Easy Evaluation of Streamer Discharge Criteria · PDF fileEasy Evaluation of Streamer...

Easy Evaluation of Streamer Discharge Criteria Göran1ABB AB, Corporate Research*Corresponding author: Abstract:devised, where analytical criteria for the occurrence of streamer discharges in strong electric fields are evaluated. This is highly useful when designing high voltage power transmission systems aprovided by a gas, e.g. air or SF Keywords: Streamer, Breakdown, Flashover 1. Introduction

transmission systems two important trends influencing the engineering efforts are: (i) The steadily increasing voltage levels introduced to reduce resistive losses and (ii) Compacting, i.e. tryingpossible. From an insulation point of view, however, these two requirements are conflicting in the sense that making clearance distances between conductors at different voltage levels smaller, while at the same time also these voltage differences, obviously makes the occurrence of electric discharges and flashoverssee Fig. 1,happen clearly requires a very thorough design work; critical areas suffering from excessive stress mustdesign modifications. Doing this by experiments is very timenumerical simulations of the electric field distribution provide a highly useful tool.Unfortunately, to translate the result from an into a statement whether a discharge or flashover will occur or not.

a discharge is the ionization of the insulating gas[1]ionization of free charges (electrons) in gas. It is equal to the ionization rate minus the recombination and attachment rates. strongly dependent on the electric field

depending on the gas.

Easy Evaluation of Streamer Discharge CriteriaGöranABB AB, Corporate Research

*Corresponding author:

Abstract:devised, where analytical criteria for the occurrence of streamer discharges in strong electric fields are evaluated. This is highly useful when designing high voltage power transmission systems aprovided by a gas, e.g. air or SF

Keywords: Streamer, Breakdown, Flashover

1. Introduction

In recent development of efficient power transmission systems two important trends influencing the engineering efforts are: (i) The steadily increasing voltage levels introduced to reduce resistive losses and (ii) Compacting, i.e. tryingpossible. From an insulation point of view, however, these two requirements are conflicting in the sense that making clearance distances between conductors at different voltage levels smaller, while at the same time also these voltage differences, obviously makes the occurrence of electric discharges and flashoverssee Fig. 1,happen clearly requires a very thorough design work; critical areas suffering from excessive stress mustdesign modifications. Doing this by experiments is very timenumerical simulations of the electric field distribution provide a highly useful tool.Unfortunately, to translate the result from an into a statement whether a discharge or flashover will occur or not.

The basic mechanism behind the creation of a discharge is the ionization of the insulating gas[1]. An important quantity ionization of free charges (electrons) in gas. It is equal to the ionization rate minus the recombination and attachment rates. strongly dependent on the electric field

eff (depending on the gas.

Easy Evaluation of Streamer Discharge CriteriaGöran ErikssonABB AB, Corporate Research

*Corresponding author:

Abstract:devised, where analytical criteria for the occurrence of streamer discharges in strong electric fields are evaluated. This is highly useful when designing high voltage power transmission systems aprovided by a gas, e.g. air or SF


1. Introduction

In recent development of efficient power transmission systems two important trends influencing the engineering efforts are: (i) The steadily increasing voltage levels introduced to reduce resistive losses and (ii) Compacting, i.e. trying to make the equipment as small as possible. From an insulation point of view, however, these two requirements are conflicting in the sense that making clearance distances between conductors at different voltage levels smaller, while at the same time also these voltage differences, obviously makes the occurrence of electric discharges and flashoverssee Fig. 1,happen clearly requires a very thorough design work; critical areas suffering from excessive stress mustdesign modifications. Doing this by experiments is very timenumerical simulations of the electric field distribution provide a highly useful tool.Unfortunately, to translate the result from an into a statement whether a discharge or flashover will occur or not.

The basic mechanism behind the creation of a discharge is the ionization of the insulating gas

. An important quantity ionization of free charges (electrons) in gas. It is equal to the ionization rate minus the recombination and attachment rates. strongly dependent on the electric field

(E) and the shape of this function varies depending on the gas.

Easy Evaluation of Streamer Discharge CriteriaEriksson

ABB AB, Corporate Research*Corresponding author:

Abstract: devised, where analytical criteria for the occurrence of streamer discharges in strong electric fields are evaluated. This is highly useful when designing high voltage power transmission systems and components where provided by a gas, e.g. air or SF


1. Introduction

In recent development of efficient power transmission systems two important trends influencing the engineering efforts are: (i) The steadily increasing voltage levels introduced to reduce resistive losses and (ii) Compacting, i.e.

to make the equipment as small as possible. From an insulation point of view, however, these two requirements are conflicting in the sense that making clearance distances between conductors at different voltage levels smaller, while at the same time also these voltage differences, obviously makes the occurrence of electric discharges and flashoverssee Fig. 1,happen clearly requires a very thorough design work; critical areas suffering from excessive stress must design modifications. Doing this by experiments is very timenumerical simulations of the electric field distribution provide a highly useful tool.Unfortunately, to translate the result from an into a statement whether a discharge or flashover will occur or not.


. An important quantity ionization of free charges (electrons) in gas. It is equal to the ionization rate minus the recombination and attachment rates. strongly dependent on the electric field

) and the shape of this function varies depending on the gas.

Easy Evaluation of Streamer Discharge CriteriaEriksson


An easily implemented method is devised, where analytical criteria for the occurrence of streamer discharges in strong electric fields are evaluated. This is highly useful when designing high voltage power transmission

nd components where provided by a gas, e.g. air or SF

Keywords: High voltageStreamer, Breakdown, Flashover

1. Introduction


to make the equipment as small as possible. From an insulation point of view, however, these two requirements are conflicting in the sense that making clearance distances between conductors at different voltage levels smaller, while at the same time also these voltage differences, obviously makes the occurrence of electric discharges and flashoverssee Fig. 1, more probable. Avoiding this to happen clearly requires a very thorough design work; critical areas suffering from excessive

be identified followed by appropriate design modifications. Doing this by experiments is very time-consuming and expensive. Here, numerical simulations of the electric field distribution provide a highly useful tool.Unfortunately, to translate the result from an into a statement whether a discharge or flashover will occur or not.


. An important quantity eff, which is the rate of net production

of free charges (electrons) in gas. It is equal to the ionization rate minus the recombination and attachment rates. strongly dependent on the electric field


Easy Evaluation of Streamer Discharge CriteriaEriksson*1




High voltageStreamer, Breakdown, Flashover

1. Introduction


to make the equipment as small as possible. From an insulation point of view, however, these two requirements are conflicting in the sense that making clearance distances between conductors at different voltage levels smaller, while at the same time also these voltage differences, obviously makes the occurrence of electric discharges and flashovers

more probable. Avoiding this to happen clearly requires a very thorough design work; critical areas suffering from excessive

be identified followed by appropriate design modifications. Doing this by experiments

consuming and expensive. Here, numerical simulations of the electric field distribution provide a highly useful tool.Unfortunately, however, to translate the result from an into a statement whether a discharge or flashover will occur or not.


. An important quantity , which is the rate of net production

of free charges (electrons) in gas. It is equal to the ionization rate minus the recombination and attachment rates. strongly dependent on the electric field


Easy Evaluation of Streamer Discharge Criteria1









consuming and expensive. Here, numerical simulations of the electric field distribution provide a highly useful tool.

however, to translate the result from an into a statement whether a discharge or flashover

The basic mechanism behind the creation of

a discharge is the ionization of the insulating gas. An important quantity

, which is the rate of net production of free charges (electrons) in gas. It is equal to the ionization rate minus the recombination and attachment rates. Since the ionization rate is strongly dependent on the electric field


Easy Evaluation of Streamer Discharge Criteria

ABB AB, Corporate Research*Corresponding author: SE









however, to translate the result from an into a statement whether a discharge or flashover



of free charges (electrons) in gas. It is equal to the ionization rate minus the recombination and

Since the ionization rate is strongly dependent on the electric field



ABB AB, Corporate ResearchSE-721 78, Västerås, Sweden









however, it is not strato translate the result from an into a statement whether a discharge or flashover





) and the shape of this function varies


ABB AB, Corporate Research721 78, Västerås, Sweden


nd components where provided by a gas, e.g. air or SF6

High voltage, Electric field, Streamer, Breakdown, Flashover






it is not strato translate the result from an Einto a statement whether a discharge or flashover







ABB AB, Corporate Research 721 78, Västerås, Sweden


nd components where the insulation is 6.

