Thesis for the degree of Master of Science in Electrical Engineering, emphasize on High

Voltage Technology.

ELECTRICAL MODEL OF THE ROMAN GENERATOR

BY:

NICOLAS MORA PARRA

DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING

FACULTY OF ENGINEERING

UNIVERSIDAD NACIONAL DE COLOMBIA

BOGOTA, COLOMBIA 2009

2

Thesis for the degree of Master of Science in Electrical Engineering, emphasize on High

Voltage Technology.

ELECTRICAL MODEL OF THE ROMAN GENERATOR

BY:

NICOLAS MORA PARRA

SUPERVISOR:

Ph. D., Phil. Lic., M. Sc., Eng. FRANCISCO ROMAN

DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING

FACULTY OF ENGINEERING

UNIVERSIDAD NACIONAL DE COLOMBIA

BOGOTA, COLOMBIA 2009

3

ACKNOWLEDGEMENTS

The author would like to express his gratitude feelings to each one who participated of

this project. To his director Prof. Roman for all the support and opportunities given at

the Electromagnetic Compatibility Research Group. To his colleague and mentor Felix

Vega for all the guidance and help provided during this three years of being working

together. To his mother Clara Ines for her unconditional friendship and patience. To all

his family members who never failed on showing up. To his beloved Lina Maria. To

each member of the Electromagnetic Compatibility Research Group of the National

University of Colombia EMC-UNC. To the EMC team at the Swiss Federal Institute of

Technology EPFL and their head Prof. Farhad Rachidi. To the sponsors of the

Cattleya project.

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TABLE OF CONTENTS

1. CHAPTER 1: INTRODUCTION........................................................................... 12

2. CHAPTER 2: THE CORONA TUBE.................................................................... 16

2.1. Introduction .................................................................................................... 16

2.2. Corona tube configurations ............................................................................ 16

2.2.1. Point-to-plane corona tubes .................................................................... 20

2.2.2. Cylindrical corona tubes......................................................................... 22

2.3. Corona tube polarities..................................................................................... 22

2.3.1. Positive corona ....................................................................................... 23

2.3.2. Negative corona...................................................................................... 25

2.4. Corona inception voltage................................................................................ 27

2.4.1. Townsend breakdown mechanism.......................................................... 29

2.4.2. Streamer breakdown mechanism............................................................ 30

2.4.3. Methods for calculating the corona inception voltage............................ 31

2.5. Currents in the corona tube............................................................................. 36

2.5.1. Maxwells equations formulation for space charge dominated coronas 39

2.5.2. Unipolar charge drift formula................................................................. 40

2.5.3. Unipolar current in the point-to-plane corona tube ................................ 41

2.5.4. Unipolar current in the cylindrical corona tube...................................... 47

2.6. Arc breakdown in the corona tube.................................................................. 48

2.7. Effect of pressure modification in corona tubes............................................. 49

2.7.1. Inception voltage modification............................................................... 49

2.7.2. Unipolar charge drift modification......................................................... 51

2.7.3. Background propagation electric field modification .............................. 51

2.8. Comprehensive summary ............................................................................... 52

3. CHAPTER 3: THE FLOATING ELECTRODE.................................................... 54

5

3.1. Introduction .................................................................................................... 54

3.2. Lumped parameter model of the floating electrode........................................ 55

3.2.1. Charging phase of the floating electrode................................................ 55

3.2.2. Discharge phase of the floating electrode............................................... 61

3.3. Distributed parameter model of the floating electrode................................... 64

3.4. Pulse forming networks.................................................................................. 66

3.4.1. General transmission line theory ............................................................ 68

3.4.2. The floating electrode as a transmission line.......................................... 70

3.4.3. Charging phase of the floating electrode................................................ 72

3.4.4. Discharge phase of the floating electrode............................................... 73

4. CHAPTER 4: THE CLOSING SWITCH .............................................................. 76

4.1. Introduction .................................................................................................... 76

4.2. Self breaking switches .................................................................................... 76

4.2.1. Switching phases .................................................................................... 77

4.2.2. Switch configurations ............................................................................. 78

4.3. Breakdown voltage......................................................................................... 81

4.4. Electrical model of the switch ........................................................................ 86

4.4.1. Capacitance of the switch ....................................................................... 87

4.4.2. Resistance of the switch ......................................................................... 88

4.4.3. Inductance of the switch ......................................................................... 90

4.5. Rise-time considerations................................................................................. 91

5. CHAPTER 5: THE LOAD..................................................................................... 93

5.1. Introduction .................................................................................................... 93

5.2. Pulse discharge applications of the R. G. ....................................................... 93

5.3. HPEM applications with the R. G. ................................................................. 94

6. CHAPTER 6: SELECTED EXPERIMENTS ........................................................ 95

6.1. Introduction .................................................................................................... 95

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6.2. The R.G. at the EPFL ..................................................................................... 95

6.2.1. The corona tube ...................................................................................... 96

6.2.2. The spark gap switch .............................................................................. 96

6.2.3. Measuring system................................................................................... 98

6.3. Experiment I ................................................................................................... 99

6.3.1. Experimental setup ............................................................................... 100

6.3.2. Results and analysis.............................................................................. 100

6.4. Experiment II................................................................................................ 101

6.4.1. Experimental Setup .............................................................................. 104

6.4.2. Results and analysis.............................................................................. 104

7. CHAPTER 7: CONCLUSIONS........................................................................... 106

7.1. Introduction .................................................................................................. 106

7.2. General conclusions...................................................................................... 106

7.3. Contributions ................................................................................................ 107

7.4. Future work .................................................................................................. 107

8. REFERENCES ..................................................................................................... 109

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TABLE OF FIGURES

Figure 1.1 General scheme of the impulse current generator proposed by Roman [1]. In

the figure the main elements that compose the generator are shown. ............................ 12

Figure 2.1 a. General scheme of a corona tube. b. Circuit symbol of the corona tube.. 17

Figure 2.2 a. Corona tube without applied voltage b. Energized corona tube. Image

modified from [28]. ........................................................................................................ 18

Figure 2.3 Point-to-plane electrode configuration. Image modified from [20]............. 21

Figure 2.4 Cylindrical electrode configuration. Image modified from [27].................. 22

Figure 2.5 Typical V-I characteristic for common gaps. Image taken from [28].......... 28

Figure 2.6 Experimental data points for corona onset from wires and cylinders as a

function of wire radius. Curves show theoretical predictions obtained numerically,

predictions from Peeks formula, and predictions from the formula given by Eq. 2.10.

Image taken from [23]. ................................................................................................... 35

Figure 2.7 Experimental data points for corona onset from points and spheres as a

function of point radius. Also shown as curves are theoretical predictions obtained

numerically and also from predictions using Peeks formula for wires, and the formula

given by Eq. 2.11. Image taken from [23]...................................................................... 36

Figure 2.8 Continuity of the current between the external current and the corona tube.

Image taken from [24]. ................................................................................................... 38

Figure 2.9 Point-to-plane configuration geometry. This scheme is used to derive the

unipolar conduction current in the corona tube. Image taken from [20]........................ 44

Figure 2.10 Radial current density distributions at the plane in unipolar point-to-plane

coronas. Image taken from [20]: Curve W: current ratio as appears in Eq. 2.45, the

Warburg distribution. Curve S: Present work, Eq. 119)Eq. 2.44, Curve NO: Negative

pulseless glow in ambient air, d = 12, 13, and 14 mm, V = 22 kV (Kondo and Miyoshi,

1978).Points : Neg. Trichel, Points 0: Pos. Glow Ambient air, d = 120 mm, V = 40 kV

(Goldman, et al. 1978"). The experimental points are normalized to I near the axis. The

8

negative Trichel pulse data show a dip in the center. ................................................... 45

Figure 2.11 Image taken from [20]: The continuous ('"background'") component I, of

the corona current vs. corona voltage, for a 13-mm point-to-plane gap in ambient air,

compared with the unipolar saturation current curves. Numbers along the curves

indicate time averaged streamer currents. (Goldman and Sigmond 1981). Points 0, solid

curve: Regular streamer corona. Points *, dashed curve: Regular, periodic

streamer+spark corona. Curves S: Unipolar space-charge saturation current limits.... 46

Figure 2.12 Positive corona thresholds for I-mm diameter point, 8-cm gap in air at

various pressures indicated on the curves. Note the various thresholds listed in the

legend on the figure. Image taken from [14] .................................................................. 50

Figure 2.13 Negative point corona, 1-mm diameter point, 8-cm in air at various

pressures. Thresholds indicated in the legend. Note lack of reproducibility of points

taken with increasing and decreasing voltage during a single run. Note the difference in

the thresholds and the curves with ultraviolet triggering and with gamma-ray triggering.

