Analysis of Dielectric Response

Analysis of Dielectric ResponseMeasurement Methods and DielectricProperties of Resin-Rich Insulation

During Processingby

Anders Helgeson

Submitted in partial fulfilment of the requirements for the degree ofDoctor of Technology

TRITA EEA-0002ISSN 1100-1593

Kungl Tekniska HgskolanDepartment of Electric Power Engineering

Division Electrotechnical DesignStockholm, Sweden

Abstract

AbstractThe dielectric diagnostic methods of polarisation/depolarisation currents and recoveryvoltage, which are time domain methods, and capacitance and tan measurements at differentfrequencies, which is a frequency domain method, have been studied here. In the first part ofthe thesis, these measurement methods are analysed and evaluated. Three different types ofinsulation systems are included in the evaluation, oil/paper (high loss), resin-rich mica tape(medium loss) and cross-linked polyethylene, XLPE, (low loss). These three systemshave different dielectric response in both shape and magnitude. Conclusions are maderegarding choice of measurement method depending on the dielectric response of theinsulation material. Examples are also given of how to estimate conductivity and the dielectricresponse function in the time domain from measurements with a finite charging period.

Furthermore, relations between time domain and frequency domain and the possibilities ofFourier transforming data from one domain to the other are discussed. Fourier transforming isdone with a spline approximation technique, the Hamon approximation and by fitting basefunctions that have analytic Fourier transforms to measured data. These techniques arereviewed with special attention to the problem of estimating data outside the measurementwindow.

The second part of the thesis includes studies of the change of dielectric properties during themanufacturing stage of a composite insulation used in high voltage rotating machines. Theinsulation consists of a resin-rich mica tape with woven glass or polyester film (PET) ascarrier material. The aim was to improve the quality of the composite insulation by optimisingthe heat and pressure cycle used in the production.

A test cell has been designed and built to be able to process simple parallel plate samplesunder conditions similar to the factory process. With a maximum heating rate of 9C/min anda maximum cooling rate of 20C/min arbitrary temperature paths could be programmed. Allsamples were processed under static pressure in the MPa range.

The chemical reaction during curing of the resin-rich mica tape was studied with differentialscanning calorimetry (DSC). A simple reaction rate model was fit to the DSC measurementsmaking it possible to calculate the degree of curing during an arbitrary temperature path.

The change of dielectric response with time during curing under different temperature pathsand at a constant pressure was monitored. Both laboratory experiments and factorymeasurements have been made and based on these measurements a simple network model isproposed to explain the measured dielectric response in terms of material structure and degreeof curing.

Acknowledgements

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AcknowledgementsDuring the last five years, I have had the pleasure to work within the Electrical Insulationgroup at the Department of Power Engineering at Kungl. Tekniska Hgskolan (KTH) inStockholm, Sweden. This work resulted in my Ph.D. thesis, which is presented here. Theproject has been financially supported by ABB within the ELEKTRA program.I am very grateful to my supervisor Prof. Uno Gfvert for his guidance, never-endingenthusiasm and very inspiring discussions (no coffee break on Thursdays). This work wouldnot have been possible without him.

I would also like to thank the head of the department Prof. Roland Eriksson, who initiated thisproject and who has always been very positive and supportive throughout these years.Throughout this work, I have had the great opportunity to work together with Dr. RonBrammer, Mr. Dick Rudolfsson, Mr. Stefan Engelbertsson and Mr. Bo Ullberg at ABB-Alstom Power in Vsters, Sweden. I especially want to thank Dr. Ron Brammer and Mr.Dick Rudolfsson for their invaluable help during the preparation of this thesis.

All the DSC measurements were made at the department of Polymer Technology at KTH andI am thankful to Dr. Patrik Roseen, Mr. Henrik Hillborg and Mr. Kamyar Fateh-Alavi forassisting me.

The mechanical tests were made at ABB Corporate Research in Vsters, Sweden and Iwould like to thank Mr. Jarmo Khknen for helping me.

This work has been carried out within the Electrical Insulation group. I am very grateful topresent and former members of this group: Prof. Stanislaw Gubanski, Mr. Peter Werelius,Tech. Lic. Bjrn Holmgren, Tech. Lic. Peter Thrning, Tech. Lic. Roberts Neimanis, Mr.Hans Edin and Tech. Lic. Mats Kvarngren.

I would like to thank the rest of the staff at the department for these years, especially presentand former member of the lunch and coffee break team: Dr. Niklas Magnusson, Dr.Fredrik Stillesj, Tech. Lic. Pr Holmberg, Dr. Anders Lundgren, Dr. Anders Bergqvist,Tech. Lic. Niclas Schnborg and Ms. Anna Wolfbrandt. I would also like to thank Mrs.Marianne Ahlns, Mr. Gte Bergh and for technical assistance when building the test cell Mr.Yngve Eriksson and Mr. Olle Brnvall.

Many greetings to Roebels bar!

Finally, I want to express my deepest gratitude and love to my mother, father and girlfriend,this work would never have been possible without their support and patience. Things aredefinitely not practical but very interesting and giving. Yes I know, you have traffic lights!

Anders HelgesonStockholmApril 28, 2000

Contents

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Contents

1 Introduction_________________________________________ 11.1 Background ................................................................................................................... 11.2 Summary of the thesis ................................................................................................... 21.3 Publications ................................................................................................................... 4

2 Review of electrical insulation systems in high-voltagerotating machines ____________________________________ 5

2.1 Introduction ................................................................................................................... 52.2 Different groundwall insulation systems....................................................................... 52.3 Manufacturing process of stator coils ........................................................................... 62.4 Semi-conducting tapes, slot fillers and stress grading for stator coils .......................... 72.5 Degrading mechanisms for insulating materials ........................................................... 82.6 Today existing diagnostic methods for high-voltage rotating machines....................... 8

2.6.1 Off-line diagnostic methods .................................................................................... 82.6.2 On-line diagnostic methods ..................................................................................... 9

3 Electric properties of dielectric materials _______________ 113.1 Introduction ................................................................................................................. 113.2 Electrostatics of conductors......................................................................................... 113.3 Electrostatics of dielectrics.......................................................................................... 123.4 Electric polarisation mechanisms................................................................................ 143.5 Dielectric response in time domain ............................................................................. 173.6 Dielectric response in frequency domain .................................................................... 183.7 Time-dependence of dielectrics................................................................................... 203.8 Kramers-Kronig relations............................................................................................ 243.9 Heterogeneous dielectrics............................................................................................ 273.10 Temperature dependence of dielectrics ....................................................................... 29

4 Dielectric measurement techniques ____________________ 334.1 Introduction ................................................................................................................. 334.2 Polarisation and depolarisation currents...................................................................... 34

4.2.1 Estimation of the dielectric response function....................................................... 374.2.2 Estimation of the conductivity............................................................................... 39

4.3 Recovery voltage......................................................................................................... 404.3.1 The forward problem ............................................................................................. 41

4.3.1.1 Equations giving the analytic expression for the recovery voltage................ 424.3.1.2 Numerical calculation of the recovery voltage............................................... 434.3.1.3 Recovery voltage measurements .................................................................... 44

4.3.2 The inverse problem .............................................................................................. 474.3.2.1 Constrained minimisation of the current ........................................................ 47

4.3.3 Memory effects ...................................................................................................... 514.4 Loss factor and capacitance......................................................................................... 524.5 How to choose the best measurement method ............................................................ 554.6 Relations between time and frequency domain........................................................... 56

4.6.1 The Hamon approximation .................................................................................... 58

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4.6.2 Numerical Fourier transform ................................................................................. 604.6.2.1 Transformation from time to frequency domain ............................................ 624.6.2.2 Transformation from frequency to time domain ............................................ 64

4.6.3 Fourier transform using analytic base functions.................................................... 664.6.4 Numerical calculation of the Kramers-Kronig relations ....................................... 70

4.7 Summary ..................................................................................................................... 755 Test cell for processing resin-rich insulation _____________ 77

5.1 Introduction ................................................................................................................. 775.2 Designing the test cell ................................................................................................. 77

5.2.1 Dimensioning heating............................................................................................ 795.2.2 Dimensioning cooling............................................................................................ 81

5.3 Controlling the temperature......................................................................................... 835.4 Controlling the pressure .............................................................................................. 865.5 Measurement circuit .................................................................................................... 88

