A Compact and Portable EMP Generator Based On Tesla Transformer Technology
A Compact and Portable EMP Generator Based On Tesla Transformer Technology
A Compact and Portable EMP Generator Based On Tesla Transformer Technology
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~il~~lli\iU~1I'11IIIIIIll'I11'11111
A Compact and Portable EMP Generator
by
July 2008
To
My Father
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude and to acknowledge the assistance and
continuous support received from my supervisor, Professor Ivor Smith. I am grateful for his
me to obtain Departmental funding, without which it would not have been possible for me
to undertake the research described in this thesis. I am also grateful to him for his reviews
Special thanks are due to Dr. Bucur Novac, who was a constant source of
inspiration, expert advice and assistance during my research work. I am very fortunate to
I am very grateful to Mr. lon Brister and Mr. Charles Greenwood for their technical
assistance in designing and constructing the experimental systems. Without their help none
of the systems described would have been built. Mr. Greenwood also deserves special
mention as he not only helped me with his technical inputs but also in various other matters
I am also grateful to Dr. Sean Braidwood for his assistance in the initial phase of the
research work and for his readiness to help me with his expert advice even after he returned
to Australia.
I would also like to thank Dr. Marko Istenic, Mr. Peter Senior, Dr. ling Luo and Mr.
Rajesh Kumar colleagues in the Pulsed Power Group at Loughborough University for
Finally, but most importantly, I would like to thank my wife Sweta, our son Saumil
and my entire family for their constant support and patience during this period.
ii
Abstract
ABSTRACT
electronic systems and modem bio-medical technology. In response to the current trend, a
simple, compact, and portable electromagnetic pulse (EMP) radiating source has been
rise-time pulses at voltages exceeding 0.5 MY. For this type of application pulsed
transformer technology offers a number of significant advantages over the use of a Marx
generator, e.g. design simplicity, compactness and cost effectiveness. The transformer is
operated in a dual resonance mode to achieve a high energy transfer efficiency, and
although the output voltage inevitably has a slower rise-time than that of a Marx generator,
this can be improved by the use of a pulse forming line in conjunction with a fast spark-gap
switch. The transformer design is best achieved using a filamentary modeling technique,
that takes full account of bulk skin and proximity effects and accurately predicts the self
One main objective of the present research was to achieve a high-average radiated
power, for which the radiator has to be operated at a high pulse repetition frequency (pRF),
with the key component for achieving this being the spark-gap switch in the primary circuit
of the pulsed transformer. Normally a spark-gap switch has a recovery time of about ten
milliseconds, and a PRF above 100 Hz is difficult to achieve unless certain special
techniques are employed. As the aim of the present study is to develop a compact system,
the use of a pump for providing a fluid flow between the electrodes of the spark gap is
iii
Abstract
ruled out, and a novel spark-gap switch was therefore developed based on the principle of
corona-stabilization.
sensor, with data being collected by a suitable fast digitizing oscilloscope. Post-numerical
processing of the collected data was necessary to remove the ground reflected wave effect.
Measurements of the radiated electric field at 10 m from the radiating element indicated a
Much of the work detailed in the thesis has already been presented in peer reviewed
iv
Contents
CONTENTS
ACKNOWLEDGMENTS .................................................................... ii
1. INTRODUCTION ..................................................................... 1
References .............................................................................. 7
2.3.4 Mutual inductance of two parallel and coaxial circular loops ........... 16
v
Contents
solver ................................................................... 25
..................................................................... 29
References ............................................................................ 47
vi
Contents
vii
Contents
viii
Contents
ix
List of Figures
LIST OF FIGURES
Figure 1.1 Block diagram for a direct switching system ................... ..... ......... 2
Figure 1.2 4-stage Mane generator ......... ......... ...... ... .......... ..... ... ... ... 3
Figure 1.3 Lumped circuit diagram for a Tesla transformer........ .......... .... .. ...... ... 4
Figure 2.12 Equivalent filamentary circuit as in Figure 2.10, but with secondary turn
................................................................................................ 38
x
List ofFigures
Figure 2.17 Schematic representation ofthe transformer (not to scale) ................... .40
Figure 2.20 Variation ofprimary inductance with the number offilaments ............... .42
Figure 2.24 Current distribution in single-turn primary with a secondary coil pitch of
. Figure 2.25 Current distribution in single-turn primary with a secondary coil pitch of
Figure 2.26 Maxwell 2D model of helically wound high-voltage pulse transformer ofFigure
Figure 3.1 Inductively coupled primary and secondary circuits of a Tesla transformer
................................................................................................ 56
Figure 3.2 Energy transfer efficiency and energy transfer time as afunction of coupling
values of Q ....................................................................................... 62
Figure 3.7 2D electric field plot of the transformer with metallic parts ............... 70
xi
List ofFigures
Figure 3.8 2D electric field plot of the transformer without metallic parts ............... 71
Figure 3.12 DC-DC converter and battery pack inside RF shielded box ............... 77
Figure 3.13 Protection circuit ofpower supply againstfull voltage reversal ...... 78
Figure 5.1 Equivalent circuit ofa generic voltage divider ................................. 108
Figure 5.6 Uncoupled primary circuit discharge voltage waveform ........................ 117
Figure 5.7 Determination ofmutual inductance between two coils ........................ 118
Figure 5.9 Arrangement for calibration of in-built capacitive voltage sensor using
xii
List ofFimres
Figure 5.15 Capacitor bank discharge voltage and secondary voltage: experimental and
Figure 5.17 Capacitor discharge voltage waveform, CS-SG, commercial switch ...... 129
Figure 5.19 Charging voltage waveform for a PRF of 1 kHz ................................. 131
Figure 5.20 Charging voltage waveform for a PRF of 1.25 kHz ........................ 132
Figure 5.21 Charging voltage waveformfora PRF of2 kHz ................................. 133
Figure 5.22 Variation of corona-current with time, in the burst mode ............... 134
Figure 5.24 Variation of corona-current with charging voltage at different SF6 pressure
Figure 5.25 Variation ofself-breakdown voltage with SF6 pressurefor CS-SG ...... 137
Figure 5.26 Charging voltage waveform with a PRF of 200 Hz, capacitor bank charged to
Figure 6.2 2D electricfieldplotfor PFL using Maxwell2D FEM software package ... 144
Figure 6.4 Testing of FSG for high pressure withstand ................................. 147
xiii
List of Figures
Figure 6.6 Layout ofTesla transformer, PFL and FSG ................................. 148
Figure 6.14 Typical voltage waveform obtained with a FCV divider ...... .................. 157
Figure 6.15 A typical output voltage waveformfrom Tesla transformer ............... 158
Figure 6.17 Arrangement for measurement ofradiated electric fields ............... 159
Figure 6.19 Radiatedfield waveform corrected for ground influence measured 10 m from
Figure 6.21 Radiatedfield waveform corrected for ground influence measured 15 m from
Figure 6.23 Radiated field waveform corrected for ground influence measured 20 m from
xiv
List of Figures
xv
List of Acronyms
LIST OF ACRONYMNS
CB Capacitor Bank
EM Electromagnetic
HV High-Voltage
MV Mega-Volt
MW Mega-Watt
PC Personal Computer
RF Radio Frequency
SW Switch
xvi
1.lntroduction
1. INTRODUCTION
Over the last few decades, there has been considerable progress in the development of
high peak power microwave and radio-frequency sources. Such sources are required in a
variety of applications that include transient radar [1.1], mine detection [1.2],
communication systems [1.3], industrial materials processing [lA] and modem bio-medical
field [1.5, 1.6, and 1.7]. The advantages of using wideband waveforms for radar include
better spatial resolution and target information recovery from reflected signals being
intercepted more easily than with narrowband signals [1.1]. A new and quite different area
in conjunction with chemotherapy [1.7], where it has been observed that the effectiveness
pulse (EMP) radiation, which raises the necessity of investigating the effect that is
produced on all these systems and also of designing suitably hardened equipment.
Sources capable of producing high peak power radio-frequency (RF) pulses can be
broadly classified into two categories, based respectively on: i) direct switching technology
field.
pulses generated by a fast discharge circuit, with the basic system elements required being
shown in Figure 1.1. The initial energy supply is a low power electrical source which
supplies energy to the pulsed power generator in continuous form. Here the energy is
1
1. Introduction
compressed in time and transformed to the higher power level suitable for supplying to the
radiating element.
The class of device based on bremsstrahlung radiation requires both a high vacuum
and (often) a strong external magnetic field (in addition to an electric field), in addition to
the fast discharge circuit used in direct switching technology. Magnetrons, magnetically
insulated line oscillators (MILOs), backward wave oscillators, and vircators all belong to
switching technology is much lower than one using bremsstrahlung radiation (due to
absence of the vacuum and the external magnetic field), which is extremely important when
focus of the work presented in this thesis is towards the design of a simple, compact and
portable source based on direct switching technology for i) single-shot operation and ii)
The high-voltage pulsed power generator represented by Figure 1.1 can be based on
either Marx generator or pulse transformer technology, both of which have inherent
advantages and disadvantages. Marx generators use an assembly of capacitors that are DC
charged in parallel and then switched into a series configuration, with the resulting output
2
1. Introduction
voltage across the load given by the product of the initial charging voltage and the number
RV R R R R
c c c
R R R R
Load
Figure 1.2. A 4-stage Marx generator.
increased voltage level being generated across a load [1.9]. The inherent advantages of this
arrangement in comparison with a Marx-type circuit, are its simplicity, compactness and
lower expense, due to the reduced number of components (in particular switches) that are
required. In addition, because of the increased number of components Marx generators can
suffer from increased losses and breakdown faults as the PRF is increased. The
although this can be mitigated by the use of a pulse forming line (PFL) and a spark-gap
Transformers used for the generation of extremely high voltages require by necessity
significant insulation both between the primary and secondary windings and within the
secondary winding, and high primary currents and fast pulses often preclude the use of
ferromagnetic materials in the transformer core. Under these conditions, achieving high
3
I. Introduction
magnetic coupling between the primary and secondary circuits is extremely difficult, and
generally speaking transformers operating with low magnetic coupling will have a poor
energy transfer efficiency. This problem is, however, alleviated by operating the
transformer in a pulsed resonant mode [1.10, 1.11], with tuned primary and secondary
circuits, when maximum energy is transferred to the load a few resonant half cycles
The use of coupled high frequency resonant circuits is essentially the basis of the so-
called Tesla transformer, named after Nikola Tesla who designed and built high-voltage
generators using such resonant techniques. A typical lumped circuit diagram for a Tesla
transformer is shown in Figure 1.3, and the important design features associated with these
transformers are
a) An air-core.
b) A high degree of insulation between the primary and secondary windings, resulting
switch Rs
Cs
Ls
4
1. Introduction
Due to the above reasons, pulse transformers are commonly preferred to Marx generators
for use in high-voltage pulsed power generation, with a block schematic for a possible EMP
Radiating Elements
(
~_~A.~ ______
,
Inverter + Capacitor PFL
Battery
Pack ~
HVPower
Supply
Bank +
Spark gap ~
Tesla
transformer ,. +
Fast
Antenna
switch Switch
In the work described in this thesis a pulse transformer was designed using a
filamentary modelling technique [1.12, 1.13], with the primary winding divided into an
assembly of square filaments taken along the current path and the secondary winding
considered as an assembly of filamentary rings (see Chapter 2). A set of first-order ordinary
differential equation for this arrangement can then be produced and solved numerically.
Due to the transient nature of the current, the effective inductances (self-inductances of the
primary and secondary windings, and the mutual inductance between them) will all be
different from their DC values, but they can be readily calculated by an energy method
using the filamentary currents previously obtained [1.14]. Chapter 2 deals with the
various pulsed power generators, and the performance of the transformer is critical from the
5
1. Introduction
point of view of the overall EMP generator performance. A general review of the Tesla
transformer together with design detail and construction of the transformer and its
As part of this work, repetitively operated spark-gap switches were required, with a
PRF extending to several hundred hertz, and capable of high-voltage switching. The
performance of the spark-gap dictates the system efficiency, and factors involved in the
repetitive operation of spark-gap switches and their design are presented in Chapter 4. A
Chapter 5 deals with all aspects of the measurement of the primary and the secondary
Design aspects of the radiating elements, viz. PFL, fast switch, and antenna, together
with measurements of the fast voltage pulse at the output of the fast switch and the radiated
Suggestions for possible future work are outlined in the conclusions in Chapter 7.
6
1. Introduction
References:
[1.2] C. E. Baum et a!., "JOLT: A highly directive, very intensive, impulse-like radiator,"
[1.3] F. J. Agee et al., "Ultra-wideband transmitter research," IEEE Trans. Plasma Sei.,
processing of materials", J. Phys. D: Appl. Phys. Vol. 34, pp. R55-R75, July 2001.
[1.8] E. Kuffel and W.S. Zaengl., High Voltage engineering, 2nd ed Oxford, Newnes, 1999
[1.9] W J Sarjeant and R E Dollinger, High-power electronics, Tab Books Inc., USA, 1989
[1.10] C. R. J. Hoffmann, "A Tesla transformer high-voltage generator," Rev. Sci. Instrum.,
[1.11] M. Denicolai, "Optimal performance for Tesla transformer," Rev. Sci. Instrum., vol.
73,no.9,pp.3332-3336,Sep.2002.
method for the analysis of current diffusion and heating in conductors in railguns and
homopolar generators", IEEE Trans. Magnetics, Vol. 25, No. I, pp.610-615, 1989.
7
I. Introduction
[1.14] J Luo, B M Novae, I R Smith and J Brown, "Fast and accurate two-dimensional
pp. 955-963,2005.
8
2. Computational Technique
2. COMPUTATIONAL TECHNIQUE
2.1 Introduction
High-voltage pulse transformers are key elements in pulsed power sources employed
in a wide range of activities, and they can be used in either a repetitive or a single-shot
mode· of operation. The forces acting on the conductors of the transformer may be
extremely high, due to the flow of several mega-amperes of current even for a very short
duration, and there may be undesirable conductor movement, as well as melting and
critical from the point of view of the overall transformer behaviour, and a knowledge of the
ohmic heating and the electromagnetic forces are necessary from the design point of view.
It is therefore important to have a numerical code that describes the basic physics involved
in the system, and which is able to predict accurately the overall performance and to
identify possible ways of improving the efficiency. In the absence of such a code, it would
resulting in considerable cost and a long development time. In the past numerical codes
have been developed to predict the beha~iour of the conductors, but they were very
complex and had to be run on large computers, leading to a situation where the cost of the
numerical process became comparable with that of an actual experimental programme. The
thesis utilises a recently developed simple but nonetheless accurate numerical code, that
9
2. Computational Technique
In the case of electromagnetics, the transient behaviour of both the electric current
and the magnetic field is governed by Maxwell' s equations. However, for low frequency
problems Maxwell's equation can be reduced to the diffusion equations, which are used to
describe the classical physics of electric current and magnetic field penetration into a
conductor. The diffusion equations can be solved using either classical numerical methods
or finite element methods, once the boundary conditions are specified. If however a system
can be modelled with lumped parameters, it is more convenient to approach the problem
used to solve electromagnetic (EM), thermal and dynamic problems. It is a very useful tool,
costing considerably less than advanced codes such as ANSYS®. The basic work on
filamentary modelling was reported by the Institute of Solid State Physics, Japan [2.1, 2.2]
and was carried forward by Loughborough University, where the technique was used in
various situations [2.3 - 2.9]. A simple and fast approach to determine the inductances of a
high-voltage air-cored transformer was described by J.Luo et. al. in [2.3]. This approach
was further developed to predict accurately the resistances of the transfornier winding
taking, account of skin and proximity effects and is discussed later in section 2.5.
assembly of ordinary differential equations. It can be applied to systems where the temporal
and spatial variation of the current distribution carry most of the information about the
system. The origin of the technique can be traced to Maxwell's work on calculating
10
2. Computational Technique
inductance [2.10]. With the passage of time, more complicated and refined theoretical
models have been developed to predict data more accurately and so match the experimental
data more closely. Examples of the accuracy that can be achieved are provided in [2.2]
which relates to megagauss magnetic field generation, [2.5] that deals with explosively
compression, with both skin and proximity effects being included in [2.1 I].
To obtain a filamentary model for a linear problem in which the direction of the
1
current can be assumed, the conductor is divided into an assembly of filaments taken along
the current path. The filaments obtained must be sufficiently small for the current
distribution in their cross sections to be regarded as uniform, i.e. the dimensions of each
filament must be much less than the equivalent skin depth. The number of filaments
calculating the parameters for a small number of filaments, and then repeating the process
with a progressively greater number, until the difference between successive calculations is
less than say I %. When an appropriate number is obtained, the ohmic resistance of each
filament is calculated from the cross-sectional area, the length of the filament and the
temperature dependent resistivity. The self inductance of each filament and the mutual
inductance between every possible filamentary pair can be calculated from the geometry
using well-known formulae, and the original EM problem is reduced to simple circuit
consideration. If the current in each filament is defined as a state variable, the circuit
equations can be written as a set of linear first-order differential equations (ODEs) that are
solved for the circuit currents. The set of equations thus formed takes the matrix form
[2.ll].
11
2. Computational Technique
LI
d M I'N] [11]
M ',I ML,I,' ....,, M',N, I, = [VI]
V, (2.1)
dt .. , ... . . . . . . . ..
[
MN,I M N,' LN IN VN
where N is the number of filaments, .!!... is the time derivative, /j (i = 1 .. N) is the current
dt
in the ith filament, Vj (i = 1 .. N) is the complete inductive voltage term in the circuit
containing the i th filament and Mj,j (i,j = 1 .. N) is the mutual inductance between the i th and
For the work presented in this thesis, movement of the conductors is not considered
(due to the low discharge current) as the device is a static system and the inductance matrix
is therefore constant throughout the duration of an experiment. Equation (2.1) can then be
written as
-I
L\ Ml,2 Ml,N
MN,I MN,2
or in abbreviated form
dI_ -1 .V
-- M (2.3)
dt
Equation (2.2) can be solved using either the Runge-Kutta or the Gear method, depending
on the stiffness of the equation. Any non-linear considerations that arise in the numerical
into an equivalent linear equation. The electric and magnetic field distribution, the energy
12
2. Computational Technique
deposited in the conductors, the electromagnetic forces between the conductors etc. can all
Many EM situations have rotational symmetry about one axis. This enables the
models developed to be defined in a cylindrical co-ordinate system (p, z, 8), with symmetry
maintained about the z-axis. In general, the effect of an arbitrary current path through the
Most device used in EM applications make use of coils in a circular form, (e.g.
helical flux compression generators, helically or spirally wound transformers etc.), which
can all be divided into convenient circular rings to generate a set of ODEs.
I3
2. Computational Technique
(2.4)
where
·2 4ap
k =---'-=-2- (2.5)
(a+p) +z'
and K(k) is the complete elliptic integral of the first kind, given by [2.13]
!!.
