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

Goteborg, Sweden, 2007

Multipactor in Low Pressure Gas

and in Nonuniform RF Field Structures

Richard Udiljakc©Richard Udiljak, 2007

ISBN 978-91-7291-885-6Doktorsavhandlingar vid Chalmers tekniska hogskolaNy serie nr 2566ISSN 0346-718X

Department of Radio and Space ScienceChalmers University of TechnologySE–412 96 GoteborgSwedenTelephone +46–(0)31–772 10 00

Cover: Susceptibility chart for multipactor in a waveguide iris for fivedifferent height/length-ratios.

Printed in Sweden byReproserviceChalmers Tekniska HogskolaGoteborg, Sweden, 2007

Multipactor in Low Pressure Gas

and in Nonuniform RF Field Structures

RICHARD UDILJAKDepartment of Radio and Space ScienceChalmers University of Technology

Abstract

Resonant electron multiplication in vacuum, multipactor, is analysedfor several geometries where the RF electric field is nonuniform. In par-ticular, it is shown that the multipactor behaviour in a coaxial line isboth qualitatively and quantitatively different from that observed withthe conventionally used simple parallel-plate model. Analytical esti-mates based on an approximate solution of the non-linear differentialequation of motion for the multipacting electrons are supported by ex-tensive particle-in-cell simulations. Furthermore, in a microwave iris theelectrons tend to perform a random walk in the axial direction of thewaveguide due to the initial velocity distribution. The effects of this phe-nomenon on the breakdown threshold are analysed. The study showsthat the threshold is a function of the height-to-length ratio of the irisand for a fixed value of this ratio, the multipactor susceptibility chartscan be generated in the classical engineering units. Using the parallel-plate concept, the multipactor threshold in low pressure gases has beenanalysed using a model for the electron motion that takes into accountthree important effects of electron-neutral collisions, viz. the frictionforce, electron thermalisation, and impact ionisation. It is found thatall three effects play important roles, but the degree of influence de-pends on parameters such as order of resonance and secondary emissionproperties. In addition, a new method for detection of multipactor ispresented. By applying a weak amplitude modulation to the input signaland performing a fast Fourier transform on the detected signal, accurateand unambiguous measurement results can be obtained. It is demon-strated how the method can be used in both single and multicarrieroperation.

Keywords: Multipactor, discharge, breakdown, microwave discharge,nonuniform fields, coax, iris, low pressure gas, detection methods.

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iv

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 Methodfor 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. Plas-mas, 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 IEEETrans. 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, partII: Particle-in-Cell simulations”, accepted for publication in Phys.Plasmas

v

Conference contributions by the author (not included in this thesis):

[G] R. Udiljak, G. Li, D. Anderson, P. Ingvarson, U. Jordan, U. Jostell,A. Kryazhev, M. Lisak, V. E. Semenov, “Suppression of Multi-pactor 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. Se-menov, “Detection of Multicarrier Multipaction using RF PowerModulation”, 4th International Workshop on Multipactor, Coronaand Passive Intermodulation in Space RF Hardware, 8-11 Septem-ber, 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 ex-periment”, NATO Advanced Research Workshop on Quasi-OpticalControl of Intense Microwave Transmission , 17-20 February, 2004,Nizhny-Novgorod, Russian Federation.

[J] R. Udiljak, D. Anderson, M. Lisak, J. Puech, V. E. Semenov, “Mi-crowave breakdown in the transition region between multipactorand corona discharge.”, RVK 05, June 14 - 16 juni, Linkoping

[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 equip-ment.”, RVK 05, June 14 - 16 juni, Linkoping

[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 PassiveIntermodulation 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 InternationalWorkshop on Multipactor, Corona and Passive Intermodulationin Space RF Hardware, 12-14 September, 2005, ESTEC, Noord-wijk, The Netherlands.

[N] C. Armiens, B. Huang, R. Udiljak, D. Anderson, M. Lisak, U. Jostell,and P. Ingvarsson, “Detection of Multipaction using AM signals”,

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

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Contents

Publications v

Acknowledgement xi

Acronyms xiii

1 Introduction 1

2 Multipactor in vacuum 5

2.1 Single Carrier . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Basic theory . . . . . . . . . . . . . . . . . . . . . 62.1.2 Hybrid modes . . . . . . . . . . . . . . . . . . . . . 172.1.3 Factors effecting the threshold . . . . . . . . . . . 182.1.4 Methods of suppression . . . . . . . . . . . . . . . 202.1.5 Effect of random emission delays and initial veloc-

ity spread . . . . . . . . . . . . . . . . . . . . . . . 232.2 Multicarrier . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Design guidelines . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Single carrier . . . . . . . . . . . . . . . . . . . . . 292.3.2 Multicarrier . . . . . . . . . . . . . . . . . . . . . . 29