6 Measurements on resin-rich insulation _________________ 936.1 Introduction ................................................................................................................. 936.2 Preparing test samples for a dielectric response measurement ................................... 936.3 Preparing test samples for a differential scanning calorimetry (DSC) measurement . 956.4 Woven glass as carrier material................................................................................... 96

6.4.1 The uncured system ............................................................................................... 976.4.2 The fully cured system........................................................................................... 986.4.3 Different degrees of curing.................................................................................... 99

6.4.3.1 Degree of curing, 0.4 ............................................................................. 1006.4.3.2 Degree of curing, 0.6 ............................................................................. 1016.4.3.3 Degree of curing, 0.8 ............................................................................. 1026.4.3.4 Degree of curing, 0.9 ............................................................................. 1036.4.3.5 Loss, tan, plotted against degree of curing .............................................. 104

6.4.4 Dielectric response along the surface of the resin-rich mica tape ....................... 1056.4.4.1 The uncured system...................................................................................... 1066.4.4.2 The fully cured system ................................................................................. 107

6.4.5 Curing under different temperature paths............................................................ 1086.4.5.1 Curing under constant temperature .............................................................. 1096.4.5.2 Curing under a temperature ramp................................................................. 1116.4.5.3 Curing under Factory temperature path .................................................... 1136.4.5.4 Curing under Soft temperature path.......................................................... 1156.4.5.5 Curing under Hard temperature path ........................................................ 118

6.4.6 Mechanical measurements................................................................................... 1206.4.7 Factory measurement during pressing of stator bar............................................. 122

6.4.7.1 Temperature measurement ........................................................................... 1236.4.7.2 Capacitance and tan measurement.............................................................. 1256.4.7.3 Tip-up measurement after pressing the test stator bar.................................. 1276.4.7.4 Summary ...................................................................................................... 128

6.5 PET-film as carrier material ...................................................................................... 1286.5.1 The uncured system ............................................................................................. 1296.5.2 The fully cured system......................................................................................... 1306.5.3 Different degrees of curing.................................................................................. 131

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6.5.3.1 Degree of curing, 0.1 ............................................................................. 1326.5.3.2 Degree of curing, 0.3 ............................................................................. 1336.5.3.3 Degree of curing, 0.5 ............................................................................. 1346.5.3.4 Degree of curing, 0.8 ............................................................................. 134

6.6 Summary ................................................................................................................... 1357 Modelling a resin-rich insulation system _______________ 137

7.1 Introduction ............................................................................................................... 1377.2 Modelling the curing of a resin ................................................................................. 137

7.2.1 Resin-rich mica tape with woven-glass as carrier material ................................. 1397.2.1.1 Different degrees of curing........................................................................... 1417.2.1.2 Curing under constant temperature .............................................................. 1427.2.1.3 Curing under temperature ramp ................................................................... 1437.2.1.4 Curing under Factory, Soft and Hard temperature paths ................... 144

7.2.2 Resin-rich mica tape with polyester film as carrier material ............................... 1457.2.2.1 Different degrees of curing........................................................................... 146

7.3 Temperature dependence for the resin-rich mica tapes............................................. 1477.3.1 The resin rich-mica tape with woven glass as carrier material............................ 149

7.3.1.1 Degree of curing, 0 ................................................................................ 1497.3.1.2 Degree of curing, 0.4 ............................................................................. 1507.3.1.3 Degree of curing, 0.6 ............................................................................. 1517.3.1.4 Degree of curing, 0.8 ............................................................................. 1517.3.1.5 Degree of curing, 0.9 ............................................................................. 1527.3.1.6 Degree of curing, 1 ................................................................................ 1537.3.1.7 Comparison between different degrees of curing......................................... 1537.3.1.8 Comparison between fully cured samples cured under Factory, Soft and

Hard temperature paths ............................................................................. 1557.3.2 The resin rich-mica tape with polyester film (PET) as carrier material .............. 157

7.3.2.1 Degree of curing, 0 ................................................................................ 1577.3.2.2 Degree of curing, 0.1 ............................................................................. 1587.3.2.3 Degree of curing, 0.3 ............................................................................. 1597.3.2.4 Degree of curing, 0.5 ............................................................................. 1597.3.2.5 Degree of curing, 0.8 ............................................................................. 1607.3.2.6 Degree of curing, 1 ................................................................................ 1617.3.2.7 Comparison between different degrees of curing......................................... 161

7.4 Modelling of the dielectric response from resin-rich mica tapes .............................. 1637.4.1 Network model .................................................................................................... 1637.4.2 Dielectric response of the resin-rich mica tape with woven glass as carrier

material ................................................................................................................ 1657.4.2.1 Assumption epoxy-resin: dipole + conductivity .......................................... 1677.4.2.2 Assumption epoxy-resin: LFD + conductivity............................................. 169

7.4.3 Dielectric response of the resin-rich mica tape with PET-film as carrier material............................................................................................................................. 171

7.5 Summary ................................................................................................................... 1748 Conclusions _______________________________________ 177

8.1 Analysis of dielectric measurement methods............................................................ 1778.2 Study of resin-rich mica tape insulation.................................................................... 178

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9 Future work_______________________________________ 18110 Bibliography ______________________________________ 183Appendix A: Orientational polarisation ___________________ 191Appendix B: Analytic solution to the Kramers-Kronig relations

for the Curie-von Schweidler model___________ 193Appendix C: Numerical Fourier transform by approximating

data with cubic splines ______________________ 197Appendix D: The time domain solution to the network model_ 201Appendix E: List of symbols_____________________________ 205Appendix F: Pictures of the test cell ______________________ 209

Introduction Chapter 1

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1 Introduction

1.1 BackgroundThis Ph.D. thesis is the result of a project, which started in the beginning of 1995 at thedepartment of Power Engineering at Kungl. Tekniska Hgskolan in Stockholm, Sweden. Theidea was to study fundamental dielectric properties of insulation in high-voltage rotatingmachines with diagnostic methods developed at the department. The existing methods at thattime were in the frequency domain, a high voltage measurement system for capacitance andtan at variable frequency [98] and in the time domain, a high voltage polarisation anddepolarisation current and recovery voltage measurement system [45]. It was of interest tounderstand these measurement methods, their practical advantages and disadvantagesespecially regarding diagnostics of rotating machines. An aim was to link this understandingto existing diagnostic methods, which are in use today.

The idea from the beginning was to build up a new measurement system mainly toincorporate the time and frequency domain systems into one system but also to build upknowledge in sensitive low voltage and low current measurement techniques [37]. In parallelwith this, it was natural to study techniques to extract material characterising parameters fromthe measurement data and to transform data between domains. From the very beginning, theidea was to test the diagnostic methods by measuring insulation systems with differentdielectric response in both shape and magnitude.

The motivation for studying rotating machines in particular is that the electrical insulation is avery important part in the whole machine. The condition of the insulation will determine botheconomical and technical lifetime as well as quality of the power generated or distributed.Different types of pre-impregnated (resin-rich) mica tapes are commonly used as groundwallinsulation. Their final mechanical, thermal and electrical properties are closely related to themanufacturing stage and especially the curing process of the epoxy resin. For example, ifinternal stresses are built in during curing this can result in poor binding between layers or tothe insulated conductor. When such problems occur in a coil, it will most likely not passfactory tests or it will have a reduced service life.

The first step in studying the manufacturing step was to scale down the factory process to aneasy to handle test cell in the laboratory. A test cell was constructed and built in the workshopand could press parallel plate samples under constant pressure and a pre-programmedtemperature path. The first specifications for the test cell were to match heating and coolingrates as found in the factory process. Several rather complex steps involving cycling thepressure were skipped and after discussions it was decided that static pressure would beenough as long it was so high that the resin was evenly distributed between each insulationlayer and that possible excess resin between layers was pressed away.

Chapter 2 Review of electrical insulation systems in high-voltage rotating machines

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Two questions were raised from the beginning. What would the change in dielectric responselook like during curing and was it possible to read out a specific degree of curing from adielectric response measurement. A rather simple idea was stated of how the rate of cross-linking in the epoxy-resin would affect the quality of the insulation in terms of built in tensionand adhesion between layers and towards the conductor. A high rate of curing would quicklyfreeze the system leaving out possible cross-links whereas a slower rate would ensure thatall cross-links would find each other. However, one limiting economical factor in theproduction of stator coils turned out to be time and especially time in the press. So, was itpossible to find an optimal curing process from both an economical and insulation qualitypoint of view?