K(k) =J d<p (2.6)
o ~1-k'sin2<p
and E(k) is the complete elliptic integral of the second kind, given by [2.13]
",
E(k) = Nl-k'sin'<p·d<p (2.7)
o
The relation between the magnetic flux density B and magnetic vector potential A is
given by B= VxA, and the flux density can be specified as [2.12]
_Pol f
Bp(a,p,z)- z [ -K(k) + a' + p'"E(k)
+ z' ] (2.8)
2tr p-V(a+p)'+z2 (a-p) +z
and
P I z [ a 2 _ p2 _ Z' ]
B,(a,p.z) = 0 ~ K(k)+, 2 E(k) (2.9)
2tr (a+p)'+z' (a-p) +z
14
2. Computational Technique
The forces acting on a small loop of wire in a magnetic field are precisely like those
acting on an electric dipole in an electric field, and any electrical circuit may be considered
as a mesh of such loops. The force between two parallel coaxial loops of wire of radii a and
and
,.
f
F, = I,B/a,b,c) bdB = 27fbl,Bp(a,b,c) (2.11)
o
15
2. Computational Technique
Substituting into equations (2.10) and (2.11) from equations (2.8) and (2.9) gives:
Jlol·rb [ a'-b'-c' ]
Fp(a,b,c) = ~ K(k)+, ,E(k) (2.12)
(a+b)' +c' (a-b) +c
The coefficient of mutual inductance M12 between two current loops of infinitesimal
cross-sectional area is defined as the flux tP12 that is produced in circuit 1 by a unit current
where d~ is an incremental element of loop 1 and d is the distance between loop 1 and
loop 2. The above equation demonstrates the reciprocal property of mutual inductance i.e.
M21 = M12. For a circular loop the vector potential A is entirely in the 8-direction, hence it
has same value for all the elements of loop 2 and is parallel to each element. Thus from
equation (2.14)
(2.15)
(2.16)
where k, K, and E are defined by equations (2.5), (2.6) and (2.7) respectively. Though
equation (2.16) has been developed for conductors of infinitesimal cross-sectional area, it
16
2. Computational Technique
provides a good approximation in other cases when the mean radii of the loops are used in
the calculations.
The coefficient of self-inductance of a circuit element L can be defined as the flux <P
produced by unit current flowing in the element, and can be expressed as [2.12]
(2.17)
where A is the vector potential due to a unit current in the circuit, di and if are line co-
[2.14]
2
8rm 7 a ( In-+-
L=Jlorm [ In---+-- 8rm 1 )] (2.18)
a 4 8rm2 a 3
(2.19)
17
2. Computational Technique
L = lI~r 8r -.!.]
[In a+hm
(2.20)
.-om 2
where rm is the mean radius of the ring, and a and h are the radial and axial dimensions of
For a circular ribbon of radius rm and width h (axial length), the self-inductance can be
where
h
q=- (2.22)
4rm
Due to the slow rise-time of currents occurring in the high-voltage pulse transformers
considered in the thesis, the above low-frequency formulae were used when calculating self
inductances.
2.3.6 Resistance
Since the movement and temperature rise of the conductors are not considered in the
thesis (as the current in the high-voltage transformer is very low), only DC resistance
1 2Jrr
R =1'/(Pd,T)- =-1'/(Pd,T)
m
(2.23)
s s
where r m is the mean radius of circular loop of length 1, S is the conductor cross-sectional
area, and 1'/(Pd' T) is the resistivity of the conductor at a density Pd and temperature T.
18
2. Computational Technique
For completeness this section presents the properties of z-current circuits, despite the
Consider a rail gun, where the upper and lower rails can be divided into a number of
filaments of infinitesimally small cross-sectional area in such a way that each filament in
the upper rail has a corresponding one in the lower rail. The effect of the conductors
carrying a z-current distribution will be investigated, under the assumption that the
magnetic field is stationary and that the current is uniformly distributed throughout the
(2.24)
where Bo is the circular component of B, and r is the radius of a circular path at right angle
to the conductor, with its centre on the axis of the conductor.
19
2. Computational Technique
I "r(z) 1
L= f.lof f -dr·dz
21r 0 r
'r(z)
(2.25)
where I is the length and ir(Z) and °r(z) are the outer radius of the inner conductor and inner
radius of the outer conductor respectively. For a cylindrical structure the self-inductance
(2.26)
(2.27)
The mutual inductance between two identical parallel straight conductors of length I,
From equation (2.23) the resistance of a cylindrical conductor of length Iz, inner
20
2. Computational Technique
(2.29)
In EM problems, skin and proximity effects have a major influence on the system
performance, with both resistance and inductance being frequency dependent. The tendency
of any time-varying current is to flow mostly through the outer surface of the conductor and
this is termed skin effect. The depth of the conductor from the outer surface through which
the majority (63%) of the current flows [2.11) is termed the skin depth and is given by
(2.30)
where f.Io is the magnetic permeability of free space, ais the conductivity of the material of
the conductor, and f is the frequency of the time-varying current. Due to skin effect the
effective cross-sectional area of the conductor decreases, thereby increasing the resistance
The proximity effect also increases the effective resistance and is associated with the
magnetic coupling between two conductors which are close together. If each carries a
current in the same direction, the regions of the conductors in close proximity are cut by
more magnetic flux than the remote regions. As a result the current distribution is not
uniform throughout the cross-section, with a greater proportion being carried by the remote
region. If the 'currents are in opposite directions, the region in close proximity will carry the
greater density of current. The effect decreases as the distance between the conductors
increases and eventually, when the conductors are wide apart, the inductive coupling and
21
2. Computational Technique
Skin and proximity effect can both be modelled very accurately using the filamentary
There are various ways to determine the lumped inductance of a high-voltage pulse
Once the filamentary currents of a high-voltage pulse transformer have been obtained
by solving equation (2.2), the magnetic energy stored in all the filaments can be calculated
at any time during the capacitor bank discharge. The total stored energy in a winding is the
summation of the magnetic energy associated with each element, which is equal to the total
energy stored in the corresponding lumped component i.e. LI' . Thus the self-inductance
2
(2.31)
where, Np is the number of filaments, and Ij and Ij are the currents through i'h and t
filament respectively.
(2.32)
22
2. Computational Technique
where, N, = Np + N.
and the mutual inductance between the primary and secondary winding is
N, N
I I
1::1 j=N +1
MuIJ)
M= ' (2.33)
~ 1~
L./, (-L./,)
1::1 N i=1
Although the above method can predict accurately the self and mutual inductances,
even for dynamic cases which may arise due to current redistribution during a fast transient
solution viz. during early the stage of design, the modelling can be simplified by assuming
that the current is distributed uniformly in the winding i.e. all the filamentary currents are
equal. This eliminates the process of solving ordinary differential equations (ODEs), and
the inductances can be obtained by a simple summation, which is a much faster process.
The inductance can also be calculated using the formulae in [2.15], with the coil
assumed to be an equivalent current sheet. Thus the calculated value will inevitably be
inaccurate if the frequency is high and the current is not uniformly distributed. In addition,
the method is less accurate when calculating the inductance of a large loosely-wound coil.
calculated by equating the overall ohmic energy dissipated to the sum of the filamentary
losses as
23
2. Computational Technique
(2.34)
where R;DC and I; are the dc resistance and the current of the ith filamentary ring, n is the
mode (discussed in Chapter 3) affects both the resonant effects and the performance of the
of the winding and C is the inter-turn capacitance. The effect of a nearby conductor or the
ground is not taken into account and the model will fail to work when nearby conductors or
R L
c
Figure 2.3 Equivalent circuit of an inductor
metallic component. The distributed stray capacitance modelling can be achieved by use of
the Maxwell 2D electrostatic solver [2.17) to obtain the node-to-node lumped stray
24
2. Computational Technique
capacitance as in Figure 2.4, and this model can then be reduced to the form shown in
Ct3
t--f--II----'T---t / -
C;N.2)g
Ground
1 N
2.7.1 Stray capacitance calculation using Maxwe1l2D electrostatic field solver [2.17]
is modelled as a set of coaxial planar loops while a spirally wound coil is modelled as the
25
2. Computational Technique
Axi symmetric
boundary spiral-strip
(coil axis) winding
Open
boundary Open
boundary
mandrel
1·......._ _ Axisymmetric
boun (coil axis)
(a) (b)
Figure 2.6 Maxwe1l2D axisymmetric model (a) helical. and (b) spiral-strip type winding
The number of loops or rectangular strips is the number of turns of the winding, while the
distance between them i.e. the centre-to-centre distance, is the pitch of the winding. The
axis of the winding is treated as the axisymmetric boundary (for cylindrical coordinates),
while the other edges are treated as an open boundary, as shown in Figure 2.6. In
increase in the number of segments of the polygon increasing the accuracy of the result.
Since with any increase in the number of segments the distance between the two conductors
segments per circumference to be used in the model, which can be found in such a way that
even after doubling the number of segments the effective change in capacitance is less than
say 1.5%.
26
· 2. Computational Technique
Figure 2.7, and with the outside boundary taken as the reference, the net charge on each
conductor is
The capacitance matrix in equation (2.36) gives the relation between the charge Q
and potential V for n conductors and ground. If one volt is applied to conductor 1, and zero
volts to the other conductors, the above capacitance matrix reduces to the form
27
2. Computational Technique
The diagonal elements in the matrix (viz. C(1,I) are the sum of all the capacitances
from one conductor to all other conductors, these terms representing the self-capacitance of
the conductor. Each is equal to the charge on a conductor when one volt is applied to that
conductor and the other conductors including ground are set to zero volts.
The off-diagonal terms in each column are numerically equal to the charges induced
on other conductors in the system when one volt is applied to that conductor. For instance,
in column one of the above capacitance matrix, C(1.2) is equal to - C12. This is equal to the
charge induced on conductor 2 when one volt is applied to conductor I and zero volts are
applied to conductor 2. The terms are simply the negative values of the capacitances
between the corresponding conductors (the mutual capacitances). In column one of the
capacitance matrix the off-diagonal terms represent the capacitances between conductor I
and the other conductors; in column two they represent the capacitance between conductor
2 and the other conductors and so forth. It will be noted that the capacitance matrix is
symmetric about the diagonal, which indicates that the mutual effects between any two
of electrostatic field simulations. In each case, one volt is applied to a single conductor and
zero volts to all the other conductors. Therefore, for an n-conductor system, n field
The energy stored in the electric field associated with the capacitance between two
conductors is
28
2. Computational Technique
%=~fDI'EJdV
,
(2.37)
where the integration extends over the whole volume outside the conductors. Wij is the
energy in the electric field, associated with flux lines that connect charges on conductor i to
those on conductor j, Dj is the electric flux density associated with the case in which one
volt is placed on conductor i, and Ej is the electric field associated with the case in which
(2.38)
(2.39)
where the N x N admittance matrix Yij represents the node-to-node capacitances, and Ij and
Vj are the node currents and voltages respectively. In the above representation each turn of
the coil is denoted as a node. In an actual case, only the coupling of the !Ch turn with the (k-
2), (k-I), (k+I), and (k+2) turns are considered. All others are ignored, as their capacitance
is low.
_ Y"Y,x
Y.q-Yxx - - - (2.40)
Y"
29
2. Computational Technique
Y,I Y,(N-I) ]
where Y" = [
YN1 YN(N-n
1
2N
Y... ,and
YcN-I)N
Y2 (N-I) 1
l(N-I)(N-I)
Chapter 3) using the filamentary technique is discussed below for two types winding, i)
multi-turn primary winding of thin copper sheet, but here only a single-turn will be
considered. The single-turn primary winding is basically a hollow cylinder having a narrow
slot along its width (or length). The secondary winding is made from thin copper wire as
30
2. Computational Technique
frequency of operation is low the magnetic field diffuses easily through the thin winding. In
such situations, the primary winding can be represented by a single layer of filaments and
the secondary winding is assumed to be an assembly of rings as shown in Figure 2.9. If, the
effective skin depth is less than half of the conductor thickness, it is necessary to divide the
31
2. Computational Technique
Rss
o~ __________________ ~~ __________________ ~
z
e,
Figure 2.9 Filamentary representation of a helically wound high-voltage pulse
transformer, where rp is the outer radius ofprimary; tp is the thickness of primary winding;
lp is the width of primary copper strip, RSL and Rss is the largest and smallest radii of
secondary winding; superscripts p and s are for primary and secondary windings,
If the coil is closely wound, i.e. if there is little spacing between any two adjacent
turns, proximity effect cannot be neglected, in such a case the secondary winding has to be
further divided into a number of filaments. Another consideration for the number of
filaments in a conductor is the number along its width (lp as in Figure 2.9), with the number
chosen being such that, even after increasing it by 10%, the net change in inductance is less
than 1%. Also if the coil is symmetric about its horizontal centre plane, i.e. perpendicular to
voltage transformer of Figure 2.9 is shown in Figure 2.10, where the secondary turns are
32
2. Computational Technique
considered as an assembly of rings and there is only one current path in the, secondary
RJl2
Is Rs,
Ip'
RPNp Mp-s
IJ\rp
Secondary Circuit
Ip Rb Lb Cp
Primary circuit
Figure 2.10 Equivalent Jilamentary circuit of Figure 2.9, p and s subscripts are for primary
and secondary circuits, Rb and Lb are the spark-gap switch and capacitor bank inductance
and resistance, and RI and LI are the load resistance and inductance respectively.
The primary winding is divided into Np filaments, where Np = np X mp i.e. the product of
the number of rows and the number of columns (as shown in Figure 2.9). The helical
number of filaments in the model is N, = Np + N,. From Figure 2.10 a set of first order
(2.41)
where, i =1 ... Np
d1 N N dIN dIQ
T _ ' +RI + ~ Rs I + ~ M~-,-j + ~ M~-p-j +-' =0
'"'ld
. t " L
k-.N,,+I
k' L
l:Np+1
'I dt L1",1 ij d t C$ (2.42)
where, (i = Np + 1)
33
2. Computational Technique
dQp
-=1 (2.43)
dt p
and
dQ, =1 (2.44)
dt '
In the above equations, Vo is the initial voltage of the primary capacitor Cp , and Qp is its
charge. ifS denotes the mutual inductance between the filaments in the primary and
secondary windings. Mi/ is the self-inductance of the lh filament. The primary current,
Ip = t
w
I, , the primary and secondary filamentary resistances are given by Rp, = 2i'Z'r,
~
and
Rs. = 2i'Z'r. respectively, where rl and rk are the radii of ;'h and kfh filament in the primary
aa,
2
I·t· i'Z'd
p p and a, =__ 0 • The mutual inductance between two filaments separated by an
np·mp 4
axial distance dij is calculated using equation (2.16) while the self-inductance can be
calculated using equation (2.19) to (2.21), depending on the cross-section of the conductor.
Case II: A detailed approach. The primary winding has a similar representation to
Figure 2.11, and the current distribution in each turn is different, with the current density
being the same for all turns. In the equivalent filamentary circuit diagram of Figure 2.12 the
primary winding is divided into Np filaments, as above, and each secondary turn is divided
into q filaments, Ns being the number of turns. The total number of filaments in the
From Figure 2.12 a set of first order differential equations can be written as
34
2. Computational Technique
R Ls
IS(N .. l) 2 ~.-l)q+l (N.-l)q+2
••••
·••• ·•••
Ls~
Is
1st turn IS 2q 2nd turn N •. q N,th turn
Ip Rb Lb ~ Is RI
"
Primary circuit Secondary Cirruit
Figure 2.12 Equivalent filamentary circuit as in Figure 2.10, but with secondary turn
wherei=2 ... Np
(2.47)
(2.48)
35
2. Computational Technique
(2.49)
dQp
-=1 (2.50)
dt p
dQ, =1 (2.51)
dt '
a multi-turn secondary. In some cases the primary turn is also part of the outermost
-
HV
input
}
-
Figure 2.13 Spirally wound high-voltage autotransformer.
36
2. Computational Technique
Figure 2.14 A spiral-strip type high-voltage pulse transformer (only secondary winding is
shown in Figure 2.15, and the corresponding equivalent electric circuits for the primary and
lowl!f r
37
2. Computational Technique
, , ,
Ip , Rp, Lp, Rs' , Ls, LSN~
Is' Rs,
'Y " ...... ....
Rp,
7 Mp" , , , ,
Ls, Ls Ns
Is' Rs, Rsi Ls, RS~f
lp, Mp' ' \ Lp, ••••
•;N"\. ~2Np ••
Ip~
Rp~ .1
Lp~
i'J
M p. s Rs,'"
.
Ls, ..
••
. . ..
LS Nf
IS' ~~~~
Rs, !:'!~- •••• !-~N'
"
1.r 11
IP Rb Lb Cp 11
Is R, L, Cs
Primary circuit
Secondaty Cirruit
From Figure 2.14 the circuit equation for the filaments can be written as
(2.52)
d1 N, N, dI N, d1 Q
L - ' +R1 + '" Rs'I+ '" MN_f - '" M,-p_f +-' =0
'dt /, L...',
k=1
L...
J=Np+1
if dt L...
}=I
if d t es (2.53)
where,(i=Np+1....N,)
dQp
-=1 (2.54)
dt p
dQ, =1 (2.55)
dt '
Most of the terms are defined in equations (2.40) ..., (2.43), except for N, = Np + N, and
N = ns x ms x N" where Ns is the number of secondary turns, and ns and ms are the number
N,
of rows and columns respectively. The total secondary current is given as I, = 2· L I,
i=Np+l
38
2. Computational Technique
N,
and the primary current is 1p =2· L I, ; the cross-sectional area of a primary filament is
;:::1
the coil is symmetric about its horizontal plane (as shown in Figure 2.13), only the upper
filamentary currents are ne~ded in the model, which reduces the computation time.
However, when calculating the mutual inductances both the upper and lower part must be
considered.
Based on the computation model detailed in section 2.8, several program routines
were presented to represent the transformer windings and these were written in Mathcad, a
A unique feature of the present 2D code is its ability to predict correctly the
resistance (and also inductance) of any pulsed power device during transient conditions i.e.,
it includes skin and proximity effects as discussed earlier. To illustrate this feature of the
present code some examples are given here. The fictitious transformer shown in Figure
2.17 will be considered, with the single-turn copper sheet primary having a radius of
43 mm, a length of 60 mm and a thickness of 0.1 mm. The secondary winding has a plastic
cylindrical former with a mean diameter of 80 mm, and it is wound with 0.3 mm copper
wire and has 4 turns. For purpose of illustration, results for a few cases are presented for
39
2. Computational Technique
tp = O.lmm
Dp=86mm
Ds=8Omm
i·.,"
d=O.3mm-4
The filamentary circuit representation of the transformer is shown in Figure 2.l2. The
circuit of Figure 2.1S is used to analyse the performance of the transformer. The circuit on
the primary side consists ofa capacitor Cp of300 nF, a spark-gap switch and the single-turn
connected to the output terminals. The primary capacitor is charged to a certain voltage and
The self-inductances of the primary and secondary windings, together with their
mutual inductances, were calculated after solving equations (2.46) - (2.52), and the results
are compared in Table 2.1. The calculations were for (SOx 10) filaments for the primary and
14S filaments representing each secondary turn, giving a total of 1392 filaments. The
variation of inductance with the number of filaments in the primary winding is shown in
Figure 2.20. It can be noted that with an increase in the number of filaments from SO to 140
40
2. Computational Technique
s Rp-O.lO R,-2.0n
1 p-300nF
M
Ls
Figure 2.18 Test circuit for the transformer, Cp primary capacitor, Lb inductance of
Secondary inductance
Mutual inductance
41
2. Computational Technique
The effective resistance of the secondary winding during discharge of the capacitor is
calculated using equation (2.34), with the time varying ratio of this effective resistance to
the dc resistance of the coil being shown in Figure 2.21 for various pitches, Figure 2.22
showing the corresponding time varying primary current. The current distribution in the
secondary winding is shown in Figure 2.23 for the I sI turn only and the corresponding
110
100
t
\
80 ~ r-t-...
