly Coupled Multi-Conductor Transmission Lines”, submitted to IEEE Transactions on Power Delivery.

VII. H. Tavakoli, D. Bormann, G. Engdahl, “Fault Localization in Transformer Windings

using Time-Domain Representation of Response Functions”, submitted to European Transactions on Electrical Power.

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Contents ABSTRACT

ACKNOWLEDGMENTS

LIST OF PUBLICATIONS

1 INTRODUCTION ....................................................................................................................................... 1

1.1 BACKGROUND AND AIM ....................................................................................................................... 1 1.2 OUTLINE OF THE THESIS ....................................................................................................................... 2

2 FREQUENCY RESPONSE ANALYSIS (FRA) OF POWER TRANSFORMERS .............................. 3

2.1 FREQUENCY RESPONSE MEASUREMENTS ............................................................................................. 3 2.1.1 Impulse Response Method............................................................................................................... 3 2.1.2 Frequency Sweep Method ............................................................................................................... 4

2.2 MECHANICAL FAULTS IN A TRANSFORMER .......................................................................................... 4 2.3 DIAGNOSING MECHANICAL FAULTS IN A TRANSFORMER WITH THE HELP OF FRA .............................. 4

3 CORE MATERIAL MODEL AND MEASUREMENTS ........................................................................ 7

3.1 BACKGROUND ...................................................................................................................................... 7 3.2 COMPLEX-PERMEABILITY MODEL........................................................................................................ 8 3.3 DETAILED HYSTERESIS MODEL ............................................................................................................ 9

3.3.1 Static Hysteresis .............................................................................................................................. 9 3.3.2 Excess Losses ................................................................................................................................ 11

3.4 MEASUREMENTS AND MODEL ADJUSTMENTS .................................................................................... 11 3.5 COMPARISON BETWEEN MODEL AND MEASUREMENTS ...................................................................... 13

4 HIGH FREQUENCY WINDING MODEL ............................................................................................ 17

4.1 TRANSFORMER WINDING MODEL FOR HIGH FREQUENCY APPLICATIONS .......................................... 17 4.2 THREE DIFFERENT RESOLUTIONS OF THE MODEL............................................................................... 17 4.3 CALCULATION OF THE CAPACITANCES ............................................................................................... 19 4.4 CALCULATION OF THE INDUCTANCES AND RESISTANCES ................................................................... 20 4.5 STATE SPACE MODEL FOR THE WINDING ........................................................................................... 21 4.6 COMPARISON BETWEEN THE THREE MODELS ..................................................................................... 25

5 FREQUENCY RESPONSE MEASUREMENTS................................................................................... 27

5.1 IMPEDANCE MEASUREMENT DEVICE AND DIMENSIONING OF THE EXPERIMENTAL SETUP ................. 27 5.2 IMPEDANCE MEASUREMENT RESULTS................................................................................................ 28

6 MODEL VERIFICATION ....................................................................................................................... 31

6.1 COMPARISON OF MODEL WITH MEASUREMENTS................................................................................ 31

7 INTERPRETATION................................................................................................................................. 35

7.1 EXPLANATION OF THE DIFFERENT OSCILLATION MODES ................................................................... 35 7.1.1 Radial Resonance Modes .............................................................................................................. 35 7.1.2 Azimuthal Resonance Modes......................................................................................................... 37

7.2 RADIAL AND AZIMUTHAL RESONANCES FOR A WINDING WITH MORE THAN ONE DISC..................... 38

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8 RELUCTANCE NETWORK METHOD FOR INCLUSION OF THE SKIN AND PROXIMITY EFFECTS............................................................................................................................................................. 41

8.1 SKIN AND PROXIMITY EFFECTS .......................................................................................................... 41 8.2 USING RELUCTANCE NETWORK METHOD (RNM) TO CALCULATE THE FREQUENCY DEPENDENT

INDUCTANCE AND RESISTANCE MATRICES ....................................................................................................... 42 8.3 RNM FOR HIGH FREQUENCIES ........................................................................................................... 43

8.3.1 Principle of the Reluctance Network Description ......................................................................... 44 8.3.2 Calculation of the Reluctances for High Frequencies................................................................... 46

8.4 EXTENSION OF RNM TO LOW FREQUENCIES ...................................................................................... 49 8.4.1 Summary of the Generalized Approach......................................................................................... 49 8.4.2 Exact Solution of a Single-Slab Problem ...................................................................................... 51

8.5 EXTENSION TO VARIABLE GAP WIDTH............................................................................................... 54 8.6 COMPARISON WITH FEM CALCULATIONS .......................................................................................... 56

8.6.1 Results for Rectangular Conductors ............................................................................................. 57 8.6.2 Results for Round Conductors....................................................................................................... 60

8.7 CONCLUSIONS ABOUT RNM ............................................................................................................... 62

9 TIME DOMAIN REFLECTOMETRY (TDR)....................................................................................... 63

9.1 COMBINATION OF FRA AND TDR ...................................................................................................... 63 9.2 BASIC IDEAS BEHIND THE TDR TECHNIQUE ....................................................................................... 63 9.3 SIMPLIFIED MODEL OF A TRANSFORMER WINDING AND THE EFFECT OF DISPERSION ........................ 64 9.4 THE TRANSFORMER WINDING MODEL USED FOR TDR SIMULATIONS ............................................... 66 9.5 THE EFFECT OF CHOICE OF RISE-TIME FOR THE APPLIED VOLTAGE................................................... 66 9.6 SIMULATIONS WITH AND WITHOUT WINDING DAMAGES .................................................................... 68 9.7 CONVERSION OF IN-IMPEDANCE MEASUREMENTS TO DTDR SIGNALS .............................................. 75

10 SUMMARY, CONCLUSIONS AND FUTURE WORK........................................................................ 77

REFERENCES.................................................................................................................................................... 79

APPENDIX A ...................................................................................................................................................... 85

APPENDIX B ...................................................................................................................................................... 89

LIST OF SYMBOLS .......................................................................................................................................... 95

LIST OF ACRONYMS .................................................................................................................................... 103

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1 Introduction In this Chapter, a short background information about power transformers is given, and the aim, outline and structure of the thesis are presented.

1.1 Background and Aim

A power transformer is an electric device employed in electric power systems for transmission and distribution of electric power, and it is used when there is a need for a voltage transformation. In a transformer, electric energy is transferred between different electrical circuits by the use of electromagnetic induction.

Power transformers are very large and expensive so preventive replacement of old ones with new ones, in order to increase the system reliability, is often not economically justified; therefore, they are supposed to be used for maximum number of years, and they are also supposed to be in continuous operation for reliable and uninterrupted power supply. These goals can only be achieved through proper maintenance and condition monitoring, so that deteriorations in the transformer, which may lead to a failure, can be detected at an early stage.

A number of different methods are routinely used for condition monitoring and diagnostics of transformers, for instance oil analysis, winding resistance measurements, winding transfer ratio measurements, dielectric response measurements (DFR) and frequency response analysis (FRA). This thesis concentrates on the method of FRA, which is one of the more recent and advanced diagnostic methods. A fundamental problem with FRA is the fact that despite the advancements and improvements of this method since its invention in 1978, there is still no general agreement about how to draw reliable conclusions from it regarding the transformer health. In particular, the interpretation of FRA measurement results has still not been standardized.

So the goal of this thesis is to try to improve the understanding of the information contained in FRA measurements. This has been done in two ways: (1) by considering the FRA method for (much) higher frequencies than what is standard, and (2) by developing a new method in which FRA and “Time Domain Reflectometry” (TDR) are combined.

The usual upper frequency limit for FRA measurements is around 2 MHz, and this limit has been increased here by a factor of ten in order to investigate new high frequency phenomena and to examine whether this high frequency region has the potential to facilitate the detection of winding deformations. It was found that, for the disc geometry employed here, two classes of internal resonance modes exist for higher frequencies, so-called “radial” and “azimuthal” oscillation modes, and measurements show that these modes are very sensitive to small changes in the disc geometry.

Time Domain Reflectometry (TDR) is a method traditionally used for detection and localization of changes in transmission lines. Time-domain diagnostic methods have scarcely been applied to power transformers at all in the literature since the late 1970s, after the low-voltage impulse method [1] had declined in popularity. In the new time domain method described in this thesis, principles of FRA and TDR have been combined to create an easy and intuitive approach for detection and localization of winding faults, where frequency response

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measurements are visualized in the time domain in order to facilitate their interpretation. Thereby mechanical changes in transformer windings can be identified and located in a simple manner.

As tools for carrying out the steps mentioned above, models for the magnetic core and the winding have been developed and verified by comparison to measurements.

It may be mentioned that during the work on this thesis an attempt has been made also to combine the method of FRA with the formal reliability analysis of fleets of power transformers or individual units (published as paper no. III cited in the List of Publications on page ix above). However, this work is only loosely connected to the main body of this thesis and therefore not included here.

1.2 Outline of the Thesis

The thesis is structured as follows: Chapter 2 presents a general overview of FRA and measurement techniques, mechanical

faults in transformers and the way these faults can be detected by FRA. In Chapter 3, which deals with the core material model, the developed simple complex

permeability model, a detailed hysteresis model and the measurements are presented and compared to each other. The complex permeability model has been developed by the author with the support of Dr Bormann, the detailed hysteresis model has been developed by Dr Ribbenfjärd, and the measurements have been performed by the author.

Chapter 4 presents the developed lumped element winding models, the formulas for the lumped element parameters, and the state space equation which is used for calculating the currents and voltages in the model. This Chapter is concluded by a comparison between the impedance of three models with different levels of discretization in the frequency spectrum. The winding models have been developed by the author with the support of Dr Bormann.

Chapter 5 presents the winding measurement set-up and the performed measurements (carried out by the author supported by Dr Bormann), and in Chapter 6, the winding models are verified by comparison with measurements.

Chapter 7 deals with the explanation and interpretation of the different oscillation modes of a single disc (accomplished by the author in cooperation with Dr Bormann), and is concluded by analyzing the state of the modes when a winding consists of more than one disc.

In Chapter 8, for calculation of frequency dependent inductances and resistances, a reluctance network method, which includes the impact of the skin and proximity effects, is deduced and compared to finite element calculations. The theory behind the reluctance network method has to a considerable extent been developed by Dr Bormann with the support of the author, and the implementation of it in MATLAB and creation of finite element models for the purpose of comparison have been carried out by the author.

In Chapter 9, the time domain method TDR, its benefits, and the way it can be combined with FRA for diagnosis of transformer winding faults are presented. Also, the developed winding models are used in the time domain to simulate the detection, localization and identification of some different kinds of winding faults. This has been performed by the author, supported by Dr Bormann.

And finally, Chapter 10 contains a summary, conclusions and some suggestions for future work.

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2 Frequency Response Analysis (FRA) of Power Transformers

In this Chapter, the method of FRA and the measurement techniques are explained. Further, the different mechanical faults in transformers and the way they can be detected by FRA are reviewed.

2.1 Frequency Response Measurements

Frequency Response Analysis (FRA) is a powerful method for characterizing a system by analyzing its frequency response, which is uniquely defined by the system parameters; this means that FRA can be used to either design a system or to analyze an existing one. It is the phase and magnitude response of a system when subjected to sinusoidal inputs, and it has become a popular method for evaluating the mechanical condition of the windings and the clamping structure of power transformers [2−16].

The first technical work describing the possibility of using the FRA technique for diagnosing mechanical faults inside power transformers was published by Dick and Erven [17] in 1978. Since then, FRA has been gaining popularity among researchers and utilities as a potential method to detect mechanical changes inside power transformers.

The frequency range for FRA is generally from 10 Hz up to 2 MHz and the evaluation is based on the fact that the frequency response of a transformer is defined by its capacitance and inductance distributions, which are determined by the geometrical construction of the transformer and characteristics of materials used. Therefore, mechanical deformations change the capacitive and inductive parameters, yielding deviations in the FRA spectrum. This means that FRA is basically a comparative method, in which a fingerprint measurement taken at an earlier stage is compared with a measurement taken at a later stage, perhaps after relocation or during a maintenance operation. Then the changes in characteristics of the response are analyzed to detect mechanical changes inside the transformer.

Frequency response can either be measured directly by sweeping the frequency (sweep frequency method) or be estimated from impulse response measurements. Both methods have advantages and disadvantages. For example, the impulse response method needs less measuring time, but it is very noise sensitive. On the other hand, the frequency sweep method takes a little longer time for the measurements, but it is not so noise sensitive.

2.1.1 Impulse Response Method

In the impulse response method, an impulse voltage that has adequate frequency content is applied to the test object and both the applied voltage and some resulting response voltage or current are simultaneously measured. This method is based on the definition of the transfer function which says that the transfer function doesn’t depend on the applied signal when the system is linear and time invariant. Then both of the measured signals are numerically transformed into the frequency domain using Fast Fourier Transform (FFT). The ratio

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between the FFT of the response signal and the applied signal is the frequency response of the corresponding transfer function.

This method has been used by many researchers for diagnosing mechanical faults in power transformers [5]. Limitation of the excitation source that can produce enough energy in the whole frequency band of interest, reduced energy level of injected impulse at higher frequencies limiting the upper limit of the calculated frequency response, and the need for good noise prevention techniques are some of the disadvantages of the impulse response method.

2.1.2 Frequency Sweep Method

In this method, a sinusoidal exciting voltage is applied and the magnitude and phase of the response voltage or current are measured at different frequencies. This means that this is a direct method for determining the frequency response, since the final result is already available after sweeping the predefined frequency range. This is the method which has been used in this thesis.

2.2 Mechanical Faults in a Transformer

A transformer can be damaged due to a variety of reasons. Some of these reasons are: insulation degradation, transportation, installation, and the forces inside it due to the interaction of the current and the (leakage) magnetic flux density according to:

F I B (1) where F is the force, I is the current in the winding and B is the magnetic flux density. According to Eq. (1) there will be heavy mechanical stresses in the transformer in case of a sudden short circuit fault, as the current flowing through the winding at that time is enormous. Eq. (1) also means that there will be two kinds of force vectors generated by the axial component of the leakage flux density (radial force) and by the radial component of the leakage flux density (axial force). The radial forces tend to squeeze the inner winding and expand the outer winding, resulting in circlet buckling in the former and circlet deformation in the latter due to imbalance of the radial pressure. In contrast, the axial forces tend to displace the windings axially in relation to each other, perpetrating pressure on the clamping structure. It may bend conductors between rigid axial spacers, and during winding movements the insulation between the turns could be abraded, which can lead to short circuiting and damaging of the windings in the same layer, the same disc, different layers or different windings. Short circuit faults can cause great harms, because if the clamping pressure is not capable to counteract the involved forces, significant winding deformation or even break down of windings can happen almost immediately, often convoyed with shorted turns.

2.3 Diagnosing Mechanical Faults in a Transformer with the Help of FRA

It is generally said that FRA has the capability of identifying faults of the types: core movement, winding deformation, winding movement, broken or loose winding or clamping structure, partial collapse of the winding and short-circuited turns or open circuit windings. Interpretation of FRA results, when searching for mechanical changes inside a transformer, has neither been standardized nor fully agreed among researchers yet. Therefore, one may

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find several different types of transfer functions (in-impedance, transfer-impedance, voltage transfer ratio, etc.), different measurement techniques and different ways of interpreting FRA results. But as mentioned before, FRA is essentially a comparative method and therefore, a fingerprint response measurement of the same transformer which is going to be diagnosed or a sister unit should be available for comparison with the present measurement. When the measurement is compared with the reference set, then the changes in the frequency response which could be identified as mechanical faults are as follows: Abnormal shifts in the existing resonances Emergence of new resonances or evaporation of the existing ones Considerable changes in the overall shape of the frequency response. By comparison of the new and the old measurements, or by comparison of the measurement results from different phases of the same transformer (since normally not all phases are affected in the same way by the fault [18]), an expert can identify possible faults.

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3 Core Material Model and Measurements In this Chapter, which is largely based on [19], a simple core material model is developed. For efficient magnetic field calculations in electrical machines and transformers, the hysteresis and eddy current losses in laminated electrical steel must be modelled in a simple and reliable way. Therefore, in this chapter, a frequency dependent complex permeability model and a more detailed model (describing hysteresis, classical eddy current effects, and excess losses separately) are compared with single sheet measurements. It is discussed under which circumstances the simple complex-µ model is an adequate substitute for the more detailed model.

3.1 Background

Recent research has resulted in detailed models of the magnetic hysteresis and loss mechanisms in a wide frequency range [20−22]. Although these models provide a good description of magnetic material properties or of simple reluctance circuits based on them, they are too demanding numerically to be incorporated into a full-scale magnetic field simulation of a realistic geometry, as with a FEM or FDM calculation tool. In other words, while such a detailed simulation of the H-B relation of a single or a few interacting cells is still perfectly feasible, simulating thousands or ten thousands of them simultaneously may be inconvenient or impossible.

Moreover, in many practical situations a detailed description is not required either (as is the case in this thesis). Often the goal is to obtain a good estimate of some local or global quantity containing much less information than the detailed local H-B relation, such as the local losses causing dangerous hot spots, or simply the total losses in a machine relevant for cooling or economic reasons. For such applications it is desirable to use a simple model of magnetic hysteresis and losses, which can easily be incorporated in field calculation tools but which at the same time is sufficiently close to reality, within the frequency range of interest for the specific application. Such a model is the description of laminated magnetic materials by a suitable frequency dependent complex permeability, which is the most general linear description of a local and isotropic H-B relation. If desired, it can easily be extended to a nonlocal and/or anisotropic H-B relation by turning µ from a scalar function of position into a distance dependent integral kernel and/or tensor, respectively [23].

In this thesis, it is discussed to which extent results from measurements on strips of electrical steel, obtained with a single-sheet tester, can reliably be described by a simple complex permeability function of frequency. Both the resulting H-B curves and the effective complex permeability are compared to the measured data at different frequencies. For comparison, simulation results obtained with a much more detailed model of the magnetic hysteresis, eddy current and excess losses are also reported.

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3.2 Complex-Permeability Model

Reduced to its simplest terms, hysteresis introduces a time phase difference between B and H. B is assumed to lag H by a constant angle θh called the hysteresis angle. In such a description, harmonics introduced by saturation are ignored, and the hysteresis loop becomes an ellipse whose major axis is inclined by an angle θh relative to the H-axis. Using complex field

components B and H , a low-frequency complex permeability including hysteresis can be defined as

hjh 0 r

ˆˆ e

ˆB

H . (2)

In addition to this, eddy currents in the lamination sheets introduce frequency dependence. The well-known procedure [24] for deriving the effective frequency dependent complex permeability is briefly sketched below. Faraday’s law

t

B

E (3)

and Ampere’s law

t

D

H J , (4)

in combination with the constitutive relations

J E , D E , hB H , (5)

and time-harmonic assumption lead to

2h

ˆ ˆj j 2H H H . (6)

For lower frequencies when wave propagation can be ignored (i.e., ), one has 2

hˆ ˆj .