1.2 Summary of the thesisThis chapter contains a brief summary of the contents of this thesis. The summary is made inorder to focus on the contributions, which are made.

Chapter 1 gives the background to this project and a summary of what is found in this Ph.D.thesis, followed by a list of publications.

In chapter 2, a short review discusses common features concerning insulation systems anddiagnostic methods for rotating machines. Many factors interact and contribute to the ageingof the groundwall insulation material. Important decisions concerning the condition of thecomplete machine are often made based on the results from the diagnostic methods. It isimportant for a user of a generator or a motor to know and understand what can go wrong inthe insulation and how this can affect the properties of the generator or the motor. One of theultimate goals with diagnostics is to estimate the residual life of the insulation. Diagnostics ofgenerators or motors are especially interesting since a failure in either may cost a lot of moneyin loss of production.

In chapter 3, basic concepts of dielectric materials are discussed. A brief introduction is madeto electric polarisation in a static electric field and to different polarisation mechanisms.Models for heterogeneous materials are discussed to raise a few questions how to model thecomposite insulation in rotating machines. This chapter continues by showing that the electricbehaviour of a dielectric material can be described either in the time domain (, , f(t)) or inthe frequency domain (, , ()). If the material is linear, the information obtained in eitherdomain will be equivalent. The Kramers-Kronig relations are introduced and the mastercurve shifting technique is discussed for modelling the temperature dependence.

Fundamental relations between three different measurement methods are studied in chapter 4.The three methods are polarisation/depolarisation currents and recovery voltage in the timedomain and capacitance/tan in the frequency domain. In this chapter, three differentelectrical insulation systems will be studied with the three different methods mentioned aboveat room temperature. The insulation systems are oil/paper insulation, resin-rich mica tapeinsulation and cross-linked polyethylene (XLPE) insulation. These three insulation systemsare commonly used today in high voltage applications and they all have different dielectricresponse ranging from higher, oil/paper, towards lower loss, XLPE. It will be shown whattype of information is realistic to extract not only from each method but also from eachinsulation system. Conclusions are made about choice of method depending on insulationsystem response and on the type of information and physical mechanism wanting to be

Introduction Chapter 1

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diagnosed. Furthermore, relations between time domain and frequency domain and thepossibilities of Fourier transforming data from one domain to the other are discussed. Fouriertransforming is done with a spline approximation technique, the Hamon approximation and byfitting base functions that have analytic Fourier transforms to measured data. Thesetechniques are reviewed, especially the problem of estimating data outside the measurementwindow.

In chapter 5, all the design steps of the test cell are described. The basic idea behind buildinga test cell was to simulate the manufacturing process of the electrical insulation in rotatingmachines. The insulation system studied here is based on a pre-impregnated (resin-rich) micatape that has to be cured under a given temperature and pressure profile in order to reachmaximum electrical, thermal and mechanical performance. The test cell was built to matchthese demands.

In chapter 6, the dielectric response in the frequency domain has been measured on resin-richmica tape insulation used as electrical insulation in rotating machines. Two different types ofresin-rich mica tapes have been studied here. The first tape has a carrier material made ofwoven glass and the second one has a polyester film (PET). Low voltage measurements, 5 to200 V, in the frequency range 10 mHz to 1 MHz were made using the test cell described inchapter 5.

The main idea was to use non-destructive dielectric response measurements to study themanufacturing stage of resin-rich mica tape insulation. Simple parallel plate samples werepressed and cured under varying conditions in order to investigate the relation between themeasured dielectric response and the degree of curing. The dielectric response was measuredbefore, during, and after the processing cycle. Several different temperature paths were usedincluding processing under constant temperature. Experimental results were compared to afull-scale factory measurement.

In order to try verifying the assumption that a slow temperature path compared to a fastgave better mechanical properties a simple mechanical test of the adhesion between thecopper conductor and the insulation were performed.

The chemical reaction during curing of the two resin-rich mica tapes was also studied withdifferential scanning calorimetry (DSC). Making a DSC measurement and relating thismeasurement to the measurement of an uncured sample would determine the degree of curing.

In chapter 7, a simple Arrhenius activated reaction model was used to describe the progress ofcuring, change in , for the two resin-rich mica tapes studied. The parameters in the reactionmodel were estimated from differential scanning calorimetry (DSC) measurements. Whenthese parameters were known the degree of curing could be calculated for an arbitrarytemperature path which made it possible to relate degree of curing, , to the measureddielectric response, capacitance and tan.

The master curve approach is used here to study the temperature behaviour, thermalactivation, and how this is changing with degree of curing, . It is expected that the thermalactivation will be reduced with degree of curing, which relates both to the change in viscosityand the glass transition temperature during the curing process.

To explain the dielectric response from the resin-rich mica tapes a circuit network model isproposed consisting of the materials building up the tapes. Mica and the two tape carriermaterials woven glass and polyester (PET) are supposed to be unchanged during the curing


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process and will only have a temperature dependence. The dielectric response of the resin willchange with both degree of curing, , and temperature. Certain features like barrier effectscan be explained with the geometrical configuration.

In chapter 8, the conclusions from this Ph.D. thesis are made. Suggestions for further studiesand future work are mentioned in chapter 9.

1.3 PublicationsThroughout the work in this project the following publications have been written1. A. Helgeson: Calculation of the Dielectric Response Function from Recovery Voltage

Measurements, Master Thesis (in Swedish), A-EEA-9408, Dept. Electric PowerEngineering, Kungl. Tekniska Hgskolan, Stockholm, Sweden, 1994.

2. U. Gfvert, S. M. Gubanski, A. Helgeson: Calculations of Thermally StimulatedDepolarisation Current from Isothermal Dielectric Response Data, Proc. of the 1995International Symposium on Electrical Insulating Materials, Tokyo, Japan, 1995.

3. A. Helgeson, U. Gfvert: Calculation of the Dielectric Response Function from RecoveryVoltage Measurements, Proc. of the 1995 Conference on Electrical Insulation andDielectric Phenomena, Virginia Beach, Virginia, USA, 1995.

4. A. Helgeson: Dielectric Properties of Machine Insulation studied with DielectricResponse, Licentiate Thesis, Kungl. Tekniska Hgskolan, TRITA EEA-9704, ISSN1100-1593, Stockholm, Sweden, 1997.

5. A. Helgeson, U. Gfvert: Dielectric Response Measurements in Time and FrequencyDomain on High Voltage Insulation with Different Response, Proc. of the 1998International Symposium on Electrical Insulating Materials, Toyohashi, Japan, 1998.

6. A. Helgeson, U. Gfvert: Dielectric Response During Curing of a Resin-Rich InsulationSystem for Rotating Machines, Proc. of the 1999 Conference on Electrical Insulation andDielectric Phenomena, Austin, Texas, USA, 1999.

Review of electrical insulation systems in high-voltage rotating machines Chapter 2

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2 Review of electrical insulation systems inhigh-voltage rotating machines

2.1 IntroductionThis short review discusses common ideas concerning insulation systems and diagnosticmethods for rotating machines [24], [10]. The review will give a background to the diagnosticmethods discussed in chapter 4.

The most common purpose of insulation diagnostics is to detect ageing [63]. Many factorsinteract and contribute to the ageing of the groundwall insulation material. Importantdecisions concerning the condition of the complete machine are often made based on theresults from the diagnostic methods. It is important for a user of a generator or a motor toknow and understand what can go wrong in the insulation and how this can affect theproperties of the generator or the motor [27], [82]. One of the ultimate goals with diagnosticsis to estimate the residual life of the insulation. Diagnostics of generators or motors areespecially interesting since a failure in either may be expensive due to loss of production.

