On simulation of surface discharges at variable …13103/FULLTEXT01.pdfOn simulation of surface discharges at variable voltage frequency Om simulering av yturladdningar vid variabel

On simulation of surface discharges at variable voltage frequency

Om simulering av yturladdningar vid variabel spänningsfrekvens

Nadja Jäverberg

Stockholm 2007 Thesis work

The Royal Institute of Technology School of Electrical Engineering

Division of Electromagnetic Engineering

Foreword & Acknowledgements

“God made solids, but surfaces were the work of the devil”.

Wolfgang Pauli This report accounts for the thesis work “On simulation of surface discharges at variable voltage frequency” performed between January and July 2007 under supervision of Hans Edin. It was performed during 20 weeks as a part of Master of Science education in Electrical Engineering, major in Electrotechnical Design at the division of Electromagnetic Engineering, The Royal Institute of Technology. I would like to express my gratitude to my supervisor, Hans Edin, for proposing the project, for guidance and motivation. I thank Cecilia Forssén and Nathaniel Taylor, Ph. D. students at the division of Electromagnetic Engineering, for valuable advices and support. And last but not the least I would like thank my husband Magnus, who without the first knowledge of what a capacitor is, managed to read through this rapport and come up with most valuable comments. Nadja Jäverberg Stockholm 2007

I

II

Abstract Insulation diagnostics is a very important tool in optimization of electric installations’ maintenance. One of the possible measures of insulation deterioration that can be used for diagnostic purposes are partial discharges. This thesis work describes an attempt to model a resistive-capacitive network for simulating partial surface discharges in Matlab. Unfortunately this attempt proved to be a failure due to an unexpectedly considerable dependency of high voltage capacitances on surface resistivity. Another attempt described here was performed in COMSOL Multiphysics, a finite-element based program for simulation of physical processes. The main drawback with COMSOL Multiphysics model is long simulation times. It proved to be possible to simulate discharges in COMSOL Multiphysics. Here surface resistance was modeled with the help of a resistive layer. Discharges were simulated by changing conductivity of the mentioned layer. Here another problem was discovered: very long simulation times when using non-linear, electric field dependent expressions for conductivity. All the simulations, both in Matlab and COMSOL Multiphysics, were performed on a computer with Intel dual-core processor: 2.13 GHz, 0.99 GB of RAM.

Sammanfattning Isolationsdiagnostik är ett redskap som är av stor betydelse för underhållsoptimering av elektriska anläggningar. Ett av de möjliga mått på isolationsförsämring som kan användas i diagnosticeringssyfte är partiella urladdningar. Det här examensarbetet beskriver ett modelleringsförsök av ett resistivt-kapacitivt nätverk för simulering av partiella yturladdningar i Matlab. Tyvärr blev försöket misslyckat på grund av ett oväntat stort beroende av högspänningskapacitanser på ytresistiviteten. Ytterligare ett försök genomfördes i COMSOL Multiphysics, ett program baserat på finita elementmetoden ämnat för simuleringar av fysikaliska processer. Den huvudsakliga nackdelen med COMSOL Multiphysics modellen är långa simuleringstider. Det visade sig vara möjligt att simulera urladdningar i COMSOL Multiphysics. Här modellerades ytresistansen med hjälp av ett resistivt skikt. Yturladdningar simulerades genom att ändra det resistiva skiktets konduktivitet. Här upptäcks ytterligare ett problem: mycket långa simuleringstider vid användandet av olinjära konduktivitetsuttryck som beror på det elektriska fältet. Alla simuleringar, både i Matlab och COMSOL Multiphysics, utfördes på en dator med Intel dual-core processor: 2.13 GHz, 0.99 GB of RAM.

III

List of Symbols U – potential with respect to the earthed electrode Ui – potential in the ith node with respect to the earthed electrode Ch – capacitance to the high voltage side Cp – disc capacitance Ck – gas-gas capacitance along the disc surface r – radius r’ - resistance rdisc – radius of the dielectric disc ddisc – thickness of the dielectric disc ltot – distance between the high voltage electrode and the outer disc edge ε – permittivity ε0 – permittivity of free space εr – relative permittivity or dielectric constant rel – electrode radius hel – electrode height rgnd – radius of the earthed electrode N – number of nodes lcyl – cylinder height lk – thickness of the resistive layer ρ – resistivity Rk – surface resistance RD – discharge resistance Z – impedance t – time T – period σ – conductivity ω – angular frequency Je – external current density Qj – current source V – voltage E – electric field strength D – electric displacement

IV

Contents Foreword & Acknowledgements ......................................................................................... I Abstract ............................................................................................................................. III Sammanfattning ................................................................................................................ III List of Symbols ................................................................................................................. IV 1. Introduction..................................................................................................................... 1

1.1 Background............................................................................................................... 1 1.2 Aim ........................................................................................................................... 1 1.3 Method ...................................................................................................................... 2

2 Theory .............................................................................................................................. 3 2.1 Gaseous ionization mechanisms ............................................................................... 3 2.2 Surface discharge modeling with equivalent circuits ............................................... 5

3 Experimental background data ........................................................................................ 8 4 Modeling of surface discharges ..................................................................................... 12

4.1 Experimental set-up used in this work.................................................................... 12 4.2 Attempt 1: Resistive-capacitive network ................................................................ 14

4.2.1 Dimensioning the capacitive part of the resistive-capacitive network ............ 14 4.2.2 Dimensioning the resistive part of the resistive – capacitive network............. 20 4.2.3 Testing the capacitive-resistive model in frequency domain........................... 21 4.2.4 Testing the capacitive-resistive model in time domain.................................... 23 4.2.5 Conclusions on Attempt 1 (calculations in frequency domain)....................... 27 4.2.6 Conclusions on Attempt 1 (calculations in time domain)................................ 27 4.2.7 Conclusions on Attempt 1................................................................................ 27

4.3 Attempt 2: COMSOL Multiphysics model............................................................. 28 4.3.1 General information on the model ................................................................... 28 4.3.2 Modeling a discharge in COMSOL Multiphysics model ................................ 30

5 Conclusions.................................................................................................................... 38 6 Recommendations for future work ................................................................................ 39 References......................................................................................................................... 40 Bibliography ..................................................................................................................... 40

APPENDIX................................................................................................................... 41 Appendix: Calculations in frequency domain for dimensioning of capacitances from high voltage side ........................................................................................................... 43 Appendix: Calculations in time domain ....................................................................... 44 Appendix: Matlab Algorithm........................................................................................ 46 Appendix: Matlab codes ............................................................................................... 47

1. Introduction

1.1 Background One of the most important parts of any electrical installation is electric insulating material. Its’ main function is to isolate two different electric potentials. A good electric insulation material has to be able to withstand

1. electrical stress (i.e. appropriate value of dielectric permittivity, low dielectric loss, high breakdown strength)

2. thermal stress (i.e. high fire point, low thermal expansion coefficient, etc.) 3. mechanical stress 4. ambient conditions (i.e. insulation material cannot be a health hazard, has to be

resistant to weather conditions, etc.) Normally insulation material is exposed to all the above mentioned stresses which in general depend on one another, e.g. both electrical and mechanical properties depend on temperature. This results in ageing which as a rule worsens insulation properties. As many electrical installations, such as generators, transformers, power cables, etc., are designed to last many years it is imperative to be able to predict lifetime for insulation materials. A lot of research has been performed on the insulation diagnostic methods as it is a valuable tool in optimization of electric installations’ maintenance. Partial discharges (PD) can be studied as a measure of insulation deterioration. By definition “partial discharge” is “a localized electrical discharge that only partially bridges the insulation between conductors and which may or may not occur adjacent to a conductor”.1 A special case of partial discharges is surface discharges that appear at the boundary of different insulation materials. Surface discharges can be explained with the help of Paschen’s law if the electric field can be assumed uniform. It is possible to enhance the probability of surface discharges to occur with the help of ultraviolet ray irradiation. Surface discharges depend on a variety of factors, such as temperature, humidity, setup geometry, materials used for insulation, etc. More on surface discharges can be found in the theoretical section.

1.2 Aim The aim with this thesis work is to simulate partial surface discharges at variable frequency of the applied voltage and make a comparison to experimental results.

1 Definition according to IEC Standard 60270 (3rd ed., 2000) Partial Discharge Measurements. International Electrotechnical Commision (IEC) , Geneva, Switzerland

1

1.3 Method This simulation is to be performed with the help of a resistive-capacitive network in Matlab. Using a capacitive-resistive lumped circuit model for describing surface discharges showed great promise in earlier research [1], [2]. This attempt along with a model using simple equivalent circuit [3] is described in short in the theoretical section. The ambition of this work is to perform simulations for variable frequency of the applied voltage numerically in Matlab adding high voltage side capacitances and finally compare Matlab solutions to experimental results.