Fig. 1: Laminate infinite in z direction, with a width in y direction much larger than its thickness 2b, exposed to

a H field in z direction. For analysis of the magnetic field in a laminate, the simple geometry illustrated in Fig. 1 is appropriate. The magnetic field is applied in the z direction, hence the only component of the magnetic field strength is Hz which varies only in the x direction, Hz = Hz (x). In one dimension, Eq. (6) reduces to

x

y( )zH x

2b

9

2

22

ˆzz

HH

x

, (7)

which has the general solution

ˆ ˆ1 2( ) e ex x

zH x A A . (8)

The field strength on the both sides of the laminate is assumed to be H0. For the reason of symmetry the following condition is obtained

0( ) ( )z zH b H b H . (9)

The final expression for the magnetic field strength then becomes

0

ˆcosh( )

ˆcoshz

xH x H

b

. (10)

The effective, complex permeability of a lamination is given as the average magnetic flux density B in the laminate normalized to the surface magnetic field strength H0 ,

eff eff effˆ j h h0 0

ˆ1 tanh( )ˆ ˆ( )d

ˆ2

b

z

b

B bH x x

H H b b

. (11)

This expression accounts for the effect of hysteresis without saturation, and the effect of eddy currents. It is assumed here that additional (or “excess”) losses are either negligible or have a similar frequency dependence so that they can be incorporated in the expression (11) for eff .

3.3 Detailed Hysteresis Model

Later in this Chapter, some results obtained with a more detailed model of the magnetic hysteresis, eddy current and excess losses will be reported, so therefore, this more detailed model, which has been developed by Dr. David Ribbenfjärd in [21], is described here in short (for a thorough description and explanation, see [21]).

The total hysteresis is a combination of three different phenomena, namely, static hysteresis, eddy current effects and excess eddy currents. For the detailed hysteresis model, the following approach has been used. The static hysteresis is modeled using Bergqvist’s lag model [25, 26], the classical eddy currents are modeled using Cauer circuits [22, 27], and the excess losses are modeled using an approach by Bertotti [20].

3.3.1 Static Hysteresis

The Bergqvist’s lag model of static hysteresis starts from the idea that the magnetic material consists of a finite number of pseudo particles np, i.e., volume fractions with different magnetization. The total magnetization is then a weighted sum of the individual magnetization of all pseudo particles.

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Fig. 2: An-hysteretic curve (left), play operator (middle), and resulting hysteresis curve (right); figure taken

from [21]. The hysteresis curve for one particle is introduced by applying a “play operator” with a play equal to the “pinning strength” k (which will determine the width of the hysteresis curve) on the an-hysteretic curve, see Fig. 2, where m is the magnetization of the actual pseudo particle, and η is the back field i.e. the field that will give the magnetization m if no hysteresis is present.

Using a population of pseudo particles with different pinning strengths allows constructing minor loops. An individual pinning strength λik is assigned to every pseudo particle, where k is the mean pinning strength, and λi is a dimensionless number for particle i. The total magnetization is then given by a weighted superposition of the contributions from all pseudo particles (Fig. 3).

Fig. 3: Weighted superposition of the contributions from pseudo particles describes a minor loop; figure taken

from [21]. The expression

an ss

2( ) arctan

2

HM H M

M

(12)

is used for the an-hysteretic magnetization, where Ms is the magnetization saturation and χ is the susceptibility at H = 0. For infinite number of pseudo particles, the total magnetization of the material is then given by

an an

0

( ) ( ) ( )dkM cM H M P H

, (13)

where c is a constant that governs the degree of reversibility, and the integral describes the hysteretic behaviour (irreversible part). Pλk is a play-operator with the pinning strength λk, and ς(λ) is a density function describing the distribution of the pseudo particles. Finally, the magnetic flux density is obtained from B = μ0(H+M).

m m

H H

k

k2k

21k 22k 23k

HHHH

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3.3.2 Excess Losses

Excess losses are caused by microscopic eddy currents induced by local changes in flux density due to domain wall movements. For the detailed model, an approach described by Bertotti [20] is used. In this approach, a number of active correlation regions are assumed randomly distributed in the material. The correlation regions are connected to the micro-structure of the material like grain size, crystallographic textures and residual stresses. In Bertotti’s model, the resulting contribution to the magnetic field strength is given by

0 0excess 2

0 0

4 2 d d1 1 sign

2 d d

n V G bw B BH

n V t t

, (14)

where w is the width of the laminate and 2b, as before, its thickness. G is a parameter depending on the structure of the magnetic domains. n0 is a phenomenological parameter related to the number of active correlation regions when the frequency approaches zero, whereas V0 determines to which extent micro-structural features affect the number of active correlation regions.

The parameters n0 and V0 are by definition frequency independent, but they are expected in reality to depend on the amplitude of the B field [28]. Since the precise form of this dependence is unknown, their values are usually adjusted empirically for a given amplitude. In the simulations reported here one set of (empirically determined) values is used, although the amplitude of the B field varies slightly in the measurements.

3.4 Measurements and Model Adjustments

The magnetic measurements were carried out using a Single Sheet Tester. It consists of two equal U-shaped yokes placed face-to-face to each other (Fig. 4). The magnetic sheet to be tested is placed between the yokes and most of the flux is forced through it due to its high permeability. For the measurement of the flux in the test material, a coil is surrounding the strip which is connected to a flux meter. The magnetic field strength is measured with a Hall probe placed close to the surface of the sample and connected to a Tesla meter. A sinusoidal H field was applied to the sample; the H and B field values were measured for 100 periods and numerically filtered. Thereafter, the mean values at different phase angles of the B and H fields were calculated and used for the experimental verification of the complex-µ model.

Fig. 4: Cross sectional view of the Single Sheet Tester.

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The measured (mean-value) H-B curve is then approximated with a complex-µ ellipse characterized by the permeability meas . This is done by matching both the peak values Hp , Bp

and the area A of the ellipse to the measured results (i.e. the measured H-B curve and the complex-µ ellipse will have the same area and peak values). This is of course appropriate as long as the shape of the measured H-B curve is close to an ellipse, i.e., if saturation effects are not too pronounced. The power loss per cycle, the area A within the H-B curve, is given by the integral

measmeas meas meas

0

dd d

d

T HA B H B t

t , (15)

where Bmeas and Hmeas are time dependent fields (correlated trough meas ) defined as

approximations of the time dependent measured B and H fields, respectively, and T is the duration of a period. If the measured H field is assumed to vary sinusoidally, then one can use the approximation

jmeas p p( ) Re e cos( )tH t H H t , (16)

with the derivative

measp

d ( )sin( )

d

H tH t

t . (17)

The approximation of the measured B field becomes then

j jmeas meas p meas meas p p meas measˆ( ) Re e Re j e cos( ) sin( ) .t tB t H H H t t

(18) By inserting Eq. (18) and (17) into Eq. (15) one gets

meas 2p

A

H

. (19)

Furthermore, from the relation meas p pˆ H B one obtains

2

2 2 2 pmeas meas meas

p

ˆB

H

, (20)

which implies

2

2pmeas meas

p

B

H

. (21)

Both meas and meas are functions of frequency.

13

-400 -300 -200 -100 0 100 200 300 400-1.5

-1

-0.5

0

0.5

1

1.5

H

B

(a)-400 -300 -200 -100 0 100 200 300 400-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

H

B

(b)

Fig. 5: H-B curves from measurements (blue) and complex-µ model (green) withmeas measmeas

ˆ j , for

(a) f = 50 Hz and (b) f = 400 Hz. Fig. 5 compares the measured H-B curves with complex-µ ellipses, generated with the adapted meas at frequencies f = 50 Hz and 400 Hz.

eff as defined in Eq. (11) is a function of frequency and of a vector x = (r , θh , b2)

containing the model parameters. It is adjusted to measured data by numerically minimizing the expression

eff meas

2

1

( ) ( )ˆ ˆ, i i

N

i

x (22)

with respect to x. meas ( )ˆ

i are the “measured” complex permeability values, defined by

Eq. (19) and (21), at N different frequencies ωi = 2πfi, i = 1, …, N. Measurements at N = 9 different frequencies ranging from 50 Hz to 2 kHz (see Fig. 6 and Fig. 7 below) were performed on a 100 mm × 3.2 mm strip of the non-oriented magnetic material M600 with a thickness of 2b = 0.5 mm.

3.5 Comparison between Model and Measurements

Since the measurement setup was quite sensitive to noise, the measurements had to be numerically filtered. Adjusting eff to the filtered data using Eq. (22), the following model

parameter values were obtained: μr = 3366, θh = 0.477 rad, and b2 = 0.243 Sm, i.e., = 3.89×106 S/m which is somewhat larger than the true DC conductivity dc = 3.33×106 S/m since excess losses were included in the classical phenomenological form (11). In Fig. 6, the real and imaginary parts of the “measured” complex permeability are compared at different frequencies with the adjusted eff .

H [A/m]

B [

Tes

la]

H [A/m]

B [

Tes

la]

14

100

102

104

106

108

0

500

1000

1500

2000

2500

3000

Frequency [Hz]

'eff

/0

''eff

/0

'meas

/0

''meas

/0

Fig. 6: Real and imaginary parts of the “measured” complex permeability (symbols) and of the fitted

permeability function (curves), normalized by 0. The agreement is quite satisfactory considering the simplicity of the model, especially at higher frequencies. The deviation between meas and eff at the lowest frequencies is

probably due to saturation effects which are not properly taken into account by the expression (11) for eff , see for instance the measurement at 50 Hz (Fig. 5(a)), where the

amplitude had to be chosen large enough for the signal not to be covered by noise. Below, the H-B hysteresis curves are shown for all measured frequencies. Measurement,

simple model, and detailed model are represented by solid green lines, dashed blue lines and dotted red lines, respectively.

-400 -300 -200 -100 0 100 200 300 400-1.5

-1

-0.5

0

0.5

1

1.5f = 50 Hz

H

B

-400 -300 -200 -100 0 100 200 300 400-1.5

-1

-0.5

0

0.5

1

1.5f=100

H [A/m]

B [

Tes

la]

H [A/m]

B [

Tes

la]

15

-400 -300 -200 -100 0 100 200 300 400-1.5

-1

-0.5

0

0.5

1

1.5f = 200 Hz

H

B

-400 -300 -200 -100 0 100 200 300 400-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1f = 400 Hz

B

-400 -300 -200 -100 0 100 200 300 400-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1f = 500 Hz

B

-400 -300 -200 -100 0 100 200 300 400-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8f = 800 Hz

B

-400 -300 -200 -100 0 100 200 300 400-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8f = 1000 Hz

B

-400 -300 -200 -100 0 100 200 300 400-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6f = 1250 Hz

H

-500 -400 -300 -200 -100 0 100 200 300 400 500-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6f = 2000 Hz

B

Fig. 7: H-B curves from measurements (solid green line), detailed model (dotted red line) and complex-µ

model (dashed blue line) with µ calculated from expression (11), at different frequencies ranging from 50 Hz to 2 kHz.

H [A/m]

B [

Tes

la]

H [A/m]

B [

Tes

la]

H [A/m]

B [

Tes

la]

H [A/m]

B [

Tes

la]

H [A/m]

B [

Tes

la]

H [A/m]

B [

Tes

la]

H [A/m]

B [

Tes

la]

16

The above way of defining a “best fit” of ellipses to the more complicated H-B hysteresis relations approximately preserves both H and B amplitudes and magnetic losses in the whole frequency range. This is illustrated in the Fig. 7, where the measured H-B curves are compared with the corresponding complex-µ ellipses and the detailed model at different frequencies. As can be seen, the simple model agrees very well with the measurements as long as saturation is not too strong, which means for low amplitude fields and/or for frequencies higher than about 200 Hz.

17

4 High Frequency Winding Model In this Chapter, a winding model based on a lumped element approach with three different levels of discretization has been developed. The developed models are analyzed using state space analysis in the frequency domain and the impedances for the three different models are compared to each other to detect the new phenomena emerging for higher frequencies as the model discretization is made finer and finer. Parts of this chapter are based on [29].

4.1 Transformer Winding Model for High Frequency Applications

In the field of transformer winding modeling, various approaches and tools are available. Among the most common tools are lumped element circuits [5, 30−32]. Usually all the turns of one or two discs are lumped together into one inductive element (“segment”) of the model, which leads to a decreased computation time but also to a reduction of the model’s upper frequency limit, typically to values around some 100 kHz.

In this thesis, lumped element models with much higher resolution (up to 4 segments per turn) are used, in order to increase the upper frequency limit.

Each lumped element represent a part, a section of the physical geometry with similar quantities like magnetic flux, electric potential, resistance and etc, and these lumped elements are connected together to represent the whole geometry. For power transformers, the windings are divided into finite sections represented by lumped resistance, inductance, and capacitance, where each section should be small enough so that it can be assumed that the current through it is constant and not influenced by the displacement current which will be noticeable at higher frequencies.

Up to a few hundreds of kilo hertz, the displacement current will not be so remarkable and can be approximated to zero, so that a winding can merely be modeled by means of its self and mutual inductances and resistances alone. But at higher frequencies, the aforementioned approximation is no longer valid, and the displacement currents from a section to other sections or to conductive bodies have to be accounted for, to make the model realistic, and this is done by means of capacitors. The total capacitance for a particular section is then split into two halves and located at both ends of the section. As mentioned above, the complete winding model is made by connecting all the sections together.

In this thesis, the transformer winding is a single phase continuous disc winding (with the order of turns as shown in Fig. 9), composed of quadratic discs as in Fig. 8. Also, since the low voltage winding is on a much lower voltage than the high voltage winding, it (the LV-winding) is replaced by ground in the models developed here.

4.2 Three Different Resolutions of the Model

Three different model resolutions are studied: in the models labeled 1, 2, and 3, each turn in the discs is modeled by one, two, or four segments, respectively. This implies that there are one, two, or four turn-to-turn capacitances between any two neighboring turns in a particular disc of the HV winding, respectively (see Fig. 8 from left to right). Further, there are one, two,

18

or four disc-to-disc capacitances between any two turns facing each other in two neighboring discs, respectively. Also, at the innermost and outermost turn of every disc, there are in addition one, two, or four capacitances to ground, respectively (these capacitances are not shown in Fig. 8 due to the lack of space).

Fig. 8: The three different levels of discretization with the nodes numbered in an increasing sequence from one

winding end to the other. The resolution increases from model 1 to model 3, and the disc-to-disc capacitances between neighboring discs are oriented perpendicular to the plane and are not shown here.

Fig. 9: The cross section of the continuous disc winding.

Fig. 10: One segment and its electrical circuit equivalence. It has been argued in the literature on the basis of simulations and measurements that the impulse voltage distribution in transformer windings, and the winding impedances for higher frequencies can be analyzed using air-core self and mutual inductances neglecting the iron core without serious error (see for instance [33−34]). This implies that the complex permeability model developed in Chapter 3 does not need to be included in the calculations of

Lself Rseg

h

i /2

ks

1 2 n w

htot

od

i

Ground

id

Ground Plane of symmetry

2n+1

2

1

3

2n-1 mutual inductances

model 22n

4

5

1

2

n+1

n

3

mutual inductances

model 1

~ U

I

4n+1

2 3

4 1

5

6

4n

4n-2

model 3

mutual inductances

self inductance resistance

4n-1

7

8

9

o / 2

19

inductances and losses. In order to further facilitate the calculation of self and mutual inductances, the discs have been chosen to be of quadratic shape here. The whole HV winding thus consists of straight segments which are all either parallel or perpendicular to each other, so that the self and mutual inductances can be calculated by simple analytic formulas [35]. Every segment is modeled by one resistance in series with one self inductance, as it is depicted in Fig. 10, and there are mutual inductances between any two parallel segments of the whole winding. For simplicity, self inductances and resistances are shown in Fig. 8 on one segment only, and some of the mutual inductances are indicated by arrows. In case of model 1, also the connection of the voltage source for impedance measurement for one single disc is shown.

Finally, all the inductance and capacitance parameters (i.e., all model parameters except for the damping resistances) are estimated from the physical winding geometry, and not fitted to measurements, and they are discussed in Sections 4.3 and 4.4. The models are analyzed by solving their state space equations [30−32] in both the frequency- and time domain, which is discussed in Section 4.5 of this chapter and in Chapter 9, respectively.

4.3 Calculation of the Capacitances

The capacitance depends on the geometry of the conductors and on the permittivity of the dielectric material between them. Here, the relation for the capacitance between two planar surfaces, capacitance = permittivity area / distance, is used [36]. The total capacitance between two turns Ctt in the quadratic disc is then approximately given by

i o itt i 0

i

24

2

hC

, (23)

where h is the height of the conductor, τi is twice the insulation thickness, εi is the relative permittivity of the conductor insulation, i is the “inner” length of one side of the square

discs, o is the “outer” length of one side of the square discs, and the addition of 2τi to h

accounts for the fringing effect [37]. The total capacitance between two discs, if they are close enough to each other and if air is used as insulation between them, is approximately given by

2 2o i

dd air 0i ks

( / 2) ( / 2)4C

(24)

where εair is the relative permittivity of air (=1) and τks is the distance between two discs. The total capacitance between one disc and the outer ground wall is given by

o totog air 0

o

4h

Cd K

(25)

and the total capacitance between one disc and the inner ground wall is given by

i totig air 0

i

4h

Cd K

(26)

20

where K is the total number of discs in the winding, htot is the total height of the winding, and d0 and di are the outer and inner distances between the winding and the ground respectively.

4.4 Calculation of the Inductances and Resistances

For the calculation of the self and mutual inductances, formulas in [35] are used. The self inductance Lself of each straight segment with the length , height h and width w in the disc is

0self

2ln 1

2 0.2235( )L

w h

. (27)

The mutual inductance M between two segments which are perpendicular to each other is zero, whereas for two parallel segments of length , separated by a distance x (as in Fig. 11), it is given by

2 2

0 ln 1 12

xM

x x x

. (28)

Fig. 11: Two filaments with negligible cross section area with same lengths.

When the segments are parallel but have different lengths, as in Fig. 12, the mutual inductance is given by

2 m p m q p qM M M M M , (29)

Fig. 12: Two filaments with negligible cross section area with different lengths.

where for example Mm+p is the mutual inductance between two straight wires both having the length m+p and being placed relative to each other as in Fig. 11, and which for the symmetric case p = q reduces to

m p pM M M . (30)

x p

m

q

x

21

Of course, in the formulas for the mutual inductances it is assumed that the conductors have very small cross section areas, which is just an approximation. The resistance Rseg of each segment is assumed to be of the form

0seg

1 1

2( )

fR

wh w h

, (31)

where is the conductivity of the conductor and f is the frequency. The first term is the DC resistance [38] (which has a vanishing effect for the frequencies dealt with here) and the second term accounts for the skin effect at higher frequencies [39−40]. Since proximity losses are not included in the model, a numerical factor α > 1 has been introduced and adjusted so that a realistic level of resonance damping is obtained.