2.2 Different groundwall insulation systemsShellac micafolium: Here mica flakes are bound together with shellac to form sheets, whichare wrapped and hot pressed around the slot section of the winding. This system containsmany voids due to evaporation of volatiles in the shellac during the process stage. The manyvoids make this system suffer from heavy partial discharge erosion and poor heat transfer.This kind of insulation system was used from the beginning of 1900 to the sixties [76].Asphalt-bonded mica tape (1.8 kV/mm): The entire coil is insulated with asphalt-bondedmica tape. Asphalt-mica is a thermoplastic insulation system. The asphalt-bonded mica tapesystem can be vulnerable to tape separation due to thermal cycling (this is especially true forlarge hydropower generators which have problems with oil vapour). This insulating systemwas used from the forties to the seventies [76].Synthetic resin bonded mica tape (2.4 kV/mm): This system consists of small mica flakesthat are deposited on glass fibre backing tape see Figure 2.1. Once the tape has been woundon the conductors, the synthetic resin is cured at elevated temperature and pressure. Thistechnique offers the possibility of manufacturing nearly void-free insulation that canwithstand high dielectric stress. This insulation system has been in use since the sixties [7],[16], [76], [83].Silicone Rubber: Silicone rubber is used especially in applications that need to withstand hightemperatures. Even though there is no mica in this system, it can withstand partial discharges


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(PD) well. Silicone rubber is vulnerable to mechanical damages and this restricts the voltagelevel to 6 kV. Silicon rubber can be used together with mica to increase the voltage level.However, the costs for such an insulating system are higher than for an epoxy system [76].

Figure 2.1 Resin-rich mica tape with woven glass as carrier material used as groundwallinsulation in rotating machines. In the background, different types of strandand turn insulation are shown. (Picture from ABB-Alstom Power AB).

2.3 Manufacturing process of stator coilsManufacturing of stator coils for high-voltage rotating machines when using mica tapes andsynthetic resins can be divided in two different categories. The first alternative is to use aresin-rich mica tape with the epoxy-resin in a B-stage already in the tape and the second is touse a dry mica tape that is first vacuum pressure impregnated (VPI) and then cured. There areadvantages and disadvantages with both ways of producing coils.

To produce a coil the first step is to form geometrically the coil from the desired number ofconductors all individually insulated see Figure 2.2. It is crucial that the right dimensions areobtained and that the conductors bond together.

When using resin-rich mica tapes the coil is most often wrapped in half-lap, either by hand orwith a taping machine with a tension of about 40-60 N see Figure 2.2. It is important to applythe right number of turns to get the correct insulation thickness so that the coil will fit into thestator core. There is a compression ratio of about 25-30% after pressing. The coil is loadedinto a hot press, around 100C, with suitable release films. Two temperature levels are used,the first to soak the whole coil with the resin and the second to cure the insulation this isdone at, for example, 170C for 30-40 min. Pressure is applied to ensure that the resin iscorrectly distributed throughout the coil and that the right geometrical dimensions arereached. A good press procedure is important to avoid internal stresses being built in duringcuring, which can result in poor binding between layers or to the insulated conductor. It isalso important to avoid wrinkles in the tape especially at corners since this could otherwiselead to electric field enhancement. The coils are installed and connected in the stator coreafter pressing and a post-curing phase will take place, normally for around 12 hours at 140C.During the post-curing phase the coil ends are cured, usually under pressure from shrink tape.


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Figure 2.2 Cross section of a statorbar, showing the different layers for insulation, strand,turn and main (groundwall), slot corona protection and end corona protectionused as stress grading. (Picture from ABB-Alstom Power AB).

When using the VPI technique the coil is wrapped with a dry mica tape. The tension andamount of overlapping will affect the impregnation process [17], [102], [103]. The mica paperis mostly of uncalcined type, which has a high porosity, making it easier to impregnate.Depending on the size of the impregnation tank a whole stator or for large machines singlestator bars will be impregnated. In case of impregnation of a whole stator, the coils are putinto the stator core and the whole winding is completed. Then predrying takes place 1-12hours at 100C after which the stator is lifted into the VPI tank and vacuum (


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2.5 Degrading mechanisms for insulating materialsThermal: As organic materials form a major part of rotating machine insulation, thermalfactors influence the ageing. It is common to express the thermal lifetime with the Arrheniusrelationship, Life(L)=A*exp(B/T) where T is the temperature and A, B are material constants[76], [91].Electrical: Electrical degradation can take the form of partial discharge activity [11], [76]: Internal partial discharges in the bulk of the groundwall isolation can occur and can

become very serious if the strands of the conductor become loose.

High intensity of external partial discharge in the slot can by time erode the groundwallinsulation. However, before that the slot corona protection will be deteriorated andharmful ozone levels reached.

Partial discharges in endwing areas are often less life threatening, although they candamage surface coatings and produce ozone.

Mechanical: During normal operation the generator or motor stator coils are exposed toelectromechanical forces that can result in erosion of the ground insulation if the coil is notfirmly anchored in the slot. Under short circuit conditions, very strong electromechanicalforces can lead to cracking of the endwinding portion of the stator coils [63], [76].Ambient: Motors and generators are often exposed to severe environmental conditions such asmoisture, oil and dirt. Problems can occur when, for example, oil dissolves bonding materialsand leads to the development of cracks in the groundwall insulation. A common problem isthe combination of oil and dirt, which can block ventilation ducts and can cause overheating[76].Duty: The mode of operation has an important influence on the service life of the generator ormotor. Frequent load cycling can contribute to delaminating of the ground insulation resultingin increased internal partial discharge activity [76].

2.6 Today existing diagnostic methods for high-voltagerotating machines

Different diagnostic methods that are in use today will be reviewed here to get a widerperspective and to understand the importance and complexity of the electrical insulation in arotating machine. A few of these methods like return voltage, insulation resistance,polarisation index and dissipation factor measurements are closely related to methodsdiscussed later in this thesis. Having this in mind when reading those parts of the thesis mightintroduce some new ideas although this is not the main focus of the thesis.

2.6.1 Off-line diagnostic methodsVisual Inspection: Visual inspection of the end windings and the ends of the slots is importantbecause it reveals some types of deterioration. When the field poles or the entire rotor isremoved, it is common to check the tightness of the slot wedges [27], [58].


9

Diagnostics Based on Operation History: The operating hours of a stator winding have beenconsidered a standard index for insulation life. Start and stop of a generator or motor andfrequent load cycling accelerates the ageing of stator insulation [63], [58].Return Voltage: A DC-voltage is applied over the ground insulation of the generator or motorfollowed by a short-circuit and finally after opening the short-circuit the return voltage can bemeasured. This is a very simple method to investigate whether the insulation resistance andthe time constants of the insulation system change with ageing [3].Insulation Resistance: Insulation resistance is in North America defined as the resistancemeasured by applying a negative DC-voltage for one minute. It is useful to measure theresistance because it can give an indication if the insulation is wet and contaminated.However, partially wet insulation is not detected with this method. To detect such a problem itis necessary to make a dissipation factor measurement (tan). The measured insulationresistance value is strongly temperature dependent [27], [33], [58], [84], [85].Polarisation Index: The polarisation index is defined as the ratio of the insulation resistance atten minutes to that at one minute. The polarisation index indicates the same sort of defects aswere mentioned for the insulation resistance with the exception that the polarisation index isless temperature dependent [27], [58], [84], [85].Direct-Voltage Ramp Test: Here the resistance of the ground insulation is measured as afunction of a direct voltage ramp. This is a non-destructive method to detect cracks andfissures [27].Dissipation Factor Tip-Up: Here the loss factor tan as a function of voltage is measured. Thisshould result in a horizontal line since tan is independent of voltage, but when partialdischarges occur there is a tip-up in the curve. This gives an indication if there is a likelihoodof partial discharges under normal operation of the generator or motor [27], [33], [58], [84],[85].Electromagnetic Probe: Sometimes it can be difficult to perform a tip-up measurement for asingle coil. A way to overcome this problem is to use an electromagnetic probe and a radionoise meter to detect partial discharges associated with a coil in a specific stator slot. Aproblem here is to find the right frequency to tune the probe to [76].

2.6.2 On-line diagnostic methodsPartial Discharges: Partial discharge measurements can be divided into three parts [62] Partial discharge pulse analysis can be applied in generators to distinguish slot discharges

from external discharges on the end parts of the winding [11]. Slot discharges togetherwith coil vibrations are very energetic and will cause damage.

Partial discharges can be measured with a movable radio frequency antenna. With the helpof the antenna, it is possible to find the coils with the highest discharge activity.

Partial discharges can be measured by acoustic methods. The acoustic methods will revealfor example, vibrating coils. An advantage with acoustic methods is that their sensitivity isnot dependent on the capacitance of the test object.


