Multipactor in Low Pressure Gas and in ... - of Richard Udiljak

Thesis for the degree of Doctor of Philosophy
Multipactor in Low Pressure Gas
and in Nonuniform RF Field
Structures
Richard Udiljak
Department of Radio and Space Science
Chalmers University of Technology
Göteborg, Sweden, 2007

Multipactor in Low Pressure Gas
and in Nonuniform RF Field Structures
Richard Udiljak
c○Richard Udiljak, 2007
ISBN 978-91-7291-885-6
Doktorsavhandlingar vid Chalmers tekniska högskola
Ny serie nr 2566
ISSN 0346-718X
Department of Radio and Space Science
Chalmers University of Technology
SE–412 96 Göteborg
Sweden
Telephone +46–(0)31–772 10 00
Cover: Susceptibility chart for multipactor in a waveguide iris for five
different height/length-ratios.
Printed in Sweden by
Reproservice
Chalmers Tekniska Högskola
Göteborg, Sweden, 2007

Multipactor in Low Pressure Gas
and in Nonuniform RF Field Structures
RICHARD UDILJAK
Department of Radio and Space Science
Chalmers University of Technology
Abstract
Resonant electron multiplication in vacuum, multipactor, is analysed
for several geometries where the RF electric field is nonuniform. In particular,
it is shown that the multipactor behaviour in a coaxial line is
both qualitatively and quantitatively different from that observed with
the conventionally used simple parallel-plate model. Analytical estimates
based on an approximate solution of the non-linear differential
equation of motion for the multipacting electrons are supported by extensive
particle-in-cell simulations. Furthermore, in a microwave iris the
electrons tend to perform a random walk in the axial direction of the
waveguide due to the initial velocity distribution. The effects of this phenomenon
on the breakdown threshold are analysed. The study shows
that the threshold is a function of the height-to-length ratio of the iris
and for a fixed value of this ratio, the multipactor susceptibility charts
can be generated in the classical engineering units. Using the parallelplate
concept, the multipactor threshold in low pressure gases has been
analysed using a model for the electron motion that takes into account
three important effects of electron-neutral collisions, viz. the friction
force, electron thermalisation, and impact ionisation. It is found that
all three effects play important roles, but the degree of influence depends
on parameters such as order of resonance and secondary emission
properties. In addition, a new method for detection of multipactor is
presented. By applying a weak amplitude modulation to the input signal
and performing a fast Fourier transform on the detected signal, accurate
and unambiguous measurement results can be obtained. It is demonstrated
how the method can be used in both single and multicarrier
operation.
Keywords: Multipactor, discharge, breakdown, microwave discharge,
nonuniform fields, coax, iris, low pressure gas, detection methods.
iii

Publications
This thesis is based on the work contained in the following papers:
[A] R. Udiljak, D. Anderson, P. Ingvarson, U. Jordan, U. Jostell, G. Li,
M. Lisak, L. Lapierre, J. Puech, and J. Sombrin, “New Method
for Detection of Multipaction”, IEEE Trans. Plasma Sci., Vol. 31,
No. 3, pp. 396-404 , June 2003.
[B] R. Udiljak, D. Anderson, M. Lisak, V. E. Semenov, and J. Puech,
“Multipactor in low pressure gas”, Phys. Plasmas, Vol. 10, No. 10,
pp. 4105-4111, Oct. 2003.
[C] R. Udiljak, D. Anderson, M. Lisak, V. E. Semenov, and J. Puech,
“Improved model for multipactor in low pressure gas”, Phys. Plasmas,
Vol. 11, No. 11, pp. 5022-5031, Nov. 2004.
[D] R. Udiljak, D. Anderson, M. Lisak, J. Puech, and V. E. Semenov,
“Multipactor in a waveguide iris”, accepted for publication in IEEE
Trans. Plasma Sci.
[E] R. Udiljak, D. Anderson, M. Lisak, V. E. Semenov, and J. Puech,
“Multipactor in a coaxial transmission line, part I: analytical study”,
accepted for publication in Phys. Plasmas
[F] V. E. Semenov, N. Zharova, R. Udiljak, D. Anderson, M. Lisak,
and J. Puech, “Multipactor in a coaxial transmission line, part
II: Particle-in-Cell simulations”, accepted for publication in Phys.
Plasmas
v

Conference contributions by the author (not included in this thesis):
vi
[G] R. Udiljak, G. Li, D. Anderson, P. Ingvarson, U. Jordan, U. Jostell,
A. Kryazhev, M. Lisak, V. E. Semenov, “Suppression of Multipactor
Breakdown in RF Equipment”, RVK 02, June 10-12, 2002,
Stockholm, Sweden.
[H] R. Udiljak, D. Anderson, U. Jostell, M. Lisak, J. Puech, V. E. Semenov,
“Detection of Multicarrier Multipaction using RF Power
Modulation”, 4th International Workshop on Multipactor, Corona
and Passive Intermodulation in Space RF Hardware, 8-11 September,
2003, ESTEC, Noordwijk, The Netherlands.
[I] J. Puech, L. Lapierre, J. Sombrin, V. Semenov, A. Sazontov,
N. Vdovicheva, M. Buyanova, U. Jordan, R. Udiljak, D. Anderson,
M. Lisak, “Multipactor threshold in waveguides: theory and experiment”,
NATO Advanced Research Workshop on Quasi-Optical
Control of Intense Microwave Transmission , 17-20 February, 2004,
Nizhny-Novgorod, Russian Federation.
[J] R. Udiljak, D. Anderson, M. Lisak, J. Puech, V. E. Semenov, “Microwave
breakdown in the transition region between multipactor
and corona discharge.”, RVK 05, June 14 - 16 juni, Linköping
[K] D. Anderson, M. Buyanova, D. Dorozhkina, U. Jordan, M. Lisak,
I. Nefedov, T. Olsson, J. Puech, V. Semenov, I. Shereshevskii,
R. Tomala, and R. Udiljak, “Microwave breakdown in RF equipment.”,
RVK 05, June 14 - 16 juni, Linköping
[L] V. E. Semenov, N. Zharova, R. Udiljak, D. Anderson, M. Lisak,
J. Puech, and L. Lapierre, “Multipactor inside a coaxial line”,
5th International Workshop on Multipactor, Corona and Passive
Intermodulation in Space RF Hardware, 12-14 September, 2005,
ESTEC, Noordwijk, The Netherlands.
[M] R. Udiljak, D. Anderson, M. Lisak, V. E. Semenov, and J. Puech,
“Microwave breakdown in low pressure gas”, 5th International
Workshop on Multipactor, Corona and Passive Intermodulation
in Space RF Hardware, 12-14 September, 2005, ESTEC, Noordwijk,
The Netherlands.
[N] C. Armiens, B. Huang, R. Udiljak, D. Anderson, M. Lisak, U. Jostell,
and P. Ingvarsson, “Detection of Multipaction using AM signals”,

5th International Workshop on Multipactor, Corona and Passive
Intermodulation in Space RF Hardware, 12-14 September, 2005,
ESTEC, Noordwijk, The Netherlands.
vii

viii

Contents
Publications v
Acknowledgement xi
Acronyms xiii
1 Introduction 1
2 Multipactor in vacuum 5
2.1 Single Carrier . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Basic theory . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Hybrid modes . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 Factors effecting the threshold . . . . . . . . . . . 18
2.1.4 Methods of suppression . . . . . . . . . . . . . . . 20
2.1.5 Effect of random emission delays and initial velocity
spread . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Multicarrier . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Design guidelines . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Single carrier . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Multicarrier . . . . . . . . . . . . . . . . . . . . . . 29
3 Multipactor in low pressure gas 35
3.1 Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.2 Multipactor boundaries . . . . . . . . . . . . . . . 37
3.1.3 Main results . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Advanced Model . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Analytical formulas for argon cross-sections . . . . 47
3.2.3 Multipactor boundaries . . . . . . . . . . . . . . . 49
ix

3.2.4 Key findings . . . . . . . . . . . . . . . . . . . . . 52
4 Multipactor in irises 59
4.1 Model and approximations . . . . . . . . . . . . . . . . . . 60
4.2 Multipactor regions . . . . . . . . . . . . . . . . . . . . . . 64
4.3 Comparison with experiments . . . . . . . . . . . . . . . . 65
4.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5 Multipactor in coaxial lines 69
5.1 Analytical study . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.1.2 Multipactor resonance theory . . . . . . . . . . . . 73
5.1.3 Main findings . . . . . . . . . . . . . . . . . . . . . 81
5.2 Particle-in-cell simulations . . . . . . . . . . . . . . . . . . 82
5.2.1 Numerical implementation . . . . . . . . . . . . . . 82
5.2.2 Simulations . . . . . . . . . . . . . . . . . . . . . . 83
5.2.3 Comparison with experiments . . . . . . . . . . . . 88
5.2.4 Main conclusions . . . . . . . . . . . . . . . . . . . 91
6 Detection of multipactor 93
6.1 Common Methods of Detection . . . . . . . . . . . . . . . 93
6.1.1 Global methods . . . . . . . . . . . . . . . . . . . . 94
6.1.2 Local methods . . . . . . . . . . . . . . . . . . . . 99
6.2 Detection using RF Power Modulation . . . . . . . . . . . 100
6.2.1 Single carrier . . . . . . . . . . . . . . . . . . . . . 105
6.2.2 Multicarrier . . . . . . . . . . . . . . . . . . . . . . 105
6.2.3 Main achievements . . . . . . . . . . . . . . . . . . 106
7 Conclusions and outlook 111
References 115
Included papers A–F 123
x

Acknowledgement
I wish to thank Prof. Dan Anderson and Prof. Mietek Lisak for accepting
me as a PhD student and for guidance and support in my daily
work. I also want to thank Prof. Vladimir Semenov at the Institute
of Applied Physics in Nizhny Novgorod, Russia, for fruitful discussions
and for his patience with all my questions. Thank you Prof. Lars Eliasson,
Director at the Institute of Space Research in Kiruna, for providing
both financial and moral support making my PhD candidate appointment
possible. A very warm thank you also to Jerome Puech for many
interesting discussions about space related microwave problems and to
his employer, Centre National d’ Études Spatiales, for financial support.
Thanks to my friends in Toulouse and especially Dr. Omar Houbloss and
Raquel Rodriguez. I thank my fellow members of the National Graduate
School of Space Technology and my colleagues at Chalmers and especially
Dr. Pontus Johannisson, Dr. Ulf Jordan and Dr. Lukasz Wolf for
beneficial discussions and lots of support with Linux and LaTex. Many
thanks also to our secretary Monica Hansen for guiding me through the
administrative jungle. I want to thank my dear mother, Monica, for
encouraging and supporting me and my family when 24 hours a day
wasn’t enough and my father, brother and sisters for believing in me. I
am also grateful to my unofficial mentors: my father-in-law Lars-Göran
Östling, my friends Jörgen Otbäck and Anders Wilhelmsson, and my
brother-in-law Nicklas Östling. Most of all I thank my wife Malin and
our daughters Janina and Lizette for encouragement and support during
this time.
xi

xii

Acronyms
AM amplitude modulation
DC direct current
DUT device under test
EDDM electron density detection method
ESA european space agency
FFT fast fourier transform
NLSQ non-linear least square
PIC particle-in-cell
PSK phase-shift keying
QPSK quadrature phase-shift keying
RF radio frequency
SEY secondary electron yield
SMA sub miniature version a
TGR twenty gap crossings rule
TEM transverse electric and magnetic field
TWTA travelling wave tube amplifier
UV ultra violet
VSWR voltage standing wave ratio
WCAT worst case assessment tool
xiii

xiv

Chapter 1
Introduction
Resonant secondary electron emission RF discharge or multipactor was
discovered and studied by Philo Taylor Farnsworth in the early 1930’s.
The phenomenon was then used as a means to amplify high frequency
signals as well as to serve as a high frequency oscillator. Using his multipactor
tubes, Farnsworth succeeded in developing the first electronic
television system. The success stimulated other researchers to investigate
the phenomenon and one of the first detailed analyses were done
by Henneberg et al. [1] in the mid 1930’s. Gill and von Engel [2] made
an even more detailed study, both theoretical and experimental, where
they, among other things, showed the importance of the secondary electron
yield on the development of the vacuum discharge. In a follow up
paper [3] Francis and von Engel studied not only the initial stage of the
electron multiplication, but also the saturation stage. They showed that
the electron space charge effect could be one of the major causes for the
discontinued electron growth. Other researchers continued the work and
a basic overview of these results can be found in the two review papers
by Gallagher [4] and Vaughan [5].
During the past 20-30 years, multipactor has mainly been studied
due to the adverse effects it can have on microwave systems operating
in a vacuum environment. It can disturb the operation of high power
microwave generators [6] and electron accelerators [7], but, above all, it
can cause severe system degradation and failure of satellites, which are
difficult or impossible to repair after launch [8]. Satellites operate under
vacuum conditions and the most common means of communication with
the Earth is microwave transmission. Microwave frequencies are required
as the ionosphere is not transparent for low frequency radiowaves. In
1

addition, it is difficult to make compact, light weight, and high gain
antennas for low frequency transmission. Many microwave components
are hollow metallic structures that guide the electromagnetic power. A
free electron inside the device will experience a force due to the electric
field and since there is no gas or other material stopping the electron,
it can accelerate to a very high velocity. Upon impact with one of
the device walls, the energetic electron can knock out other electrons
and under certain circumstances this procedure is repeated continuously
until the electron density is large enough to counter-act the effect of the
applied electric field and a steady state is achieved. A consequence of
this can be that the incident power is reflected instead of transmitted
to the intended load. Since many satellites lack sufficient protection
against reflected power in order to save weight, such reflected power can
cause severe damage to the high power stage of the system.
When a satellite is launched it carries fully charged batteries and
in the beginning, before the solar panels are deployed, they are the
only source of electric power. The capacity of these batteries is usually
low and may only last a couple of days, since a satellite in operation
will normally only lack access to power from the solar panels for a few
hours, at most, and consequently the batteries are made small in order to
save weight. If the solar panels are not deployed before the batteries are
exhausted, the satellite is permanently lost. Thus, a new satellite is often
taken into operation quickly after being put into orbit. A concern then
is that the satellite components may not be completely vented and there
is a risk for ordinary corona breakdown, which is more prone to occur
at intermediate pressures than at high and very low pressures. A corona
discharge is usually much more detrimental than multipactor and for a
certain range of pressures, the breakdown threshold for corona is lower
or much lower than for multipactor. Basic theory for ordinary corona
microwave discharge, when the mean free path of the electrons is smaller
than the characteristic length of the device, is well known [9]. However,
the intermediate range, between very low pressure and vacuum, has
received little attention and therefore one of the main topics of this thesis
is devoted to explaining what happens with the breakdown threshold at
these pressures.
Theoretical studies of the multipactor phenomenon have to a great
extent been performed using a one-dimensional model with a spatially
uniform approximation of the electromagnetic field. However, many
common RF devices involve structures where the field is inhomogeneous,
2

where breakdown predictions based on such simple models will not be
reliable. Examples of important geometries in microwave systems where
the field is inhomogeneous are waveguides, coaxial lines, irises and septum
polarisers. An important effect due to the non-uniform field that is
not present when the electric field is spatially uniform, is the so called
ponderomotive or Miller [10] force, which tends to push charged particles
towards regions of low field amplitude. This can have both a qualitative
and a quantitative effect on the multipactor regions. Analytical study of
resonant multipactor in a non-uniform field is not a trivial matter and
most researchers have resorted to numerical methods of investigation.
However, in this thesis different aspects of multipactor in structures
where the field is inherently inhomogeneous are investigated using analytical
methods and the results are compared with numerical simulations
as well as with experimental data found in the literature.
Many microwave systems of today operate in multicarrier mode,
which means that several signals at different frequencies are transmitted
simultaneously. In contrast to the single carrier mode, the electric
field envelope of the multicarrier system varies constantly. In most cases
this is advantageous from a multipactor point of view, as the changing
amplitude will destroy the resonance condition and thus suppress the
discharge. In systems where the frequency separation is small, however,
there is a risk that the signals will interfere constructively for a large
number of field cycles and the amplitude will remain fairly constant,
thus allowing a discharge to develop. In such cases, the microwave engineer
will have to try to find the worst case scenario and design the
component with respect to that case or perform tests that guarantee
that the part fulfils the requirements. Some attention will be given to
these aspects in this thesis, which can be useful for the engineer when
making multipactor free multicarrier microwave designs.
The thesis is organised as follows. A general introduction to basic
theory of multipactor in vacuum is given in chapter 2 as well as some
guidelines when it comes to multipactor free design. It will serve as a
base when continuing with the analysis of multipactor in low pressure
gas in chapter 3, where first a simple model is presented, which only
considers the friction force of the gas molecules. It is then followed by
a more advanced model, which includes also the effect of impact ionisation
as well as thermalisation of the electrons. Starting with waveguide
irises in chapter 4, vacuum discharge in structures where the field is
non-uniform is considered. It is followed by a detailed analytical and
3

numerical study of the phenomenon in a coaxial line in chapter 5. Finally,
chapter 6 is devoted to different means of detecting multipactor
with special focus on detection by means of RF power modulation.
4

Chapter 2
Multipactor in vacuum
Multipactor normally occurs at microwave frequencies, i.e. at 300 MHz -
30 GHz. When discovered by Farnsworth in the 1930’s, he applied the
technique to amplify an electric current. Others have also tried to find
useful application of the phenomenon, e.g. in multipactor duplexers and
switches [11] and in electron guns [12,13]. However, during the last 30
years it has mainly been studied due to its detrimental effects on microwave
components. It has been found to cause electric noise, which
reduces the signal to noise ratio, a very serious problem if it occurs
e.g. in a communication satellite where the signal power is limited and
counter-measures are difficult or impossible to implement. It can also
detune microwave cavities, commonly used as e.g. resonators in filters,
thus reflecting the incoming power back to the power amplifier. If the
system does not have an appropriate power protection device, the amplifier
may suffer permanent damage. Another concern is heating, which
is a result of the power dissipated to the device walls as the multipacting
electrons strike the walls. Furthermore, the discharge can also cause
direct physical damage to the component with the risk of permanently
changing the electric properties of the device. However, the risk of such
direct damage seems low, especially for metallic components. In cases
where damage has been reported, it is not certain that it was caused by
multipactor [14,15]. Multipactor is known to be able to trigger ordinary
gas discharges [16–18], either by increasing the outgassing from the component
or just by starting the breakdown at a pressure and a voltage
where corona is not expected, since gas breakdown can be sustained at a
much lower voltage than what is needed to initiate breakdown directly.
Corona discharges are much more energetic and are known to be able
5

to physically damage microwave components. Many researchers thus
suspect that the observed damage was due to a multipactor induced gas
discharge.
This chapter will present the basic theory of vacuum multipactor
between two metallic parallel plates with an applied homogeneous, harmonic
electric field. It is divided into two major parts, one describing
the single carrier case and another devoted to multicarrier multipactor.
2.1 Single Carrier
One of the first communication satellites, Telstar I, operated in single
carrier mode. It had a capacity of 12 simultaneous telephone conversations
[19] and the solar panels provided a power of only 15 watts. Today,
satellites operate in multicarrier mode with powers of several kilowatts
and new satellites are being designed for tens of kilowatts. Thus, the
single carrier mode is seldom found in real applications. Nevertheless,
the single carrier case is important as it has been thoroughly studied
over the years and by making certain assumptions, the multicarrier case
can be approximated by the single carrier state and design and testing
can be done based on the simpler situation.
2.1.1 Basic theory
There are two main kinds of multipactor, the single-surface and the
double-surface types. Single-surface multipactor can occur in structures
with nonuniform field or with crossed electric and magnetic fields [20],
where the electron, accelerated by the electric field, returns to the original
surface due to the circular motion caused by the magnetic field.
This thesis, however, will focus on double-surface or parallel plate multipactor,
but some attention will be given to single-sided multipactor in
the case of a coaxial line.
A multipactor discharge starts when a free electron inside a microwave
device is accelerated by an electric field. In a strong field the
electron will quickly reach a high velocity and upon impact with one of
the device walls, secondary electrons may be emitted from the wall. If
the field direction reverses at this moment, the newly emitted electrons
will start accelerating towards the opposite wall and, when colliding with
this wall, knock out additional electrons. As this procedure is repeated,
the electron density grows quickly and within fractions of a microsecond
a fully developed multipactor discharge is obtained (see Fig. 2.1).
6