Dashes represent data which are not reproducible because of current fluctuations. Image

taken from [14]. .............................................................................................................. 50

Figure 2.14 Circuit model of the H. V. source connected to the corona tube. This model

can be used for simulations. ........................................................................................... 53

Figure 3.1 a. General scheme of the R. G. The equivalent capacitance between the F. E.

and ground is shown. b. The equivalent circuit of the R. G. with the F. E. capacitor

included is shown. .......................................................................................................... 55

Figure 3.2 R.G. model during the charging phase......................................................... 56

Figure 3.3 Equivalent circuit for analyzing the R. G. during the charging phase. ........ 57

Figure 3.4 Current and voltage in the F. E. for the example considered....................... 60

Figure 3.5 Current and voltage in the F. E. when the continuous discharge process of

the generator is considered. ............................................................................................ 61

Figure 3.6 Equivalent circuit of the R. G. during the discharge phase.......................... 62

9

Figure 3.7 Current waveform simulation and spectral density for the example

considered....................................................................................................................... 66

Figure 3.8 Distributed parameter cell of the F. E. ......................................................... 67

Figure 3.9 Interconnection of multiple lossless cells simulating the F. E.. ................... 67

Figure 3.10 Differential element of the F. E. when considered as a transmission line . 68

Figure 3.11 Transmission line geometries and their distributed parameters. Image taken

from [31]......................................................................................................................... 71

Figure 3.12 Equivalent of the F. E. as a transmission line of length l and characteristic

impedance Zo. ................................................................................................................. 72

Figure 3.13 Equivalent circuit of the R. G. during the discharge phase........................ 73

Figure 3.14: Waveforms at the load of the R. G. during the discharge of the F. E. Image

taken from [31]. .............................................................................................................. 74

Figure 4.1 Schematic representation of the switching phases in a gas filled switch.

Image modified from [31]. ............................................................................................. 77

Figure 4.2: Typical design of gas spark gap. Image modified from [32] ....................... 79

Figure 4.3 Common electrode configurations for gap tests. Image taken from [32]. ... 80

Figure 4.4 Parallel plate spark gap. ............................................................................... 81

Figure 4.5 Field enhancement factor for common electrode configurations. Image taken

from [32]......................................................................................................................... 83

Figure 4.6 Electrical model of the spark gap................................................................. 87

Figure 6.1 Picture and diagram of the R. G. constructed by Eng. Felix Vega at the

EPFL laboratory. Image modified from [53].................................................................. 95

Figure 6.2 Experimental setup used for the verification of the PRF model .................. 98

10

Figure 6.3 Experimental and theoretical results for the PRF in the RG...................... 101

Figure 6.4 General scheme of the experimental setup for measuring the current that

flows from the high voltage source toward the corona tube......................................... 102

Figure 6.5 Equivalent circuit of the experimental setup during charging phase ......... 103

Figure 6.6 Experimental and theoretical results for the PRF in the R. G. with series

resistor. ......................................................................................................................... 105

Figure 6.7 Experimental and theoretical results for the voltage on the corona plate in

the R. G. with series resistor......................................................................................... 105

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LIST OF TABLES

Table 3.1 Parameters of a R. G. with a point-to-plane corona tube example................ 59

Table 4.1 Dielectric strength factor for typical gases.................................................... 82

Table 6.1 Parameters of the R. G. used in the experiments at the EPFL....................... 97

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1. CHAPTER 1: INTRODUCTION

In 1996, Roman presented the general scheme for constructing a repetitive and constant

impulse current generator based on the properties of floating electrodes that are stressed

with very intense electrical fields [1]. The general scheme of the fast impulse generation

system is explained in Figure 1.1.

Figure 1.1 General scheme of the impulse current generator proposed by Roman [1]. In the figure the

main elements that compose the generator are shown.

As it is shown in Figure 1.1, the high voltage (H. V.) source is connected to the H. V.

electrode. A floating electrode (F. E.) [2, 3] with a protrusion on its top surface is placed

below the H. V. electrode. The term floating electrode is used to describe the fact that

the electrode is not attached to a potential reference; therefore it can acquire any

potential between the ground plane and the H. V. source. The system composed by the

H. V. electrode and the F. E. is referenced by the author in this work as the corona

tube, due to the fact that electrical coronas will occur between the F. E. and the H. V.

electrode. The F. E. and the electrode before the load make a gas switch. The cathode of

the switch is connected to ground through a load that ideally should be a very low

inductive resistor.

13

When the H. V. source of positive or negative polarity is connected to the H. V.

electrode, an electrostatic field distribution is generated between the terminals of the

corona tube. Due to the presence of the protrusion inside the corona tube, there will be

an electric field enhancement nearby the protrusion that modifies the Laplacian

electrostatic field distribution.

The theory that explains the effects of stressing the F. E. with intensive electric field

was presented by Roman in [2, 3]. If the amplified electric field over the protrusion is

high enough to reach the critical electric field for the onset of corona discharges,

electrical coronas will start to flow between the F. E. and the H. V. electrode.

Before the initiation of electrical coronas, the F. E. has no net charge. The electrical

potential of the F. E. is defined by its location in the Laplacian electric field distribution.

After the onset of electrical coronas inside the corona tube, the F. E. acquires a net

charge of the same polarity of the H. V. electrode and will start to increase its potential

with respect to ground. If the potential of the F. E. exceeds the breakdown voltage of the

gas switch, the latter will close and the F. E. will discharge to ground.

During the switch breakdown, almost all the electric charge that was stored in the F. E.

is delivered to the load in the form of a fast current impulse. If the corona discharge

mechanism can be sustained, breakdown will occur again and a continuous charge-

discharge process is established. Therefore, a repetitive impulse current generator can

be built based on this mechanism. This system is known as the Roman Generator and

will be referenced as R. G. anywhere in this document.

Based on this mechanism, different implementations of the R. G. have been constructed

[4, 5, 40, 41, 52-55]. Every version of the R. G. was an improved model depending on

the application for which it was constructed.

The first R. G. reported in [54-55] was able to produce impulses of 1.5 [kA] with a rise-

time of 10 [ns]. This R. G. was used to test the response of low voltage protective

devices under subsequent stroke-like impulse current. Diaz presented in [5], a high

current R. G. that was able to produce pulses up to 10 [kA] and rise-time in the order of

tens of [ns] on very low impedances In [40], Mora et al. presented a sub-[ns] R. G. that

was able to produce on a 100 [] resistor a 10 [A] current pulse with a rise-time of 600

14

[ps]. As it was described in [41], the R. G. was used with a discone antenna to radiate

electromagnetic impulses. Vega et al. [52] presented the design of a meso-band high

power electromagnetic radiator [34] based on the sub-[ns] R. G. [40] and a switched

oscillator. In [53] Vega will present a modified version of the R. G. that is able to

produce impulses of up to 100 [A] in 200 [] with sub-[ns] rise-time.

In three reports of R. G. prototypes [1, 5, 40], the theoretical background about physical

aspects of electrical coronas production the F. E. that are stressed with intense electrical

fields was presented. In [5] detailed design considerations of the R. G. were reported. In

[5] a circuit model was proposed to simulate the current output of the R. G. The circuit

model response was in good agreement with laboratory measurements.

The electrical models of the R. G. proposed in previous works are only useful for

simulating a previously fabricated R. G. A model that includes physical considerations

of the electrical coronas production has not been included in any of the previous works.

Furthermore, the elements that compose the R. G. have always been considered as a

whole unit and therefore, some aspects related to the design of a proper way to deliver

the energy have been neglected.

This work presents a complete electrical model of the R. G. that considers separately

each part of the impulse generation system to produce a consisting theory for predicting

the output of the R. G. Several reports on electrical coronas [6-30, 42] were consulted

and synthesized for characterizing and predicting the behavior of the corona tube.