10

On-line measurements can be made with a rotor mounted scanner. The rotor mounted scannerprovides continuous scanning of each slot or tooth in the stator for partial discharges,acoustic emission, vibrations, temperature and the dimension of the air gap between the rotorand the stator.

Measuring Vibrations: With a Vibro-meter the air gap between the stator and the rotor as wellas stator vibrations can be measured on-line [76].Temperature: Resistance temperature detectors (RTD) are normally used by embedding themwinding of a generator. They will give an indication if the temperature in the winding ishigher than normal, causing accelerated thermal ageing [76].Thermography: An infrared imaging technique to measure the temperature in a generator witha resolution of 1 K. This method is suitable for on-line measurements [82].There are several diagnostic methods to choose from and they all reveal different kinds ofinformation. It is difficult to perform AC-measurements on a large object due to the largedisplacement current. In today's insulating materials, mica is an important component becauseof its endurance to partial discharges. Ironically, this feature of mica insulation makes lifetimeprediction more difficult. There is also today a trend towards on-line monitoring fordiagnostic purposes either on a periodic or a continuous basis.

Electric properties of dielectric materials Chapter 3

11

3 Electric properties of dielectric materials

3.1 IntroductionMaterials can be divided, with regards to their electric properties, into three main classes,conductors, semiconductors and insulators. In this chapter, the focus will be on insulators(dielectric materials) but a few observations about conductors will be mentioned to make thisshort review complete. The information presented here should be seen as backgroundknowledge for understanding the dielectric properties observed in measurements and modelsfound in this thesis. This information can be found in a number of different textbooks but theintention was to gather this information and present it in a logical and easy accessible way[12], [18], [49], [64], [66], [74], [75], [88].

3.2 Electrostatics of conductorsIf a conductor is put in a static electric field the fundamental property of the conductor impliesthat the electric field inside the conductor must be zero. Hence, all the charges in theconductor must be located on its surface. The boundary condition at the surface of theconductor states that the electric field must be normal to the surface at every point.

It is well known that any electric field applied to a conductor will cause a free flow ofcharges, which is called current density. This current density, J, will according to Ohms lawbe proportional to the applied electric field, E.

( )2mA EJ = (3.1)where is the conductivity of the conductor. The electron was discovered in 1897 by J. J.Thompson and in 1900 P. Drude proposed a model for electric conductivity of metals basedon kinetic gas theory [9]. The basic idea in the Drude model is that there exist freely movingelectrons (no interaction) and positive immobile ions built up by the nucleus and its coreelectrons. The electrons collide with the ions and bounce off in random directions. However,the precise source of scattering does not matter for the qualitative and often quantitativeunderstanding of metallic conduction. If n is the number of free moving electrons per unitvolume and the average time between collisions the conductivity can be expressed as

( )m1 nem

ne2

drift == (3.2)

where e is the charge of one electron and m is the mass of one electron. Instead of talkingabout average time between collisions, it is common to introduce mobility for the electrons.This is just a different notation but does not change anything in the assumptions made byDrude.

Chapter 3 Electric properties of dielectric materials

12

This approach to model conductivity with a drift of charge carriers in an electric field does notonly work for metals but is also an important component when modelling semiconductors anddielectric liquids [51]. It could be stated that true conduction in both conductors and indielectrics arises from free movements of charged particles. Other interesting phenomenawhich contributes to the current density but are either conduction processes or polarisationprocesses are electrochemical reactions, diffusion and convection of charge densities. Theseprocesses can for example be found in dielectric liquids but also in resins during curing.

3.3 Electrostatics of dielectricsIn the case of a dielectric material, there exist no free charges since positive and negativecharges are all bound. The molecules in a dielectric material are usually distributed in such away that they make the dielectric material overall neutral. When a dielectric material is put ina static electric field, the electric field will penetrate into the material because bond chargescannot move freely. This should be compared to a conductor where an external field cannotpenetrate since free moving charges will rearrange themselves so that the internal field will bezero. The internal electric field in a dielectric material will change the material in such a waythat the electronic and ionic structure of the molecules on a microscopic scale is shifted. Thisresults in a change of molecular charge density. A multipole expansion of the potential fromeach molecule with this changed charge density can then be done [9],[49]. It turns out that thedominant molecular multipole with the applied field is the dipole, pn, since most of themolecules are neutral, see Figure 3.1.

-qpn = qdn

di10

+q

-q

di+1+q

Figure 3.1 Molecules are forming dipoles from an applied electric field shifting electronicand ionic charge densities on an atomic scale.

If every dipole location and corresponding charge magnitude were known, a microscopicmodel could be built. This microscopic model would have electric and magnetic fields, whichare very fast varying fields due to the rapid variation in charge distribution on an atomic scale.However, since this is not practical and often not interesting the behaviour of these dipoles isdescribed on a macroscopic scale averaging thousands of molecules and atoms [9],[49]. Thisis done by introducing the macroscopic electric polarisation vector P which is defined asdipole moment per unit volume.

( )2vN1i

i0Yi

0YmC

1lim

lim

=

=

=

ppP (3.3)

where pi is the dipole moment from the i:th dipole in the unit volume v and N the number ofdipoles per unit volume. If the molecules have a net charge of ei and if there is a macroscopicexcess of free charges the charge density on a macroscopic level can be written as


13

( )3excessvN1i

i0YmC e

1lim +

= =

(3.4)

As mentioned above most of the molecules are neutral which results in that the averagemolecular charge is zero.

If the dielectric material is now looked at on a macroscopic level, it is possible to build up theelectric field or the potential by linear superposition of the contribution from macroscopicallysmall volume elements V. If we then treat each V as infinitesimal, the potential at x can bewritten as [18]

( ) ( ) ( )[ ] ( )V Vd -

-

41

0

=xx

xPxx

V(3.5)

which is actually the potential from a charge distribution (P). This is an electrostaticcase which means that E = resulting in that the fundamental postulate for free spacemust be modified to take into account the electrical polarisation arising in the dielectricmaterials. The first Maxwell equation is therefore written as

[ ] ( )20

FmC 1 PE = (3.6)

From this the electric displacement is defined as

( )20 mC PED += (3.7)If the dielectric material is linear and isotropic the electric polarisation is directly proportionalto the electric field intensity and can be written as

( )20 mC EP e= (3.8)where e is the dimensionless electric susceptibility. In a linear dielectric material the electricsusceptibility is constant and independent of the magnitude of the electric field intensity andin an isotropic dielectric material the electric susceptibility is constant and independent of thedirection of the electric field intensity. The electric displacement can now be expressed as

[ ] ( )2r00 mC 1 EED =+= e (3.9)where r is the relative permittivity. If now the dielectric material is also homogenous whichmeans that the electric susceptibility is constant and independent of position in the dielectricmaterial the divergence equation can be written as

( )2r0

FmC = E (3.10)

The conclusion of this is that the electric field intensity in a dielectric material (linear,isotropic and homogenous) is reduced by a factor 1/r compared to that in free space [18]. Thepolarisation of the atoms and molecules in the dielectric material gives rise to electric fieldintensities inside the material, which partly cancels out the applied electric field intensity.This is used in for example capacitors where the capacitance is increased by r if a dielectricmaterial is inserted between the electrodes. The electrostatic energy in a material is expressedas


14

( )J dv E21dv

21W

V

20r

Ve == ED (3.11)

This shows that a dielectric material can store more energy than free space.

3.4 Electric polarisation mechanismsElectric polarisation in dielectric materials can be produced by many mechanisms in thematerial. But before looking into these mechanisms it is important to realise the differencebetween electric polarisation and electric conductivity. It can be stated that [51] "Polarisation arises from a finite displacement of charges in a steady electric field"

"Conduction arises from finite average velocity of motion of charges in a steady electricfield"

The electric polarisation will therefore never contribute to a continuous conduction currentunless a very high electric field is applied over the dielectric material. This is good to have inmind when reviewing the following simplified classification of electrical polarisationmechanisms [21], [25], [40], [51], [54], [88] Electronic (optical) polarisation: This occurs when an electric field, EL, in the optical

region is applied to an atom with fixed position. The positive nucleus and the negativeelectron cloud will be shifted relative to each other. The atom requires a dipole moment p,which is proportional to the electric field.