42
2. Computational Technique
10
,.,.
g
~
~
..
'!:l
"
.g
-"
~
~
6
.~ 4
.." ,
\\;'
."
-:."
:'l
u I\... /""'
"'
~
2
"':":r:t~ ..-~ ;':4·:'V.""~'::'::':': ... _~106''.1;''''".l':.l.'' :,;,".1;".1;".1;.... ".1;"::.".:.":.:. 1.t~1.:t:::.:::.:.l.:.~.
o 05 1 15 25
time (iJ.s)
Figure 2.21 Time varying normalised secondary coil resistance, solid line pitch 0.6 mm,
If r\\ 1/\
/ 1\ /
V
\/
-=
43
2. Computational Technique
(a) (b)
Figure 2.23 Current distribution in the secondary conductor with a pitch of 0.6 mm (a) at
(a) (b)
Figure 2.24 Current distribution in single-turn primary with a secondary coil pitch of
0.6 mm; tp is thickness and lp is length of single turn (a) at 20 ns and (b) at 220 ns,
The reason for such a high resistance of the secondary coil at the beginning of the
transient, is clear from Figure 2.23 (a). Initially the magnetic field is unable to diffuse into
44
2. Computational Technique
the conductor and most of the current flows through the outer edges. Later on, due to
diffusion of the magnetic field, the current distribution evens out as seen in Figure 2.23 (b)
which results in a fall in the effective resistance. The effect of temperature rise is not
considered in the above case. The inductive effect of the secondary winding is clearly
reflected in the primary current distribution. As the pitch of the secondary coil is quite
small (0.6 mm), i.e. comparable to the diameter, there is only one 'hump' in the primary
current distribution of Figure 2.24. For a secondary coil pitch of 10 mm, the reflections of
all four turns in the primary current distribution are clearly visible, in Figure 2.25.
Figure 2.25 Current distribution in single-turn primary. with a secondary coil pitch of
IOmm.
Figure 2.26 in Maxwell 2D. The round conductors of the secondary winding are modelled
as polygons with 36 segments per circumference, and the primary turn is considered as
ground in the capacitance matrix. With all these considerations the effective secondary
45
2. Computational Technique
5.5 pF.
single-turn primary
cylindrical
former
axis of
helical winding
~---.....I
Figure 2.26 Maxwell 2D model of helically wound high-voltage pulse transformer of
Figure 2.17
2.10 Summary
technique, where the conductor is divided into small filaments taken along the current path.
The various inductances and resistances can be calculated accurately using the energy
method. The distributed stray capacitance of the transformer is modelled using Maxwell
2D, the axisymmetric electrostatic solver, and the lumped stray capacitance is calculated by
the node reduction method of section 2.7.1.2. It will be shown later that the dynamic
I
performance of the high-voltage pulse transformer has been successfully predicted using
filamentary modelling.
46
2. Computational Technique
References:
[2.3] J Luo, B M Novae, I R Smith and J Brown, "Fast and accurate two-dimensional
[2.4] B M Novae, I R Smith and M Enache, "Accurate modeling of the proximity effect in
helical flux-compression generators", IEEE Trans. Plasma ScL, vol. 28, pp.l353-1355,
2000.
[2.5] B M Novae, I R Smith, M Enache and H R Stewardson, "Simple 20 model for helical
flux-compression generators", Laser and Particle Beams, vol. 15, pp. 379-395,1997.
capacitor discharge induction launcher model", IEEE Trans. Magnetics, vol. 31, pp .. 599-
603, 1995.
[2.7] B M Novae, I R Smith, M C Enache and P Senior, "Studies of a very high efficiency
compression in a-pinch geometry", IEEE Trans. Plasma SeL, vol. 32, pp.l896-1901, 2004.
.electromagnetic action through metallic shields", IEEE Trans. Magnetics, vol. 39, pp. 305-
309,2003.
47
2. Computational Technique
method for the analysis of current diffusion and heating in conductors in rail guns and
[2.12] W R Smythe, Static and dynamic electricity, McGraw-Hill Book Co. 3rd Edition,
1968.
tables, edited by Milton Abramowitz and Irene A. Stegun., New York, Dover, 1965
Technica, 1958
[2.16] S Mei and Y I Ismail, "Skin and proximity effects with reduced realizable RL
circuits", IEEE Trans. VLSI Systems, Vol. 12, No. 4, pp. 437-447, April 2004.
using the finite element method", IEEE Trans. Electromagnetic Compatibility, vol. 43, pp.
88-93, 2001.
48
3. Tesla Transformer Design
3.1 Introduction
As stated in Chapter 1, a Tesla transformer was selected as the means for driving the
pulsed power generator of Figure 1.1. Since the overall performance of the proposed
generator is critically dependent on the performance of the Tesla transformer, every aspect
of the design and the associated parameters is reviewed in this Chapter. A brief account of
presentation of the circuit theory of the transformer in section 3.3, with various design
considerations being presented in section 3.4. Section 3.5 contains a description of the
Over the years, the design and construction of high-voltage generators has presented
a major challenge for scientists. The pioneering work in the field of high-voltage generator
is due to Nikola Tesla (1856 - 1943), the holder of more than 100 patents, who was born in
Smiljan, on the Austria-Hungary border and studied in Austria before moving to Paris.
There he worked for Continental Edison Company, where he invented the induction motor.
current machines, fluorescent lights, lightning discharges, and electrical resonance. He also
performed experiments intended to develop a system for the wireless transfer of electrical
power between remote sites. Among his inventions, the resonance transformer entitled
"Apparatus for transmitting electrical energy" in his patent [3.1] is better known these days
Tesla transformers have been in use since the early 20th century. As the damped high
frequency oscillation of the transformer voltage output is similar to the typical switching
49
3. Tesla Transformer Design
and arcing transients in power system, they are widely used as a voltage source for the
routine testing of ceramic insulators for puncture withstand. They have subsequently found
of insulators [3.2], the generation of high-voltage pulses for use in accelerators [3.3], the
study of the physics of lightning discharges [3.4] and in the film industry for producing
special effects.
In addition to its many practical applications the Tesla transformer is widely used by
hobbyists.
The pioneering work of Tesla is well reported in a series of his patents [3.1, 3.5, and
3.6]. In [3.1] he mentioned the use of an air-core transformer to generate high voltage at
high frequency for the production of light. The ensuing patent [3.5] presents an improved
version of the circuitry described in [3.1]. The first mention of the use of resonance
appeared in patent [3.6], which was devoted to the transfer of electrical energy to lamps and
Oberbeck [3.7] developed a mathematical model for the Tesla transformer, in which
the device was considered as two coupled circuits tuned to resonate at the same frequency.
Subsequently, Drude [3.8] showed that the maximum output voltage is achieved in the
secondary circuit by tuning the primary and secondary circuits to resonate at the same
frequency and with a coupling coefficient of 0.6. The practical approach of Terman [3.9]
relates the energy transfer efficiency to both the quality factor and the coupling coefficient.
In a more generalised approach than that of Drude [3.8], Finkelstein [3.1 0] presented
the dual resonance condition under which complete energy transfer from the primary circuit
50
3. Tesla Transformer Design
to the secondary circuit takes place. He went on to show that for any coupling coefficient
other than 0.6, the instant at which the complete energy transfer takes place is delayed. The
design he proposed used a spiral-strip type secondary winding wound on an acrylic former,
with the inter-turn insulation provided by multi-layered Mylar. Mylar film was also used to
insulate the single-turn primary winding from the multi-turn secondary winding. De-ionised
water insulated the transformer within its housing, thereby enabling it to withstand an
output voltage up to 1 MV. The transformer was used with a variety of loads, including
exploding wires and high-voltage water filled capacitors and also to test a water-filled spark
gap ..
transformers to arive accelerators having a beam output in the range 0.5-5 MeV. The
primary circuit of the transformers consisted of several turns of metallic strip shaped into a
conical helix, and the secondary windings had several hundred turns wound in a single
layer, with the voltage between turns being less than 3 kV. The parameters were selected to
provide a voltage step-up ratio of lOO-ISO, thereby achieving a MV output with an initial
charge of 10-50 kV on the primary capacitor bank. Dual resonance operation was employed
for maximum efficiency, with the resonant frequencies being several tens of kilohertz.
Hoffmann [3.11] used a secondary winding of 1130 turns wound in a single layer on
a Lucite former, with the conically shaped primary winding having 4 turns of aluminium
strip. The overall coupling coefficient was 0.37 and the design reduced both the capacitive
coupling and the electric field stress between the primary and secondary windings. The
transformer windings were housed in a metallic vessel filled with SF6 and, with an initial
voltage of 13 kV on the primary winding, voltage levels of 1.5 MV were measured at the
51
3. Testa Transformer Design
secondary terminals. It was noted that the conductive walls of the vessel had an effect on
the value of the non-shielded circuit parameters, resulting in a slight reduction in both the
Boscolo [3.12] used a similar design to Hoffmann, but with the transformer mounted
in a metallic vessel filled with N2 (or a SF6-N2 mixture) at a pressure of 25 bar. The load
capacitance was a meter long coaxial pulse forming line (PFL) of 200 pF capacitance, with
the walls of the transformer housing forming the outer conductor of the PFL. That part of
the inner conductor of the PFL located near the transformer was made from wire to negate
Rohwein [3.13] employed a robust design for the transformer that was used to drive
the TRACE I electron beam generator at Sandia National Laboratory. The primary
capacitance was a 14.5 ftF capacitor and the load was a 8.3 nF water-filled PFL. The
coupling coefficient of 0.55 between the primary and secondary windings gave a maximum
voltage at the PFL of 565 kV for an initial voltage of 20 kV. Rohwein developed a
shielding technique that reduced the field enhancement along the edges of the windings and
thereby prevented damage from discharges due to corona or breakdown. Another design
[3.14] which also had concentric ring cages built into both the core and the case of the
transformer to shape the electric field near the margins of the secondary coil achieved an
output of3 MV. The pulse energy was high, and with an initial voltage of 100 kV (±50 kV)
on the primary capacitor bank the stored energy was about 4.6 kJ. The transformer was
tested in two modes of operation, an off-resonance mode, and a dual resonance mode. In
the off-resonance mode with a 1.1 nF load, an initial voltage of 100 kV resulted in a
maximum load voltage of 2.2 MV with an energy transfer efficiency of 58%. Dual
52
3. Testa Transformer Design
resonance was achieved with a 0.76 nF load, when the maximum voltage obtained was
3 MY and the energy transfer efficiency was 91 %. A novel feature of the design was that a
multi-channel rail-gap switch connected the primary capacitor bank to the transformer
circuit.
Cook and Reganito [3.15] designed a transformer that operated in the autotransformer
mode. The secondary was spirally wound, with the driving point positioned at the junction
of the primary and secondary windings. It was used to charge a 250 kV water-filled
Blumlein system but, as the switch connecting the primary capacitor with the transformer
was unable to handle any reverse current flow, operation needed to be in an off-resonance
mode in order to optimise the peak voltage step-up across the load.
Reed [3.16] showed that a peak voltage increase of about 18% over that obtained
under conditions of maximum efficiency (i.e. tuning the primary and secondary circuits to
resonate at the same frequency and a coupling coefficient of 0.6) could be achieved by
employing off-resonance tuning with a suitable coupling coefficient. This analysis was
further generalised by Phung [3.17], who provided a set of equations that enabled all tuning
Bieniosek [3.18] presented an analysis for a triple resonance transformer circuit i.e.
three circuit in resonance, used to improve the output efficiency of the MEDEA II electron
accelerator [3.19]. The stray capacitance of the spiral secondary winding was comparable to
that of the load capacitance, and as a consequence a significant quantity of energy remained
stored in the stray-capacitance and was not delivered to the load. With the addition of a
suitable inductor between the transformer secondary winding and the load, the circuit was
made to operate in a triple resonance mode, thereby increasing the energy transfer
53
3. Tesla Transformer Design
efficiency to the load. Due to this design change, there was a reduction in the peak voltage
across the secondary winding. De Queiroz [3.24] further extended the analysis of
A compact and repetitive Tesla transformer based pulsed accelerator was developed
separately at the Institute of Eiectrophysics (IEP) and the Institute of High Current
Electronics (IHCE) [3.20, 3.21], both in Russia. The pulsed power source of these
forming line with an open steel magnetic core to increase the coupling coefficient to almost
unity. The secondary winding was wound on a conical former and the voltage step-up ratio
was very high. This ensured that the charging voltage of the primary capacitor bank could
be less than a kilo-volt, enabling low voltage capacitors and switches to be used in the
primary circuit, which helped to reduce the cost and also in solving problems related to
. 350 kV and operating repetitively using a pressurised hydrogen spark-gap switch. The
primary winding was of conical form, so that it was remote from the high-voltage output
end of the secondary and the capacitive loading and voltage stress between the output
terminal and the primary winding were minimised. The entire assembly was inside a
Korioth [3.23] proposed a design with a 'super low' inductance primary (SLIP), for
use in dual resonant transformers, with the aim of making the primary set-up of capacitor
bank and switch compact and of low inductance. One unit used two 6" long, 8" diameter
coaxial cylinders, with twelve 2 nF primary capacitors and a hydrogen spark-gap switch
54
3. Tesla Transformer Design
placed between a slit in the conductive cylinders. The inductance was 200 nH and the
to tune the primary and the secondary coil to resonate at same frequency and then to
increase the coupling coefficient to 0.6. Also, in order to minimise the loss it is essential for
the maximum voltage at the secondary to be obtained in the shortest possible time.
The basic Tesla transformer can be regarded as the two inductively coupled damped
resonant circuits [3.25] as shown in Figure 3.1, where subscripts P and S identifY the
respectively and M denotes the mutual inductance between the two circuits. The resistances
represent the resistive loss in the circuits, which in practice is mainly the time dependent
loss in the primary circuit due to the spark-gap switch. A more accurate approach would be
the stray-capacitance of the secondary winding. However, since the load capacitance is
generally sufficient to lower the oscillation (LC) frequency to well below the self-resonance
figure associated with the distributed reactance of the unloaded winding, the· lumped
parameter assumption can predict the transformer performance with sufficient accuracy for
Applying Kirchhoff's law to the circuit of Figure 3.1 with the switch closed, for the
primary circuit
f·
I di d'
- Ip dt+ R'
plp+ Lp -p+ M -I,= 0 (3.\)
Cp dt dt
55
3. Tesla Transformer Design
f
- 1 idt+Ri +L -'+M---L=O
di di (3.2)
C , ' " 'dt dt
switch Rs
r- ep Lp
•
M •
Ls
~
Figure 3.i. inductively coupled primary and secondary circuits ofa Tesla transformer
. dqp.,
I =-- (3.3)
p,' dt
q dq d 2q d2
--1!...+R -p +L --p +M~=O (3.4)
Cp p dt p dt 2 dt 2
(3.5)
(3.6)
-1]
[D 2+-R,L, D +L,C,'
- M D2
+- q q =0 (3.7)
L, p
56
3. Testa Transformer Design
D2 + Rp D + m2 ]q +kft'D 2q =0 (3.8)
[ L P P L '
P P
(3.9)
where
k= M (3.10)
~LpL,
and
m; = (LpC pt (3.11)
m; =(L,C, r'
liP and ms are the angular resonance frequencies of the uncoupled primary and secondary
circuits, and k is the coupling coefficient (0 < k < I). Equations (3.8) and (3.9) can be
solved numerically (by a Runge Kutta method), subject to the initial conditions that at t =
0, qp = qo, qs = 0, and Dqp = Dqs = 0, where qo is the initial charge on the primary
capacitor. We will consider two special cases (i) in which the resistive loss in the circuit is
neglected i.e. Rp = Rs = 0 and a complete analytical solution is possible and (ii) the primary
and secondary resonant frequencies are matched liP = ms and, a low-loss approximate
Even though the lossless circuit is impractical, the analysis provides the maximum limit to
the actual transformer performance. Substituting Rp = Rs = 0 into equations (3.8) and (3.9)
57
3. Tesla Transformer Design
v:; (t) = v.
ft
'~
Lp
2k
(l-T)'+4k'T
sin (w p +w)
2
't x sin (w' -wP t )
2
(3.12)
, 0)'
where, Tisthe tuning ratio T=---f [3.11] and
0),
(l+T)-~(I-T)' +4k'T
2(1- k')
(3.13)
(1+T)+~(l-T)' +4k'T
2(l-k')
where, wp and w, are the angular resonant frequencies of the primary and secondary circuits
when coupled. Their value are always real and clearly w, > wp. From equation (3.12) it can
Wp+Ws
be seen that the secondary voltage has a high frequency oscillation --"--"- which is
2
W -w
amplitude modulated by a second but lower frequency oscillation' p.
2
Equation (3.12) shows that the maximum voltage across the secondary capacitance
can be expressed as
v, Il:. 2k
'VLp ~(l-T)'+4k'T
which can be achieved only if both the sine terms in equation (3.12) are simultaneously
(Wp +w,)t=(2n+I)7!'
(3.14)
(w, -wp)t = (2m+I)7!'
where n and m are integers. The earliest time that the maximum secondary voltage will
occur is when m = 0 and thus t = 7!'/ (w,- wp); substituting and rearranging equation (3.14)
then yields
58
3. Tesla Transformer Design
(3.15)
where Vmax is the maximum voltage achieved across the secondary capacitance during the
time period t :5: ,,/ (ws - wp.!. Satisfying the condition of equation (3.14) gives
4k'T
17 (1- T)' + 4k'T
The complete energy transfer takes place, i.e. 17 = 1, when T = 1 or the resonant
frequencies are matched. The maximum voltage across the secondary capacitance is then
V; = v,
,
-, when . 1+2n
JE
Lp
k -:-"--'---,-
2n +2n+1
coefficient k, equation (3.12) was used with T = 1 (matched resonance),to produce the
It is evident from Figure 3.2 that with a coupling coefficient of less than 0.6, the
energy transfer time is delayed, as shown by Finkelstein [3.9]. For a coupling coefficient of
0.43 < k < 0.53, the performance is poor, with almost 13% of the energy remaining in the
primary circuit for T = 1. There are other discrete values of k such as 0.385, 0.28, 0.22, etc.
59
3. Tesla Transformer Design
for which complete energy transfer takes place if T = 1 [3.25]. When corresponding results
are plotted for T# 1.0 the performance of the transformer becomes even worse. Hence it is
clearly desirable to design a Tesla transformer with a matched resonance and to have a
.. .....