4.5 State Space Model for the Winding

The circuit model for the three different winding models in Fig. 8 and Fig. 9 of the single winding consists of K·ni winding sections resulting in K(ni + 1) nodes and K·ni inductive branches and associated capacitances and resistances, where ni = 2i−1·n, for i = 1, 2, 3 for model 1, 2 and 3 respectively, and where n is the number of turns in one disc. K is the total number of discs used in the winding and hence one will arrive at the following two matrix equations by considering the voltage difference between the nodes of inductive branches and the current conservation at the nodes:

d

dt Γ I C V , (32)

T d

dt Γ V L I R I . (33)

Here V and I are the vectors containing the voltages at the nodes and the currents in the inductive branches, respectively

1

2

( 1) ( 1) 1

i

ii

Kn

K n K n

V

V

V

V

V ,

1

2

( 1)

1

i

ii

K n

Kn Kn

I

I

I

I

I . (34)

The matrix Г connects the currents and voltages and consists of 1, −1 and 0, and ГT is the transpose of Г

( 1)i iK n n

S 0 0

0 S 0 0

Γ 0

S 0

0 0 S

, where

( 1)

1 0 0

1 1 0 0

0

0 1 1

0 0 1i in n

S

(35)

22

and 0 is a zero matrix. The resistance matrix for the whole winding R is a diagonal matrix

disc

disc

disc

disci iKn Kn

R 0 0

0 R 0 0

R

0 0 R 0

0 0 R

, (36)

composed of the resistance matrix for individual discs Rdisc , where

seg,1

seg,2

disc

seg, -1

seg,

0 0

0 0 0

0 0 0

0 0i

ii i

n

n n n

R

R

R

R

R

, (37)

is composed of the resistances of all segments in one disc (for example, Rseg,j is the resistance of segment j (see Eq. (31))). The inductance matrix for the whole winding L is composed of smaller matrices

disc 12 13 1

21 disc 23 2

disc

1 2 disci i

K

K

K K Kn Kn

L L L L

L L L L

L

L

L L L

, (38)

where the off-diagonal matrices Lkj are ni×ni matrices for the mutual inductance between disc k and j, and the matrix in the diagonal Ldisc is the inductance matrix for a single disc being composed of the mutual inductances M (Eqs. (28)−(30)) and self inductances Lself (Eq. (27)) of the segments in one disc

self,1 12 13 1

21 self,2 23 2

disc

self, 1

1 2 self,

i

ii i

K

K

n

K K n n n

L M M M

M L M M

L

M M L

L

, (39)

(for example, Mkj and Lself,j are the mutual inductance between segment k and j, and the self inductance of segment j, respectively). The total capacitance matrix C in Eq. (32) is

23

( ) ( ) ( )DD DD

( ) ( ) ( ) ( )DD DD DD

( ) ( ) ( ) ( )DD DD DD

( ) ( ) ( )DD DD ( 1) ( 1)

2

2

i i

i i i

i i i i

i i i i

i i i

K n K n

C C C 0 0

C C C C

C 0 0

C C C C

0 0 C C C

, (40)

where, as mentioned before, i = 1, 2, 3 for model 1, 2 and 3 respectively. C(i) is the “specific” capacitance matrix for model i, and for model 1, the roughest model, it is

(1) (1) (1)tt ig tt

(1) (1) (1)tt tt tt

(1) (1) (1)tt tt tt

(1)

(1) (1) (1)tt tt tt

(1) (1) (1)tt tt tt

(1) (1) (1)tt tt og

1 10 0

2 21 3

0 02 20 2 0 0

0 2 0

3 10

2 21 1

0 02 2

C C C

C C C

C C C

C C C

C C C

C C C

C

( 1) ( 1)n n

(41)

where (1) (1) (1)

tt tt ig ig og og, ,C C C C C C , (Ctt is the turn-to-turn capacitance (Eq. (23)) and

Cog /Cig are capacitances to ground (Eqs. (25)−(26))). For model 2, the next finer model, it is

(2) (2) (2)tt ig tt

(2) (2) (2)tt ig tt

(2) (2) (2)tt tt tt

(2) (2) (2)tt tt tt

(2)

(2) (2) (2)tt tt tt

(2) (2) (2)tt tt tt

(2)tt

1 10 0 0

2 20 0 0 0

1 30 0 0

2 20 0 2 0

0 0 2 0 0

3 10 0 0

2 20 0

C C C

C C C

C C C

C C C

C C C

C C C

C C

C

(2) (2)tt og

(2) (2) (2)tt tt og

(2 1) (2 1)

0

1 10 0 0

2 2 n n

C

C C C

(42)

where (2) (2) (2)

tt tt ig ig og og/ 2, / 2, / 2C C C C C C . For model 3, the finest model, it is

24

(3) (3) (3)tt ig tt

(3) (3) (3)tt ig tt

(3) (3) (3)tt ig tt

(3) (3) (3)tt ig tt

(3) (3) (3)tt tt tt

(3) (3) (3)tt tt tt

(3)

1 10 0 0 0 0

2 20 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0

1 30 0 0 0 0 0 0

2 20 0 0 0 2 0 0 0 0

C C C

C C C

C C C

C C C

C C C

C C C

C

(3) (3) (3)tt tt tt

(3) (3) (3)tt tt tt

(3) (3) (3)tt tt og

(3) (3) (3)tt tt og

(3) (3) (3)tt tt og

(3) (3)tt tt o

0 0 0 0 2 0 0 0 0

3 10 0 0 0 0 0 0

2 20 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

1 10 0 0 0 0

2 2

C C C

C C C

C C C

C C C

C C C

C C C

(3)g

(4 1) (4 1)n n

(43)

where (3) (3) (3)

tt tt ig ig og og/ 4, / 4, / 4C C C C C C . The matrix ( )DDiC in Eq. (40) accounts for the

capacitive coupling between two neighbouring discs and it has the form

( )dd

( )dd

( )DD

( )dd

( )dd

( 1) ( 1)

10 0

20 0

0 0

10 0

2i i

i

i

i

i

i

n n

C

C

C

C

C

(44)

where ( )

dd dd /iiC C n for all three models i.e. i = 1, 2, 3 (Cdd is the disc-to-disc capacitance

according to Eq. (24)). The pre-factors 1/2 in the capacitance matrices are due to the fact that the capacitances

connected to the first and the last nodes in a disc account only for a half segment, and the pre-factors 1, 2 and 3/2 in the diagonals are due to the fact that, if there is no capacitance to ground, the sum of the elements in a row/column must be equal to zero.

When an external voltage source is connected to a node (k), its node voltage is no longer unknown (k = 1 here). The voltage at that node Vk and its time derivative dVk /dt should therefore be separately inserted in (32) and (33) as inputs accompanied by the corresponding columns of matrices C and ΓT. Eq. (32) and (33) are then transformed to

d d

d d kVt t

Γ I C V O (45)

T d

dkVt

P Γ V L I R I . (46)

Here, O consists of one column taken out from the C matrix corresponding to index k, and P consists of the k:th column taken out from ΓT (transpose of Γ). In addition, appropriate changes should be made according to the connection of the terminals of the discs and node grounding [41]. By rearranging the terms in these equations and putting them in one matrix equation, Multi Input Multi Output (MIMO) state space model of the lumped parameter circuit can be formulated as:

25

d

d kVt

X A X B (47)

where

VX

I,

1

1 T 1

0 C ΓA

L Γ L R and

1

1

d

dt

C OB

L P

. (48)

The state vector X consists of all the nodal voltages (except the applied one) and inductor currents of the lumped circuit. By taking the Fourier transformation of the equation system and selecting all state variables as outputs, one will arrive at

1( )( ) j

( )kV

X

TF II A B , (49)

where is the angular frequency, II is the identity matrix with the same size as A, and

1

1

j

C OB

L P. (50)

TF() contains all the transfer functions of the nodal voltages (except the applied one) and inductor currents with respect to the applied voltage Vk.

4.6 Comparison between the Three Models

The first comparison between the three models is made for a single disc winding, and the new phenomena that emerge with increasing model resolution are studied. The disc in the models consist of n = 10 turns of varnished copper wire with the conductivity σ = 5.8×107 S/m, the conductor height h = 7 mm and width w = 3 mm. The inner sides of the square disc have a length of i = 1.2 m, and the gap between any two neighboring conductors (turns) i.e. twice

the insulation thickness is τi = 0.4 mm, and there is no ground wall which means that do and di are set to infinity in the calculations (see Fig. 9 for a geometrical illustration of the parameters). These are also the dimensions of the experimental setup which will be described in Chapter 5. The reason for this choice of dimensions will be explained in the same Chapter.

The magnitudes and phases of the calculated in-impedances Z(f)=U(f)/I(f) (see Fig. 8) are compared to each other in Fig. 13(a) and (b), respectively. It can be seen that several resonances occur: the first, pronounced impedance maximum is the fundamental resonance of the winding, due to the total inductance and series capacitance of the whole disc. As it will be argued in Section 7.1, the three following resonances can be interpreted as “radial” resonance modes, and the two after that, which form a pronounced impedance minimum above 10 MHz and do not appear in the lowest-resolution model 1, as “azimuthal” resonance modes.

As can be seen in Fig. 13, for model 1 the impedance becomes purely capacitive i.e. it becomes of the form Z = (jωChf)

-1 after the radial modes i.e. for frequencies higher than about 6 MHz (Chf is the winding model capacitance at high frequencies). This is not a physical reality and it means that model 1 is for sure not valid for that part of the frequency spectrum. For model 2 and 3, the impedance becomes purely capacitive in the end of the frequency spectrum after the azimuthal modes, which as expected would mean that with finer discretization the model becomes valid for higher frequencies.

26

106

107

100

102

104

Frequency [Hz]

Impe

danc

e am

plitu

de [

]

model 1model 2model 3

106

107

-100

-50

0

50

100

Frequency [Hz]

Impe

danc

e ph

ase

[deg

rees

]

model 1model 2model 3

Fig. 13: (a) Impedance amplitude and (b) impedance phase for the different models.

“radial” modes

“azimuthal” modes

fundamental coil resonance

(a)

(b)

27

5 Frequency Response Measurements In this Chapter, the measurement device and the experimental setup are introduced and the choice of dimensions for the setup is explained. Also, the different measurements performed are described.

5.1 Impedance Measurement Device and Dimensioning of the Experimental Setup

A network analyzer Bode 100 from Omicron Electronics [42] (frequency range 1 Hz –40 MHz) was used for the impedance measurements (see Fig. 14).

Fig. 14: Impedance measurement device Bode 100 from Omicron Electronics. As mentioned before, the shape of the discs is chosen quadratic so that all the self and mutual inductances can be calculated by simple analytic formulas from [35]. The location of the resonances of the winding in the frequency spectrum depends on the physical geometry and material properties of the winding, and generally, larger dimensions of the winding leads to larger inductance and capacitance values, which in turn leads to the resonances occurring for lower frequencies.

The measurement device can measure up to the frequency of 40 MHz, but the measurements will be more sensitive to the effects of environmental noise and measurement cables for the higher part of this frequency range. Consequently, the measurements will be more disturbed and unreliable for this high-frequency part of the frequency range, and due to this fact, the geometrical size of the discs had to be chosen so that all the interesting phenomena and resonances occur for frequencies below approximately 20 MHz.

The most effective way to satisfy this requirement is to design the square discs with large side lengths. So the constructed discs consist of n = 10 turns of varnished copper wire with rectangular cross section (7 mm 3 mm). The inner sides of the square discs have a length of

28

1.2 m, and the gap between any two neighboring conductors (turns) is varying between 0.4 mm (= twice the insulation thickness) and about 1 mm because of manufacturing irregularities. The cross section dimensions of the varnished copper wires (7 mm 3 mm) are regular wire dimensions used in power transformers, and the number of turns can not be chosen too high (the discs will be to heavy and impractical to handle), or too low (there will be too few resonances), so n = 10 turns seemed reasonable and was chosen.

10 units of these discs were manufactured by the transformer manufacturing company Nordtrafo AB [43] (see Fig. 15).

Fig. 15: Separate disc units manufactured by Nordtrafo AB.

5.2 Impedance Measurement Results

The first measurements were performed on each of the ten different single disc units separately (Fig. 16).

Fig. 16: Frequency response measurement on a single disc. By comparison between the impedances of the different discs, it could be affirmed that due to the manufacturing irregularities mentioned above, the impedances of the discs differ from

29

each other more and more as the frequency is increased. This seems consistent, because for higher frequencies, the capacitances between the turns in the discs play a larger part in shaping the impedance. Since a capacitance is per definition sensitive to small distance changes between two conductors, and since (as mentioned earlier) the gap between any two neighboring conductors (turns) is varying between 0.4 mm and about 1 mm (because of manufacturing irregularities), hence the discs have different capacitive features for higher frequencies, leading to different high frequency impedances. In Fig. 17, the measured impedance for four of the discs is plotted to illustrate the differences.

106

107

100

102

104

Frequency [Hz]

Impe

danc

e am

plitu

de [

]

Fig. 17: Impedance measurement for four different discs.

It can from Fig. 17 be seen that for frequencies up to approximately 300 kHz, the discs have the same impedance value. This is the so called inductive regime where the impedance behaves as Z = jωLlf (where Llf is the inductance of a disc for low frequencies), and since the inductance is not so sensitive to small irregularities in the physical geometry, hence the discs have all the same impedance. But as the frequency is approached to the fundamental coil resonance frequency, the capacitance starts to play in, and it can be seen that from now on the impedances do not coincide with each other. The “radial” and “azimuthal” modes are still there for all of the discs, but they do not have exactly the same location, shape and amplitude.

Measurements were then performed on units with a varying number of discs connected together. The distance between two neighbouring discs τks was changed between approximately 1 mm to 5 cm, and different types of connections between the discs were tried, and all these for comparison between models and measurements. It was found that as the distance between the discs is decreased, the models deviate more and more from the measurements, and the reason for this will be discussed and explained in Chapters 6 and 8. Next, the model simulations will be compared to measurements for verification.

fundamental coil resonance

“radial” modes

“azimuthal” modes

31

6 Model Verification In this Chapter, model 3 is verified by comparison with measurements. The model is compared with measurements for a winding with one, three, six and nine discs respectively, and it is shown that the resonances predicted by the model also occur in the measurements.

6.1 Comparison of Model with Measurements

The impedance magnitudes of model 3 and measurement for one single disc are compared in Fig. 18, where it can be seen that they have an overall satisfactory agreement. Also, the comparison shows that the radial modes and the high-frequency azimuthal modes, which model 1 is unable to produce and which model 2 produces partially, are no model artifacts but real physical phenomena.

105

106

107

102

104

Frequency [Hz]

Impe

danc

e am

plitu

de [

]

simulationmeasurement

Fig. 18: Comparison between measurement and model 3 for a single disc. In Chapter 7, the physical meaning of the radial and azimuthal resonances will be explained and discussed.

In the measurements, the radial modes are shifted toward somewhat higher frequencies compared to those in the simulations, but they are fully recognizable. Such a shift is expected to occur due to the proximity effect which has not been taken into account in the model calculations reported here, and which is going to be discussed in Chapter 8.

faz

32

Next, the impedance magnitudes of model 3 and measurements for a three, six and nine disc winding, with the distance between two discs τks being approximately 6 mm, are compared in Fig. 19 to Fig. 21, respectively.

105

106

107

102

104

Frequency [Hz]

Impe

danc

e am

plitu

de [

]


Fig. 19: Comparison between measurement and model 3 for a three disc winding.

105

106

107

102

104

106

Frequency [Hz]

Impe

danc

e am

plitu

de [

]


Fig. 20: Comparison between measurement and model 3 for a six disc winding.

33

105

106

107

102

104

106

Frequency [Hz]

Impe

danc

e am

plitu

de [

]


Fig. 21: Comparison between measurement and model 3 for a nine disc winding.

Even here in Fig. 19−Fig. 21, the comparisons show an overall satisfactory agreement between model 3 and measurements, confirming the general accuracy of the model. But, as was the case in Fig. 18, there are shifts between some of the resonances reproduced by the model and the measured resonances. The reason for this, as for the resonance shifts for one single disc, is expected to be the proximity effect which is neglected in the models and which is going to be discussed in Chapter 8.

35

7 Interpretation This Chapter, which is mainly based on [29], deals with the radial and azimuthal resonances. The features of these resonances are explained, and it is shown that they are present also in a winding with several discs.

7.1 Explanation of the Different Oscillation Modes

As it was mentioned in Section 4.6, the three resonances somewhere between 1 MHz to 8 MHz in Fig. 13 for a single disc are called radial resonances and the ones somewhere between 10 MHz and 20 MHz are called azimuthal resonances, and it could be seen in Fig. 18 that these resonances also occur in the measurement, which means that these are physical realities and not model artifacts, and now the physical meaning of these resonances will be investigated and explained.

7.1.1 Radial Resonance Modes

Those resonances whose node voltages vary rapidly in the radial direction, but slowly in the azimuthal direction are called “radial”. By radial and azimuthal directions, the ρ and φ directions in polar coordinates are meant, respectively (see Fig. 22 for model 3).

Fig. 22: Geometry of the disc, showing the definitions of coordinates φ and ρ.

In Fig. 23 – Fig. 25, instantaneous node voltages for a sinusoidal excitation voltage U are depicted for different resonance frequencies, each at two different instants of time during an oscillation period, obtained from simulations of model 3.

1

2

4n

4n-2

4n+1

4n-1

3

4 5

φ = 0

φ = π/2 φ = π

φ = 3π/2

ρ

6

φ ρ0

ρ1

36

A linear voltage profile along the whole winding (which is the low-frequency limiting behavior) is subtracted, so that the values at both end nodes of the winding are equal to zero. The green line depicts the geometry of the winding disc and the location of the nodes, and the thin horizontal red line shows the zero level of the voltage as a reference. Black lines connect voltage levels in radial direction, and vertical blue lines indicate the correspondence between voltage levels and nodes.

The voltage distribution in the disc for the first, second and third radial resonance of Fig. 13 is depicted in Fig. 23, Fig. 24, and Fig. 25, respectively. fk (k = 1, 2, or 3 ) denotes the frequency for that particular resonance, and 1/fk is the corresponding period time. The radial resonances appear as standing voltage waves which can approximately be described by the formula

0 0 1rad,

1 0

( , , ) cos(2 ) sin 22k k k kV t f t A k B

(51)

for ρ0 < ρ < ρ1 , where ρ0 and ρ1 are the inner and outer “radii” of the disc, respectively (see Fig. 22). The resonance voltage amplitudes Ak and Bk are damping dependent. Note that the approximate expression (51) is independent of . It can be seen in Fig. 23–Fig. 25 that the approximation (51) is best for low resonance order k. The amplitude Bk is close to zero for k = 1 and increases with increasing resonance order k.

Fig. 23: Voltage profile of the first radial resonance (k = 1), at times t = 0.7/f1 (left) and t = 1.4/f1 (right).

Fig. 24: Voltage profile of the second radial resonance (k = 2), at times t = 0.4/f2 (left) and t = 0.8/f2 (right).

Vrad,k

Vrad,k

37

Fig. 25: Voltage profile of the third radial resonance (k = 3), at times t = 0.3/f3 (left) and t = 0.6/f3 (right).

7.1.2 Azimuthal Resonance Modes

For “azimuthal” resonances, just like the “radial” ones, the node voltages vary rapidly in the ρ direction, but the difference is that there are also significant node-voltage variations in the φ direction. This pattern can be seen in Fig. 26 and Fig. 27 which depict the instantaneous node voltages for the two dominant azimuthal resonances, appearing in Fig. 13 and Fig. 18 as pronounced minima close to faz, at two different instants of time. Again, model 3 has been employed and a linear voltage profile has been subtracted.