Chapter 2 covers all the relevant aspects to design a corona tube and to predict its

behavior.

The available electrical models of the R. G. [1, 3, 5] are lumped parameter circuits. Due

to the fact that sub-[ns] pulses can be generated with the R. G., the model it could be

needed to include a distributed parameter model for some parts of the system. This kind

of model could lead to a better understanding and design of the entire system.

Therefore, the revision of the model of the F. E. as a lumped and distributed circuit

element is presented in Chapter 3. Expressions are given to predict the R. G. response

based on a RLC circuit model and a transmission line model.

Chapter 4 explains the main characteristics of the gas switch and its circuit equivalent

15

according to what was consulted in several pulsed power references [31-33, 43, 44].

In chapter 5 some possible applications of the R. G. are introduced depending on the

kind of load that is being connected.

Finally, in Chapter 6 some selected experiments that were used to study the R. G. are

presented

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2. CHAPTER 2: THE CORONA TUBE

2.1. Introduction

The system conformed by the H. V. electrode, the F. E., and the gas in between both

(namely the gap), is presented in this chapter as the corona tube of the R. G. The name

of corona tube was chosen by the author because of the similarities of the working

principles of this part of the R. G. with the vacuum tubes used in past decades in

electronics.

In Sections 2.2 and 2.3, the principal configurations for corona tubes are presented. In

the following Sections 2.4, 2.5, and 2.6 of this chapter, the voltage current

characteristics of corona tubes are studied. Finally, the effect of modifying the pressure

inside a corona tube is presented in Section 2.7.

In Section 2.8 the electric model of the corona tube is presented according to what was

concluded in Sections 2.2 to 2.7.

2.2. Corona tube configurations

One of the major fundamental differences between breakdown in the uniform or

quasiuniform field and that in the nonuniform field is that the onset of a detectable

ionization in the uniform field usually leads to the completion of the transition and the

establishment of a complete breakdown [28]. In the nonuniform field the case is entirely

different and various visual manifestations of ionization and excitation processes can be

viewed before the complete voltage breakdown occurs. These manifestations have long

been called coronas. A corona tube can be viewed as any system whose nonuniform

field configuration leads to the development of corona discharges inside. In order to

create the discharge, nonuniform field distributions must be generated with properly set

electrodes.

Thus, a corona discharge system will consist of active electrodes or surfaces surrounded

by ionization regions where free charges are produced; low field drift regions where

charged particles drift and react; and low field passive electrodes, mainly acting as

charge collectors [17].

Any sort of electrodes inside a gas volume, capable of producing a corona discharge are

defined in this document as a corona tube. Figure 2.1 a shows the general scheme of a

17

corona tube. The electrode having the positive potential is defined as the anode of the

tube. The electrode having the negative potential is defined as the cathode of the tube.

Figure 2.1 b shows the circuit symbol defined by the author to represent the corona tube

in this document. As it will be presented in the coming Sections of this chapter,

additional to the anode and cathode of the tube, there are other parameters that must be

also taken into account in the design of corona tubes but are not included in this

diagram for simplicity.

Consider the corona tube shown in Figure 2.2 a. that is going to be energized with a

voltage source. In the absence of electric field inside the tube, there will be no preferred

direction for the motion of charges, and the behavior of the gas will be governed by

classical thermodynamics. Therefore, if a galvanometer is placed in the external circuit

of the tube before turning on the source, no current will be registered as shown in Figure

2.2 a.

Figure 2.1 a. General scheme of a corona tube. b. Circuit symbol of the corona tube.

When the H. V. electrode of the generator is energized, a motion of charged particles

present in the gas starts from anode to cathode due to applied electric field force. When

both negative and positive charges are present between the electrodes, the positive

particles will move toward the cathode and the negative particles toward the anode. The

movement of charges will induce negative charge accumulation in the cathode and

decrease of the negative charge in the anode. This will be accompanied by the flow of

charge in the external circuit and current will be recorded, even though the charge did

not emerge form one of the electrodes or was absorbed by the other [28]. Figure 2.2 b.

illustrates the latter process.

18

Figure 2.2 a. Corona tube without applied voltage b. Energized corona tube. Image modified from [28].

If the positive and negative charged particles in the gas travel a distance dl + and dl ,

respectively, in a time interval dt, the total change of surface charge density in the

electrodes will have two components s and

s , that satisfies [28]:

Eq. 2.1: ][Clednlednsss

In Eq. 2.1 n + and n correspond to the volumetric densities of the positive and negative

particles respectively, and e the electron charge. The conduction current density is the

time derivate of q:

Eq. 2.2: ])[( 2mAvnvne

dtlden

dtldenJJ

dtdJ CCsC

Where v + and v are the drift velocities of the positive and negative particles. It is

important to recognize that according to Eq. 2.2 the current flowing in the external

circuit has as many components as the various types of charged particles with different

densities, velocities, or charges [28].

In order to obtain the drift velocities of the particles, the acceleration due to the applied

electric field is used. In the majority of the practical cases most of the positive ions

present in a gas are singly ionized. Therefore, theoretically the force acting on charged

particles in a gas is eE. The particles will experiment acceleration

Eq. 2.3: ][ 2sm

mEe

dtvda

In Eq. 2.3 m is the mass of the accelerated particle. If the particle is initially at rest and

moves through vacuum, the velocity v will be given by the integral of a . This is not

19

the case on a typical corona tube, because it is filled with gas and the accelerated

particles will certainly collide with other gas particles. This will start ionization

processes that modify not only the velocity but the electric field configuration of the

corona tube during time.

The velocity of charged particles strongly depends on the electric field configuration of

the tube and the gas discharge physic processes occurring during the multiple collisions

that occur in the gas. The former will be treated in this chapter in order to obtain some

conclusions about the voltage-current characteristics of corona tubes. The latter is an

extensive subject that has been treated by many of the authors referenced in this text and

therefore it is not going to be developed in this work. The author suggests [28-30] as the

main references for understanding gas discharge physics.

As it will be shown later for the breakdown criteria, and as it has been shown in this

chapter, any derivation of an expression for electron acceleration and multiplication

must be based on the knowledge of the electric field intensity in the gap. This function

is not known for many irregular configurations of technical developments that have

been done with the R. G. [1-5]. In this work, the point-to-plane and the coaxial cylinder

configurations will be considered. The point-to-plane configuration was chosen because

it has been frequently used in fundamental research on coronas in which large

nonuniformity of the field is desired. The cylinder configuration has also been chosen

because of its accessibility to exact field analysis and its relative significance in practice

and research [28].

According to Waters in [29] besides the cylinder and point-to-plane arrangement, there

are other commonly used electrode configurations. These are going to be mentioned in

this work but are not analyzed:

a. Sphere gaps: widely used for measurement of high voltages of direct, alternating

or impulse form.

b. Sphere - plane gaps: They offer a convenient means, with increasing gap length/

sphere diameter ratio, of progressing from pseudo-uniform field to an

asymmetrical field configuration.

c. Rod rod gaps: frequently employed in H. V. engineering for chopped-voltage

20

testing and insulation co-ordination. In this case the rod has squared Section.

Other profiles are hemispherically or conically tipped cylindrical rods, or

confocal paraboloids.

d. Concentric sphere-hemisphere gaps: offer a simply evaluated field distribution,

but edge effects and difficulties of manufacture limit their use.

e. Conductor-plane gaps: this term is used to describe a configuration employing a

cylindrical cross Section conductor whose axis is parallel to a plane electrode.

f. Conductor-conductor gaps formed by parallel cylinders have an obvious

relationship to power transmission configurations.

In order to derive the electric field distribution, an effort will be addressed to obtain the

analytical solution for the point-to-plane configuration and the cylindrical configuration.

Any such account must recognize that the initial field distribution will be grossly

modified by the accumulation of space charges produced during the breakdown process.

Nevertheless, this apparently complicating influence can lead to a simplification in the

breakdown behavior since any high electric field which is initially present near the

electrodes is rapidly reduced [29].