••
· •
••
~.V
.- ~
1
i'/ V
0.8
....... . . ..,
•
·
"'"I'v-" T=l
0.8
••
•• T m O.8 ,,- ".
..,.
/,
""
<1
..../ 0
",
l..../"
~
••
• ....... . ........ 0.6
.."
~
... 'il
...,.
•• ~
••• II
••
••,
0.4 · ,, 0.4
~ ... ..... .... .. ,
......
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
coupling coeffICient
Figure 3.2. Energy transfer efficiency and energy transfer time as a jUnction of coupling
will have losses in both the primary and secondary circuits. Ohmic resistance losses will be
60
3. Tesla Transformer Design
greatly enhanced by skin and proximity effects as the resonant frequency increases. In
addition, the dielectric loss in the capacitors will increase with the increase in frequency
which will appear as an effective series resistance. Perhaps the major contribution will
With the quality factor of the primary and secondary circuits defined as [3.26]
Q = (i)pLp
p Rp
(3.17)
Q, = (i),L,
R,
a simple expression for the secondary voltage across the secondary capacitance for a
V(t)= Vo
, 2 V~e(-j,)[cos(~)-cos(~)]
Lp ..)I-k .JI+k
(3.18)
2
where T
4QpQ,(l-k ) the d ampmg
--''''-'-'----'-, .. time constant. T h··· . .
IS approximatIOn IS qUite accurate
(i)(Qp +Q,)
fork<0.6andQ> 10.
Using equation (3.18), the variation of energy transfer efficiency with coupling
coefficient was plotted for different values of quality factor. It is evident from Figure 3.3
that the efficiency decreases with a decrease in the quality factor, with the decrease in
efficiency being greater for lower values of the coupling coefficient. Hence it is desirable to
design a transformer with matched resonance, coupling coefficient of about 0.6, and a Qp of
61
3. Testa Transformer Design
.. , .....
0.9 .'
.. ...... . ...... .
.......... .' " Q=IOO·. _~ -
0.8
.......... ...
/""
",.- -- "'"'""-.
"
....
",.
... ,/
/
/
.1j
El
~
0.7
- ,/ /"" --- "- ,/ Q = 50 "\. V'" , /
-.
"
/
0.6
/ Q=20
~.-'
0.5
Figure 3. 3. Variation of energy transfer efficiency with the coupling coefficient for various
The design specification ofthe transformer to drive the EMP radiator is as follows:
transformer secondary circuit, and as a result the complete system will be bulky.
62
3. Tesla Transformer Design
3) Voltage gain ~ IS. In order to minimise the electrical stress on the primary side it
is desirable that the charging voltage of the primary circuit should be less than
30kV.
5) Charging time of secondary load capacitance < 500 ns. This is to assist the
duration [3.24].
To achieve the above design specification, it is desirable to operate the Tesla transformer in
a dual-resonant mode with a coupling coefficient of about 0.6 and a matched resonance
circuit i.e.
I I
--.====/l}
~L,C, '
For a dual resonant mode, the maximum secondary voltage appears during the second
peak of the voltage waveform, with the charging time for the load capacitance being the
time from the instant of closure ofthe primary spark-gap switch closure to the second peak
of the secondary voltage waveform. The design specification requires the charging time to
be < 500 ns, and the matched resonant frequency should therefore be more than 2 MHz.
A set of circuit parameters for the Tesla transformer that satisfies the above
63
3. Testa Transformer Design
For an ideal case, with 100% energy transfer efficiency, the maximum voltage gain VG is,
and to achieve a voltage of 500 kV across the secondary, the primary capacitor needs to be
charged to 25 kV. In an actual circuit there will be resistive losses arising from the effective
circuit resistance although other significant contribution are due to the discharge action of
the spark-switch in the primary circuit, and the dielectric loss of the capacitors. These loss
mechanisms provide the effective resistance terms in equation (3.17), and from Figure 3.3
it can be concluded that a quality factor of 20 or higher is desirable for the primary circuit.
The inductance calculations for the Tesla transformer were performed using the
filamentary technique discussed in Chapter 2. From the discussion in section 3.2, most
64
3. Tesla Transformer Design
high-voltage transfonners are designed with the low-voltage primary winding on the
outside of a concentric secondary winding. For a low inductance primary circuit of 200 nH,
a single-turn of copper sheet is used, with the following dimensions being decided after
number of iterations:
The calculated self-inductance of the single-turn primary winding with these dimensions is
180 nH.
a) Helical winding
b) Spiral winding
For a helical winding of copper wire, it is necessary to wound the wire on a conical
fonner, both to ensure good insulation and to maximise the magnetic coupling with the
primary winding. The details of a suitable winding being decided after a number of
iterations are:
• No. of turns 31
from which the calculated self-inductance of the secondary coil is 85 J.lH, the effective
65
3. Tesla Transformer Design
For a spiral winding, the secondary coil is wound with copper sheet. The details of a
• No. of turns 25
from which the calculated self-inductance of the secondary coil is 70 !lH, the effective
Even though the spiral winding resistance is much lower than that of the helically
wound coil, it has an inherent disadvantage in that its stray-capacitance is more than the
amount of energy is used in charging this stray capacitance. On balance, the helically
A transformer winding was constructed with the dimensions given in section 3.4.1.
The primary was wound on a cylindrical plastic mandrel of outer diameter 218 mm, with
the copper sheet sandwiched and heat bonded between two 125!lm layers of Mylar-
The helical secondary was wound on a plastic conical former with a coil height of
66
3. Testa Transformer Design
67
3. Tesla Transformer Design
As the secondary winding experiences a high electric stress, the winding is immersed in
transformer oil. A schematic view of the transformer together with the pulse forming line
The metallic base plate closes one end of the arrangement and also helps in grading
the electric field. But it also has a detrimental effect, due to the induced eddy currents
decreasing the effective inductance and thereby reducing the gain and energy transfer
efficiency [3.27]. A similar effect is also noticed when the conductors of the PFL are placed
at the other end ofthe transformer, and in order to minimise both effects the metallic parts
PFLinner
conductor
conductor
Conical
Laminated --rr-''-----_
sheet
plate
68
3. Testa Transformer Design
Electric field calculations of the above transformer arrangement, with and without the
commercially available software package that utilises finite element modelling. Figures 3.7
and 3.8 are 2D plots of the electric field distribution, calculated when the voltage at the coil
termination was assumed to be 500kV. Rotational symmetry was assumed in the model,
with each turn of the winding represented by a conductive ring. As expected the electric
field produced by changing magnetic field in secondary i.e. E = -VV - 8AJOt, where V is
the scalar potential and A is the vector potential, gave a negligible value.
The effect of the metallic base plate and PFL conductors in grading the electric field
will be seen, as will be the fact that the field along the edge of the primary winding strip is
highest and that it decreases with the addition of metallic parts. Mylar-polyethylene
lamination of the copper sheet helps in withstanding the field stress, as the copper sheet is
not then in direct contact with air. As is evident from the figure, throughout most of the
space between the windings the field strength is less than IS kV/mm, except at points
adjacent to the wires where it reaches about 20 kV/mm. The use of transformer oil assists in
With the above arrangement of the transformer windings, the effective self and
mutual inductances and the stray-capacitance of the secondary winding were calculated
using the techniques discussed in Chapter 2, with the results being presented in Table 3.2.
The fiIamentary model of Figure 3.9 was used in calculating the inductances of the
transformer.
69
3. Tesla Transformer Design
2.0000e+007
1. 9000e+007
1.8000e+007
1. 7000e+007
1. 6000e+007
1 . 5000e+007
1. 4000e+007
1. 3000e+007
1. 2000e+007
1. 10000+007
1.0000e+007
9 . 0000e+006
8.0000e+006
7.0000e+006
6 . 0000e+006
5 . 00000+006
4 . 0000e+006
3.00000+006
2 .0000e+006
1.00000+006
0.00000+000
Figure 3.7. 2D electric field pLot of the transformer with metallic parts
70
3. Tesla Transformer Design
2.0000.+007
1 . 9000.+007
1. 8000.+007
1. 70000+007
1. 60000+007
1. 50000+007
1. 40000+007
1 . 30000+007
1 . 2000H007
1 . 1000H007
1. 0000H007
9 . 0000e+006
8 . 0000e+006
7 . 0000H006
6 . 00000+006
5 . 00000+006
4 . 00000+006
3 . 00000+006
2.00000+006
1. 0000.+006
3 . 6818.-002
Figure 3.8. 2D eLectric fi eLd pLot of the transform er withollt metallic parts
71
3. Tesla Transformer Design
r p
1" n' p.,
Lj
60 00 0 00
Mm
,
MP-'
do 3 "
0 0
00 ............ 0
R SL L'm
o 0 oi-
M~ N:" Rss
,,
o~ ____________________ ~~ ____________________ ~
z
t,
transformer, where rp is the outer radius ofprimary; tp is the thickness ofprimary winding;
Ip is the width of primary copper strip, RSL, and Rss are the largest and smallest radii of
secondary winding; superscripts p and s are for primary and secondary windings,
72
3. Testa Transformer Design
The capacitors used for the primary capac itor bank (CB) should sati sfy the conditi ons:
• Low inductance
Ceram ic capacitors are most suited to meet these requirements, and in addition they are less
expensive than Mica capacitors. Mo rgan Electronics 2 nF, 30 kV ceramic capacitors with a
As the inductance requirement is low o n the primary side, the capacitors were
The capacitor bank, CB is connected to the primary winding of the Tesla transformer
throu gh a short flat transmission line and a self-break spark-gap switch, as shown in Figure
3.1 I. A tri ggered spark-gap switch was not used, as the aim was to develop a s imple and
compact system , and the addition of a tri gger mechani sm would lead to a bulki er and
complex system.
73
3. Testa Transformer Design
74
3. Testa Transformer Design
The Tes la transformer was operated in either (i) a single-shot mode, or (ii) a repetitive
electrode spark-gap switch (model SG-11 2M, R. E. Beverl y, USA) was used, in which the
pressuri sing to 150 kPa. The inductance of the switch as specified by the manufacturer is
For repetitive operati on a novel spark-gap sw itch working on the principle of corona-
stabilisation was employed. A detailed discuss ion of this switch will be presented in
Chapter 4. The load for the transformer is a coaxial oi l-fi ll ed PFL integrated into the
3.6.
75
3. Testa Transformer Design
deve lo p a portabl e system that can be operated in the fi eld, a battery based power suppl y
was used. Depend ing on the mode of operatio n i.e. single-shot or re petitive, the battery s ize
(power) and HV DC power sup ply di ffe r, si nce the power require ment for re petitive
3.5.1 Ballery
Rechargeable batteries are preferred to di sposable one, due to their long term cost
effectiveness . Vari ous types are avai lable commerciall y, including sealed lead acid (SLA),
nickel metal hydride (N iMH), ni ckel cadmium (N iCd), and lithium ion [3.28]. Sealed lead
acid batteries were chosen due to thei r lower initial cost, even though all the others are
more compact.
For an operating vo ltage o f 500 kV, and taking into account losses in the converter
and Tesla transformer, the energy drawn from the batteries per pulse is expected to be
between 2 and 15 J .
Fo r a low PRF (5 Hz or less) mode the average power consumption from the battery
may reach 40-60 W, and if the source is ope rated at thi s rate for 60 minutes the supply
capac ity w ill need to be abo ut 50 Wh o Hence the battery pack consisted of two series-
connected SLA mai ntenance free batteries (YUASA make) each rated at 12 V, 4 Ah. For a
highe r PRF i.e. greater than 400 Hz, the power consum ption is far greater, and the batte ry
76
3. Testa Transformer Design
The DC- DC converter may be considered as the initial stage of the pulsed power
generator, stepping up the low battery voltage (12-24 Y) to the 30 kY range as it charges
For low PRF (or single-shot) operation , the average output power of the converter
used in the system . This converter is designed to charge a small capacitance of a few tens of
nF.
The combination of the battery pack and the DC- DC converter was enclosed in a RF
shielded box, as shown in Figure 3. 12, in order to protect the converter from the high
Figure 3. J2. DC- DC converter and battery pack inside a RF shielded box
77
3. Testa Transformer Design
commerciall y and an inverter (model SM4273, Switch Mode Ltd, UK) which converts
12 V DC to 240 V AC together w ith a capac itor charging power supply (model 202,
As the load of the CB is highl y inductive the DC- DC converter and the capacitor
chargin g power supply woul d normally experience the full vo ltage reversal. To protect
against thi s a high-vo ltage crowbar diode was employed as shown in Figure 3. 13 [3.29].
RI R2
HV AV'
Power - ,-
Supply CB =~
Diode
_L
Figure 3.13. Pro/ec/ion circui/ ofpower supply agains/ foll vol/age reversal
3.6 Summa ry
A 0.5 MV Tes la transformer was designed to operate in a dua l resonant mode. The
primary winding consisted of a single-turn of copper sheet, and the 3 I-turn secondary
wind ing was wound helica ll y on a conical fo rmer and immersed in oil , to max imise
co upling wi th the primary wind ing and to ensure good insulation. Electric fi eld calculations
showed that the electri c fie ld generated is well with in the limit of breakdown. A
rechargeable SLA battery together with suitable hi gh-voltage DC charger, formed as the
prime power supp ly of the system, as the aim is to design a portable system. Operation of
78
3. Tesla Trans form er Design
References:
[3.2] N. Hardt, D. Koenig, "Testing of insulating material s at high frequencies and high
voltage based on the Tesla transformer principle", Conference record of the 1998 IEEE
[3.3] E. A. Abramyan, "Transformer type accelerator for intense electron beams", IEEE
[3.4] D. J. Malan, The physics of lightning, English Universities Press, London, 1963
[3.5] N. Tesla, Means for generating electric current, Patent 514 168, Feb. 1894
[3.6] N. Tesla, Apparatus for transmitting electrical energy, Patent 1119732, Dec. 1914
FIREBALLS", E xtract from TCBA NEWS , Volume 8, #3, 1989, Accessed on 25/08/2005
at http://home.dmv.com/-tbastianlball.htm
Te legraphie," Annalen der Physik, Vo!. 13, pp. 512-561 , 1904, Accessed on 25/08/2005 at
http://www.coe.ufrj .br/-acmq/teslalmagnifier.html.
[3. 11] C. R. J. Hoffmann , "A Tesla transformer high-voltage generator", Rev. Sci. Intrum. ,
79
3. Tesla Transformer Design
[3. 12] I. Boscolo et. aI. , " Tesla transformer accelerator for the production of intense
relativistic elect ron beams", Rev. Sc i.lnstrum ., Vol. 46, pp. 1535-1538, 1975
[3. 13] G. J. Rohwein, "TRACE I, A transformer-charged e lectron beam acce lerator", IEEE
[3. 14] G. J. Rohwein , "A three megavolt transformer for PFL pul se chargin g", IEEE Trans.
water Blumlein", IEEE Trans. Electron Device, Vol. 26, pp. 1512-1517, 1979
[3. 16] J. L. Reed, " Greater voltage gain for Tesla-transformer accelerators", Rev. Sci.
[3. 17] B. T. Phung et aI., 7'h Internatio nal Symposi um on Hi gh Voltage Engineering, Vol.
[3 .1 8] F. M. Bieniosek, "Triple resonance transformer circuit", Rev. Sci. In strum., Vol. 61,
[3.19] F. M. Bieniosek et.al., "MEDEA 11 two-pu lse generator development", Rev. Sci .
International Pul sed Power confere nce, USA, pp. 73 -77, 199 1
generator", 11th IEEE International Pulsed Power Conference, Baltimore, USA , pp. 730-
735 , 1997
80
3. Tesla Trans(ormer Design
[3. 22] M. C. Scott, "A 350 kV dual resonant transformer for charging a 40pF PFL at kilo-
hertz rep-rates", 10'h IEEE International Pulsed Power Conference, Albuquerque, USA, pp.
1466-147 1, 1995
[3 .23] J. L. Korioth et. aI., "A novel super low inductance primary ring utili zed in a pul se
dual resonant tuned transformer", 12th IEEE Internati onal Pulsed Power Conference,
[3.24] A. C. M. de Queiroz, "Multiple resonance networks", IEEE Trans. Cir. Sys. I, Vol.
[3 .25] M. Denicolai, "Optimal performance for Tesla transformers", Rev . Sci. Intrum. , Vol.
[3.26] W. J. Sarjeant and R. E.Do llin ger, "H igh-power electron ics", TAB BOOKS In c.,
USA ,
[3.27] G. J. Rohwein, "Design of pulse transformers for PFL chargin g", 2 nd IEEE
International Pul sed Power Conference, Lubbock, USA, pp. 87-90, 1979
[3.28] I. Buchmann, "Batteries in a portable world", 2 nd edition, Cadex Electron ics Inc.,
1997, USA
[3.29] Lambda Americas Inc ., App lication Note 5 17, http: //www.lambda-
81
4. Spark-gao Switching Process
This chapter deals with the physical process that occurs in the types of spark-gap
switches that find extensive use in the field of pul sed power, with a comprehensive stud y
made of their repetitive performance. Fin ally the design of a novel spark-gap switch for
4.1 Introduction
switches fall into the category of closi ng sw itches, which can be sub-di vided into a) so lid-
state switch es e.g. thyri stors, transistors, IGBTs etc. and b) spark-gap switches which
includes liquid-gap and gas-gap devices. With their relatively simple design and low cost,
vo ltage withstand capab ility (as hi gh as a few MV) and a hi gh charge transfer capability. In
add ition, spark-gap switches have a fast closure time of between sub-nanoseconds and a
few nanoseconds, and their operation can be synchronized with other circuit elements by
A gas switch can be considered to be electrica ll y closed when under a high electric
fie ld stress the insulating gas between the electrodes becomes conducting and a plasma
chan nel develops. T hi s transition of the switch fro m the insulating to the conductin g state
can be exp lained in two ways, namely the Townsend model and the streamer model.
82
4. Spark-gap Switching Process
The pioneering work of J. S. Townsend (19 10) forms a usefu l basi s for a theoretical
required between the e lectrodes and, although such a source can be created by irradiation,
for any given volume of gas there ex ists a low but constant source of free e lectrons du e to
the ionisi ng effect of cosmic particle interaction with neutral gas atoms. In the presence of
an electric field between the electrodes, the free electrons gain suffic ient energy to collide
with neutral gas atoms and so ionise them . An electron ava lanche may then occur, due to
the initial creation of a free electron subsequently freeing numerous other e lectrons before
being absorbed by the anode. The existence of e lectron avalanches does not however lead
maintain a steady current flow, whose magn itude in most cases is very low. T he Townsend
breakdown mechani sm states the requirement of a secondary ioni sation process that enables
sustai nin g discharge can develop as a region of high condu ctivity. T he Townsend
breakdown model can be developed for a uniform field between two charged electrodes and
steady-state conditi ons i.e. the current at any plane between the electrodes is constant in
time. The electron current at the cathode 10 generated by an external radiati on source
increases with the distance from the cathode, due to the impact ioni sation of the gas by the
electrons. The growth of electron cu rrent in space is described by the steady state continuity
eq uation [4. 1]
d1
- '= a l (4.1)
dx '
83
4. Spark-gap Switching Process
where I , is the e lectron current, and a is the ionisation coeffi cient, which is the number of
electrons produced by an electron as it travels unit distance in the directi on of th e fi eld and
is known also as Townsend ' s first ion isat ion coe ffi cient. The so lution of equation (4 .1 ) is
I, = I 0ea., (4.2)
where 10 is the externally produced current at the cathode (x = 0) . As the ion current is zero
at the anode (x = d) , the externa l current in the circuit I will be equal to the electron current
I, at the anode, or
included, and secondary electrons can result from three cathodic processes: i) e lectron
emi ssion due to the impact of pos itive ions (y effect), ii) photoelectric emi ssion from the
cathode (8 effect), iii) photo ioni sation of the gas (T] effect).