In contrast to the radial resonances which are spread out in frequency, the azimuthal resonances are “clustered” (at least when viewed on a logarithmic frequency scale) around a characteristic frequency faz slightly above 10 MHz. They cannot be described by a simple formula like that for the radial resonances (51), but their common characteristics is approximated by the expression

0 1az az az( , , ) cos(2 ) 1 cos( )

2V t A f t

. (52)

This fundamental behavior is indicated in Fig. 26 and Fig. 27 by dotted lines. Individual azimuthal resonance modes differ from it by additionally superposed short-wavelength modulations. The mode in Fig. 26 resembles more closely to the fundamental expression (52) than the one in Fig. 27.

Fig. 26: Voltage profile of the first azimuthal resonance, at times t = 0.25/faz (left) and t = 0.7/faz (right).

Vaz

Vrad,k

38

Fig. 27: Voltage profile of the second azimuthal resonance, at times t = 0.4/faz (left) and t = 0.65/faz (right).

7.2 Radial and Azimuthal Resonances for a Winding with More Than One Disc

As it was reported in Chapter 6 and illustrated in Fig. 19, Fig. 20 and Fig. 21, measurements were also performed for a winding with more than one disc. In Fig. 28, measurements are shown (again) for a winding consisting of one, three, six and nine discs respectively, connected together in a continuous way as in Fig. 9.

105

106

107

102

104

106

Frequency [Hz]

Impe

danc

e am

plitu

de [

]

1 disc3 discs6 discs9 discs

Fig. 28: Measurements on windings with one, three, six, and nine discs respectively.

It can be seen that even for a several-disc-winding (and irrespective of the number of discs in it), the radial and azimuthal resonances exist, and occur around the same frequencies as for a single disc (see the dotted rings in Fig. 28). This supports the picture that these resonances are internal oscillations in every individual disc.

Also, it can be noticed that the radial resonances (at least the first two) and the azimuthal resonances tend to get smoothed out as the number of discs in the winding is increased. This is expected to be due to the small manufacturing differences leading to the positions, shapes and amplitudes of the resonances for different discs being different (see Fig. 17), and when

Vaz

Radial modes

Azimuthal modes

39

the number of discs in the winding is increased, these relatively different resonances superimpose on each other, leading to a smoothening-out. Another reason for this smoothening-out could be the proximity losses which enter the picture when several current carrying conductors are placed close to each other (see Section 8.1).

Furthermore, measurements and simulations show that especially the azimuthal resonances are very sensitive to small changes in the winding geometry (e.g. mechanical winding deformations). For instance, in the measurements on different single discs of identical design but with small manufacturing differences (see Fig. 17), the relative strength and shape of the two dominating azimuthal resonances varied noticeably, each disc thus having its individual “finger print”.

It is also worth to mention that the new resonances which emerge in the frequency range 2×105−2×106 for a winding with more than one disc are “global” oscillations along the total length of the winding between different discs, called “axial” resonances here . A hallmark of the “global” resonances is this observation that they shift considerably in frequency when the number of discs in the winding is increased.

41

8 Reluctance Network Method for Inclusion of the Skin and Proximity Effects

In this chapter, the impact of the skin and proximity effects on the inductances and losses for a winding is explained, and a method based on the reluctance network approach is described to compute the frequency dependent inductance and resistance matrices. This method (which has been developed in [44−45]) employs a network of complex reluctances describing the flux paths around the conductors. The use of this method in this chapter is limited to the conductor arrangements of multi-conductor transmission lines surrounded by a shield, since these are easier to start with than a winding, and its extension to windings is left for future work. The chapter is concluded with verification of the results reproduced by this method by comparison with finite-element method calculations. This chapter is to a large extent a description of what has been stated in [44−45].

8.1 Skin and Proximity Effects

As it was mentioned earlier in Section 4.4, the formulas for the inductances are for thin filaments and are moreover frequency independent. This is a good approximation as long as the distance between two conductors is much larger than their largest cross sectional dimension, and as long as the frequency is low enough. But when the frequency is high and the two conductors are close to each other, the so called skin effect and proximity effect will be present respectively. Skin effect – when a time-varying current flows in a conductor it creates a time-varying magnetic field which in turn induces eddy currents i.e. induced currents that counteract the original current. The consequence is that the total current tends to be confined to the surface of the conductor. This effect becomes stronger as the frequency is increased and the effective current carrying area of the conductor becomes restricted to a thin layer below the surface, which is called skin depth and is defined by

0 r

1

f

(53)

The simulation in Fig. 29 (a) shows the influence of skin effect on the distribution of the current- and magnetic flux densities. It can be noticed that due to the skin effect, the current density becomes non-uniform in the radial direction ρ, but it is still uniform in the azimuthal direction φ when the conductor has cylindrical symmetry. Proximity effect – in contrast to the skin effect, proximity effect is about eddy currents which are induced in a conductor due to a time-varying magnetic field produced by the currents in the other conductors in the vicinity. Due to the proximity effect, the current- and magnetic flux density distributions become unsymmetrical in both ρ and φ directions (see Fig. 29 (b)).

42

(a) (b)

Fig. 29: Distribution of magnetic flux density (arrow plot) and current density (surface plot) for circular

conductors; in (a) skin effect is present only, while in (b) both the skin- and proximity effects are present (for the currents in the conductors flowing in the same direction). This figure is borrowed from [46].

The impact of skin effect on the self inductance is that the internal inductance of the conductor decreases, but since the internal inductance is a very small part of the total self inductance of a single conductor, the influence of skin effect on the self inductance can be neglected. The skin effect introduces of course additional, frequency dependent losses. These losses have already been incorporated in Eq. (31)

But the proximity effect (in combination with the skin effect) is more serious, and this is due to the fact that when the frequency is very high and the conductors are really close to each other, the current is not only confined to the surface of the wires but it is distributed around the axis in conformity with the law of distribution of the charges in the corresponding electrostatic problem [35]. This means that if current flows in opposite/parallel directions in two parallel conductors, the current density in each conductor is a maximum at the nearest/farthest points of the cross sections of the conductors. This has the effect of a reduction/expansion of the effective spacing of the conductors, which means that the mean distance between the effective current carrying areas will be smaller/larger than the distance between the centers of the two conductors which is the distance used in the thin filament approach (see Fig. 11 and Fig. 12). This in turn means that the actual mutual inductances for turns close to each other will deviate from the ones calculated in expressions (28) – (30). The proximity effect will also introduce additional, frequency dependent losses, which would be equivalent with the resistance matrices in Eqs. (36) and (37) gaining off-diagonal frequency dependent components. It is argued here that the deviations between model and measurements in Fig. 18 to Fig. 21 are most probably due the absence of these high-frequency phenomena in the formulas used to calculate the inductances and resistances in Sections 4.4 and 4.5.

8.2 Using Reluctance Network Method (RNM) to Calculate the Frequency Dependent Inductance and Resistance Matrices

The solution proposed here for calculation of the frequency dependent inductance and resistance matrices is the so called Reluctance Network Method (RNM); this method is started by considering the cross section of one side of a single quadratic disc embedded in a reluctance network, as depicted in Fig. 30.

43

Fig. 30: The cross section view of one side of a single disc, showing the conductors, conductor-insulation,

exaggerated air gaps and different reluctances. The dimension of the air gaps (due to the manufacturing uncertainty and irregularities) between the conductors is exaggerated in the figure above to make is possible to illustrate the problem picture.

As it will be shown below, it is possible to find an easy analytical expression for the reluctance in a gap in which the length of the channel is much larger than its width, i.e. in between the turns in Fig. 30. But this is not the case for the reluctances “outside” the turns ( and ) since there is no easy way to define an area which the magnetic flux density passes through and a length which it travels along. This means that it is easier to start the development of this method with a configuration containing narrow gaps only. This would be a configuration in which the conductors are surrounded by a shield; a multi-conductor transmission line. What follows below in the rest of the chapter is a description of the derivations and results in [44−45]: the method is first developed for higher frequencies, and it is then extended to lower frequencies. After that, it is applied to some simple example geometries, and its accuracy is verified by comparisons with Finite-Element Method (FEM) field calculations.

8.3 RNM for High Frequencies

Consider an arrangement of parallel conductors of length aligned in x direction, with arbitrary cross section in the y-z plane. On these AC currents of frequency ω are imposed, denoted by Ik with k = 1,…,n , where n is the total number of conductors. A very simple example with n = 2 conductors is pictured in Fig. 31. A slightly more complex and general example is discussed further below (Fig. 32). The convention that positive current directions always point out of the plane (red symbols) is adopted here. For clarity, is kept as a parameter in the equations below; in order to obtain per-length expressions, one simply has to divide by .

1R 2R 3R

32 1

1I 2I

(a)

m,1 m,2

1I 2I

(b) Fig. 31: Cross-sectional view of n = 2 parallel conductors, oriented perpendicularly to the plane. (a) Reluctance

network with chosen branch orientations, (b) circulating mesh fluxes.

conductor

I1 I2 In-1 In h

τi

magnetic flux

44

8.3.1 Principle of the Reluctance Network Description

Assume for the moment that the conductors are perfectly conducting (or, alternatively, that the frequency tends to infinity), so that the skin depth vanishes and magnetic flux exists only outside the conductors. The magnetic coupling between these conductors is described by a network of reluctances iR (i = 1,…,s), characterizing “flux channels” formed by the gaps between the conductors as well as the “open space” around them. In other words, the iR characterize all the available distinct magnetic flux paths [47]. The iR of the “flux channels” are determined by the geometry and material parameters of individual gaps between conductors, whereas those of the “open space” also depend on the availability and nature of some external current return path. Each network branch corresponds to one of these flux paths and reluctances, and each network mesh surrounds one of the conductors.

For each reluctance branch an (arbitrary but fixed) orientation is chosen, defining the direction of positive flux i in the branch.

At any given moment, the currents Ik flowing through all the conductors and the “branch fluxes” i in the reluctance network can be combined into vectors,

1

n

I

I

I and 1

s

. (54)

Since magnetic flux is conserved locally (i.e., at each node of the network, the total inflowing flux is zero), any possible flux state can be built up by superposition of “mesh fluxes” m,k , k = 1,…,n , circulating counter-clock-wise in the meshes around the n conductors (Fig. 31b, Fig. 32b).They are combined here into a new vector m with n elements, and a s n matrix is defined as follows: Definition of the matrix : (55)

For any given branch index i and conductor index k, has an element ∂ik = +1 (or −1) if the branch i belongs to the mesh around conductor k and is oriented in parallel (or opposite) to the circulation direction, and ∂ik = 0 if it does not belong to mesh k.

can be interpreted as the “connectivity matrix” of the graph which is dual to the reluctance network. In the simple example of Fig. 31, the total number of reluctance branches is s = 3, and the matrix is given by:

1 2

1

2

3

1 01 1 .0 1

k i

(56)

Using , the relation between the branch and mesh fluxes can be written as a simple matrix equation,

m . (57)

The line integral of the magnetic field H along any branch i is proportional to the flux Φi along that branch, with iR as proportionality constant. Summing up these branch contributions around any mesh k in the network yields the line integral of H on a closed loop

45

around conductor k, which according to Ampère’s law is equal to the enclosed current Ik . Expressing the reluctances iR as a diagonal matrix

1 0

0 s

R

RR = , (58)

this can again be written as a matrix relation,

TI R , where T denotes the matrix transpose of . Together with (57) this results in

T

mI R . (59)

Finally, by virtue of Faraday’s law the induced voltage drops between opposite ends of the conductors are given by the time derivatives of the mesh fluxes, md dtU , which in

frequency domain leads to mj U ZI . (60)

with

1Tj

Z R . (61)

Finally, when losses in the conductors are included the reluctances become complex functions of frequency (see next section), and the total series impedance matrix of the conductor arrangement acquires a resistive contribution and can simply be expressed as

1Tj j

Z R L R . (62)

In the second example shown in Fig. 32 where n = 5 and s = 8, the outermost conductor (k = 5) forms a shield surrounding the other conductors. Its cross section is multiply connected, so that the contour describing its mesh flux Φm,5 (dashed line in Fig. 32b) consists of two parts.

k = 1

3

2

4 5

i = 1

2 3

4

5

6 7

8

(a)

k = 1

3

2

4 5

(b) Fig. 32: Cross-sectional view of a more general arrangement of n = 5 parallel conductors. (a) Reluctance

network with chosen branch orientations, (b) circulating mesh fluxes.

46

For this example, the matrix is given by:

1 2 3 4 5

1

2

3

4.

5

6

7

8

1 0 0 0 11 1 0 0 01 0 1 0 00 1 0 0 10 1 1 0 00 0 1 1 00 0 0 1 10 0 0 0 1

k i

(63)

Note that the sum over each row except the last is zero, such that any “common mode” voltage (i.e., U1 = … = Un) corresponds to current flow in the shield only (i.e., Ik = 0 for k ≠ n), and the shield current obeys the relation j n s nI U R where s is the index of the

outermost flux path surrounding the shield (in the present case, s 8). On the other hand, if the shield is short-circuited (i.e., if Un = 0 is imposed) all flux outside

the shield is suppressed at nonzero frequencies, and so flux path s can be disregarded altogether. For the calculation of reluctances of the “inner” flux paths the shield is treated in the same way as the other conductors. Under these conditions, the remaining conductor voltages U1,…,Un−1 and currents I1,…,In−1 are related to each other by a “reduced” Z matrix which does not depend on sR , and the shield acts as common current return path. The numerical examples in Section 8.6 below will be limited to this situation (it may be called the “differential mode” situation).

8.3.2 Calculation of the Reluctances for High Frequencies

At a finite frequency which is still so high that the skin depth in all conductors is small compared to their thickness, the flux penetrates only slightly into the conductors, and the above picture of separate “flux channels” which do not interfere with each other is still valid.

Eq. (62) has provided an explicit expression of the impedance matrix in terms of the gap reluctances iR , but one still needs to calculate the latter for a given geometry. In the limit of high frequencies, one can focus on a single gap and model its geometry locally as shown in Fig. 33. Orienting the channel i in such a way than the positive flux direction is to the right, all quantities related to the conductor above the gap have been labelled here with a subscript “+,i” and those below the gap with “−,i”. Later on, conductor specific quantities will instead sometimes be labelled by the original conductor index k, which obeys the condition ∂ik = +1 or ∂ik = −1 for the conductor above or below the gap, respectively, according to definition (55) of the matrix .

It is assumed that the medium in the gap is perfectly insulating (g,i = 0) and has a finite permeability g,i > 0. The conductors above and below the gap are given finite conductivities ,i > 0 and permeabilities ,i > 0, respectively, which leads to flux penetration into the conductors characterized by finite skin depths δ,i (see further below).

47

conductor ( , , ,, ,i i i )

insulator ( g,i )ia

iw

,i

,i

,id

,id

conductor ( , , ,, ,i i i )

“flux channel” g,i

,i

,i x

z

y

skin layer

skin layer

Fig. 33: Local geometry of the gap between two conductors.

x

z

y g,iR

,iR

,iR

g ,i

,i

,i

i

Fig. 34: Parallel reluctances of insulating gap and skin layers in the adjacent conductors.

Assume for the moment that the gap is straight and that its width ai is constant along its whole length wi. The case of varying gap width will be discussed below in Section 8.5. For a long and/or narrow gap (aiwi) and for strong skin effect (δ,i d,i, where d,i are the thicknesses of the two conductors in direction perpendicular to the channel, see Fig. 33), the conductor cross sections can be approximated locally as infinite half planes, and it can be assumed that the current densities in the conductors and all the fields are oriented parallel to the surfaces everywhere (in the x-y plane) and are varying in z direction only. The current density is chosen to point in x direction and the magnetic field to point in y direction. Since the quasi-static magnetic coupling between the conductors is of interest here, solutions of Maxwell’s equations where the charge density ρ is zero everywhere are looked for. The only non-vanishing components of the electric and magnetic fields are then Ex and Hy, and Maxwell’s equations reduce to the one-dimensional problem given by

d

jd

xy

EH

z , (64)

d

jd

yx x

HE J

z . (65)

Since there are no surface currents, both Ex and Hy must be continuous in z direction. Current density and electric field are assumed to be related by Jx = Ex, so (64) and (65) can be combined into a single equation for Hy. Neglecting the displacement current jωεEx (which has

48

a noticeable effect only at much higher frequencies than the ones of interest here), one ends up with the well-known simple diffusion equation [48] for Hy inside a conductor,

22

2

dj

dy

y y

HH H

z , (66)

where

1 jj

, with 2

. (67)

For simplicity, denotes the total permeability here and in the following, including the permeability of free space 0, and it may have a different value in each conductor. The conductivity may also have a different value in each conductor. κ is the complex wave number for magnetic field penetration into the conductor at frequency ω and δ the corresponding real skin depth. Since the gap is insulating, Hy is constant in this region, according to Ampère’s equation (65). The fundamental solutions of Eq. (66) within the conductors are exponential functions e z .

In order to calculate the reluctance of the channel, one has to relate the total flux i through it to the magnetic field Hy in the gap. One can start with the contribution of the upper conductor, assumed for the moment to occupy the region z > 0, so that the solution to (66) has the simple exponential form

,

,0( ) e i zy yH z H . (68)

Here Hy,0 is the magnetic field at the conductor surface z = 0 and inside the gap i (where it is independent of z), and , ,1 ji i are the complex wave numbers in the conductors

above and below gap i, as defined in (67). As a consequence of (68), the magnetic flux +,i inside the upper conductor (which is roughly confined within a distance δ+,i from the surface) is given by

, ,0

, ,,0

d i yi i y

i

Hz H

. (69)

The corresponding expression for the magnetic flux −,i inside the lower conductor is analogous. Inside the gap the magnetic field is constant and equal to Hy,0 , so that the flux in the gap is given by g, g, ,0i i i ya H . Together with the fluxes (69) in the adjacent skin layers

it adds up to the total flux in the channel,

, ,, g, , g, ,0

, ,

i ii i i i i i y

i i

a H

. (70)

The reluctance iR of the channel is now defined as the proportionality constant between i

and the line integral of the magnetic field along the channel wi Hy,0 , and thus

1

, ,g,

, ,

i iii i i

i i

wa

R . (71)

49

Note that this result can be interpreted as a “parallel connection” 11 1 1, g, ,i i i i

R R R R of

individual reluctances

,,

,

iii

i

w

R , g,g,

1ii

i i

w

a

R , ,,

,

iii

i

w

R , (72)

which refer to insulating gap and adjacent skin layers, as indicated in Fig. 34.