2

2 Theory

2.1 Gaseous ionization mechanisms The aim with this section is to provide some information on discharges that take place in air. Air is a very good insulator at normal temperature and pressure with a conductivity of 10-16 – 10-17 A/cm2 at low fields. This level of conduction originates from radiation, both cosmic and radioactive. [4] A discharge implies electrical breakdown of air which can be described by a change of the gas from insulator to a conductor. For an electrical breakdown to take place there have to be electrons available. This is achieved through ionization processes. There is a number of ionization processes in gases, here follows a short recollection of them. Ionization processes: Ionization due to electron impact: Consider a free electron in a medium consisting of atoms. It will accelerate in the applied electric field until it collides with an atom in its path. If the electron energy is larger than the atom ionization energy the atom may become ionized. One of the parameters that can be used for quantitative description of this phenomenon is the Townsend primary ionization coefficient, α. It is defined as the number of ionization collisions made by an electron in moving a unit distance along the direction of the electric field. A stepwise ionization is also possible when the first collision only excites an atom and the next one ionizes it. Photoionization Ionization of an atom caused by photon absorption. Photon energy has to be larger or equal to the ionization energy of the atom. Thermal ionization Rising temperature results in the velocity increase of atoms in the gas, leading to higher kinetic energies. Eventually the temperature will become high enough to allow ionization through collisions. But this is not the only mechanism that takes place. Photoionization is also possible due to thermal emissions. Both processes, ionization through collisions and photoionization, produce electrons. This allows ionization by electron impact. Penning ionization Ionization caused by metastable excited atoms due to collision processes.

3

Ionization due to positive ions Positive ions can accelerate in the applied electric field causing ionization due to collision processes. Deionization processes: Processes opposing ionization are called deionization processes. They attempt to reduce the amount of electrons in the environment. Electron – ion recombination can happen in a number of ways that are classified in accordance to how the extra kinetic energy of the electron is removed, such as:

1) radiative recombination: the excess energy is released as a photon; 2) dissociative recombination (takes place in molecular gases): the excess energy is

transformed into vibration energy, following by dissociation of a molecule; 3) three-body recombination: the excess energy is transferred to a third body.

Electrical breakdown: Consider a simple case with uniform electric field: plane parallel gap. In the absence of electric field ionization and deionization processes are in equilibrium. This situation changes with increasing strength of the applied electric field. At a certain critical value of the electric field, known as breakdown electric field, part of electrons obtained through ionization will be accelerated so fast that recombination will not take place; thus creating free electrons needed for electrical breakdown. This critical value for air is approximately 3 kV/mm. [5] Experimental data shows that the breakdown electric field changes with gap distance, it reduces with increasing plate separation. This implies that the breakdown electric field should extend over a certain critical length; the shorter the length the higher value of the electric field is required. This allows formulating two necessary conditions for an electrical breakdown in air in a plane parallel gap:

1) the electric field should exceed the breakdown electric field, 2) the electric field should extend over a certain critical length, this length depends

on the magnitude of the electric field. Mechanism of electrical breakdown:

1) A neutral molecule of air is ionized creating an electron and a positive ion. The applied electric field accelerates them and hinders from recombination.

2) This single electron starts an avalanche through ionization due to electron impact. 3) Avalanche grows. Space charge in the avalanche head modifies the electric field

in its proximity. 4) Avalanche converts into a streamer discharge as soon as electric field from the

space charges reaches a critical value. This happens when the number of carriers reaches approximately 108. [6]

4

5) There are two possibilities here: a) the complete breakdown takes place (the streamer bridges the gap) b) or a number of streamers start propagating from the same place at the

electrode as the first one (streamer stem). Currents flowing in the streamers cause temperature to rise. This leads to thermal ionization when temperature reaches certain critical level, increasing stem conductivity and converting streamer discharge to a leader discharge. At this point of time conductivity is high enough to allow electrical potential to extend all the way to the head of the leader channel. This causes strong non-uniform electric field at the leader head and as a result streamers start appearing from the head of the leader channel. When the gap is bridged the current through it increases and the applied voltage collapses.

All of the above-mentioned mechanisms apply to a gaseous discharge. However one has to consider a number of other mechanisms when it comes to a surface discharge; such as surface emission, surface trapping, etc. Surface emission can be described as follows: electrical discharge propagating along a surface generates photons, which in their turn irradiate the surface and cause emission of electrons from the surface. Surface is also as a rule not perfect, in a sense that impurities and defects are most likely present. This results in surface states that act as traps for charged particles, i.e. surface trapping. Charge present on the surface also affects the surface discharge, more on this in section 4.1.

2.2 Surface discharge modeling with equivalent circuits Modeling surface discharges is a very complex problem due to electric field behavior both before and during a discharge. In order to simplify the description of discharges equivalent circuits were used. Here follows a short description of two equivalent circuits based on articles [1], [2] and [3]. Simple equivalent circuit The first simple equivalent circuit to be considered was obtained by Busch (1971) by simplifying a more complicated circuit proposed earlier by Skanavi (1958) [3]. In the article [3] it is used for describing surface discharges close to the inception voltage in the experimental set-up with cylindrical symmetry depicted in Figure 1a. The experimental set-up consists of two cylindrical electrodes of radius x0 and dielectric disc of thickness 2a with a radius larger than x0. The frequency of alternating voltage is chosen to be 50 Hz. For the schematic of the simple equivalent circuit please refer to Figure 1b. It is important to note here the assumption of all electrical conduction occurring in a gas layer of thickness w next to the dielectric disc.

5

CΔx CΔx CΔx

C1/Δx

R’Δx R’Δx R’Δx

AC

C1/Δx C1/Δx

Chamber

2x0

wx

Electrode

Electrode

R1

R1

AC

a) c)

CΔx CΔx CΔx

R’Δx R’Δx R’Δx

AC

b)

Dielectric disc 2a

θ

Figure 1 a) Experimental set-up for studying surface discharges; b) simple equivalent circuit; c) generalized simple equivalent circuit The circuit elements are computed as:

( )( )

0

0

'

disc

Rw x x

x xC

a

ρθ

ε θ

=⋅ ⋅ +

⋅ ⋅ +=

(1)

Here Δx is a length element along the radius of the dielectric disc, θ is an angle. R’ is surface resistivity, C is capacitance per unit length of the dielectric disc of half width ( ) and ρ is the specific resistivity of the gas in the test chamber. discd = a During the experiments the length of the first discharge in a cycle is measured. These measurements together with data from spectral analysis reported by Brzosko et al in 1979 [3] confirm the above mentioned assumption about electrical conduction. This assumption is valid both before and during the discharge. The resistivity of the gas layer next to the dielectric disc should of course change drastically depending whether a discharge occurs or not. The simple equivalent circuit is then generalized, as seen in Figure 1c, by adding a longitudinal capacitance C1. It is modeled as follows:

( )1 0gasC w xε θ= ⋅ ⋅ ⋅ + x

m

(2)

It is shown that this capacitance can be neglected for 0.02 w m

6

Authors of the article conclude that the simple equivalent circuit is useful for investigation of surface discharges at the inception voltage. It is important to note that calculations concerning the simple equivalent circuit were performed analytically. An attempt is also made for the problem to be solved numerically with the help of Fourier expansion. With conclusion that such analysis should be treated cautiously with regard to the resulting physical parameters. [3] A capacitive-resistive lumped circuit model for the surface discharge Another suggested circuit is described in articles [1] and [2]. Experimental set-up in this case looks as presented on Figure 2a; it consists of two asymmetrically placed parallel electrodes with a dielectric disc in between. The corresponding circuit used is depicted in Figure 2b. Here the authors split the circuit in two parts: one for discharging part and one for non-discharging part. The following assumptions are made:

1) dielectric plate is infinitely long 2) discharge advances only in x – direction 3) cross-sectional area of the streamer stem is constant and does not depend on time

All the calculations are performed analytically; conditions for discharge advancing, stopping and transition from polbüschel2 to gleitbüschel are formulated as follows:

1) the electric field at the streamer tip remains constant as long as the streamer is advancing (advancing condition)

2) the streamer stops advancing if the tip current is lower than a given critical value, this implies disappearance of the conductive stem. (stopping condition)

3) transition from polbüschel to gleitbüschel occurs when the total amount of energy dissipated in the whole stem reaches a critical value. (transition condition)

Figure 2 a) Common arrangement for studying surface discharges; b) capacitive-resistive lumped circuit model used for describing surface discharges

2 One sometimes talks about “polbüschel” and “gleitbüschel” instead of “streamer” and “leader” respectively when surface discharges are concerned. [1]

7

The following differential equations are solved:

11

1 1

22

2 2

'

1

s

V r ix for discharging parti VCx tV i dtx C for non discharging part

i VCx t

∂ ⎫− = ⋅ ⎪⎪∂⎬∂ ∂ ⎪− = ⋅⎪∂ ∂ ⎭

∂ ⎫− = ⋅ ⋅ ⎪∂ ⎪ −⎬∂ ∂ ⎪− = ⋅ ⎪∂ ∂ ⎭

∫ (3)

Here C is the capacitance per unit length between the streamer and electrode B; r’ is the resistance per unit length of the streamer stem and Cs is the series capacitance per unit length. It is also important to note that these parameters are constant and independent of both coordinate x and time t. Boundary and initial conditions are assumed as follows:

1) dielectric plate is initially uncharged. 2) a step voltage is applied at time t = 0. 3) potential equals to zero infinitely far away from the electrodes 4) potential is continuous at the tip of the streamer 5) electric field at the tip of the streamer in the non-discharging part is constant as

long as the streamer is advancing The authors conclude that this model explains the tendency of the experimental data qualitatively very well.