"
1, (0) = 10 +0 JI, (x) dx (4.4)
o
and the solution of equation (4.1) with the boundary condition given by equati on (4.4) is
ax
I, (x) = Ioe (4.5)
1- 0 (ea" -I )
a
I ea"
1 =1, (d) =- --"'--- (4.6)
1- (j) ( ea" - I)
a
84
4. Spark-gap Switching Process
(4.7)
then I is undefined, which implies that a finite value of current can be obtained for 10 = 0,
and is defined as breakdown. Equation (4 .7) is Townsend's breakdown criteria, and can be
interpreted as the condition that must be met if the process of ionisation is to become self-
sustaining.
Townsend ' s breakdown mechani sm concurs with the Paschen curve, which is a plot
of the breakdown voltage and the product of the gap spacing d and the gas pressure p (or
den sity N). Experimentally it has been proved that a / p (and also 0) / p) is a unique
re-written as
Since E = VI d, where V is the breakdown voltage of the uniform field gap, the above
V = j'(pd) (4.9)
showing that V is a unique function j' of the product of pressure and gap spacing for a
given gas and electrode material , which relationship is known as Paschen ' s Law.
where A and B 'are constants for a given type of gas. 1f ion impact is taken as the most
important secondary ionisation process, experiments have shown that y is a slowly varying
85
4. Spark-gap Switching Process
function of E / p over a wide range, and a fter using equation (4.4) and solving for V
V= Bpd (4 . 11 )
C+ ln(pd)
/
/
I
/
\ ./
,/"
°-KI 10 100
pd (forr-cm)
Equation (4. 11) enables the Paschen curve shown in Fig ure 4.1 to be generated fo r air at
stand ard pressure and te mperature. It can be seen that there is a min imum of the brea kdown
vo ltage, which is a un ique property of many gases and electrode materials, and is given by
Vm1n = Bx/f .C) at pd = /f -C) using equati on (4 . 11 ). There is a region to the left of the
min imum where the rati o of the gap di stance to the mean free path of electrons decreases,
thereby reducing the probability of collisions. To maintain the ioni sing collision at a value
86
4. Spark-gap Switching Process
suffic ient to cause breakdown, higher electron energies are required and the breakdown
voltage is increased. To the ri ght of the minimum, the electron mean free path decreases,
and hi gher e lectri c fi eld s are necessary to provide the electrons between colli sions with
suffi cient energy to cause ionisation. At hi gher pressures additi onal effects due to
irregul arities in the cathode surface have to be taken into account, which cause fi e ld
intensi fi cation and lead to lower breakdown voltages than those given by the Paschen
curve.
Townsend's breakdown model does not account for the form ative time lag of the
order of IO ns when gaps with a hi gh pd value are overstressed by a fast voltage pulse. Thi s
lag is the time interval between the start o f an initi ating electron avalanche and the
duration wasn' t possib le at that time and later on during 1930 it was made possible with the
values, due to the development of ionised plasma trail s. Thi s phenomenon could not be
explai ned by Townsend ' s mechani sm, and leads to the development of a furth er model of
the breakdown mechani sm, known as the streamer mode l, fo llowin g the work of Loeb
Streamer models describe the development of the discharge fro m a single electron
avalanche due to photoioni sation of the gas in the gap. The hi gh electron multipl ication
factor of the ava lanche leads to the development of a space charge fi eld at the fron t of the
avalanche, and there is a concentrati on of posi ti ve ions at the trail ing edge of th e avalanche
87
4. Spark-gap Switching Process
as positive ions are heavier than electrons. If the avalanche attain s a critical size before it
reaches the anode, the resulting fast ionisation process leads to the formation of streamers
whi ch bridge the gap with a plasma channel. The main di ffe rence between the two theories
is the stage at which departures occur from the deve lopment of the exponential avalanche.
In the models by Loeb and Meek this happens when a si ngle avalanche reached the anode,
whereas in the Reather model it occurs when the avalanche is at the middl e of the gap.
In the models by Loeb and Meek the avalanche is considered to be critical when the
rad ial field due to the ava lanche is equal to the external field just in front of the anode. The
concentration of positive ions at the trailing edge of the ava lanche attracts and absorbs
auxiliary electron avalanches that have been initiated by photoionisation of the gas. A
pl asma stream is thu s formed whi ch expands rapid ly towards the cathode.
According to Reather, on its way to the anode the avalanche reaches a critical
dimension such that secondary electrons begin to be generated just ahead of the ava lanche,
by photoionisation of the gas due to ion ising radiation generated in the avalanche. There is
a group of e lectrons just ahead of the avalanche and in a hi gh electric field region due to th e
field enhancement caused by the avalanche space charge. The hi gh electron multiplication
factor causes a space charge region to develop, which grows rapid ly to the dimension of the
parent avalanche. This extends the space charge front till it reaches the anode, with the
progression termed a streamer, and on reaching the anode a similar process begins to occur
at the cathode end . The photo-electrons generated are accelerated towards the avalanche,
extend ing the ion sheath of the parent ava lanche towards the cathode. Breakdown occurs
immediately upon the space charge region reaching the cathode, with Reather's breakdown
88
4. Spark-gap Switching Process
where C is a dimensionless constant and the critical di stance de must be less than or equal
The streamer propagation velocity is at least two orders of magnitude greater than the
electro n dri ft velocity. As the streamer model requires onl y the form ation of a single critical
avalanche, the formative times are shorter than those from the Townsend model. The
branched grow th of the subsequent discharge chan nel can be explained by the local field
distortion caused by the space charge and the variable spatial distribution of the auxi liary
Townsend ' s breakdown criteri on of equation (4.7) was developed for a uni fo rm fie ld
the spatial electric field distribution wi ll no longer be un iform and non- uniformi ty may also
result from the bui ld-up of positive ions and electron space charges. Due to the field
variation the ioni sation coefficient wi ll also vary, and Townsend's breakdown criteri on has
to be modi fied. If the ioni sation coeffi cient a is a functi on of the spati al coordinate x, then
(4. 13)
a [e'jad< - I1
w =I (4 .14)
89
4. Spark-gap Slvitching Process
di stribution and the dependence of a on E are known. The space charge concentratio n in a
non-uniform fi eld will disrupt the progress of discharge, in contrast to the uniform fi eld
conditi ons. The space charge will lead to parti al breakdown of the insulating gas w ithout
the formation of a complete breakdown, and the breakdown voltage will be hi gher than that
required for a uniform field. In the case of a non-uniform fi eld the applied vo ltage polarity
al so plays an important rol e, due to the effect of fi eld strength on the secondary ioni satio n
Some atoms (or molecules) with empty spaces 10 the ir outermost orbit have a
tendency for these empty spaces to become fill ed with free electrons to form negative ions.
Such gases are known as electronegative gases, and 02, SF6 and other halogen compo und s
are typica l exampl es. On attachment of these electrons to the gas molecules they are
removed from the ionisati on process, as the mo bility of a negative ion is much lower than
that of an electron . The breakdown voltage is therefore hi gher for electronegative gases,
where the coeffi cient of attachment /3 is de fined as the average number of attaching
co ll is ions made by one electron moving unit length in the direction of the fi eld. Also the
(4. I 6)
90
4. Spark-gap Switching Process
For high pd products the actual breakdown voltage does not follow the Paschen
curve, with the deviation depending on factors such as the gap separatio n, e lectrode area
and electrode material. The breakdown vo ltage is lower than expected and its rate of
change with increasing pressure is reduced. At higher pressure there are other factors which
• Dependence of the breakdown voltage on the electrode material and surface fini sh,
• The breakdown voltage for a compressed gas decreases for a larger electrode area .
• Spark gap conditi oning shots are necessary, si nce the breakdown voltage increases with
the number of shots before reaching a saturation level. With a larger electrode areas
and a higher applied field the number of conditioni ng shots also increases.
These effects and the discrepancies from the Paschen law variation cannot be explained
If chargin g pulses with rise-times less than the stat istical delay time are app lied to the
spark-gap, it is poss ible to exceed th e DC se lf-breakdown voltage level before the gap
closu re takes place. The statistical delay time is defined as the time required for the
appearance of an initiating electron to begin the avalanche or streamer process. If the pulse
rise-time is of the order of a few tens of nanoseconds, it is comparable to the formative time
lag (see section 4. 1.2). The advantage of apply in g such a fast vo ltage pulse is that the
spark-gap can sustain a voltage between 2-3 times the DC self-breakdow n level. Another
91
4. Spark-gap Switching Process
All the different breakdown mechani sms described in the previous section result in
the formation of a weakly ionised gas channel between the two electrodes. This section
describes the development of thi s channel until it electrically closes the electrode gap.
Energy from the electri c field is transferred into kinetic motion of the electrons, ions
and neutral gas molecules, which results in a rapid rise of temperature of the channel to
thousands of Ke lvin . As a result, the weakly ioni sed gas channel is converted to a hi ghl y
conductin g narrow pl asma channel. Due to the correspondin g increase in the condu ctivity,
current from the external circu it flows th ro ugh the channel, dissipating more energy and
giving ri se to a rapid expansion of the channel and the generation of a shock wave in the
media surro unding the channel. Drabkina [4.4] and subsequentl y Braginskii [4.4] proposed
model s for the shock wave expansion of the spark channels, both of which made
assumptions concerning the radial expansion of the spark channel but provided results in
good agreement with experim ental data. Bragin skii proposed a formulae for the channel
where 1(1} is the current throu gh the channel, u is the electrica l conductivity of the channel,
The switch closure process consists of various phases, each requiring a finite ti me,
and with the total time refe rred to as the switchi ng time. One phase is ca lled the ' resistive
92
4. Spark-gap Switching Process
phase' , in which the spark resistance decreases by many orders of magnitude as the
discharge channel radius increases due to thermal ionisation. It is very di ffi cult to quantify
the resistance accurately, as it is dynamic and is due to various complex processes that are
statistical in nature. T here are empirical formul ae valid fo r pa rticul ar cases, but the most
simpli stic approach is to consi der the discharge channel as havin g a cylindrical geometry.
where / is the gap length, r the channel rad ius and a- the electrical conductivity, and
/
R(I) = (4. 19)
(4;~' ) [j
1/3
J(I)213 dl]
When the electri ca l co nducti vity of the spark cha nnel becomes high, the switch behav iour
is more suitably described by its inductance, w ith thi s phase bein g referred to as the
inducti ve phase. The value of the inductance is determi ned by the overall switch geometry,
2/ I ) /
L=2x 10- 7 ( In -;:-- (4 .20)
where re is the cha nnel rad ius and / is the length of the channel. For a spark-gap switch with
(4.2 1)
93
4. Spark-gap Sw itching Process
Together, the res istive and inductive phases can be considered to control the ri setime of a
switch, with the 10-90% switch rise time I, bein g expressed as [4. 1]
(4.22)
where IR is the time for the resistive ph ase (or the time from the establi shment of a
conducting channel to the po int at whi ch the switch resistance is equal to the inductive
switch impedance) and IL is the time of the inductive phase (or that porti on of the rise time
dominated by the sw itch inductance). Another empirica l formula due to G iri et al [4.5] that
was used to specify the nanosecond rise time of switches used in hi gh power impulse
transmitters is
(4 .23)
where p is the density of the gas in the gap in gcm·3, Z is the impedance of the load c ircuit
conditi ons wi thin the gap must recover to their pre-breakdown state. If the time between
two consecutive di scharges is insuffi cient, the switch w ill pre-fire at well below its se lf-
Initiall y a high temperature plasma column remams in the gap even after the
discharge current has ceased to fl ow, and the temperature and e lectron concentratio n are
still suffic ientl y high to reignite the gap on application of a reduced vo ltage. Such a
breakdown will occur when there is thermionic emi ssion from the electrodes and even at an
app lied vo ltage of 100 V or less. If the external voltage is not reapplied, recombination and
94
4. Spark-gap Swilching Process
attachment process will deionise the plasma co lumn in about 100 fls and only a hot neutral
gas col umn will remain in the gap. Due to the presence of free charge carriers the gap
Another recovery process occurs with coo ling o f the hot neutral gas column from
several thousand Kelvin to ambient temperature. During this time the breakdown vo ltage
depends on the gas density, and the factors affecting the rate of thermal conduction and
convection are important for gap recovery. The thermal diffusivity of the gas is the most
important factor affecting the thermal conduction , but also significant are the gap spacing,
Recovery of the gap, without any aidi ng mechani sm, occurs with the decrease in the
gas temperature to ambient, which takes about 10 ms for most gases except hyd rogen [4.6].
The trend in repetitive pulsed power is towards higher peak power and higher
repetition rate (more than 400 Hz), i.e. higher average power systems. For high peak power
applications the high-pressure spark-gap sw itch offers the best cho ice, due to the
advantages menti oned in section 4. 1. They are inexpensive, tri ggerab le and simple to use
and have a wide operating range. One of the maj or shortcomings of 2-electrode spark-gap
switches is however that their repetiti ve operati on is restricted to less than 100 Hz, due to
the slow recovery process discussed above. Heat from the previous discharge is the main
factor in slowing the recovery process. There are however various ways of im prov ing the
The use of hi gh pressure (about 1.4 MPa) hyd rogen gas allows an order of magnitude
improvement in the recovery time, without any gas fl ow [4.6]. Hydrogen has both a hi gh
95
4. Spark-gap Switching Process
molecular speed and a high thermal diffusivity, which helps in reducing the recovery time
of the electrode gap. However, being so li ght is quite difficult to provide a leak-proof
The recovery time can a lso be improved by purging the insulating gas through the
gap, thereby cooling and removing the residue from the previous di scharge [4.7] . Higher
voltages also lead to higher gas flow requirements, since the voltage rating of a spark gap is
generally determined by the product of the gap pressure and electrode spacing, which also
determines the mass of gas that must be removed or cooled by the flowin g process.
Increased voltage thus implies an increased gas flow , and a higher repetition rate leaves less
time between pulses for the gap to cool, requiring a higher rate of cooling gas flo w [4.1] .
As the flow rate requirement is quite hi gh (35 Nm 3/h) an air flow pump / compressor is
Triggering a spark-gap switch at well below its self-breakdown voltage also improves
the recovery time [4.6] , but the addition of a triggering circuit to the overall system also
Another way of improving the recovery time of a spark-gap switch is by the use of
corona stabilisation, which requires the presence of both a non-uniform electric field and an
electronegative gas [4.8,4.9]. A highly non-uniform fi eld is achieved by designing the two
maj or electrodes (high-voltage and ground electrode) of the spark-gap to have needle-plane
geometry, and with these conditions breakdown of the spark-gap is preceded by a corona
di scharge in the hi gh field region [4 .1 0]. Corona encircles the hi ghly stressed electrode and
locks the field around it to a corona onset value. Breakdown of the gap will occur only
96
4. Spark-gap Switching Process
when s ufficient space charge is generated to increase sufficiently the fi eld. As it takes time
for the process to develop, full voltage recovery is allowed and pre-fire prevented.
T he above conditions are for a DC charged or slowl y ri sing vo ltage pul se. For pul sed
charge operation of a spark-gap switch, i.e. when there is a dead-time between successive
voltage pul ses, the voltage recovery process is restricted following partia l density recovery
by the residual ion popul ation. Appli cati on of a DC bi as vo ltage between the electrodes
during the inter-pulse period minimizes the effect of this and thus improves the recovery
time [4. 11] . The influ ence on the voltage recovery time of the low gas density region due to
the switching arc can be reduced by employ ing electrode geometri es which possess
breakdown voltage-pressure characteri stics that show little dependence on pressure above
400 kPa. Thi s results in a hi gh percentage of vo ltage recovery fo r onl y a parti al recovery in
As the aim is to develop a simple, co mpact and portabl e EMP radiator the spark-gap
switch design should be simple and the use of accessories such as a triggering system, or a
gas-flow pump etc. are undesirable. Of the various techn iques fo r increas ing the PRF,
corona stab ilization provides the best poss ible choice and is relati vely simple to implement.
The anomalous breakdown behav iour of SF6 in the presence of a non-uniform fi eld
was reported in 193 9 by Pollock and Cooper [4. 10] and verifi ed later in 1953 by Works and
Dak in [4.9]. Thi s techni que has previously been utilised to improve the repetition rate of a
spark-gap sw itch [4.8 , 4. 11 ], but the switch described here is substantiall y different.
97
4. Spark-gap Switching Process
Voltage
(a)
/ /(b)
/
P, Pressure
Figure 4.2. Voltage-pressure characteristics of spark-gap switch (a) breakdown curve (b)
As stated, corona stabilization occurs in the presence of a hi ghl y divergent fi eld and
an electronegati ve gas and is generated by the use of electrodes with needl e-plane
geo metry. Figure 4.2 curve (a) shows a typi ca l breakdown voltage-pressure characteristics
fo r a swi tch under these conditions and, for slowly ri sing vo ltages, breakdown o f the
electrode gap is preceded by a corona discharges up to the pressure PI. The onset of corona
discharge is shown in Fi gure 4.2 by the corona-onset curve (b). As the voltage is ra ised,
corona surround s the highl y-stressed electrode as shown in Figure 4.3 and locks the fi eld
around it to the corona onset va lue. The space-charge resulting from th e corona activity
effectively shields the stressed electrode from the remainder of the gap and the other
electrode. The space-charge dri fts from the stressed electrode into the rema in ing part of the
gap, which is a low-field region and enhances the fie ld in this region. This needs to develop
98
4. Spark-gap Switching Process
to a sufficient degree before breakdown will occur. As the process is time dependent, full
vo ltage recovery can be allowed to take place and premature switch closure prevented.
initiated at a point on the cathode surface and develops towards the anode in a continuously
increas ing field where ionisation is high. A positive ion space charge trail s the path of the
ava lanche, due to its low mobility. As the electric field intensity is high near the anode,
most of the free electrons created are absorbed in the anode. Negative ions are formed
The electric field is enhanced in the gap by the formation of a positive ion space
charge near the anode. Photons released by excited molecul es in the primary avalanch e give
rise to secondary electrons, which are acce lerated in the en hanced field region and create
secondary ava lanches, thus promoting propagation of the discharge in the gap, along a
streamer channel. Corona discharges at the anode prior to breakdown of the gap can be
categorised by their electrical, physical and visual characteristics in the order of increasing
field intensity as: burst corona, onset streamer discharge, positive glow di scharge, and
Burst Corona discharge is due to ionisation activ ity at the anode surface, where the highly
energetic incoming electrons lose their energy prior to their absorption by the anode.