8.4 Extension of RNM to Low Frequencies

8.4.1 Summary of the Generalized Approach

With decreasing frequency the skin depth increases until it exceeds half the conductor thickness, i.e., skin layers of opposite sides of the conductor start overlapping and the above picture of individual “flux channels” is not applicable anymore. However, it will be argued below that even for low frequencies a reasonable approximation can still be obtained by generalizing Eqs. (62) and (71) in the following way:

1Tint intj j

Z R L Z Z R R , (73)

where R is still a diagonal matrix, but its elements are modified by additional factors θ,i to the inverse skin layer reluctances 1

,iR (compare (72)):

1

, ,, g, ,

, ,

i iii i i i i

i i

wa

R . (74)

It will be shown below that these factors have the form

, ,, tanh( )

2i i

i

d , (75)

coming from the usual skin effect formula for slabs [48] with finite thicknesses d,i. At high frequencies (i.e., for δ,i d,i), θ,i tend to 1 so that the previous expression (71) is recovered. An alternative way to write (75), which will be more convenient further below, uses conductor indices k instead of subscripts :

,, tanh( )

2k k i

k i

d , (76)

where dk,i denotes the thickness of conductor k perpendicular to the adjacent channel i (i.e., dk,i = d+,i or d−,i , and k = +,i or −,i , etc. for ∂ik = 1 or −1, respectively).

intZ and intZR are diagonal matrices, which have been added since the original expression

(62) in combination with (74) leads to a wrong low-frequency limit for the internal impedances of individual conductors. The element int ,kZR is the internal impedance of

conductor k, resulting under the assumption that (62) together with (74) are valid at all

50

frequencies. It is calculated from these equations by setting all permeabilities in the system equal to zero, except for k (which physically means that conductor k is completely immersed in a perfectly diamagnetic medium, so that all flux is confined to the interior of that conductor). In that limit, all matrix elements of

T 1( ) R tend to zero except for

T 1( )

kk

R

1

1,

with 0ik

i kk i

i k

w

(77)

(which is the inverse of the sum of all the skin layer reluctances around conductor k), so that

int, 1,

with 0

j

ik

kk

k i k ii

Zw

R . (78)

The summation is around the mesh surrounding the conductor k. (Note that the sum over the wi without the 1

,k i factors is just the cumulative length of the channels which surround

conductor k, i.e., the approximate circumference of the conductor cross section.) At low frequencies, 1

, ,2k k i k id and so expression (78) is proportional to jω, implying a

vanishing DC resistance which is physically incorrect. In order to fix this problem, intZR has

been subtracted in (73) and instead a diagonal matrix Zint whose elements are the true internal impedances Zint,k has been added . It will be argued below that a reasonable approximation for Zint,k is provided by an expression very similar to (78),

int,,

with 0

j

ik

kk

k i k ii

Zw

, (79)

where only 1

,k i is replaced by the new quantity

,, tanh( )

4k k i

k i

d . (80)

At high frequencies (δ,i d,i) where both 1,k i and ,k i tend to 1, int,kZ and int ,kZR cancel each

other. In the limit ω → 0, expressions (79), (80) lead to the correct DC resistance for conductors with rectangular or circular cross section, and therefore they are adopted here as a general rule. For instance, for a rectangular cross section with side lengths p and q, numbering the channels around it by i = 1,…,4, one has dk,1 = dk,3 = w2 = w4 = p and

dk,2 = dk,4 = w1 = w3 = q, as well as 1, ,4k i k k id at low frequencies. Eq. (79) then reduces to

int, 2

j kk

k

Zpq

k pq

, (81)

which is the correct DC resistance Rdc of a conductor of length with cross-sectional area pq.

51

8.4.2 Exact Solution of a Single-Slab Problem

In order to justify the modified reluctance formula (74) for conductors of finite thickness, the particular two-dimensional geometry shown in Fig. 35 is studied. A conducting slab of length , width w, and thickness d with conductivity and permeability (since there is only one conductor, the conductor index is dropped) is sandwiched between two insulating sheets of thicknesses a1 and a2 with permeabilities g,1 and g,2 , respectively. The boundary conditions in z direction are given by perfect diamagnetic media ( = 0) and those in y direction by perfect magnetic conductors ( = ∞), so that when a current is flowing in x direction through the conductor, the magnetic flux is confined within the insulating sheets as well as the skin layers in the conductor, and it sees zero reluctance when going around the side faces of the slab. As a consequence, the reluctance network corresponding to this geometry has the simple structure shown in Fig. 36.

conductor ( , , )

insulator ( g,2 )

w

x

z

y

insulator ( g,1 )

0

0

,1( )

,2 ( )

1a

d

2a

skin layer

skin layer

2d

2d

0

z

Fig. 35: Simple single-slab geometry.

x

z

y

0

0

I

g,2R

,2R

,1R

g,1R

mg ,2

,2

,1

g ,1

Fig. 36: Reluctance network of the single-slab geometry.

The reluctances are calculated in a similar fashion as in Section 8.3.2 above. Now, two surfaces of the slab have to be take into account, so the origin of z is chosen in the middle of

the slab and the functions 12sinh e ez zz and 1

2cosh e ez zz are used as

52

fundamental solutions of (64), (65) (instead of eκz). The general solution in a slab of thickness d extending from z = −d/2 to z = d/2 can then be written in the form

cosh( ) sinh( )( )

cosh( 2) sinh( 2)y

z zH z C C

d d

, (82)

sinh( ) cosh( )( )

cosh( 2) sinh( 2)x

z zE z C C

d d

, (83)

with arbitrary coefficients C. These coefficients have to be determined for a given total current I through the slab. According to (82), (83), the field values at the slab surfaces,

( 2)y yH H z d and ( 2)x xE E z d , are related to the C in the following simple

way:

yH C C , (84)

x

CE C

, (85)

with sinh( 2)

tanh( 2)cosh( 2)

dd

d

. (86)

At low frequencies, when the skin depth ( ,1 ,2 , see Fig. 35) is larger than half the

conductor thickness d, the two skin layers overlap and Φ+,1, Φ−,2 can no longer be calculated separately. However, the total magnetic flux Φc = Φ+,1 + Φ−,2 in y direction inside the conducting slab can be obtained by integration of (82),

2

c

2

2d

d

y

d

Cz H

. (87)

Similarly, the total current I flowing in x direction through the slab is obtained by integration of (83) (or, alternatively, using Ampère’s law (65)):

2

2

d 2d

x y y

d

I w z E w H H wC

. (88)

The total fluxes Φg,1, Φg,2 in the insulating layers below (z < −d/2) and above (z > d/2) the slab, respectively, are given by

g,1 1 g,1 1 g,1ya H a C C and g,2 2 g,2 2 g,2ya H a C C . (89a,b)

The voltage drop U in the slab can be obtained by integration of the electric field along a closed contour which runs in x direction along the conductor (for instance, at one of its surfaces) and returns in the field free region below (z < −d/2 −a1) or above (z > d/2 +a2) the system. According to Faraday’s law, both choices of contour must lead to the same result,

53

since the total flux in y direction (both inside and outside of the slab) must vanish due to flux conservation,

g,1 c g,2 0 . (90)

Here, the contour below is closed and the following result is obtained

g,1j xU E

1 g,1jC

a C C C

1 g,1

1j j

Ca C C

. (91)

Now new quantities are defined

1

1 1 g,1

wa

R , 1

2 2 g,2

wa

R . (92)

Since these expressions are special cases of (74) for the present geometry, they are interpreted as generalized channel reluctances. Re-writing (91) as

11

1

j

U C wC C

R , (93)

and inserting (87), (89a,b) into (90), the following is obtained

1 11 20 C C C C R R

1 1 11 2 22C C C R R R . (94)

The second line in (94) makes it possible to express C+ − C− in (93) by C− alone, and thereby via (88) by the current I. The final result is

1 2

1 1j

2

UZ

I w

R R

. (95)

Now 11 in this simple case, which implies that

T1 2R R R , and so (95) is of the

form (73) with

int

j

2Z

w

R and int

j 1

2Z

w

. (96)

Furthermore, the expression for intZR coincides with the limit g,1, g,2 → 0 of

T 1j ( ) R

and as a consequence the expression for Zint is the exact internal conductor impedance for the simple geometry of Fig. 35. It does contain the “inverted” θ factor as in (79) but, in contrast to (80), no additional factor of 2 in the denominator of the tanh argument. The physical

54

explanation for this is that even in the limit g,1, g,2 → 0, the conductor in Fig. 35 is not embedded in a perfect diamagnetic medium from all sides, but magnetic conductors to the left and right are still present. These absorb all flux in y direction, and so there are no skin layers on the lateral vertical faces of the conductor in this case.

The above argument can be generalized to an arbitrary number of conducting slabs, stacked onto each other in z direction and separated by insulating sheets (see Appendix A). Again the results appear to be consistent with Eqs. (73)–(80), which shows that there is no additional interference between the individual flux channels apart from the one described by the θ factors in (73) and (74), at least in this specific geometry (and presumably in general).

8.5 Extension to Variable Gap Width

So far it has been assumed that the gap widths ai between conductors are constants. In general they may vary along any given channel, i.e., in y direction of Fig. 33, Fig. 34. However, it may still be possible to find simple analytical expressions for the complex reluctances iR ; for instance, under the assumption that the length scale of this variation is large compared to the gap width itself, the only change in the basic expression (74) is that the gap width becomes a function ai(y) and the factor wi is replaced by an integral with respect to y.

Some basic cases are shown below in Fig. 37. For simplicity the subscript i labelling the gap has again been omitted. Along with the earlier case (a) of parallel conductor surfaces (= constant gap width), (b) flat surfaces which are slightly inclined with respect to each other, and (c) locally cylindrical conductor surfaces with given radii of curvature r+ , r− are considered.

w

(a) (b) (c)

R

R

( )a y

amax

amin

a

Rx

z

y

amin

rr

Fig. 37: Some basic geometries of the gap (white space) between neighbouring conductors (shaded). The

channels are vertically oriented here. The current direction (x-direction) is perpendicular to the plane, as before.

(a) Planar, parallel conductor surfaces: This is the basic case (74) discussed earlier where a = const and so

1

g

wa

R . (97)

55

(b) Planar but inclined conductor surfaces: Here a varies along the gap as a(y) = amin + (amax − amin)y/w for 0 < y < w, and so

1

2g

2 10

1d ( ) ln 1

w w Ky a y

K K

R

with

1 min gK a

,

2 g max minK a a . (98a–c)

When amin, amax are equal, this reduces to case (a).

(c) Locally cylindrical conductor surfaces with curvature radii r+ , r− : The largest contribution to the channel reluctance comes from its narrowest region, which is assumed to be located approximately in the middle. The origin of y is chosen at that point, so that

2

min

1 1( )

2

ya y a

r r

in the vicinity and

2

1

122

g

2 1 2

2arctan( )1d ( )

KwwK

w

y a yK K

R

with

1 min gK a

,

g2

1 1

2K

r r

. (99a–c)

One or both of the radii r+ , r− may be infinite. When both are infinite, (99a–c) again reduces to case (a). Alternatively, one of the radii r+ , r− may be negative, corresponding to a situation where both surfaces are bent in the same direction (examples are the gaps between conductors and shield in Fig. 39 below). Note that as long as the flux channel is tight enough in the middle (i.e., for those frequencies for which |K1|w2 K2), the arctan above may be replaced by π/2 so that the total channel reluctance does not depend on the length w of the channel anymore but is determined by width and curvature radii of this “high-reluctance bottleneck” only.

From the formal derivation in Sections 8.3 and 8.5 it is apparent that the RNM should be asymptotically exact in the limit where all lateral dimensions characterizing a given flux channel i (i.e., the maximum gap width ai,max and the skin depths δ,i) are much smaller than both its length wi , the typical scale at which ai(y) varies, and the lateral conductor thicknesses d,i . Roughly speaking, it is expected to be exact in the limit of high frequencies and narrow gaps between the conductors, which is precisely the limit where FEM calculations are most difficult to perform. This will be illustrated below for two simple example geometries.

56

8.6 Comparison with FEM Calculations

The predictions of the extended reluctance network method, summarized by Eqs. (73)–(80) and (97)–(99a–c) above, have been verified by comparison with finite-element field computations, using the commercial FEM tool COMSOL.

Comparisons were made for two different test geometries, which are illustrated below. The first example is a 33-matrix of identical rectangular conductors, surrounded by a rectangular magnetic shield (Fig. 38). The shield thickness exceeds the magnetic skin depth within it at all frequencies considered here (1 Hz to 10 MHz), so there never is any magnetic flux in the outer space surrounding the shield.

The second example is a symmetric arrangement of seven identical round conductors, surrounded by a magnetic shield with a circular inner cross section boundary, so that a number of insulator-filled pockets are created between the conductors and towards the shield (Fig. 39).

In both examples the values k = 6107 S/m and μk = μ0 for all conductors, μg,i = μ0 for all

gaps, and μshield = 1000 μ0, shield = 107 S/m for the shield have been used. These values were

chosen so as to roughly represent copper conductors with an iron shield or armour, but the precise numbers are unimportant for the points made here. In all examples shown below, inductances or resistances per length have been plotted.

k = 1 4 7

2 5 8

3 6 9

shield

a' w' d'

d"

a"

w"

Fig. 38: Cross section of rectangular-conductor arrangement (not to scale). Conductors and shield are shaded, the reluctance network is shown in blue. In this example, a' = a" = 0.2 mm, d' = 7.3 mm, w' = d' + a' = 7.5 mm,

d" = 3.3 mm, w" = d" + a" = 3.5 mm, and the shield is much thicker than the skin depth at all considered frequencies.

57

k = 1

23

4

56

7

a"

a'

a'r'

r"

w'

w"

r'

shield

Fig. 39: Cross section of circular-conductor arrangement (not to scale). Conductors and shield are shaded, the reluctance network is shown in blue. In this example, a' = a" = 0.05 mm, r' = 2.6 mm, r" = 3r' + a' + a" = 7.9 mm,

w' = 1.4 r' = 3.6 mm, w" = 3.5 r' = 9.1 mm, and the shield is much thicker than the skin depth at all considered frequencies.

8.6.1 Results for Rectangular Conductors

Fig. 40 presents a comparison between RNM and FEM results for the geometry of Fig. 38 with rectangular conductors. Graph (a) shows the self inductance and resistance of conductor 1 (divided by ), whereas (b) and (c) show its mutual inductances and resistances with conductors 2 and 9, respectively. Since the magnetic coupling between the conductors is very strong due to the high-permeability shield, self and mutual inductances have almost the same values. For that reason, differences L11 L12 and L11 L19 have been plotted here (which can be interpreted as “differential mode inductances”) instead of L12 and L19 themselves. As a consequence, the plotted inductance curve displays somewhat higher values for the pair of distant conductors in (c) than for the pair of close conductors in (b), although the actual mutual inductance is of course larger in the latter case. Similar calculations have been carried out for all self and mutual impedances; the graphs all look very similar and therefore only a selection is reproduced here.

The deviation between RNM and FEM at high frequencies (above ca. 105 Hz) is caused by an insufficient resolution of the FEM mesh, particularly in the high-permeability shield where the skin depth is very small. The circles represent the FEM results with quadratic order elements and a mesh size of 68912 elements (about the highest resolution which the employed desktop computer could handle), whereas the small dot symbols in (a) and (b) represent FEM results with a lower resolution (37860 elements). The convergence of the FEM results with increasing mesh resolution towards the RNM prediction shows that the latter is accurate at high frequencies.

58

At high frequencies, both diagonal and off-diagonal elements of the R matrix are dominated by circulating currents in the short-circuited high-permeability shield, and so are almost identical. Due to the skin effect they grow as 1 2 with increasing frequency. At very high frequencies, self and mutual inductances level off to their asymptotic limits corresponding to complete flux expulsion from conductors and shield.

At lower frequencies, between 102 and 103 Hz, the mutual inductance curves have a knee caused by the factors in (74). Fig. 40 thus confirms that the extension (74) of (71) is quite accurate at low frequencies. It is not exact, though, but only an adequate approximation in that frequency regime. Below the knee the skin depths are larger than half the conductor thicknesses, so that the flux completely penetrates the conductors in that frequency region. The self inductance L11 continues to grow proportionally to 1 2 with decreasing frequency, since it is dominated by the flux in the growing skin layer in the magnetic shield. The resistance R11 levels off towards the DC resistance value (81) of about 0.7 mΩ/m, whereas all off-diagonal elements of the R matrix tend to zero.

100

101

102

103

104

105

106

10-8

10-7

10-6

10-5

Indu

ctan

ce p

er le

ngth

(H

/m)

Frequency (Hz)10

010

110

210

310

410

510

6

10-3

10-2

10-1

100

Res

ista

nce

per

leng

th (

/m)

11/L 11 /R

dc/R

deviations due to finite mesh

(a)

59

100

101

102

103

104

105

106

10-8

10-7

Indu

ctan

ce p

er le

ngth

(H

/m)

Frequency (Hz)10

010

110

210

310

410

510

6

10-3

10-2

10-1

100

Res

ista

nce

per

leng

th (

/m)

100

101

102

103

104

105

106

10-7

10-6

Indu

ctan

ce p

er le

ngth

(H

/m)

Frequency (Hz)10

010

110

210

310

410

510

6

10-3

10-2

10-1

100

Res

ista

nce

per

leng

th (

/m)

Fig. 40: Comparison between RNM and FEM calculation for the geometry of Fig. 38 with rectangular conductors. A selection of matrix elements of the L and R matrices obtained from Eq. (73) are plotted as

functions of frequency in the range 1 Hz–10 MHz. Full/dotted line: resistance/ inductance per length according to RNM. Dots/circles: results from FEM calculations with rough/fine mesh.

(b)

11 12 /L L 12/R


(c)

11 19 /L L 19/R


60

8.6.2 Results for Round Conductors

Fig. 41 presents a comparison between RNM and FEM results for the geometry of Fig. 39 with round conductors. The approximation (99a–c) has been used to calculate the reluctances for RNM. The appropriate choice of the channel lengths is not as natural as in the previous example; the values w' = 1.4 r' = 3.6 mm and w" = 3.5 r' = 9.1 mm have been chosen here, which are of the right order of magnitude and lead to a satisfactory agreement between RNM and FEM. Due to the arctan in the expression for the reluctance, the results are not too sensitive to the exact choice.

As before, graph (a) shows the self inductance and resistance of conductor 1 (divided by ), whereas (b) and (c) show its mutual inductances and resistances with conductors 2 and 7, respectively. Similar observations as in Section 8.6.1 can be made; the overall agreement is good also in this case, although the low-frequency agreement for the mutual inductance L12 in (b) is somewhat inferior to that for the rectangular conductors, due to the additional approximations made to treat variable gap widths.

The high-frequency deviations between RNM and FEM are less pronounced than for the rectangular conductors, probably because the channels are much wider on the average than in the previous example and therefore do not require as fine a mesh resolution. The channels are narrow only in the vicinity of the conductor-conductor and conductor-shield near-contact points, making it possible to achieve a higher mesh resolution in these regions with the available computing resources. In this case, the DC resistance of a single conductor is about 0.8 mΩ/m.

100

101

102

103

104

105

106

10-7

10-6

10-5

10-4

Indu

ctan

ce p

er le

ngth

(H

/m)

Frequency (Hz)10

010

110

210

310

410

510

6

10-3

10-2

10-1

100

Res

ista

nce

per

leng

th (

/m)


Rdc/

R11/ L11/

(a)

61

Fig. 41: Comparison between RNM and FEM calculation for the geometry of Fig. 39 with round conductors. A

selection of matrix elements of the L and R matrices obtained from Eq. (73) are plotted as functions of frequency in the range 1 Hz–10 MHz. Full/dotted line: resistance/ inductance per length according to RNM.