Figure 2.1: Initial stage of parallel plate multipactor, where a free electron
is accelerated by the electric field and is forced into one of the
plates, where it causes emission of secondary electrons.
The motion of an electron in vacuum with an applied electric field
can be studied by means of the equation of motion,
m¨x = eE (2.1)
where m (≈ 9.1 × 10 −31 kg) and e (≈ −1.6 × 10 −19 C) are the mass and
charge of the electron, x the direction of motion, and E the electric field.
Multipactor requires an alternating field and in the parallel-plate model
a spatially uniform harmonic field E = E0 sin ωt is assumed. Solving
Eq. (2.1) with this field yields expressions for the velocity, ˙x, and the
position, x,
˙x = − eE0
cos ωt + A (2.2)
mω
x = − eE0
sin ωt + At + B (2.3)
mω2 where A and B are constants of integration, which will be determined by
the initial conditions. By assuming that an electron is emitted at x = 0
with an initial velocity v0 when t = α/ω, fully constrained expressions
for the velocity and position are obtained, viz.
˙x = eE0
(cos α − cos ωt) + v0
(2.4)
mω
x = eE0
mω2 � � v0
sinα − sin ωt + (ωt − α)cos α + (ωt − α) (2.5)
ω
For resonant multipactor to occur it is necessary for the electron to
reach the other device wall (x = d) when ωt = Nπ + α, where N is an
7

odd positive integer (N = 1,3,5...). Applying this resonance condition
to Eq. (2.5), the following expression is obtained for the amplitude of
the harmonic electric field.
E0 =
mω(ωd − Nπv0)
e(Nπ cos α + 2sin α)
(2.6)
An important quantity when studying multipactor is the impact velocity,
since this determines the secondary electron yield. It can be found
by inserting ωt = Nπ + α in Eq. (2.4), which yields
Multipactor boundaries
vimpact = 2eE0
mω
cos α + v0
(2.7)
When constructing the multipactor boundaries, i.e. the boundaries of
the regions in parameter space where multipactor can occur, an assumption
will have to be made concerning the initial velocity. In reality, the
initial velocity of the emitted electrons will follow some kind of distribution
and a common choice is the Maxwellian distribution [21],
�
(v − vm) 2�
f(v) ∝ exp −
2v 2 T
(2.8)
where suitable parameters for the mean velocity, vm, and for the rmsvalue
(or thermal spread), vT, have to be chosen. When performing
particle-in-cell (PIC) simulations, such a distribution can be used to
more accurately describe the initial velocity of the emitted electrons.
However, for an analytical solution a simpler assumption will have to
be made. There are two common approaches, one which assumes that
the electrons are emitted with a constant initial velocity, v0, regardless
of the impact velocity. The other assumes that the ratio between the
impact and initial velocities is equal to a constant, k = vimpact/v0. Both
these approaches will be used and compared in this chapter, but in the
following chapter, which deals with multipactor in low pressure gas, the
constant k approach will be used only for the simple model while the
constant initial velocity approach will be used for the more advanced
model as that assumption is more physically correct. The reason why
the constant k model has been used to such a great extent is the fact
that it can successfully be fitted to experimental data. The cause of this
success will be explained in the subsection on hybrid modes below.
8

In addition to fulfilling the resonance condition, which resulted in
Eqs. (2.6) and (2.7), the secondary electron yield, SEY or σse, must be
greater than or equal to unity. For most materials the secondary yield as
a function of the impact velocity (or the impact energy, W = mv2 /(2|e|))
has the same shape (see Fig. 2.2), even though the absolute values vary
very much between different materials. The impact energy where the
secondary yield first reaches unity is called the first cross-over point and
is denoted W1, after that the yield increases and reaches a maximum
at Wmax and the energy at which the yield drops below unity again
is called the second cross-over point, W2. Below Wzero no secondary
yield is obtained [22, 23]. However, some researchers have published
measurements of the SEY, which indicate that it is possible that the
SEY does not drop to zero below a minimum impact velocity [24–28].
On the contrary, it can increase after reaching a minimum yield and even
reach a yield close to unity for very low impact velocities. A yield close
to unity implies that the electron does not produce any secondaries, but
rather that the electron bounces off the surface. This could have an
important effect on the multipactor threshold and development, but in
this thesis, the model by Vaughan [22] has been used unless otherwise
specified.
By setting the impact velocity, Eq. (2.7), equal to the first cross over
point (converted to velocity, v1) and taking Eq. (2.6) into account, the
resonant phase, α, can be found as a function of ωd,
tan α = 1
2
� �
2ωd − Nπ(v1 + v0)
. (2.9)
v1 − v0
Using this result, the amplitude can be plotted as a function of ωd or
fd using Eq. (2.6) (or Eq. (2.7)).
One final thing that need to be confirmed before drawing the multipactor
charts is the non-returning electron limit. If the secondary
electrons are emitted before the electric field reverses, the electrons will
be retarded by the field and if the velocity is low, they are likely to
return to the wall of emission and thus being lost as their energy is too
low to produce new secondaries. The limit can be found by solving the
following system of equations:
�
˙x = 0
(2.10)
x = 0
An analytical solution to this system of equations is not possible and in
order to establish the non-returning electron limit, either a numerical
9

σ se [−]
2.5
2
1.5
1
0.5
W 1
W max
Secondary electron yield
σ se =1
0
0 500 1000 1500 2000 2500
Primary electron energy [eV]
Figure 2.2: Secondary electron yield as a function of the impact energy. Plotted
using the formula for secondary electron yield presented in
Ref. [22]. Parameters used are: Wmax = 400 eV, σse,max = 2,
and Wzero = 10 eV.
10
W 2

solution or some kind of approximate solution will have to be used. In
Ref. [21] an approximate formula for the non-returning electron limit is
given and re-writing it for the constant v0 approach yields,
�
16v0
αmin = −
(2.11)
5v0 + 3vimpact
Using Eqs. (2.6), (2.9), and (2.11) with v1 equal to the velocity corresponding
to the unity secondary electron yield, the lower multipactor
threshold can be plotted. However, multipactor breakdown is possible
also for impact velocities greater than v1, in fact, for all impact velocities
between the first and second (v2) cross-over points, the phenomenon can
occur, i.e. for
v1 < 2Vω cos α + v0 < v2 , (2.12)
where Eq. (2.7) has been re-written using the oscillatory velocity Vω =
eE0/mω. Thus, in order to construct the complete multipactor boundaries,
the thresholds for a number of different energies between these two
points should be determined and then the envelope of all the thresholds
will be the complete multipactor susceptibility zone (see Fig. 2.3). Furthermore,
each order of resonance, N, will have its own zone and, as
shown in Fig. 2.3, the zones become narrower with increasing N. This
type of chart, based on the assumption of constant initial velocity, will
be referred to as a Sombrin chart, since J. Sombrin is one of the major
advocates of this assumption [29].
Using the other approach, with a constant ratio between the impact
and initial velocities, k = vimpact/v0, the formulas for the resonant phase,
Eq. (2.9), the amplitude, Eq. (2.6), and the non-returning electron limit,
Eq. (2.11), will have to be slightly re-written:
tan α = 1
k − 1
E0 =
e( k+1
k−1
αmin = −
� kωd
v1
mω 2 d
− (k + 1)N π
2
Nπ cos α + 2sin α)
�
16
8 + 3(k − 1)
�
(2.13)
(2.14)
(2.15)
Using these formulas, multipactor charts similar to the one in Fig. 2.3
can be produced, cf. Fig. 2.4. When this is used, the charts are commonly
referred to as Hatch and Williams charts, as they were the first
11

Voltage [V]
10 4
10 3
10 2
10 0
N=1
N=3
N=5
N=9
N=7
10 1
Frequency − Gap product [GHz⋅mm]
Figure 2.3: Multipactor susceptibility chart based on the constant initial velocity
approach. Parameters used are: W0 = 3.68 eV, W1 =
23 eV, W2 = 1000 eV, and Nmax = 9.
12

who produced charts of this type [30]. Characteristic for the Hatch
and Williams charts are that the multipactor zones are wider than the
Sombrin charts for increasing voltage. This occurs since a constant
vimpact/v0 implies that v0 increases as the impact velocity increases.
When e.g. Wimpact = 3000 eV, this means that for k = 2.5 the initial
energy W0 = 480 eV, clearly an unrealistic initial velocity.
Voltage amplitude [V]
10 4
10 3
10 2
10 0
N=1
N=3
N=5
N=9
N=7
10 1
Frequency − Gap product [GHz⋅mm]
Figure 2.4: Multipactor susceptibility chart produced with the constant k
assumption. Parameters used are: k = 2.5 (corresponding to
an initial W0 = 3.68 eV when Wimpact = W1), W1 = 23 eV,
W2 = 1000 eV, and Nmax = 9.
By knowing the secondary electron emission characteristics of a material
as given by the parameters W1, W2, and W0 or k, multipactor
charts for that material can be designed. However, Woode and Petit
[31] performed a large series of multipactor experiments during the
1980’s and used the Hatch and Williams charts to fit the experimental
data. By tuning the k and W1 parameters for each zone, they were able
to produce multipactor charts that fit the experimental data quite well
(cf. Fig. 2.5). The problem with this empirical approach is that it has
to assume different values for the first cross-over point for each zone in
order to obtain good fitting. This is clearly an unphysical approach and
will not contribute to an improved understanding of the phenomenon,
13

even though it may be sufficient from an engineering point of view. On
the other hand, the higher order modes have a narrower phase-focusing
range (see below), which makes it difficult to compensate for e.g. initial
velocity spread, and thus a secondary yield of unity may not be sufficient
to sustain a discharge. Consequently, an impact energy somewhat
higher than the first cross-over point will be needed when constructing
the lower multipactor threshold for the higher order modes. This will
be discussed further in the subsection “Effect of random emission delays
and initial velocity spread.”
Voltage amplitude [V]
10 4
10 3
10 2
10 0
10 1
Frequency − Gap product [GHz⋅mm]
Figure 2.5: Hatch and Williams charts for aluminium together with measurement
data by Woode and Petit [31].
When the microwave engineer assesses the risk of having a multipactor
discharge, it is usually not the boundary of the individual breakdown
region that is considered. Typically, the lower envelope of the all
the zones is taken as the “design threshold” (cf. Figs. 2.3 and 2.4). By
setting the phase, α, in Eq. (2.7) to zero, the lowest field amplitude to
achieve a certain vimpact is obtained. Thus the lower envelope, which
is the same for both the Sombrin Chart and the Hatch and Williams
chart, is given by,
E0 = (v1 − v0)mω
. (2.16)
2e
14

Phase-focusing
In the previous subsection a mechanism called phase-focusing [1,5,32]
was mentioned and in multipactor theory this is an important concept.
In order for an electron to be a part of the discharge, it must have
a phase close to the resonant phase as given by Eq. (2.9) or (2.13).
Due to delays between the impact and emission of a new electron or
a spread in the initial velocity, an electron will always acquire a small
phase error. Inside the phase-focusing range, such an error will decrease
as the electron traverses the electrode gap. In other words, the phases
of the electrons will tend to converge towards the resonant phase, thus
keeping all electrons close together. Outside the range of phase-focusing,
the error will grow with each passage and after one or a few transits
the electron will be lost. In order for a discharge to occur under such
circumstances, the impact energy has to be large enough to produce a
secondary yield sufficiently above unity to compensate for the incurred
losses.
To see in what range the phase focusing mechanism is active, a small
phase error can be introduced in Eq. (2.5) while keeping the amplitude
and phase constant and setting x = d. The ratio between the final and
initial error is called the stability factor, G [33], and the condition for
stable phase is:
|G| < 1 (2.17)
By setting |G| = 1, the phase range within which the phase is stable can
be obtained. An interesting observation here is that even though the
lower multipactor threshold for the constant k theory and the constant
initial velocity model are identical, the range of stable phases varies substantially
[34]. This can be seen clearly from the analytical expressions
for the phase stability limits, which for constant k theory reads,
φR = arctan( 2
πN (vimpact − v0
vimpact + v0
)) (2.18)
φL = − arctan( 2
) (2.19)
πN
and for the constant initial velocity approach,
φR = arctan( 2
) (2.20)
πN
φL = − arctan( 2
πN (vimpact + v0
vimpact − v0
)) (2.21)
15

where φL and φR are the left and right limits respectively. This difference
is illustrated graphically in Fig. 2.6. However, when v0 ≪ vimpact both
approaches yield the same phase stability limits.
Voltage [V]
10 3
10 2
10 1
Unstable phase range (constant v 0 )
Stable phase range (constant v 0 )
Unstable phase range (constant k)
Stable phase range (constant k)
N=1
10 0
Frequency − Gap product [GHz⋅mm]
Figure 2.6: Lower multipactor thresholds in vacuum for the first 3 orders
of resonance (N = 1, 3, and 5). The curves for the constant k
model are plotted slightly offset as the curves otherwise overlap.
Parameters used are: W1 = 23 eV , W0 = 3.68 eV, and k = 2.5.
Saturation
In order to sustain a multipactor breakdown, the secondary electron
emission yield must be greater than or equal to unity. If the yield is less,
the electron number will quickly decrease and the discharge disappears.
With a σse greater than unity the electron number will grow rapidly with
each impact and if no saturation mechanism is considered the number
of electrons after a time t, if the field frequency is f, will be:
N=3
N=5
Ne(t) = Ne(0)(σse) 2ft
N (2.22)
The rapid growth of the number of electrons can be illustrated with
an example. Suppose σse = 1.5 and f = 2 GHz, then the number of
16

electrons after 20 ns for the first order of resonance with one initial
electron will be more than 10 14 .
In a very short time, the number of electrons will grow to very high
values and it is clear that some kind of saturation mechanism will become
active. Two main saturation processes have been described in the
literature. The first is the space charge effect [3], which is the most
obvious effect. Within the electron bunch the individual electrons will
repel each other causing a change in phase of those electrons and if the
phase error is too large, the probability of losing electrons increases and
eventually the effective secondary yield will be equal to unity and saturation
has occurred. The second type of saturation process [35,36] can
set in if the discharge takes place inside a resonant cavity. Due to a high
Q-value, the electric field strength is high and thus the risk for a discharge
will increase. If a multipactor discharge is started, the electrons
traversing the gap make up an alternating current, which loads the cavity.
Loading the cavity means that the Q-value will decrease and thus
also the electric field strength. It is clear that this is a self-suppressing
effect. As the multipactor current increases, the field strength decreases
and with it the impact velocity of the electrons leading to a lowered
secondary emission yield. Eventually the secondary yield reaches unity
and saturation has been reached.
2.1.2 Hybrid modes
It may seem somewhat contradictory to assert that the model based on
a constant initial velocity is more physically correct than the constant
k theory, when the latter approach can be better fitted to experimental
data. However, as briefly mentioned previously, the reason for this
paradox is the hybrid modes. Some of these modes were identified by
Refs. [29,37,38] and a general treatment is given in [39]. The modes can
be found by allowing N in the resonance condition for Eq. (2.5) to be
a sequence of odd half-cycles of the electric field, {N1,N2,N3...}, where
N1 = N and the remaining Nn ≥ N for the hybrid modes between the
N th and (N + 2) th zones. Each such sequence will result in a narrow
multipactor zone located between the main multipactor areas. The lowest
order hybrid mode in the parallel-plate case is the {1,3} mode, which
means that the transit time in one direction takes 1/2 RF-cycle and the
return transit takes 3/2 RF-cycles. This mode is then also associated
with two different resonant phases, viz. α1 = 0 and α2 = π/3 [39].
This mode can be found between the two first classical resonance zones
17

(cf. Fig. 2.7). When taking the envelope of these zones, the differences
in the right boundaries of the main zones, between the constant initial
velocity and the constant k approaches, become negligible. This can be
seen clearly in Fig. 2.7 [40]. The existence of the hybrid modes requires
Figure 2.7: Vacuum multipactor with the main zones as well as a few hybrid
zones [40]. With the envelope of the hybrid zones included
the resemblance between the Sombrin chart and the Hatch and
Willams chart (cf. Fig. 2.5) is striking.
phase stability, just as for the classical zones discussed above. However,
the width of each hybrid zone is very small and thus it is very sensitive
to an initial velocity spread. On the other hand, there are many hybrid
zones very close to one other and this spread will result in a mixing or
overlapping of the resonances [39].
2.1.3 Factors effecting the threshold
There are many different aspects that need to be considered when determining
the multipactor threshold. The most important and most obvious
ones are type of material, gap size, and amplitude and frequency
of the electric field. These are all part of the basic theory as described
above. Apart from these there are other more or less important factors.
The supply of primary electrons does not effect the theoretical
threshold, which can be determined with methods described earlier in
this thesis. Nevertheless, a weak source of seed electrons can result in
an apparent higher threshold during testing. In a typical test setup for
determination of the breakdown amplitude, an electric field is applied
18

and the field strength is increased at regular intervals. If no electron
is in an advantageous position, i.e. has a suitable phase from a multipactor
point of view, when the right amplitude is set, a discharge will
not occur. As the amplitude is increased further, the impact velocity
of any free electrons, also those that are not in a favourable position,
will be high and the secondary yield will be an additional source of free
electrons. Thus the chances of getting a breakdown increases until it
eventually occurs. For experimental use, a hot filament or a radioactive
source can be used to produce a sufficient amount of free electrons to
achieve reliable measurement results [14].
Another factor that can have a significant effect on the threshold
is contamination. Both the first cross-over point, W1, and the maximum
secondary yield, σse,max, can be drastically affected. A lowered
W1 means that a discharge can occur at a lower voltage and an increased
σse,max can result in a faster growth of the total number of electrons.
In Ref. [31] a detailed analysis of the impact of different types of contaminants
was made. It was noted that the plastic bags, which were
normally used to protect the microwave components from dirt, were the
main source of contamination. A threshold reduction of up to 4 dB
was found. Also dust and fingerprints were a direct source of a lowered
threshold. In the report [31] it was recommended that cleaned
microwave parts for space use should be handled with cotton gloves and
stored in hard plastic boxes.
Microwave parts which have not been properly vented before power
is applied can also have a threshold that is different from the expected
multipactor threshold. If there is too much gas, corona breakdown may
occur, and within a certain range of pressures, close to the minimum of
the so called Paschen curve, the breakdown threshold can be significantly
lower than in the multipactor case. In the pressure range corresponding
to the transition region between corona and multipactor, a higher
threshold can sometimes be expected. More details about this will be
presented in chapter 3.
Other factors that can have an indirect effect on the multipactor
threshold are the voltage standing wave ratio, VSWR, and the temperature.
If the VSWR is greater than what was intended with the design,
the peak field strength in the system will also be greater than expected
and thus a discharge may occur at a lower power level than assumed. An
increased temperature can lead to increased outgassing from the device
walls resulting in concerns similar to those of improper venting.
19

2.1.4 Methods of suppression
Many of the factors mentioned in the previous section that affect the
multipactor threshold can also be utilised to suppress the discharge. The
without doubt easiest method of avoiding a breakdown is to pressurise
the component. The field strength required to achieve breakdown at
atmospheric pressure is in general much higher than at low pressures or
in vacuum. However, such a method is seldom feasible for components
that will be used e.g. in space, where the external environment is a high
vacuum. A small leakage can lead to slow venting of the component and
thus risking severe corona discharge when the pressure reaches the range
where the minimum breakdown field occurs.
Another way of suppressing multipactor is to amplitude modulate
the main carrier [21, 41]. If both signals are sinusoidal, the total field
can be written:
Etot = E1 sinω1t + E2 sin ω2t (2.23)
This means that the envelope of the signal will vary according to (see
also Fig. 2.8):
�
Eenv = E2 1 + E2 2 + 2E1E2 cos (ω1 − ω2)t (2.24)
When the total field strength is well above the multipactor threshold
(see Fig. 2.8), the secondary electron yield will increase quickly according
to Eq. (2.22). However, as soon as the voltage drops below the
threshold again, the electron loss will be large and according to Ref. [21]
all electrons will be lost in just a few RF cycles. However, whether or
not this is true also depends on the secondary yield properties of the
electrode material. For materials with a very high maximum secondary
yield, the number of electrons gained while above the threshold can be
greater than the losses incurred while below. In such a case, no suppression
is achieved and in some cases, the discharge may even become
more powerful than before the modulation carrier was added [42] (cf.
Fig. 2.9). Thus in order to successfully suppress a multipactor discharge
using amplitude modulation, it is vital that the material has a low maximum
secondary yield (preferably less than about 1.5). Due to the risk
of contamination, which can greatly increase the maximum secondary
yield, great care should be taken to assure a high level of cleanliness if
this method of suppression is to be used.
To AM-modulate the carrier is probably not feasible in most cases, as
it would require extra hardware to produce the AM-signal. However, the
20