3 Experimental background data The experimental set-up to be used in this work is shown in Figure 3. The detailed description of it is provided in the next Section: Modeling of surface discharges. During the experiment partial discharges are first detected, namely their magnitude and the phase at which they occurred, then they are counted and sorted with respect to their magnitude and to the time at which they took place, finally they are plotted with all acquired data mapped over to one period. In other words a PD pattern represents in effect a summary over entire number of periods, for which an experiment is performed, mapped over to one period. It shows the magnitude of charge in pC in a partial discharge that occurred at a certain phase in degrees. More details on the measuring of PD patterns can be found in [7].

8

For this work a number of PD patterns were recorded for a certain range in frequencies, namely between 0.1 Hz and 100 Hz, over a number of periods. Figure 4 depicts a PD pattern taken at the frequency 10 Hz, the measurement was performed at 4 kV sinusoidal voltage over 500 periods.

Figure 3 Experimental set-up

Phase (deg)

Cha

rge

(pC

)

SDPC0104.DAT 10 Hz qmax

= 7111 pC 4 kV 500 periods

0 90 180 270 360

-6000

-4000

-2000

0

2000

4000

6000

0 0.61.21.72.32.93.44 4.55.15.76.26.87.47.98.59

Figure 4 A PD pattern taken for the frequency of 10 Hz

9

A lot of information can be extruded from a single PD pattern; here some examples will be mentioned:

1. Total charge per cycle with respect to the phase, Figure 5 2. Total number of partial discharges with respect to the phase, Figure 6 3. A distribution of the partial discharge amplitude (for positive and negative PDs),

Figure 7 It is also possible to get frequency dependence if one has a number of PD patterns, as for instance total number of PDs per cycle with respect to frequency, Figure 8.

0 90 180 270 360

-200

-150

-100

-50

0

50

100

150

200

Phase (deg)

Cha

rge

per c

ycle

(pC

)

Total charge (hqs) 10 Hz SDPC01

Figure 5 Total charge per cycle with respect to the phase

0 90 180 270 360

-0.1

-0.05

0

0.05

0.1

Phase (deg)

PD

s pe

r cyc

le

Total number of PDs (hn) 10 Hz SDPC01

Figure 6 Total number of partial discharges with respect to the phase

10

0 1000 2000 3000 4000 5000 6000 7000 80000

0.02

0.04

0.06

Charge (pC)

PD

s pe

r cyc

le

Positive PDs (Nqpos) 10 Hz SDPC01

0 1000 2000 3000 4000 5000 6000 7000 80000

0.1

0.2

0.3

0.4

Charge (pC)

PD

s pe

r cyc

le

Negative PDs (Nqneg) 10 Hz SDPC01

Figure 7 Distribution of the partial discharge amplitude

10-1

100

101

102

0

1

2

3

4

5

6

7

Frequency (Hz)

PD

s pe

r cyc

le

Total number of PDs SDPC01

Figure 8 Total number of PDs per cycle with respect to frequency

11

4 Modeling of surface discharges

4.1 Experimental set-up used in this work The experimental set-up analyzed in this work is shown in Figure 9a below. It consists of

• an electrode (radius , height ) 33 10elr m−= ⋅ 210 10elh m−= ⋅

• a polycarbonate disc (radius , thickness , 26 10discr −= ⋅ m −= ⋅ 3.3r31 10discd m = ) ε

• an earthed electrode (radius ) 25.5 10gndr m−= ⋅

lcyl

Cp Cp Cp

Ch Ch Ch

Ck Ck Ck Ck

1 2 N N+1

Electrode

Electrode

Dielectric disc

Earthed electrodeEarthed electrode

Dielectric disc

O

z

O

z

a b

r r

High potentialHigh potential

Small wedge

Figure 9 a) Experimental set-up (cylindrical symmetry), b) discretization model consisting of capacitances Applying alternating voltage to the high voltage electrode produces streamers on the surface of the dielectric disc. These streamers occur according to the same mechanism as described above in electrical breakdown in gases. The main difference is the criterion for the avalanche to streamer discharge transition. The critical value for breakdown electric field is much lower in this case as the electric field is non-uniform. The discharges are more likely to start in the small wedge between the high voltage electrode and the dielectric disc as the electric field is strongest there. A radial discharge pattern, symmetrical about the high voltage electrode, is expected due to rotational symmetry. There may be an exception though in case if a gleitbüschel or leader channel is formed. Applying voltage to the high voltage electrode will result in deposition of surface charges on the dielectric disc. This causes “memory” consequences that affect discharges coming after the first one, assuming that the dielectric disc is uncharged prior to the first discharge. Back discharges are a typical result of these “memory” effects. They occur due to the residual surface charge distribution on the dielectric disc when the applied alternating voltage is decreasing to zero after passing the maximum amplitude. Back discharges neutralize part of the surface charges deposited on the dielectric disc.

12

In this work the following assumptions are made:

1) before the first discharge takes place the dielectric disc is uncharged 2) all electrical conduction occurs in the gas layer of height lk next to the dielectric

disc 3) the disc is of finite size and the outermost edge of the disc is grounded

Calculations are to be performed numerically in Matlab using resistive-capacitive network. More on this in section 4.2. There are two main reasons why implementing the capacitive-resistive model in Matlab is of interest:

1) saving simulation time, i.e. it would be possible to run simulations faster in Matlab in comparison to, for instance, COMSOL Multiphysics.

2) advantage with such a model would be the easiness with which discharges can be simulated. The idea was to exchange surface resistances to smaller values depending on the strength of electric field and potential on the disc surface. This also means that back discharges can be simulated. For details please refer to Appendix: Matlab Algorithm.

In this report two attempts of modeling surface discharges are described:

• Matlab model, which unfortunately proved to be much harder to realize than first anticipated (Here modeling of the resistive-capacitive network prior to simulating surface discharges is described thoroughly)

• model implemented in COMSOL Multiphysics.

13

4.2 Attempt 1: Resistive-capacitive network

4.2.1 Dimensioning the capacitive part of the resistive-capacitive network The task is to obtain the potential distribution on the disc surface with respect to distance from the electrode and time and to model partial surface discharges. In this attempt the experimental set-up is modeled as a resistive-capacitive network in Matlab. The first step to this model is to assume the dielectric disc to be a perfect insulator. This assumption allows to disregard the disc resistance leaving just the capacitive part of the network, see Figure 9b and Figure 10. The model is obtained by discretizing the disc into N+1 segments of length dr. Here “Ch” are the capacitances to the high voltage side (electrode), “Cp” are the disc capacitances and “Ck” – gas-gas capacitances along the disc surface. Boundary conditions are set to:

• high voltage Uh in the node 0 next to the electrode • ground in the last (N+1) node.

The second boundary condition introduces a systematic error into the calculations as will be seen in Figure 13 and Figure 14.

Figure 10 Capacitive network Easiest to model are the disc capacitances, here a following distribution is assumed:

2

,2

1

gndp tot p

disc disc

tot

tot

r r drC Cd d

here l is the distance from the electrode to the outermost edge of the disc

ldrN

ε π ε π⋅ ⋅ ⋅ ⋅ ⋅ ⋅= ⇒ =

=+

(4)

14

The gas - gas capacitances along the disc surface are modeled as the capacitances between coaxial cylinders of equal height lcyl spaced at equal distances dr as shown in Figure 9b.

( )0

, 1

21

ln

0 :1:

cylk n

el

el

lC

r n drr n dr

here n N

π ε+

⋅ ⋅ ⋅=

+ + ⋅⎛⎜ + ⋅⎝ ⎠

=

⎞⎟ (5)

Thus there is only one degree of freedom left for determining the potential distribution on the disc surface, namely capacitances to the high voltage side, Ch. Keeping in mind that this model should produce the same result as any other model, the potential distribution obtained with the COMSOL Multiphysics model3 was used in order to calculate capacitances Ch. Some COMSOL Multiphysics results for U = 1 kV and f = 10 Hz are shown in Figure 11 and Figure 12. Results from Figure 12 are used for comparison. This is performed by using Kirchhoff’s laws which can be rewritten for calculating the capacitances in the following way (this can be achieved by rewriting equation system found in Appendix: Calculations in frequency domain for dimensioning of capacitances from high voltage side):

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

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

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

1 2

1

1

1

1 2 1 2 11

1 1

1

k k p k k hh

h

k k p i k i kh

h i

k k p N k Nh

h N

C C C U C U C UC

U U

C i C i C i U C i U C i UC i

U U

C N C N C N U C N UC N

U U

1i+ −

−

+ + ⋅ − ⋅ − ⋅=

−

+ + + ⋅ − + ⋅ − ⋅=

−

+ + + ⋅ − ⋅=

−

(6)

This gives the distribution of high voltage side capacitances over the distance between the electrode and the outer disc radius. It can be approximated with a rather satisfactory result (see Figure 13, Figure 14) with help of the following expression:

1917

, 5

140 10 64 102.5 10h apprC

r

−−

−

⋅= +

+ ⋅⋅

(7)

As seen from Figure 14 the approximation error does not go above 15% except for the outer most edge of the disc. This can be readily explained as a fault caused by approximation formula (7): Ch,appr never goes to zero at the outermost disc edge while the distribution of high voltage side capacitances obtained from COMSOL Multiphysics does so. Using formula (7) naturally leads to a systematic error, implying that all the results obtained by Matlab model at the outermost disc edge will not be trustable. It has however

3 This will be explained in detail in the next section.

15

no impact on usability of the resistive-capacitive network for simulating partial surface discharges since the latter will appear in the close proximity to the electrode.