Symbols: results from FEM calculations.

100

101

102

103

104

105

106

10-7

Indu

ctan

ce p

er le

ngth

(H

/m)

Frequency (Hz)10

010

110

210

310

410

510

6

10-3

10-2

10-1

100

Res

ista

nce

per

leng

th (

/m)

(L11 L17)/

R17/

(c)


100

101

102

103

104

105

106

10-7

Indu

ctan

ce p

er le

ngth

(H

/m)

Frequency (Hz)10

010

110

210

310

410

510

6

10-3

10-2

10-1

100

Res

ista

nce

per

leng

th (

/m)(L11 L12)/

R12/

(b)


62

8.7 Conclusions about RNM

In this chapter, it has been shown that for calculating L and R matrices of a multi-conductor transmission line, the extended reluctance network method is in quite good agreement with FEM down to the lowest frequencies, and it is even superior to the latter at high frequencies. It is expected to become exact in the limit of high frequencies and narrow gaps between the conductors, which is precisely the “difficult” limit for FEM calculations.

The challenge for applying this method to a transformer winding is, as it was stated earlier, to find an adequate reluctance representation for the areas “outside” the turns, i.e. the regions where there is no well-defined gap. Examples are the area between the LV and HV windings, or between a winding and the transformer tank. A possible solution could be a combination of FEM and RNM where the reluctances for the “outer” regions are calculated using FEM and implemented in the RNM network and this is left for future work.

First attempts to apply the RNM technique to transformer windings have indicated that the resonance frequency shifts observed in Fig. 18−Fig. 21 are mainly due to the proximity effect discussed above, so the old description without proximity effect is still considered to be qualitatively correct. Furthermore, there is an overall satisfactory agreement between model and measurements in Fig. 18−Fig. 21. Hence, the winding calculations in the rest of this thesis will continue to be based on the equations derived in Section 4.4 for the calculation of inductances and resistances.

63

9 Time Domain Reflectometry (TDR) Time Domain Reflectometry (TDR) has been used for localization of discontinuities in transmission lines in diverse applications. In this chapter, its potential as an alternative to Frequency Response Analysis (FRA) for the detection and localization of mechanical changes and damages in transformer windings is investigated. It is suggested that the idea of TDR can be used to visualize the results of frequency response measurements in time domain, where they are easier to interpret. What is reported in this chapter is a description of the content in [52−53].

9.1 Combination of FRA and TDR

As it was stated in Chapter 2, FRA is a powerful method for characterizing a system, analyzing its response to an electrical stimulus of varying frequency. It is a popular method for assessing the mechanical integrity of windings in power transformers and electrical machines. Since measurements are carried out in frequency domain, their reproducibility and signal to noise ratio are very good. Its main disadvantage is however the fact that despite a wealth of experimental and analytical investigations, the interpretation of the FRA results has neither been standardized [54] nor fully agreed among researchers yet (e.g., [5, 41, 55−58]). There is no general consensus about how the deviations in the FRA spectrum are correlated to different types of mechanical damages, and to their location along the winding (see for instance [13, 59−64]). Consequently, one finds in the literature different types of transfer functions (in-impedance, transfer-impedance, voltage transfer ratio, etc.), different measurement techniques, and different ways of interpreting FRA results.

Time Domain Reflectometry (TDR), on the other hand, is a well-established method for detecting and locating faults in transmission lines and cables by observing reflections of injected signals in time domain. Its advantage is the easy and intuitive interpretation of the measurement data.

In this Chapter, an attempt is made to combine the advantages of both the FRA and TDR methods, as a step towards an easier and more reliable detection and localization of faults in transformer windings. This new method is based on the principles of TDR, in which frequency response measurements are visualized in the time domain, in order to facilitate their interpretation.

9.2 Basic Ideas behind the TDR Technique

TDR has traditionally been used as a technique for determining the characteristics of transmission lines by observing reflections of injected signals. It is a powerful tool for the analysis of electrical or optical transmission media such as coaxial cables [65] and optical fibers [66], or for the measurement of soil characteristics in geology and soil science [67]. The principle of TDR consists in injecting a pulse or transient signal of some kind (like a step signal or wave packet) into one end of the transmission line, recording reflections of it at later times, and analyzing their delay and possibly changes in shape. When the injected signal reaches the end of the transmission line or any point of change of wave impedance along it, all or part of the

64

signal is reflected back. By analyzing the magnitude, duration and shape of the total reflected signal, the nature of the wave impedance variation can in some cases be determined. For instance, in case of a single impedance discontinuity, the size of the change can be determined from the magnitude of the reflected signal, and its location from the delay of the reflection relative to the original signal.

The potential of the TDR method for transformer winding faults diagnostics has, to the author’s knowledge, so far not been systematically analyzed in the literature. In this chapter, an attempt is made to provide such an analysis. The main problem is the following: Since transformer windings usually are intrinsically inhomogeneous (due to their layer or disc structure, their division into main and tap winding, to taps and interleaving, etc.) and since signal reflection occurs at all changes of wave impedance, TDR as defined above would not only detect reflections from mechanical damages and deteriorations, but also those from normal geometrical irregularities in the winding. The latter can be removed by considering the difference between TDR measurements before and after mechanical changes have occurred. In other words, the “pathological” reflections from the mechanical changes are separated from other reflections that are “normal”; the difference between the TDR measurements contains information only about the mechanical changes. This procedure is thus comparative in the same sense as the FRA method above, and will be called differential TDR (DTDR) here.

9.3 Simplified Model of a Transformer Winding and the Effect of Dispersion

In TDR, it is considered to be an advantage to be able to send in a narrow pulse trough the injection-point of the system which travels forth in the conductor, reaches its end, and travels back to the injection-point without any serious dispersion. Dispersion in this context means that components with different wavelengths travel at different speeds and thus a pulse gets smoothed out with time. Any dispersion in the system is undesired since the more narrow and localized the pulse remains while travelling along the conductor all the way forth and back, the more narrow and localized will the reflections due to any possible mechanical change (which manifest itself as a impedance change) be, which in turn makes it easier to find the exact position of the presumed mechanical change along the winding. Further, another possible problem with a significant dispersion would, in addition to the smoothening of the pulse, be a pronounced loss of pulse magnitude, which could lead to the reflected pulse getting too weak to be measurable.

When the potential of the TDR method for diagnostics of transformer winding integrity is investigated, the differences between windings and cables have to be considered. A crucial difference between a cable, which is a specific kind of a transmission line, and a transformer winding is the fact that for a transmission line, ideally each infinitesimal segment only has a self-inductance and a capacitance to ground, whereas for a winding, due to the fact that the conductor is winded tightly, there will be mutual inductances between all turns and capacitances between neighboring turns in addition to the self inductances and ground-capacitances.

Using a lumped-element approach, a lossless transmission line can be modeled by a ladder network consisting of self-inductances and capacitances to ground, as depicted in Fig. 42(a), where l and cg are inductances and ground-capacitances per unit length respectively, and ∆x is a differential length of the transmission line.

65

cg∆x

l∆x

x

i

(a)

cg∆x cs∆x

l∆x i

(b)

Fig. 42: Equivalent circuits of a transmission-line (a) and of a transformer winding (b). For a transformer winding, the lumped-element network depicted in Fig. 42(a) has to be modified so as to include the mutual inductances between turns and the capacitances between adjacent conductors. The lumped-element network for a simplified case, where each disc in the HV winding is lumped together into one element, can be seen in Fig. 42(b) (the LV winding is assumed to be open and at low potential everywhere, so that it can be replaced by ground), where cg is the ground-capacitance of each disc and cs is the capacitance between adjacent discs, both per unit length. Further, l∆x is the self-inductance of each element, and the dashed lines illustrate their mutual inductances.

The solutions for the voltage u and current i along the transmission line and the transformer winding are derived in [68, 69] and in Appendix B, where it is shown that the wave velocities in the transmission line and the winding are

g

1v

lc , and 2 2 2 2 2s s

0g g g

1 c cv v

lc c c (100)

respectively ( is the length of conductor per disc). The wave velocity in the transmission line is independent of the frequency, but for the wave velocity in the transformer winding, it can be seen that it has the value v0 = (lcg)

−½ only for waves of very low frequencies. As the driving frequency ω increases, the velocity of propagation v decreases and vanishes at a critical frequency ωcr = ( 2

slc )−½ beyond which no propagation is possible within the winding. If the

driving frequency exceeds the critical frequency, the wave velocity v in the latter part of Eq. (100) becomes imaginary and can be rewritten as

2 2 2s0

g

j jc

v vc

. (101)

The voltage in the winding will now be of the form (see [68, 69] and Appendix B)

j /0e et xu u . (102)

This indicates that above the critical frequency, no wave propagation can exist in the winding. Instead, there is merely an exponential decay of the voltage as one progresses into the winding.

It can be concluded that for a lossless transmission line, since the wave velocity is frequency independent (up to very high frequencies) and thus the different frequency components propagate with the same speed, the pulse used for TDR may be chosen arbitrarily narrow in a wide frequency range; it will travel along the line, be reflected at any impedance change, and travel back to the observer without any change in its shape. For a transformer winding, in contrast, the situation is entirely different since the wave velocity is frequency dependent and hence the different frequency components of a signal travel with different speeds. Components with frequencies higher than the critical frequency will not even propagate at all.

66

As a consequence, there may be a considerable change in the shape of the TDR-pulse as it travels along the winding; it may lose its narrowness and magnitude, which makes the localization of an impedance change in a winding more difficult, although it may still be possible to a certain extent if the impulse shape is suitably chosen. Further below it will be investigated how the effects of dispersion can be minimized by an appropriate choice of the pulse width. But before that, the computer model used for this study is presented.

9.4 The Transformer Winding Model Used for TDR Simulations

For the simulations in this chapter, the most accurate model developed in Chapter 4 (model 3) has been used. Also, a reasonable agreement between this model and measurements up to 20 MHz was observed in Chapter 6, which means that the model should be able to correctly predict the propagation of pulses with frequency content up to about 20 MHz, with essentially only a time rescaling factor of order unity due to the shift of resonance frequencies.

9.5 The Effect of Choice of Rise-Time for the Applied Voltage

The above mentioned model has been used to simulate a winding consisting of 60 discs with the same dimensions as in the experimental set-up in Chapter 5, and with the distance from the inner turns to ground (di) being 50 mm, the outer ground wall being neglected (do → ∞), and the vertical distance between two neighboring discs (ks) being 6 mm (see Fig. 9). The impedance amplitude seen from the terminal of the winding is plotted in Fig. 43. It was previously argued in Section 9.3 that when a narrow pulse is injected into a winding, the different frequency components of the pulse travel with different speeds, the pulse suffering from dispersion and getting smoothed out. The state space equations (Eq. (47)) of the model have been solved in the time domain using an ode-solver in MATLAB, with an applied external voltage of Gaussian shape, Vapp(t) = (10 V) × exp(−0.5×(t / t1)

2), with different values of the parameter t1 chosen in the range between 0.4 µs and 2.5 µs. The Fourier transform is of the form Vapp(ω) exp(−0.5×(ω / ω1)

2) with ω1 = 1/t1. The resulting absolute and normalized voltage along the winding for different values for t1 are depicted in Fig. 44 and Fig. 45, respectively, at an instant of time where the peak of the wave has reached about the middle of the winding. It can be observed that the pulse amplitude and width both decrease with decreasing t1, but when t1 is decreased below somewhere around 1 µs, the pulse continues to lose in amplitude (Fig. 44) without getting markedly narrower anymore (as seen most clearly in Fig. 45). The bell-shaped Fourier transform of the applied voltage Vapp(ω) decreases rapidly with frequency; the major frequency content of the applied voltage Vapp(t) is confined to frequencies below about 2ω1. For t1 = 1 µs and ω1 = 1/t1, the significant part of the frequency content of Vapp(t) is located below about 3× 105 Hz. Thus, by inspecting the in-impedance of the winding shown in Fig. 43, it can be concluded that the regular series of resonances (the local minima) of the impedance extending up to around 3 × 105 Hz are the ones building up the travelling pulse in the winding. It can also be observed that the three or four first resonances are the most pronounced ones.

The normalized voltage along the winding for an even shorter applied voltage pulse (which contains more high-frequency components) is depicted in Fig. 46, where it can be observed that the higher frequency components introduced in the system only cause local oscillations along the winding. These are “internal” oscillations within individual discs which were called azimuthal resonances in Section 7.1.2, located at frequencies above 10 MHz and causing the

67

most pronounced impedance minima in both simulations and measurements (see Fig. 18 to Fig. 21 and Fig. 43).

Hence it can be concluded that when applying TDR to a transformer winding, the frequency content of the applied voltage should preferably be confined to the frequency range where the first few pronounced resonances of the winding in-impedance occur, to get the pulse as steep as possible without an excessive loss of magnitude. Also, the simulation models need to be valid only in a frequency range which covers the first group of resonances, implying that every disc in the winding can be lumped together into one element of the model. On the other hand, localization of damages with a higher accuracy than this minimum pulse width (which in this case is of the order of 10 discs) will not be possible.

104

105

106

107

102

104

106

Frequency [Hz]

Impe

danc

e am

plitu

de [

]

Fig. 43: Simulated in-impedance amplitude for a winding with sixty discs.

0 10 20 30 40 50 60

0

1

2

3

4

5

6

7

8

9

10The voltage along the winding

Vol

tage

(V

)

disc number

Fig. 44: The voltage along the winding for different values for t1.

t1 = 2.5 µs

t1 = 0.4 µs

t1 = 1 µs

68

0 10 20 30 40 50 60

-0.2

0

0.2

0.4

0.6

0.8

1

The normalized voltage along the winding

Nor

mal

ized

vol

tage

disc number

Fig. 45: The normalized voltage along the winding for different values for t1.

0 10 20 30 40 50 60-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

vol

tage

disc number

The normalized voltage along the winding

0 1 2 3

0

0.01

0.02

0.03

Fig. 46: The normalized voltage along the winding showing azimuthal oscillations

9.6 Simulations with and without Winding Damages

The same winding configuration as in the previous section has been simulated in the time domain, before and after a set of different “damages” were introduced. A step voltage (somewhat smoothed, i.e., with a finite rise time of the order of 1 µs) was applied on one end of the winding, while the other winding end was grounded and the injection-point current was

t1 = 2.5 µs

t1 = 1 µs

t1 = 0.4 µs

Very short and high-frequent applied voltage

69

recorded. The reason why a step voltage was used instead of a pulse voltage as in Section 9.5 is explained at the end of this section.

The introduced damages were of three types (radial buckling, conductor bending and loose winding conductors), and the location of each fault was varied along the winding. Radial buckling (RB), caused by the radial forces acting on the winding during a short circuit fault, is a compression of the inner winding (usually the LV winding) or part of it, leading to a local change in the ground capacitance of the HV winding [5]. In the model, this implies a decrease in Cig for those discs of the HV winding which face the buckled region of the grounded LV winding (Fig. 47(a)). In the examples below it is assumed that buckling is restricted to one corner of one disc of the square winding, resulting in a doubled distance between the HV winding and ground for that corner, which corresponds to a local decrease of Cig to half of its original value.

Loose Winding Conductors (LWC) and Conductor Bending (CB) are instead caused by axial forces compressing the winding, and may also occur in one or several winding sections. The axial forces can for instance result in a temporary tilting of the conductors, leaving the turns in one or several discs loose when the forces are removed, which in turn leads to a local decrease in Ctt (Fig. 47(b)). They may also bend the winding conductors in the region between the radially placed insulation spacers, resulting in an average decrease in separation distance between two neighboring discs and hence a local increase in Cdd (Fig. 47(c)). For the damaged sections, it is assumed that the distance between the turns in a disc is increased to half of the conductor width for the LWC case, and that the separation distance between two discs is reduced by a factor of ½ for the CB case. These changes correspond to a local reduction of Ctt to about 6% of its original value or a local doubling of Cdd , respectively, in the model. Since these relatively small geometrical changes mainly influence the capacitive distribution, the changes in the inductances were neglected [59].

increasedspace

increased space

grou

nd

decr

ease

d sp

ace

grou

nd

(a) (b) (c)

Fig. 47: (a) Radial buckling (RB), (b) loose winding conductors (LWC), and (c) conductor bending (CB). The locations for the three different fault types were selected as follows: for the RB case, the buckling was chosen to be close to the 10th, 30th, or 50th disc, respectively; for the LWC case, the turns in the 10th, 30th, or 50th disc, respectively, were chosen to be loose; for the CB case, the distance between the 10th and 11th, 30th and 31st, or 50th and 51st discs, respectively, was reduced. The difference ∆I = Ibefore– Iafter between the currents in the injection-point before and after introducing the different damages is plotted in Fig. 48 to Fig. 50 as a function of the time after applying the voltage step.

70

0 0.5 1 1.5 2 2.5 3

x 10-5

-5

0

5x 10

-6

I

(A)

time (s)

0 0.5 1 1.5 2 2.5 3

x 10-5

-5

0

5x 10

-6

I

(A)

time (s)

0 0.5 1 1.5 2 2.5 3

x 10-5

-5

0

5x 10

-6

I

(A)

time (s)

Fig. 48: DTDR for buckling (RB) close to the 10th, 30th, and 50th disc, respectively.

Reflection from the change in the 10th disc

Reflection from the winding end





71

0 1 2 3 4

x 10-5

-3

-2

-1

0

1

x 10-5

I

(A)

time (s)

0 1 2 3 4

x 10-5

-2

-1

0

1

x 10-5

I

(A)

time (s)

0 1 2 3 4

x 10-5

-2

-1

0

1

x 10-5

I

(A)

time (s)

Fig. 49: DTDR for loose turns (LWC) in the 10th, 30th, and 50th disc, respectively.







72

0 1 2 3 4

x 10-5

-4

-2

0

2

x 10-5

I

(A)

time (s)

0 1 2 3 4

x 10-5

-4

-2

0

2

x 10-5

I

(A)

time (s)

0 1 2 3 4

x 10-5

-4

-2

0

2

x 10-5

I

(A)

time (s)

Fig. 50: DTDR for decreased space (CB) between the 10th and 11th, 30th and 31st, and 50th and 51st discs,

respectively.


Reflection from the change between the 50th and 51st discs


Reflection from the change between the 30th and 31st discs

Reflection from the change between the 10th and 11th discs


73

For the RB case (Fig. 48), the reflections have roughly the shape of single Gaussian pulses; the reason for this is that the damages in this case are local changes of the capacitance Cig to ground, which implies that the reflected current pulse is proportional to the time derivative of the incoming step voltage. The reflections from damage in the 50th disc and from the winding end partially overlap.

In Fig. 49 and Fig. 50, which represent the LWC and CB cases respectively, the reflections have roughly the shape of derivatives of Gaussian pulses (almost the same pattern for both cases, but with different signs and amplitudes). Similarly as above, the reason for this is that the damages now are local changes of the series capacitance between different parts of the winding (Ctt or Cdd), which implies that the reflected current pulse is proportional to the second time derivative of the incoming step voltage. As above for the RB case, the more the damage is located near the winding end, the more the reflections from the damage overlap with those from the winding end, which makes them harder to detect; a possible remedy of this problem might be to perform separate measurements with signal injection from either end of the winding. It can be concluded that for a step-like applied voltage, a localized RB damage can be distinguished from the two other cases by its different reflection shape. A discrimination between the LWC and CB damages is more difficult since their reflection shape is the same. In the simulations here, their reflections had opposite signs, but if the damage for the CB case had been an increase instead of a decrease of the inter-disc space, they would have the same sign and thus be indistinguishable.