, Electric field, Streamer, Breakdown, Flashover.






it is not straE-field calculation

into a statement whether a discharge or flashover


. An important quantity is the effective , which is the rate of net production





721 78, Västerås, Sweden


the insulation is

, Electric field,






it is not straightforward field calculation



is the effective , which is the rate of net production







the insulation is

, Electric field,


to make the equipment as small as possible. From an insulation point of view, however, these two requirements are conflicting in the sense that making clearance distances between conductors at different voltage levels smaller, while at the same time also increasing these voltage differences, obviously makes the occurrence of electric discharges and flashovers




ightforward field calculation





Since the ionization rate is strongly dependent on the electric field E





the insulation is

, Electric field,


to make the equipment as small as possible. From an insulation point of view, however, these two requirements are conflicting in the sense that making clearance distances between conductors at different voltage levels

increasing these voltage differences, obviously makes the occurrence of electric discharges and flashovers









Since the ionization rate is E, so is





the insulation is

, Electric field,



increasing these voltage differences, obviously makes the occurrence of electric discharges and flashovers,









Since the ionization rate is , so is





the insulation is

, Electric field,



increasing these voltage differences, obviously makes the

, more probable. Avoiding this to

happen clearly requires a very thorough design work; critical areas suffering from excessive








Since the ionization rate is , so is



721 78, Västerås, Sweden, [email protected]


[email protected]

high voltage test equipment

formation of a discharge is that somewheresince lead to a macroscopically detectable discharge. For this to happen an electron avalanche must be formethe electric field causes increasing rate. independent of electrodes, will form provided the total number of electrons becomes larger than some critical number. avalanchacceleration, i.e. condition for a selfwritten

pwhere usually tsay 18.4 in what followscalled the streamer inception criterionfulfilled stationary streamer (or corona) discharge extending from the electrode, where usually


[email protected]

Figure 1

high voltage test equipment Obviously, a necessary condition for

formation of a discharge is that somewheresince lead to a macroscopically detectable discharge. For this to happen an electron avalanche must be formethe electric field causes increasing rate. independent of electrodes, will form provided the total number of electrons becomes larger than some critical number. avalanchacceleration, i.e. condition for a selfwritten

where the integral is performed along the particular field line where usually tsay 18.4 in what followscalled the streamer inception criterionfulfilled stationary streamer (or corona) discharge extending from the electrode, where usually


[email protected]

Figure 1high voltage test equipment

Obviously, a necessary condition for

formation of a discharge is that somewheresince thelead to a macroscopically detectable discharge. For this to happen an electron avalanche must be formed, i.e. each primary electronthe electric field causes increasing rate. independent of electrodes, will form provided the total number of electrons becomes larger than some critical number. avalanchacceleration, i.e. condition for a selfwritten [1,2]

where the integral is performed along the articular field line

where usually tsay 18.4 in what followscalled the streamer inception criterionfulfilled stationary streamer (or corona) discharge extending from the electrode, where usually


[email protected]

Figure 1high voltage test equipment

Obviously, a necessary condition for formation of a discharge is that somewhere

the primary electrons lead to a macroscopically detectable discharge. For this to happen an electron avalanche must be

d, i.e. each primary electronthe electric field causes increasing rate. independent of electrodes, will form provided the total number of electrons becomes larger than some critical number. If we further assume that such an avalanche will extend acceleration, i.e. condition for a self

[1,2]

=


eff usually taken to be in the range 15 say 18.4 in what followscalled the streamer inception criterionfulfilled it guarantees the formation of a stationary streamer (or corona) discharge extending from the electrode, where usually


[email protected]

Figure 1. Very high voltage test equipment

Obviously, a necessary condition for formation of a discharge is that somewhere. This is, however, not sufficient

primary electrons lead to a macroscopically detectable discharge. For this to happen an electron avalanche must be

d, i.e. each primary electronthe electric field causes increasing rate. independent of electrodes, will form provided the total number of electrons becomes larger than some critical

If we further assume that such an e will extend

acceleration, i.e. condition for a self

[1,2]

=


(E) > 0. The critical number aken to be in the range 15

say 18.4 in what followscalled the streamer inception criterion

it guarantees the formation of a stationary streamer (or corona) discharge extending from the electrode, where usually


[email protected]

Very high voltage test equipment

Obviously, a necessary condition for formation of a discharge is that

. This is, however, not sufficient primary electrons

lead to a macroscopically detectable discharge. For this to happen an electron avalanche must be

d, i.e. each primary electronthe electric field causes increasing rate. independent of electrodes, will form provided the total number of electrons becomes larger than some critical


acceleration, i.e. along condition for a self


) > 0. The critical number aken to be in the range 15



[email protected]

Very long discharges created with a high voltage test equipment




d, i.e. each primary electronthe electric field causes increasing rate. A selfindependent of charge injection from the electrodes, will form provided the total number of electrons becomes larger than some critical


along condition for a self-sustained avalanche can be

(

where the integral is performed along the articular field line under consideration




[email protected]

long discharges created with a high voltage test equipment




d, i.e. each primary electronthe electric field causes

A selfcharge injection from the

electrodes, will form provided the total number of electrons becomes larger than some critical

If we further assume that such an e will extend in the direction of electron

along an electric field linesustained avalanche can be

( )

where the integral is performed along the under consideration




[email protected]

long discharges created with a high voltage test equipment.




d, i.e. each primary electronthe electric field causes further

A self-sustcharge injection from the


If we further assume that such an in the direction of electron

an electric field linesustained avalanche can be

( ) >



say 18.4 in what follows The above criterion is called the streamer inception criterion


long discharges created with a


. This is, however, not sufficient primary electrons themselves


d, i.e. each primary electronfurther

sustcharge injection from the




>



The above criterion is called the streamer inception criterion




. This is, however, not sufficient themselves


d, i.e. each primary electron further ionization at an

sustained discharge, charge injection from the










. This is, however, not sufficient themselves


acceleionization at an

ained discharge, charge injection from the





) > 0. The critical number aken to be in the range 15 -




Obviously, a necessary condition for formation of a discharge is that eff

. This is, however, not sufficient themselves will not


accelerated in ionization at an





where the integral is performed along the under consideration and only

) > 0. The critical number - 20, let us

The above criterion is called the streamer inception criterion. If it is



Obviously, a necessary condition for eff > 0

. This is, however, not sufficient will not


rated in ionization at an




an electric field line, the sustained avalanche can be

where the integral is performed along the and only

) > 0. The critical number Ccrit20, let us

The above criterion is . If it is



Obviously, a necessary condition for the > 0







, the sustained avalanche can be

(1)


crit is 20, let us




the > 0







, the sustained avalanche can be

(1)


is 20, let us



the high voltage to the grounded electrode, the fulfillment of (1) will lead to an immediate breakdown across the gap. If, on the other hand,

still a chance for a comphappen ifstreamer head is so high that the ionization forward. This process, called streamer propagation, field

Here electrodes, line, Epositive electrode and discharge starting from the negative electrode.

where the conductivestreamer head enmore or less independently of the external field.

electrostatic field to evaluate the aboveprcommercial field solver is to analyze a large number of field lines at the same time and to postresult in an instructive way. inpossible to analyze a limited number of field lines

Multiphysics there has been no easily implemented method to perform the integrals along field lines, although done by formulating the problem in terms of an additional partial diffesolved [3introducedMultiphysicsbe computed and analyzed in a mostraightforward and implementation is simple and a large number of field lines can be evaluated simultaneously, both

eff(Eeff (E

If the high voltage to the grounded electrode, the fulfillment of (1) will lead to an immediate breakdown across the gap. If, on the other hand,

eff (still a chance for a comphappen ifstreamer head is so high that the ionization forward. This process, called streamer propagation, field

Here electrodes, line, E0 0.5 kV/mm for a discharge starting from the positive electrode and discharge starting from the negative electrode.