Amplitude
1.5
1
0.5
Multipactor threshold
Amplitude Modulation, E0/E1=0.4
Main signal
Modulation signal
Sum signal
Envelope
0
0 1 2 3 4 5
Time
6 7 8 9 10
Figure 2.8: Two signals and their sum signal (absolute values). The envelope
varies and is partly above and partly below the multipactor
threshold.
typical bandpass filtering of a PSK (Phase-Shift Keying) signal causes
modulations in the time-domain. The QPSK (Quadrature Phase-Shift
Keying) signal in Fig. 2.10 has unity amplitude before filtering. Afterwards,
the peaks are higher than the original amplitude, but the troughs
can sometimes go down almost to zero amplitude. Comparing this with
the AM-suppression, it is clear that an electron avalanche that is initiated
during the peak periods, will be extinguished as the amplitude
falls close to zero. However, for a typical PSK-signal, the duration between
phase shifts (which normally coincides with the troughs) is several
hundreds of RF cycles. Thus for most microwave systems, there will be
ample time for a discharge to develop. But, when the amplitude drops
below the threshold, the electron bunch will disappear and when the
amplitude increases above the threshold again, there may not be any
seed electrons present to restart the electron avalanche. Thus, the system
will have sporadic discharges, which, if they do not occur too often,
may not seriously degrade the signal.
A very common way of suppressing a vacuum discharge is to apply
a coating [26–28], a surface treatment [43], or a film [44] that has a
high first cross-over point as well as a low maximum secondary electron
21

Noise (dBm)
Power #1 (dBm)
Match #1 (dB)
Power #2 (dBm)
−65
−70
−75
−80
−85
Amplitud Modulation, w2/w1=1.16)
0 1 2 3 4 5 6 7 8 9 10
x 10 4
48
47.5
47
46.5
0 1 2 3 4 5 6 7 8 9 10
x 10 4
40
20
0
0 1 2 3 4 5 6 7 8 9 10
x 10 4
−20
−11
−11.5
−12
0 1 2 3 4 5 6 7 8 9 10
x 10 4
Time (ms)
Figure 2.9: Multipactor experiment with two carriers with E2/E1 = 0.36
and ω2/ω1 = 1.16. Due to a high maximum secondary yield,
multipactor suppression is not possible (the material used in the
experiment was plain aluminium, which can have a σse,max ≈ 3).
When the modulation signal is applied, the magnitude of the
multipactor noise increases significantly.
Signal amplitude [−]
1
0.5
0
−0.5
−1
Square Root Raised Cosine filtered QPSK signal
1 1.5 2 2.5
Time [µ s]
3 3.5
Figure 2.10: Example of a QPSK modulated signal after bandpass filtering.
22

emission yield. So far, no practical coating with a σse,max below unity
has been found. However, alodine is a commonly used surface coating for
space-bound microwave devices made of aluminium. It increases the first
cross-over point to around 60 eV and reduces σse,max to about 1.5, even
though the actual values vary much between samples. The concern with
a material with very good anti-multipactor properties is contamination.
A few fingerprints or a very small layer of dust can drastically alter the
properties of the material and make it prone to discharges.
By applying a DC electric or magnetic field, the electron trajectory
can be disturbed and the important resonance condition can be
destroyed, thus making multipactor impossible. Simulations [45] have
shown that an external DC magnetic field applied in the direction of
wave propagation in a rectangular waveguide can efficiently suppress
multipactor. A drawback with the method is the extra components required
to produce the magnetic or electric field and thus the method
may not be feasible for e.g. space applications, where extra weight is
undesirable.
The most efficient way of avoiding multipactor is to make a design
where the mechanical dimensions are such that a power much higher
than the nominal power is required to start a discharge. However, that
may lead to large and heavy designs, which are to be avoided in space
systems, and thus one may have to resort to one or several of the above
mentioned methods of multipactor suppression.
2.1.5 Effect of random emission delays and initial velocity
spread
In the above analysis of multipacting electrons in a harmonic electric
field, it was assumed that all secondary electrons were emitted with a
fixed initial velocity, v0. However, as briefly mentioned previously, the
electrons are actually emitted with a distribution of velocities and the
Maxwellian distribution is often used in simulations. Apart from the
spread in initial velocities, there is also a finite time between impact of
the primary electron and emission of the secondaries. Since this time in
most cases is very small compared to the RF period, it was neglected in
the previous analysis. Nevertheless, this time will cause a small phase
error and if the resonant phase is close to the phase stability limits as
given by Eqs. (2.18) - (2.21), the phase error may result in an increased
electron loss.
A detailed analysis of the effect of random secondary delay times and
23

andom spread in emission velocities was done by Riyopoulos et al. [33].
They found that by including the effects of these random parameters,
the effective secondary electron yield, σ ∗ se, was reduced to a number in
the range σse/2 < σ ∗ se < σse. This means that the effective secondary
electron yield will be a function not only of the impact velocity, but also
of the resonant phase as well as the phase spread caused by the spread
in initial velocities and secondary delay times. Another study, which
supports this result, investigated the effect on the different resonance
zones for different values of the maximum SEY due to initial velocity
spread [46]. It was found that, except for the first order mode, a realistic
thermal spread of the initial electrons raised the multipactor SEY
requirement from unity to above unity. For the higher order modes a
SEY greater than approximately 1.5 was necessary to compensate for
the losses incurred. In addition, with increasing velocity spread, the
multipactor zones started to overlap. The increased SEY requirement
will result in an increased threshold for the higher order modes and can
explain the success with increasing the first cross-over point in the Hatch
and Williams charts when fitting experimental data (see Fig.2.5).
The importance of the spread in initial velocities can be seen when
constructing multipactor charts for a constant initial velocity without allowing
compensation for electron losses outside the phase stability range.
In Fig. 2.11 zones bounded by solid lines indicate the region where multipactor
can take place under this assumption. The dashed lines make
up wider zones that encompass the other zones and are identical to the
zones shown in Fig. 2.3.
By including a higher secondary electron yield and a spread in initial
velocities, the multipactor zones will become wider than the solid
line zones. A σse greater than unity, which will be the case when the
impact velocity is greater than the first cross-over point, will compensate
for some of the losses incurred due to phase instability. A spread
in initial velocities will widen the range of possible resonant phases (cf.
Eq. (2.9)) and the left and right limits will not be as sharp as indicated
by the solid line multipactor zones in Fig. 2.11. This widening of the
multipactor zones has been taken into account to a certain extent in the
traditional analytical approach, which is indicated by the wider dashed
line zones in Fig. 2.11. However, the widening should not only be towards
the left side, but also towards the right [39]. Furthermore, the
sharp lower left corner of each dashed line zone is misleading, as that
indicate a point where the secondary electron emission is unity and the
24

Voltage [V]
10 4
10 3
10 2
10 0
10 1
Frequency − Gap product [GHz⋅mm]
Figure 2.11: Multipactor charts based on the same parameters as in Fig. 2.3.
The solid line zones indicate the zones within which phasefocusing
is active. The dashed line zone is produced by including
also unstable phases until the non-returning electron limit.
phase is very unstable, thus making a discharge impossible. A more
correct boundary would be a rounded shape, which starts in the lower
left corner of the solid line zone and smoothly joins the dashed left hand
side [47]. This is confirmed by experiments [30,48], cf. Fig. 2.12, which
shows measurement data from one of the early multipactor experiments
by Hatch and Williams [48]. A similar rounded shape can also be seen in
numerical simulations and examples of this is shown in chapter 5, which
includes PIC simulations of multipactor in a coaxial line.
2.2 Multicarrier
Modern satellites operate in multicarrier mode, i.e. several signals at different
frequencies exist simultaneously in the microwave and electronic
systems. An example of such a system is Sirius 3, which is one of the
Nordic satellite [49]. It has 15 channels in the frequency range 11.7 -
12.5 GHz and each channel has a bandwidth of 33 MHz. Assume that
each channel has a power of 200 W. Then the maximum instantaneous
power of the system, the peak power, is equal to 45 kW. The peak power
increases with the square of the number of carriers. Such a high instan-
25

Figure 2.12: Multipactor experiment [48] showing the expected rounded off
lower left corner of the first multipactor zone [47].
26

Amplitude [V]
15
10
5
0
0 5 10 15 20
Time [ns]
Figure 2.13: In-phase multicarrier
signal. The signal oscillates
rapidly, which
makes the signal envelope
appear clearly in
this time resolution.
Amplitude [V]
15
10
5
0
0 5 10 15 20
Time [ns]
Figure 2.14: Random phase multicarrier
signal.
taneous power is very unlikely in a real system since it will occur only
when all the signals are in phase as illustrated by Fig. 2.13. The most
likely scenario, if the carriers are not phase locked, is that the phase of
each carrier is a random number and will result in a signal with much
lower maximum instantaneous power as illustrated in Fig. 2.14.
The signals in Figs 2.13 and 2.14 are characterised by all carriers
having the same amplitude and a constant frequency. Consider a signal
with N carriers, each carrier having the same amplitude E0, but different
phases φn and with a frequency spacing ∆f. The period of the envelope
will then be T = 1/∆f and the envelope is given by
Eenv = E0
�
��
� N−1
�
�
cos (n2π∆ft + φn)
n=0
�2
+
� N−1
�
n=0
sin(n2π∆ft + φn)
(2.25)
A more realistic signal would have different amplitudes for each carrier
and the frequency spacing would not be constant. The envelope of
such a signal can be found from
�
��
� N−1
Eenv =
�
�
En cos (knω0t + φn)
n=0
�2
+
� N−1
�
n=0
En sin(knω0t + φn)
�2
�2
(2.26)
27

where kn is a factor determining the frequency spacing
kn = fn/f0 − 1, n = 0,1,...,N − 1 , (2.27)
f0 is the lowest carrier frequency and ω0 = 2πf0. When assessing the
worst case scenario from the multipactor point of view, it is important
to study a whole envelope period. For arbitrarily spaced frequencies, the
envelope period, T, can be found by solving the following Diophantine
systems of equations:
T = ni
, ni ∈ N i = 1,2,...,N − 1 (2.28)
∆fi
where N is the number of carriers, f0 is the signal with the lowest frequency
and ∆fi = fi − f0. The envelope period will be the solution
with the smallest possible integers. For equally spaced carriers, the solution
becomes n1 = 1, n2 = 2,...,nN−1 = N − 1, which is implies that
T = 1/∆f, like before.
When studying multicarrier multipactor it is common to make certain
simplifications that will allow using single carrier methodology to
asses also the multicarrier case, e.g. the mean frequency of all the carriers
is used as the design frequency. Thus, most of what has been said
about single carrier multipactor will then be valid also for the multiple
signals case.
2.3 Design guidelines
From an industrial point of view it is important not only to understand
the physics of multipactor, but also how the theoretical and experimental
results should be applied when making multipactor-free microwave
hardware designs. In Europe, most space hardware designers follow the
standard issued by ESA [50]. This standard includes both the single
and the multicarrier cases, but for the latter it is stated that the design
guidelines are only recommendations. Most research support these
recommendations, but not enough tests have been performed to verify
the theoretical findings. When using the standard it is important to be
aware of the fact that it is primarily based on the parallel-plate model
with a uniform electric field. Design with respect to this approach for
other geometries is normally a conservative and safe way. However, in
many common microwave structures, the geometry is such that losses
of electrons is much higher than in the parallel-plate case. Thus the
28

multipactor threshold in geometries such as coaxial lines, waveguides
and irises, can be higher or much higher than that obtained using the
plane-parallel model.
2.3.1 Single carrier
In the ESA standard [50] components are divided into three categories
or types. Type 1 is a well vented component where all RF paths are
metallic and the secondary electron emission properties are well known.
This type of component has the lowest design margins with respect to
multipactor and, depending on the type of test, range from 3-8 dB.
The second type of component may contain dielectrics with established
multipactor properties and the component should be well vented. Also
depending on the type of test, the design margins range from 3-10 dB.
All other components are categorised as type 3 and the design margins
range from 4-12 dB.
When designing with respect to multipactor a complete electric field
analysis is performed and regions with high voltages and critical gap
sizes are identified. Using the frequency-gap size product, the multipactor
threshold can be found in a susceptibility chart for the material
in question. A susceptibility chart in the ESA standard is basically an
envelope of the multipactor zones as shown in e.g. Fig. 2.5. If a margin
larger than the largest design margin, 12 dB, is found, no testing is
required. However, in most cases, the component will have to be tested
and methods for detecting multipactor will be discussed in chapter 6.
2.3.2 Multicarrier
In the multicarrier case, only components of type 1 are covered by the
recommendations given in the ESA standard. Type 2 and type 3 components
will require further research before they can be included in the
standard. In the single carrier case, the level that is compared with the
multipactor threshold in the susceptibility chart is the amplitude of the
signal and no ambiguities exist. For multicarrier designs, the traditional
way of designing was to set the design margin with respect to the peak
power of in-phase carriers, shown in Fig. 2.13. This design method is
still allowed by the ESA standard and for type 1 components the design
margins range from 0-6 dB depending on the type of testing that will
be performed. However, as previously mentioned, in-phase carriers for
non-phase locked signals is extremely unlikely and thus the standard
29

allows for another design margin, which is set with respect to the so
called P20 power level. The P20 level corresponds to the “peak power of
the multicarrier waveform whose width at the single carrier multipaction
threshold is equal to the time taken for the electrons to cross the multipacting
region 20 times” [50]. This level is illustrated in Fig. 2.15.
Figure 2.15: An example where the in-phase peak power is above the single
carrier threshold, while the P20 level is more than 4 dB below the
same threshold. The peak voltage is 128.4 V, the single carrier
threshold is 91 V, and the P20 voltage is 57 V. Signal data: 12
carriers, equally spaced, fmin = 1.545 GHz, ∆f = 24 MHz and
each carrier amplitude is 10.7 V. Material properties: W1 =
23 eV and σse,max = 3.
In the case when a design is made with respect to the P20 level, the
design margins range from 4-6 dB depending on the type of testing. A
problem with the P20 level is that it is not a trivial problem to find the
peak power level for 20 electron gap crossings. This power level is usually
referred to as the worst case scenario, even though it may not always be
the worst case from a multipactor point of view. A number of different
30

ways of finding the worst case scenario have been proposed, e.g. using
parabolic or triangular phase distribution in the equally spaced carriers
case (cf. Ref. [51]). Some of the better methods for finding the worst
case scenario will be described in the following subsections after a brief
discussion about the 20 gap crossings rule (TGR).
Twenty gap crossings rule
The TGR was proposed in Ref. [14] in 1997 and in its original version
it reads:
“As long as the duration of the multicarrier peak and the mode order
of the gap are such that no more than twenty gap-crossings can occur
during the multicarrier peak, then multipaction-generated noise should
remain well below thermal noise (in a 30 MHz band).” [14]
The rule is a result of an analysis of simulated multi-carrier multipactor
discharges. Comparison with experiments showed great deviations,
where the simulated noise could be as much as 75 dB greater
than the measured noise level. In the experiments, a minimum of 99
gap-crossings were required before the produced noise was detectable
above the noise floor of -70 dBm. Of course, there may be bit errors
even at lower noise levels, but as the number of electrons grows exponentially
with the number of gap crossings (see Eq. (2.22), there is a
huge difference between 20 and 99 gap-crossings.
However, the TGR is certainly a good first attempt to lower the
requirements for multi-carrier multipactor. It is a fairly conservative
method and thus the risk of applying it should be quite limited. However,
more appropriate guidelines should be based on an unambiguous
theoretical concept, which can take the material properties into account.
Then, when performing simulations and experiments to verify the idea,
it is of paramount importance to make sure that the actual material
properties of the test samples are well known and that these properties
are also being used in the simulations. Due to the large difference
in secondary emission properties between different materials, it would
seem reasonable that for a material with a low σse,max one would allow
more gap crossings than, in the opposite case, for a material with a high
σse,max.
31

Boundary function Method
One of the best engineering methods for finding the worst case scenario
for equally spaced carriers, the boundary function method, was originally
designed by Wolk et al. [51]. Unfortunately the used function was
found empirically while studying the worst case scenario with an optimisation
tool, and is thus not physically founded. A consequence of this is
that under certain circumstances, the boundary function produces poor
results. It was also limited to work only for equally spaced carriers. However,
as part of the present thesis work, this method has been further
developed, and it has been found that the original boundary function
approximately describes a function that tries to squeeze all the energy
of the multicarrier signal during one envelope period into a specified,
shorter, time period. This works just as well in both the equally spaced
and the non-equally spaced carrier cases and can be summarized by the
following formulas:
⎧ �
FV (TX) =
⎪⎨
FV,max = N�
⎪⎩ FV,min =
TH
TX
Ei
i=1
� N�
E
i=1
2 i
N�
E
i=1
2 i
(2.29)
Here TX is the time period of interest, which is often set to T20, i.e.
the time it takes the electrons to traverse the gap 20 times. TH is the
period of the envelope and Ei is the voltage amplitude of each carrier.
FV (TX) is the design voltage and is shown as two symmetric curved lines
in Fig. 2.15. The design voltage can never exceed the in-phase voltage,
given by FV,max, and if all power is distributed evenly over the entire
envelope period the voltage amplitude will be FV,min, which is indicated
by a dashed line in Fig. 2.15.
The main advantage with the boundary function method is its simplicity.
It is also very reliable, although a little conservative and this is
especially true for non-equally spaced carriers, where the P20 level can
be much lower than FV . The method has been implemented as an auxiliary
method in WCAT, which is a software tool originally developed by
the present author and Genrong Li as part of a Master’s Thesis [52] at
32

Saab Ericsson Space. It has since been upgraded with additional functionality
by the present author as well as by Mariusz Merecki as part of
a Master’s Thesis [40] at Centre National d’ Études Spatiales, Toulouse,
France. Fig. 2.16 shows the graphical user interface of the present version
of WCAT and an example when the worst case of non-equally spaced
carriers have been assessed using the built in genetic algorithm.
200
150
100
50
Threshold =192
Hyb. Threshold =158
Env Threshold =139
Boundary Function
FVmin
In−Phase Envelope
Envelope
← Fv =129
10 3
10 3
10 2
10 2
10 1
10 1
0
0 5 10 15 20 25 30 35 40 100
0 5 10 15 20 25 30 35 40 100
Figure 2.16: Assessment of worst case scenario for multicarrier multipactor
using WCAT, Worst Case Assessment Tool. The example shows
a case with 10 non-equally spaced carriers with varying amplitude.
Optimisation methods
A more direct method of finding the worst case scenario is to use some
kind of optimisation tool. In WCAT several different methods of finding
the worst case scenario are implemented. One uses the non-linear least
square (NLSQ) functionality of Matlab to find the set of phases that will
fit as much energy as possible inside a time period TX ≤ TH, where TH
is the envelope period. Another method generates a variable number of
sets of random phases and the corresponding envelopes are compared to
find the worst case. This method is better than the NLSQ optimisation
method when it comes to finding the worst case for non-equally spaced
33

carriers, because the NLSQ method requires a good seed phase in order
to find a good local minimum. By combining these methods and using
the result of the best random phase approach as a seed phase for the
NLSQ optimisation, the best results are found. The problem with a
random phase approach is that for a large number of carriers, the number
of phase-sets needed to achieve a good result becomes discouragingly
high [52].
By using a genetic algorithm, a great improvement in finding the
worst case scenario in a short time has been achieved, especially for the
non-equally spaced carrier case. This type of optimisation scheme has
the advantage of being able to find not only a good local minimum as
in the NLSQ case, but the actual global minimum can be found. The
genetic algorithm was implemented by Merecki [40] and in addition to
improving the optimisation part of the software, he also implemented
the threshold of the hybrid modes (see Fig. 2.16) as well as many other
useful functions.
The main problem with multipactor in microwave systems is the
electric noise that is generated, which degrades the signal to noise ratio.
Thus the worst case may not always be the maximum power within the
T20 period. Depending on the material properties some other case may
produce a substantially larger amount of energetic electrons. Therefore,
WCAT also includes the possibility of finding the phase-set that will
produce the largest amount of electrons.
In addition to assessing the risk for a vacuum discharge, it can also
be of value to investigate if a multipactor free design may have a risk
of corona breakdown if the component is not thoroughly vented when
it is brought into operation. In WCAT this is analysed using the mean
carrier frequency and comparing the corona threshold with the in-phase
peak voltage. If the minimum of the Paschen curve is greater than this
voltage, then the corona margin is displayed in the output window of
WCAT. If the opposite is true, the pressure range within which there is
risk for corona discharge is presented (see Fig. 2.16).
34