If the damages are sufficiently localized, their rough location (dm) can be inferred by comparing the time delays of the reflections from the modified disc (tm) and from the winding end (twe). Assuming that the pulse propagation speed is roughly constant along the winding, the location of the damage can be estimated as dm dwe × tm / twe where dwe is the total length of the winding. For example, applying this relation to the second curve in Fig. 48, dwe × tm / twe = 60 × (1.41 × 10–5) / (2.78× 10–5) = 30.4 should be roughly equal to dm = 30, which obviously is the case.

As promising as the above appears, a severe limitation may be the smallness of the difference between responses with and without damage. Since the magnitudes of reflections ∆I from the changes for all three cases are of the order of 0.5 – 1.8 × 10–5 A, and the total current I in the injection-point (before or after the damage) is of the order of 5 × 10–4 A, the ratio ∆I / I is about 1–4%. This means that the reflections are of the same order as the usual absolute measurement accuracy in transformer diagnostics, and thus may be hard to detect unambiguously. Interpretation of frequency domain FRA spectra of course suffers from the same problem.

Finally, the question of the applied signal shape is addressed. When examining the effect of choice of rise-time on the propagating pulse in Section 9.5 a Gaussian voltage pulse was chosen, whereas when simulating DTDR for damage detection in the present Section a step-like shape whose time derivative is a Gaussian pulse was chosen. Since the time evolution of the system is linear, this means that also the reflected current ∆I(t) in the former case is proportional to its time derivative in the latter. Whereas the information content is exactly the same in both cases, the step-like applied voltage thus leads to reflections with fewer oscillations (at least for the RB case), which in the author’s opinion facilitates interpretation. This is exemplified in Fig. 51 to Fig. 53, where a selected number of DTDR simulations have been performed again, with a Gaussian voltage pulse (rise time of the order of 1 µs). Comparing the 1st graph in Fig. 48 with Fig. 51, the 1st graph in Fig. 49 with Fig. 52, and the 1st graph in Fig. 50 with Fig. 53, it is seen that the step-like applied voltage leads to reflections with fewer oscillations.

74

0 0.5 1 1.5 2 2.5 3

x 10-5

-5

0

5

x 10-6

I

(A)

time (s)

Fig. 51: DTDR for buckling (RB) close to the 10th disc with a Gaussian shaped applied voltage.

0 1 2 3 4

x 10-5

-4

-2

0

2

4x 10

-5

I

(A)

time (s)

Fig. 52: DTDR for loose turns (LWC) in the 10th disc with a Gaussian shaped applied voltage.

0 1 2 3 4

x 10-5

-5

0

5

x 10-5

I

(A)

time (s)

Fig. 53: DTDR for decreased space (CB) between the 10th and 11th discs with a Gaussian shaped applied voltage.





Reflection from the change between the 10th and 11th discs


75

9.7 Conversion of In-Impedance Measurements to DTDR Signals

As already mentioned in Section 9.1, there is no general agreement in the literature about how to interpret changes in the response characteristics of an arbitrary winding directly in the frequency domain. If these changes are instead converted into time domain by a Fourier transformation, interpretation along the lines of Section 9.6 becomes possible.

The procedure is as follows. The Fourier transform of the difference between currents in the injection-point before and after damage, ∆Ĩ(ω) = Ĩbefore– Ĩafter , is obtained as the product of Vapp(ω) and ∆Ỹ(ω), where Vapp(ω) is the Fourier transform of the applied voltage at the injection-point, and ∆Ỹ(ω) = Ỹbefore– Ỹafter is the difference between the in-admittances (inverse of in-impedances) of the winding before and after damage, measured in frequency domain. After that, ∆Ĩ(ω) is inverse-Fourier transformed to provide ∆I(t). Note that the frequency dependent in-impedance of a winding has to be measured instead of the more conventional end-to-end FRA transfer function. From the latter the DTDR signal cannot be deduced. The main advantages of this procedure are summerized:

The measurements are performed in the frequency domain, leading to better measurement reproducibility and signal-to-noise ratio than impulse measurements.

Instead of measuring the more conventional end-to-end FRA transfer function (which requires connections from the measurement instrument to both winding ends), the in-impedance (which requires only one connection to the signal injection point) is measured. This in turn leads to simplified measurement procedures and thus possibly to better reproducibility.

Simple and intuitive interpretation: the type and location of possible mechanical damages may be determined by inspection of ∆I(t), in the way explained in Section 9.6 above. It is not known how to achieve this directly with the frequency domain data.

77

10 Summary, Conclusions and Future Work In this thesis, an attempt has been made to elevate the understanding of the information contained in FRA measurements by evaluating the FRA method for (much) higher frequencies than what is standard and by developing a new method in which FRA and TDR are combined. Appropriate models have been created to accomplish these tasks:

A simple, frequency dependent complex-µ model of magnetic core material has been developed and adjusted to measurements. Its real and imaginary parts were compared to measurements in a wide frequency range. The agreement was found satisfactory, especially for higher frequencies, which makes the complex-µ model a very convenient starting point for the estimation of flux distribution and losses in complicated core geometries.

Furthermore, H-B curves from the measurements, the simple complex-µ model and a detailed hysteresis model were compared for different frequencies. Again the results from the complex-µ model were found to agree well with measurements at higher frequencies. At low frequencies and high field amplitudes the complex-µ model deviates from measurements and detailed hysteresis model, since it does not take saturation effects properly into account. This is, however, not expected to affect its usefulness for loss estimation.

Further, a winding model based on the lumped element approach has been developed. The

model has three steps of discretization, and these three models can be simulated in both frequency and time domains.

When simulated in frequency domain, the models are used to predict the impedance of the winding in a wide frequency range (which is akin to FRA). The frequency domain simulations have been compared to measurements and the model has been verified. Two classes of internal resonance modes of a single disc have been identified, the “radial” modes at high frequencies and the “azimuthal” modes at even higher frequencies (above around 10 MHz for the coils investigated in this thesis), and have been studied by measurements and model simulations. The radial modes are characterized by a rather constant voltage amplitude within every turn, whereas the azimuthal modes describe electrical oscillations of different parts of the same turn against each other and therefore can only be seen in models with more than one segment per turn. Moreover, measurements show that these modes (especially the azimuthal modes) are highly sensitive to small changes in the disc geometry.

The application of these mode signatures to winding fault detection could be subject of future research, but it should be kept in mind that it is extremely difficult to achieve a sufficient measurement reproducibility on real power transformers in that frequency range.

When simulated in time domain, the models are used to predict the response of the winding to an injected impulse (which is akin to TDR). It has been shown that the distortion of the injected signal, caused by the effect of dispersion, can be limited to an acceptable level if the frequency content of the applied signal is confined within the right frequency window. Also, three different fault types with different locations along the winding have been investigated, and it has been shown that these faults can be detected and to some extent located, and one of the fault types can be distinguished from the two others. Furthermore, it has been shown that frequency dependent in-impedance measurements can be numerically converted to time

78

domain signals, which allows a simple and intuitive interpretation. In future work, it remains to apply this method to a real damaged transformer, and to investigate its potential for detection and localization of other fault types.

Also, since the proximity effect was not included in the calculation of losses and inductances, a reluctance network method has been presented. Its original purpose was to reliably compute the frequency dependent inductance and resistance matrices characterizing the transformer winding in a wide frequency range, and as a starting point, it has been developed here for shielded multi-conductor transmission lines. The method has been demonstrated for some simple example geometries, where its accuracy has been verified with harmonic FEM field calculations. It remains to extend this method to the full geometry of transformer windings in future work.

79

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Effects in Multi-Conductor Transmission Lines”, submitted to IEEE Transactions on Magnetics.

45. D. Bormann, H. Tavakoli, “Reluctance Network Theory of Skin and Proximity Effects in Strongly Coupled Multi-Conductor Transmission Lines”, submitted to IEEE Transactions on Power Delivery.

46. Nilanga Abeywickrama, “Effect of Dielectric and Magnetic Material Characteristics on Frequency Response of Power Transformers”, Doctoral Thesis, Chalmers University of Technology, Gothenburg, Sweden 2007.

47. J. Turowski, M. Turowski, M. Kopec., “Method of Three-Dimensional Network Solution of Leakage Field of Three-phase Transformers”, IEEE Transactions on Magnetics, Vol. 26, No. 5, pp. 2911-2919, Sept. 1990.

48. See any theoretical textbook on electrical engineering or on electromagnetic wave propagation in dissipative media, for instance [49] p. 268, [50] p.438, or [51] p. 204.

49. Karl Küpfmüller, “Einführung in die theoretische Elektrotechnik”, 6th ed., Springer Verlag Berlin/Göttingen/Heidelberg, 1959.

50. Richard B. Adler, Lan Jen Chu, Robert M. Fano, “Electromagnetic Energy Transmission and Radiation”, John Wiley & Sons, Inc., New York, 1960.

51. A. Greenwood, “Electrical transients in power systems”, 2nd ed., John Wiley, 1991.

52. H. Tavakoli, D. Bormann, G. Engdahl, “Time Domain Reflectometry for Fault Localization in Transformer Windings”, submitted to International Journal of Applied Electromagnetics and Mechanics.

82

53. H. Tavakoli, D. Bormann, G. Engdahl, “Fault Localization in Transformer Windings using Time-Domain Representation of Response Functions”, submitted to European Transactions on Electrical Power.

54. P. Picher et al., Final Report of Cigré WG A2.26, “Mechanical Condition Assessment of Transformer Windings using Frequency Response Analysis (FRA)”, CIGRE Technical Brochure No. 342, April 2008. (67 pages) [ISBN 978-2-85873-030-8]

55. L. Satish, S. K. Sahoo, “An Effort to Understand What Factors Affect the Transfer Function of a Two-Winding Transformer”, IEEE Transactions on Power Delivery, Vol. 20, No. 2, pp. 1430−1440, Apr. 2005.

56. Zhongdong Wang, Jie Li, D. M. Sofian, “Interpretation of Transformer FRA Responses− Part I: Influence of Winding Structure”, IEEE Transactions on Power Delivery, Vol. 24, No. 2, pp. 703−710, April 2009.

57. D. M. Sofian, Zhongdong Wang, Jie Li, “Interpretation of Transformer FRA Responses− Part II: Influence of Transformer Structure”, IEEE Transactions on Power Delivery, Vol. 25, No. 4, pp. 2582−2589, October 2010.

58. J. A. S. B. Jayasinghe, Z. D. Wang, P. N. Jarman, A. W. Darwin, “Winding Movement in Power Transformers: A Comparison of FRA Measurement Connection Methods”, IEEE Transactions on Dielectrics and Electrical Insulation, vol. 13, No. 6, pp. 1342−1349, December 2006.

59. P. Karimifard, G. B. Gharehpetian, S. Tenbohlen, “Localization of Winding Radial Deformation and Determination of Deformation Extent Using Vector Fitting-Based Estimated Transfer Function”, European Transactions on Electrical Power, Vol. 19, No. 5, pp. 749−762, July 2009.

60. E. Rahimpour, D. Gorzin, “A New Method for Comparing the Transfer Function of Transformers in Order to Detect the Location and Amount of Winding Faults”, Electrical Engineering (Archiv fur Elektrotechnik), Vol. 88, No. 5, pp. 411-416, May 2005.

61. S. K. Sahoo, L. Satish, “Discriminating Changes Introduced in the Model for The Winding of a Transformer Based on Measurements”, Electric Power Systems Research, Vol. 77, No. 7, pp. 851−858, May 2007.

62. K. Ragavan, L. Satish, “Localization of Changes in a Model Winding Based on Terminal Measurements: Experimental Study”, IEEE Transaction on Power Delivery, Vol. 22, No. 3, pp. 1557−1565, July 2007.

63. P. Karimifard, G. B. Gharehpetian, S. Tenbohlen, “Determination of Axial Displacement Extent Based on Transformer Winding Transfer Function Estimation Using Vector-Fitting Method”, European Transactions on Electrical Power, Vol. 18, pp. 423−436, May 2008.

64. P. Karimifard, G. B. Gharehpetian, “A New Algorithm for Localization of Radial Deformation and Determination of Deformation Extent in Transformer Windings”, Electric Power Systems Research, Vol. 78, pp. 1701−1711, 2008.

65. R. Eriksson, R. Papazyan, G. Mugala, “Localization of Insulation Degradation in Medium Voltage Distribution Cables”, First International Conference on Industrial and Information Systems, pp. 167−172, 8−11 August 2006.

66. Y. Sato, K. Aoyama, “Optical Time Domain Reflectometry in Optical Transmission Lines Containing In−Line Er−Doped Fiber Amplifiers”, Journal of Lightwave Technology, Vol. 10, issue 1, pp. 78−83, January 1992.

67. A. Cataldo, L. Tarricone, F. Attivissimo, A. Trotta, “A TDR Method for Real−Time Monitoring of Liquids”, IEEE Transactions on Instrumentation and Measurement, Vol. 56, issue 5, pp. 1616−1625, October 2007.

http://ieeexplore.ieee.org/xpl/RecentCon.jsp?punumber=4216542

http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=19

83

68. R. Rüdenberg, “Performance of Travelling waves in Coils and Windings”, AIEE Transactions, Vol. 59, pp. 1031−1040, December 1940.

69. R. Rüdenberg, “Electrical Chock Waves in Power Systems”, Harvard University Press, Cambridge, Massachusetts, 1968.

85

Appendix A

RNM for a Stack of Slabs In this Appendix, the reluctance network formulas are proven to be valid also for a configuration with several conducting slabs, stacked on each other in z direction with insulating sheets in between. The derivations in this Appendix are borrowed from [45].

The solution for a single-slab geometry (Section 8.4.2) is with out much difficulty generalized here to a stack containing an arbitrary number n of slabs. The slabs are labelled by k = 1,…,n and the insulating sheets by i = 0,…,n, in the way shown in Fig. 54. As before, the stack ends in a perfectly diamagnetic ( 0) layer above the uppermost insulating sheet (not shown in the picture).

Since Eqs. (82)–(89a,b) are valid separately for every slab, the interface fields ,y kH and

,x kE , the fluxes Фc,k and Фg,i, as well as the total currents Ik can again be expressed in terms of

coefficients kC and kC .

x

z

y

0

2I

g,2R

,2R

,1R

g,1R

g ,2

,2

,1

g,1

1I

,1R

,0R

g,0R

,1

,0

g,0

,2R,2

3I

1a

2d

2a

0a

1d

c,2

c,1

w

Fig. 54: Geometry of a stack of slabs, generalizing Fig. 35, Fig. 36. Also shown are the different flux paths in the insulating gaps and skin layers, together with their respective reluctances.

86

Assuming as before that the connections to the conductors are made from below the stack, the following are obtained for the voltages Uk:

1 g,0 ,1j xU E , (A1)

2 g,0 c,1 g,1 ,2j xU E , (A2)

etc., so that in general

1

g, 1 c, g, 1 ,1

j jk

k i i k x ki

U E

. (A3)

(By definition, the empty sum for k = 1 is equal to zero.) Using (85), (87), (89a,b) this can be re-expressed as

1

1 g, 11

j 2k

i ik i i i i i

i i

U a C C C

1 g, 1j j k kk k k k k k

k

a C C C C

1j k

k kk k

C

. (A4)

In the same way, the condition of zero total flux can be re-expressed as follows:

g, 1 c, g,1

0n

i i ni

1 g, 11

2n

i ii i i i i

i i

a C C C

g,n n n na C C . (A5)

Since the field Hy is constant across any given gap k and equal to the adjacent surface fields

,y kH and , 1y kH , (84) implies the following matching condition for kC and kC :

, , 1k k y k y kC C H H

1 1k kC C , for 1,..., 1k n . (A6)

Eq. (A6) allows in particular to express the terms 2 iC in the sums of (A4), (A5) in terms of

differences i iC C , since

2 i i i i iC C C C C

1 1i i i iC C C C , (A7)

so that the sums of (A4), (A5) can be re-arranged into the following forms:

87

11

1

j 1j

kk

k i i i k ki k k

U w C C C

R , (A8)

1 11

1

0n

i i i n n ni

C C C C

R R , (A9)

with the coefficients 1

iR defined by

1 1 10 0 g,0

1

aw

R ,

1 1 1g,

1

i i i ii i i

i i

aw

R for 1,..., 1i n ,

and 1g,

n nn n n

n

aw

R . (A10)

The iR have exactly the form of the channel reluctances defined in (74), adapted to the particular geometry and labelling convention of the stack in Fig. 54. Furthermore, due to (88) one can simply write

2k kI wC . (A11)

Finally, defining as in (96)

int ,

j

2k

k kk

Zw

R and int ,

j 1

2k

kk k

Zw

, (A12)

and moving the second term on the r.h.s. in (A8) to the left, one obtains

1int, int , 1

1

jk

k k k k k i i ii

U Z I Z I w C C

RR . (A13)

The r.h.s. of this equation is now denoted by Qk,

11

1

jk

k i i ii

Q w C C

R for 1,...,k n , (A14)

and all Qk are collected in a column vector Q. In order to prove (73), it then remains to show that

T jQ I R . (A15)

This can be achieved in the following way. First, it is noted that for the particular stack geometry of Fig. 54,

88

1 2

0

1

1 0 01 1

,0 1 01

0 0 1

k n i

n

(A16)

so that

0 1 1

1 1 2 2T

2 2 3

1

1 1

0 0

0 0 .

0 0n

n n n

R R R

R R R R

R R R

R

R R R

R (A17)

To facilitate the argument, one can resort to the following trick. One can choose some (arbitrary) 1nC

and 1nC with 1 1n nC C

n nC C , so that the condition of zero total flux

(A9) takes the form (A14) with Qn+1 = 0, whereby the validity of (A14) is extended to 1k n . Defining in addition Q0 = 0, one has for all 0,...,k n :

11 1 1jk k k k kQ Q w C C R . (A18)

This in turn implies that for all k = 1,…,n , the k:th component of the l.h.s. of (A15) can now be written as

1 1 1 1k k k k k k kQ Q Q R R R R

1 1 1k k k k k kQ Q Q Q R R

1 1j jk k k kw C C w C C

j 2 jk kw C I , (A19)

where the last two equalities follow from (A6) and (A11), respectively. But the end result is just the r.h.s. of (A15) — which completes the proof of (73) for the particular stack geometry of Fig. 54.