Here, we don´t consider leader propagation, where the conductivestreamer head enmore or less independently of the external field.

Although the computation of the background electrostatic field to evaluate the abovepreviously commercial field solver is to analyze a large number of field lines at the same time and to postresult in an instructive way. in-house developed possible to analyze a limited number of field lines

In previous versions of COMSOL Multiphysics there has been no easily implemented method to perform the integrals along field lines, although done by formulating the problem in terms of an additional partial diffesolved [3introducedMultiphysicsbe computed and analyzed in a mostraightforward and implementation is simple and a large number of field lines can be evaluated simultaneously, both

E) has its maximum, out to the surface where E) = 0.

If effthe high voltage to the grounded electrode, the fulfillment of (1) will lead to an immediate breakdown across the gap. If, on the other hand,

(E) < 0 along parts of the field line there is still a chance for a comphappen ifstreamer head is so high that the ionization forward. This process, called streamer propagation, field

Here U electrodes, line, U0

0.5 kV/mm for a discharge starting from the positive electrode and discharge starting from the negative electrode.

Here, we don´t consider leader propagation, where the conductivestreamer head enmore or less independently of the external field.

Although the computation of the background electrostatic field to evaluate the above

eviously commercial field solver is to analyze a large number of field lines at the same time and to postresult in an instructive way.

house developed possible to analyze a limited number of field

simultaneouslyIn previous versions of COMSOL

Multiphysics there has been no easily implemented method to perform the integrals along field lines, although done by formulating the problem in terms of an additional partial diffesolved [3introducedMultiphysicsbe computed and analyzed in a mostraightforward and implementation is simple and a large number of field lines can be evaluated simultaneously, both

) has its maximum, out to the surface where ) = 0.

eff (Ethe high voltage to the grounded electrode, the fulfillment of (1) will lead to an immediate breakdown across the gap. If, on the other hand,

) < 0 along parts of the field line there is still a chance for a comphappen if the electric field just in front of the streamer head is so high that the ionization effectively moves the streamer head forward. This process, called streamer propagation,

along the field line satisfies

U is the voltage difference between the electrodes,

1 0.5 kV/mm for a discharge starting from the

positive electrode and discharge starting from the negative electrode.

Here, we don´t consider leader propagation, where the conductive streamer head enmore or less independently of the external field.


eviously commercial field solver is to analyze a large number of field lines at the same time and to postresult in an instructive way.



Multiphysics there has been no easily implemented method to perform the integrals along field lines, although done by formulating the problem in terms of an additional partial diffesolved [3]. introduced Multiphysicsbe computed and analyzed in a mostraightforward and implementation is simple and a large number of field lines can be evaluated simultaneously, both

) has its maximum, out to the surface where

E) >the high voltage to the grounded electrode, the fulfillment of (1) will lead to an immediate breakdown across the gap. If, on the other hand,

) < 0 along parts of the field line there is still a chance for a comp

the electric field just in front of the streamer head is so high that the

effectively moves the streamer head forward. This process, called streamer propagation, can take place provided the ave

along the field line satisfies

=

is the voltage difference between the electrodes, =

10-30 kV is an empirical constant, and 0.5 kV/mm for a discharge starting from the


Here, we don´t consider leader propagation, where the hot

that the field enhancstreamer head enmore or less independently of the external field.


eviously not commercial field solver is to analyze a large number of field lines at the same time and to postresult in an instructive way.



Multiphysics there has been no easily implemented method to perform the integrals along field lines, although done by formulating the problem in terms of an additional partial diffe

Using the Particle Tracing Module in version 4.3 of COMSOL

Multiphysics, such field line integrals can now be computed and analyzed in a mostraightforward and implementation is simple and a large number of field lines can be evaluated simultaneously, both


) > 0 along the entire field line, from the high voltage to the grounded electrode, the fulfillment of (1) will lead to an immediate breakdown across the gap. If, on the other hand,



effectively moves the streamer head forward. This process, called streamer

can take place provided the avealong the field line satisfies

is the voltage difference between the =30 kV is an empirical constant, and

0.5 kV/mm for a discharge starting from the positive electrode and discharge starting from the negative electrode.

Here, we don´t consider leader propagation, hot plasma channel becomes so that the field enhanc

streamer head enables the streamer to propagatemore or less independently of the external field.

Although the computation of the background electrostatic field Eto evaluate the above

not been commercial field solver is to analyze a large number of field lines at the same time and to postresult in an instructive way.




Using the Particle Tracing Modulein version 4.3 of COMSOL

such field line integrals can now be computed and analyzed in a mostraightforward and implementation is simple and a large number of field lines can be evaluated simultaneously, both


0 along the entire field line, from the high voltage to the grounded electrode, the fulfillment of (1) will lead to an immediate breakdown across the gap. If, on the other hand,





>

is the voltage difference between the

30 kV is an empirical constant, and 0.5 kV/mm for a discharge starting from the


Here, we don´t consider leader propagation, plasma channel becomes so

that the field enhancables the streamer to propagate

more or less independently of the external field.Although the computation of the background

E is straightforward, tto evaluate the above

been commercial field solver is to analyze a large number of field lines at the same time and to postresult in an instructive way.


simultaneously. In previous versions of COMSOL






) < 0 along parts of the field line there is still a chance for a complete flashover.




>

is the voltage difference between the is the length of the field


positive electrode and Edischarge starting from the negative electrode.




is straightforward, tto evaluate the above-mentioned criteria has

been commercial field solver software. is to analyze a large number of field lines at the same time and to post process and visualize the result in an instructive way.

house developed codespossible to analyze a limited number of field

In previous versions of COMSOL

Multiphysics there has been no easily implemented method to perform the integrals along field lines, although done by formulating the problem in terms of an additional partial differential equation to be





) < 0 along parts of the field line there is lete flashover.






E0 1.2 kV/mm for a discharge starting from the negative electrode.




is straightforward, tmentioned criteria has

been available in standard software.

is to analyze a large number of field lines at the process and visualize the

result in an instructive way. codes

possible to analyze a limited number of field

In previous versions of COMSOL Multiphysics there has been no easily implemented method to perform the integrals along field lines, although it can in principle be done by formulating the problem in terms of an

rential equation to be Using the Particle Tracing Modulein version 4.3 of COMSOL

such field line integrals can now be computed and analyzed in a mostraightforward and efficientimplementation is simple and a large number of field lines can be evaluated simultaneously, both







+



1.2 kV/mm for a discharge starting from the negative electrode.




is straightforward, tmentioned criteria has available in standard software.


result in an instructive way. Even in specialized it has only been


In previous versions of COMSOL Multiphysics there has been no easily implemented method to perform the integrals

it can in principle be done by formulating the problem in terms of an


such field line integrals can now be computed and analyzed in a mo

efficientimplementation is simple and a large number of field lines can be evaluated simultaneously, both




the electric field just in front of the streamer head is so high that the associated



.





that the field enhancement at the ables the streamer to propagate


is straightforward, tmentioned criteria has available in standard software. The challenge


Even in specialized it has only been






efficient implementation is simple and a large number of field lines can be evaluated simultaneously, both




the electric field just in front of the associated


can take place provided the avealong the field line satisfies [2]





ement at the ables the streamer to propagate


is straightforward, tmentioned criteria has available in standard

The challenge is to analyze a large number of field lines at the

process and visualize the Even in specialized it has only been






way. The implementation is simple and a large number of field lines can be evaluated simultaneously, both



) < 0 along parts of the field line there is lete flashover. This may

the electric field just in front of the associated


can take place provided the ave[2]







is straightforward, the ability mentioned criteria has available in standard











) < 0 along parts of the field line there is This may

the electric field just in front of the associated field


can take place provided the average







he ability mentioned criteria has available in standard












the electric field just in front of the field


rage

(2)





ement at the ables the streamer to propagate,

more or less independently of the external field. Although the computation of the background

he ability mentioned criteria has available in standard







such field line integrals can now be computed and analyzed in a more





the electric field just in front of the field


rage

(2)



1.2 kV/mm for a


ement at the ,

Although the computation of the background he ability

mentioned criteria has available in standard






rential equation to be Using the Particle Tracing Module in version 4.3 of COMSOL

such field line integrals can now re


in 2D and 3D. Hopefully, this will result in a much faster and more reliable optimization process of insulation desigequipment.

way to apply the massless particle tracing technique in order to compute the integrals, as well as an example of how the results can be presented in an intuitive manner. Some applications involving 2D and 3D geomprovided

2

some words should be said about the effective ionization function depending on the insulating gas. For air it can be approximated by a quadratic formula

bar), and 0.3 1/(mm bar). in bar.therefore is the net production of charge carriers to occur.