3 Multipactor in low pressure gas 35

3.1 Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.2 Multipactor boundaries . . . . . . . . . . . . . . . 373.1.3 Main results . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Advanced Model . . . . . . . . . . . . . . . . . . . . . . . 443.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.2 Analytical formulas for argon cross-sections . . . . 473.2.3 Multipactor boundaries . . . . . . . . . . . . . . . 49

ix

3.2.4 Key findings . . . . . . . . . . . . . . . . . . . . . 52

4 Multipactor in irises 59

4.1 Model and approximations . . . . . . . . . . . . . . . . . . 604.2 Multipactor regions . . . . . . . . . . . . . . . . . . . . . . 644.3 Comparison with experiments . . . . . . . . . . . . . . . . 654.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Multipactor in coaxial lines 69

5.1 Analytical study . . . . . . . . . . . . . . . . . . . . . . . 705.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . 705.1.2 Multipactor resonance theory . . . . . . . . . . . . 735.1.3 Main findings . . . . . . . . . . . . . . . . . . . . . 81

5.2 Particle-in-cell simulations . . . . . . . . . . . . . . . . . . 825.2.1 Numerical implementation . . . . . . . . . . . . . . 825.2.2 Simulations . . . . . . . . . . . . . . . . . . . . . . 835.2.3 Comparison with experiments . . . . . . . . . . . . 885.2.4 Main conclusions . . . . . . . . . . . . . . . . . . . 91

6 Detection of multipactor 93

6.1 Common Methods of Detection . . . . . . . . . . . . . . . 936.1.1 Global methods . . . . . . . . . . . . . . . . . . . . 946.1.2 Local methods . . . . . . . . . . . . . . . . . . . . 99

6.2 Detection using RF Power Modulation . . . . . . . . . . . 1006.2.1 Single carrier . . . . . . . . . . . . . . . . . . . . . 1056.2.2 Multicarrier . . . . . . . . . . . . . . . . . . . . . . 1056.2.3 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 accept-ing me as a PhD student and for guidance and support in my dailywork. I also want to thank Prof. Vladimir Semenov at the Instituteof Applied Physics in Nizhny Novgorod, Russia, for fruitful discussionsand for his patience with all my questions. Thank you Prof. Lars Elias-son, Director at the Institute of Space Research in Kiruna, for providingboth financial and moral support making my PhD candidate appoint-ment possible. A very warm thank you also to Jerome Puech for manyinteresting discussions about space related microwave problems and tohis employer, Centre National d’Etudes Spatiales, for financial support.Thanks to my friends in Toulouse and especially Dr. Omar Houbloss andRaquel Rodriguez. I thank my fellow members of the National GraduateSchool of Space Technology and my colleagues at Chalmers and espe-cially Dr. Pontus Johannisson, Dr. Ulf Jordan and Dr. Lukasz Wolf forbeneficial discussions and lots of support with Linux and LaTex. Manythanks also to our secretary Monica Hansen for guiding me through theadministrative jungle. I want to thank my dear mother, Monica, forencouraging and supporting me and my family when 24 hours a daywasn’t enough and my father, brother and sisters for believing in me. Iam also grateful to my unofficial mentors: my father-in-law Lars-GoranOstling, my friends Jorgen Otback and Anders Wilhelmsson, and mybrother-in-law Nicklas Ostling. Most of all I thank my wife Malin andour daughters Janina and Lizette for encouragement and support duringthis time.

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xii

Acronyms

AM amplitude modulationDC direct currentDUT device under testEDDM electron density detection methodESA european space agencyFFT fast fourier transformNLSQ non-linear least squarePIC particle-in-cellPSK phase-shift keyingQPSK quadrature phase-shift keyingRF radio frequencySEY secondary electron yieldSMA sub miniature version aTGR twenty gap crossings ruleTEM transverse electric and magnetic fieldTWTA travelling wave tube amplifierUV ultra violetVSWR voltage standing wave ratioWCAT worst case assessment tool

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xiv

Chapter 1

Introduction

Resonant secondary electron emission RF discharge or multipactor wasdiscovered and studied by Philo Taylor Farnsworth in the early 1930’s.The phenomenon was then used as a means to amplify high frequencysignals as well as to serve as a high frequency oscillator. Using his mul-tipactor tubes, Farnsworth succeeded in developing the first electronictelevision system. The success stimulated other researchers to investi-gate the phenomenon and one of the first detailed analyses were doneby Henneberg et al. [1] in the mid 1930’s. Gill and von Engel [2] madean even more detailed study, both theoretical and experimental, wherethey, among other things, showed the importance of the secondary elec-tron yield on the development of the vacuum discharge. In a follow uppaper [3] Francis and von Engel studied not only the initial stage of theelectron multiplication, but also the saturation stage. They showed thatthe electron space charge effect could be one of the major causes for thediscontinued electron growth. Other researchers continued the work anda basic overview of these results can be found in the two review papersby Gallagher [4] and Vaughan [5].