2.2.1. Point-to-plane corona tubes

According to Nasser in [28] the sphere-plane arrangement of electrodes with a radius

chosen according to the degree of nonuniformity desired is one of the electrode

configurations that lends itself quite satisfactorily for experimental and theoretical

studies of fundamental nature. Since the sphere has to be mounted in position by some

means and the voltage has to be applied to it by a lead or a connection, a shaft has to be

connected to it. This is used both as an electric lead as well as a mechanical support.

This leads to the hemispherically capped cylinder, the standard electrode, shown in

Figure 2.3 and used frequently in fundamental research on coronas.

As presented in [29] the hyperboloidplane gap is also reported in the literature, and can

be identified as point-to-plane configuration. The advantage of working with such gaps

is that they allow an analytical solution of the Laplace equation.

Thus, point-to-plane configuration consists on a hemispherically or hyperboloidally

21

capped cylinder acting as an active electrode, separated a known distance from a passive

conductive plane acting as charge collector. When highly stressed, such configuration is

able to support a self-sustained discharge where a Laplacian electric field confines the

ionization processes to regions close to high field electrodes or insulators, as already

presented at the introduction of Section 2.2 [17].

Figure 2.3 shows the standard point-to-plane discharge gap as presented by Sigmond in

[20]. Some terminology is introduced for further use during this work.

The ionization region in Figure 2.3 is the volume where all ionization processes are

confined. According to [20, 28-30], only here the electric field is high enough to make

positive the difference between the ionization coefficient and the attachment

coefficient . The ionization region is regarded as a self sustained discharge that passes

current at a practically constant value cV of the voltage fall across it or, equivalently, at a

constant value cE of the field in this region. The ions of the ionization region are then

injected into the drift region, where their average density will be always much larger

than that of ions of the opposite polarity. The drift region is usually regarded as a fairly

passive resistance in series with the ionization region discharge and as the main reason

for the exceptional stability of coronas [20, 28].

Figure 2.3 Point-to-plane electrode configuration. Image modified from [20].

The parameters which define the electric properties of the point-to-plane corona tubes

are the point radius r, the point-to-plane separation distance d, the type of gas in the gas,

and the pressure p of the gas inside the gap.

22

2.2.2. Cylindrical corona tubes

Cylindrical electrode configuration has been frequently used in available literature

because of the exact analytical results that can be obtained, due to symmetry of the

electric field distribution. It can be found since the works of Townsend, many analytical

formulas to describe the electric field distribution in vacuum and charge dominated

breakdown processes. In [28] Nasser does a complete analysis of the electric field

distribution with and without space charge accumulation.

Figure 2.4 shows a typical cylindrical configuration of the corona tube as presented in

[27]. Notice that the ionization region, and drift region are also defined in this figure as

they were defined for point-to-plane configuration.

Figure 2.4 Cylindrical electrode configuration. Image modified from [27].

The parameters which define the electric properties of the cylinder corona tube are the

inner and outer radii r1 and r2 respectively.

2.3. Corona tube polarities

When the corona tube is connected to a voltage source, the discharge process occurring

inside will depend on the polarity of the coronating electrode. Two different kinds of

processes can be distinguished for a corona tube: positive corona and negative corona.

Positive corona occurs in the tube when the positive electrode is responsible for the

ionization due to the high no uniformity of the field in its surroundings. Positive corona

is sometimes regarded as anode corona [28]. On the contrary negative corona or cathode

corona occurs when the cathode is the responsible of the ionization process inside the

23

tube.

Several references can be found addressing the development of either positive or

negative corona in different configurations. The major references are found in Leonard

Loebs research group (includes Loeb, Kip, Trichel, Meek, English, Morton) papers at

the department of physics in the University of California [6-13, 18, 21]. After

Townsend, they can be recognized as the pioneers of the research of not only corona

development but the complete gas discharge process. The first works on the basic

mechanisms of positive and negative coronas in air at atmospheric pressure began

around 1937 with the work of Kip and Trichel. The many research studies that have

been performed utilize the point-to-plane standard geometry, but the mechanisms are

equally applicable to other geometries.

Meek and Craggs also published a very complete textbook regarding the complete gas

discharge process [29]. In [29] Sigmond was invited to write a chapter on electric

coronas and Waters was invited to write about nonuniform breakdown. Nasser

published a very complete textbook on the same subject [28]. All the contents of this

Section in this work are taken from many of these references. In the sake of simplicity,

only the relevant information is going to be presented. Readers can go further inside the

topics by consulting [6-13, 21, 28-29].

2.3.1. Positive corona

Generally speaking, if a corona tube is designed to work on the positive polarity, when

connected to a DC voltage source, several gas processes occur while the voltage is

increased. Depending on the applied voltage, different current regimes can be found to

occur between the tube terminals. This current regimes range from the [pA] at the very

first beginning of charge movement at low voltages (tenths to hundredths of [V]) to [A]

after the onset of the corona regime (some [kV]). In the following Sections, more formal

development is going to be presented for the corona inception voltage, the current-

voltage curves of the corona tubes, and the complete breakdown development. In this

Section the physic processes occurring inside the tube are to be briefly introduced as

they are not necessary to understand the complete functioning of the R. G.

According to [29] a representative positive corona tube may typically pass through the

24

following stages as the voltage is increased:

a. Field intensified dark current

b. Burst pulses or preonset streamer corona (only in electron attaching gases)

c. Positive glow (only in electron attaching gases) and/or pre-breakdown streamer

corona

d. Spark breakdown

A characteristic feature of this type of discharge is that positive space charges

completely dominate throughout the discharge gap, even in strongly electron attaching

gases [29]. The cathode is isolated from the ionization region by the drift region, which

delays or blocks the cathode secondary processes.

In [9] English describes the corona in positive point-to-plane configuration. He did a

characterization of the current flowing out in the external circuitry connected to the

plane cathode while increasing the potential in between point and plane. During this

process he could see an intermittent pulsed current regime starting at the threshold

voltage Vg. The current pulses are due to the building up of positive ion space charge

formations by electron avalanches. In this regime a blue glow adhering closely to the

point (burst pulses) appears. Faint luminous filaments (preonset streamers) can also be

seen. This current depends on the number of external triggering electrons introduced

into the gap.

Then, appears a steady state burst corona, starting at an inception potential Vo. In this

phase there is a blue glow adhering closely to the point. The streamer formation is

prevented by space charge weakening of the field. This current is independent of

external ionization.

If potential is increased, breakdown streamers appear. These streamers are similar to

preonset but, are larger and more brilliant. This happens when the potential is large

enough to overcome the effect of space charge. These streamers convert into a spark

when they cross the whole gap and initiate secondary processes in the cathode. Nasser

explains in detail the streamer formations in [28]. In the sake of simplicity, readers are

25

referred to [28].

According to Kip in [12, 13], there is an ohm region in the V-I curves for positive

corona. The positive discharges for lower potentials consist of field intensified

avalanches that give currents of about 10e-9 [A] order. At potentials of Vg the preonset

phenomena characteristic of the Geiger regime is observed. Then the current abruptly

increases to the order of 1e-8 [A], and random pulses of different magnitudes are

observed. When the voltage increases to Vo (about 500 [V] over Vg), the current

suddenly jumps to a high value of about 1e-7 [A]. Above this point the corona

phenomena show current increase linear with the potential for some 2000 [V], after

which the current increases more rapidly.

The physical processes involved on a positive corona discharge are the same processes

involved in the explanation of streamer formation. Detailed explanation around this can

be found in [28].

2.3.2. Negative corona

The same analysis that was performed for positive corona can be done for negative

corona. As presented in [29], a representative negative corona tube must follow the

following processes while the voltage is increased in the terminals:

a. Field intensified dark current

b. Trichel pulse corona (observed in electron attaching gases only)

c. Steady negative glow (found only with attaching gases)

d. Negative Streamers

e. Spark Breakdown

The most characteristic feature of the negative corona is that the cathode lies on the

ionizing zone, thus securing for this region a prompt supply of secondary cathode

electrons. Positive space charge dominates the ionization region, while the drift region

will have a weak or strong negative space charge according to the importance of

electron attachment.

26

In the negative corona, in stead of having bursts, the intermittent pulses are regarded as

Trichel pulses. They are named after their discoverer [6]. In [6] Trichel explains the

general processes in the negative point-to-plane corona tube. He explains that negative

corona is composed by discrete pulses named after as Trichel pulses. The magnitude

and frequency of the Trichel pulses are function of the mean current, the point size and

the applied pressure. The frequency is independent of the gap length.