3 3

viz. Tracingafter the other, in two different Studies. First, the Electrostaticsproblem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem2, evaluating the streamer criteria integrals for each field line.

3.2 Electrostatic background field

procedure we use an axisymmetric model of a high voltage (105 kV) conductor penetrating an openingsurrounded by a flange that complicates the field pattern and the solution. After running Study 1,


In the present contribution we describe the way to apply the massless particle tracing technique in order to compute the integrals, as well as an example of how the results can be presented in an intuitive manner. Some applications involving 2D and 3D geomprovided

2. Effective ionization function

Before proceeding to the solution procedure some words should be said about the effective ionization function depending on the insulating gas. For air it can be approximated by a quadratic formula

Here,

bar), and 0.3 1/(mm bar). in bar.therefore is the net production of charge carriers to occur.

3. Solution procedure 3.1 General

The model includes two physical interfaces, viz. Tracingafter the other, in two different Studies. First, the Electrostaticsproblem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem2, evaluating the streamer criteria integrals for each field line.


For the purpose of demonstrating the procedure we use an axisymmetric model of a high voltage (105 kV) conductor penetrating an openingsurrounded by a flange that complicates the field pattern and the solution. After running Study 1,


In the present contribution we describe the way to apply the massless particle tracing technique in order to compute the integrals, as well as an example of how the results can be presented in an intuitive manner. Some applications involving 2D and 3D geomprovided

Effective ionization function


Here,

bar), and 0.3 1/(mm bar). in bar. therefore is the net production of charge carriers to occur.

Solution procedure

.1 General

The model includes two physical interfaces, viz. ElectrostaticsTracingafter the other, in two different Studies. First, the Electrostaticsproblem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem2, evaluating the streamer criteria integrals for each field line.


For the purpose of demonstrating the procedure we use an axisymmetric model of a high voltage (105 kV) conductor penetrating an openingsurrounded by a flange that complicates the field pattern and the solution. After running Study 1,


In the present contribution we describe the way to apply the massless particle tracing technique in order to compute the integrals, as well as an example of how the results can be presented in an intuitive manner. Some applications involving 2D and 3D geomprovided.



Here, k bar), and 0.3 1/(mm bar).

eff therefore is the net production of charge carriers to occur.

Solution procedure

.1 General

The model includes two physical interfaces, Electrostatics

Tracing, see Fig. 2after the other, in two different Studies. First, the Electrostaticsproblem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem2, evaluating the streamer criteria integrals for each field line.


For the purpose of demonstrating the procedure we use an axisymmetric model of a high voltage (105 kV) conductor penetrating an opening in a grounded wall. The hole is surrounded by a flange that complicates the field pattern and the solution. After running Study 1,


In the present contribution we describe the way to apply the massless particle tracing technique in order to compute the integrals, as well as an example of how the results can be presented in an intuitive manner. Some applications involving 2D and 3D geom



) =

k = 1.6 mm bar /kVbar), and 0.3 1/(mm bar).

eff (E) > 0 for therefore is the net production of charge carriers to occur.

Solution procedure

.1 General


, see Fig. 2after the other, in two different Studies. First, the Electrostatics problem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem2, evaluating the streamer criteria integrals for each field line.


For the purpose of demonstrating the procedure we use an axisymmetric model of a high voltage (105 kV) conductor penetrating an

in a grounded wall. The hole is surrounded by a flange that complicates the field pattern and the solution. After running Study 1,

in 2D and 3D. Hopefully, this will result in a much faster and more reliable optimization process of insulation desig




=

= 1.6 mm bar /kVbar), and 0.3 1/(mm bar).

) > 0 for therefore is the criticanet production of charge carriers to occur.

Solution procedure


, see Fig. 2after the other, in two different Studies. First, the

problem is solved as a stationary problem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem2, evaluating the streamer criteria integrals for each field line.







Before proceeding to the solution procedure some words should be said about the effective ionization function effdepending on the insulating gas. For air it can be approximated by a quadratic formula


) > 0 for critica

net production of charge carriers to occur.

Solution procedure


, see Fig. 2. These are run separately, one after the other, in two different Studies. First, the

problem is solved as a stationary problem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem2, evaluating the streamer criteria integrals for







Before proceeding to the solution procedure some words should be said about the effective

eff (Edepending on the insulating gas. For air it can be approximated by a quadratic formula


) > 0 for critical field strength needed


Solution procedure

The model includes two physical interfaces, and These are run separately, one

after the other, in two different Studies. First, the problem is solved as a stationary

problem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem2, evaluating the streamer criteria integrals for








E). It behaves differently depending on the insulating gas. For air it can be approximated by a quadratic formula

= 1.6 mm bar /kVbar), and 0.3 1/(mm bar). p

) > 0 for E > 2.6 kV/mm, which l field strength needed


Solution procedure

The model includes two physical interfaces, and

These are run separately, one after the other, in two different Studies. First, the





in 2D and 3D. Hopefully, this will result in a much faster and more reliable optimization process of insulation designs in high voltage




It behaves differently depending on the insulating gas. For air it can be approximated by a quadratic formula

= 1.6 mm bar /kV2, is the pressure given > 2.6 kV/mm, which

l field strength needednet production of charge carriers to occur.

The model includes two physical interfaces, and Charged Particle






in 2D and 3D. Hopefully, this will result in a much faster and more reliable optimization

ns in high voltage




It behaves differently depending on the insulating gas. For air it can be approximated by a quadratic formula

.

, = 2.2 kV/(mm is the pressure given

> 2.6 kV/mm, which l field strength needed


The model includes two physical interfaces, Charged Particle


problem is solved as a stationary problem in Study 1, providing the electrostatic background field. Then, the time dependent Charged Particle Tracing problem is run in Study 2, evaluating the streamer criteria integrals for





ns in high voltage




It behaves differently depending on the insulating gas. For air it can be approximated by a quadratic formula [4

.

= 2.2 kV/(mm is the pressure given





problem is solved as a stationary problem in Study 1, providing the electrostatic background field. Then, the time dependent

is run in Study 2, evaluating the streamer criteria integrals for





ns in high voltage

In the present contribution we describe the way to apply the massless particle tracing technique in order to compute the integrals, as well as an example of how the results can be presented in an intuitive manner. Some applications involving 2D and 3D geometries are


It behaves differently depending on the insulating gas. For air it can be

[4]:











ns in high voltage

In the present contribution we describe the way to apply the massless particle tracing technique in order to compute the integrals, as well as an example of how the results can be presented in an intuitive manner. Some

etries are




> 2.6 kV/mm, which l field strength needed for a









ns in high voltage


etries are



(3)


> 2.6 kV/mm, which for a








ns in high voltage


etries are



(3)


> 2.6 kV/mm, which for a






in a grounded wall. The hole is surrounded by a flange that complicates the field

displayed in the Model Builder

theNote the regions where value 2.6 kV/mm required for net charge production.

3.3 Particle Tracing

calculated we can turn to the evaluation of the streamer integrals. Charged Particle Tracingthis node we define the 4.

originally intended for calculating trajectories of moving charged partimagnetic fields. However, electric field lines can be identified as the trajectories of charged particles having zero mass. Under the top node (Charged Particle Tracingunder the Massless

selecting those surfaces from which the field lines are starting. This is done under the node. Also, the number of field lines ("particles") is entered. Preferably one selects all sureither high or zero potential in order to make sure that all field lines are included in the analysis.

Figure displayed in the Model Builder

the electric fielNote the regions where value 2.6 kV/mm required for net charge production.


Once the background electric field is calculated we can turn to the evaluation of the treamer integrals.