Chapter 3
Multipactor in low pressure
gas
Microwave discharges can occur in both gas and vacuum. In vacuum, the
phenomenon is usually called multipaction or multipactor and the theory
for such vacuum microwave breakdown was presented in the previous
chapter. In a gas it is normally called corona discharge or gas breakdown
and it can occur when the electron mean free path between collisions
with molecules is smaller than the characteristic dimensions of the vessel.
An applied microwave electric field can widen the velocity distribution
of the free electrons and thus make more electrons energetic enough to
ionise the gas. If the production of electrons exceeds the loss through
diffusion, attachment, and recombination, the electron density will grow
exponentially and microwave gas breakdown will occur.
When the mean free path between collisions is of the same order as
the device dimensions, classical theory for microwave gas and vacuum
discharge can not be used. Diffusion loss can no longer be assumed,
like in the gas breakdown case, since that requires a mean free path
several times shorter than the characteristic length of the component.
Nevertheless, the electrons will meet a resistance due to collisions with
the neutral gas molecules and thus pure vacuum can not be assumed
either.
Low pressure multipactor has received comparatively little attention.
However, a few studies, theoretical as well as experimental, have revealed
some parts of the complicated picture. Vender et al. [17] performed
PIC-simulations to study the electron density development and showed
that at sufficiently low pressures, the gas discharge is initiated by a
35

multipactor discharge. Using a Monte Carlo algorithm, Gilardini [53]
made quite a general study of the phenomenon and presented breakdown
voltages normalised to the first cross-over point of the material for a
wide range of dimensionless variables. He also paid special attention
to a particular and realistic case, namely multipactor in low pressure
argon [54]. This was done partly in an effort to compare the simulations
with the experimental results of Höhn et al. [18].
In paper B of this thesis, low pressure multipactor was studied using
an analytical model that takes into account only the friction force due
to collisions between the electrons and the neutral gas particles. The
main theory and results from this study will be presented in the first
section below. In addition to the friction force, the collisions will also
cause a random velocity spread of the electrons that results in a higher
average impact energy. Furthermore, due to the long distance between
molecules, the electrons are free to accelerate to very high velocities and
upon impact with a gas molecule or atom the energy is sufficient to cause
ionisation. In paper C of this thesis a more detailed analysis has been
done, where all these effects have been considered and the used model
as well some highlights from the results are presented in the section
“Advanced Model” below.
3.1 Simple Model
In a first attempt to understand the behaviour of multipactor in a low
pressure gas, a simple analytical model was used, which takes only the
friction force of the collisions with neutrals into account. By deriving explicit
expressions for the multipactor threshold, qualitative comparison
with experimental results [18] as well as results from computer simulations
[53,54] could be made.
3.1.1 Model
The differential equation governing the behaviour of the electrons in
a low pressure gas is given by the equation of motion, Eq. (2.1), but
augmented to include also the effects of collisions:
m¨x = eE − mνc ˙x (3.1)
where νc = σcn0v is the collision frequency between the free electrons
and the neutral particles. σc is the collision cross-section, n0 the neutral
36

gas density and v the electron velocity. The collision cross-section is
generally a function of the electron velocity, but in order to avoid a
non-linear differential equation, σc is assumed to be a constant.
As in the vacuum case, a spatially uniform harmonic field E =
E0 sin ωt is assumed. Solving Eq. (3.1) with the same initial conditions
as for the vacuum case, i.e. that an electron is emitted from x = 0 with
an initial velocity v0 when t = α/ω, the position and velocity of the
electron can be found:
where
x = 1
νc
α
νc(
(1 − e ω −t) )(v0 + Λ[ω cos α − νc sin α])
+ Λ
ω [ω(sin α − sin(ωt)) + νc(cos α − cos(ωt))] (3.2)
α
νc(
˙x = v0e ω −t) α
νc(
+ Λ[e ω −t) (ω cos α − νc sin α)
− ω cos(ωt) + νc sin(ωt)] (3.3)
Λ =
eE0
m(ω 2 + ν 2 c )
(3.4)
The resonance condition requires that an electron emitted when t =
α/ω should reach the other electrode, at x = d, when ωt = Nπ + α,
where N is an odd positive integer as in the vacuum case. Applying
this condition to Eqs. (3.2) and (3.3) yields expressions for the required
electric field and the impact velocity:
E0 =
m
e (ω2 + ν 2 c)(d + v0
νc
(e− Nπνc
ω − 1))
Nπνc
Nπνc
− − (1 + e ω )sin α + ((1 − e ω ) ω
νc
2νc + ω )cos α
(3.5)
Nπνc
Nπνc
− −
vimpact = v0e ω + Λ(1 + e ω )(ω cos α − νc sin α) (3.6)
In order to draw the multipactor boundaries, an expression for the
non-returning electron limit is needed. Like in the vacuum case no explicit
analytical expression for this can be found and thus the limit will
be obtained numerically instead. However, as a rough approximation,
the limit given by Eq. (2.11) can used.
3.1.2 Multipactor boundaries
When constructing only the lower multipactor threshold in vacuum,
which depends on the first cross-over energy, the threshold value under
the assumption of a constant initial velocity is the same as with
37

the assumption of a constant ratio k = vimpact/v0 between impact and
initial velocities. This is true also in the presence of collisions, when the
above simple model is used and thus the constant k approach will be
used in the following expressions. Combining Eqs. (3.5) and 3.6 under
this assumption yields an expression for the resonant phase, viz.
tan α = ω2 [βΦ + γ] + 2ν 2 cvimpact(Φ − k)
ξω(Φ + 1)
(3.7)
where Φ = exp(−νcNπ/ω), β = kdνc + (k + 1)vimpact, γ = kdνc −
vimpact(k + 1), and ξ = [kdνc + (k − 1)vimpact]νc.
Equation (3.7) can be used together with Eq. (3.5) to plot the multipactor
threshold in a low pressure gas as a function of gap size, pressure,
or frequency. However, by multiplying the expression for the amplitude
of the electric field with the gap size, d, an expression for the voltage as a
function of the frequency-gap size and the pressure-gap size products can
be obtained. This approach will be used in a subsequent section, where
a more advanced model is used to analyse the phenomenon. Figure 3.1
shows the lower multipactor threshold in low pressure air for different
pressures. The graphs are based on Eqs. (3.5) and (3.7) only and do not
consider the non-returning electron limit nor the phase stability limits.
From Fig. 3.1 it is clear that the multipactor threshold increases
with increasing pressure. By sweeping the pressure instead of the frequency
and comparing the changing threshold with the corona breakdown
threshold, an understanding can be obtained of how the transition
between these two types of discharges can occur. Fig. 3.2 shows that
the multipactor threshold first increases until a certain point, where it
intersects the curve corresponding to the corona threshold. It will then
follow this curve towards the minimum of the Paschen curve. This is
just a qualitative picture and the sharp intersection would in reality be
a smooth transition.
Figure 3.1 does not consider the non-returning electron limit, nor
the phase stability limits. As explained in the previous chapter, phase
focusing is needed to maintain the generated electrons in a close bunch,
since an electron with a too large phase error will be lost. By introducing
a small phase error in Eq. (3.2) and keeping the amplitude and phase
constant while setting x = d, the error after the passage can be found.
The ratio between the final and initial error is the stability factor, G,
and when the absolute value of this factor is less than one, the phase
focusing effect is active. With the present model, the expression for the
38

Voltage (V)
10 3
10 2
10 1
Multipaction threshold curves in low pressure air (d=0.1 m)
p=0.1 Pa
p=1 Pa
p=10 Pa
Vacuum
10 −3
Frequency (GHz)
Figure 3.1: Multipactor chart showing the lower multipactor threshold at
different pressures. i.e. each curve is based on an impact velocity
corresponding to W1 = 23 eV. Phase stability and the
non-returning electron limit are not considered, only resonance.
Parameters used are: σc = 6.9 × 10 −20 m 2 , k = 2.5, N = 1 and
d = 0.1 m.
10 −2
39

Voltage (V)
10 3
10 2
10 1
10 −2
Threshold curves in low pressure air (fd=1GHz ⋅ mm)
Multipactor in air
Multipactor in vacuum
Corona in air
10 −1
10 0
Pressure (Pa)
Figure 3.2: Thresholds for multipactor and corona discharges in air as functions
of pressure together with the multipactor vacuum threshold
as a reference level. Parameters used are: σc = 6.9 × 10 −20 m 2 ,
W1 = 23 eV, k = 2.5, N = 1, f × d =1 GHz·mm and d = 0.1 m.
40
10 1
10 2

stability factor becomes:
G = (νc
ω
ω
(CΦ − 1) + C (Φ − 1))tan α + 1 − C
νc
ω
νc
(1 − CΦ)tan α − (CΦ + 1)
(3.8)
where C = (k + 1)/(k − Φ). In Fig. 3.3 the phase limits, where |G| = 1,
have been plotted together with the non-returning electron limit. Both
the positive and negative phase error limits tend to decrease with increasing
pressure. However, the limit for non-returning electrons increases,
which is a more important limit when the electron impact energy exceeds
the first cross-over energy, thus reducing the width of the multipactor
zone.
Phase limit (degrees)
40
20
0
−20
−40
−60
−80
10 −3
−100
Postive phase error
Negative phase error
Non−ret. el. in vacuum (Semenov et. al.)
Numerical non−returning electron limit
10 −2
Phase Limits
10 −1
Pressure (Pa)
Figure 3.3: Phase limits in a low pressure gas based on the simple analytical
model. The dashed line and the solid line (dashed at the end)
show the upper and lower phase limits beyond which a phase
error will start growing. The dash-dot line is the phase below
which emitted electrons will not be able to escape from the wall
of emission. The dotted line is the phase limit obtained using
Eq. (2.15) and is an approximation of the dash-dot line in the
vacuum case. Parameters used are: σc = 6.9 × 10 −19 m 2 , N = 1,
k = 7.6, d = 0.1 m, and W1 = 23 eV.
10 0
10 1
41

3.1.3 Main results
The main result found in paper B is that a higher microwave power is
required to initiate breakdown in a low pressure gas, since the collisions
tend to slow down the electrons. By combining the low pressure multipactor
graph with the corona threshold curve, it was concluded that
with increasing pressure, the required threshold will first increase and,
after reaching a plateau, it will make a smooth transition to the low
pressure branch of the Paschen curve. This behaviour is confirmed by
the investigations made by Gilardini [53] for materials with a low first
cross-over point, close to the ionisation energy of the gas, and for N = 1,
i.e. for the first order of multipactor.
For materials with a higher first cross-over energy and for higher
order multipactor, Gilardini found no initial increase in the multipactor
threshold, instead a monotonically decreasing breakdown voltage was
seen. A possible explanation for the differences between the result obtained
by the simple analytical model and the result of Gilardini is also
presented in paper B and it is suggested that the reason is that for materials
with a higher W1 and for N > 1 the contribution of electrons from
impact ionisation decreased the required W1 (see Fig. 3.4). However,
as will be seen in the next section, electron contribution from impact
ionisation is not the only reason for this behaviour. The collisions will
also cause an electron velocity spread, which will result in a larger total
impact velocity and thus a lower voltage is needed to achieve the
necessary first cross-over energy.
A comparison was made with experiments by Höhn et al. [18] in low
pressure argon as well as with PIC-simulations by Gilardini [54] in the
same gas (see Fig. 7 in paper B). However, the fd-product chosen by
these authors was located in the middle of the right boundary of the
first multipactor zone, an area dominated by the hybrid modes [39].
The simple model used in the presented analytical approach is not applicable
to these modes and consequently the behaviour found in the
experiments and simulations could not be confirmed. Furthermore, the
impact energy of the electrons at this fd-product is several times higher
than the ionisation energy of argon and thus a significant contribution
of electrons from collisional ionisation would be expected. In addition,
the required W1 would be reduced due to the electron velocity spread
and thus a behaviour similar to curves (b) or (c) in Fig. 3.4 should be
expected and it is also what is found.
To further analyse multipactor in a low pressure gas, a better ana-
42

Voltage (V)
Threshold curves in a low pressure gas
Multipactor in a low pressure gas without ionisation
Multipactor in vacuum
Corona
a
b
c
Pressure (Pa)
Figure 3.4: Qualitative form of the dependence of breakdown threshold with
pressure in the region between which multipactor and corona,
respectively, dominate the breakdown process: (a) the friction
force due to collisions with neutrals dominates, (b) electron velocity
spread reduces the required W1 and collisional ionisation
contributes significantly to the total number of electrons, (c) intermediate
situation.
43

lytical model is obviously needed. The model must, in addition to the
friction force, be able to take collisional ionisation into account as well
as collision induced velocity spread of the electrons. In the next section
a more advanced model that includes all these effects will be presented.
3.2 Advanced Model
The simple model used in the previous section provided important qualitative
understanding of the multipactor threshold behaviour in a low
pressure gas. However, due to the inherent limitations of the model,
some of the results found by other researchers could not be confirmed.
This section will present an improved model for multipactor in a low
pressure gas and it is based on paper C of this thesis. As a representative
gas, the noble gas argon will be used in the included examples.
3.2.1 Model
Just as in the simple model, the basic geometric configuration is electron
motion between two parallel plates perpendicular to the x-direction.
During the passage, no electron loss, only generation through collisional
ionisation, will occur. Using the differential equations for the total electron
momentum and for the change in the number of electrons, one can
derive the following equation for the electron drift acceleration:
du
dt
= eE
m − u(νc + νiz). (3.9)
where u is the drift velocity and νiz the ionisation frequency. In general,
the collision and ionisation frequencies are functions of the electron velocity.
However, by assuming that νc and νiz are constants, Eq. (3.9)
becomes a first order linear differential equation. Multipactor requires
an alternating driving electric field and as in the previous model a harmonic
field E = ˆxE0 sin ωt is used, where ˆx is the unit vector, ω the
angular frequency, and t the time. Assuming the electric field to be
homogeneous, the drift velocity will be parallel to the field, u = ˆxu,
and the vector notation for E and u can be dropped in the following
analysis. By setting ν = νc + νiz, u = ˙x, and du/dt = ¨x, Eq. (3.9) can
be written,
¨x = eE
− ˙xν. (3.10)
m
44

Since the equation has the same form as Eq. (3.1), it will also have the
same solutions and as the initial conditions are identical, the formulas
for the resonant field amplitude and the impact velocity will be identical.
However, it should be noted that instead of νc one will have ν and v0
should be replaced by u0. An important difference is that the velocity
in the previous model was only directed in the x-direction, but now
there is also a thermal velocity component, vt, i.e. the total velocity
is v = u + vt. With these new designations, the expressions for the
resonant field amplitude and the impact velocity become:
E0 =
m
e (ω2 + ν2 )(d + u0
ν (Φ − 1))
(1 + Φ)sin α + ((1 − Φ) ω
ν
2ν + ω )cos α
(3.11)
uimpact = u0Φ + Λ(1 + Φ)(ω cos α − ν sinα) (3.12)
where Φ = exp (−Nπν/ω) has been introduced for simplicity. Λ is given
by Eq. (3.4) as before, but νc should be replaced by ν.
In order to construct the multipactor boundaries, the same approach
as in the simple model case is taken and an expression for the resonant
phase is obtained by combining Eqs. (3.11) and (3.12), which yields,
tan α = ω2 [ρΦ + χ] + 2ν 2 (Φu0 − uimpact)
(dν + uimpact − u0)νω(1 + Φ)
(3.13)
where ρ = dν + uimpact + u0 and χ = dν − uimpact − u0 have been
used for convenience. The reason why the expression looks somewhat
different from Eq. (3.7) is that the constant initial velocity approach has
been used instead of the assumption of a constant ratio between impact
and initial velocities. This will also affect the expression for the phase
stability factor, which in this case becomes
G = (Φ − 1)(ν2 + ω 2 )sin αΛ − Φνu0
ν((1 + Φ)(ν sin α − ω cos α)Λ − Φu0)
(3.14)
So far, the differences between the simple and the more advanced
model are fairly trivial. However, the parameters used (νc and νiz)
are not constants, they depend to a great extent on the total electron
velocity, which is the vector sum of the drift and thermal velocities. The
thermal velocity will have a random direction and therefore the average
total velocity will be equal to the drift velocity. However, the total
(average) energy, ɛ, will still depend on both velocities and it becomes,
ɛ = mv2
2
= m
2 (u2 + 〈vt 2 〉) (3.15)
45

where 〈vt 2 〉 represents the average of the square of the magnitude of
the thermal velocity. In paper C, a differential equation for the thermal
velocity is derived, viz.
d〈vt 2 〉
dt + (νcδ + νiz)〈vt 2 〉 = u 2 (νc(2 − δ) + νiz) (3.16)
where δ is the energy loss coefficient. By assuming that νc, νiz and δ are
constants, like before, Eq. (3.16) can be solved explicitly and with the
initial condition 〈vt 2 (t = α/ω)〉 = 0, the thermal impact velocity, when
ωt = Nπ + α, can be found and thus the total impact velocity can be
determined. However, the expression is very complicated and will not
be reproduced here.
The total impact velocity will determine the secondary electron emission
yield. For vacuum multipactor as well as in the previous simple
model for low pressure multipactor, the impact velocity was perpendicular
to the electrodes. In such a case, the secondary yield depends only
on the impact velocity. However, for angular incidence, which will be
the case now with the random three dimensional thermal velocity component,
the yield will be a function not only of the impact energy but
also of the angle of incidence. To account for the angular incidence the
expressions given in Ref. [22] have been used and for ease of reference
they are reproduced here,
ɛmax(θ) = ɛmax(0)(1 + θ 2 /π) (3.17)
σse,max(θ) = σse,max(0)(1 + θ 2 /2π) (3.18)
η = ɛimpact − ɛ0
ɛmax(θ) − ɛ0
σse = σse,max(θ)(η exp 1 − η) k
(3.19)
(3.20)
where θ is the impact angle with respect to the surface normal. ɛmax is
the impact energy when the secondary emission reaches its maximum,
σse,max. ɛimpact is the total impact energy and ɛ0 is the energy limit
for non-zero σse. The formulas are valid for “a typical dull surface”,
according to Ref. [22]. The coefficient k is given by k = 0.62 for η < 1
and k = 0.25 for η > 1.
In vacuum multipactor, the only source of new electrons is secondary
yield from each impact. When the phenomenon takes place in a gas,
another potential source of new electrons is impact ionisation of the
gas molecules. The ionisation threshold of most gases of interest is
46

in the range 10-20 eV. This is well below the first cross-over point of
most materials and thus when the electron energy is sufficient to initiate
multipactor, it is also enough to ionise the gas molecules. In this model,
the contribution from impact ionisation is included by modifying the
breakdown condition from σse = 1 to
σse + 〈νiz〉Nπ
µ = 1 (3.21)
ω
where 〈νiz〉 is the average ionisation frequency and µ is an ionisation
factor, which ranges from 0 − 1 and indicates the fraction of the electrons
from ionisation that is able to become a part of the multipacting
bunch. Determination of the correct value of µ is not a trivial problem
and for simplicity a constant µ = 0.75 is used. This is quite a rough
approximation, but for materials with a low first cross-over point, it will
be shown that the ionisation contribution is fairly small and the exact
value for µ is not so important. On the other hand, for materials with
a high first cross-over energy, the importance of µ can not be neglected
and thus a detailed investigation of µ should be performed, but due to
the complexity, it will be left as future work.
Apart from µ, there are other parameters, which need to be determined
with good accuracy in order to obtain useful quantitative results.
The energy loss coefficient, δ, which is used in Eq. (3.16), is a small
quantity and for pure elastic collisions, the value is equal to 2m/M,
where m is the electron mass and M is the mass of the argon atom [55].
It is also a function of the electron energy and for inelastic collisions,
which will occur when the electron energy is greater than a few eV, the
value is about 10 to 100 times larger than the elastic value [56]. However,
the value is still quite small and will not have major effect on the
low pressure multipactor threshold and for simplicity a value 10 −3 will
be used, which is about 37 times greater than the elastic value. The
remaining parameters, νc and νiz, are very important for the threshold
and a detailed description of these values will be given in the next
section.
3.2.2 Analytical formulas for argon cross-sections
For most materials of interest, the first cross-over energy is in the range
20-70 eV and it is this value that determines the lower multipactor
threshold. Thus it would be of value to have expressions for the collision
and ionisation cross-sections that give an accurate description of
47

these quantities in the range from 0 eV to about 100 eV. For the electronargon
collision cross section, the data given in Ref. [57] is used and it
covers the range up to 20 eV. An analytical formula has been devised,
which approximates the given data quite well in the measurement region
(cf. Fig. 3.5). Outside the measurement points, the cross-section
for very low energy electrons has been set to converge towards the geometrical
cross-section of the argon atom. For high energy electrons, the
cross-section is set to fall off with the same rate as for the last few eVs.
The analytical formula is given by the expression,
σc = (
1.68
+
1 + (8ɛ) 3
ɛ
1 + (0.07ɛ) 2 )1.5 × 1.15 × 10 −20 [m 2 ] (3.22)
where ɛ is the total electron energy, given by Eq. (3.15).
Total collision cross section [m 2 ]
10 −18
10 −19
10 −20
10 −3
10 −21
Buckman and Lohman
Analytical approximation
10 −2
10 −1
10 0
Electron energy [eV]
Figure 3.5: Absolute total collision cross-section for electrons scattered from
argon. The stars indicate measurement data by Buckman and
Lohmann [57] and the solid line is the analytical approximation
given by Eq. (3.22).
The ionisation cross-section increases rapidly for electron energies
slightly above the ionisation threshold and thus it is of importance to
have an accurate description in this range, especially since this is close
to the first cross-over energy of many materials. A simple function that
48
10 1
10 2

accurately describes the cross-section for the entire measurement range
can be given by (cf. Fig. 3.6)
σiz = q1
ln ɛ/ɛi
�
ɛ/ɛi + 0.1(ɛ/ɛi) 2
[m 2 ] (3.23)
where ɛi is the ionisation threshold of argon and q1 = 4.8 × 10 −20 m 2 .
Ionisation Cross Section [m 2 ]
10 −19
10 −20
10 −21
10 −22
Ionisation cross section for Argon
10 2
Electron energy [eV]
S.C.Brown
H.C.Straub
Analytical approximation
Figure 3.6: Ionisation cross-section for electron-argon collisions. The circles
and stars indicate measurement data by S. C. Brown [58] and
Straub et al. [59] respectively and the solid line is an analytical
approximation given by Eq. (3.23).
3.2.3 Multipactor boundaries
In the following section the above model will be used to determine the
multipactor boundaries. The best accuracy is attained when solving
the basic differential equations numerically while using good approximate
formulas for the different parameters. However, such computation
takes very long time, since both the initial and the final multipactor
conditions have to be fulfilled. Faster computation can be achieved
by using different approximations, e.g. constant parameters as shown
in Eqs. (3.10)-(3.12), Eq. (3.13), and Eq. (3.14). Two different implementations
are used in paper C, one purely numerical and one semi-
10 3
49

analytical. Attempts were made to find a purely analytical implementation
as well, but due to the strong non-linearities in the functions for the
cross-sections, no accurate such implementation could be found. Details
concerning the two implementations are presented in paper C and will
not be reproduced here.
As mentioned in chapter 2, when constructing the complete multipactor
zones, the multipactor thresholds corresponding to impact velocities
between the first and second cross-over points are determined
for a specific order of resonance within the phase range from the nonreturning
electron limit to the upper phase stability limit. The zone
for that order of resonance is then the envelope of all these curves (cf.
Fig. 2.3). However, to explore the basic effects on the multipactor phenomenon,
it is sufficient to study the threshold corresponding to unity
SEY. Thus, in most of the following charts, only the lower multipactor
threshold will be considered. However, in keeping with the multipactor
tradition, the complete zones will be presented as well.
One concern that appears when making low pressure multipactor
charts is the parameters which should be used on the chart axes. Classical
vacuum multipactor charts use engineering units with voltage as a
function of the frequency-gap size product, like in Fig. 2.3. By multiplying
Eq. (3.11) by the gap size, d, to get the voltage and rearranging
Eqs. (3.12), (3.13), and (3.14), these expressions can all be written as
functions of two natural parameters, viz. fd and pd, i.e. the frequencygap
size and the pressure-gap size products. Thus, for a given pd the
multipactor zones can be constructed in the classical engineering units
as shown in Fig. 3.7. Note that three different pd-values are used, one
for each zone. The chosen values are close to the limit of stability of the
numerical implementation for each zone.
In Fig. 3.7 both the analytical (semi-analytical) and the numerical
implementations are used to plot the thresholds. Very good agreement
between the two implementations is found and therefore the faster analytical
version is used to produce all other figures. The most striking
first impression of the graphs in Fig. 3.7 is the difference in behaviour
between the first and the higher order modes. The first order mode
shows an increased threshold, which is a consequence of the friction
force experienced by the electrons due to collisions with neutrals. This
is in agreement with the model presented in paper B, which only considered
the friction force. However, for higher order modes, the result
is the opposite. Instead of an increased threshold as in the friction
50