According to Loeb in [10] for the negative point-to-plane corona tube increasing the

potential proportionally increases the current as in an Ohms law regime. This is

accompanied by an increase of the Trichel pulse frequency and increment of sequential

bursts, but no increase in magnitude or basic character of pulses. These pulses have

durations of about 400 [ns] [9].

The discharge appears as a bright bluish purple button glow of some millimeters of

diameter close to the point. In contrast to the positive which gives the impression of

being adhered to the point, the negative gives the impression of being detached. This

glow is separated by a narrow dark space. When the voltage increases to some [kV], and

current to some [A], the luminosity of the discharge seems to increase. But the general

assembly conserves.

Lama presents in [15] a summarized model of the Trichel pulse formation developed by

Loeb [10]: In time sequence, the pulse is initiated by an electron ejected from the

cathode surface by some mechanism such as field emission or positive ion

bombardment, and proceeds by Townsend ionization. The positive ions left in the wake

of the electron avalanche serve to increase the ionization field, leading to a rapid

buildup of the current.

The positive ions further provide an additional source of electrons through

bombardment of the cathode surface. The electron avalanche is choked off in a very

short time by the negative space charge which forms by electron attachment just outside

the ionization region and reduces the field in that region below the avalanche threshold.

The rise time of the pulse is extremely short, on the order of [ns]. The electron

avalanche then remains off until the negative space charge is removed by the electric

field at a sufficient distance (this is regularly regarded as the "clearing length" and

"clearing time") for the field in the ionization region to regain its critical value. At high

27

frequency, many space-charge clouds will be simultaneously in transit.

In negative corona, the creation of negative ions occurs only in electron attaching gases

as Oxygen. The electrons that are accelerated in the high field regions go to the glow

regions and stay attached to Oxygen. When this occurs, the Trichel begins to choke off

[10].

Depending on the corona polarity, geometry, and gas, the voltage current characteristic

of the corona tube will change. Therefore, in the sake of an electric model of the corona

tube, a theoretical derivation of some parameters of the tube must be obtained. The most

relevant parameters here derived for corona tubes are the corona inception voltage, the

current in the corona tube and the arcing limit of the corona tube. The following

Sections deal with the theoretical derivation of these parameters for cylinder and point-

to-plane configuration.

2.4. Corona inception voltage

Before starting the discussion of the calculation of the corona inception voltage, a

qualitative description of the voltage-current relation for a common gap is given in

order to present the overall possibilities of discharges before arc breakdown in a corona

tube.

Typical V-I characteristic for common gaps presents a typical V-I relation for a

common gap as presented in [28]. This characteristic is obtained using DC voltages.

The current values also depend on the external circuit of which the corona tube is only a

part.

When the voltage is raised and the current is observed, random current pulses of less

than 1e-16 [A] will be the first manifestation of current in the tube. The particles

involved until this point are free electrons present in the gap that are produced in the

volume by external ionization. It is also possible to have electron produced by

photoemission in the cathode. Under a constant radiation level the current will increase

with voltage until it reaches a plateau known as the saturation current. In this phase, all

the electrons emitted from the cathode and/or produced in the gas are collected [28].

If the voltage is increased, the current will maintain for some range until it starts

increasing again. This increase is exponential and this phase is known as the

28

Townsend discharge region because in this region ionization by collision starts to occur

as the electric field is increased.

Further increase in the voltage will lead to an over exponential increase in the current.

This abrupt transition is known as breakdown. There are to mechanisms responsible of

such breakdown, the Townsend mechanism, and the streamer mechanism that will be

presented in the following Sections. The breakdown is characterized by an increase in

several order of magnitude of the current with almost no increase of voltage. The

voltage across the gap that starts the breakdown is commonly regarded as the inception

voltage of the gap. This value is very important in the calculations of the charging

current of the R. G. and therefore there is special interest in calculating the inception

voltage of the corona tube.

Figure 2.5 Typical V-I characteristic for common gaps. Image taken from [28]

The current beyond breakdown is regarded as self sustained because it becomes

independent of the external ionizing source [28, 29]. If the current is allowed to increase

further, there are some regions of the discharge that may be recorded before the

complete arc breakdown of the tube occurs. This typically occurs in non uniform field

29

breakdown. In uniform field breakdown, as it was stated at the beginning of this

chapter, the creation of self sustained discharges (often by Townsend mechanism)

usually leads to the completion of the transition and the establishment of a complete arc

breakdown. In the other hand, when streamer mechanism is the responsible of the self

sustained discharge, various visual manifestations of ionization and excitation processes

can be viewed before the complete voltage breakdown occurs. Such manifestations

correspond to what many authors regard as the corona, subnormal glow, normal glow,

and abnormal glow of the gaps [9, 10, 28-29]. Many aspects of the discharge in these

regions where briefly discussed in the past Section. Nevertheless, references are given

for further research in this topic.

Finally, when the current is allowed to increase further, another transition occurs and a

new form of discharge, known as the arc because of extreme brightness [28], develops.

The following lines of this Section will deal with both the Townsend mechanism,

typical in uniform field breakdown and streamer mechanism that is common in non

uniform field breakdown. At the end of this Section, different methods proposed to

estimate the inception voltage are presented.

2.4.1. Townsend breakdown mechanism

The transition to a self sustained current has been presented for uniform field gaps

according to the general picture first proposed by J. S. Townsend in the early 1900s.

Under the general term Townsend mechanism one should imagine the successive

development of electron avalanches between the electrodes, in which every avalanche

produces one or more successor avalanches until the channel conductivity has reached a

value high enough to make the current theoretically infinite and practically limited by

the outer circuit [28].

The analytical treatment of the development of avalanches is well described in [28, 29].

According to Townsend theory of breakdown, if a uniform discharge gap is considered,

the current i between anode and cathode written in terms of the first and second

ionization coefficients and is:

Eq. 2.4: )1(1

dd

oo ee

nn

ii

30

In Eq. 2.4 n and no are the number of electrons and the initial number of electrons,

respectively. Thus, i and io are the current in the gap and the initial current respectively.

According to Eq. 2.4 the current value starts to grow indefinitely when the singularity is

satisfied. Therefore the Townsend breakdown criterion is:

Eq. 2.5: 1)1( de

It is important to note that Eq. 2.5 is the simplified version of the complete criterion. In

this, the photon absorption coefficient, and the attachment coefficient have been

neglected.

Townsend breakdown criterion states that each one of the initial no electrons must

produce a successor by cathode emission from a colliding avalanche created by such

electron. This will make the current independent of io and maintain itself. Although this

mechanism is not always responsible for breakdown, it lends itself easily to theoretical

analysis that in many cases has been verified in experimental data [28]. The most

important example is the calculation of the breakdown voltage by the use of the well

known Paschens law. According to this law, the breakdown voltage of the uniform

configuration is a unique function of the product of pressure and electrode separation

for a particular gas and electrode material. Paschens law will be treated in chapter 4

when describing the breakdown voltage of the switch in the R. G.

2.4.2. Streamer breakdown mechanism

Streamer breakdown mechanism often occurs in uniform fields for high values of the

product of pressure and distance pd. When avalanches develop inside a uniform gap, the

space charge traveling in the tip of the avalanche creates an electric field that somehow

can be comparable to the applied electric field. Therefore, the field enhancement

contributes to further ionization from photoelectrons that are formed from previously

emitted photons from the avalanche. This ionization can be self sustained and another

way of having breakdown appears as the streamer mechanism.

According to Meek [21] the breakdown of a uniform field is considered to occur by the

transition of an electron avalanche proceeding from cathode to anode into a self-

propagating streamer, which develops from anode to cathode to form a conducting

filament between the electrodes. A streamer will develop when the radial field about the

31

positive space charge in an electron avalanche attains a value of the order of the external

applied field. For then photoelectrons in the immediate vicinity of the avalanche will be

drawn into the stem of the avalanche and will give rise to a conducting filament of

plasma, and a self-propagating streamer proceeds towards the cathode. Further

information about the streamer propagation can be found in the analysis performed by

Nasser in [28].