Charged Particle Tracingthis node we define the

. The

originally intended for calculating trajectories of moving charged partimagnetic fields. However, electric field lines can be identified as the trajectories of charged particles having zero mass. Under the top node Charged Particle Tracing

under the Massless

The particle tracing procedure starts by selecting those surfaces from which the field lines are starting. This is done under the node. Also, the number of field lines ("particles") is entered. Preferably one selects all sureither high or zero potential in order to make sure that all field lines are included in the analysis.

Figure displayed in the Model Builder

electric fielNote the regions where value 2.6 kV/mm required for net charge production.




The originally intended for calculating trajectories of moving charged partimagnetic fields. However, electric field lines can be identified as the trajectories of charged particles having zero mass. Under the top node Charged Particle Tracing

under the Massless


Figure 2displayed in the Model Builder

electric fielNote the regions where value 2.6 kV/mm required for net charge production.




The Charged Particle Tracingoriginally intended for calculating trajectories of moving charged partimagnetic fields. However, electric field lines can be identified as the trajectories of charged particles having zero mass. Under the top node Charged Particle Tracing

under the Massless formulation.


2. The general structure of the model displayed in the Model Builder

electric fielNote the regions where value 2.6 kV/mm required for net charge




Charged Particle Tracingoriginally intended for calculating trajectories of moving charged partimagnetic fields. However, electric field lines can be identified as the trajectories of charged particles having zero mass. Under the top node Charged Particle Tracing

under the Particle Propertiesformulation.

The particle tracing procedure starts by selecting those surfaces from which the field lines are starting. This is done under the node. Also, the number of field lines ("particles") is entered. Preferably one selects all sureither high or zero potential in order to make sure that all field lines are included in the

The general structure of the model displayed in the Model Builder

electric field is found as is shown in Fig. 3Note the regions where value 2.6 kV/mm required for net charge





Particle Propertiesformulation.



d is found as is shown in Fig. 3Note the regions where value 2.6 kV/mm required for net charge


Once the background electric field is calculated we can turn to the evaluation of the treamer integrals. This is done by running the






d is found as is shown in Fig. 3Note the regions where value 2.6 kV/mm required for net charge

Once the background electric field is calculated we can turn to the evaluation of the

This is done by running the Charged Particle Tracingthis node we define the subnodes

Charged Particle Tracingoriginally intended for calculating trajectories of moving charged particles in electric and/or magnetic fields. However, electric field lines can be identified as the trajectories of charged particles having zero mass. Under the top node Charged Particle Tracing



The general structure of the model displayed in the Model Builder.

d is found as is shown in Fig. 3Note the regions where E value 2.6 kV/mm required for net charge


This is done by running the Charged Particle Tracing problem in Study 2. In

subnodes

Charged Particle Tracingoriginally intended for calculating trajectories of

cles in electric and/or magnetic fields. However, electric field lines can be identified as the trajectories of charged particles having zero mass. Under the top node Charged Particle Tracing (cpt)

Particle Properties


The general structure of the model

d is found as is shown in Fig. 3 exceeds the critical

value 2.6 kV/mm required for net charge


This is done by running the problem in Study 2. In

subnodes


cles in electric and/or magnetic fields. However, electric field lines can be identified as the trajectories of charged particles having zero mass. Under the top node

(cpt)Particle Properties



d is found as is shown in Fig. 3exceeds the critical




subnodes shown in



(cpt)) one can in fact Particle Properties







shown in

Charged Particle Tracing originally intended for calculating trajectories of


one can in fact choose







shown in

feature is originally intended for calculating trajectories of



The particle tracing procedure starts by selecting those surfaces from which the field lines are starting. This is done under the node. Also, the number of field lines ("particles") is entered. Preferably one selects all surfaces at either high or zero potential in order to make sure that all field lines are included in the






shown in Fig.




The particle tracing procedure starts by selecting those surfaces from which the field lines are starting. This is done under the Inletnode. Also, the number of field lines ("particles")

faces at either high or zero potential in order to make sure that all field lines are included in the


d is found as is shown in Fig. 3. exceeds the critical




Fig.



one can in fact choose the

The particle tracing procedure starts by selecting those surfaces from which the field

Inletnode. Also, the number of field lines ("particles")



. exceeds the critical




Fig.



one can in fact the

The particle tracing procedure starts by selecting those surfaces from which the field

Inlet node. Also, the number of field lines ("particles")


conductor and grounded wallboundaries of regions where with grey curves.

"particles" under the we want the particle trajectories to be identical to the field lines we should define the velocity as being parallel to the normalized field in the definition of according to velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the opposite wall. Under the the wall condition particle does not bouncvolume.

we then define the required integrals along each field line. This is done by introducing

Particle Tracing

Figure


The n


Under the we then define the required integrals along each field line. This is done by introducing

Figure

Particle Tracing

Figure


The n



Figure

Particle Tracing

Figure conductor and grounded wallboundaries of regions where with grey curves.

The next step is to define the velocity of the "particles" under the we want the particle trajectories to be identical to the field lines we should define the velocity as being parallel to the normalized field in the definition of according to velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the opposite wall. Under the the wall condition particle does not bouncvolume.


Figure Particle Tracing

Figure 3. conductor and grounded wallboundaries of regions where with grey curves.

ext step is to define the velocity of the "particles" under the we want the particle trajectories to be identical to the field lines we should define the velocity as being parallel to the normalized field in the definition of according to velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the opposite wall. Under the the wall condition particle does not bounc


Figure 4. Particle Tracing

. Electric field between high voltageconductor and grounded wallboundaries of regions where with grey curves.

ext step is to define the velocity of the "particles" under the we want the particle trajectories to be identical to the field lines we should define the velocity as being parallel to Ethe normalized field in the definition of

=velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the opposite wall. Under the the wall condition particle does not bounc

Under the Charged Particle Tracingwe then define the required integrals along each field line. This is done by introducing

. Definitions made in the Particle Tracing node

Electric field between high voltageconductor and grounded wallboundaries of regions where

ext step is to define the velocity of the "particles" under the Particle Propertieswe want the particle trajectories to be identical to the field lines we should define the velocity as

E. Here it is convenient to use the normalized field in the definition of

/|velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the opposite wall. Under the the wall condition Freezeparticle does not bounc

Charged Particle Tracingwe then define the required integrals along each field line. This is done by introducing

Definitions made in the node.

Electric field between high voltageconductor and grounded wallboundaries of regions where

ext step is to define the velocity of the Particle Properties

we want the particle trajectories to be identical to the field lines we should define the velocity as

. Here it is convenient to use the normalized field in the definition of

| |, see Fig. velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the opposite wall. Under the

Freezeparticle does not bounc


Definitions made in the

Electric field between high voltageconductor and grounded wall. boundaries of regions where




, see Fig. velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the opposite wall. Under the Wall

Freeze to make sure that the particle does not bounce back into the gas



Electric field between high voltage Field lines in red and

eff (




, see Fig. velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the

Wallto make sure that the

e back into the gas



Electric field between high voltageField lines in red and

(E) > 0 are marked




, see Fig. velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the

Wall node we choose to make sure that the

e back into the gas




) > 0 are marked




, see Fig. 5. Bvelocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the

node we choose to make sure that the

e back into the gas

Charged Particle Tracingwe then define the required integrals along each field line. This is done by introducing Auxiliary



) > 0 are marked

ext step is to define the velocity of the Particle Properties node.



By using a velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate therequired for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the


e back into the gas

Charged Particle Tracingwe then define the required integrals along each

Auxiliary

Definitions made in the Charged


) > 0 are marked

ext step is to define the velocity of the node.



y using a velocity of the order of 1 m/s, and knowing the typical distances between positive and negative electrodes, one can easily estimate the time required for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the


e back into the gas

Charged Particle Tracing node, we then define the required integrals along each

Auxiliary

Charged


) > 0 are marked

ext step is to define the velocity of the node. If


. Here it is convenient to use the normalized field in the definition of v

y using a velocity of the order of 1 m/s, and knowing the typical distances between positive and negative

time required for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the


e back into the gas

node, we then define the required integrals along each

Auxiliary

Charged

Electric field between high voltage Field lines in red and

) > 0 are marked

ext step is to define the velocity of the If


. Here it is convenient to use v

y using a velocity of the order of 1 m/s, and knowing the typical distances between positive and negative

time required for a "particle" to reach the opposite electrode. One also has to determine what happens to a "particle" once it reaches the


e back into the gas

node, we then define the required integrals along each

Charged

made in the

Dependent Variableswant to calculate, see Fig. 4.