Figure 11 Equipotential lines at t = 25 ms (contour plot)

Figure 12 Potential on the disc surface with respect to distance from the electrode over one period (T = 100 ms)

16

0 0.01 0.02 0.03 0.04 0.05 0.060

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

-13 Capacitances from high voltage side

Distance from the electrode [m]

Cap

acita

nce

dist

ribut

ion

[F/m

]

Approximated distributionCalculated distribution, C

Figure 13 Capacitance distribution, Ch, from high voltage electrode to disc surface

0 0.01 0.02 0.03 0.04 0.05 0.06-70

-60

-50

-40

-30

-20

-10

0

10

20Approximation error in %


(Ch,

calc

- C

appr

)/Cap

pr

Figure 14 Approximation error for capacitance distribution Next step is using all the above mentioned capacitances for calculating potential distribution on the disc surface. This is done by putting up the following equation system using Kirchhoff’s current law in the frequency domain (for details please refer to

17

Appendix: Calculations in frequency domain for dimensioning of capacitances from high voltage side):

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

1 2

1 1

1

1 2

1: 1 2 1 1 2 1 1

: 1 1

: 1

This can be rewritten as: , where , ,...,

k k h p k k h h

k i k k h p i k i h

k N k k h p N h h

Node C C C C U C U C C U

Node i C i U C i C i C i C i U C i U C i U

Node N C N U C N C N C N C N U C N U

A U b U U U

− +

−

− + + + ⋅ + ⋅ = − + ⋅

⋅ − + + + + ⋅ + + ⋅ = − ⋅

⋅ − + + + + ⋅ = − ⋅

⋅ = = [ ]TNU

h (8)

Here it can be noted that the cylinder height lcyl is chosen in such a way as to produce minimal condition number for matrix A of the equation system (8) in order to provide reasonable values for capacitances. Here it is chosen to be ( )71 10cyll m−= ⋅ giving condition number of approximately 13. Equation system (8) is solvable as it consists of N equations and N unknowns. Solving it gives very good correspondence between capacitive network and COMSOL Multiphysics models see Figure 15 and Figure 16.

0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

500

600

700

800

900

1000Potential distribution on the disc surface


Pot

entia

l dis

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tion

[V]

Approximated (capacitive network)Comsol

Figure 15 Potential distribution on the disc surface

18

0 0.01 0.02 0.03 0.04 0.05 0.06-20

-10

0

10

20

30

40

50

60

70Approximation error in %


(Uca

lc -

UCo

mso

l)/Uca

lc

Figure 16 Difference in % between the capacitive network and COMSOL Multiphysics models

19

4.2.2 Dimensioning the resistive part of the resistive – capacitive network Now that capacitive network is dimensioned it is important to remember that the dielectric disc does have a finite surface resistance, the bulk resistance is neglected. This fact has to be taken into account in the model. Here the surface resistance is of interest as surface discharges are to be simulated; it is modeled as if a concentric resistive disc with

1rε = of thickness lays on top of the polycarbonate disc that is a perfect insulator with kl3.3rε = of thickness . Resistance in accordance with a very well-known formula is

represented in the following way: discd

2kk

l dRA r

rl

ρ ρπ

⋅ ⋅= =

⋅ ⋅ ⋅ (9)

These resistances are connected in parallel with the “Ck” – gas-gas capacitances along the disc surface in the resistive-capacitive network, see Figure 17.

Figure 17 Resistive-capacitive network

In analogy with equation system (8) potential distribution on the disc surface in the frequency domain is calculated as follows:

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

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

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

1 2

1 1

1

1 1 1 1 1 1 1 1: 1 2 1 1 2 1 1

1 1 1 1 1 1 1 : 1 1 1

1 1 1 1 1 1 : 1

hk k h p k k h

i ik k k h p k h

N Nk k k h p h

Node U U UZ Z Z Z Z Z Z

Node i U U U UZ i Z i Z i Z Z i Z i Z i

Node N U U UZ N Z N Z N Z N Z N Z N

− +

−

⎛ ⎞ ⎛ ⎞− + + + ⋅ + ⋅ = − + ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

⎛ ⎞⋅ − + + + ⋅ + ⋅ = − ⋅⎜ ⎟⎜ ⎟+ +⎝ ⎠

⎛ ⎞⋅ − + + + ⋅ = − ⋅⎜ ⎟⎜ ⎟+⎝ ⎠

h

i h (10)

Here: 1

1 1

kk

k k

h ph p

RZj R C

Z Zj C j

ω

ω ω

=+ ⋅ ⋅ ⋅

= =C⋅ ⋅ ⋅ ⋅

(11)

20

4.2.3 Testing the capacitive-resistive model in frequency domain With all the elements of the capacitive-resistive model defined, the Matlab model is tested against the COMSOL Multiphysics model. Here an attempt is made to model both with and without high voltage side capacitances. For results please refer to Figure 18 through Figure 22. Here results from COMSOL Multiphysics were extracted, but only the data for each 20th point were used due to memory usage consideration.

0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

500

600

700

800

900



Pot

entia

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tion

[V]

Approximated (cap-res network with Ch)Approximated (cap-res network w/o Ch)Comsol

Figure 18 Comparison of the Matlab and COMSOL Multiphysics models for ρ = 1*108 Ω*m

0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

500

600

700

800

900



Pot

entia

l dis

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[V]


Figure 19 Comparison for ρ = 1*107 Ω*m

21

0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

500

600

700

800

900



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0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

300

400

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900



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[V]



22

0 0.01 0.02 0.03 0.04 0.05 0.060

100

200

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400

500

600

700

800

900



Pot

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[V]



4.2.4 Testing the capacitive-resistive model in time domain The capacitive-resistive model is tested now in time domain. The following equation system (12) is obtained for time domain calculations (for details please refer to the Appendix: Calculations in time domain). This equation system is based on the Kirchhoff’s laws and finite-difference scheme. This is solved in Matlab, unfortunately due to numerical complications a really low value for time step has to be used. The time step is directly proportional to the resistivity, the lower the resistivity – the lower the time step. This implies that simulations can only be performed for high resistivities ( ) within reasonable time limit. Results can be seen in Figure 23 through Figure 26.

71 10 mρ ≥ ⋅ Ω⋅

( ) ( ) ( )

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

( ) ( ) ( ) ( )

1 2 1 1 1 2 2

1 2 1 1 1 2 2 1 11 2 2 1

1 1 1 1

k k h p k

hk k h p k k h h h

k k k k

ki i ki ki hi pi i ki i

Node 1 :

C C C C U t t C U t t

U tt t tC C C C U t C U t t C C U t t U tR R R R

Node i :

C U t t C C C C U t t C U t t− + + +

− + + + ⋅ + Δ + ⋅ + Δ ≈

⎛ ⎞ ⎛ ⎞Δ Δ Δ≈ + − + + + ⋅ + − ⋅ − ⋅ Δ − + ⋅ + Δ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⋅ + Δ − + + + ⋅ + Δ + ⋅ + Δ ≈

( ) ( ) ( ) ( ) ( ) ( )( )1 1 1 11 1

(12)

ki i ki ki hi pi i ki i hi h hki ki ki ki

kN

t t t tC U t C C C C U t C U t C U t t U tR R R R

Node N :

C U

− + + ++ +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞Δ Δ Δ Δ≈ − ⋅ + + − + + + ⋅ + − ⋅ − ⋅ + Δ −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⋅ ( ) ( ) ( )

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

1 1

1 11

N kN kN hN pN N

kN N kN kN hN pN N hN h hkN kN kN

t t C C C C U t t

t t tC U t C C C C U t C U t t U tR R R

− +

− ++

+ Δ − + + + ⋅ + Δ ≈

⎛ ⎞ ⎛ ⎞Δ Δ Δ≈ − ⋅ + + − + + + ⋅ − ⋅ + Δ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

23

0 0.01 0.02 0.03 0.04 0.05 0.060

200

400

600

800

1000

1200

Distance [m]

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Potential distribution with respect to distance at t = 25 [ms], ρ = 1e+8 [Ohm*m]

TD modelComsol

0 0.01 0.02 0.03 0.04 0.05 0.06-40

-20

0

20

40

60

80

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ComsolTD model

0 0.01 0.02 0.03 0.04 0.05 0.06-1000

-900

-800

-700

-600

-500

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-100

0

Distance [m]