Voltage [V]
10 2
Numeric stable phase
Numeric unstable phase
Analytic stable phase
Analytic unstable phase
pd=15 Pa mm
10 0
pd=7 Pa mm
Frequency − Gap product [GHz⋅mm]
pd=5 Pa mm
Figure 3.7: Lower multipactor thresholds in low pressure argon for three different
fixed pd-values, one for each zone. The dotted lines represent
the multipactor zones for vacuum multipactor. Parameters
used are: W1 = 23 eV , W0 = 3.68 eV, σse,max(0) = 3, ɛ0 = 0.
51

only model, the inclusion of ionisation and thermal spread leads to a
decreased threshold with increasing pressure.
Thresholds as in Fig. 3.7 can be found for all impact velocities between
W1 and W2 and by constructing the envelope of all these curves
for each order of resonance, the complete, classical multipactor zones can
be found for a given pd-product for each zone. This is done in Fig. 3.8,
which shows the complete zones for three different pd-values. The drawback
with this chart is that the model does not account for the hybrid
zones and thus the right boundary of each zone will not accurately reflect
the true multipactor threshold for those fd-values. The model can,
however, be extended to include also the hybrid modes, but due to the
increased complexity, this is left as future work
Voltage [V]
10 3
10 2
pd=10 Pa mm
10 0
pd=4 Pa mm
Frequency−Gap product [GHz⋅mm]
pd=2 Pa mm
Figure 3.8: Multipactor susceptibility zones in low pressure argon (solid lines)
together with vacuum zones (dotted lines) for comparison. Parameters
used are: W1 = 23 eV, W2 = 1000 eV, W0 = 3.68 eV,
σse,max(0) = 3 and ɛ0 = 0.
3.2.4 Key findings
Among the main results is that the friction force dominates the low
pressure multipactor threshold for materials with a low first cross-over
52
10 1

energy for the first order of resonance, as indicated by Fig. 3.7. However,
this figure only shows the behaviour for a given pd-product and in order
to see what happens when the gas density increases, the threshold can be
plotted as a function of the pd-product. This has been done in Fig. 3.9
for a material with a low first cross-over energy and, as expected, the
threshold increases with increasing pressure for the lowest order mode,
N = 1, and after reaching a maximum, the threshold starts to decrease
again. For higher order modes, the threshold decreases monotonically
as the gas becomes dense enough to affect the multipacting electrons.
This behaviour is identical to that found by Gilardini [53] in his Monte
Carlo simulation of low pressure multipactor. He also observed that for
materials with a higher first cross-over point, the threshold does not
increase with increasing pd, instead it falls off monotonically, which is
the behaviour shown in Fig. 3.10. The main reason for this difference
in behaviour is the contribution of electrons from collisional ionisation,
which increases drastically when the electron energy is well above the
ionisation threshold.
Normalised Voltage
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
10 −2
10 −1
10 0
Pressure × gapsize [Pa⋅mm]
Figure 3.9: Normalised multipactor thresholds for varying pd. The thresholds
are normalised with respect to the vacuum threshold. Curves for
the three first orders of resonance are shown. Parameters used
are: W1 = 23 eV, W0 = 3.68 eV, σse,max(0) = 3, ɛ0 = 0, fdN=1 =
0.6 GHz·mm, fdN=3 = 2.4 GHz·mm, and fdN=5 = 4.2 GHz·mm.
For higher order of resonance, N > 1, Gilardini found no difference
in the basic behaviour regardless of material. The threshold falls off
N=1
N=5
N=3
10 1
53

Normalised Voltage
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
10 −2
10 −1
10 0
Pressure × gapsize [Pa⋅mm]
Figure 3.10: Normalised multipactor thresholds for varying pd. The thresholds
are normalised with respect to the vacuum threshold.
Curves for the three first orders of resonance are shown for
a material with a first cross-over point more than 7 times
greater than in Fig. 3.9. Parameters used are: W1 = 170 eV,
W0 = 4 eV, σse,max(0) = 1.3, ɛ0 = 0, fdN=1 = 1 GHz·mm,
fdN=3 = 3.2 GHz·mm, and fdN=5 = 5 GHz·mm.
54
N=5
N=1
N=3
10 1

Voltage [V]
10 2
µ=0
µ=0.75
10 0
Frequency − Gap product [GHz⋅mm]
µ=0.75
µ=0
Vacuum multipactor
Figure 3.11: Multipactor thresholds in low pressure argon for the two lowest
order modes (N = 1 and N = 3) with µ = 0.75 and µ = 0
respectively. Parameters used are: W1 = 23 eV, W0 = 3.68 eV,
σse,max(0) = 3, ɛ0 = 0, pd = 15 Pa·mm for N = 1, and pd =
7 Pa·mm for N = 3.
directly from the vacuum threshold without showing any maximum and
the same behaviour is seen in Figs. 3.9 and 3.10. Even with a low W1,
where the contribution of electrons from collisional ionisation is low, a
monotonically decreasing threshold is obtained. The cause of this distinct
lowered threshold is the partial thermalisation of the electrons due
to the collisions. The velocity spread results in a total impact velocity,
which is greater than the drift velocity alone and thus for the same
secondary electron emission, a lowered impact drift velocity is possible.
Even though the friction force requires a higher voltage to achieve the
same impact drift velocity, the thermalisation effect dominates, with a
lowered threshold as a result. This becomes clear in Fig. 3.11, where it is
apparent that it is not the electrons from ionisation that constitute the
main reason for a decreased threshold, rather it is a consequence of the
partial thermalisation. In the case with a high W1, the thermalisation
effect is also important, but without the contribution from collisional
ionisation, the behaviour would not be the same, which can be seen in
Fig. 3.12.
To summarise the key findings from the more advanced model, it
55

Voltage [V]
10 3
10 0
µ=0
µ=0.75
µ=0.75
Vacuum multipactor
Frequency − Gap product [GHz⋅mm]
Figure 3.12: Multipactor thresholds in low pressure argon for the two lowest
order modes (N = 1 and N = 3) with µ = 0.75 and µ = 0
respectively. In this case, the second cross-over energy is about
7 times greater than in Fig. 3.11. Parameters used are: W1 =
170 eV , W0 = 4 eV, σse,max(0) = 1.3, ɛ0 = 0, pd = 25 Pa·mm
for N = 1, and pd = 15 Pa·mm for N = 3.
56
µ=0

can be said that there are three main effects that affect the low pressure
multipactor threshold. The friction force tends to increase the threshold
as a higher electric field is needed to reach the necessary impact velocity.
The thermalisation, on the other hand, increases the total impact energy
and thus a lower electric field is needed to achieve the required impact
velocity. For materials with a low first cross-over point, the first effect
dominates for the first order of resonance, while for higher order modes,
the latter plays the main role. In addition to these two effects, the model
also includes contribution from impact ionisation to the total number
of electrons. This addition also tend to lower the multipactor threshold
and the effect becomes very prominent for materials with a high first
cross-over energy as a consequence of the concomitant high ionisation
cross-section.
57

Chapter 4
Multipactor in irises
A common microwave component is the waveguide iris, which is often
used as a shunt susceptance for the purpose of matching a load to the
waveguide. There are many different types of irises, but a typical configuration
consists of a step-like, short length, reduction of the waveguide
height. Similar structures also appear in other configurations, e.g. as
apertures in array antennas, as coupling slots in directional couplers,
and as irises in waveguide filters. As the field strength in the iris can
be very high and the gap height is small, there is a pronounced risk of
having a multipactor discharge.
So far in this thesis, all the models considered have been based on
the plane-parallel model with a spatially uniform harmonic electric field.
In general, most theoretical studies of the multipactor phenomenon have
been limited to this or similar approximations. However, many RF devices
involve more complicated electric field structures where predictions
based on the parallel-plate model are not applicable. This is true for
e.g. the waveguide iris, where the electric field will be a combination
of several different electromagnetic modes, most of which typically are
evanescent. However, due to the short length of the iris, these modes
will be of importance. Nevertheless, in this analysis, which is described
in more detail in paper D, it is only the importance of the random drift
due to initial velocity spread of the secondary electrons that has been
considered. Thus, as far as the field is concerned, a spatially uniform
harmonic field based on the parallel-plate model is used.
Experiments [60, 61] as well as numerical studies [61, 62] of multipactor
in an iris have shown that the discharge threshold increases with
decreasing length of the iris. It has been suggested that the reason for
59

the increased threshold is losses of electrons out of the iris region [60]. In
this analysis, we show that one of the contributing factors to this electron
loss is a random drift due to the axial component of the initial velocity
of the secondary emitted electrons. Other loss mechanisms, which are
due to the inhomogeneity of the field, tend to further enhance the losses
and these effects will be more pronounced for small gap lengths. This
means that by taking only losses due to the random drift into account,
a conservative increase of the breakdown threshold should be obtained.
4.1 Model and approximations
The geometry used in the model is the 2-dimensional structure shown
in Fig. 4.1. The iris has a gap height h in the y-direction, a length l
in the z-direction and is assumed to be fitted into a waveguide with a
height that is much greater than h. The harmonic electric field E is
assumed uniform in the gap, as a simple approximation of the actual
field. There are two main reasons for choosing a uniform field. Firstly,
the deterministic model developed for the parallel plate case, which is
described in chapter 2, can be used to describe the basic behaviour
of an electron trajectory inside the gap. Secondly, the effect of the
initial velocity spread of the secondary electrons along the z-axis on the
multipactor threshold can be analysed separately from the drift force due
to inhomogeneities in the electric field. In addition, it gives a convenient
base for comparing the results with those of the parallel plate model.
By assuming a uniform E-field in the y-direction, the electron motion
along the z-direction is not affected by the field. The motion in this
direction, the drift motion, will occur with a fixed velocity vz between
the impacts. Lets assume that a seed electron is emitted inside the gap at
the coordinate z0, −l/2 < z0 < l/2, at one of the walls. As the electron
traverses the gap and hits the opposite side of the iris, it has become
displaced a distance ∆z in the z-direction. This drift is determined by
the velocity in the z-direction, vz , together with the transit time, tg,
and is given by ∆z = vztg. For a fixed mode order, N, and frequency,
f, of the field, each transit time is the same and is given by,
tg = Nπ
, (4.1)
ω
where ω = 2πf.
The electron trajectory in the z-direction will perform a random
walk with a change of velocity, vz, after each impact. When the impact
60

h
l
Figure 4.1: The geometry used in the considered model.
coordinate is outside the iris area |z| > l/2, i.e. one of the gap edges
has been passed, the electron trajectory is lost. The probability of survival,
p(k) (see Fig. 4.2), for the electron trajectory decreases with the
number of transits, k, and for a general one-dimensional random walk
problem, with the jump size governed by a continuous distribution function,
Φk(z), an explicit solution for this, the first passage time problem,
is not always possible. However, in paper D it is explained that the
asymptotic behaviour of p(k) is determined by the largest eigenvalue γ0
of the expansion of Φk(z), i.e.
E
y
p(k) ∝ γ k 0 . (4.2)
A detailed description of how to determine p(k) is given in paper D
together with approximate solutions for γ0 when the normalised iris
length, η = l/(vTtg), is either very small or very large. This summary,
however, will focus on the effect the random electron drift has on the
multipactor susceptibility zones.
Each seed electron inside the iris gap will start to multiply with the
successive wall collisions. Due to the stochastic losses, the number of
electrons will sometimes become large and sometimes small. However, if
on average the generation of electrons due to wall collisions is larger than
the loss over the gap edges, there is a finite probability that a sufficiently
z
61

Probability of survival
1
0.8
0.6
0.4
0.2
0
l=2 mm
l=8 mm
l=16 mm
50 100 150 200 250 300 350 400 450 500
Number of collisions
Figure 4.2: The probability of survival, p(k), for an electron emitted in the
center of the iris gap, z = 0, for three different iris lengths. Parameters
used: f = 1 GHz, N = 1, and WT = 2 eV (corresponding
to vT, the rms-velocity of the Maxwellian distribution
of initial velocity in the z-direction).
strong discharge will appear. The generated number of electrons over
the initial number of electrons after k collisions is given by,
Ne
N0
≡ g(k) = p(k)σ k se. (4.3)
Depending on the start position of the seed electrons, the initial behaviour
of Ne can vary. If the start position is close to the iris edge,
the average electron number will first decrease and then if σse is large
enough, it will start increasing again. But if the start position is in
the center, it may first start to increase, but after a number of transits,
it will start decreasing (cf. Fig. 4.3). Eventually, it is the asymptotic
behaviour of p(k) that will determine whether or not there will be a
discharge. Thus from Eq. (4.2) and Eq. (4.3) one can conclude that the
asymptotic change in the electron number is given by,
g(k) ∝ (σseγ0) k . (4.4)
Thus the average number of electrons will grow if
62
σseγ0 > 1 (4.5)

N e /N 0 [−]
3
2.5
2
1.5
1
Multipactor in iris
σ = 1.159
σ = 1.151
0.5
0 50 100
Number of collisions
150 200
Figure 4.3: The growth in electron number as a function of the number of
gap crossings for two different SEY-coefficients. Parameters used:
f = 1 GHz, N = 1, l = 2 mm, and WT = 2 eV.
or equivalently
σse > 1/γ0 > 1. (4.6)
This implies that the secondary electron yield must be greater than a
value that is larger than unity (1/γ0 > 1) to have growth of the number
of electrons. This modified breakdown criterion is the only difference
between the model considered here and the conventional resonance theory
of multipactor inside a plane-parallel gap (where σse > 1 is used
when determining the threshold). The condition for σse, Eq. (4.6), can
be converted into a range of impact energies,
W1 < Wmin < Wimpact < Wmax < W2, (4.7)
where the impact energies Wmin and Wmax are determined by σse =
1/γ0. Consequently, using Wmin and Wmax instead of W1 and W2, respectively,
in the parallel-plate model, multipactor regions that account
for the electron losses due random drift can be obtained.
63

4.2 Multipactor regions
Using the above described model and by employing a natural scaling
parameter of η, viz. l · f, multipactor charts in the traditional engineering
units (voltage vs. frequency-gap-size product) can be devised
for a specific value of the ratio of gap height and iris length, h/l. Figure
4.4 shows an example of this, where the multipactor regions have
been constructed for 5 different h/l-ratios.
Peak voltage [V]
10 4
10 3
10 2
10 0
0.001
1
3
5.2
6.3
Multipactor regions for different height/length ratios
10 1
Frequency gap height product [GHz ⋅ mm]
Figure 4.4: The first four multipactor susceptibility zones for a microwave
iris with five different height/length-ratios. Parameters used are:
W0 = 2 eV (y-direction), WT = 2 eV (z-direction), W1 = 85.6 eV,
and σse,max = 1.83 (material properties for alodine [50]).
The transit time decreases with increasing frequency according to
Eq. (4.1) and thus the distance traversed in each step becomes smaller,
which implies that the probability of surviving k steps increases. This
causes the multipactor zones to shrink towards higher frequencies with
increasing h/l as is evident in Fig. 4.4. The transit time is also a function
of the mode order, which increases the transit time for higher order
modes. This counteracts the decrease due to increasing frequency and
consequently a behaviour similar to that of the first resonance zone can
be observed also for the higher order modes.
For materials with a low maximum SEY, like in Fig. 4.4, the ability
64

to compensate for electron losses is not very good and thus the zones
will disappear for relatively small values of h/l. However, in the opposite
case for a material with a large SEY, like e.g. aluminium, multipactor
will be possible also for relatively thin irises.
4.3 Comparison with experiments
By comparing the current model with experimental data [61], good qualitative
agreement can be observed (see Fig. 4.5). As the h/l-ratio increases,
the threshold increases until, beyond a certain limit, no multipactor
is possible. Since only the electron losses due to the random drift
are accounted for, the model predicts the existence of a discharge beyond
the limit found in the experiments. Consequently, from an engineering
point of view, this is a conservative measure of the increased threshold
and thus it should be safe to apply it when designing multipactor free
microwave hardware.
Peak Voltage [V]
3500
3000
2500
2000
1500
Multipactor in iris
1000
Current model a)
Current model b)
Current model c)
Current model d)
Measur. −presentation
Measur. −proceedings
500
0 1 2 3 4
Height/Width [−]
5 6 7 8
Figure 4.5: The multipactor threshold as a function of h/l for different parameters
of the current model. For comparison, measurement
data from [51] is included. Parameters used in a): h = 1.2 mm,
f = 9.56 GHz, WT = 2 eV, W0 = 2 eV, W1 = 59.1 eV, and
σse,max = 2.22, (i.e. W1 = Wf2 and σse,max = αmax in table A-6
for silver in [50])). Modified parameters in: b) W0 = WT = 4 eV,
c) W1 = 40 eV, and d) W1 = 80 eV.
65

The step-like behaviour of the increasing threshold is due to the fact
that several multipactor zones are involved (f · h ≈ 11.5 GHz·mm).
Starting with mode number N = 7 for ’current model a)’, the lower
threshold for the parallel-plate case is found and as the h/l-ratio increases,
the zone corresponding to N = 7 shrinks until, with an almost
sudden voltage step, the next threshold, being determined by the N = 5
zone, is reached. Finally the last N = 3 zone determines the threshold
before it also vanishes.
In addition to the experimental comparison, Fig. 4.5 brings forward
the importance of different parameters of the current model as well as of
the used model for SEY [22]. By increasing the initial velocity (’current
model b)’), the overall threshold decreases as a lower field strength will
be sufficient to reach the same impact velocity (cf. Eq. (2.7)). The effect
of an increased thermal spread, WT, is that the electron losses increases
and the threshold starts to increase for lower h/l-values (also shown
in ’current model b)’). By lowering the first cross-over point (’current
model c)’), the parallel-plate threshold decreases, since an additional
zone, N = 9, comes into play. However, as it shrinks away, the threshold
increases in a sudden step to the same level as in case ’a)’ and then it
follows ’a)’ except that the steps occur at higher h/l-values. An increased
first cross-over point, case ’d)’, shows a change of behaviour opposite to
’c)’, except for the parallel-plate threshold as it is still the N = 7 zone
that determines this threshold.
In the current model a uniform electric field has been used. Due to
the geometry of the iris, the actual electric field will tend to be curved
outwards at the edges of the slot instead of being straight (cf. Fig. 4.1).
Since the field amplitude is higher in the centre of the iris than at the
edges, the Miller force [10], which is proportional to the negative gradient
of the square of the electric field amplitude, will tend to push the
electrons out of the iris. This effect is most important for the higher
order resonances, where several RF-cycles are required to cross the gap.
In addition to the Miller force, the curved electric field will have a component
in the z-direction, which, in particular for the first order mode,
will drive the electrons toward the iris edges. This means that the electron
losses will be greater than in the case of a uniform field, which
will lead to an even further increase of the multipactor threshold. This
effect should be more pronounced for thin irises and could explain why
the current model predicts the existence of a discharge beyond a certain
h/l-ratio where experiments cannot detect it (cf. Fig. 4.5).
66