The general criterion of having streamer propagation is the development of space charge

whose field becomes comparable to the applied field in a gap. Meek proposed an

equation to estimate the streamer growth possibility known as the Meekss criterion for

streamer growth [29]:

Eq. 2.6: 5.0)(

)()( 0nxkEe x

dxx

In Eq. 2.6 is the first ionization coefficient of the gas, is the attachment coefficient,

k is a constant, n is the gas density, and xE is the field in the direction of the avalanche.

2.4.3. Methods for calculating the corona inception voltage

Depending on the electrodes configuration, and the polarity, Eq. 2.5 to Eq. 2.6 can be

used to derive a critical electric field that could cause breakdown to start. The

mechanisms involved can include electron avalanches or streamers. A very deep

explanation of the breakdown in each polarity is done in [23, 28]. In [23] a complete

summary of the theories presented around corona inception is presented. In [28] corona

is explained step by step when the gap is subjected to impulse voltages and DC

voltages.

Several methods have been proposed in the literature [23, 25, 28-30] for calculating the

corona inception voltage depending on the geometry of the corona tube and the polarity.

In [29] Waters presents a complete chapter on nonuniform breakdown. In this work,

several formulas are presented for calculating the inception voltage for different

arrangement of electrodes and polarities, as derived by other authors. All this formulas

correspond to specific works, and none of them are applicable to other geometries.

There have also been many experimental investigations of the voltage for the onset of

32

corona from wires or cylinders and points or spheres as a function of wire radius,

voltage polarity, gas temperature and gas pressure (e. g. Peek (1929), Kip (1938), Loeb

(1965), Grunberg (1973), Nasser and Heiszler (1974), Waters and Stark (1975), Bhm

(1976), DAlessandro and Berger (1999) and Moore et al (2000)) [23].

The corona inception voltage has been reported to be almost the same value for negative

and positive polarity, even though the breakdown mechanisms involved in each are

different. In [8], English makes a systematic study of corona onsets for the point-to-

plane configuration for positive and negative polarity, and the effect of point material

and radius of curvature on both polarities onset potential. English presents the surprising

equality of the positive and negative onset potential. This result is also presented by

Loeb in [10].

According to Lowke and DAlessandro in [23], in a review of Goldman and Goldman in

1978, it is implied that the onset of negative corona is determined from the Townsend

breakdown criterion and positive corona from the streamer breakdown criterion. As it

has already been presented, onset fields for the two polarities from the two criteria are

believed to be approximately equal.

Peek in 1929 developed an empirical equation for the electric field cE at the surface of

a cylinder for the onset of corona as a function of the wire radius, and relative air

density [16, 20, 23]. This basic equation for cylinders is known as Peeks formula. This

formula has been commonly used and reformulated for other geometries [16, 23]. Here

in the sake of obtaining only one criterion, this formula will be redefined based on a

charge accumulation concept.

For both the Townsend breakdown criterion, and the streamer criterion, can be written

in the same form

Eq. 2.7: ][)(

CQedr

the integral Eq. 2.7 is taken over the ionization region near the wire or point, and Q is

constant. For the Townsend breakdown criterion

33

Eq. 2.8: ][1 CQ

For the streamer criterion, Q is the number of electrons in the avalanche necessary for

particles to produce space charge fields of the order of the field necessary for ionization.

The evaluation of Eq. 2.7 for both criteria, values of Q that lie in the range of 1e4 ions

to 1e8 can be obtained. Of course, as the range is about four orders of magnitude, there

is not a clear rule to calculate the critical electric field. Recently in [23] several

experiments where carried in which a value of Q = 1e4 led to inception in almost every

configuration tried.

As derived in [23] corona inception for cylinder configuration can be calculated with:

Eq. 2.9: ])[)(

ln1(

cmkV

rBNNE

QEE

oo

ooc

Here oo NE , are the values of the electric field E, and the gas density N, at the threshold

for net positive ionization (i. e. ). The factor oN

N ensures that E is the field for

which ionization equals attachment for conditions of temperature and pressure other

than the standard conditions of 293 [K] and 1 [Bar] for which oo NE , are defined. B is a

constant chosen to fit experiments, and r is the inner cylinder radius.

Eq. 2.9 is Peeks equation, but with coefficients in terms of discharge parameters.

Substituting the values

]1[1951.2],[1208.2],[175.98],[72.24 322 cmeN

cmVeB

cmVe

NE

cmkVE o

o

oo

and using a value of Q=1e4 [C] a modified Peeks formulas is obtained [23]:

Eq. 2.10: ])[4.01(72.24cmkV

rEc

Here r is the inner cylinder radius taken in [cm]. The coefficient 0.4 could be changed to

0. 3, as in original Peeks equation, by choosing a value of Q = 200.

34

For the point-to-plane geometry, in [23] an expression is also derived from a similar

perspective for the critical field for inception:

Eq. 2.11: ])[03.035.01(72.24cmkV

rrEc

In Eq. 2.11 r is the tip radius taken in [cm] again.

For calculating the inception voltage in the cylinder corona configuration, one can use

the expression for calculating the electric field at the surface of the inner cylinder [28]

and replace Eq. 2.10:

Eq. 2.12: ])[ln()4.01(72.24)ln( kVrRr

rrRrEV co

In Eq. 2.12 r is the inner cylinder radius taken in [cm], and R is the outer cylinder radius

taken in [cm].

Following the same procedure for calculating the inception voltage from the electric

field at the surface of the tip [7, 15], in a point-to-plane configuration:

Eq. 2.13: ])[4ln()03.035.01(36.12)4ln(21 kV

rdr

rrrdrEV co

Here d is the point-to-plane separation measured in [cm], and r is the tip radius taken in

[cm].

Figure 2.6 and Figure 2.7 present a comparison between the critical field as calculated

from Peeks formula, and Eq. 2.10 and Eq. 2.11, and the measurements performed by

other authors for each geometry.

There are other approaches where the corona gap geometry dependence may be taken

into account by the somewhat unrigorous but very successful engineering concept of the

equivalent (active electrode) radius eR simply defined by Les Renardieres research

group as [17]:

Eq. 2.14: ][max

mEV

R gape

35

The field

Eq. 2.15: ][)( 222

max

mV

rRV

rRErE egape

will adequately simulate the field just outside the real electrode surface of radius of

curvature R, and the integrals in Eq. 2.5 to Eq. 2.6 can be calculated once for any given

set of parameters.

Figure 2.6 Experimental data points for corona onset from wires and cylinders as a function of wire

radius. Curves show theoretical predictions obtained numerically, predictions from Peeks formula, and

predictions from the formula given by Eq. 2.10. Image taken from [23].

These calculations led Berger to propose a modification of the Peeks formula [22] for

the corona inception field iE , valid for air at NTP [17]:

Eq. 2.16:

])[341(38.2

])[166.01( 45.0

mMVHeE

mMV

REE

H

eHi

In Eq. 2.16 H is the absolute humidity in ][ 3mg . The corona inception voltage is then:

Eq. 2.17: ][MVREV eio

36

Figure 2.7 Experimental data points for corona onset from points and spheres as a function of point

radius. Also shown as curves are theoretical predictions obtained numerically and also from predictions

using Peeks formula for wires, and the formula given by Eq. 2.11. Image taken from [23].

For any given geometry, eR is easily determined as an analytical or computational

function of the actual minimum radius R of the electrode curvature and the gap distance

d. For example the coaxial configuration and the point-to-plane one has respectively

[17]:

Eq. 2.18:

])[21ln(5.0

])[1ln(

mRdRR

mRdRR

e

e

2.5. Currents in the corona tube

Current in the corona tube can appear as a conductive current and as a displacement

current. The conductive current density cJ appears due to the motion of charged

particles inside the tube as presented in the beginning of the chapter in Eq. 2.2:

Eq. 2.2: ])[( 2mAvnvne

dtlden

dtldenJJ

dtdJ CCsC

As it was already presented, the drift velocity of the charge carriers is dependent on the

37

electric field from Eq. 2.3.