Auxiliary Dependent Variablesthe field line length integral having different streamer integrals for streamers emerging from the positive electrode and one for streamers starting at the negative electrcorresponding to the positive electrode the integrand which is 1 when positive electrode and 0 when other integral is defined in an analogous way for the definitions we can separate between streamers starting from either the positive or the negative electrode.

a time dependent simulation, solving for the Charged Particle TrFig. 6some final time t_endwhere |v

4.

streamer integrals computed, it remains to present the result in an informative and easily understandable way.

corresponding to regielectrodes cana net electron production close to the positive electrode is shown while those close to the negative

Figure made in the


The integrand is called Auxiliary Dependent Variablesthe field line length integral having different streamer integrals for streamers emerging from the positive electrode and one for streamers starting at the negative electrcorresponding to the positive electrode the integrand which is 1 when positive electrode and 0 when other integral is defined in an analogous way for the definitions we can separate between streamers starting from either the positive or the negative electrode.

The final step consists of running Study 2 as a time dependent simulation, solving for the Charged Particle TrFig. 6some final time t_endwhere v| ~ 1 is the normalized velocity

. Post processing

Now when we have the solution and the streamer integrals computed, it remains to present the result in an informative and easily understandable way.

First, the corresponding to regielectrodes cana net electron production close to the positive electrode is shown while those close to the negative

Figure made in the


The integrand is called Auxiliary Dependent Variablesthe field line length integral having different streamer integrals for streamers emerging from the positive electrode and one for streamers starting at the negative electrcorresponding to the positive electrode the integrand which is 1 when positive electrode and 0 when other integral is defined in an analogous way for the negative electrode. According to these definitions we can separate between streamers starting from either the positive or the negative electrode.

The final step consists of running Study 2 as a time dependent simulation, solving for the Charged Particle TrFig. 6. The integration is performed from some final time t_end can easily be estimated as where D

~ 1 is the normalized velocity

Post processing


First, the corresponding to regielectrodes cana net electron production close to the positive electrode is shown while those close to the negative

Figure 5made in the Particle Properties


The integrand is called Auxiliary Dependent Variablesthe field line length integral having different streamer integrals for streamers emerging from the positive electrode and one for streamers starting at the negative electrcorresponding to the positive electrode the integrand which is 1 when positive electrode and 0 when other integral is defined in an analogous way for

negative electrode. According to these definitions we can separate between streamers starting from either the positive or the negative electrode.

The final step consists of running Study 2 as a time dependent simulation, solving for the Charged Particle Tr

. The integration is performed from some final time

can easily be estimated as D is the dimension of the object and


Post processing


First, the corresponding to regielectrodes cana net electron production close to the positive electrode is shown while those close to the negative electrode can be seen in Fig. 8

5. Definitions of the normalized velocity Particle Properties


The integrand is called Auxiliary Dependent Variablesthe field line length integral having different streamer integrals for streamers emerging from the positive electrode and one for streamers starting at the negative electrcorresponding to the positive electrode the

eff which is 1 when positive electrode and 0 when other integral is defined in an analogous way for

negative electrode. According to these definitions we can separate between streamers starting from either the positive or the negative


. The integration is performed from some final time

can easily be estimated as is the dimension of the object and


Post processing


First, the effective ionization function corresponding to regielectrodes can a net electron production close to the positive electrode is shown while those close to the

electrode can be seen in Fig. 8

Definitions of the normalized velocity Particle Properties


The integrand is called Auxiliary Dependent Variablesthe field line length integral having Rdifferent streamer integrals for streamers emerging from the positive electrode and one for streamers starting at the negative electrode. For the integral corresponding to the positive electrode the

(E) which is 1 when positive electrode and 0 when other integral is defined in an analogous way for



. The integration is performed from some final time t =



Post processing


effective ionization function corresponding to regi

be plotted. In Fig. 7a net electron production close to the positive electrode is shown while those close to the



Dependent Variables, one for each integral we want to calculate, see Fig. 4.

The integrand is called Auxiliary Dependent Variablesthe field line length

R = 1. different streamer integrals for streamers emerging from the positive electrode and one for streamers starting at the

ode. For the integral corresponding to the positive electrode the

) is multiplied with a factor which is 1 when E positive electrode and 0 when other integral is defined in an analogous way for



. The integration is performed from = t_end



Post processing


effective ionization function corresponding to regi




, one for each integral we want to calculate, see Fig. 4.

The integrand is called Auxiliary Dependent Variablesthe field line length L

= 1. different streamer integrals for streamers emerging from the positive electrode and one for streamers starting at the


is multiplied with a factor decreases away from the

positive electrode and 0 when other integral is defined in an analogous way for


The final step consists of running Study 2 as a time dependent simulation, solving for the Charged Particle Tracing

. The integration is performed from t_end



Now when we have the solution and the streamer integrals computed, it remains to present the result in an informative and easily

effective ionization function corresponding to regions close to the two




, one for each integral we want to calculate, see Fig. 4.

The integrand is called Auxiliary Dependent Variables

is give= 1. Then we define two

different streamer integrals Sfor streamers emerging from the positive electrode and one for streamers starting at the





The final step consists of running Study 2 as a time dependent simulation, solving for the

acing . The integration is performed from

t_end. A reasonable value for can easily be estimated as

is the dimension of the object and ~ 1 is the normalized velocity


effective ionization function ons close to the two



Definitions of the normalized velocity node

, one for each integral we

The integrand is called Source RAuxiliary Dependent Variables

is giveThen we define two

S given by (1): One for streamers emerging from the positive electrode and one for streamers starting at the






variables only, see . The integration is performed from

. A reasonable value for can easily be estimated as

is the dimension of the object and ~ 1 is the normalized velocity





Definitions of the normalized velocity node.


Source RAuxiliary Dependent Variables heading. First,

is given Then we define two

given by (1): One for streamers emerging from the positive electrode and one for streamers starting at the



positive electrode and 0 when E increases. The other integral is defined in an analogous way for




. A reasonable value for can easily be estimated as

is the dimension of the object and ~ 1 is the normalized velocity that is used.



be plotted. In Fig. 7 a net electron production close to the positive electrode is shown while those close to the


Definitions of the normalized velocity


Source R heading. First, by such an

Then we define two given by (1): One

for streamers emerging from the positive electrode and one for streamers starting at the



increases. The other integral is defined in an analogous way for




. A reasonable value for can easily be estimated as t_end

is the dimension of the object and that is used.



the areaa net electron production close to the positive electrode is shown while those close to the




under the heading. First,

by such an Then we define two







variables only, see . The integration is performed from t

. A reasonable value for t_end




the areaa net electron production close to the positive electrode is shown while those close to the

electrode can be seen in Fig. 8.











variables only, see t = 0 to

. A reasonable value for t_end = D



effective ionization function eff ons close to the two

the area with a net electron production close to the positive electrode is shown while those close to the











variables only, see = 0 to

. A reasonable value for D/v,

is the dimension of the object and v =


eff (E) ons close to the two

with a net electron production close to the positive electrode is shown while those close to the











variables only, see = 0 to

. A reasonable value for ,

=


) ons close to the two

with a net electron production close to the positive electrode is shown while those close to the

time stepping in Study 2

the regions with net electron production are located and hence where we discharges to be initiated.

electron prelectrode

Figure


From Figures 7 and 8the regions with net electron production are located and hence where we discharges to be initiated.

Figure


Figure



Figure


Figure 6time stepping in Study 2


Figure electron production electrode.

6. Settings for running the time stepping in Study 2


Figure 7. oduction

Settings for running the time stepping in Study 2


. The region having a positive net oduction

Settings for running the time stepping in Study 2


The region having a positive net oduction eff

Settings for running the time stepping in Study 2.