99
4. Spark-gap Switching Process
During this process, positive ions are created in the vicinity of the anode, with the number
building to form a positive space charge and so suppress the discharge . The free electrons
then move to another part of the anode, and the resulting discharge current consists of very
small positive pulses, each corresponding to the spread of the ionisation over a small area
of the anode, and its subsequent suppression by the positive space charge produced.
Onset streamer discharge results from the development of the discharge . The formation
of positive ion space charge near the anode surface enhances the field in its immediate
vicinity and attracts subsequent electron ava lanches. A streamer channel thus develops,
resulting in the onset of a streamer discharge and considerable positive ion space charge
being formed in the low field region. The cumulative effect of the success ive electron
ava lanches and the absorption of free electrons at the anode results in the eventual
formati on of res idual space charge in front of the anode. The local electric field there drops
below the critical value for ionisation and suppresses the streamer discharge. A dead time is
thus required for the applied field to remove the positive ion space charge and restore the
conditions necessary for the development of a new streamer. The discharge develops in a
pul sating mode, producing a positive current pul se of large amp litude but relatively low
repetition rate.
Positive glow discharge In this mode, a thin luminous layer develops near the anode
surface, where intense ionisation activity takes place. The discharge current is basically a
direct current, on which a small alternating current component is superimposed with a high
repetition rate in the range of hundreds of kilohertz. The field is such that the positive ion
space charge is rapidly renewed from the anode, thus promoting surface ionisation
activities. The field intens ity is not sufficiently hi gh to allow the development of the
100
4. Spark-gap Switching Process
discharge and streamer formation. The role of the negative ions is to suppl y the necessary
eventually reappear, and lead to breakdown of the gap . The discharge is similar to the onset
streamer, although the streamer current is more intense. The development of a breakdown
streamer is directly related to the effective removal of the positive space charge by the high
A spark-gap switch for repetitive operation was developed based on the corona
stabili sation technique. As the requirement for a highly divergent field is essential, the
switch had a needle- plane geometry, with the high voltage electrode of the switch therefore
consists of needles, while the ground (load) electrode is a plane plate, as shown in Figure
4.4. The number of needles is decided by the charge transfer and the field distribution in the
gap. [t should not be too high to lower the peak electric field in the gap, which will degrade
the corona-stabilisation performance [4.8], and there is also a minimum number necessary
to accommodate the required charge transfer capability. The optimum number of needles
was determined experimentally and for the experimental switch twelve needles were used
with a tip diameter of 0.6 mm. Both the needles and load electrode were of brass, with the
two electrode plates separated by a 25 mm thick Perspex insulator. The gap between the
HY needles electrode and the load electrode is fixed at 3 mm . With SF 6 gas at ambient
the CS-SG to 90 kPa. Another criterion for the switch design was that the inductance of the
switch should be as low as possible and certainly should not exceed 20 nH (which is the
101
4. Spark-gap Swilching Process
same value as that of the commercial switch used for low PRF or single-shot operation),
which puts a restriction on the overall size of the switch. In general, the probability of
surface breakdown increases for a given voltage with any decrease in the size of the
insulator component. Hence the Perspex insulator, is ridged along the discharge path, to
provide an increased resistance to su rface discharge. Care was taken to avo id a hi gh-
concentration of electric field near the ridging. As the pressure inside the CS -SG is quite
low (90 kPa), nylon bolts were used with a factor of safety of2, and complete sealing of the
switch is provided by two O-rings. A view of the CS-SG is shown in Figure 4.5 , wh ich was
designed so that it cou ld replace the commercial single-shot switch without changing the
primary circuit set-up of the Tesla transformer. A field plot of the CS-SG is shown in
Figure 4.6 and is evident that the field inten sity is very at the tip of the pin.
12 Needles as
HV electrode Load ElectrOde (Brass)
102
4. Spark-gap Swilchillg Process
4 .0000e+O06
J . 8000e+O06
J . 6000e+Q06
l . 4000e+(106
J . ZOOOe+Q06
l . OOOOe+Q06
Z. 8000e+Q06
2 . 60DOe+Q06
2.4000e+O06
2.2000e+O06
2 . 0000~06
1 . 8000et{106
l. 6OOOe+O06
1. 4OOOe+Q06
1.2000uC06
1.OOOOH006
8.0000e+OOS
6. OOOOe-+OOS
4 .0000 e+QOS
2.0COOe+OOS
O.OOOOe+OOO
103
4. Spark-gap Switching Process
4.5 Summary
together with relevant details concerning gaseous breakdown and spark channel
development. Emphasis was given to the switch rise time, since this is useful in designing
The factors affecting the switch recovery time and various possible ways to improve
this were discussed. Finally a low-inductance switch design for high PRF operation was
outlined, which utilises the corona stabilisation technique as this is the most simple to
implement and does not require accessories such as a triggering system or gas-flow pump.
104
4. Spark-gap Switching Process
References:
[4.1] G. Schaefer, M. Kristiansen, and A. Guenther eds., Gas Discharge Closing Switches,
International IEEE Pulsed Power Conference, vol. 2, pp. 1171-1174, June 1999
[4.5] V. Giri et aI., "Design, Fabrication, and Testing of a Paraboloidal Reflector Antenna
and Pulser System for Impulse-Like Waveforms", IEEE Trans. Plasma Science, vol. 25, pp.
[4.6] S. L. Moran and L. W. Hardesty, "High repetition rate hydrogen spark gap", IEEE
[4.7] G. J. J. Winands et aI., "Long lifetime triggered spark-gap switch for repetitive pulsed
power applications", Rev. Sci. Instrum., vol. 76, 085107, 6 pages, 2005
[4.10] H. C. Pollock and F. S. Cooper, "The effect of pressure on the positive point-to-
plane discharge in N2, O2, CO2, S02, SF6, CChF2, A, He, and H2", Physical Review, vol.
105
4. Spark-gap Switching Process
[4.11] S. J. MacGregor et ai., "Factors affecting and methods of improving the pulse
106
5. Voltage Diagnostics & Results
This chapter begins by describing various types of voltage diagnostics that were
implemented to investigate the performance of the pulsed power generator. The period of
the voltage pulse to be investigated is in the range of few hundreds of nano-seconds, and it
can therefore be regarded as relatively slow. A brief review of the basics of voltage dividers
is presented, with both commercial and in-house dividers being used to measure the high
voltage output of the generator. A brief description of the data acquisition is also provided.
Finally the results of high-voltage measurements for both single-shot and repetitive
The basic definition of a voltage divider is a device which reduces the high input
voltage to a level that can be measured directly by other devices such as a voltmeter or an
in series to which the high voltage is applied, with the output voltage being taken across Z2
The attenuation factor of the divider depends on the ratio of ZI and Z2, which is
ideally a constant, i.e. independent of frequency. Another important factor for any voltage
divider is that it should not load the source, i.e. it should draw negligible current, and
therefore its input impedance must be much greater than the source impedance. Voltage
dividers can be classified broadly as: (a) resistive dividers and (b) capacitive dividers.
107
5. Voltage Diagnostics & Results
~I
input VI
z, output
V,
Resistive voltage dividers consist of two series connected resistors RI and R2 replacing Z,
v,= R, 1 (5.1)
f'; RI +R, l+RI/R,
Measurements by resistive dividers are very accurate for DC or at low frequency, with an
error of less than 1%. However, for high frequency applications the impedance of the
resistors changes drastically, thereby affecting the attenuation factor. and the results
obtained become unreliable. Also for high voltage measurement the resistor has to be
physically large so as to withstand breakdown and surface flash over, and as a consequence
the capacitance to ground becomes significant, which again alters the attenuation factor.
The high frequency response of a resistive divider can be improved by the addition of
108
5. Voltage Diagnostics & Results
capacitive arms across the resistive arms, as shown in Figure 5.2 and such dividers are
termed RC-compensated. For these dividers the condition Rp, = R,C, has to be satisfied,
C,
(5.2)
C,+C,
v;
2
For low frequency operation most of the current flows through the resistive arms whereas
for high frequency operation it flows mainly through the capacitive arms.
Capacitive voltage dividers consist only of two series connected capacitors, when the
v; C, I
(5.3) .
V; C, +C, l+C,/C,
For high frequency operation a capacitive divider is much more accurate than a resistive
divider and, for an ideal divider, the attenuation factor is again independent of frequency.
109
5. Voltage Diarmostics & Results
However in practice, the finite resistance of the leads connecting to the voltage source and
to the display device, together with stray inductance, impose a limitation on the high
frequency application.
In many applications, capacitive dividers are designed as an integral part of the system,
thereby avoiding high-voltage leads and minimising the loading effect on the system.
The rise time of the response of any voltage divider is normally defined as the time
taken for the output response to rise from 10% (tlO) to 90% (t90) of the steady-state value,
The rise time ofthe divider can also be represented mathematically, by assuming that
for high frequency application it acts as a low-pass filter of resistance R and capacitance C,
when the output voltage v(t) following a step-function change at the input is
The rise time of a divider is also related to its upper frequency limit, i.e. the upper -3 dB
point Am. At this frequency the impedances due to the resistive and capacitive network are
equal and the amplitude of the output voltage has fallen by 50% of its initial value, thus
110
5. Voltage Diagnostics & Results
I
R (5.5)
0.35
t =-- (5.6)
,. hdB
Equation 5.6 is widely used for dividers, oscilloscopes, etc., wherever a rise time has to be
specified. It is applied wherever a step response reaches its final value in the shortest
possible time without any overshoot. For high frequency applications the divider response
practice, an overshoot of not more than 10% of the final value is acceptable for equation
This section identifies the commercial high-voltage dividers that were used in
The Agilent 10076A high-voltage probe is a resistive divider rated for I kV DC with
a quoted bandwidth of 250 MHz. The attenuation ratio is 100: I, the input resistance is
pulsed, with a bandwidth of 75 MHz. The attenuation ratio is 1000: I, the input resistance is
111
5. Voltage Diagnostics & Results
and 100 kV pulsed, with a quoted bandwidth of 90 MHz. The attenuation ratio is 2000: 1,
the input resistance is 400 MO, and the input capacitance is 12 pF.
application. It is rated for 1 MV pulsed, but has a bandwidth of only 30 MHz. The input
capacitance is 20 pF.
5.2.5 Oscilloscopes
Several digital oscilloscopes were used for monitoring and recording the voltage
• Tektronix TDS 654C oscilloscope, bandwidth 500 MHz, sampling rate 5 GS/sec
• Tektronix 30348 oscilloscope, bandwidth 300 MHz, sampling rate 2.5 GS/sec
All of the above oscilloscopes have a data storing facility, with the data stored on floppy
discs.
An in-built capacitive voltage sensor was constructed and used to monitor the output
voltage of the Tesla transformer. This operated in the V-dot mode so that its output is
proportional to the time derivative of the input signal. It was integrated into the coaxial oil
112
5. Voltage Diagnostics & Results
pulse fonning line (discussed later in Chapter 6) as shown in Figure 5.3 and has the
Constructionally the divider consists of a copper strip separated from the inner wall of the
outer conductor of the PFL by a thin layer of Mylar insulation, thereby fonning a capacitor
in which w is the width and I is the length of the copper strip, d is the thickness of the
Mylar insulation between the copper strip and the outer conductor ofthe PFL and Cnn is the
relative pennittivity of the Mylar. Cl is the capacitance between the inner conductor of the
PFL and the copper strip separated by the oil and is given by [5.2]
C
," WIT7,1
('O-d)ln
r. -d
(5.8)
where ro is the inner radius of the outer conductor of the PFL, rl is the outer radius of the
inner conductor, Co is the pennittivity of free-space and Coil is the relative pennittivity of the
oil.
The high-voltage input V;(t) to the circuit of Figure 5.4 is obtained by integrating the
recorded signal S(t) representing the derivative of the low-voltage output of the sensor as
(5.9)
113
5. Voltage Diagnostics & Results
BNC
PFL outer
PFL inner conductor
conductor
HV
Vi
Equation 5.9 is true only if the RC time constant (where R is the coaxial cable terntination
to the oscilloscope and is equal to 50 Q and C = Cz) of the voltage sensor is less than the
time period of the signal, V;(t) [5.3]. This is referred to as the "V-dot" mode of operation, as
the signal requires further integration to yield the input voltage waveform. For a copper
114
5. Voltage Diagnostics & Results
5.3 nF, giving an RC time constant of 265 ns. The resonant frequency of the Tesla
transformer circuit· is 2 MHz (section 3.4) which corresponds to a 500 ns period and is
As the sensor utilises the system capacitance to attenuate the input signal, there is no
possibility of loading the circuit, and the measured value is extremely accurate. In practice
the V-dot mode was preferred to a capacitive voltage divider due to the high level of
electromagnetic noise involved in the experiment, since the unwanted noise picked up is
highly attenuated by integration of the signal. The advantages of V-dot capacitive sensors
over resistive sensors are manifold; they do not shunt the load with a lower impedance,
they are immune to surface breakdown, and they require neither balancing of the resistive
This section presents the results of voltage measurements made on the pulsed power
parameters of the Tesla transformer and the calibration of the in-built capacitive voltage
A commercial spark-gap switch was used for single-shot operation. The self-
breakdown voltage was varied by changing the nitrogen gas pressure in the gap, and the
Primary circuit: A discharge technique was used to determine the parameters ofthe Tesla
transformer primary circuit shown in Figure 5.5. The capacitor bank, Cp of 24 nF was
115
5. Voltage Diagnostics & Results
charged to 18 kV and discharged into the primary winding with the secondary winding
removed. The spark-gap switch was pressurised to 13.79 kPa to give a self-breakdown
voltage of about 18 kV and the voltage was monitored by the PVM-6 Northstar voltage
probe. The corresponding discharge voltage waveform (i.e. the voltage between points A
and B Figure 5.5) was recorded on a 300 MHz Tektronix oscilloscope, giving the result
switch
B
LcB+SW
bank, Lp inductance of primary winding, LcB +sw inductance of capacitor bank, short
T'
L =-- (5.10)
p 21rCp
116
5. Voltage Diagnostics & Results
where T, the time period of the voltage wavefonn, is obtained from Figure 5.6 as 456 ns.
The self-inductance of the primary circuit is therefore 220 nH. The inductance of the
single-turn primary turn primary as calculated by the filamentary technique (Chapter 2) was
180 nH, with the 40 nH difference being due to the commercial switch (20 nH specified),
the capacitor bank arrangement and the short flat transmission of Figure 3.11. The
20
~1
2 r
10
r 1\
f\
\
f
\
o
\
-10
\; w
V
V
tl t2
o 500 1000 1500 2000
time (ns)
An important design parameter of a Tesla transfonner is the 'quality factor' of the primary
circuit, or
(5.11)
117
5. Voltage Diagnostics & Results
(5.12)
VI, V2 and tI, t2 are indicated on Figure 5.6. From this data the quality factor of the primary
circuit is obtained as 24.7, which is well within the design criterion that it should be greater
Secondary circuit: The self-inductance of the helically wound secondary winding was
measured as 85 J.lH using an LCR bridge. The mutual inductance between the primary and
transformer as in Figure 5.7. In case (A) Lab "= Lp + Ls + 2M and in case (8) Lcd = Lp + Ls-
2M, so that the mutual inductance between the two coils is M = (Lab - Lcd) / 4.
a
e
- - ' . M.
Lp Led
Lab
d
b
(A) (B)
The coupling coefficient k calculated using equation (3.10) is 0.55, close to the design
value of 0.6 (Table 3.1). The capacitance of the secondary circuit is 65 pF, determined as
described in section 2.7.1. The uncoupled frequency of the secondary circuit is 2.14 MHz,
118
5. Voltage Diagnostics & Results
so that the tuning ratio (section 3.3.1) of 2.2/2.14 = 1.02, which is acceptable as unity
tuning ratio is desired for maximum energy transfer efficiency (Figure 3.2).
In-built capacitive voltage sensor calibration: In-situ calibration of the in-built capacitive
voltage sensor, described in section 5.3.1, was carried out using the Northstar Megavolt
probe with the pulsed power generator circuit of Figure 5.8. The experimental arrangement
Rs
M
switch •
•
Lp
Ls
~B+SW
119
5. Voltage Diagnostics & Results
Figure 5.9. Arrangement for calibration of in-built capacitive voltage sensor using
The output terminal of the in-built capacitive voltage sensor was fed to a 300 MHz
The output voltage of the sensor is proportional to the time derivative of the input signal, so
that post numerical processing is necessary to determine the high-voltage output of the
transformer. The primary capacitor bank (CB) was charged to different voltages, with the
Northstar PVM-6 probe used to monitor its discharge voltage. Results provided by the
Megavolt probe when used to monitor the output voltage of the Tesla transformer are
compared with the integrated responses of the in-built capacitive voltage sensor in
120
5. Voltage Diarmostics & Results
1\
10
\ 100
I~ (\, f\ "..
V 1 "V
-10 \l J -100
\
\
\/
-200
r.t
- o 2:10 4ll 600 800 1000 -200 o 200 400 600 800 1000
time (1'11) time (Ill)
Sec.JUbrywltage
300
- - - - - ~ - N orthstar Megavolt
integrated signa!, in-built probe
200
lOO
.-. f\
§
~
""!i!
-lOO
0
'\ / \ ~
~
-200
\lJ
.
,~
'.
121
5. Voltage Diagnostics & Results
\
200
!; 10
1/\
tJ /\ ... ~
t 0
/' \ I \
V
I V
-10
\\ I' J -200
-400
-20
-200 o 200 400 600 800 1000
-200 o 200 4lJ 600 800 1000
time (M)
time (ns)
Secondary voltage
300
- - - - - .• N orthstar Megavolt
integrated signal, in-built probe
200 f.'\.,
I
,
"~
.fJ!\
:',
'
'
lOO
-\
,
~
.-- ./
54
.s
~
":>-
0
0
".~ ~ ...
-100 \~ .'.
-200 \ J
V
o 200 400 600 800 1000
Time inns
122
5. Voltage Diagnostics & Results
200
~
'\
100
\
r I1 \ ~ l \
1\
V
~
\ \ / '"V
-10 \
\V
I \
V
/ -100
->Xl
J Y
-
~
200 4lJ
time (lIS)
600
'''' "'" o lOO ""
timt(llI)
<!Xl
"" 1000
Secondary voltage
300
----_ .. N orlhstar Megavolt
integrated signal, in-built probe
~L
200
'I
100
. .
11\
• • / 1\
~
~
j'; / \
~ 0
""is! ~ ~,
-lOO •
-200 \
o 200
V 400 600 800 1000
time (lIS)
123
5. Voltage Diagnostics & Results
I'
20
200
r..
1
!' V\ ~
0
-10
t I V J -200
\~ -41)0
- o 200 4)0 600 800 1000 -200 0 200 4JO
time (M)
600 800 1000
time (ns)
Secondary voltage
300 .!
..
I I
- - - - - -. Northstar Megavolt
integrated signal. in-built probe
.•.,.."..
200 .
100
R... "•
,
• /\
;; 0 k::. fi
. •
.
\
t ~ ~"
-100
'\::..
-200
-300
\J
.....