Eq. 2.3: ][ 2sm

mEe

dtvda

The electric field inside the corona tube is the superposition of the electrostatic

Laplacian field plus the space charge field. The space charge field begins to distort the

electrostatic potential distribution once it becomes at least 10% of the vacuum

configuration of the field. This is willing to happen when the number of ions created by

collision exceeds 1e6 [17]. Therefore, one can express the electric field as:

Eq. 2.19: ][mVEEE v

In Eq. 2.19 vE denotes the electrostatic Laplacian field, sometimes regarded as the

vacuum field [24], and E denotes the space charge created field. Thus, the

displacement current density can be calculated as the derivative of the electric field E:

Eq. 2.20: ][)(

2mAJJ

tEE

tEJ DvD

vooD

where vDJ is the vacuum displacement current density, and DJ is the space charge

caused (SSC) displacement current density [24]. The conductive and the displacement

current density add up to the total current density J:

Eq. 2.21: ][ 2mAJJJ DC

Because of the continuity law we see that the total current density J, which flows into

the electrode system, is equal to the external current i that feeds the corona tube, as

shown in Figure 2.8. Therefore integrating the density in one of the electrodes of the

corona tube, one has:

Eq. 2.22: ][ASdJiS

Thus, the total current flowing from the external circuit is equal to the displacement

38

currents and the conduction current, as derived in Eq. 2.2 and Eq. 2.20:

Eq. 2.23: ][Aiiii DvDC

The conduction current on the corona tube depends on the geometry and gas properties.

It can be calculated from the integration of the conduction current density at the

electrodes surface:

Eq. 2.24: ][ASdJiSelectrode

cC

The vacuum displacement current can be calculated from the derivative of voltage

across the terminals of the corona tube and the corona tube capacitance as:

Eq. 2.25: ][Adt

dVCi tubetubevD

In Eq. 2.25 tubeC is the corona tube capacitance, and tubeV is the voltage across the tube.

For the SSC calculation, the Ramo-Shockley theorem can be used to obtain:

Eq. 2.26: tube

vtube

D AdVdtdV

Vi ][1

Figure 2.8 Continuity of the current between the external current and the corona tube. Image taken from

[24].

In Eq. 2.26 vV is the modified potential profile due to the space charge advance, and is

the volumetric charge density. A complete formal derivation of Eq. 2.23 can be found in

Section 2.3 of [24]. Here only the important results are shown before presenting the

39

complete derivation of the conduction current in the corona tube.

In the R. G., the major contributor of the corona tube current is the conduction current.

Therefore, in the following Sections, the derivation of the conduction current will be

dealt for the point-to-plane geometry and the cylinder geometry. The displacement

currents are not going to be derived further in this chapter, only brief conclusions about

their nature are to be presented in the last Section of this chapter.

2.5.1. Maxwells equations formulation for space charge dominated coronas

When subjected to nonuniform fields as it has already been implied throughout this

chapter, ionization currents can no longer be calculated with the Townsend mechanism

conclusions [18].

All the calculations performed for gas discharge analysis deal with charged particle

flow. The theoretical and computational treatment of charge particle flow is not easy to

obtain. The main complication is the space-charge field, which makes the evolution of

the density distribution )(ti of any charged species i dependent on the total charge

distribution i

i tt )()( throughout the gap. The general equation the distribution of

charged particles, and thus for the discharge current, will be a nonlinear integro-

differential, and can at present only be solved analytically for very special, highly

symmetric geometries under quite drastic simplifying assumptions [20].

At very low corona currents the charge space charge accumulation is not a dominating

phenomenon; therefore, the drift region field can be calculated from the geometrically

determined Laplacian field distribution, satisfying Laplaces equation:

Eq. 2.27: ][],[0 22

mVVE

mVV

When the current is increased, the space charge will perturb this field distribution, and

finally, in the saturation limit, dominate it completely.

To solve the problem only unipolar ions drifting without diffusion, with constant

mobility in a gas, in combined space-charge and externally-generated electric fields

will be considered. This type of flow dominates low current electrical coronas, where

40

the ionization processes usually are confined to very small volumes near the electrodes,

while most of the discharge volume is filled with ions drifting in low electric fields [20].

Because the drift region in a corona discharge will often be dominated by ion species of

one sign and of mobilities sufficiently equal and constant to be considered as a single

type of ion [29]. Neglecting diffusion, the equations that govern the drift of stationary

ions with constant mobility are:

Eq. 2.28:

][

][

][0

][

22

3

2

mVV

mVVE

mAJ

mAEvJ

o

Eq. 2.22 give by eliminating ,,J

and E

:

Eq. 2.29: 0)()()( 222 VVV

This homogeneous, nonlinear fourth-order partial differential equation contains only the

potential distribution )(rV . The constants , and o and indeed the very presence of

space charges, must be introduced through the boundary conditions [20].

Eq. 2.29 has exact analytical solutions coaxial cylinders as will be shown later. It also

has an exact solution for concentric spheres, and for parallel planes. For the point-to-

plane configuration, little effort has been done due to the complexity. According to [20]

in 1963 Felid discussed the full Eq. 2.29 and gave particular solutions for certain space

charge saturated cases, and later Atten gave a general method for its numerical solution.

These general methods, however, are not very helpful for quick estimates of point to

plane and similar geometries, and will not be further discussed here.

2.5.2. Unipolar charge drift formula

The drift of ions of one sign with constant mobility , charge density ),( tr , current

density vJ

, no diffusion, and subjected to an electric field ),( trE

is considered.

41

The rate of change of along the path of the ions is [20]:

Eq. 2.30: ][ 3mAv

tdtd

From the continuity equation one has that:

Eq. 2.31: ][)( 3mAvvvJ

t

Thus, by eliminating the drift velocity and the electric field from Poissons equation:

Eq. 2.32: ][)()( 32

mAEEvv

tdtd

o

The solution of this ordinary differential equation can be done by separation of

variables. This yields the unipolar charge drift formula as derived by Sigmond [20, 29]

and Waters for deriving the current in point-to-plane configuration and cylinder

configuration:

Eq. 2.33: ])[()(

1)(

1 3

0 Cmtt

tt oo

It describes exactly the spread-out of the charge density along the path of a cloud of

unipolar ions drifting with constant mobility, in arbitrary, time dependent electric fields

[20]. With Eq. 2.33 the ion path is not needed, only the age of the ions is the initial

condition.

2.5.3. Unipolar current in the point-to-plane corona tube

In the point-to-plane geometry, the Laplacian field line divergence between a

hyperboloid or parabolic point and a plane starts as in cylindrical geometry and ends up

as in plane geometry. An abundant supply of ions at the point will cause the field and

drift lines there to diverge more strongly because of space charge repulsion,

approaching the spherical case. Therefore, a constant space charge dominated electric

field is expected throughout most of the discharge gap and to rise somewhat towards

both electrodes [20].

42

As the initial density )( ot near the ionization zone in all geometries is much greater

than at the larger electrode, the plane:

Eq. 2.34: ][)( 3mC

TT op

In Eq. 2.34 T is the ion transit time along the field line of length L, from the point to the

plane. The average velocity along this field line will be dependent on the average field

along the gap with:

Eq. 2.35: ][smEv

Therefore, the corresponding transit time:

Eq. 2.36: ][sEL

vLT

is the minimum one and thus insensitive to small deviations from the constant field

conditions. Replacing Eq. 2.36 in Eq. 2.34 one has that nearby the plane, the unipolar

charge density in the line is equal to:

Eq. 2.37: ][ 3mC

LEo

p

With Eq. 2.37 the current density at the plane can be calculated with:

Eq. 2.38: ][ 2mAE

LEvJ popC

According to the arguments presented, one could assume that the average field and the

field nearby the plane electrode are equal. Waters in [26] have shown the rough

approximation that in the point-to-plane configuration both fields are equal,

proportional to the applied voltage V, and inversely proportional to the distance L:

Eq. 2.39: ][)(

mV

LVVV

EE op

43

In Eq. 2.39 oV is the corona inception voltage, whose calculation was presented in

Section 2.4. Replacing relationship Eq. 2.39 in the expression for the current density at

the plane is modified as:

Eq. 2.40: ][))((

23 mA

LVVV

vJ oopC

Consider the point-to-plane configuration shown in Figure 2.9. To find the current

density distribution )(RJ p over the plane electrode one must find L(R) . Following the

example of Sigmond [20], if one assumes that the length of a field line ending at R is the

same in the space charge free and space charge dominated cases. For a hyperboloid to

plane geometry the field lines are ellipses with cos

d and R as the two half axis. To a

fair approximation the distance L along the field:

Eq. 2.41: ][)tan21(cos

22

22 mddRL

Replacing Eq. 2.41 in Eq. 2.40:

Eq. 2.42: ][)tan21())((

)( 223

23 m

Ad

VVVvJ oopC

The first experimental indication of the peculiarities of unipolar space charge saturated

ion drift was obtained by Warburg [20, 26, 29] in 1899. He reported that the current

density distribution )(CJ over the plane in stationary point-to-plane discharges closely

followed the formula

Eq. 2.43: omCC mAJJ 63],[cos)0()( 2

With m=4. 82 for positive corona, and m=4. 62 for negative corona. This so called

Warburg distribution has been amply confirmed. The value of m is usually set equal to 5

[20, 26, 29]. For greater than 63 the current density usually falls rather abruptly to

zero, indicating that field lines so far from the axis do not connect with the ionization

region [20].