The region having a positive net eff (E) > 0 close

Settings for running the

From Figures 7 and 8 we now know where

the regions with net electron production are located and hence where we discharges to be initiated.

The region having a positive net ) > 0 close


we now know where the regions with net electron production are located and hence where we



we now know where the regions with net electron production are located and hence where we


Settings for running the particle tracing

we now know where the regions with net electron production are located and hence where we might

The region having a positive net ) > 0 close to the positive

particle tracing

we now know where the regions with net electron production are

might

The region having a positive net to the positive

particle tracing


expect


particle tracing


expect


particle tracing


expect


electron production electrode

for streamer inceptionaddition, the criteriabreakdown across the gap is to be expected.

to streamers starting from the positive electrode and satisfying (1). The magnitude of the integral S within the region having net charge production.

inception condition (1) for discharges starting from the positive electrode.color coded.

Figure

electron production electrode

For this to happen

for streamer inceptionaddition, the criteriabreakdown across the gap is to be expected.

Figure 9to streamers starting from the positive electrode and satisfying (1). The magnitude of the integral

is color coded and the lines are only shown within the region having net charge production.

Figure


Figure electron production electrode

For this to happenfor streamer inceptionaddition, the criteriabreakdown across the gap is to be expected.



Figure


Figure electron production electrode.




Figure inception condition (1) for discharges starting from the positive electrode.color coded.

Figure 8. The regionelectron production


Figure 9 shows the field lines that correspond to streamers starting from the positive electrode and satisfying (1). The magnitude of the integral


Figure 9. inception condition (1) for discharges starting from the positive electrode.

The regionelectron production


shows the field lines that correspond to streamers starting from the positive electrode and satisfying (1). The magnitude of the integral


Field lines satisfying the streamer inception condition (1) for discharges starting from the positive electrode. Magnitude of streamer integral

The regionelectron production eff




Field lines satisfying the streamer inception condition (1) for discharges starting from the

Magnitude of streamer integral

The regioneff (E) > 0 close

For this to happen, however, for streamer inception hasaddition, the criteria (2) breakdown across the gap is to be expected.





The regions having a positive net ) > 0 close

, however, has

(2) must be satisfied if breakdown across the gap is to be expected.





having a positive net ) > 0 close

, however, to be fulfilled. In

must be satisfied if breakdown across the gap is to be expected.





having a positive net ) > 0 close

, however, the to be fulfilled. In






having a positive net to the

the criteria (1) to be fulfilled. In






having a positive net to the negative

criteria (1) to be fulfilled. In






having a positive net negative









must be satisfied if




Magnitude of streamer integral S is



must be satisfied if


is color coded and the lines are only shown


is

propagation condition (2) for discharges the positive electrode. It is interesting to observe that streamers are not initiated at all points on the electrode surface where the electric field exceeds the cri2.6 kV/mm. In Fig. 10corresponding field lines whiccondition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for discharges starting at the negative ebe seen in Figs. 11 and 12

inception condition (1) for discharges starting from the negative electrode.

Figure

propagation condition (2) for discharges the positive electrode. It is interesting to observe that streamers are not initiated at all points on the electrode surface where the electric field exceeds the cri2.6 kV/mm. In Fig. 10corresponding field lines whiccondition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for discharges starting at the negative ebe seen in Figs. 11 and 12

Figure inception condition (1) for discharges starting from the negative electrode.

Figure

propagation condition (2) for discharges the positive electrode.

It is interesting to observe that streamers are not initiated at all points on the electrode surface where the electric field exceeds the cri2.6 kV/mm. In Fig. 10corresponding field lines whiccondition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for discharges starting at the negative ebe seen in Figs. 11 and 12


Figure propagation condition (2) for discharges the positive electrode.



Figure 10. propagation condition (2) for discharges the positive electrode.


Figure 11. inception condition (1) for discharges starting from the negative electrode.

. Field lines satisfying the streamer propagation condition (2) for discharges the positive electrode.


. Field lines satisfying the streamer inception condition (1) for discharges starting from the negative electrode.

Field lines satisfying the streamer propagation condition (2) for discharges the positive electrode.



Field lines satisfying the streamer propagation condition (2) for discharges




It is interesting to observe that streamers are not initiated at all points on the electrode surface where the electric field exceeds the cri2.6 kV/mm. In Fig. 10corresponding field lines whiccondition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for discharges starting at the negative ebe seen in Figs. 11 and 12.



It is interesting to observe that streamers are not initiated at all points on the electrode surface where the electric field exceeds the cri2.6 kV/mm. In Fig. 10 corresponding field lines whiccondition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for discharges starting at the negative e



It is interesting to observe that streamers are not initiated at all points on the electrode surface where the electric field exceeds the cri

is shown the corresponding field lines which satisfy the condition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for discharges starting at the negative e


Field lines satisfying the streamer propagation condition (2) for discharges starting from

It is interesting to observe that streamers are not initiated at all points on the electrode surface where the electric field exceeds the cri

is shown the h satisfy the

condition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for discharges starting at the negative electrode can


Field lines satisfying the streamer starting from

It is interesting to observe that streamers are not initiated at all points on the electrode surface where the electric field exceeds the critical value

is shown the h satisfy the

condition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for

lectrode can



It is interesting to observe that streamers are not initiated at all points on the electrode surface

tical value is shown the

h satisfy the condition (2) for breakdown across the gap. Note that in this case all field lines that satisfy (1) also fulfill (2). Finally, a similar pair of figures for

lectrode can






lectrode can

Field lines satisfying the streamer

inception condition (1) for discharges starting from the





lectrode can


propagation condition (2) for discharges starting from the negative electrode.

5.

well in 2D. Below we also show that it can easily be applied to 3D geometries as well. To illustrate this we rotate the 2D geometry used above to generate a corresponding 3D model. Then, in order to element, a block protruding from the inner high voltageelectric field is solved for and the particle tracing is carried out generating 1000 field lanalyze as is seen in Fig. post processing can be employed as for the 2D case shown earlier. As Fig. 15from the positive to the negative electrode.

Figure propagation condition (2) for discharges starting from the negative electrode.

. A

As demonstrated above, the method works

well in 2D. Below we also show that it can easily be applied to 3D geometries as well. To illustrate this we rotate the 2D geometry used above to generate a corresponding 3D model. Then, in order to element, a block protruding from the inner high voltageelectric field is solved for and the particle tracing is carried out generating 1000 field lanalyze as is seen in Fig. post processing can be employed as for the 2D case shown earlier. As Fig. 15from the positive to the negative electrode.

Figure

Figure propagation condition (2) for discharges starting from the negative electrode.

A 3D example

As demonstrated above, the method works well in 2D. Below we also show that it can easily be applied to 3D geometries as well. To illustrate this we rotate the 2D geometry used above to generate a corresponding 3D model. Then, in order to element, a block protruding from the inner high voltage electric field is solved for and the particle tracing is carried out generating 1000 field lanalyze as is seen in Fig. post processing can be employed as for the 2D case shown earlier. As Fig. 15 from the positive to the negative electrode.

Figure

Figure 1propagation condition (2) for discharges starting from the negative electrode.

3D example

As demonstrated above, the method works well in 2D. Below we also show that it can easily be applied to 3D geometries as well. To illustrate this we rotate the 2D geometry used above to generate a corresponding 3D model. Then, in order to introduce a truly three dimensional element, a block protruding from the inner high

conductor is added, see Fig. 13electric field is solved for and the particle tracing is carried out generating 1000 field lanalyze as is seen in Fig. post processing can be employed as for the 2D case shown earlier. As

the field lines giving rise to a flashover from the positive to the negative electrode.

Figure 1

12. propagation condition (2) for discharges starting from the negative electrode.

3D example

As demonstrated above, the method works well in 2D. Below we also show that it can easily be applied to 3D geometries as well. To illustrate this we rotate the 2D geometry used above to generate a corresponding 3D model. Then, in

introduce a truly three dimensional element, a block protruding from the inner high



13. 3D geometry used in the simul

. Field lines satisfying the streamer propagation condition (2) for discharges starting from the negative electrode.