( )Cm LopticalEp = (3.12)where optical is the electronic polarisability of the atom. Because of the electron shellconfiguration, the polarisability will most likely be anisotropic but constant in thefrequency region studied in this thesis. The electronic polarisability alone is found in non-polar substances and ranges from around 10-41 Fm2 to 10-39 Fm2. The low values arefound for noble gases because they have a completely filled outer electronic shell, whichscreens the nuclei from any external field. The high values are found for alkali metals withonly one electron in the outermost shell. This electronic polarisation alone or interactingwith other mechanisms is present in most materials.

Molecular (optical and infra-red) polarisation: This occurs when an electric field, EL, isapplied to a molecule that has before the field was applied a total dipole moment that iszero. These types of molecules, polar substances, are built up by atoms that are interactingleading to the chemical bonds between atoms and due to symmetry also a distribution ofelectrons that gives a zero total dipole moment. In this case, the applied electric field willinduce a dipole moment due to elastic displacements of charges. Since there is such agreat difference between the mass of electrons and nuclei there is one group of normalmodes which is related to the displacements of electrons relative the nuclei, forfrequencies in the optical range, and one group of normal modes which is related to thedisplacement of nucleus, for frequencies in the infra-red region. Displacement of a thenucleus also includes displacement of electrons, which is inevitable because of theirinteraction.


15

The total dipole moment, p, for polar substances in the frequency range studied in thisthesis can be written as

( ) ( )Cm Lred-infraoptical Ep += (3.13)where optical is the electronic polarisability and infra-red is the atomic polarisability of theatom. Both these polarisabilities will most likely be anisotropic but also independent offrequency, electric field and temperature. Examples of polar substances are carbon dioxideCO2, benzene C6H6 and many others both in solid, liquid and gas phase. At sufficient lowtemperatures many dipolar materials, see below, also start to behave polar like since thethermal energy will be insufficient to turn the dipoles within a reasonable time. The mostrepresentative materials of the polar substances are the ionic crystals that may showconsiderable atomic polarisability. Most crystals of salts like for example sodium Na+Cl-are examples of ionic crystals. Compared to most other dielectric materials, most salts onmelting become ionic conductors.

Orientational (optical and infra-red and dipole) polarisation: This occurs when an electricfield is applied to a dipolar material. The molecules in the dipolar material, which have apermanent dipole moment, tend to align themselves with the applied field. It is importantto emphasise that the electric field has only a small effect on the tendency to alignment andthat the thermal effect causing the chaotic rotational motion is dominant. In zero electricfield, the molecules will be randomly oriented and the material has no net electricpolarisation. The effect of a static electric field on a group of weakly interacting moleculeswas first studied by Debye [21] see Appendix A. The static polarisability of a dipolarmolecule can be written as a sum of orientational, optical and infra-red polarisability

( )2red-infraopticalB

2dipole

tot Fm T3k++

=

p (3.14)

where pdipole is the permanent dipole moment of the molecule and T is temperature. Sincethe assumption made is that the groups of molecules studied are weakly interacting, thelocal electric field, EL, acting of one molecule will be the same as the applied externalelectric field, EA. The macroscopic polarisation can therefore be written as

( ) ( )2AtotAs0 mC VN1 EEP

== (3.15)

where N is the total number of molecules in the volume V. By changing the temperature,T, and measuring the static relative permittivity, s, it is possible to calculate the permanentdipole moment of the molecule, pdipole, and the sum of its electronic and atomicpolarisability, optical+infra-red. So far, interaction has been neglected between molecules,which implies that these models will be inadequate for solid dielectric materials atsufficiently low frequencies where inter-molecular (particle) distances can be very small. Adiscussion about effects of particle interactions is a topic outside the scope of this thesis.This is a difficult subject but just to mention a historical important step the Clausius-Mossotti relation will be mentioned. What they did was to assume that the N isotropicallyinteracting molecules (particles) were confined within a dielectric sphere of volume V.Since the polarisation of a sphere is uniform, it is expected that the mean local electricfield, EL, will be the same for all molecules. The local electric field will be the sum of theexternal electric field, EA, and the average electric field exerted on one molecule from its


16

(N-1) neighbours. Using the total induced dipole moment for a sphere in a uniform staticelectric field, the following Clausius-Mossotti relation can be derived

( )( ) tot0A

L

s

s

3VN

EE

21

=

+

(3.16)

The vector notation is dropped since EL and EA are parallel.

Other expressions relating the local electric field to the external electric field have beenderived by Lorentz-Lorenz, Debye, Onsager, Kirkwood and Frhlich and can be found inmany standard textbooks [21], [25], [51], [66], [88], [96].An example of a polar material is water where the atoms (H2O) are arranged in a triangleresulting in a net dipole moment of pdipole=1.84 D (units in Debye: 1 D=3.335610-30 Cm).However, not all molecules are necessarily dipolar, see above, this is very intimately linkedwith the nature of the chemical bonds between the atoms.

Hopping charge carrier polarisation: Another important form of electric polarisation, foundmostly in solids for both bulk and surface, is polarisation due to hopping charge carriers.This type of mechanism is something in-between on one hand induced and permanentdipoles and on the other hand free moving charges [51].

Tota

l ener

gy of p

artic

le

Distancex1 x2

pEAcosW

R12R21

Figure 3.2 A double potential well representing a model for hopping charge carriers. Anexternal electric field, EA, is applied which changes the probability oftransition, R21 > R12.

Hopping charge carriers (electrons or ions) spend most of their time in localised sites, x1 orx2, see Figure 3.2. They are strongly dependent on temperature (thermal vibrations) andoccasionally they make a jump over the potential barrier W. The probability of a hoppingtransition, R12 or R21, is determined by the distance between two possible locations and theheight of the potential barrier W (tunnelling may also occur). If an electric field is appliedover the solid dielectric material the probability of transition will change, R21 > R12. It isimportant to notice that it is not the electric field itself that will cause the charge carrier tojump; it is still the thermal vibrations. If a charge carrier can jump through the whole of thedielectric material then it will contribute to DC current conduction. However, the mobilityof the jumping charges will in that case be much lower than free carrier conduction.Hopping charge polarisation is found in electronic and ionic conductors and in and onhumid materials like for example glass, mica and granular materials such as sand and soil[51], [53], [54].


17

Not all dielectric materials involve all the polarisation mechanisms described here. Thedifferent mechanisms are also characterised by specific time constants, which may differ bymany orders of magnitude. They also depend differently on temperature.

There are also other types of dielectric materials for example electrets that have a nonzero netdipole moment and an electric polarisation vector P even when there is no external electricfield [12]. Another group of dielectric materials is the ferroelectrics, which also exhibits a netdipole moment without an external electric field. Ferroelectrics show hysteresis when theelectric polarisation, P, is plotted versus the electric field, E. Ferroelectricity usuallydisappears above a certain temperature called the transition temperature [12], [64].

3.5 Dielectric response in time domainAll classical macroscopic electromagnetic phenomena will follow the set of four equations forelectric and magnetic fields E and B set up by J. C. Maxwell in 1865 [74], [75].

( ) ( )( )law Ampres

t 0

law Faradays t

law Coulombs

+==

==

DJHB

BED(3.17)

These coupled equations can be seen as the equations giving the electric and magnetic fieldseverywhere in space provided all the sources, free charge and free current densities and Jare given. The constitutive relations, expressing material properties, connect E and B with Dand H as

MBHPED 0 =+=0

1

(3.18)

where P is the polarisation, bond charges, and M is the magnetisation, bond currents, ofthe material [49]. The dielectric materials studied here are assumed linear, homogenous,isotropic and they have zero magnetisation.

If a given electric field E is applied over a dielectric material both the free and bondcharges will give rise to sources inside the material in form of charge and current densitieswhich will give rise to a magnetic field B according to Maxwells equations. The currentdensity in the dielectric material will be given by Ampres law as

( )2current onPolarisaticurrentnt displaceme Vacuum

0current Induced

mA tt

+

+= PEEH

(3.19)

The first part, the induced current, is the contribution from the materials volume conductivity, the second part is the vacuum displacement current and the third part is the polarisationcurrent.