4.4 Main results
This analysis has shown that the random electron drift along the iris
length due to the initial velocity of the secondary electrons tends to
significantly increase the multipactor threshold in a waveguide iris as
compared to predictions based on the classical two parallel-plate model.
Inherent in the presented model is the scaling parameter h/l, which
makes it possible to produce useful multipactor susceptibility charts in
the traditional engineering units. An increase in the h/l-ratio results in
a shrinkage of each multipactor resonance zone. For each zone, the zone
reduction effect is more pronounced for lower frequencies. A consequence
of this is that the multipactor free region in the parallel plate model at
low frequency-gap-height products grows with increasing h/l-ratio.
67

Chapter 5
Multipactor in coaxial lines
The coaxial line is a very common and important component in microwave
systems. It is a transmission line that consists of an inner
cylindrical conductor and a coaxial outer conductor. A constant crosssection
is maintained by means of a dielectric medium, which is contained
between the conductors. In some space applications, as well as in
other systems, the dielectric medium has been partly omitted in order to
save weight or to reduce the dielectric losses. In a vacuum environment,
the line may become evacuated, which makes it exposed to the risk of a
multipactor discharge.
The electric field in a coaxial line is nonuniform, which makes analytical
analysis difficult, since the the equation of motion for an electron becomes
a non-linear differential equation. However, multipactor in a coaxial
line has been studied experimentally [63] and numerically [64–67]. In
these studies it was found that two different types of resonant multipactor
can occur, namely a two-sided discharge between the outer and
the inner conductors and the one-sided analogue on the outer conductor.
In an attempt to understand the effect of varying the relative inner
radius, i.e. varying the characteristic impedance of the coaxial line,
different scaling laws were suggested in these studies. Another study focused
on the current due to the multipacting electrons and treated this
as a radially oriented Hertzian dipole in order to determine the electric
field generated by the multipactor discharge [68].
In paper E resonant multipactor in a coaxial line is analysed by
means of an approximate analytical solution of the non-linear differential
equation of motion, which in a large range of microwave frequencies
and amplitudes agrees very well with the numerical solution. As
69

support for the qualitative analytical results, paper F presents PICsimulations
of the phenomenon in the same geometry. The advantage
of PIC-simulations is the ability to include aspects, which are stochastic
in nature, like e.g. the initial velocity spread of the secondary electrons.
This summary will focus on the results of the study and for a
more detailed description of the model and approximations used, see
papers E and F.
5.1 Analytical study
By finding an analytical solution of the equation of motion, general properties
of the multipactor can be found, which may be difficult to identify
when numerically studying the phenomenon. In addition, the time of
computation can be radically reduced when using explicit analytical expressions
instead of a numerical scheme. In this section an approximate
analytical solution of the non-linear differential equation that governs
the electron motion in the nonuniform field between the inner and outer
conductor of a coaxial line is found. Using these expressions, the effect
on the multipactor resonances and thresholds is studied. The validity
of the expressions is then confirmed by solving the differential equation
numerically.
5.1.1 Model
The cylindrical coaxial line has an outer radius Ro and an inner radius Ri
(see Fig. 5.1). The applied field is the fundamental TEM-mode, which
means that the electric field, E, is radially directed and the amplitude
will be inversely proportional to the distance from the centre of the line.
There will be no dependence on the angle around the coaxial axis, which
means that the problem can be studied as a one dimensional problem,
provided that the effect of the magnetic field is neglected and only a
cross-section of the coaxial line is considered. In vacuum, the equation
of motion for an electron in an electric field can be written
mr ′′ = −qE (5.1)
where m is the mass of the electron, q the unit charge, and E the instantaneous
strength of the electric field. The radial position of the electron
is designated r and r ′′ is the second time derivative of the position.
Assuming a time harmonic electric field, E = Eo(Ro/r)sin (ωt), where
70

Figure 5.1: The geometry used in the considered model.
Eo is the field amplitude at the outer conductor, and introducing the
notation Λ = qEoRo/m, Eq. (5.1) can be written:
r ′′ = − Λ
sin ωt (5.2)
r
The relation between the field amplitude and the voltage amplitude is
given by Uc = EoRo ln (Ro/Ri). Since the field is inhomogeneous and
stronger near the centre conductor, there will be a net average force that
slowly, compared to the fast harmonic oscillations, pushes the electron
towards the outer conductor. This force is called the ponderomotive or
Miller force [10] and it tends to push the electrons away from regions with
high amplitudes of the RF electric field. By separating r(t) according
to r(t) = x(t) + R(t), where x(t) is the fast oscillating motion and R(t)
the slowly varying motion (the time averaged position), an approximate
solution of Eq. (5.2) can be derived (see paper E) where the position
and velocity of the electron are given by:
r(t) ≈ Λ
ω2 �
sin(ωt)
+
(5.3)
and
�
C1(t − C2) 2 + Λ2
2ω 2 C1
C1(t − C2) 2 + Λ2
2ω 2 C1
r ′ (t) ≈ 1
�
C1(t − C2) +
R(t)
Λ
�
cos (ωt)
ω
, (5.4)
71

where R(t) is the average position,
�
R(t) =
C1(t − C2) 2 + Λ2
2ω 2 C1
. (5.5)
The constants of integration, C1 and C2, are determined by the initial
conditions, which for an electron starting at the outer conductor are
r(t = t0) = Ro and r ′ (t = t0) = −v0. Using Eq. (5.3), the position of an
electron emitted from the outer conductor with no initial velocity has
been plotted in Fig. 5.2. The accuracy of the expression is evident from
the comparison with the numerical solution.
Position [mm]
10
9
8
7
6
5
Multipactor in coax
0 1 2 3 4
Time [ns]
5 6 7 8
Figure 5.2: Motion of an electron emitted from the outer conductor of a coaxial
line. The solid line corresponds to the analytical expression
Eq. (5.3), the dotted line is a numerical solution of the differential
equation Eq. (5.2) (almost covered by the solid line) and the
dashed line is the average motion according to Eq. (5.5). Parameters
used: Vc = 1200 V, f = 3 GHz, W0 = 0 eV (the initial
electron energy), Ro = 10 mm, and Ri = 5 mm.
An important result can be obtained by only looking at the average
position, Eq. (5.5). The minimum of this equation, Rmin, is the small-
72

est achievable radial position for an electron emitted from the outer
conductor, provided that the oscillations are not too large. The expression
for the minimum of R(t) will be a function of the field amplitude,
the frequency, the initial electron velocity as well as the initial phase,
α = ωt0:
Rmin = ΛRo(Λ 2 + 2Λ 2 cos α 2 + 4Λcos αv0ωRo + 2v 2 0ω 2 R 2 o) −1/2
(5.6)
However, for v0 = 0 a much more compact expression, which is independent
of field amplitude and frequency, is obtained,
Rmin ≈
Ro
� 1 + 2(cos α) 2
≥ Ro
√3 . (5.7)
This means that if the radius of the inner conductor, Ri, is smaller than
58% of the outer radius, Ro, then two sided multipactor is not possible
when the initial velocity is low and the oscillations are small.
5.1.2 Multipactor resonance theory
In a coaxial line, both double-sided and single-sided multipactor (on the
outer conductor) are possible. First, double-sided discharge will be considered
and typical for this is that the one way transit time corresponds
to an odd integer of half RF field periods. However, in a coaxial line,
the transit time is normally longer for electrons emitted from the outer
conductor than for electrons emitted from the inner conductor. Thus,
the sum of two transits must be considered and the condition for this is
that it should be an integer number of RF periods. This is the resonance
criterion and in addition to this the phase-focusing effect should be active,
which for the parallel-plate case is given by Eqs. (2.18)- (2.21). It
is instructive to compare the coaxial case with the parallel-plate case,
since in the limit when the Ri ≈ Ro the coaxial and parallel-plate models
should give the same results. For the parallel-plate case, when the
initial velocity is neglected (v0 = 0), the phase stability range is given
by the following inequalities [39],
πk < λ < � (πk) 2 + 4, (5.8)
where k is an odd positive integer. The normalised gap width, λ, is
defined by
λ = ωd/Vω = m(ωd) 2 /eU, (5.9)
73

where d stands for the gap width, Vω = eEω/mω is the amplitude of
the electron velocity oscillations in the spatially uniform RF field, Eω is
the RF field amplitude inside the gap, and U is the voltage between the
conductors. In addition, for an electron avalanche to start, the impact
velocity should be between the first and the second cross-over points
(cf. Eq. (2.12)), which for zero initial velocity is given by,
v1 < 2Vω < v2 . (5.10)
Due to the asymmetry in the electron motion, the simple analytical
analysis that is feasible in the parallel plate case is not applicable for
the coaxial line, despite the fact that the approximate electron position
and velocity are known explicitly (Eqs. (5.3) and (5.4)). One way of
finding the resonance zones in the parameter space is to compute a series
of successive electron trajectories, searching for the conditions when it
converges to a periodically repeated sequence [69]. In Fig. 5.3 one can
see the result of such a method, where stable resonance zones have been
found numerically. To allow simple comparison with the parallel-plate
case, the normalised gap size has been used and in terms of the coaxial
line it becomes,
λ = m(ωd)2
eUc
= G(Ro − Ri) 2
R2 , (5.11)
o ln (Ro/Ri)
which coincides with Eq. (5.9) when Ro/Ri is close to unity. The convenient
parameter, G = ωRo/Vω,o, has been introduced as representing
a normalised outer radius and Vω,o = qEo/mω.
When Ro/Ri is close to unity the parallel-plate and coaxial models
give similar results. When the ratio becomes larger, the zones deviate
from the straight lines predicted by Eq. (5.8) and the deviation is more
pronounced for the higher modes. When the ratio becomes too large,
all two sided resonances disappear. This is a consequence of the Miller
force and for the higher order modes, where the approximate analytical
solution is very accurate, the prediction (Eq. (5.7)) is that the double
sided resonances should disappear roughly at Ro/Ri = √ 3 ≈ 1.73. Figure
5.3 shows that this is indeed true. The first order mode, however, is
not accurately described by the analytical expression and the numerical
calculations show that this mode can exist for values of Ro/Ri as high
as 4.
In Fig. 5.4 the double-sided discharge regions have been computed
numerically for Ro/Ri = 1.4. The classical multipactor zones have upper
and lower thresholds that satisfy Eq. (5.10) in the following sense: since
74

λ
22
20
18
16
14
12
10
8
6
4
7 8
5 c
4
3 3 c
6
Double sided multipactor, numerical data
2 h
5
1 c
1 1.5 2 2.5 3 3.5 4
R /R [−]
o i
Figure 5.3: Normalised gap width according to Eq. (5.11) vs. Ro/Ri. The
solid straight lines form the classical zones according to Eq. (5.8).
The dots (blue) indicate stable resonances where the sum of two
transits equal an odd number of RF-cycles. The crosses (red) are
for sums equal to an even number of RF-cycles. The sum of the
transit time in RF-cycles for some distinct zones are indicated.
The lowest order hybrid modes are marked with an ’h’ and the
classical resonances with a ’c’. The dashed vertical line indicates
Ro/Ri = √ 3 ≈ 1.73.
75

there are different oscillatory velocities depending on whether the electron
starts on the outer or the inner conductor, an average value must
be computed. By setting Vω,o = qEo/mω = v1/2 the corresponding voltage,
VRo, can be derived. Similarly, by setting Vω,i = qEi/mω = v1/2
the voltage VRi can be obtained. The lower threshold voltage is then
approximately equal to:
Uth ≈ VRo + VRi
. (5.12)
2
The upper threshold is computed in a similar manner, only with v2
instead of v1. The hybrid modes, which in general have a lower average
impact velocity will require a stronger electric field to reach an energy
equal to the first cross-over point. Consequently, a threshold higher than
the lower envelope is obtained for these zones (cf. Fig. 5.4).
Amplitude of Conductor Voltage [V]
10 4
10 3
10 2
1 c
2 h
Double sided multipactor: Z c = 20Ω
3 c
10 0
4
Frequency [GHz]
Figure 5.4: Numerically obtained double sided multipactor chart. The
dashed lines are the approximate lower and upper envelopes given
by Eq. (5.12). The sum of the transit time in RF-cycles for each
zone is indicated. Parameters used: W1 = 23 eV, W2 = 2100 eV,
W0 = 0 eV, Ro = 10 mm, and Ri = 7.2 mm.
If the inner radius is sufficiently small, single-sided multipactor becomes
the dominating scenario. Single-sided multipactor is less complicated
than its double-sided counterpart, as the complicated asymmetry
does not appear in this case. This allows an analytical analysis based
76
7
6
10 1

on the approximate solution for the electron trajectory, Eq. (5.3). The
analytical expression is accurate when the oscillations are small and a
consequence of this is that accurate stable phase multipactor regions are
only found for N ≥ 2, i.e. when the duration of the trajectory is at least
2 RF-periods.
By analysing the resonance and stability conditions, one can show
that single-sided breakdown will have not only one region of stable resonant
phase, but rather two stable regions can be found. One somewhat
wider region with resonance close to zero and another, which is resonant
close to π/4. It can be shown that these regions, in the case when
v0 = 0, are approximately given by:
and
0 < αR < α1
α2 < αR < α3
(5.13)
(5.14)
where
α1 ≈ 4
(5.15)
Nπ
α2 ≈ π 1
− (5.16)
4 Nπ
α3 ≈ π 1
+ (5.17)
4 Nπ
For increasing N, the second region converges to αR = π/4, which according
to Eq. (5.7) corresponds to Rmin ≈ Ro/ √ 2. In Fig. 5.5 the
resonant stable phase, αR, has been plotted as a function of N. Except
for the lowest order resonance (N = 1), the regions of phase stability for
the numerically and analytically obtained phases agree very well.
To obtain the multipactor threshold, it is necessary to know the
impact velocity, which is given by
vimpact ≈ 2Vω,o cos α + v0
(5.18)
The lower boundary shown in Fig. 5.6 is obtained for the maximum
impact velocity for each mode, i.e. vimpact = 2Vω,o when v0 = 0. In the
same figure one can also identify a second set of regions with a higher
breakdown threshold. These zones correspond to the second stable phase
region, Eq. (5.14). Since the phases in this region are close to π/4,
the impact velocity is vimpact ≈ √ 2Vω,o. This value is also indicated in
Fig. 5.6, but it should not be expected to serve as an exact envelope of the
77

Phase [degrees]
60
50
40
30
20
10
α 3
0
0 5 10
N (rf−cycles)
15 20
α 2
α 1
Figure 5.5: Stable resonant phase for single-sided multipactor. Analytically
obtained stable phases are shown as diamonds (red) and the ones
obtained numerically are indicated with dots (blue). Eqs. (5.15) -
(5.17) are shown as solid lines. The dashed line indicates α =
π/4 .
zones as there are phases that are smaller than π/4 (cf. Fig. 5.5), which
will yield a higher impact velocity and consequently a lower threshold.
Furthermore, in the numerical solution of Eq. (5.2) for the first order
mode, an impact velocity of as much as four times Vω,o can be observed.
This results in a threshold much lower than the envelope. The maximum
impact velocity of the second zone is slightly lower than 2Vω,o, resulting
in a somewhat higher threshold. The following higher order modes then
quickly converge to the analytical limit (cf. Fig. 5.6).
When the initial velocity is zero, the only parameters left to vary are
G and the ratio Ro/Ri. Since the characteristic impedance in ohms of
a coaxial line in vacuum is given by Z ≈ 60ln (Ro/Ri), it follows that
only two parameters remain to be varied, viz. G and Z. By following
trajectories for different values of G and Z, stable phase points were
found in this parameter space and the result is plotted in Fig. 5.7, which
was produced using a numerical solution of the equation of motion (a
version of this figure using the analytical expressions can be found in
paper E).
The straight lines in Fig. 5.7 on the right hand side are regions of
stable single-sided resonances. The fact that these appear as straight
78

Amplitude of Conductor Voltage/ln(R 0 /R i ) [V]
10 5
10 4
10 3
10 2
10 0
Single sided Multipactor: Z c =100 Ω
10 1
Frequency x R o [GHz ⋅ mm]
Figure 5.6: Single-sided multipactor breakdown regions based on the numerical
solution of Eq. (5.2). Regions corresponding to N = 1 − 22
RF-periods are shown. The regions with an initial phase, α, close
to zero are produced using dots (blue) and the regions with α close
to π/4 are indicated by crosses (red). The dash-dot line is the
approximate lower envelope given by Vω,o = v1/2 and the dashed
line is given by Vω,o = v1/ √ 2. Parameters used: W1 = 23 eV,
W2 = 2100 eV, W0 = 0 eV, and Z = 100 Ω (corresponding to
e.g. Ro = 10 mm and Ri = 1.88 mm.)
10 2
79

G
100
90
80
70
60
50
40
30
20
10
0
0 0.5 1 1.5 2
Z/Z (Z =50 Ω)
0 0
Figure 5.7: The normalised parameter G vs. normalised characteristic
impedance Z. Each mark represents a stable phase solution and
an effort has been made to suppress polyphase modes in order
to clearly show the behaviour of the main resonance modes. The
chart was obtained by numerically solving the equation of motion.
Stars mark (blue): double-sided multipactor, dots (red):
single-sided multipactor with 0 < αR < 20 o , and crosses (green):
single-sided multipactor with αR > 20 o . The dashed line indicates
Ri,min = Ro/ √ 3 and the dash-dot line Ri,min = Ro/ √ 2.
80

lines indicates that there is no dependence on Z, which implies that a
simple scaling law exists in the single-sided case, viz.
P ∝ (ωRo) 4 Z. (5.19)
In Fig. 5.6 this law has been used to normalise the axes, but voltage
is used on the ordinate instead of power. The chosen normalisation
of the axes in Fig. 5.6 is general and using the analytical solution of
Eq. (5.2) presented above, it can be shown that this normalisation is
valid also for non-zero initial velocity. It is important, however, to be
careful when scaling to a different radii ratio, since for smaller values
of the characteristic impedance, the single-sided multipactor zones may
not exist at all.
In the double-sided case, it is evident that G is a function of Z.
Consequently a more complicated scaling law should be expected. For
small values of Z, however, the coaxial case becomes similar to the
parallel-plate geometry, where the resonance voltage can be written as
function of the frequency-gap-size product. For the coaxial case, this
scaling law becomes
P ∝ (ω(Ro − Ri)) 4 1
. (5.20)
Z
and for the first order resonance this scaling law is quite accurate (cf.
Fig. 5.3), but for the higher order modes it quickly loses its validity with
increasing Z.
5.1.3 Main findings
A qualitative comparison with experiments [63] shows good agreement
with the present analysis. The experimental data shows an increase in
the multipactor threshold for increasing radii-ratio Ro/Ri. It was also
found that the first multipactor zone became narrower for increasing
Ro/Ri. These features are in agreement with the results of this study
as shown in Figs. 5.3 and 5.7. By mapping the data of Fig. 5.3 into the
voltage vs. frequency-gap-size space, used in the experiments, a clear
threshold increase compared with the parallel-plate case can be seen as
well. Even though the experiments used quite large values of Ro/Ri,
no case of single-sided multipactor was observed. This can be explained
by the fact that a material with a low first cross-over point was used,
where the initial velocity will play an important role when the applied
voltage is not high enough. More importantly, only the first order mode
was studied and in this case, as shown in Fig. 5.3, multipactor will be
81