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ComsolTD model

0 0.01 0.02 0.03 0.04 0.05 0.06-50

-40

-30

-20

-10

0

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20

30

Distance [m]

Pot

entia

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[V]


ComsolTD model

Figure 23 Potential distribution at different times (T/4, T/2, 3T/4, T) for ρ = 1*108 Ω*m

0 0.01 0.02 0.03 0.04 0.05 0.060

200

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600

800

1000

1200

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ComsolTD model

0 0.01 0.02 0.03 0.04 0.05 0.06-50

0

50

100

150

200

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ComsolTD model

0 0.01 0.02 0.03 0.04 0.05 0.06-1200

-1000

-800

-600

-400

-200

0

Distance [m]

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ComsolTD model

0 0.01 0.02 0.03 0.04 0.05 0.06-160

-140

-120

-100

-80

-60

-40

-20

0

20

Distance [m]

Pot

entia

l dis

tribu

tion

[V]


ComsolTD model

Figure 24 Potential distribution at different times (T/4, T/2, 3T/4, T) for ρ = 1*107 Ω*m, T = 100 ms

24

0 0.02 0.04 0.06 0.08 0.1 0.12-150

-100

-50

0

50

100

150

Time [s]

Pot

entia

l dis

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[V]

Potential distribution with respect to time at r = 0.002 [m]

TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-50

-40

-30

-20

-10

0

10

20

30

40

50

Time [s]

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entia

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TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-25

-20

-15

-10

-5

0

5

10

15

20

25

Time [s]

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entia

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tribu

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[V]


TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-20

-15

-10

-5

0

5

10

15

20

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entia

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TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-15

-10

-5

0

5

10

15

Time [s]

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entia

l dis

tribu

tion

[V]


TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-5

-4

-3

-2

-1

0

1

2

3

4

5

Time [s]

Pot

entia

l dis

tribu

tion

[V]


TD modelComsol

Figure 25 Potential distribution as a function of time at a distance r = 2, 4, 6, 8, 10, 20 mm away from the eletrode for ρ = 1*108 Ω*m

25

0 0.02 0.04 0.06 0.08 0.1 0.12-100

-50

0

50

100

150

200

Time [s]

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[V]


TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-50

-40

-30

-20

-10

0

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20

30

40

50

Time [s]

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TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-30

-20

-10

0

10

20

30

Time [s]

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l dis

tribu

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[V]


TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-20

-15

-10

-5

0

5

10

15

20

Time [s]

Pot

entia

l dis

tribu

tion

[V]


TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-15

-10

-5

0

5

10

15

Time [s]

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entia

l dis

tribu

tion

[V]


TD modelComsol

0 0.02 0.04 0.06 0.08 0.1 0.12-5

-4

-3

-2

-1

0

1

2

3

4

5

Time [s]

Pot

entia

l dis

tribu

tion

[V]


TD modelComsol

Figure 26 Potential distribution as a function of time at a distance r = 2, 4, 6, 8, 10, 20 mm away from the eletrode for ρ = 1*107 Ω*m

26

4.2.5 Conclusions on Attempt 1 (calculations in frequency domain) The capacitive-resistive model fits rather well with COMSOL Multiphysics results for resistivities higher or equal to . For 81 10 mρ = ⋅ Ω ⋅ 71 10 mρ = ⋅ Ω⋅ the capacitive-resistive model fits satisfactory with COMSOL Multiphysics results. The resistivity here is so high that the system behaves as a pure capacitive model. It is important to notice here that one cannot disregard the high voltage side capacitances. For resistivities lower than results from the Matlab model diverge gfrom the COMSOL Multiphysics results. One can also notice that with diminishresistivity the influence of high voltage side capacitances reduces. It is possible to completely disregard them for 41 10 ρ

71 10 mρ = ⋅ Ω⋅ reatly ing

m≤ ⋅ Ω ase shown in Figure 22 corresponds to a purely resistive model.

⋅ . The last c

4.2.6 Conclusions on Attempt 1 (calculations in time domain) Comparing Matlab results to the results obtained from COMSOL Multiphysics it is easy to notice that Matlab model cannot describe behavior of the system at zero-crossings and in the proximity to the electrode. This is most unfortunate as the first discharges will most certainly occur close to the electrode.

4.2.7 Conclusions on Attempt 1 High voltage side capacitances Ch are required to be present in the model for describing system response in the pure capacitive case (high resistances or high frequencies). In pure resistive case (low resistances or low frequencies) it is possible to disregard Ch completely and only use disc capacitances, gas-gas capacitances along the disc surface and surface resistances. It is obvious that the above dimensioned capacitive-resistive model cannot be used for simulating partial surface discharges, as it does not describe the system correctly in the proximity to the electrode. The main reason why Matlab model gives such poor results is that high voltage side capacitances change whenever surface resistances change. It is theoretically possible to recalculate high voltage side capacitances for each surface resistance in question for stationary conditions. However when discharges start occurring different parts of the disc surface will have different resistivities (during a discharge) and recalculating the high voltage side capacitances will be practically impossible. This leads to the next attempt to simulate the discharges directly in COMSOL Multiphyscis. This is described in the next section Attempt 2.

27

4.3 Attempt 2: COMSOL Multiphysics model

4.3.1 General information on the model Draw mode: The experimental set-up (see Figure 9a) is drawn in COMSOL Multiphysics as shown in Figure 27. The resistive layer has 0.1 mm thickness and thus is hard to see, an area close to the electrode is zoomed in. Chosen module for analysis is 2D – axial symmetry, AC/DC – Meridional Electric Currents (emqvw). All distances are in meters.

Figure 27 COMSOL Multiphysics, draw mode Solver Parameters: For comparison with Matlab model solutions in both frequency and time domain were obtained. For solution in frequency domain Time-harmonic, Electric Currents/Stationary analysis was used. Here the governing equation solved in COMSOL Multiphysics is:

( )( )0e

r jj V Jσ ωε ε−∇⋅ + ∇ − = Q

28

For solution in time domain Transient Electric Currents/Time dependent analysis was performed. Here the governing equation solved in COMSOL Multiphysics is:

( ) ( )0 r ej

VV J Q

tε ε

σ∂ ∇

−∇⋅ −∇⋅ ∇ − =∂

Boundary settings are chosen in accordance with the necessary analysis while subdomain settings stay the same. Here it is important to note that all analysis was performed for isotropic materials implying the constitutive equation: 0 rD Eε ε= , there are no current sources ( ) and

external current density equals to zero (

0jQ =

0eJ = ). Subdomain Settings: The disc (subdomain 1) is modeled as a perfect insulator with zero conductivity

0σ = and relative permittivity equaling 3.3rε = (typical value for polycarbonate). Air (subdomain 2) has conductivity 0σ = and relative permittivity 1rε = . Finally, resistive film has variable conductivity σ depending on the case to be modeled and relative permittivity 1rε = . Boundary Settings: The symmetry line is set as axial symmetry 0r = . The electrode is set as electric potential either as ( )0 sinV V tω= ⋅ ⋅ for time dependent analysis or in phasor notations as ffrequency domain analysis. Surfaces on the interface between air/resistive film and resistive film/dielectric disc are set to continuity

0V V= or

( )1 2 0n J J⋅ − = . The rest of the boundaries are set to ground . 0V = Here all the simulations were performed for U = 5 kV and f = 10 Hz.

29

4.3.2 Modeling a discharge in COMSOL Multiphysics model One of the difficulties in simulations is how to present the resulting data for comparison with experiment. Experiment with the set-up shown in Figure 9a was performed for a certain range in frequencies, between 0.1 Hz and 100 Hz, over a number of periods and a standard PD-pattern was obtained. However, it is important to keep in mind that partial surface discharge is a stochastical process, i.e. it may or may not occur even if all the requirements for a discharge are fullfilled. The same stochastic behaviour has to be introduced into COMSOL Multiphysics model. Simulations have to be performed in this case over a number of periods (at least 10) in order to gather the results needed for statistics.

First step on the way to the above formulated goal is to simulate a single discharge. This is done by modifying conductivity of the resistive layer in the proximity to the high voltage electrode, as a wedge discharge is most likely to take place. The idea is to set high conductivity over a certain length of the layer, in this case 10-2 S/m, while maintaining low conductivity, here 10-9 S/m, over the remaining part of the resistive layer. This can be done with the help of the following logical expression:

( )9 210 10 crr lσ − −= + ⋅ < (13)

Here it is important to note that expression (13) does not depend on time. This is of course impossible situation in practice, but it is an important step in simulations. Setting conductivity to expression (13) and running simulations over a quarter of the period (25 ms) with lcr = 0.02 m and time step dt = 10-4 s, the obtained results are presented in Figure 28 through Figure 32. Simulation time: 11.172 s.