Assume now that the dimensions of the dielectric material are small compared to the changeof the electric field E in space. The room coordinates can then be dropped and all fieldquantities can be seen as only functions of time. The electrical polarisation P can then bedivided into two parts, one part representing rapid polarisation processes and one partrepresenting slow polarisation processes [45]. The rapid part follows the applied electric


18

field whereas the slow part is built up from a convolution integral between the appliedelectric field and a function called the dielectric response function, f(t). The dielectricresponse function represents the memory effects in a dielectric material and this functionwill be discussed further in chapter 3.7. The electric polarisation can now be written as

( ) ( ) ( ) ( ) ( ) ( ) ( )2onpolarisati Slow

t

0onpolarisati Rapid

ee mC dt ftttt

+=+= EEPEP 000 (3.20)The distinction between rapid and slow polarisation processes is not fundamental butdepends on the relevant time scale of our observation. Rapid polarisation processes is meantprocesses which are faster than the time scale of the applied electric field E(t).The total current density J(t) through a dielectric material in an electric field E(t) can then beexpressed as.

( ) ( ) ( ) ( ) ( ) ( )2current onpolarisati Total

t

0

e0current Induced

mA dtft1t

t

++

+=

=

EEEJ (3.21)

It is seen from the equation above that the conductivity , the high-frequency component ofthe relative permittivity and the dielectric response function f(t) will characterise thebehaviour of the dielectric material. This gives in the time domain the possibility to apply anelectric field, measure the current density and then try to estimate parameters that characterisethe material.

3.6 Dielectric response in frequency domainAssume again that the dielectric material is linear, homogenous, isotropic and non-magnetic.Now only time-harmonic electromagnetic fields, fields that have a time dependence that canbe described with one frequency, are considered. The electromagnetic fields can then bewritten in a complex form where the physical field is represented by the real part. This willresult in that time derivatives can be written as the function itself times i [94].The current density in the frequency domain will then be written according to Ampres lawas

( )2current onPolarisati

currentnt displaceme Vacuum

0current Induced

mA ii PEEH ++= (3.22)

If now the same separation of the electric polarisation P in rapid and slow processes isdone as in chapter 3.5 the electric polarisation P will be written in the frequency domain as

( ) ( ) ( ) ( ) ( ) ( ) ( )2ee mC f EEPEP 000 +=+= (3.23)The convolution integral describing slow polarisation processes in the time domain will be inthe frequency domain represented by a product. This important simplification will makecalculations easier and faster. Now the dimensionless frequency-dependent complex electricsusceptibility () is introduced and is defined as [51]


19

( ) ( ) ( ) ( ) dtf(t)efi0

ti === - (3.24)The real and the imaginary part of the complex electric susceptibility () can be seen as thecosine and the sine transform of the dielectric response function f(t).

( ) ( ) ( )

( ) ( ) ( )dt sin tf=

dt cos tf=

0

0

(3.25)

Carrying out the inverse cosine and sine transforms will return f(t). f(t) can be determinedfrom either the real or imaginary part of the electric susceptibility.

( ) ( ) ( )

( ) ( ) ( )( )s1

d sin 2=tf

d cos 2=tf

0

0

(3.26)

This is interesting because this shows that the Fourier transform is a link between thefrequency-dependent electric susceptibility () and the time-dependent dielectric responsefunction f(t). Given one of the functions then the other can be determined.The total current density J() in a dielectric material under a time-harmonic excitation E()can be expressed as

( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )

( )

( ) ( ) ( )2part Resistive

loss Dielectricloss Conduction

part Capacitive

e

mA i1i

iii

E

EEEEJ

00

000

+++=

=+++=

=

e

(3.27)

From this expression it is seen that there is one part of the current density J() which is inphase and one part which is 90 degrees before the driving time-harmonic electric field E().The part of the current that is in phase with the driving field is associated with the energylosses in the dielectric material. Two types of energy losses are seen in the material. The firsttype, which is due to the conduction (free charges) in the material, gives rise to ohmic losses.The second type, which is due to electric polarisation in the material, gives rise to what iscalled dielectric losses. Dielectric losses occur due to the inertia of the bound charges whenthey are accelerated in the driving field. The part of the current that is 90 degrees before thedriving field, displacement current, is associated with the capacitance of the material.

In many situations, it is more convenient to talk about the complex relative permittivity,which is defined as follows [51]


20

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )

1

ii

+=

++==

=

0

0 EJ

e

(3.28)

It is seen from the equation above that the conductivity , the high-frequency component ofthe relative permittivity and the electric susceptibility () characterise the behaviour ofthe dielectric material under time-harmonic excitation. This equation shows, as in the timedomain, that it is possible in the frequency domain to make measurements that characterisethe material. Under the assumptions that the dielectric material is linear, homogenous andisotropic, the measured information in either the time domain or frequency domain are equal.The information found in one domain can be transformed via the Fourier transform of f(t) or() to the other.

3.7 Time-dependence of dielectricsThe inevitable inertia of all physical processes in nature is one of the most obvious reasonsfor a time-dependent dielectric response. No dielectric material is able to directly followarbitrarily changing forces (driving electric field). This can be compared to the instantaneousresponse of free space.

To go a step further in the modelling of electric polarisation in time-dependent fields it isnecessary to better define the dielectric response function f(t) [51]. If an electric field with theamplitude is applied over a dielectric material for t seconds the time-dependent electricpolarisation can be expressed as

( )20 mC f(t)tE=P(t) (3.29)where f(t) is the dielectric response function. Causality demands that there should be noreaction before action therefore

( )s1 0


21

Pola

risat

ion

Time

E1t

E2

t E3t

tt1

t2

t3

Dielectric response function,f(t), of the material

Figure 3.3 An excitation of a dielectric material from three different delta functions ofheight E1,2,3 (V/m) and duration t. The total polarisation at time t is accordingto the principal of superposition the sum of polarisation from each deltafunction.

When the number of delta functions goes to infinity, the total electric polarisation can beexpressed as a convolution integral between the dielectric response function and the appliedelectric field.

( ) ( ) ( )20

0 mC dtEf=P(t) (3.33)The lower limit of the convolution integral is set to zero since f(t)=0 for t


22

only for a number of polar liquids but not for solid materials. Attempts to produce solidDebye-like material by diluting the dipolar part to remove interactions do not give resultsince the magnitude of loss most often decreases more rapidly than the strength of interaction[51].

log(D

iele

ctric

re

spon

se fu

nct

ion

)

log(Time)

Debye

Curie - von Schweidler

"General response"

t=

Slope -nSlope -m

Figure 3.4 Different types of dielectric response functions, f(t), in time domain. For polarliquids response functions like the Debye function are commonly found. Insolid dielectrics, the response functions are more of the fractional power lawtype as seen in the General response and Curie von Schweidler type offunctions.

The dielectric response function of the Curie-von Schweidler model is valid for manydielectric materials over a wide range of times [51].

( ) ( )s1 tAtf n= (3.36)It is important to notice that this model diverges at zero time if n>1. This means that themodel with n>1 is not applicable for arbitrarily short times. If n


23

long times for n m

Figure 3.5 (A) Frequency response of a Debye model typically found in gases and diluteliquids with non-interacting dipoles. (B) Wider dipolar peak found in solidmaterials like polymers.

The wider dipolar peak corresponds in time domain to the dielectric response functions foundin equations (3.37) and (3.38). The response function in (3.37) can not be Fourier transformedbut the response function in equation (3.38), which behaves asymptotically the same, can beFourier transformed to [90]

( ) ( )( )( )

( )( )

( ) 2n01m0

in1

i1n1

i1m1

An1

n

n1

n

m1

m


24

the time domain Curie-von Schweidler model found in equation (3.36). The Fourier transformof equation (3.36) is

( ) ( ) 1


25

( )

( )

=

a

022a

a

022a

dxx

(x) lim2=

dxx

(x)x lim2

(3.44)

This form of the Kramers-Kronig relations are the ones with the most physical significancesince it is only possible to measure data at zero and positive frequencies. The Kramers-Kronigrelations are a direct consequence of the assumption that the dielectric material studied herecan be described by a system that obeys causality. A general meaning of causality is that thereis no reaction before there is an action in the system, see equation (3.30).An other interesting relation which follows directly from the Kramers-Kronig relations is thatthe real part of the electric susceptibility for static conditions, =0, can be written as

( ) +

+

=

==

a

0a

a

a-a

d(lnx) (x)lim2dxx

(x)lim10 (3.45)

which relates the total area under the imaginary part of the complex electric susceptibilityplotted against natural logarithm of the frequency to the value of the real part of the complexelectric susceptibility at zero frequency. This shows that to every polarisation mechanismthere must exist a corresponding dielectric loss peak somewhere in the frequency spectrum.The real part of the complex electric susceptibility is almost independent of frequency whenthe dielectric loss is small, see Figure 3.7.

