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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.
4
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
6
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
7
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
11
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
12
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
16
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