Figure 28 Electric potential at t = 25 ms (surface plot)

30

Figure 29 Conductivity with respect to distance from the electrode

Figure 30 Electric potential with respect to distance from the electrode for various times (from 0 to 25 ms)

31

Figure 31Electric field strength, r - component, with respect to distance from the electrode

Figure 32 Current, r - component, with respect to distance from the electrode

32

As stated earlier that is an impossible scenario. First discharge will occur only after some certain time t0 after voltage is applied, when the electric field increases above a critical value. Trying to add a time dependency to the conductivity leads to the following expression (14):

( ) ( ) ( )( ) ( ) (

9 2

2

10 ( 10 2 20 & 21.1

10 2.3 22 & 23 0.1 21 & 22 )

crr l t t ms t ms

t t ms t ms t ms t ms

σ −= + < ⋅ ⋅ − ⋅ > < +

+ − ⋅ + ⋅ > < + ⋅ > < ) (14)

Conductivity in this case is low, 10-9 S/m, in entire resistive layer until time t0 = 20 ms; it then goes drastically up to a high value of 10-2 S/m over a certain length lcr in the vicinity of high voltage electrode, remaining low over the rest of the layer. The results of this simulation can be seen in Figure 33 through Figure 38. One can see on the plots behaviour characteristical for a discharge. The electric potential start propagating to the head of the discharge as soon as conductivity goes up; there is no voltage drop over the part of the resistive layer with high conductivity, please refer to Figure 35 and Figure 36. The electric field collapses after reaching the critical value, Figure 37. The current should first increase and then, after the electric field collapses, decrease. Unfortunately due to a numerical problem the current has a peak.

Simulation time: 68.359 s

Figure 33 Electric potential (surface plot) at t = 25 ms

33

Figure 34 Conductivity with respect to time at r = 2 mm away from the electrode

Figure 35 Electric potential with respect to time at r = 2 mm away from the electrode

34

Figure 36 Electric potential with respect to distance from the electrode

Figure 37 Electric field, r - component, with respect to time at r = 2 mm away from the electrode

35

Figure 38 Current, r - component, with respect to time at r = 2 mm away from the electrode However there is a big minus with this approach, namely the point in time when conductivity changes has to be chosen beforehand. Instead it has to depend on the electric field. A naive approach to correct this oversight would be to try the following expression (15):

( ) ( )9 210 10r cr r crE E E Eσ − −= ⋅ < + ⋅ > (15) Expression (15) depends directly on the electric field; it sets conductivity to a low value, 10-9 S/m, when r – component of the electric field is below the critical value, and to a high value, 10-2 S/m, otherwise. Unfortunately simulation time increases immensely, it becomes far too long: 2 % of the simulation of 25 ms takes approximately 97000 s. This is far too long for a thesis work. Instead a more sophisticated expression for conductivity, expression (16), is tested:

910 exp r

cr

EE

σ − ⎧ ⎫= ⋅ ⎨ ⎬

⎩ ⎭ (16)

This simulation runs much faster, but it still takes over 10 hours. Trying out different values for Ecr takes far too long time as the time step has to be reduced drastically (10-7 s in this case). Modifying expression (16) provides expression (17) that looks promising. It does maintain conductivity high even when the electric field collapses for as long as the current stays high enough. Unfortunately implementing it in COMSOL Multiphysics proves to be tricky as expression (17) is recursive, i.e. conductivity depends on itself

36

(thanks to Ohm’s law: J Eσ= ⋅ ). Not to mention very long simulation times that will doubtless occur.

910 exp r r

cr cr

E JE J

σ − ⎧ ⎫= ⋅ +⎨ ⎬

⎩ ⎭ (17)

The main problem with COMSOL Multiphysics simulation is time consumption. In order to reduce simulation time two attempts were made in order to exchange the thin resistive film for a boundary described by 1) PDE and 2) weak form. They are not presented here as they did not give good correspondence with the original model described in this work. The most likely reason is the lack of connection to the environment. This is regrettably as far as I had time to investigate considering the time limit on the thesis work and the fact that the lion part of the time was spent on the Matlab model as well as trying to optimize the COMSOL Multiphysics model.

37

5 Conclusions

1. Regrettably resistive-capacitive network, suggested in this work, is not fit for simulation of discharges. The most likely reason for deviation between Matlab and COMSOL Multiphysics models is the high voltage side capacitances changing with respect to resistivity.

2. The gas - gas capacitances along the disc surface can in fact be neglected as claimed in [3]. All the simulation graphs in this work are obtained with Ck:s present however.

3. It is possible to simulate first discharge in COMSOL Multiphysics; unfortunately dependency on electric field is hard to implement. It does work to simulate by changing conductivity in time over a certain length of the resistive layer.

4. Two unsuccessful attempts on optimization of the COMSOL Multiphysics model were made in order to minimize simulation times: using PDE and weak form for describing the resistive layer.

5. Simulation in COMSOL Multiphysics did not provide with sufficient statistics for comparing to the experimental results as simulations could only be performed over a quarter of a period (25 ms) within manageable time limits.

38

6 Recommendations for future work Both models, Matlab model and COMSOL Multiphysics model, are still promising regardless of the failures in simulation discharges described in this work. It should be possible to find another way to simulate high voltage capacitances for the Matlab model. Regarding COMSOL Multiphysics model it should be possible to exchange the resistive layer for an appropriate PDE or a weak form specifying the connection to environment. This approach will decrease the number of mesh elements immensely at the same time decreasing the number of degrees of freedom.

39

References [1] A. Kanematsu, G. Sawa, M. Ieda (1983) An Analytical Approach to Empirical Law on the Transition Voltage from Polbüschel to Gleitbüschel using a Transmission Line Model. Japanese Journal of Applied Physics, 22(8), pp. 1271-1276 [2] A. Kanematsu, G. Sawa, M. Ieda (1983) An Analysis of Various Properties of Surface Discharge Using a Transmission-Line Model. Japanese Journal of Applied Physics, 22(12), pp. 1906-1909 [3] J. Ashkenazi et al. (1982) Potential – field solutions and dielectric surface discharges. J. Phys. D: Appl. Phys., 15(1982), pp. 1849 – 1871. [4] E. Kuffel, W.S. Zaengl, J. Kuffel (2000) High Voltage Engineering: Fundamentals. 2nd ed., Oxford: Butterworth – Heinemann, ISBN 0 7506 3634 3 [5] V. Cooray (ed.) (2003) The Lightning Flash. London: Institution of Electrical Engineers, IEE Power and Energy Series, ISBN 0 8529 6780 2. [6] J.M. Meek, J.D. Craggs (1978) Electrical Breakdown of Gases. Chichester: John Wiley & Sons, Ltd, ISBN 0 4719 9553 3. [7] H. Edin (2001) Partial Discharges Studied with Variable Frequency of the Applied Voltage. Ph. D. Thesis, Stockholm, Royal University of Technology, Department of Electrical Engineering. TRITA – EEK – 0102, ISSN 1100 – 1593.

Bibliography [1] G. Engdahl et al. (2005) Elektroteknisk Modellering. Stockholm: Kungliga Tekniska Högskolan, Elektrotekniska System, A – ETS/EEK – 0502. [2] G. Engdahl et al. (2005) Elektroteknisk Modellering. Stockholm: Kungliga Tekniska Högskolan, Electrical Engineering, A – ETS/EEK – 0507.

40

APPENDIX

41

42

Appendix: Calculations in frequency domain for dimensioning of capacitances from high voltage side Since applied alternating voltage is sinusoidal it is possible to use jω - method. Using the capacitive circuit shown in Figure 10 and applying Kirchhoff’s laws provides the following:

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

1 1 1 1

1 1

1

1: 1 1 1 2

: 1

: 1

k h h h p k

k i i h h i p i k i i

k N N h h N p N k

Node j C U U j C U U j C U j C U U

Node i j C i U U j C i U U j C i U j C i U U

Node N j C N U U j C N U U j C N U j C N U

ω ω ω ω

ω ω ω ω

ω ω ω ω− +

−

⋅ ⋅ − + ⋅ ⋅ − = ⋅ ⋅ + ⋅ ⋅ −

⋅ ⋅ − + ⋅ ⋅ − = ⋅ ⋅ + ⋅ + ⋅ −

⋅ ⋅ − + ⋅ ⋅ − = ⋅ ⋅ + ⋅ + ⋅

2

N

Reorganizing this gives equation system (8):

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

1 2

1 1

1

1: 1 2 1 1 2 1 1

: 1 1

: 1

k k h p k k h h

k i k k h p i k i h

k N k k h p N h h

Node C C C C U C U C C U

Node i C i U C i C i C i C i U C i U C i U

Node N C N U C N C N C N C N U C N U

− +

−

− + + + ⋅ + ⋅ = − + ⋅

⋅ − + + + + ⋅ + + ⋅ = − ⋅

⋅ − + + + + ⋅ = − ⋅

h

This can be rewritten as: [ ]1 2, where , ,..., T

NA U b U U U U⋅ = =

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

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

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

1 2 1 1 2 0 . . . . . 0

. . . . . . . . .

0 0 . 1 1 0 . 0

. . . . . . . . .