(), "()

log()

(0)

1"()

2"()

p1 p2

1()

2()

r 1

r 2

Figure 3.7 Two different polarisation mechanisms, p1 and p2, with corresponding losspeaks. The real part of the electric susceptibility at zero frequency representsin a log-log plot the total area under the imaginary part of the electricsusceptibility. This leads to the real part of the electric susceptibility beingalmost constant in-between two loss peaks.

The Kramers-Kronig relations are very useful for checking measurement data of the complexrelative permittivity in the frequency domain. Measurement data from any test object shouldobey the Kramers-Kronig relations otherwise, there is reason to suspect that something iswrong with the measurement or the measurement set-up [30], [51].


26

When making frequency domain measurements the complex relative permittivity is measured.Recalling equation (3.28), the complex relative permittivity can be expressed as

( ) ( ) ( )

++=

i0

(3.46)

where () is the complex electric susceptibility. The high-frequency component of therelative permittivity, , is according to Figure 3.7 chosen to r1 or r2 depending on whichpolarisation mechanism is measured. Applying the Kramers-Kronig relations to this measuredset of data it is seen that a constant value, like , and a 1/ behaviour, like /(0), will notcontribute to the result of the transform. This can be shown as follows [90]

( ) [ ] 0a

aln limxln limdxx

lim1a

a

a

a-a

=

+

==

=

+

+

a-a (3.47)for a constant and

( ) 0a

alnlimx

xln limdxx(xlim

1a

0-

a0

a

a-

0a

=

+

=

=

=

+

+

a

a

(3.48)

for a 1/ behaviour. These results have the consequence that in order to get the true Kramers-Kronig compatible pair from measured data the high-frequency component of the relativepermittivity, , must be subtracted from the real part of the complex relative permittivity andthe pure DC conductivity, /0, must be subtracted from the imaginary part of the complexrelative permittivity.

What the Kramers-Kronig relations are really saying is that there is a relationship between thereal and the imaginary part of the complex electric susceptibility. The complex relativepermittivity is the Fourier transform of the dielectric response function. For a few dielectricresponse functions, it is possible to express the Kramers-Kronig relations analytically. Oneexample is the Curie-von Schweidler model, see Appendix B, where the Kramers-Kronigrelations can be expressed as [106]

( ) ( )( )( ) 1


27

analytic expression for both the Fourier transform and the Kramers-Kronig relations. Thistype of curve fitting is an easy way of avoiding calculating time consuming transforms [30].

3.9 Heterogeneous dielectricsMost insulation systems found in practical applications are composites or mixtures of severaldifferent dielectric materials. The calculation of the dielectric properties of such a medium is aproblem of both theoretical and practical importance. The principal aim is to calculate therelative permittivity of the mixture in terms of the relative permittivities of the constituents,their relative amounts and their spatial distribution. It is important to realise that eachdielectric material has a dielectric response and when putting these materials together the totalresponse will not only reflect each material but also the way they are put together. Sometimesthe total dielectric response describes one physical process but the materials in the mixturehave dielectric responses revealing totally different physical processes. This can be the casewhen for example a high viscosity liquid with ionic conduction is mixed with a solid, whichhas no conduction [2].Two simple systems are found in Figure 3.8 representing a capacitor filled with two differentdielectrics in two different ways representing a parallel and series case [41], [88], [96].

21

1 2

Cp1 Cp2Cs1

Cs2

(A) (B)

h

S1 S2h1

h2

S

Figure 3.8 A capacitor which is filled with two dielectrics, 1 and 2, but distributed in twodifferent ways (A) in parallel and (B) in series.

In the first case, Figure 3.8 (A), there are two dielectrics where each form a cylinder with across section, of an arbitrary but uniform cross section, with its axis parallel to the appliedelectric field. This case of parallel dielectrics can be generalised and the effective relativepermittivity for N different dielectrics in parallel can be written as

( )=

=

N

1iiieff w (3.51)

where wi is the volume ratios of the ith dielectric material in the capacitor in (A). In the secondcase, Figure 3.8 (B), there are two dielectrics that instead are in parallel and in the samemanner as above can this case be generalised. For N different dielectric materials in series theeffective relative permittivity is expressed as

=

=

N

1i i

i

eff

w1 (3.52)

where also here wi is the volume ratios of the ith dielectric material in the capacitor in (B).


28

However, in many practical cases composite dielectrics are complex (statistical) mixtures ofseveral dielectrics, which are both in parallel and in series. In this case the equivalentdiagrams in Figure 3.8 are not sufficient but the true value of the effective relativepermittivity should lie somewhere between the values determined by equations (3.51) and(3.52). This is formulated in the Wiener inequality [88], [96]

( ) =

=

N

1iiieffN

1i i

i

ww

1(3.53)

There exist many semi-empirical formulas for the calculation of the effective relativepermittivity of statistic mixtures [81]. A few will be mentioned here just to illustrate theircomplexity. Lichtnecker-Rother have derived the formula for logarithmic mixing

( ) ( )( )=

=

N

1iiieff logwlog (3.54)

which works well for foams and porous material. Landau-Lifshitz derived for statisticmixtures the following formula [66]

( ) ( )( )=

=

N

1i

31ii

31eff w (3.55)

and Maxwell-Wagner derived the well known formula for a binary mixture consisting ofdielectric spheres, relative permittivity 1, distributed uniformly in a dielectric of relativepermittivity 2. The effective relative permittivity can then be written as

( )( )( )

( )

+

+

+

=

12

12

12

12

2eff

2w1

22w1

(3.56)

where w is the volume concentration of the dielectric spheres. From this expression, the well-known Maxwell-Wagner effect (dispersion) can be derived by making the spheres conductiveand the surrounding dielectric media insulating.

In Figure 3.9 the different ways of modelling the effective relative permittivity for twodielectric materials, as mentioned above, are plotted as a function of their volume ratios in themixture. This illustrates in a good way the Wiener inequality where the extreme cases aregiven by the series and the parallel models.

The models presented above are semi-empirical formulas that work well especially forstatistic mixtures. In this thesis two different resin-rich mica tapes have been studied whichare used as electrical insulation in rotating machines. These tapes consist of epoxy, mica and acarrier material which is polyester film (PET) or woven glass. Both tapes have a giveninternal periodic structure, which is of a dimension not vanishing small compared with thefinal insulation dimensions. Therefore it is possible to identify a small unit cell which can bedescribed by a network of series and parallel capacitors which contains the dielectric materialsthe tape is built up from [39].


29

Serie

Parallel

Lichtnecker-Rother

0% A100% B

100% A0% B

eff

B

A

Landau-Lifshitz

Maxwell-W agner

Figure 3.9 Effective relative permittivity for several different ways of mixing two dielectricmaterials A and B with relative permittivity A and B as a function of theirvolume content in the mixture.

From a more general point of view it turns out that a H-structure like the one found in Figure3.10 is interesting not only for resin-rich mica tapes but also for other insulation combinationsfound in high voltage apparatus. In chapter 7, there will be a more in-depth discussion and theeffective relative permittivity for the network shown in Figure 3.10 will be solved both intime and frequency domain.

h1

S1 S11 2

2 1h1hG

C1

C1

C2

C2

Cp2

1

2 3

4

Figure 3.10 Example of an H-network formation used to describe the dielectric behaviourof resin-rich mica tape insulation.

3.10 Temperature dependence of dielectricsIn general, the dielectric losses caused by dipole mechanisms reach a maximum at certaindefinite temperatures Tm [96]. The increase in temperature and resulting decrease in viscositywill affect the dielectric losses originating form rotational movements. On one hand thedegree of dipole orientation will increase with temperature, total polarisation will increase,which will increase the dielectric losses but on the other hand there is a reduction in energy toovercome the resistance of the viscous medium (internal friction) for the rotating dipolesdecreasing the dielectric losses with temperature.

For a quite large group of solid dielectric materials, it is found that the shape of the dielectricresponse (spectral shape) does not change very drastically with temperature. This is at least


30

true for temperature ranges over which the material does not change its internal

Analysis of Dielectric Response

Documents

time domain methods

change of dielectric

measured dielectric

frequency domain method

degree of curing

dielectricproperties

degreeof curing

composite insulation