possible for quite large Ro/Ri-values. In the simulations by Sakamoto et
al [64], the experimental results were confirmed. In addition, single-sided
multipactor was observed, which confirms the result of this theoretical
study.
Among the more important results of this part of the study are the
following. The analytical approximate solution of the nonlinear differential
equation of motion for an electron in a coaxial line. The dual regions
of stable phase, Eqs. (5.13) - (5.17), which explain why single-sided multipactor
will be possible also for smaller values of Ro/Ri. The scaling
law for single-sided multipactor, Eq. (5.19), simplifies the presentation
of multipactor prone regions of the single-sided case. The limit formula
for the transition from double- to single-sided multipactor, Eq. (5.7), is
an interesting feature for future experiments to confirm. The reduced
threshold for the first order zone of single-sided multipactor, which must
be taken into account when constructing the lower boundary of all the
zones. Finally, this analysis shows that the behaviour of resonant multipactor
is significantly affected by the nonuniform field and it shows the
benefits of analytically studying different geometries to understand the
basic behaviours before performing numerical simulations.
5.2 Particle-in-cell simulations
In this part of the coaxial study, which is based on paper F, extensive
PIC-simulations have been performed in order to verify the analytical
results as well as to investigate the importance of initial velocity spread
and different maximum secondary emission. One of the advantages with
PIC-simulations is the ability to include parameters that are stochastic
in nature. The stochastic properties of some of the parameters as well
as the actual value of the maximum secondary emission coefficient may
have significant effects on the multipactor threshold as well as on the
existence of a discharge. This has previously been shown in the case
of plane-parallel geometry [46] and it is demonstrated that the effect is
similar also in coaxial geometry.
5.2.1 Numerical implementation
The geometry and field is described in the analytical section. The code
uses normalised parameters such as G and λ and the SEY follows the
model by Vaughan [22]. The initial velocities of the secondary electrons
82

are assumed to have a Maxwellian distribution, i.e.
f(vx,vy,vz) ∝ vn
v exp
�
− 1 v
( )
2 vT
2
�
(5.21)
where v is the absolute value of the initial velocity, vn its normal component
with respect to the surface of emission, and vT is the thermal
initial velocity spread. Another of the used parameters related to this
is the normalised spread of initial electron velocity defined as vT/vmax,
where vmax is the impact velocity for maximum SEY.
Calculations were performed for 2-D arrays of different sets of the
normalised parameters (e.g. ρ = Vω/vmax vs. λ with the other parameters
fixed) and each run corresponds to one particular point in one of
these arrays. Each run was primed with 200 seed electrons, uniformly
distributed over initial phase, and the run was terminated when either
the number of particles exceeded 4500 or when 200 RF-periods had
elapsed. The run was also terminated in case the number of electrons
dropped below 10 before 200 RF-cycles had passed. At the end of each
run, the following parameters were recorded:
• Number of RF-periods needed to exceed 4500 particles. If 4500
particles were not attained within 200 RF-cycles, this parameter
was set to 200.
• Number of electrons at the end of each run.
• Heating asymmetry, i.e. the ratio between the average power deposited
on the inner conductor and the average power deposited
on the outer conductor.
• Average electron growth rate (over the 10 last RF-periods), normalised
with respect to the RF-period.
5.2.2 Simulations
To facilitate comparison between the theoretical result presented in
Fig. 5.7 a simulation was made in the same parameter space. Figure 5.8
shows the number of electrons obtained after 200 RF-cycles. As expected,
the lower order resonances (i.e. at lower and leftmost G-values)
indicate high electron numbers, since more impacts with the conductors
will occur during the same number of RF-periods. Due to this fact, a
parameter space was chosen, which did not include any points in the
83

upper right region of the figures, where the electron growth is very slow.
There is good agreement in the general behaviour and the transition
from double-sided to single-sided multipactor occurs at more or less the
same impedances for the different zones in both the PIC-simulation and
the theoretical data, since the non-zero initial velocity in the PIC-data
is small relative to the oscillatory velocity.
G
100
90
80
70
60
50
40
30
20
10
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Z/Z (Z =50 Ω)
0 0
Figure 5.8: Number of electrons after 200 RF-cycles. The white dots are
points of 2-sided multipactor, the green and yellow dots 1-sided
multipactor, and the white dashed lines correspond to Ri,min =
Ro/ √ 2 (left) and Ri,min = Ro/ √ 3 (right) - all from Fig. 5.7.
Parameters used: σse,max = 1.6, W1 = 50 eV, vT/vmax = 0.01,
ρ = Vω,o/vmax = 0.5, and f = 1.5 GHz.
The straight lines on the right hand side of Fig. 5.8 reveal that this
should be single-sided multipactor. This is confirmed by looking at the
ratio of power deposited on the inner and the outer conductors, which
directly identifies the type of discharge. In Fig. 5.9, the dark blue areas
are regions where most or all power is deposited on the outer conductor,
which implies single-sided multipactor on this conductor. The orange
84
4500
4000
3500
3000
2500
2000
1500
1000
500

and yellow areas indicate that a similar amount of power is deposited
on both conductors and thus the double-sided scenario dominates.
G
100
90
80
70
60
50
40
30
20
10
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Z/Z (Z =50 Ω)
0 0
Figure 5.9: Ratio of power deposited on the inner and outer conductors due
to electron impacts, logarithmic scale log 10(Pinner/Pouter). The
same parameters are used as in Fig. 5.8.
Since the theoretically obtained points agree in general with the PICsimulations,
this confirms the validity of the scaling laws, Eqs. (5.19) and
(5.20). The two different types of modes for single-sided discharge, one
with a phase α ≈ 0 and the other with α ≈ π/4 can also be seen. The
former type of mode is discontinued before reaching the first dashed line
from the right hand side and the latter before the other dashed line in
Fig. 5.7. This is also evident in the PIC-data, especially for values of G
between 30 and 40 in Fig. 5.9 where they are fairly well separated and
only the lower of the paired bands extend into the region between the
dashed lines, as predicted.
In Figs. 5.8 and 5.9 the multipactor threshold can not be identified,
since the oscillatory velocity is kept constant. In Figs. 5.10 and 5.11,
however, the oscillatory velocity has been swept for different G-values
while keeping the ratio Ri/Ro constant and equal to 0.7, i.e. Z = 21.4 Ω
(Z/Z0 = 0.428). Each figure is produced for a different maximum SEY.
When σse,max is low, the ability to compensate for losses is weak and the
zones are well defined and fairly narrow (cf. Fig. 5.10). With increasing
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
85

σse,max the zones become wider and zones previously suppressed by the
losses can appear (cf. Fig. 5.11). This behavior is very similar to that
noted for the parallel plate geometry [46]. The lower (left) envelope is
G
50
45
40
35
30
25
20
15
10
5
0.2 0.4 0.6 0.8 1
V /V
ω max
1.2 1.4 1.6 1.8
Figure 5.10: Number of electrons after 200 rf-cycles. The vertical straight
lines indicate the lower and upper theoretical envelopes according
to Eq. (5.12). Parameters used: Ri/Ro = 0.7, f = 1.5 GHz,
σse,max = 1.3, vT/vmax = 0.01, and W1 = 50 eV.
obeyed, but the upper can be exceeded since a non-zero initial phase will
yield a lower impact velocity and a concomitant higher upper threshold.
In Figs. 5.10 and 5.11 the velocity spread is quite low, vT/vmax =
0.01. When increasing this ratio, a greater portion of the electrons
can have a large negative initial phase, which leads to overlapping of
the multipactor regions when σse,max is large (see Fig. 5.12). On the
other hand, the increased velocity spread also increases the losses, which
especially affects the higher order resonances, since the phase-focusing
effect gets weaker with increasing mode order. If σse,max is low, the
losses are not sufficiently compensated for and this leads to suppression
of the higher order modes (cf. Fig. 5.13). This is a result similar to
that found in the parallel plate case [46]. The corresponding behaviour
is seen also for single-sided multipactor (see paper F).
For single-sided multipactor it was noted that for the first order
mode, the envelope of the breakdown zones Vω,o = v1/2 was not obeyed.
This is confirmed by the PIC-simulations, where the first order mode
86
4500
4000
3500
3000
2500
2000
1500
1000
500

G
50
45
40
35
30
25
20
15
10
5
0.2 0.4 0.6 0.8 1
V /V
ω max
1.2 1.4 1.6 1.8
Figure 5.11: Same as Fig. 5.10 only with σse,max = 2.0.
G
50
45
40
35
30
25
20
15
10
5
0.2 0.4 0.6 0.8 1
V /V
ω max
1.2 1.4 1.6 1.8
Figure 5.12: Number of electrons after 200 rf-cycles. The vertical straight
lines indicate the lower and upper theoretical envelopes according
to Eq. (5.12). Parameters used: Ri/Ro = 0.7, f = 1.5 GHz,
σse,max = 2.0, vT/vmax = 0.1, and W1 = 50 eV.
4500
4000
3500
3000
2500
2000
1500
1000
500
4500
4000
3500
3000
2500
2000
1500
1000
500
87

G
50
45
40
35
30
25
20
15
10
5
0.2 0.4 0.6 0.8 1
V /V
ω max
1.2 1.4 1.6 1.8
Figure 5.13: Same as Fig. 5.12 only with σse,max = 1.3.
clearly passes this limit (see Fig. 5.14).
5.2.3 Comparison with experiments
As mentioned in the introduction to this chapter, PIC-simulations can
include aspects of the multipactor, which are difficult to analyse theoretically.
As a more realistic description is possible, quantitative comparison
with experiments becomes feasible.
In the experimental study by Woo [63] coaxial lines made of copper
were used. When taking values for the secondary electron emission properties
for copper from the ESA standard [50], the lower thresholds of the
PIC-simulations are not in very good agreement with the experimental
data. However, the secondary emission properties can vary a great deal
between different samples of the same material and contamination can
reduce the first cross-over point and increase the maximum SEY. Thus
by slightly lowering the first cross-over point in the PIC-simulations,
very good agreement is obtained (see Fig. 5.15). In paper F an alternative
method of obtaining the experimental threshold when using the
secondary emission properties given in the ESA standard [50] is presented,
based on a modification of the used model for the SEY [22].
However, this will not be discussed further in this summary.
In the experiments an increase in the multipactor threshold for de-
88
4500
4000
3500
3000
2500
2000
1500
1000
500

G
50
45
40
35
30
25
20
15
10
5
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
V /V
ω max
Figure 5.14: Number of electrons after 200 rf-cycles. The vertical straight
lines indicate Vω,o = v1/2 and Vω,o = v2/2. Parameters used:
Ri/Ro = 0.1, f = 1.5 GHz, σse,max = 2.0, vT/vmax = 0.01, and
W1 = 50 eV.
log 10 (Amplitude) [V]
3.2
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
−1.6 −1.4 −1.2 −1 −0.8 −0.6
log (Frequency) [GHz]
10
Figure 5.15: Multipactor breakdown regions for copper electrodes. The region
confined by circles (or white crosses, when inside a dark
region) is from Ref. [63]. The dark regions are obtained by the
PIC-code with: Ro/Ri = 2.3 (Z = 50 Ω), σse,max = 2.25 (from
table A-6 in [50], W1 = 27 eV, and vT = 3 eV (vT/vmax =
0.111).
4500
4000
3500
3000
2500
2000
1500
1000
500
0
4500
4000
3500
3000
2500
2000
1500
1000
500
89

creasing ratio Ri/Ro was observed. It was also found that the first multipactor
zone became narrower for decreasing Ri/Ro. This behaviour
was seen in the analytical analysis and it is evident also in the PICsimulations.
Figure 5.16 shows the result of a simulation for Z = 174 Ω
(Ro/Ri = 18.26). The agreement between simulations and experiments
is good. The multipactor zone becomes narrower and the threshold increases.
The position of the zone is slightly shifted compared with the
experimental data and the reason for this deviation may partly be explained
by an inaccuracy of the dimensions of the copper electrodes.
The main reason, however, seems to be related to the SEY-model and
the description of the initial velocity of the secondary electrons, but a
more detailed study will be necessary to settle this matter.
log 10 (Amplitude) [V]
3.2
3
2.8
2.6
2.4
2.2
2
1.8
−1.6 −1.4 −1.2 −1 −0.8 −0.6
log (Frequency) [GHz]
10
Figure 5.16: Ratio of power deposited on the inner and outer conductors
due to electron impacts (log-scale) for the same parameters and
SEY-formula as in Fig. 5.15 except that Ro/Ri = 18.26.
In the section summarizing the main findings of the analytical part,
it was mentioned that no single-sided multipactor was noted in the experiments.
This is in agreement with the PIC-simulations and shows
that no measurements were made at high enough voltage and frequency,
where single sided multipactor would be the main phenomenon (the dark
blue regions in Fig. 5.16).
90
0
−0.5
−1
−1.5
−2
−2.5
−3
−3.5
−4
−4.5

5.2.4 Main conclusions
The analytically obtained results are confirmed by the PIC-simulations.
In addition, the significance of initial velocity spread as well as different
maximum secondary electron emission have been highlighted. The
results have shown that the behaviour with respect to these conditions,
in both the single-sided and the double-sided cases, is qualitatively the
same as for the parallel plate multipactor. The results are in good agreement
with available experimental data, but the commonly used model
for SEY seems to be inadequate for large Ro/Ri-ratios, since a small,
but not negligible, deviation from the experiments was noticed.
91

Chapter 6
Detection of multipactor
In the space industry, where multipactor is a well-known problem, great
care is taken to avoid the phenomenon. According to the ESA standard
on multipactor design and test [50] only single carrier parts with a
margin of 8-12 dB (depending on the type of component) in the analysis
stage are exempt from testing. The recommended corresponding margin
for multicarrier parts is 6 dB between the peak power of the multicarrier
signal and the single carrier threshold. By performing unit acceptance
tests, the margins can be reduced to 3-4 dB in the single carrier case
and to 0 dB in the multicarrier case. In order to take advantage of the
smaller margins, accurate and reliable testing is required to verify that
the prescribed margins are actually fulfilled and successful testing relies
on accurate and unambiguous methods of detection.
This chapter starts by briefly reviewing some of the most common
methods of multipactor detection. Then a method of detection is presented,
which can be used in combination with other methods to achieve
accurate and unambiguous test results. Beginning with the single carrier
case, a theory for the underlying mechanism making the method possible
is given together with some supporting data. This is followed by a
discussion outlining how the method can be applied also in the multicarrier
case. Detection of multipactor using RF power modulation, which
is the main topic of this chapter, was analysed in paper A of this thesis.
6.1 Common Methods of Detection
There are several different ways of detecting multipactor and they can be
divided into two fairly distinct categories, viz. global and local methods
93

of detection. The global methods are characterised by being able to
discern whether or not microwave breakdown is taking place somewhere
within the tested system. However, it can not pinpoint the location of
the discharge. For flight hardware it is important to completely avoid
multipactor in the entire microwave system and consequently this type
of testing is preferred. During the product development stage, it may
be of interest to know not only that a discharge is taking place, but also
its exact position within the system. In such a case, a local method can
be useful, since it can be used to monitor a certain area inside a device.
6.1.1 Global methods
When performing systems tests on flight hardware, global methods of
detection are normally used and there are several reasons for this. The
methods can usually be applied without modifying the component, which
is advantageous as modifications can affect the electromagnetic properties
and give inadequate measurement results. In cases where the discharge
is weak, many local methods are unable to detect the phenomenon
and thus the often more sensitive global ones are a better choice. Furthermore,
it is a requirement of the ESA standard [50] that two methods
of detection should be used and at least one of them should be global.
By using two methods of detection, the risk for misinterpretations is reduced.
The most common global methods of detection will be described
in the following subsections.
Close-to-carrier noise
Multipacting electrons will be accelerated to high velocities by the electric
field and at regular intervals, 2/N times per field cycle, the electrons
will hit an electrode or device wall and experience a sudden deceleration.
The radiated power is a function of the electron acceleration and it is
described by Larmor’s formula,
P = 2 e
3
2 ˙v
4πɛ0
2
c3 (6.1)
where ˙v is the acceleration, ɛ0 the dielectric constant of vacuum, and c
the speed of light. For an applied sinusoidal electric field, the electrons
will experience a sinusoidal acceleration, except for the sudden interruptions
when colliding with the electrodes. Fig. 6.1 shows what the
acceleration may look like for first order multipactor (N = 1), where the
electrons hit the electrodes once every half cycle of the electric field.
94

Amplitude of acceleration [Gm/s 2 ]
x 10
1.5
6 C2C−Noise Timedomain
1
0.5
0
−0.5
−1
−1.5
1.76 1.78 1.8 1.82 1.84 1.86
x 10 −8
Time (s)
Figure 6.1: A qualitative plot of the electron acceleration for multipacting
electrons with N = 1 (average value for a large number of electrons).
The regular deviation from a pure sinusoid will generate harmonics,
but these will be discussed in the next subsection. Due to variations in
the time between impact and emission of new electrons, in the time it
takes to decelerate the electrons, and also in the number of electrons,
close-to-carrier noise will be generated. In fact, most noise is generated
at the carrier frequency, but this can not be seen in a measured spectrum,
as the signal amplitude masks the noise. By performing a Fast Fourier
Transform (FFT) of a sequence like the one shown in Fig. 6.1, the noise
generated close to the carrier can be seen (see Fig. 6.2). Very close to
the carrier, the noise level is even higher than what can be discerned in
Fig. 6.1 and by using a bandpass filter with high rejection at the carrier
frequency, the noise increase close to the carrier can be detected with a
spectrum analyser. In combination with a low-noise amplifier, this can
be a very sensitive method of detection.
This method of detection can be used for both single and multicarrier
signals. Care should be taken, however, when using a pulsed signal,
since such a signal will generate harmonics and if the pulse length and
type is not chosen properly, the harmonics may be generated in the
measurement band [31]. A risk with this method is that other sources of
95

(dB)
−20
−30
−40
−50
−60
−70
−80
−90
−100
Multipactor Noise Power
2 4 6 8 10 12 14 16
Frequency (GHz)
Figure 6.2: Noise spectrum of a multipactor simulation like the one in
Fig. 6.1, but with N = 3. The signal frequency is f0 = 2 GHz.
The odd harmonics as well as the peaks at odd multiples of f0/3
are clearly visible.
noise may be misinterpreted as multipactor noise. Nevertheless, by using
two different methods of detection as required by the ESA standard, this
risk is greatly reduced.
Third harmonic
The resonant behaviour of the multipactor discharge and the repetitive
acceleration and sudden deceleration of the electrons will generate noise,
which will have harmonics at the basic frequency, f0, and at odd multiples
of this frequency (cf. Fig. 6.2). Higher order multipactor will
have harmonics not only at odd multiples of the basic harmonic, but at
each odd multiple of f0/N [14] (cf. Fig. 6.2). In such cases there are
many different frequencies available for detection. However, since the
third harmonic usually is the most powerful harmonic and it is always
present, regardless of the order of resonance, it is the best choice for
detection.
Third harmonic detection is a very reliable and fast method of detection.
According to Ref. [14], it gives the fastest indication of multipactor
and that makes it the method of choice when studying mul-
96

tipactor events that are short-lived, like e.g. multicarrier multipactor,
where the discharges often are weak and of short duration. However,
the method is also sensitive to other sources of noise in the same way as
close-to-carrier noise and, thus, it is not recommended to use only third
harmonic and close-to-carrier noise detection in a test setup.
Reflected power
Mismatches in the transitions between microwave components imply
that some of the input power will be reflected. However, in a welldesigned
system, very little power is reflected. Components containing
high Q-value parts, like e.g. cavity resonators, are only well matched
at certain frequencies and minor changes in the component properties
can lead to detuning of the part. Multipactor is known to be able to
detune high Q-value components [70]. Thus the reflected power from
a component can be used as an indication of a multipactor event. To
study the absolute value of the reflected power is usually not a good way
of detection, since the input power can vary during a multipactor test
and consequently the reflected power will vary as well, as the reflected
power is a fixed fraction of the input power. This fraction is called the
return loss and is commonly measured in decibel. The return loss will be
a stable value, insensitive to power fluctuations, until the component is
detuned. Figure 6.3 shows an example where both close-to-carrier noise
and the return loss are monitored simultaneously during a multipactor
test. Both methods indicate a change around t = 50 s and thus it can
be determined with great confidence that a multipactor discharge was
initiated at that time.
The main advantage with this method is that it is quite reliable and
there is little risk that other phenomena will cause a mismatch that can
be confused with multipactor. However, for low Q-value components or
badly matched systems, the sensitivity of the method is low.
Electron monitoring
A new global method of detection was presented at the 4 th International
Workshop on Multipactor, Corona and Passive Intermodulation
in Space RF Hardware [71]. It is called the Electron Density Detection
Method, abbreviated EDDM, and uses a set of tri-axial cables as
a probe and electrons picked up by the probe are then monitored with
a high precision electron meter. The data is collected using a computer
97