0 0 0 0 0 0 0 1

k k h p k

k k k h p k

k k k h p

C C C C C

A C i C i C i C i C i C i

C N C N C N C N C N

⎡ ⎤− + + +⎢ ⎥⎢ ⎥⎢ ⎥

= − + + + + +⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥− + + + +⎣ ⎦

( ) ( )( ) ( ) ( ) ( )1 1 2 . .T

k h h h hb C C C C i C N⎡ ⎤= − + − − −⎣ ⎦

43

Appendix: Calculations in time domain

Ck2

Cp1

Ch1

Ck1

CpN

ChN

AC

Ch2

Cp2

1 2 N0 Rk1 Rk2 N+1RkN+1

CkN+1

( ) ( )

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

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

( )

1 1 1 21 1 1 1 1 2 1

1 2

1 1 21 1 1 1 1 2 1 2

2 1

11 2 1 1

i ii

hk h h h p k

k k

hk h h h p k

k k

k k h p

U t t U tdUdt t

Node 1 :U t U t dU t U t U td d dC U t U t C U t U t C C U t U t

R dt dt dt R dtdU t U t U t U t U td d dC U t U t C U t U t C C U t U t

dt dt dt dt R R

dUC C C C

+ Δ −≈

Δ

− −+ ⋅ − + ⋅ − = ⋅ + + ⋅ −

− −⋅ − + ⋅ − − ⋅ − ⋅ − = −

− + + + ⋅( )

2

1

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

( ) ( ) ( ) ( ) ( )

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

( ) ( ) ( )

2 22 1 1 1

1 2 2 1

1 1 2 21 2 1 1 2

21 1 1

1 2 2 1

1 2 1 1 1 2 2

1 1

1 1

h hk k h

k k k k

k k h p k

h h hk h

k k k k

k k h p k

t dU t U t U t dU tC U t C C

dt dt R R R R dt

U t t U t U t t U tC C C C C

t tU t U t U t t U t

U t C CR R R R t

C C C C U t t C U t t

⎛ ⎞+ ⋅ = + ⋅ − − − + ⋅⎜ ⎟

⎝ ⎠+ Δ − + Δ −

− + + + ⋅ + ⋅ ≈Δ Δ

+ Δ −⎛ ⎞≈ + ⋅ − − − + ⋅⎜ ⎟ Δ⎝ ⎠

− + + + ⋅ + Δ + ⋅ + Δ ≈

( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 2 1 1 1 2 2 1 11 2 2 1

hk k h p k k h h h

k k k k

U tt t tC C C C U t C U t t C C U t t U tR R R R

⎛ ⎞ ⎛ ⎞Δ Δ Δ≈ + − + + + ⋅ + − ⋅ − ⋅Δ − + ⋅ + Δ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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

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

( ) ( ) ( ) ( )

( )

11

11 1

1

1 1 1 1

1

:

i iki i i hi h i

ki

i i ipi ki i i

ki

ki i ki ki hi pi i ki i

ki iki ki

Node iU t U t d dC U t U t C U t U t

R dt dtdU t U t U t dC C U t U t

dt R dtin analogy to Node 1 :

C U t t C C C C U t t C U t t

t tC U tR R

−−

++ +

+

− + + +

−

−+ ⋅ − + ⋅ − =

−= ⋅ + + ⋅ −

⋅ + Δ − + + + ⋅ + Δ + ⋅ + Δ ≈

⎛ ⎞Δ Δ≈ − ⋅ + +⎜ ⎟⎝ ⎠

( ) ( ) ( ) ( ) ( )( )1 1 11 1

ki ki hi pi i ki i hi h hki ki

t tC C C C U t C U t C U t t U tR R+ + +

+ +

⎛ ⎞ ⎛ ⎞Δ Δ− + + + ⋅ + − ⋅ − ⋅ + Δ −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

44

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

( ) ( ) ( )

( ) ( )

11 1

1

1 1

1 11

:

N N N N NkN N N hN h N pN kN

kN kN

kN N kN kN hN pN N

kN N kN kN hN pNkN kN kN

Node NU t U t dU t U t dU td dC U t U t C U t U t C C

R dt dt dt R dtin analogy to Node 1 :

C U t t C C C C U t t

t t tC U t C C C CR R R

−− +

+

− +

− ++

−+ ⋅ − + ⋅ − = ⋅ + + ⋅

⋅ + Δ − + + + ⋅ + Δ ≈

⎛ ⎞ ⎛Δ Δ Δ≈ − ⋅ + + − + + +⎜ ⎟ ⎜⎝ ⎠ ⎝

( ) ( ) ( )( )N hN h hU t C U t t U t⎞⋅ − ⋅ + Δ −⎟⎠

45

Appendix: Matlab Algorithm

46

Appendix: Matlab codes Program code 1 for calculating and approximating high voltage side capacitances: clear all; close all; load Ch.txt % Potential distribution on the disc surface for f = 10 Hz Uorig0 = Ch(2:end,2); Uh = Ch(1,2); Ul = Ch(end,2); N = round(length(Ch)/20) - 1; Eps0 = 8.85e-12; Epsr = 3.3; r_jord = 5.5e-2; r_sk = r_jord; r_el = 3e-3; d_sk = 1e-3; ltot = r_sk - r_el; dr = ltot/(N+1); r1 = dr:dr:ltot-dr; U = zeros(1, N + 1); for gk = 1:N U(gk) = Uorig0(20*gk); %Potentials in nodes 1 through N end U(N+1) = Uorig0(end); %Potential in the node N+1 (GND) figure %Defining disc capacitances: Cp = Eps0*Epsr*2*pi*(r_el + r1)*dr/d_sk; %Defining gas-gas capacitances along the disc surface: lcyl = 1e-7; Ck = zeros(1, N + 1); for k = 0:N Ck(k + 1) = 2*pi*Eps0*lcyl/log((r_el + (k+1)*dr)/(r_el + k*dr)); end %Calculating high voltage side capacitances with the help of potential %distribution obtained in COMSOL Multiphysics: Ch = zeros(1, N); for g = 2:N-1 Ch(g) = ((Cp(g) + Ck(g+1) + Ck(g))*U(g) - Ck(g+1)*U(g+1) - Ck(g)*U(g-1))/(Uh - U(g)); end Ch(1) = ((Cp(1) + Ck(2) + Ck(1))*U(1) - Ck(2)*U(2) - Ck(1)*Uh)/(Uh - U(1)); Ch(N) = ((Cp(N) + Ck(N+1) + Ck(N))*U(N) - Ck(N)*U(N-1))/(Uh - U(N)); hold on %Approximating the high voltage capacitances with the following function: % % a % Ch_appr = ---------- + b % (r - r0)^n % b = 64e-17; n = 1; r0 = -2.5e-5; a = 140e-19; ya = @(r) a./(r - r0).^n + b; yanp2 = ya(r1); %Calculating approximation error: Cherr = sqrt(sum((Ch - yanp2).^2))/length(yanp2) yres = Ch - yanp2; plot(r1, yanp2, r1, Ch, '--') grid on title('Capacitances from high voltage side') xlabel('Distance from the electrode [m]') ylabel('Capacitance distribution [F/m]') legend('Approximated distribution', 'Calculated distribution, C') figure plot(r1, yres)

47

title('Approximation error for Ch') xlabel('Distance from the electrode [m]') ylabel('C_h_,_c_a_l_c - C_a_p_p_r [F/m]') grid on figure yresproc = yres./yanp2*1e+2; plot(r1, yresproc) title('Approximation error in %') xlabel('Distance from the electrode [m]') ylabel('(C_h_,_c_a_l_c - C_a_p_p_r)/C_a_p_p_r') grid on %Calculating the potential distribution on the disc surface with the help of the above defined capacitances: clear Ch yres yresproc Ch = yanp2; clear yanp absV10_0 whos A = diag(-(Ch + Cp + Ck(1:N) + Ck(2:N+1)))+ diag(Ck(2:end-1),1)+ diag(Ck(2:end-1),-1); b = -[(Ck(1)+Ch(1))*Uh, Ch(2:N)*Uh]; Uanp = [Uh; A\b'; 0]; %KA = cond(A) %Calculating condition number for matrix A figure Uorig = [Uh; U']; plot([0 r1 ltot],Uanp, [0 r1 ltot], Uorig, '--') title('Potential distribution on the disc surface') xlabel('Distance from the electrode [m]') ylabel('Potential distribution [V]') legend('Approximated (capacitive network)', 'Comsol') grid on figure Ures = Uanp(1:end-1) - Uorig(1:end-1); plot([0, r1], Ures) title('Approximation error') xlabel('Distance from the electrode [m]') ylabel('U_c_a_l_c - U_C_o_m_s_o_l [V]') grid on figure Uresproc = Ures./Uanp(1:end-1)*1e+2; plot([0, r1], Uresproc) title('Approximation error in %') xlabel('Distance from the electrode [m]') ylabel('(U_c_a_l_c - U_C_o_m_s_o_l)/U_c_a_l_c') grid on