-200 o 200 400 600 800 1000
time (ns)
124
5. Voltage Diagnostics & Results
20
200
11\
f
t
1\
\ h lA \
/I V / \ JI \V ·V
-10
~ IV -<00
o 200 <100 600 800 1000 ->l) o <00 600 800 1000
time (N) lime CIIII)
Secondary voltage
300
-----.- Northstat Megavolt •
200
integrated signal, in-built probe .1\ ",
I
100
1\ l
~
) \
~ 0
~ ~ ~.
'"
~ -100
'\
-200
-300 \ J
V
The glitches in the response of the sensors (see Figure 5.1 0-5.13) are due to partial
breakdowns occurring in the connecting cable used to connect the Tesla secondary with the
Megavolt probe, (Figure 5.9) as part of it was in air. Complete breakdown was however,
125
5. Voltage Diaf!l!ostics & Results
not observed in any of the above cases. Later, the cable insulation was reinforced and
partial breakdown was avoided, as is evident from Figure 5.14. The figure also
demonstrates a very good match between the integrated signal ofthe in-built sensor and the
An important point to be noted is that even with a primary capacitor bank charging
voltage of 27.8 kV, only 330 kV could be obtained at the secondary of the transformer,
showing that the actual voltage gain of 11.8 is much lower than the designed value. The
reason for this is undoubtedly the loading imposed by use of the Megavolt probe, with its
After being used to calibrate the in-built capacitive voltage sensor, the Megavolt
probe was therefore disconnected from the pulsed power generator circuit. The maximum
measured open-circuit secondary voltage of 550 kV then obtained (with a primary charging
voltage of29.8 kV) agreed closely with the theoretically predicted data, as shown in Figure
voltage gain for the transformer of 18.4 with a primary/secondary energy transfer efficiency
On removal of the cable connection from the transformer output to. the input of the
Megavolt probe, no partial breakdown was observed as is clear from Figure 5.15. This
series of experiments has demonstrated that there is no apparent shot-shot variation in the
126
5. Voltage Diagnostics & Results
30
20
C\,\
LA
...
~\ ..
~
~ 10 /
~
~
'"~ 0
'\
1.
/r.....• ......
...
I
\ L
....
t'-1
......
-10
"~
-20
/
o 100 200 300 400 500 600
time (ns)
Secondary voltage
600
400
I
~ R.
200 •
........
.... " I \"
~
d
~ 0
-./ '\
.'
I
/ \
'"!'!
-200
:\
-400 \ /
\ LJ
o 100 200 300 400 500 600
time (ns)
Figure 5.15. Capacitor bank discharge voltage and secondary voltage: experimental (solid
line) and theoretical (dotted line)
127
5. Voltage Diagnostics & Results
For repetitive operation the same pulsed power generator arrangement was used, but
with the commercial spark-gap switch replaced by the corona-stabilised spark-gap switch
The aim was to design a spark-gap switch which could be operated at a PRF of
500 Hz or more, with an inductance of less than 20 nH (that of the commercial spark-gap
switch), to avoid the need to redesign the Tesla circuit. It is of the utmost importance to
check the inductance of the CS-SG, which was developed after the complete single-shot
development and testing of the EMP generator, and it was not possible to check this in the
same arrangement without removal of the secondary circuit of the transformer. The
separate test set-up of Figure 5.16 was therefore used to check the inductance of the CS-
SG, with a circuit of Figure 5.5. Figure 5.17 compares the discharge voltage waveform
obtained with both the commercial spark-gap switch and the CS-SG in circuit.
Single-turn
~:-::-:cr-_''''''~' ." .
\,-<!{.,-
128
5. Voltage Diagnostics & Results
20
10
'\ ..-
\ V
o 1\ / \
-10
~
o 100 200 3()0 400 600
time (ns)
Figure 5,]7, Capacitor discharge voltage waveform, CS-SG (red line), commercial switch
(blue line)
There is an extremely close match of the two discharge voltage waveforms, and it can
be concluded therefore that the inductance of the CS-SG is the same as that of the
The repetitive charging waveforms displayed in Figure 5.18 to Figure 5.21 were
measured by the Northstar PVM-6 probe connected to the high-voltage plate of the
capacitor bank. These tests were performed with the CS-SG filled with SF6 gas at ambient
pressure, and the data presented are for a short burst mode. Figure 5.18 presents data for a
PRF of 400 Hz, and at this rate the self-break voltage repeatability is about ±2.5%. Figure
129
5. Voltage Diagnostics & Results
5.19 is for a PRF of 1 kHz, with a self-break voltage repeatability of about ±2.5%. Figure
5.20 shows the charging voltage waveform at 1.25 kHz, when the variation of self-break
voltage is within ±3.5%. Figure 5.21 is for a 2 kHz repetition rate, when the variation is still
only±5%.
20,-------,-------,-------,-------,-------,----,
o 20 40 60 80
time (ms)
130
5. Voltage Diagnostics & Results
30r------,------~------,_------~----~------~_,
20r-----~------_r------~------+_----~r_----_+_1
~ 101----
~
o 20 40 60 80 100
time (ms)
131
5. Voltage Diagnostics & Results
30
20
10 ~I 11111 f
I
1111 I1
·111111
o 10 20 30 40
time (IllS)
132
5. Voltage Diagnostics & Results
30r-------r------,r------.-------.-------.-------,
*,101---
;:;
~
01------
o 10 20 30 40 50
time (ms)
The variation of corona current (the current prior to gap breakdown) and the different
modes of corona were also noticed as the voltage was raised, as identified earlier in section
4.4.1.1. The corona current was measured by monitoring the voltage drop across a 3 MQ
high-voltage resistor connected between the load electrode and ground using the Northstar
PVM-6 probe. This allows the measurement of very small current of the order of micro-
ampere.
133
5. Voltage Diagnostics & Results
(a)
6
<'::I.
'-'
""~ 4
i'l
El
8 2
-2~--------~--------~--------~------~
o 5 10 15 ~
time (IllS)
25
20 (b)
<'::I.
'-' 15
""~ 10
I'!
El
8
5
-5
0 5 10 15 20
time (IllS)
Figure 5.22. Variation of corona-current with time, in the burst mode; (a) corona activity
134
5. Voltage Diagnostics & Results
3or--------r--------r--------r------~
(a)
20
I
I 10
5 10 15 20
time (ms)
300r---,----.----~--_r----r---_r--_,r_--~
(b)
~
:1
'-' 200
"~
":!!
S
"
lOO
°0L---~5----~10----~15----2~0--r-2L5--r-3LO--~3L5--~~
time (ms)
Figure 5.23. Different modes of corona (a) at 10 kV, the positive glow discharge mode, (b)
atl2 kV
135
5. Voltage Diagnostics & Results
Figure 5.22 presents the burst mode of corona, represented by occasional bursts, with
Figure 5.22 (a) showing the beginning of corona activity at 7.5 kV and Figure 5.22 (b) at a
higher voltage of 9 kV, with the burst appearing more frequently. Figure 5.23 (a) and (b)
illustrate the further increase of corona activity at 10 kV and 12 kV. The discharge current
of Figure 5.23 (a) has a DC component on which a small alternating current component is
superimposed with a high repetition rate, that could be seen visually as a positive glow
discharge. Figure 5.23 (b) presents the streamer discharge mode and Figure 5.24 the
800
(b)
.•· · ,,•
(a) (c)
600
<':I.
'-'
/ ••·•·• ,, I
I
I
I
·•·•· , •
<l
~
/
..~
400
•· ,"
/ ,,
c=
e
0
•••
" ..• •·• •••
200
.••• •• •
1·
~
•
.. •• .. ~
.- .
. ..
-~=..--.
'
5 10 15 20 25
voltage (kV)
Figure 5.24. Variation of corona-current with charging voltage at different SF6 pressure
136
5. Voltage Diagnostics & Results
35
30
T/
yr' V
25
vr
20
~
V
20 40 60 80 100
relative pressure (kPa)
Figure 5.25. Variation ofself-breakdown voltage with SF6 pressure for CS-SG
The variation in breakdown voltage evident in Figure 5.18 - Figure 5.21 is due to the CS-
sa operating in a self-breakdown voltage mode, when there are various phenomena taking
place in the electrode gap. These are mainly electrode heating and erosion but the effects of
gas heating and corona motion are also present. Another feature that influence the
performance is the use of a battery pack and an inverter at the input of the HYDC power
supply (for charging the capacitor bank), as the inverter generates a quasi sine wave.
The variation of the self-breakdown voltage of the cs-sa with pressure is shown in
Figure 5.25 for an electrode gap of 3 mm. Although the time delay and jitter of a switch are
often quite important, this is not so in the present application (as mentioned earlier in
Chapter I). At present the lifetime of the cs-sa has not been determined, but from work
reported elsewhere [5.4] it can be predicted that it can be used for 106 shots or even more.
137
5. Voltage Diagnostics & Results
Prior to the development of the CS-SG, the commercial spark-gap switch pressurised
with nitrogen was tested for repetitive operation, giving the performance shown in
Figure 5.26.
Figure 5.26. Charging voltage waveform with a PRF of 200 Hz, capacitor bank charged to
20kV
spark-gap switch, an unwanted effect that is due to its operation at a PRF an order of
138
5. Voltage Diagnostics & Results
5.5 Summary
The oil-insulated Tesla transformer outlined previously has been operated in the dual
resonance mode. to generate more than 0.5 MY, with a high energy transfer efficiency
The novel CS-SG developed for high PRF applications, was operated successfully in
a burst mode at a PRF of 2 kHz. The inductance of the switch is less than 20 nH.
139
5. Voltage Diagnostics & Results
References:
[5.2] R. J. Alder, "Pulse Power Formulary", North Star Research Corporation, June 2002.
[5.3] C. A. Ekdahl, "Voltage and current sensors for a high-density z-pinch experiment",
of a high repetition rate, triggered, corona-stabilized switch in air", J. Phys. D Appl. Phys.,
140
6. Radiating Elements and Operation
This chapter presents design aspects of the radiating elements of the EMP generator,
and includes the pulse forming line (PFL), the fast spark-gap switch (FSG) and the antenna.
Even though the PFL is part of the pulsed power generator, it is more accurate to term it as
the radiating element, since the resonant frequency is dependent on the PFL, which is
sometimes referred to as a tuned line. In the Pulsed Power Generator, the PFL merely acts
as a capacitor which has to be charged to a desired voltage. The design and calibration of a
fast capacitive voltage divider (FeY) is also discussed, and this was built to measure the
fast voltage pulse at the output of FSG. Finally, results of the radiated electric field
A PFL can be characterised by its conductor geometry and the type of insulating
material used. For most high-voltage, fast pulse systems a coaxial geometry [6.1] is
preferred, though other types of geometry like radial line resonators [6.2], [6.3] are also
of the coaxial PFL is that it may be readily integrated with the structure of the transformer
Various types of fluid can be used as the insulating material, though the support
structures are usually made of solid insulators such as plastic. The types of fluids used
water, transformer oil, alcohol, and glycerine. Deionised water, alcohol, and glycerine all
have a high permittivity and are often used in either pure form or when mixed together,
although due to their polar characteristics the transmission loss increases considerably for
141
6. Radiating Elements and Operation
RF signals. The transmission losses for gases are zero, but their permittivities are all very
low and close to unity. A gas filled PFL will therefore have a relatively low energy storage
density. Transformer oil offers the best choice as the dielectric medium for the present
system, as its dielectric strength is high and it has a low dissipation factor.
For a coaxial PFL, in which end effects are neglected, the maximum electric field
Emax will be at the surface of the inner conductor, and for a voltage V it is
V
(6.1)
a.ln(Ya)
where a is the outer radius of the inner conductor and b is the inner radius of the outer
conductor. The aim is to produce a simple and compact design for the EMP generator, so
that it is advisable to integrate the PFL with the Tesla transformer, i.e. to have the same
outer diameter for both the PFL and single-turn Tesla primary. From section 3.4.1 the
single-turn Tesla primary has a diameter of 220 mm, and the outer diameter of the outer
conductor of the PFL must therefore also be 220 mm. With a wall thickness of 10 mm, the
inner radius of the outer conductor is 100 mm. The partial breakdown strength of
transformer oil is about 300 kV/cm t6.5], but this is dependent on the condition of the oil
with regard to humidity and other contaminants. Assuming a factor of safety of 2 then Emax
was assumed to be about 150 kV/cm and using equation (6.1) the outer radius of the inner
In the PFL and Tesla transformer arrangement shown in Figure 6.1 the conductors
are made of aluminium. The spark gap end of the inner conductor of the PFL has a conical
shape, to increase the leakage path and so reduce the probability of surface breakdown,
since the inner (which was at a potential of more than 500 kV) and the outer (at ground
142
6. Radiating Elements and Operation
potential) conductors of the PFL are both in contact with the plastic support. An electric
field plot for the PFL using Maxwell 2D (www.ansoft.com) is shown in Figure 6.2, which
confirms that the electric field is below 150 kVfcm near the inner conductor of the PFL.
The probability of any breakdown is therefore avoided with the use of transformer oil as the
The coaxial PFL will behave as a quarter wavelength resonator when switched to an
21 = c
4f..Ji:
frequency and 8r is the dielectric constant of the medium filling the PFL. For a resonant
Figure 6.1. PFL(without oil) mounted on Tesla transformer; the conical secondary on top
143
6. Radiating Elements and Ooeratioll
E[V / m]
1. 5000.+007
1. 4250.+007
1. 35000+007
1. 27500+007
1. 20000+007
1.1250.+007
1. 05000+007
9 . 75000+006
9.00000+006
8 . 2500e+006
7 . 50000+006
6.75000+006
6.00000+00 6
5.25000+006
4.50000+006
3 . 7500.+006
3 . 0000.+006
2 . 25000+006
1. 50000+006
7 . 50000+005
PFl inner 0 . 00000+000
conductor
Figure 6.2. 20 electric field plot for PFL using Maxwe1l20 FEM software package
frequency of 100 MHz the physical length of the PFL should be 250 mm. However, loading
by both the antenna and the effective reactance of the output switch will result in a resonant
frequency somewhat lower than that predicted for an ideal quarter wavelength line.
144
6. Radiating Elements and Operation
Therefore, the actua l phys ical length of the PFL necessary for a 100 MHz resonance will
c = 27(&0&, I (6.2)
In( % )
where &0 is the permittivity of free space. With a length of 175 mm the capacitance is
To generate Rf signals effi ciently, it is essential that the FSG that connects the PFL
wi th the antenna conducts in a time at least as short as a quarter cycl e of the transient
frequency, i.e. for a central frequency of 100 MH z, the FSG rise time sho uld be about Ins .
Hi gh pressure spark-gap switches offer the best poss ible opti on to achieve the required fast
switching action, at hi gh power levels (as di scussed in Chapter 4). Typically, hydrogen
[6.3] or nitrogen [6.7] gas switches are used, tho ugh in o rder to hold-off voltages of 250-
500 kY, the pressures required with these gases is in the order o f tens of atmospheres . At
relative ly lower pressures (5-10 bar), the pro perti es of a spark gap fill ed with pure SF6 gas
As switch jitter is not an im portant criterion, the FSG was operated in the self-
breakdown mode of operation, witho ut the need fo r a tri ggering circuit, which would have
145
6. Radiating Elements and Operation
added a degree of compl ex ity to the system and increased the overall volume and weight.
With the above requirements a 2-electrode spark-gap sw itch, as shown in Figure 6.3 was
constructed and pneumati cally tested up to 15 bar. The brass electrodes are hemi spherical
in shape, with a di ameter of 25 mm . To increase the leakage path length and to reduce the
pro bability of surface breakdown, the support insulators were ridged. Figure 6.4 shows the
FSG bein g tested for hi gh pressure with stand . The inter el ectrode gap is about 7 mm , which
is sufficient to provide a voltage ri se time of about I ns or less (section 4.2), and the self-
breakdown vo ltage was varied by changin g the SF 6 pressure. As the load electrode is
connected to the antenna, it will be at a fl oating potenti al. To ensure reliabl e operati on of
the FSG, the load electrode was connected to ground through a co il (i nductance> 2 J.lH) as
shown in Figure 6.5. A schematic of the Tes la transformer integrated with the PFL and
Antenna end
146
6. Radiating Elements and Operation
147
6. Radiating Elements and Operation
6.3 Antenna
The antenna characteristics are dependant on the ratio of the signa l wave length to
the physical dimensions of the antenna structure, whether used for transmittin g
used fo r the unidirecti onal transm ission of ultrawideband pul ses, are a wavelength
(corresponding to the lowest frequency) long and the di stance between the two pl ates is at
least one half-wavelength [6.8]. Parabolic refl ectors, another class of directi ve antenna,
have aperture lengths that are several wavelengths across. Hence, at RF frequencies, with
148
6. Radiating Elements and Operation
wavelengths in the order of metres, any directional antenna structure will occupy a
significant volume.
The most compact, radiation effi c ient, antenna is the omni-directional half-
wave length dipole . For wideband operation it is best to adopt a fat dipole configuration, as
the dipole length L to di ameter D ratio affects its bandwidth; decreasi ng LID (a fatter
dipole) increases the bandwidth and a ratio of LID :::; 10 is des irabl e [6.3]. As the radiated
response of the EMP generator is ex pected to be about 100 MHz, L should be set at about
750 mm . However, the o ptimal value for L is somewhat less, due to capacitive loading
effects at the ends of the d ipo le. An overall view of the EMP generator is given in
Figure 6.7. Immediately after the FSG a portion of the antenna structure can be seen to be
In order to measure the fast ri sing vo ltage pulse developed across the FSG a fast capacitive
di vider was built, as described below. To measure the rad iated electric fie ld fro m the EMP
generator the commercial D-dot sensor described in secti on 6.4.2 was util ised.
Commerciall y available vo ltage di viders are unavail able for measurin g fast and
high-voltage pulses of the order of several hundreds of kilo-volts and with a rise time of
less than 2 ns. For measuring the output voltage pul ses of th e FSG a fast capacitive di vider
(FC V) [6.9] was therefore designed and constructed, that was able to w ithstand at least
400 kV and was sufficiently fast to measure the vo ltage rise tim e of less tha n 2 ns. Another
important criterion was that the FeV should have a coaxial structure, that would ensure
very little electric fie ld radiation. Figure 6 .8 is a schematic diagram of the FC V and an
149
6. Radiating Elements and Operation
between the central copper rod and the 50 flm copper tape wound around UHMWP E (ultra
50 flm copper ta pe and 0.2 mm copper foil separated by layers of Mylar. The design has
+ - - - ----3- Insulated
part
150
6. Radiating Elements find Operation
1
!I
UHIIWP£
JO .. _
15 1
6. Radiating Elements and Operation
ridges on the UHMWP E surface to increase the leakage current path length and so reduce
the probability of surface discharge . Two different views of the FeV are shown in Figure
6.9. As the attenuation factor achieved by the FeV of about 138 is insuffic ient to reduce the
input vo ltage to the level which can be measured directly by an osci lloscope, the output of
C2 is fed to an Agilent I0076A probe thro ugh a coaxial connector. T he o utput end of the
Fev is heavi ly shi elded by copper mesh as also is the cable of the Agilent probe, as seen in
Figure 6. 10.