44

For comparison, the Warburg distribution (2.43) can be written as [20]:

Eq. 2.44: ][)tan1)(0(cos)0()( 225

25

mAJJJ CCC

In Figure 2.10 the current ratio )0()(

C

C

JJ

as appears in Eq. 2.40 is plotted against , and

compared with the Warburg distribution Eq. 2.44, and with some recent measurements

for positive and negative coronas in ambient air. It is seen that Eq. 2.44 shows as good a

fit to the experimental data as the Warburg law does [20].

Taking the integral of Eq. 2.42 and Eq. 2.44 to calculate the current entering the plane,

one has that:

Eq. 2.45:

44.2])[0(09.242.2])[0(90.1

)( 22

EqAJdEqAJd

dSJIp

p

planeCC

Figure 2.9 Point-to-plane configuration geometry. This scheme is used to derive the unipolar conduction

current in the corona tube. Image taken from [20]

To the level of approximation here used, the difference is insignificant, and as a good

mnemonic one may state that the total corona current corresponds to the central current

density spread out over twice the squared point-to-plane distance [20] such that:

Eq. 2.46: ][)(2

)0(2 2 Ad

VVVJdI ooCC

Therefore, for a given corona configuration, Eq. 2.46 calculates the current due to

unipolar charge drift. The current in Eq. 2.46 has been confirmed experimentally by

45

choosing correct values for the mobility [26]. Lama and Gallo in [15] found

experimentally a relationship of 21

d for the current in the plane.

Figure 2.10 Radial current density distributions at the plane in unipolar point-to-plane coronas. Image

taken from [20]: Curve W: current ratio as appears in Eq. 2.45, the Warburg distribution. Curve S:

Present work, Eq. 119)Eq. 2.44, Curve NO: Negative pulseless glow in ambient air, d = 12, 13, and 14

mm, V = 22 kV (Kondo and Miyoshi, 1978).Points : Neg. Trichel, Points 0: Pos. Glow Ambient air, d =

120 mm, V = 40 kV (Goldman, et al. 1978"). The experimental points are normalized to I near the axis.

The negative Trichel pulse data show a dip in the center.

Typically for air, the ion mobility [17, 20, 26, 28]

Eq. 2.47: ][425.22

Vsmei

and for pure electron currents [20]:

Eq. 2.48: ][56.02

Vsm

e

Thus, phenomena involving electron drift exhibit a higher value of current rather than

ion drifting phenomena.

Taking into account the unipolar ion drifting assumptions that were taken to derive Eq.

46

2.46, it defines a close upper limit to the unipolar current in the given corona geometry.

If corona current in excess of this limit is measured, one may be absolutely certain that

either part of the ions are fast electrons, or part of our current is bipolar, or both. Free

electron current and streamers may very well exist in coronas below the saturation

current limit, but they must exist above it.

As long as the voltage gives too low fields to reduce electron attachment or to cause

streamers, the corona current will stay a unipolar ion one [20]. When the current is

increased, voltage levels are invariably reached which cause free electron conduction in

negative air coronas and streamer bipolar conduction in positive. Obviously, for

negative coronas electron mobility must be used and therefore, a higher current should

be measured.

Figure 2.11 Image taken from [20]: The continuous ('"background'") component I, of the corona current

vs. corona voltage, for a 13-mm point-to-plane gap in ambient air, compared with the unipolar saturation

current curves. Numbers along the curves indicate time averaged streamer currents. (Goldman and

Sigmond 1981). Points 0, solid curve: Regular streamer corona. Points *, dashed curve: Regular, periodic

streamer+spark corona. Curves S: Unipolar space-charge saturation current limits.

Figure 2.11 shows an illustration taken from [17, 20] where the continuous current pI is

plotted as function of the corona voltage for a 13-mm point-to plane in ambient air, with

the streamer current values noted along the curves. The unipolar current size and

47

distribution seem remarkably unaffected by the streamers present and do probably

dominate the field in which the streamers propagate.

The unipolar currents seem to flow also in coronas where the main current flow is by

bipolar streamer conduction [17, 20]. The unipolar conduction current in coronas is

often called the continuous or background current, as its oscillations are of high

frequencies and smeared out by the slow ion movement through the drift region [17].

2.5.4. Unipolar current in the cylindrical corona tube

As it has already been presented here, the cylindrical configuration is commonly

referred by authors [16-17, 20, 26-29] as a preferred geometry to analyze due to the fact

that the Laplacian electric field is easy to derive analytically, and that Eq. 2.29 has exact

solution. Here the unipolar current is calculated with an expression similar to Eq. 2.48:

Eq. 2.49: ])[()ln(

82 m

AVVV

rRR

I oo

sC

Eq. 2.49 is given as a surface current density due to the fact that the cylinder has a

length that must be taken into account. Therefore, larger cylinders give larger currents.

In Eq. 2.49 R is the outer cylinder radius, and r the inner cylinder radius. This equation

was derived by Townsend under the assumption of ions drifting with constant mobility

and a thin ionizing region.

As it can be seen the derived approximated expressions for unipolar current strongly

depend on the value used for the ion mobility. The representation of the drift region is

more complicated if the average ion mobility varies during the transit time of the ions

from the corona ionization region to the collecting electrode. The average mobility

might be subject to change because of a dependence upon the electric field strength,

because of the ion aging processes, which presumably could occur if the nature of the

ionic species present within the drift region changed significantly during the transit time

[27].

The unipolar assumption, which regards the corona region as being dominated by only

one type of charge, can be thought of as a physical extreme. The other physical extreme

is to regard the corona as charge free in the gap, i. e. same amount of positive charges

48

and negative charges drifting. Similar analyses as the one performed above can be done

for the bipolar case [27]. Nevertheless, only a few cases could apply.

2.6. Arc breakdown in the corona tube

When the voltage applied to the corona tube is raised further from the arcing limit, a

plasma channel is created between anode and cathode that is able to sustain a high

current value. At very large gap spacing, spark materializes from the so called

breakdown streamers. In positive corona, the spark occurs when the streamers are

capable of reaching the cathode by a retrograde ionizing wave as a result of an intensive

electron emission [28]. This is due to the high electric field in front of the space charge

of the streamer tips approaching the cathode; the electrons emitted are accelerated

toward the positive streamer front. If there is ample time and distance, they produce

large electron avalanches. If this wave is intensive enough, the current in the intensified

channel increases, becoming unstable until the total breakdown materializes [28]. A

similar process occurs for negative corona, where the retrograde ionizing wave rises

from the anode.

Therefore, the criterion for the complete breakdown or arc breakdown in either positive

or negative corona configurations is that the streamers bridge the anode and the cathode.

This means that the streamers nearby the surface of the secondary plane in point-to-

plane configuration or the cylinder in the cylindrical configuration, must be able to

propagate. If this happens, the background electric field must be approximately equal to

the propagation electric field for streamers.

The propagation electric field for streamers depends on the polarity of the drifting

charge. For positive corona this field is at NTP:

Eq. 2.50: ][500mkVEs

And for negative corona

Eq. 2.51: ][1000mkVEs

49

This values have been suggested everywhere in the literature (e. g. [17, 20, 27, 28]) and

by Les Renardieres Group and Cooray in [42]. Therefore, the Laplacian evaluation of

the field that gives a backgr