3D example





3D geometry used in the simul

Field lines satisfying the streamer propagation condition (2) for discharges starting from the negative electrode.

3D example









conductor is added, see Fig. 13electric field is solved for and the particle tracing is carried out generating 1000 field lanalyze as is seen in Fig. post processing can be employed as for the 2D case shown earlier. As an example we display in






conductor is added, see Fig. 13electric field is solved for and the particle tracing is carried out generating 1000 field lanalyze as is seen in Fig. 14post processing can be employed as for the 2D

an example we display in the field lines giving rise to a flashover

from the positive to the negative electrode.





conductor is added, see Fig. 13electric field is solved for and the particle tracing is carried out generating 1000 field l

14. Exactly the same post processing can be employed as for the 2D








Exactly the same post processing can be employed as for the 2D















conductor is added, see Fig. 13electric field is solved for and the particle tracing is carried out generating 1000 field lines to




3D geometry used in the simulation.




conductor is added, see Fig. 13. The electric field is solved for and the particle tracing

ines to Exactly the same

post processing can be employed as for the 2D an example we display in


ation.




. The electric field is solved for and the particle tracing



the field lines giving rise to a flashover

ation.




. The electric field is solved for and the particle tracing



the field lines giving rise to a flashover

propagation condition (2) for discharges starting from the positive electrode. Note the new flashover paths introduced from the protruding part of the 6. Conclusions

Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evalucriteria in a relativelyengineering point of view this will have a significant impact on the design work on high voltage components, where the development of specialized inAs was demonstrated, the method works well in both

Figure

Figure

propagation condition (2) for discharges starting from the positive electrode. Note the new flashover paths introduced from the protruding part of the 6. Conclusions

With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evalucriteria in a relativelyengineering point of view this will have a significant impact on the design work on high voltage components, where the development of specialized inAs was demonstrated, the method works well in both

Figure

Figure propagation condition (2) for discharges starting from the positive electrode.

Note the new flashover paths introduced from the protruding part of the

6. Conclusions

With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evalucriteria in a relativelyengineering point of view this will have a significant impact on the design work on high voltage components, where the development of specialized inAs was demonstrated, the method works well in both 2D and 3D geometries.

Figure 1

Figure propagation condition (2) for discharges starting from the positive electrode.


6. Conclusions

With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evalucriteria in a relativelyengineering point of view this will have a significant impact on the design work on high voltage components, where the development of specialized inAs was demonstrated, the method works well in

2D and 3D geometries.

14. The 1000 field lines generated.

Figure 15. propagation condition (2) for discharges starting from the positive electrode.


6. Conclusions

With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evalucriteria in a relativelyengineering point of view this will have a significant impact on the design work on high voltage components, where the development of specialized in-As was demonstrated, the method works well in


The 1000 field lines generated.

. Field lines satisfying the streamer propagation condition (2) for discharges starting from the positive electrode.


6. Conclusions

With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evalucriteria in a relativelyengineering point of view this will have a significant impact on the design work on high voltage components, where the development of

-house codes is no longer needed.As was demonstrated, the method works well in



Field lines satisfying the streamer propagation condition (2) for discharges starting from the positive electrode.


With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evalucriteria in a relativelyengineering point of view this will have a significant impact on the design work on high voltage components, where the development of

house codes is no longer needed.As was demonstrated, the method works well in





With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evalucriteria in a relatively simple manner. From an engineering point of view this will have a significant impact on the design work on high voltage components, where the development of





Note the new flashover paths introduced from the protruding part of the high voltage electrode.

With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has become possible to evaluate streamer discharge

simple manner. From an engineering point of view this will have a significant impact on the design work on high voltage components, where the development of





Note the new flashover paths introduced from high voltage electrode.

With the introduction of the Particle Tracing Module and in particular its ability toarbitrary quantities along field lines, it has

ate streamer discharge simple manner. From an

engineering point of view this will have a significant impact on the design work on high voltage components, where the development of




















With the introduction of the Particle Tracing Module and in particular its ability to integrate arbitrary quantities along field lines, it has







With the introduction of the Particle Tracing integrate

arbitrary quantities along field lines, it has ate streamer discharge










Note the new flashover paths introduced from




house codes is no longer needed. As was demonstrated, the method works well in

7. References 1. Kuffel, E. and Zaengl, W.S., Engineering FundamentalsOxford (2.and Boehme, H., "Streamer inception and propagation models for power devices", Proc. CEIDP 2009 Conference, Virginia Beach (Oct. 2009)3. Propagation from Electric Field Simulations", Proc. SCEE2012 Scientific Computing in Electrical Engineering, ETH Zurich4. breakdown criterion to inhomogeneous gas traps"

. References

1. Kuffel, E. and Zaengl, W.S., Engineering FundamentalsOxford (2. Pedersen, A., Christen, T., Blaszczyk, A., and Boehme, H., "Streamer inception and propagation models for power devices", Proc. CEIDP 2009 Conference, Virginia Beach (Oct. 2009)3. Christen, T., "Streamer Inception and Propagation from Electric Field Simulations", Proc. SCEE2012 Scientific Computing in Electrical Engineering, ETH Zurich4. Petcharaks, K., "Applicability of the streamer breakdown criterion to inhomogeneous gas traps"

. References

1. Kuffel, E. and Zaengl, W.S., Engineering FundamentalsOxford (

Pedersen, A., Christen, T., Blaszczyk, A., and Boehme, H., "Streamer inception and propagation models for power devices", Proc. CEIDP 2009 Conference, Virginia Beach (Oct. 2009)

Christen, T., "Streamer Inception and Propagation from Electric Field Simulations", Proc. SCEE2012 Scientific Computing in Electrical Engineering, ETH Zurich

Petcharaks, K., "Applicability of the streamer breakdown criterion to inhomogeneous gas traps", ETH Zurich Thesis No. 11192 (1995)

. References

1. Kuffel, E. and Zaengl, W.S., Engineering FundamentalsOxford (1984)



Petcharaks, K., "Applicability of the streamer breakdown criterion to inhomogeneous gas

ETH Zurich Thesis No. 11192 (1995)

. References

1. Kuffel, E. and Zaengl, W.S., Engineering Fundamentals

1984) Pedersen, A., Christen, T., Blaszczyk, A.,

and Boehme, H., "Streamer inception and propagation models for power devices", Proc. CEIDP 2009 Conference, Virginia Beach (Oct. 2009)




. References


Pedersen, A., Christen, T., Blaszczyk, A.,

and Boehme, H., "Streamer inception and propagation models for power devices", Proc. CEIDP 2009 Conference, Virginia Beach (Oct. 2009)















Pedersen, A., Christen, T., Blaszczyk, A., and Boehme, H., "Streamer inception and propagation models for designing air insulated power devices", Proc. CEIDP 2009 Conference, Virginia Beach (Oct. 2009)




1. Kuffel, E. and Zaengl, W.S., Engineering Fundamentals, Pergamon Press,

Pedersen, A., Christen, T., Blaszczyk, A., and Boehme, H., "Streamer inception and

designing air insulated power devices", Proc. CEIDP 2009 Conference,




1. Kuffel, E. and Zaengl, W.S., , Pergamon Press,






1. Kuffel, E. and Zaengl, W.S., High, Pergamon Press,



Christen, T., "Streamer Inception and Propagation from Electric Field Simulations", Proc. SCEE2012 Scientific Computing in Electrical Engineering, ETH Zurich (Sep. 2012)



High-Voltage , Pergamon Press,



Christen, T., "Streamer Inception and Propagation from Electric Field Simulations", Proc. SCEE2012 Scientific Computing in

(Sep. 2012)Petcharaks, K., "Applicability of the streamer

breakdown criterion to inhomogeneous gas ETH Zurich Thesis No. 11192 (1995)

Voltage , Pergamon Press,




(Sep. 2012)Petcharaks, K., "Applicability of the streamer

breakdown criterion to inhomogeneous gas ETH Zurich Thesis No. 11192 (1995)





(Sep. 2012) Petcharaks, K., "Applicability of the streamer

breakdown criterion to inhomogeneous gas





Petcharaks, K., "Applicability of the streamer

breakdown criterion to inhomogeneous gas

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