48

Program code 2 for comparison between COMSOL Multiphysics and Matlab models for various resistivity values (Frequency domain): clear all; close all; %Loading data from COMSOL Multiphysics for resistivity = 1e+8 Ohm*m load realVe8.txt % Potential distribution on the disc surface for f = 10 Hz load imagVe8.txt % Potential distribution on the disc surface for f = 10 Hz %Complex potential distribution U = realVe8(2:end,2) + j*imagVe8(2:end,2); Uh = abs(realVe8(1,2) + j*imagVe8(1,2)); Ul = abs(realVe8(end,2) + j*imagVe8(end,2)); ra = realVe8(:,1); figure Eps0 = 8.85e-12; Epsr = 3.3; r_jord = 5.5e-2; r_sk = r_jord; r_el = 3e-3; d_sk = 1e-3; ltot = r_sk - r_el; N = round(length(realVe8)/20) - 1; dr = ltot/(N+1); r1 = dr:dr:ltot-dr; % Taking each 20th data point: Ua = []; for g = 1:N Ua = [Ua U(20*g)]; %Potentials in nodes 1 through N end Ua = [Ua U(end)]; %Defining disc capacitances: Cp = Eps0*Epsr*2*pi*(r_el + r1)*dr/d_sk; %Defining gas-gas capacitances along the disc surface: lcyl = 1e-7; Ck = zeros(1, N + 1); for k = 0:N Ck(k + 1) = 2*pi*Eps0*lcyl/log((r_el + (k+1)*dr)/(r_el + k*dr)); end %Defining high voltage side capacitances with the help of approximating function obtained in the program code 1. b = 64e-17; n = 1; r0 = -2.5e-5; a = 140e-19; ya = @(r) a./(r - r0).^n + b; Ch = ya(r1); %Defining surface resistances: ro = 1e+8; lrk = 1e-5; f = 10; omega = 2*pi*f; Rk = ro*dr./(2*pi*[r1 ltot]'*lrk); %Calculating impedances: Zp = 1./(j*omega*Cp); Zk = (Rk./(1 + j*omega.*Rk.*Ck'))'; Zh = 1./(j*omega*Ch); %Calculating potential distribution with all capacitances and resistances present: A = diag(-(1./Zh + 1./Zp + 1./Zk(1:N) + 1./Zk(2:N+1)))+ diag(1./Zk(2:end-1),1)+ diag(1./Zk(2:end-1),-1); b = -[(1/Zk(1)+ 1/Zh(1))*Uh, 1./Zh(2:N)*Uh]; Uanp = [Uh; A\b'; 0]; %Calculating potential distribution, here high voltage side capacitances are neglected: A2 = diag(-(1./Zp + 1./Zk(1:N) + 1./Zk(2:N+1)))+ diag(1./Zk(2:end-1),1)+ diag(1./Zk(2:end-1),-1); b2 = -[Uh/Zk(1), zeros(1, N-1)]'; Uanp2 = [Uh; A2\b2; 0]; hold on Uorig = [Uh; Ua']; plot([0 r1 ltot],abs(Uanp),[0 r1 ltot],abs(Uanp2),'-.', [0 r1 ltot], abs(Uorig), '--') title('Potential distribution on the disc surface') xlabel('Distance from the electrode [m]') ylabel('Potential distribution [V]') legend('Approximated (cap-res network with Ch)','Approximated (cap-res network w/o Ch)', 'Comsol') grid on figure Ures = abs(Uanp(1:end-1)) - [Uh abs(Ua(1:end-1))]'; plot([0, r1], Ures)

49

title('Approximation error') xlabel('Distance from the electrode [m]') ylabel('U_c_a_l_c - U_C_o_m_s_o_l [V]') grid on figure Uresproc = Ures./[Uh abs(Ua(1:end-1))]'*1e+2; plot([0, r1], Uresproc) title('Approximation error in %') xlabel('Distance from the electrode [m]') ylabel('U_c_a_l_c - U_C_o_m_s_o_l') grid on

50

Program code 3 for comparison between COMSOL Multiphysics and Matlab models for various resistivity values (Time domain): clear all; close all; tic %starting the timer %Loading data from COMSOL Multiphysics: Uh = 1 kV, f = 10 Hz load Ve8r_t100.txt %Potential distribution with respect to distance from the electrode at time t = 25 ms load Ve8t_r2.txt %Potential distribution with respect to time at r = 2 mm away from the electrode load Ve8t_r4.txt %Potential distribution with respect to time at r = 4 mm away from the electrode load Ve8t_r6.txt %Potential distribution with respect to time at r = 6 mm away from the electrode load Ve8t_r8.txt %Potential distribution with respect to time at r = 8 mm away from the electrode load Ve8t_r10.txt %Potential distribution with respect to time at r = 10 mm away from the electrode load Ve8t_r20.txt %Potential distribution with respect to time at r = 20 mm away from the electrode Eps0 = 8.85e-12; Epsr = 3.3; Uh = 1e+3; r_jord = 5.5e-2; r_sk = r_jord; r_el = 3e-3; d_sk = 1e-3; ltot = r_sk - r_el; N = round(length(Ve8r_t100)/20) - 1; dr = ltot/(N+1); r1 = dr:dr:ltot-dr; %Defining disc capacitances: Cp = Eps0*Epsr*2*pi*(r_el + r1)*dr/d_sk; %Defining gas-gas capacitances along the disc surface: lcyl = 1e-7; Ck = zeros(1, N + 1); for k = 0:N Ck(k + 1) = 2*pi*Eps0*lcyl/log((r_el + (k+1)*dr)/(r_el + k*dr)); end %Defining high voltage side capacitances with the help of approximating function obtained in the program code 1. b = 64e-17; n = 1; r0 = -2.5e-5; a = 140e-19; ya = @(r) a./(r - r0).^n + b; Ch = ya(r1); %Defining surface resistances: ro = 1e+8; lrk = 1e-5; f = 10; omega = 2*pi*f; Rk = ro*dr./(2*pi*[r1 ltot]'*lrk); dt = 8e-4; %time step tstop = 0.1; %defining simulation time t = 0; M = floor(tstop/dt/10) + 1; Ut = zeros(N,1); Ares = zeros(N, M); %Calculating potential distribution in time domain: At = diag(-(Ck(1:N) + Ck(2:N+1) + Ch + Cp)) + diag(Ck(2:N), 1) + diag(Ck(2:N), -1); Adt = diag(-(Ck(1:N) + Ck(2:N+1) + Ch + Cp - dt./Rk(1:N)' - dt./Rk(2:N+1)')) + diag(Ck(2:N) - dt./Rk(2:N)', 1) + diag(Ck(2:N) - dt./Rk(2:N)', -1); bdt1 = [Ch(1) + Ck(1); Ch(2:N)']; bdt2 = [dt/Rk(1); zeros(N-1, 1)]; h = 1; for t = 0:dt:tstop Uhf = Uh*sin(omega*t); Uhfdt = Uh*sin(omega*(t + dt)); bt = Adt*Ut - bdt1*(Uhfdt - Uhf) - bdt2*Uhf; Ut = At\bt; %Ut at t = t + dt Ares(:,h) = Ut; %Storing solutions for later comparison h = h + 1; end plot(Ve8r_t100(:,1), Ve8r_t100(:,2), [0 r1 ltot], [Uh*sin(omega*tstop); Ut; 0], '--') legend('Comsol', 'TD model') title('Potential distribution with respect to distance at t = 25 [ms], \rho = 1e+8 [Ohm*m]') xlabel('Distance [m]') ylabel('Potential distribution [V]') grid on figure plot(0:dt:tstop+dt, [0 Ares(29,:)], Ve8t_r2(:,1), Ve8t_r2(:,2), '--') legend('TD model', 'Comsol')

51

title('Potential distribution with respect to time at r = 0.002 [m]') xlabel('Time [s]') ylabel('Potential distribution [V]') grid on figure plot(0:dt:tstop+dt, [0 Ares(57,:)], Ve8t_r4(:,1), Ve8t_r4(:,2), '--') legend('TD model', 'Comsol') title('Potential distribution with respect to time at r = 0.004 [m]') xlabel('Time [s]') ylabel('Potential distribution [V]') grid on figure plot(0:dt:tstop+dt, [0 Ares(86,:)], Ve8t_r6(:,1), Ve8t_r6(:,2), '--') legend('TD model', 'Comsol') title('Potential distribution with respect to time at r = 0.006 [m]') xlabel('Time [s]') ylabel('Potential distribution [V]') grid on figure plot(0:dt:tstop+dt, [0 Ares(115,:)], Ve8t_r8(:,1), Ve8t_r8(:,2), '--') legend('TD model', 'Comsol') title('Potential distribution with respect to time at r = 0.008 [m]') xlabel('Time [s]') ylabel('Potential distribution [V]') grid on figure plot(0:dt:tstop+dt, [0 Ares(143,:)], Ve8t_r10(:,1), Ve8t_r10(:,2), '--') legend('TD model', 'Comsol') title('Potential distribution with respect to time at r = 0.010 [m]') xlabel('Time [s]') ylabel('Potential distribution [V]') grid on figure plot(0:dt:tstop+dt, [0 Ares(286,:)], Ve8t_r20(:,1), Ve8t_r20(:,2), '--') legend('TD model', 'Comsol') title('Potential distribution with respect to time at r = 0.020 [m]') xlabel('Time [s]') ylabel('Potential distribution [V]') grid on toc %Measuring time needed to run the program

52

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