Noise (dBm)
−60
−70
−80
−90
0 10 20 30 40 50
Time (s)
60 70 80 90 100
Input Power (dBm)
Return Loss (dB)
46
44
42
0 10 20 30 40 50
Time (s)
60 70 80 90 100
−8
−10
−12
−14
−16
0 10 20 30 40 50
Time (s)
60 70 80 90 100
Figure 6.3: Multipactor monitored by studying close-to-carrier noise as well
as the return loss. At t ≈ 50 s the component starts to experience
multipactor discharge and becomes increasingly mismatched. Excellent
agreement with the second detection methods, close-tocarrier
noise, can be seen.
98

software and the time evolution of the electron density can be studied.
The probe does not have to be located inside or close to the area where
the discharge will take place. Even though it is advantageous for higher
sensitivity to have the probe pointing at the critical gap it is also reliable
when located outside the device under test and thus it can be used
for waveguides and coaxial transmission lines. It is not completely clear
from the paper, however, how the electrons escape from a completely
confined device like a typical waveguide or coaxial cable and one will
have to assume that there must be some small opening somewhere in
the system, where electrons can leak out.
Furthermore, the method can be used to quantitatively measure the
amount of generated electrons. however, this requires some kind of calibration
of each test setup and that may prove to be problematic. When
used in this way, the method can no longer be viewed as a global method,
which can detect a multipactor event anywhere in the system, instead it
has become a local method. Among the main advantages of the method
is the low cost involved, since no expensive microwave instruments are
needed.
Residual mass
A very slow global method of detection is to detect the gas molecules,
which are outgassed from the device walls due to the electron bombardment
during a multipactor event. The gas molecules consist of residuals
of water, air and contaminants, and using a mass spectrometer,
the different molecules can be identified. It has been noted [31] that a
detectable increase in the water spectrum can be seen during a multipactor
discharge. The major drawback of this method of detection is
its inability to detect fast multipactor transients (not enough molecules
are released from the walls) and thus it is not a suitable method for
multicarrier multipactor studies. Another disadvantage is that there is
a certain delay between onset of the discharge and indication in the instrumentation.
However, it can be useful as a diagnostic tool together
with one or two of the other described methods.
6.1.2 Local methods
In cases where it is not sufficient to only confirm the existence of a
discharge in the system, but also to determine the exact position, local
methods of detection will have to be used. The two most common local
99

methods are charge detection using a probe and optical monitoring and
these two methods will be described below.
Charge probe
A special case of the method of electron monitoring, which was described
in a previous subsection, is the use of a probe that monitors only a
certain position within the microwave device. A very common approach
used when detecting electrons inside a waveguide is to flush mount an
SMA connector and apply a positive potential to the centre pin. The
negatively charged electrons are attracted to the pin and causes a small,
but detectable, current to flow, which can serve as an indication of the
electron density.
The method is easy to implement and therefore it has been quite
common in many test setups. Unfortunately, it is also quite slow due
to the circuit used to amplify the weak current [14] and thus it is not
feasible for measuring fast multipactor events. In addition, it requires
modification of the component, which makes it useless when testing flight
hard ware. In such a case the EDDM is a better choice.
Optical detection
Optical detection is possible since the electrons that make up the multipactor
discharge can excite or ionise either the remaining gas molecules
within the device or the molecules in the device wall. It can be divided
into two groups, viz. photon detection via optical probe and photon
detection via photographs or video camera [72,73]. Both methods are
common, but the former seems to be more frequently used.
The main advantage with these methods is that they can be used to
pin-point the location of the discharge inside the device. A major disadvantage
is that they may be impossible to use for studying real parts,
as the devices may not have any suitable openings. This is especially
true for the methods based on photographic techniques.
6.2 Detection using RF Power Modulation
During verification of a test setup for multipactor at Saab Ericsson
Space, Göteborg, Sweden, an odd, spike-like phenomenon was found
in the noise generated by a discharge in a coaxial test sample. An ad-
100

ditional test sample, a resonant cavity, was manufactured and the same
type of spikes were noted again (cf. Fig. 6.4).
(dBm)
−55
−60
−65
−70
−75
−80
−85
−90
Noise Power (92.8 − 93.0 s)
92.82 92.84 92.86 92.88 92.9
Time (s)
92.92 92.94 92.96 92.98 93
Figure 6.4: Periodic spikes, which appeared during a multipactor experiment.
The main periodicity is 100 Hz and emanates from the power
supply of the high power amplifier.
A fast Fourier transform revealed that the spikes were periodic and
it was noted that the same periodicity could be found also in the input
signal after it had been amplified by the TWTA (travelling wave tube
amplifier). Due to interference from the main power supply, the signal
was amplitude modulated with a main modulation frequency of 100 Hz
and with harmonics at multiples of this frequency. The interference was
very weak and would in most cases be disregarded. In order to see if
the periodicity was present only in conjunction with multipactor events,
a large number of test runs were performed. The results were consistent
- the periodic noise only appeared when a discharge was detected.
Figs. 6.5 and 6.6 show one of the test runs where the noise is non-periodic
before onset of multipactor but periodic directly afterwards.
The AM (amplitude modulation) that was present in the input signal
was very weak and not deliberately added. A stronger AM was added to
the signal before the high power amplifier, resulting in a more distinct
modulation. This made it possible to study if there was any correlation
between the modulation strength and the corresponding peak in
101

(dBm)
(dBm)
−55
−60
−65
−70
−75
−80
−85
−90
−80
−100
−120
−140
Noise Power (32 − 52 s)
34 36 38 40 42
Time (s)
44 46 48 50 52
Fourier transform (32 − 52 s)
−160
0 50 100 150 200 250
frequency (Hz)
300 350 400 450 500
Figure 6.5: The beginning of a multipactor test sequence showing the time
before onset of the discharge. The FFT (fast Fourier transform)
gives no indication of dominant frequency components.
(dBm)
(dBm)
−55
−60
−65
−70
−75
−80
−85
−90
−80
−100
−120
−140
Noise Power (52 − 72 s)
54 56 58 60 62
Time (s)
64 66 68 70 72
Fourier transform (52 − 72 s)
−160
0 50 100 150 200 250
frequency (Hz)
300 350 400 450 500
Figure 6.6: The end of the same test sequence as in Fig. 6.5. The sudden increase
in the noise floor indicates onset of multipactor. The FFT
shows dominant frequency components at 100 Hz and multiples
thereof.
102

the FFT. It was concluded that there is a positive correlation between
the modulation depth and the strength of the detected signal at the
corresponding frequency. By taking the time average of one of the test
sequences like in Fig. 6.3 and plotting it using linear scales on both axes,
it was found that there was a more or less linear relationship between
the input power and the resulting multipactor noise power (cf. Fig. 6.7),
which can be described by the following function:
Pnoise = k · (Pinput − Pth) [W] Pinput ≥ Pth (6.2)
where k = 5.3 × 10 −11 and Pth = 25.2 W is the multipactor threshold.
Noise Power [nW]
1.4
1.2
1
0.8
0.6
0.4
0.2
Linear Plot of Noisepower vs Input Signal Power
Input Signal Power [W]
26 28 30 32 34 36 38 40 42 44
0
50 55 60 65 70 75
Time (s)
80 85 90 95 100
Figure 6.7: Time average of the test sequence shown in Fig. 6.3 with a linear
scale on both axes. The straight line has been added to show the
close to linear relationship between input power and noise power.
Note: The input power is increased every 4 seconds, which is the
reason for the step like behaviour.
The reason why the small modulation was so noticeable in the multipactor
noise (see Fig. 6.4) is that the noise signal is a function of the
difference between the input power and the multipactor threshold, i.e.
no discharge noise is generated until the multipactor threshold has been
reached. Furthermore, since the decibel scale is a relative scale, the small
increase in absolute numbers becomes very noticeable in relation to the
existing noise floor. As a comparison, the first small steps in Fig. 6.7,
103

correspond to huge increases in the decibel scale, which can be seen in
Fig. 6.8.
(dBm)
−60
−65
−70
−75
−80
−85
Noise Power (100 samples noise average)
0 10 20 30 40 50
Time (s)
60 70 80 90 100
Figure 6.8: The same sequence as in Fig. 6.7 (except that in this case the
entire sequence is shown). The small initial steps of Fig. 6.7
become huge steps on the logarithmic scale.
The above examples, which used the new detection method, relied
on close-to-carrier noise measurement data. However, the mechanism
which is utilised requires primarily that the input signal is amplitude
modulated, that the detected signal is proportional to the difference
between the input power and the multipactor threshold and that the
detected signal responds quickly to changes in the multipactor event.
The two first conditions are likely to be fulfilled by all detection methods
for multipactor, but the last will have to be verified for each method.
Results from measurements presented in [14] show that third harmonic
detection is faster than close-to-carrier noise detection, which should
make it excellent for AM detection. In general, probably any method
can be used provided that a suitable AM frequency is chosen and that
the instrument used for detection has a sampling frequency that is more
than two times larger than the modulation frequency in order to fulfil
the Nyquist criterion.
104

6.2.1 Single carrier
When using the AM detection method in the single carrier case, the
test setup implementation is quite straight forward. In practice, the
only difference between a standard multipactor test setup and one that
uses the AM detection method is that the signal data must be sent to
a computer or some other instrument that can perform a FFT. It is
also important that the signal generator is capable of producing an AM
signal, but that is a standard feature of most signal generators.
The spectrum analyser, which receives the signal, should be set to a
sampling rate that is at least two times larger than the AM signal, i.e. if
the AM signal has a frequency of 1000 Hz, then a minimum sampling rate
of 2000 samples/second should be used. However, in order to study the
shape of the modulation signal, it is better to use a sampling frequency
more than twenty times greater than the frequency of the AM. The signal
data can be stored for future processing, or if a powerful computer is
used, real time FFT can be performed, thus allowing the operator to get
an immediate indication when a discharge takes place.
6.2.2 Multicarrier
AM detection in the multicarrier case [74] is somewhat more difficult and
the method has not yet been experimentally confirmed. When performing
a multicarrier multipactor test, the phase of each carrier will have to
be adjustable in order to enable the test engineer to produce the wanted
shape of the signal envelope. The aim is to produce a signal envelope
corresponding to the assumed worst case. When the phases have been
set to their predefined values, the envelope will be periodic. If the envelope
exceeds the multipactor threshold for a time long enough to allow a
sufficient number of gap crossings, a discharge will occur, provided that
a suitable seed electron is available. When the envelope drops below
the threshold again, the electrons will disappear quickly [21] and there
will normally be no electrons left from the discharge the next time the
envelope exceeds the threshold. If no specific source of seed electrons is
available, a multipactor discharge may not occur every envelope period.
However, if an efficient electron seeding source is used, there should be
an ample amount of electrons available to initiate a discharge each time
the envelope exceeds the threshold for a sufficiently long time.
The use of AM detection of multipactor requires that the discharge
event is continuous or that it occurs regularly. Single or very sporadi-
105

cally occurring discharges will be difficult to identify in a FFT plot. By
using a source of free electrons in the test setup, e.g. a hot filament
or a UV light [14], seed electrons will be abundant and a multipactor
event is likely to occur each time the envelope exceeds the threshold for
a long enough time. If the envelope is amplitude modulated, the discharge
events will vary in strength with the AM frequency. This periodic
variation will appear as a peak in the FFT plot of the detected signal at
the same frequency. By applying a weak, 1-5% depth, synchronised AM
to the input signals, the multipactor threshold can be determined with
high accuracy and without risking ambiguous test results. A 1% AM
corresponds to less than ±0.1 dB variation in the input signal and will
have no significant effect on the measured threshold.
The detected signal must then be processed by a computer or a similar
tool in order to reveal the periodicity, as in the single carrier case.
In Fig. 6.9 an example of a possible test setup is given. In order to
achieve a good AM in the multicarrier case, all the signals should be
modulated using the same reference signal of modulation. Many signal
generators have a signal reference input, thus allowing the user to synchronise
several signal generators. To perform a successful multicarrier
experiment, the phases have to be stable in relation to each other and
thus a common reference signal will have to be used in any case. Another
way of achieving synchronised modulation could be to modulate
the gain adjustment of the high power amplifier.
In Fig. 6.9 it is suggested that the third harmonic should be monitored
and that is probably the best choice when studying multicarrier
multipactor, since third harmonic detection is fast and sensitive. Closeto-carrier
noise detection is a possible alternative, but it may not be as
sensitive and thus weak multipactor events may be overlooked.
6.2.3 Main achievements
A method of multipactor detection has been devised, which can be used
to obtain accurate and unambiguous measurement results for both single
and multicarrier multipactor. The method does not aim to replace any of
the existing methods of detection, rather it can serve as a complement
to the other methods to improve accuracy and confidence in the test
results.
Close-to-carrier noise and third harmonic detection are two fast and
sensitive methods of multipactor detection. Both methods rely on noise
generation, which makes them prone to non-multipactor generated noise.
106

Reference signal generator
Phase
control
∆φ
∆φ
M
U
X
Instrument control
&
Data storage and processing
HPA
dB
Wave form monitor
DUT
Spect.
Analys.
Vacuum
Chamber
Reflected power
Power
Meter
dB
LNA
Forward power
dB
Power
Meter
Third harmonic detection
Figure 6.9: Test setup for multicarrier multipactor measurements using RF
power modulation. Each input signal is amplitude modulated and
by phase locking the signals using a signal reference, the entire
signal envelope will be modulated. A computer can be used to
control the instruments and collect the output data, which can
then be real-time Fourier transformed to reveal any periodicity
in the detected signal.
Spect.
Analys.
107

In a typical test setup there can be many sources of noise, which could
result in ambiguous test results. On the one hand, the multipactor
threshold could be established at a too low value if non-multipactor
generated noise is misinterpreted as the result of a discharge. On the
other hand, a short-lived multipactor event could be disregarded and
lead to determination of a too high threshold based on a more distinct
indication. The AM detection method resolves this concern by only
signalling for true multipactor events.
Another advantage of the AM detection method is the fact that it
is particularly sensitive close to the multipactor threshold, since it only
responds to the signal difference between the input signal and the threshold,
as can be seen from relation 6.2. A weak amplitude modulation,
as shown in Fig. 6.10 where the single carrier signal has just passed the
multipactor threshold, will produce a very distinct modulated output
signal as indicated in Fig. 6.11. Even though the signal is very noisy,
the periodicity is very distinct. Not always will the periodicity be as
noticeable as in Fig. 6.11, but a FFT will reveal any periodicity in the
measured signal.
Amplitude [A.U.]
120
100
80
60
40
20
Input signal envelope, single carrier, 1%depth AM
Input signal
Multipactor threshold
0
0 0.5 1 1.5 2 2.5
Time
3 3.5 4 4.5 5
Figure 6.10: Example of a single carrier input signal envelope with a 1%
depth AM. The signal has barely exceeded the multipactor
threshold.
108
One of the shortcomings of the AM detection method is that it re-

Amplitude [A.U.]
2.5
2
1.5
1
0.5
Detected signal
0
0 0.5 1 1.5 2 2.5
Time
3 3.5 4 4.5 5
Figure 6.11: Qualitative form of the detected multipactor signal when the
input signal to the DUT is as shown in Fig. 6.10. The same type
of signal can be seen in Fig. 6.4, but in this case, the sampling
frequency is 200 times higher than the frequency of modulation,
which explains why the modulated signal is so prominent.
109

quires a certain time for the FFT. One can estimate that a minimum
of 5-10 times the period of the AM is needed for a reliable FFT. If the
frequency of AM is 1 kHz, then the time needed for the measurement
would be 5-10 ms. In most cases this would be acceptable, but if single
events of multipactor are to be detected, the method will not work.
110

Chapter 7
Conclusions and outlook
This thesis has presented basic theory as well as new developments concerning
the phenomenon called multipactor. Its deleterious effects on
microwave systems operating in a vacuum environment have been emphasised.
For satellites, the discharge can be catastrophic as basically no
means of repair or modification is available for parts in orbit. To avoid
the risks associated with vacuum discharges, lots of effort has been put
into studying the phenomenon and today well established engineering
methods exist, which are used by the microwave engineer when designing
hardware bound for space. However, the available methods are based
on the simple parallel-plate model, which in many instances is the worst
case. Thus, when designing important microwave features, in particular
such involving a nonuniform electromagnetic field, like e.g. irises and
coaxial lines, the parallel-plate model may not be applicable and there
is a risk of unnecessarily conservative designs. Such designs are often
large and heavy, which is a great disadvantage when it comes to devices
to be used in space. As a first attempt to establish new methods of
assessing the risk for having a discharge in such structures, a large part
of this thesis has been devoted to multipactor in irises and coaxial lines.
It has been shown that by only considering the random walk of the secondary
electrons in an iris, the threshold can be significantly increased
compared with the pure parallel-plate case. Many interesting aspects of
the phenomenon in a coaxial line have been found, e.g. the dual stable
regions of single-sided resonance and its unexpectedly low threshold
of the first order mode. In addition, the increased threshold for high
impedance coaxial lines that was found in experiments, was also found
in this study, both in the theoretical analysis and in the PIC-simulations.
111

At altitudes where most satellites operate, the pressure is very low
and for most purposes it can be approximated as a perfect vacuum.
However, during the launch phase of a satellite and during its first days
of operation as well as at times when the satellite fires its attitude and
altitude control engines, the microwave parts may not be completely
vented and thus it is important to understand what happens with the
multipactor threshold with increasing pressure. This has been one of the
main topics of this thesis. It was found that for materials with a low first
cross-over energy, for the lowest order resonance, the threshold increases
with increasing pressure until reaching a maximum, after which it starts
to decline. The reason for the increase is the friction force experienced
by the electrons when colliding with the neutral gas particles. In all
other cases, for materials with a high first cross-over energy and in general
for the higher order modes, a monotonically decreasing threshold
is noted. This behaviour can be explained by the thermalisation of the
electrons, which leads to a higher total impact energy, as well as the
contribution of electrons from collisional ionisation, which reduces the
necessary secondary emission yield and consequently also the required
impact velocity. Improved quantitative results require a more detailed
investigation of the fraction of the electrons from impact ionisation that
actually contribute to the multipactor bunch, but that is left for a future
study. Furthermore, in order to make the model useful for all frequencygap
size products, an extension of the model to include also the hybrid
modes is necessary. Due to the complexity of such a study, it was not
included in this first analysis. However, this may be an interesting topic
for a future investigation.
In addition to designing with respect to the proper thresholds, most
hardware will require some type of testing to ensure compliance with
the standard. Such tests must be of good quality to avoid ambiguities,
which could disqualify a component that is multipactor free. In this
thesis a method that gives unambiguous and highly reliable test results
was presented. It is a method based on a weak amplitude modulation of
the input signal, which becomes very distinct in the multipactor signal,
since the generated noise power is a function of the difference between
input power and the multipactor threshold. By using this auxiliary
method of detection in connection with two other detection methods,
reliable test results can be obtained.
It is quite satisfying when looking back at the chapter on future work
in my licentiate thesis [75] and realizing that the two main topics men-
112

tioned there were multipactor in nonuniform fields and irises, which were
then successfully studied during the second part of my PhD work. However,
there are still interesting projects to consider, e.g. multipactor in
irises where not only the effect of the random walk is considered, but also
the effect of the actual nonuniform field in the structure. There are many
other important structures in microwave systems, which have not been
studied with respect to multipactor discharge. Among these are crossed
irises, septum polarisers and ridged waveguides. The 20 gap-crossings
rule, which is part of the ESA standard [50], should be re-investigated
both theoretically and experimentally to make sure that the rule has
a sound theoretical base with good agreement between simulations and
experiment. As a continuation of the coaxial study, the axial dimension
could be included by considering not only a travelling wave signal but
also a standing wave. Since electrons affected by the Miller force will
drift towards positions with low field amplitude, it may not be feasible
to directly apply the peak amplitude of the standing wave in the coaxial
model presented in this thesis.
113

114

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121

122

Included papers A–F
123

Paper A
R. Udiljak, D. Anderson, P. Ingvarson, U. Jordan, U. Jostell, G. Li,
M. Lisak, L. Lapierre, J. Puech, and J. Sombrin, “New Method for
Detection of Multipaction”, IEEE Trans. Plasma Sci., Vol. 31, No. 3,
pp. 396-404 , June 2003.

Paper B
R. Udiljak, D. Anderson, M. Lisak, V. E. Semenov, and J. Puech, “Multipactor
in low pressure gas”, Phys. Plasmas, Vol. 10, No. 10, pp. 4105-
4111, Oct. 2003.

Paper C
R. Udiljak, D. Anderson, M. Lisak, V. E. Semenov, and J. Puech, “Improved
model for multipactor in low pressure gas”, Phys. Plasmas,
Vol. 11, No. 11, pp. 5022-5031, Nov. 2004.

Paper D
R. Udiljak, D. Anderson, M. Lisak, J. Puech, and V. E. Semenov, “Multipactor
in a waveguide iris”, accepted for publication in IEEE Trans.
Plasma Sci.

Paper E
R. Udiljak, D. Anderson, M. Lisak, V. E. Semenov, and J. Puech, “Multipactor
in a coaxial transmission line, part I: analytical study”, accepted
for publication in Phys. Plasmas

Paper F
V. E. Semenov, N. Zharova, R. Udiljak, D. Anderson, M. Lisak, and
J. Puech, “Multipactor in a coaxial transmission line, part II: Particlein-Cell
simulations”, accepted for publication in Phys. Plasmas