89

Appendix B

Wave Velocity in Transmission Lines and Transformer Windings In this Appendix, the relations governing the voltages and currents in a transmission line and a transformer winding are derived and the wave velocities are obtained. These derivations are largely borrowed from [68−69]. For the transmission line ladder network in Fig. 42, if u is the voltage in the nodes, and i is the current in the inductors, then the voltage difference between an arbitrary node and the node before it is

iu l x

t

, (B1)

which in the limit ∆x → 0 leads to u i

lx t

. (B2)

Further, the difference between the currents in one inductive branch and the one before is

g

ui c x

t

, (B3)

leading to

g

i uc

x t

. (B4)

By differentiating Eq. (B2) and (B4) with respect to x and t respectively, and eliminating i(x,t), one will arrive at

2 2

g2 2

u ulc

x t

, (B5)

which is the wave equation for a voltage wave (a similar one can be obtained for the current i(x,t)) propagating with the velocity

g

1v

lc . (B6)

For the case of a transformer winding, consider a tightly wound coil of individual pancake turns (or discs), which are parallel and closely spaced (Fig. 55(a)).

90

cg

cs

ground

(a)

LV

cg

cs

(b)

HV LV

Fig. 55: Tightly wound coil consisting of individual pancake turns (a), and a transformer winding in cross-

section (b). The charging current for each element of conductor length ∆x of the n:th turn of the winding, which is due to the ground-capacitance and thus flows mainly to ground, is

g, gn

n

ui c x

t

, (B7)

where cg is the ground-capacitance per unit length of conductor. Let cs be the disc-to-disc capacitance per unit length between adjacent pancake discs. Then the charging current that flows from the n:th turn to the (n −1):th turn is

1s, s s

n n nn

u u ui c x c x

t t

, (B8)

and that to the (n +1):th turn is

1s, s s

n n nn

u u ui c x c x

t t

. (B9)

The sum of these two currents,

2

s, s sn

n n n

ui c x u u c x

t t

, (B10)

is the total disc-to-disc capacitive current per unit conductor length. Here ∆u is the voltage difference between adjacent turns and ∆2u is the difference of this difference between successive turns. The voltage difference is thus that due to a length of conductor

x (B11) of one complete turn of length . Thus the second difference of voltage can be rewritten in differential rather than difference form

2 22 2 2

2 2

u uu

x x

, (B12)

which is valid if the coil contains many turns and if the detailed distribution of the phenomena along a single turn length are not of interest. The total disc-to-disc and ground capacitive

91

charging current of each turn, as given by Eqs. (B7) and (B10), is then equal the decrease −∆in in the current in the n:th turn. Thus, using Eq. (B12), one obtains

32

g s 2n n

n

u ui c x c x

t t x

, (B13)

and hence the space derivative of the current is

32

g s 2

i u uc c

x t t x

, (B14)

which shows that it depends on the change of voltage not only with time but also with space. For fast travelling pulses with steep portions, and thus pronounced changes in space along the conductor, the second term of Eq. (B14) can become very significant, particularly if the disc-to-disc capacitance cs is substantially larger than the ground capacitance cg, which is frequently the case for tightly wound coils.

The current as well as the voltage in a long coil varies from turn to turn. If the n:th turn, of self inductance λ per unit length, were separated from the rest of the coil, the voltage induced by the current in it would be

nn

iu x

t

. (B15)

The two adjacent turns coupled by the mutual inductance m per unit length further add an induced voltage

1 1n nmn

i iu m x

t

. (B16)

By subtracting in from the current in each of the adjacent turns and rearranging the terms, the following expression for the effect of the adjacent turns is deduced:

2

1 1 2 2 n nmn n n n n n

i iu m x i i i i i m x m x

t t t

. (B17)

Thus the mutual inductance of the adjacent turns contributes an induced voltage that can be thought of as consisting of two parts, one of which is proportional to effects in the n:th turn and the other due to second differences between the currents in the turns. Each succeeding turn will produce an induced voltage in the n:th turn that will have the same form as Eq. (B17), except that m will become smaller with increasing distance between turns. All induced voltages proportional to the current in in the n:th turn can be summed from Eqs. (B15) and (B17), to give

n nln

i iu m x l x

t t

. (B18)

Here l is the self-inductance of the coil per unit length derived from the total inductive effects between all turns; thus it is the self-inductance of the entire coil divided by the length of wire. In the last term of Eq. (B17) one can introduce Eq. (B11) and write

92

2 22 2 2

2 2

i ii

x x

, (B19)

when many turns are again used as basis for transformation from the difference to the differential form. For this term, only the influence of the immediately adjacent turns is taken into account, because the induced effects due to the farther ones have already been included in Eq. (B18) via the total inductance of the coil and should not be counted again. The additional inductive influence of the two adjacent turns is effective here, like that of the inter-disc capacitance treated earlier. The voltage induced in the n:th turn, in accordance with Eqs. (B17) and (B18), must be equal to the decrease −∆un of the observable voltage difference. Using Eq. (B19) one thus obtains

32

2n n

n

i iu l x m x

t t x

, (B20)

And the voltage difference along the coil is

32

2

u i il m

x t t x

. (B21)

Again it can be seen that this depends on the variation of current with time as well as with space. For fast travelling pulses with steep portions the third-order derivative may attain appreciable magnitudes. However, its influence here is smaller than the corresponding capacitive one on voltage in Eq. (B14) because the mutual inductance m between adjacent turns is small compared with the self-inductance component l per unit turn of the entire coil. This means that the second term in Eq. (B21) can be neglected as a simplification. This approximation is fully explained and justified in [68] and [69]. Thus one has

32

g s 2

i u uc c

x t t x

, and

u il

x t

. (B22)

The validity of Eq. (B22) can be extended to other cases of importance. If the coil has not only single turns as in Fig. 55(a), but a number of turns per disc as shown in Fig. 55(b), it is possible to consider average values un and in for each disc n. Correspondingly, one can now let cg be the ground-capacitance for each disc and cs the capacitance between adjacent discs, both per unit length. The definition of self-inductance l is unchanged and is now the length of conductor per disc.

By differentiating the first and second part of Eq. (B22) with respect to t and x respectively, and eliminating i(x,t), one can write

2 2 42

g s2 2 2 20

u u ulc lc

x t t x

. (B23)

A similar fourth-order differential equation holds for the current i. A simple form of solution for Eq. (B23) is

0exp(j ( / ))u u t x v , (B24)

By introducing Eq. (B24) into Eq. (B23) one will arrive at

93

2 22 2 2

g s 0lc lcv v

. (B25)

This shows the relation between the frequency ω and the velocity v of traveling waves. Eq. (B25) gives the velocity of propagation as

2 2 2 2 2s s0

g g g

1 c cv v

lc c c . (B26)

95

List of Symbols Symbol Quantity SI-Unit A Matrix used for formulating the state space equation of the

winding model

A Area of the complex-µ ellipse [m2]

1A Constant for determining the field Hz in laminate [A/m]

2A Constant for determining the field Hz in laminate [A/m]

azA Voltage amplitude for the Azimuthal resonances [V]

Ak Voltage amplitude for radial resonance k [V] a Gap width [m] a Gap width [m] a Gap width [m] ai Width of gap i [m] ai,max Maximum width for gap i [m] amax Maximum gap width [m] amin Minimum gap width [m] B Magnetic flux density vector [T] B Matrix used for formulating the state space equation of the

winding model

B Matrix used for formulating the state space equation of the winding model

B Magnetic flux density [T] Bk Voltage amplitude for radial resonance k [V]

B Complex magnetic flux density [T]

B Average magnetic flux density in laminate [T] Bmeas Approximaion for measured magnetic flux density [T] Bp Peak value of the magnetic flux density for the complex-µ

ellipse [T]

b Half of the laminate thickness [m] C Total capacitance matrix [F] C(i) “Specific” capacitance matrix for model i [F]

( )DDiC Capacitance matrix for model i, for coupling between two

neighboring discs [F]

Cdd Disc-to-disc capacitance [F] ( )dd

iC Disc-to-disc capacitance for model i [F]

Chf Winding model capacitance at high frequencies [F] Cig Disc-to-ground capacitance for inner ground [F]

96

( )ig

iC Capacitance to inner ground for model i [F]

Cog Disc-to-ground capacitance for outer ground [F] ( )og

iC Capacitance to outer ground for model i [F]

Ctt Turn-to-turn capacitance [F] ( )ttiC Turn-to-turn capacitance for model i [F]

C Constant coefficients for determining the field values in a conductor

[A/m]

kC Constant coefficients for determining the field values in conductor k

[A/m]

c Degree of reversibility [-] cg Ground capacitance per unit length [F/m] cs Series capacitance per unit length [F/m] D Electric displacement field vector [C/m2] d Conductor thickness [m] d Conductor thickness [m] d Conductor thickness [m] di Distance between inner ground and winding [m] dk,i Thickness of conductor k perpendicular to the adjacent channel

i [m]

dm Location of damage [m] do Distance between outer ground and winding [m] dwe Total length of winding [m] d,i Thickness of the conductor above/below gap i, in perpendicular

direction [m]

E Electric field vector [V/m] Ex x-component of electric field [V/m]

xE x-component of electric field at slab surfaces [V/m]

,x kE x-component of electric field at the surfaces of slab k [V/m]

F Force vector [N] f Frequency [Hz] faz Frequency for azimuthal resonances [Hz] fi Frequency for measurement i [Hz] fk Frequency for radial resonance k [Hz] G Magnetic domain structure parameter [-] H Magnetic field vector [A/m] H Magnetic field [A/m]

H Complex magnetic field [A/m] Hexcess Magnetic field due to excess effects [A/m] Hmeas Approximation for measured magnetic field [A/m] Hp Peak value of the magnetic field for the complex-µ ellipse [A/m] Hy y-component of magnetic field [A/m]

yH y-component of magnetic field at slab surfaces [A/m]

,y kH y-component of magnetic field at the surfaces of slab k [A/m]

Hy,0 y-component of magnetic field at the conductor surface [A/m] Hz z-component of magnetic field [A/m] H0 Magnetic field strength at laminate surfaces [A/m] h Conductor height [m]

97

htot Total height of the winding [m] I Current vector [A] II Identity matrix [-] I Current [A] Iafter Current in injection-point after damage [A] Ĩafter Fourier transform of Iafter [A] Ibefore Current in injection-point before damage [A] Ĩbefore Fourier transform of Ibefore [A] Ij Current in segment j [A] Ik Current in conductor k in a multi-conductor transmission line [A] i Index for pseudo particles and single sheet measurements

(Chapter 3), developed winding models (Chapter 4), gaps and reluctance branches in a multi-conductor transmission line (Chapter 8); Current (Chapter 9)

[-]; [A]

ig,n Current in the n:th turn flowing to ground [A] in Current in the n:th turn [A]

s,ni Sum of s,ni and s,ni [A]

s,ni Current flowing from the n:th turn to the next turn [A]

s,ni Current flowing from the n:th turn to the previous turn [A]

J Current density vector [A/m2] Jx x-component of current density [A/m2] j Imaginary unit [-] K Total number of discs in the winding [-] K1 Reluctance factor [H] K2 Reluctance factor [H] k Pinning strength (Section 3.3.1); index for an arbitrary node

(Section 4.5), the radial resonances (Section 7.1.1), the conductors in a multi-conductor transmission line (Chapter 8)

[Am2/s]; [-]

L Inductance matrix [H] Ldisc Inductance matrix for a single disc [H] Lkj Matrix for the mutual inductance between disc k and j [H] Lkk Self inductance of conductor k in multi-conductor transmission

line [H]

Lkk´ Mutual inductance between conductor k and k´ in multi-conductor transmission line

[H]

Llf Inductance of a disc for low frequencies [H] Lself Self inductance of a straight conductor segment [H] Lself,j Self inductance of segment j [H] l Inductance per unit length [H/m] Conductor length [m]

i Inner length of one side of the square disc [m]

o Outer length of one side of the square disc [m]

M Magnetization (Section 3.3.1 ); mutual inductance (Chapter 4) [A/m]; [H] Man An-hysteretic magnetization [A/m] Mkj Mutual inductance between segment k and j [H] Mp , Mq , Mm+p , Mm+q

Mutual inductance between two segments with the length p, q, m+p, m+q, respectively

[H]

Ms Saturation magnetization [A/m] m Magnetization of pseudo particle (Section 3.3.1); mutual [A/m]; [H/m]

98

inductance per unit length for adjacent turns (Appendix B) N Number of measurements performed with single sheet tester [-] n Number of turns in a disc (Chapters 4, 5, 7, 8), conductors in a

multi-conductor transmission line (Chapter 8) [-]

ni Number of branches in a disc for winding model i [-] np Number of pseudo particles [-] n0 Phenomenological parameter [-] O Matrix consisting of one column taken out from the C matrix

corresponding to index k [F]

P Matrix consisting of the k:th column taken out from ΓT [-] Pλk Play-operator with the pinning strength λk [-] p Length [m] Q Column vector used to prove the validity of RNM for a stack of

slabs [radAH/s]

Qk k:th element of Q [radAH/s] q Length [m] R Resistance matrix [] Rdisc Resistance matrix for one disc [] Rdc DC resistance [] Rkk Resistance of conductor k in multi-conductor transmission line [] Rkk´ “Mutual” resistance between conductor k and k´ in multi-

conductor transmission line []

Rseg Resistance of a straight conductor segment [] Rseg,j Resistance of segment j [] r Conductor radius [m] r Shield inner radius [m] r Radii of curvature [m] S Matrix consisting of 0,1 and −1 [-] s Number of gaps and branches in the reluctance network [-] TF Transfer function vector T Duration of one period of measurement [s] t Time [s] t1 Time parameter for Vapp(t) [s] tm Time delay of the reflection from the modified disc [s] twe Time delay of the reflection from the winding end [s] U Vector containing the voltage drops Uk [V] U Voltage [V] Uk Induced voltage drop between opposite ends of conductor k in a

multi-conductor transmission line [V]

u Voltage [V] un Voltage in the n:th turn [V]

mnu Induced voltage in the n:th turn due to m [V]

u0 Voltage amplitude [V]

nu Induced voltage in the n:th turn due to the current in itself [V]

V Vector containing the node voltages [V] Vapp(t) Applied voltage [V] Vapp(ω) Fourier transform of Vapp(t) [V]

azV Voltage distribution in a disc for azimuthal resonances [V]

Vj Voltage in node j [V]

99

Vk Voltage in node k [V]

rad,kV Voltage distribution in a disc for radial resonance k [V]

V0 Phenomenological parameter [A/m] v Wave velocity [m/s] v0 Maximum wave velocity in a winding [m/s] w Width of laminate (Section 3.3.2) and conductor (Chapter 4,

Section 8.4.2, and Appendix A ) [m]

w Gap length [m] w Gap length [m] wi Length of gap i [m] X State vector x Vector containing parameters of eff

Ỹafter In-admittance of winding after damage (measured in frequency domain)

[S]

Ỹbefore In-admittance of winding before damage (measured in frequency domain)

[S]

Z Impedance matrix for multi-conductor transmission line [] Zint True internal impedance matrix []

intZR Internal impedance matrix []

Z Impedance [] Zint True internal impedance [] Zint,k True internal impedance of conductor k []

intZR Internal impedance []

int ,kZR Internal impedance of conductor k []

α Numerical factor for adjusting Rseg [-] Complex wave number for magnetic penetration in laminate [m−1] Alternative way to write v when it becomes imaginary [m/s] Matrix connecting the currents and voltages [-] T Transpose of [-] Skin depth [m] δ,i Skin depth in the conductor above/below gap i [m] ε Permittivity [F/m] εair Relative permittivity of air [-] εi Relative permittivity of the conductor insulation [-] ε0 Permittivity of free space [F/m] Back field [A/m] θ Reluctance factor for skin layer [-] θh Hysteresis angle [rad] θk Reluctance factor for skin layer of conductor k [-]

,k i Alternative way to write θ,i [-]

,k i Factor similar to ,k i [-]

Reluctance factor for the skin layer above/below gap [-]

θ,i Reluctance factor for the skin layer above/below gap i [-] κ Propagation constant in a conductor [m−1] k Propagation constant in conductor k [m−1] Propagation constant in the conductor above/below gap [m−1]

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,i Propagation constant in the conductor above/below gap i [m−1]

λ Self inductance per unit length of one isolated turn [H/m] λi Number for pseudo particle i [-] µ Permeability [H/m]

eff Effective complex permeability of lamination [H/m]

eff Real part of eff [H/m]

eff Imaginary part of eff [H/m]

g Permeability of the medium in gap [H/m]

g,i Permeability of the medium in gap i [H/m]

h Low-frequency complex permeability including hysteresis [H/m]

k Permeability of conductor k [H/m]

meas Complex permeability for approximating the measured H-B curve with a complex-µ ellipse

[H/m]

meas Real part of meas [H/m]

meas Imaginary part of meas [H/m]

µr Low-frequency relative permeability excluding hysteresis [-] μshield Permeability of shield [H/m] µ0 Permeability of free space [H/m] Permeability of the conductor above/below gap [H/m] ,i Permeability of the conductor above/below gap i [H/m] ρ Radial polar coordinate (Chapter 7); volume charge density

(Section 8.3.2) [m], [C/m3]

ρ0 Inner “radius” of a disc [m] ρ1 Outer “radius” of a disc [m] Conductivity [S/m] dc DC Conductivity [S/m] g,i Conductivity of the medium in gap i [S/m] k Conductivity of conductor k [S/m] shield Conductivity of shield [S/m] ,i Conductivity of the conductor above/below gap i [S/m] ς Density function [-] τi Twice the insulation thickness [m] τks Distance between two discs [m] Vector containing the fluxes i [Wb] m Vector containing the mesh fluxes m,k [Wb] Φc Total magnetic flux inside conducting slab [Wb] Фc,k Total magnetic flux inside conducting slab k [Wb]

g,i Magnetic flux in gap i [Wb]

i Magnetic flux in branch i [Wb] m,k Magnetic flux in mesh k [Wb] ,i Magnetic flux inside the conductor above/below gap i [Wb] φ Azimuthal polar coordinate [rad] χ Susceptibility at H = 0 [-] Angular frequency [rad/s] ωcr Critical angular frequency [rad/s] ωi Angular frequency for measurement i [rad/s]

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ω1 Inverse of t1 [rad/s] Del operator [-] R reluctance matrix [A/Vs] R Reluctance [A/Vs] Reluctance between the turns for a single disc [A/Vs] Reluctance “outside” the turns for a single disc (placed

vertically) [A/Vs]

Reluctance “outside” the turns for a single disc (placed horizontally)

[A/Vs]

g,iR Reluctance of gap i [A/Vs]

iR Reluctance in gap or channel i [A/Vs]

,iR Skin layer reluctance in the conductor above/below gap i [A/Vs]

“connectivity matrix” for the reluctance network [-] T Transpose of [-] ∂ik Arbitrary element of [-] ,i Label for all quantities related to the conductor above (+) or

below (−) gap i [-]

0 Zero matrix [-] ∆I Difference between currents in injection-point [A] ∆Ĩ(ω) Fourier transform of ∆I [A] ∆x Differential length [m] ∆Ỹ(ω) Difference between the in-admittances of the winding [S]

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List of Acronyms AC Alternating current CB Conductor bending DC Direct current DTDR Differential time domain reflectometry FDM Finite difference method FEM Finite element method FFT Fast Fourier transform FRA Frequency response analysis HV High voltage LV Low voltage LWC Loose winding conductor MIMO Multi input multi output ode Ordinary differential equation RB Radial buckling RNM Reluctance network method TDR Time domain reflectometry

fra transformer model with application on time domain reflectometry

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