152
6. Radiating Elements and Oeeration
To measure the electric field radiated from the EMP generator a PRODYN free-
space AD-70 D-dot probe was used, which is a broad band differential output sensor that
responds to the time rate of change of electric displacement [6.10]. The sensor consists of
two asymptotic sensing elements mounted on the sides of a ground plate and held in
position by dielectric supports. Such sensors respond as short dipole receiving antenna for
signal wave lengths much longer than the physical size of the sensor. For a time varying
incident electric field E(/) with the frequency spectrum of the incident field being below the
frequency limit of the sensor, the output voltage signal V(I) from the sensor into a load
where E,,(I) is the component of the incident field normal to the grounded plate of the
sensor and a. is a calibration factor. The calibration factor is given by the manufacturer as
(6.5)
where EO is the permittivity of free space, A eq = 10-3 m-3 is the equivalent receiving area of
the sensor and R = 100 Q is the characteristic impedance of the sensor. A wide-bandwidth
balun (pRODYN BIB - I OOF) is used to match correctly the balanced output signal from the
sensor to the oscilloscope; in such a case R will be 50 Q. For a given calibration factor and
orienting the probe appropriately, the transient electric field at a point may be determined
by integrating the measured voltage signal from the sensor. The PRODYN AD-70 D-dot
153
6. Radiating Elements and Operation
6.5 Results
Thi s secti on describes the ca libratio n of the FCY, togeth er with the fast hi gh-
(d iscussed in secti on 5.3) when the FSG was closed, are a lso presented . Finally, rad iated
The FCY was calibrated at low-vo ltage, with the source being a pul se generator unit
The unit has a step response with a pulse ri se time of abo ut 12 ns. To mainta in the same
conditions throughout, the output of the FCY was fed to the Tektroni x TDS 654C
osc illoscope th rough the Agil ent pro be. Another coax ial cabl e too k the amplifier o utput to
the oscill oscope. A typi cal calibration data for this low-voltage case is shown in Figure
6. 12. The signa l of the FCY is not clean, due to the fact that the output voltage from the
FC Y and the Ag ilent probe unit is o nly 4 mY . The attenuation factor obtained for the FCY
The FCY was also calibrated at high-vo ltage with the hi gh-voltage so urce being a
hi gh-voltage tri gger generator un it TG-70 (Titan Pul se Science di vision) (6. 11 ] whose
154
6. Radiating Elements and Operation
output of 70 kY has a pulse ri se time of 5 ns into a 50 n load. The output voltage of the
TG-70 can be varied by pressurising with SF6 an internally located spark-gap switch. The
output of the TG-70 was connected to the input of FeV and also to the Northstar PYM-6
hi gh-vol tage probe. The output of the Fey (with the Agilent probe) and the Northstar
PVM-6 were both fed to the Tektronix TDS 654e oscilloscope, which was placed inside a
Faraday cage and powered by a battery, to reduce the effect of electromagnetic noise.
Figure 6. 13 shows typical calibration data. The attenuation factor of the FeV in thi s case
~
~ 40 ~------+_~------~--------_+----------~--------1
l
~
~ 20 ~----++------+-------+-------+-------4
-20
o 50 100 150 200
time (ns)
Figure 6.12. FCV calibration at low-voltage. FCV response solid line. delay generator
155
6. Radiating Elements and Operation
60
.. ... ~
'-..,,~ ~
40
J. ... ~"
,
~
~
,
, ....'. .....-." ........
~
~
~
'"~ 20
I ~
,
'. '.'. '..
, . •"
"
, .'
. ~
j
"
~
C)
0 .'
.' , ..
.
j
, ,
<-,
Cl
E-
-20
-40
-10 o 10 20 30 40 50
time (ns)
Figure 6.13. Calibration of FCV, FCV response solid line, Northstar probe response dOlled
line
The antenna was disconnected from the output of the FSG, and the FCY was
mounted instead, as seen in Figure 6.10. The outp ut of the FCY and the Agilent probe unit
was connected to the Tektronix 7404 oscilloscope, which was placed inside Faraday cage
to aga in provide protection from the dangers of electric field radiation. A typica l output
voltage pu lse measurement is shown in Figure 6.1 4, for wh ich the primary capacitor bank
was charged to 23 kY and the pressure in the FSG was set to 517 kPa. The pul se has a ri se
time (10% to 90%) of 2.2 ns, and it will be seen later that a much faster rise time was
156
6. Radiating Elements and Operation
obtained from the EMP generator. Th is slower rise time is due to the limitation of the FeV,
which could not be used beyond 415 kV (obtained in Figu re 6.14) due to breakdown of the
insulation.
500
n
400
300
j\
/
*' r
~
200
I r [\ {'J \ 1\
\;V\
100
V
o
) V V
-100
-2 o 2 4 6 8 10 12 14
time (ns)
The in-built capacitive voltage divider (section 5.3) was also used to measure the secondary
output voltage of the Tesla transformer when the FSG was operated and pressurised to
690 kPa. In thi s case the output cable from the div ider was double-shielded using copper
mesh and connected to the oscilloscope again placed inside the Faraday cage. A typical
measured voltage waveform when the FSG was operated is given in Figure 6.15.
157
6. Radiating Elements and Operation
400
/"..
J\
200
(\ \ /" \. / ~
"- ./ ~
"'--./
\
- 400 \
-600
o 200
\" 400 600 800 1000
time (ns)
Figure 6.15. A typical output voltage waveform fro m Tesla transformer (arrow indicates
FSG closure).
The system was mounted on a wooden platform and tested in an open space fa r
from potentiall y refl ective objects, as shown in Figure 6.16. Time-domain fi eld waveforms
were measured at various locations, using a free-space AO-70 D-dot sensor together with a
matched balun. A schematic of the measurement arrangement is shown in Figure 6 .17, with
the he ight hi and h 2 and the range R bei ng defi ned. The EMP generator and the sensor were
ori ented with respect to ground in such a way that a verticall y polarized fi eld was both
radi ated and measured. The output signal fro m the D-dot sensor was recorded on a
158
6. Radiating Elements and Operation
Figure 6.16. Open-site measurement of radiated electric field. Inset shows arrangement of
D-dot sensor
EMP Generator
h. - 33m h, - 43m
159
6. Radiating Elements and Operation
Tektroni x TDS 654C oscilloscope, located inside a Faraday cage, powered by a battery and
connected to the sensor through a 50 n hi gh-frequency coax ial cabl e. Since the output
vo ltage of the sensor is proportional to the time derivative of the inc ident electri c fie ld, post
numerica l processing was necessary to determine the radi ated e lectric fi eld . For the results
presented below, the SF6 pressure in the FSG was set to 5 17 kPa, resulting a breakdown
voltage of 400 kV ± 5%, at whi ch a rise time of 1.2 ns was observed in the radiated fi eld
emi ssion. Figures 6.1 8, 6.20, 6.22 & 6.24 shows radiated fi eld waveforms measured at
di ffere nt locations fro m the EMP generator with the data not corrected for the effect of
ground refl ections. The rate of decrease of fi eld is clearl y not pro portional to the inverse of
the range R, whi ch may have resulted from ground wave refl ections bein g superimposed on
Figures 6.19, 6.2 1, 6.23 & 6.25 are measured fi eld waveforms at various locations,
corrected for ground refl ecti ons using appropriate values fo r the ground conductivi ty and
permitti vity as descri bed in Appendi x-B. Fi gure 6.26 presents e lectri c fi eld measurements
(aga in co rrected for ground refl ections) at vari ous locations multiplied by th eir respective
di stances, when the rate of decrease of field is clearly proportional to the inverse of R. The
experiments demonstrated that the EMP generator is hi ghl y reliable, with a very good
reprod uci bility of the fie ld signals. Fi gure 6.27 presents the corres ponding frequency
spectrum , from whi ch it can be seen that the generator produces a moderate band of
radi ati on [6. 12], centred at 75 MHz and with a bandwidth of about 60 MH z, a simil ar
160
6. Radiating Elements and Operation
Assum ing a sinBdependent beam pattern, the peak radi ated power Pmax is [6.4]
,,, (E )'
Pmox = fJ mox R' . sin 3 B· dB· dt/J (6.6)
o 0 170
where Ema., is the peak amplitude of the radi ated electric fi eld measured at B= 900and 170 is
the impedance of free space; from Figure 6.26 Ema"R is 128, so that Pmax is 345 MW .
Using the same beam pattern assumption regarding, the total radiated energy per
pul se E,o,o/ from the time domain fi eld waveform E(I) measured at B= 90° is [6.4]
,•• ~ E(I)'
E /0/(1/
= f f f - R ' .sin 3B.dl .dB .dA.
'If (6.7)
o 0 0 170
A figure-of-merit (FOM) that serves as a useful measure of the system per formance
is the prod uct of the peak electric fie ld Emax and the range R at whi ch it is measured i.e.
REmax [6.8]. The va lue for the present system of 125 kY is s imil ar to or better than fi gures
reported elsewhere fo r similar omn id irecti onal EMP sources [6.3], [6.4], and [6.8].
6.6 Summary
The chapter has presented the o peration of a compact, portabl e, and repetitive EMP
generator producin g a radiated fi eld of 12.5 kY/m at 10 m. The FOM of 125 kY with the
dipole-type antenna structure compares well with oth er similar devices. The experiments
conducted have demonstrated that the system is very reli able with a very good
16 1
6. Radiating Elements and Operation
15
10
I\
:§'
~
:s 5
\1\
Jl
.~
"0
-"
~
0
) \ V
....V
-5
-2 o 2 4
\ ........r-V
6 8 10 12 14
time (IlS)
15
10
(
1\
:§'
~
:s
Jl
E
5
\
u
-"
~
0
I i\
~
I
~
~
-5
-2 o 2 4
\/ 6 8 10 12 14
Figure 6.19. Radiated field waveform corrected for ground influence measured 10 m from
source
162
6. Radiating Elements and Operation
6
h
:?
:; 4
:s
~
.f:!
...."0 2
1\
~
o
) \\ r ./'-
,..J
-2 ~ /
-2 o 2 4 6 8 10 12 14
time (ns)
10
I
8 ""'
:? 6
\
:;
:s 4
~
.E \
....
~
~
2
J 1\ ~
~
/l
0
-2
-2 o 2 4
\ 6
~
8 10 12 14
time (ns)
Figure 6.21. Radiatedfield waveform corrected for ground influe nce measured 15 mfrom
source
163
6. Radiating Elements and Opera/ion
4
./\
\
3
:§
a 2
1\
:!l
J!
.!! I \ /'
...
"G
~
) \ r
j
0
\. ~ )
-1
v
-2
-2 o 2 4 6 8 10 12 14
time (ns)
6
"'"
.g
! \
:;: 4
:si
J!
E
\ ~
...
u 2
~
0
/ \ '-"
'-.
~ If
-2
-2 o 2 4 6
1'/1
8 10 12 14
time (ns)
Figure 6.23. Radiated field waveform corrected for ground influence measured 20 m from
source
164
6. Radiating Elements and Operation
:§
I~ \
;; 2
\
:s
~
.1j
t;
I \
) \-
.!I
0
0
~ ,..... V-
-1
-2 o 2 4 6
"' ~ fJ
8 10 12 14
t ime (ns)
4
I \
]'
~ 3
:!l
o!! \
.~ 2
]
•
J 1\ J .....
o
/ V ~ t!_
'- rv-
-1
-2 o 2 4 6 8 10 12 14
time (Il$)
Figure 6. 19. Radiated field waveform corrected for ground influence measured 25 m from
source
165
6. Radiating Elemeflu and Ope ratio"
150
I
~
~
]
Jl
.l!
100
50
'" ,
~
·H
I
]
I ,~ .....
Q
-
\-....
~
0
"
l 0
"
1" -.·· . 'I
0..
\(V1~
-50
-2 o 2 4 6 8 10 12 14
'""" (ns)
locations, with the amplitude being scaled by rallge. R = 101/1 (black solid line), R = 15 m
(black dOlled line), R = 20 III (blue solid line) alld R = 25 m (blue dOlled lille)
200
150
\ h
\ .
V l v",- r... ....N
~
100 500
166
6. Radiating Elements and Opera/ion
References:
[6.1] C. E. Baum, "Jolt: A hi ghl y directive, very intensive, impul se-like radiator", Proc.
[6.2] S. L. Moran, "High repetiti on rate LC oscill ator", IEEE Transactions on Electron
Proceedin gs of the 9th IEEE Internati onal Pul sed Power Conference, Albuquerque, USA,
[6.4] K. D. Hong and S. W. Bra idwood, " Resonant antenna-source system fo r generati on
of hi gh-power wideband pul ses", IEEE Transactions on Plasma Science, Vol. 30, No. 5,
USA, 1989.
[6.6] R. J. Alder, "Pu lse Power Formul ary", North Star Research Corporation, June 2002 .
[6.7] V. P. Gubanov et. ai, "Compact 1000 pps high-vo ltage nanosecond pulse generator",
IEEE T ransacti ons on Pl asma Science, Vol. 25, No. 2, pp. 258-265, 1997
[6. 8] F. J. Agee et. ai, "U ltra-wideband transmitter research", IEEE Transactions on
[6.9] R. D. Shah et aI., "An ultra-fast probe for hi gh-vo ltage pul sed measurement", Proc.
13th IEEE Internati onal Pul sed Power Conference, pp. 1020 I023,200 I
[6. 10] PRODYN Technologies Inc. AD-70 D-dot sensor and balun, Users and Techni cal
manua l.
167
6. Radiating Elements and Operation
[6.11) Physics International Co. TG-70 High-Voltage trigger generator, Users and
environments (I EME)," IEEE Trans. Electromagn. Compat., Vo!. 46, no. 3, pp. 322-328,
Aug. 2004.
168
7. Conclusions
7. CONCLUSIONS
manu facture and test a compact, portable and repetitive EMP generator capable of radiating
high peak power pul ses. The des ign emphasis was to be on producin g a robust source, as it
was meant for appli cati ons and in vestigations outs ide a laboratory environment. All aspects
of the proj ect aims were successfull y completed and have been reported in peer-rev iewed
academic j ournals and at presti gious international conferences and sympos ia.
transform er perform ance being predi cted very accurately by the fil amentary modelling
technique. The transformer is abl e to generate pulses of more than 0.5 MY, with a hi gh
energy transfer efficiency of 82% between the primary and secondary circuits.
was developed for the repetitive operation of the EMP generator. The switch is abl e to
Various diagnostic tools, incl udi ng an in-bu ilt capaci tive divi der worki ng in a V-dot
mode and a further fast capac itive voltage di vider were developed for monitoring the
system performance. The results obtained when usin g these devices were fo und to very
Finall y, the EMP generator was tested in an open space outside the laboratory
envi ronment, where a peak radiated fi eld of 12.5 kV/m was measured at 10 m fro m the
generator, with peak radi ated powers in the order of hundreds of MW. The figure of meri t
169
7. Conclusions
of 125 kV for such a source is similar to or better than fi gures reported elsewhere for
• Modi fy in g the Tes la transformer and FSG to achieve a more compact 0.5 MV unit
• Increasing the PRF o f the novel switch to 5 kHz or more wi th the use of a trigger
• Devel opin g a compact and better matched directional antenna, with the ai m of
increas ing the amplitude of the radi ated fi eld by an order of magnitude.
• On the application fro nt for such a device is an investi gat ion of the e ffect of radi ated
In conclu sion, it is the beli ef of the author that the objecti ves of the thesis as presented have
170
8. List of Publications
Modulator Conference and High Voltage Workshop held in Las Vegas, USA on May
27-3 1,2008
International Power Modulator Conference and High Voltage Workshop held in Las
Closing Swi tch fo r EMP Application", Digest of Technical Papers, PPPS-2007, 16'1.
IEEE International Pulsed Power Conference, Albuquerque, USA, pp. 97-100, 2007.
Exploding Wire Opening Switch", Digest of Technical Papers, PPPS-2007, 16'1. IEEE
171
8. List of Publications
Generation", IEEE Transactions On Plasma SCience, Vol. 34, No. 5, October 2006,
of lET Pulsed Power Symposium, Daresbu ry, UK, 2006, pp. 62-65 .
Rad iat ion", Conference Record of the 21" international Power Modulator Symposium ,
Washington DC, USA , pp. 592-595, May 2006.
Proceedings of lEE Pulsed Power Symposium , Basingstoke, UK, 2005 , pp.311 -3/5.
"A Compact Battery-Powered 500 kV Pulse Generator for UWB Radiation", Digest of
172
Aooendix-A
APPENDIX-A
USA).
V)
o
N
'1"'1
~ 0
.0
:r:'" N
·in
>'" '-'
.0
@
.0
Cl.
>'"
-
~
~ 0)
I!)
bll ::;)
::;) (3
cd Cl)
bll .0
'-' cd
..,
I!) '-'
::;)
Cl)
Cl)
0
...
I!)
::;)
...
I!)
Cl...
Cl)
Cl)
<l.l
:...
0-
'1"'1
V)
o o
o o o o o
"<:t M N
J\:>f 'glle~IOJ\
173
Appendix-B. Ground-wave Propagation
If the electromagnetic source is near the earth's surface the measured field is
modified due to the influence of ground-reflected waves and surface wave fields [B . I, B.2,
and B.3J. The expression derived below are from these references. Both the ground-
reflected wave and surface wave field depends on the frequency of radiation j, the
cond uctivity 0", and relative-permittivity 8r of that ground. An outline of the radiating
source and the sensor for measuring the radiated field is shown in Figure B. J
source
point
A field
point
RI is the distance travelled by a direct wave travelling between the source A and a field
point B, R2 is the di stance travelled by the reflected wave travell ing between A and Band R
174
Apeendix-B. Ground-wave Propagation
is the horizontal distance between A and B. The source and sensor are at heights hi a nd hl
respectively, and the direct path length RI can be expressed in terms hi, h2 and R as
Since the thesis deals with the verticall y polari sed field only, the electri c fi eld perpendicular
to the plane of inci dence is discussed. The electric fi eld is dependent on both the rad iation
pattern and the orientati on of the sen sor, and assuming the sensor to be in the fa r- fi eld
region of the source, so eliminating any surface wave contribution, the measured e lectric
(B. I )
where Co is the speed of light, EJm the direct free space electric fi eld and R, is the plane
wave refl ecti on coeffic ient fo r a verticall y polarised incident wave, which can be expressed
as
where &0 is the permittivity of free space. The direct free space fi eld can be obtai ned by re-
(8 .3)
175
Appendix-B. Ground-wave Propagation
Thus the time domain free space field waveform can be obtained from the measured time
CB.3), and then applying an inverse Fo urier transformation to return to the time domain . For
the work in the thesis a good earth is assumed with the corresponding electrical parameters
Reference:
[B . I] A. A. Sm ith Jr. , Radio Frequency Principles and App lications, IEEE Press, New
York, 1998.
nd
[B.2] E. C. Jordan and K. G. Balmain, Electromagnetic Waves and Radiating Systems, 2
Dipole over the Earth or Sea", IEEE Transactions on Antennas and Propagation, Vol. 42,
176