During the past 20-30 years, multipactor has mainly been studieddue to the adverse effects it can have on microwave systems operatingin a vacuum environment. It can disturb the operation of high powermicrowave generators [6] and electron accelerators [7], but, above all, itcan cause severe system degradation and failure of satellites, which aredifficult or impossible to repair after launch [8]. Satellites operate undervacuum conditions and the most common means of communication withthe Earth is microwave transmission. Microwave frequencies are requiredas the ionosphere is not transparent for low frequency radiowaves. In

1

addition, it is difficult to make compact, light weight, and high gainantennas for low frequency transmission. Many microwave componentsare hollow metallic structures that guide the electromagnetic power. Afree electron inside the device will experience a force due to the electricfield and since there is no gas or other material stopping the electron,it can accelerate to a very high velocity. Upon impact with one ofthe device walls, the energetic electron can knock out other electronsand under certain circumstances this procedure is repeated continuouslyuntil the electron density is large enough to counter-act the effect of theapplied electric field and a steady state is achieved. A consequence ofthis can be that the incident power is reflected instead of transmittedto the intended load. Since many satellites lack sufficient protectionagainst reflected power in order to save weight, such reflected power cancause severe damage to the high power stage of the system.

When a satellite is launched it carries fully charged batteries andin the beginning, before the solar panels are deployed, they are theonly source of electric power. The capacity of these batteries is usuallylow and may only last a couple of days, since a satellite in operationwill normally only lack access to power from the solar panels for a fewhours, at most, and consequently the batteries are made small in order tosave weight. If the solar panels are not deployed before the batteries areexhausted, the satellite is permanently lost. Thus, a new satellite is oftentaken into operation quickly after being put into orbit. A concern thenis that the satellite components may not be completely vented and thereis a risk for ordinary corona breakdown, which is more prone to occurat intermediate pressures than at high and very low pressures. A coronadischarge is usually much more detrimental than multipactor and for acertain range of pressures, the breakdown threshold for corona is loweror much lower than for multipactor. Basic theory for ordinary coronamicrowave discharge, when the mean free path of the electrons is smallerthan the characteristic length of the device, is well known [9]. However,the intermediate range, between very low pressure and vacuum, hasreceived little attention and therefore one of the main topics of this thesisis devoted to explaining what happens with the breakdown threshold atthese pressures.

Theoretical studies of the multipactor phenomenon have to a greatextent been performed using a one-dimensional model with a spatiallyuniform approximation of the electromagnetic field. However, manycommon RF devices involve structures where the field is inhomogeneous,

2

where breakdown predictions based on such simple models will not bereliable. Examples of important geometries in microwave systems wherethe field is inhomogeneous are waveguides, coaxial lines, irises and sep-tum polarisers. An important effect due to the non-uniform field that isnot present when the electric field is spatially uniform, is the so calledponderomotive or Miller [10] force, which tends to push charged particlestowards regions of low field amplitude. This can have both a qualitativeand a quantitative effect on the multipactor regions. Analytical study ofresonant multipactor in a non-uniform field is not a trivial matter andmost researchers have resorted to numerical methods of investigation.However, in this thesis different aspects of multipactor in structureswhere the field is inherently inhomogeneous are investigated using ana-lytical methods and the results are compared with numerical simulationsas 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 transmit-ted simultaneously. In contrast to the single carrier mode, the electricfield envelope of the multicarrier system varies constantly. In most casesthis is advantageous from a multipactor point of view, as the changingamplitude will destroy the resonance condition and thus suppress thedischarge. In systems where the frequency separation is small, however,there is a risk that the signals will interfere constructively for a largenumber of field cycles and the amplitude will remain fairly constant,thus allowing a discharge to develop. In such cases, the microwave en-gineer will have to try to find the worst case scenario and design thecomponent with respect to that case or perform tests that guaranteethat the part fulfils the requirements. Some attention will be given tothese aspects in this thesis, which can be useful for the engineer whenmaking multipactor free multicarrier microwave designs.

The thesis is organised as follows. A general introduction to basictheory of multipactor in vacuum is given in chapter 2 as well as someguidelines when it comes to multipactor free design. It will serve as abase when continuing with the analysis of multipactor in low pressuregas in chapter 3, where first a simple model is presented, which onlyconsiders the friction force of the gas molecules. It is then followed bya more advanced model, which includes also the effect of impact ionisa-tion as well as thermalisation of the electrons. Starting with waveguideirises in chapter 4, vacuum discharge in structures where the field isnon-uniform is considered. It is followed by a detailed analytical and

3

numerical study of the phenomenon in a coaxial line in chapter 5. Fi-nally, chapter 6 is devoted to different means of detecting multipactorwith 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 thetechnique to amplify an electric current. Others have also tried to finduseful application of the phenomenon, e.g. in multipactor duplexers andswitches [11] and in electron guns [12, 13]. However, during the last 30years it has mainly been studied due to its detrimental effects on mi-crowave components. It has been found to cause electric noise, whichreduces the signal to noise ratio, a very serious problem if it occurse.g. in a communication satellite where the signal power is limited andcounter-measures are difficult or impossible to implement. It can alsodetune microwave cavities, commonly used as e.g. resonators in filters,thus reflecting the incoming power back to the power amplifier. If thesystem does not have an appropriate power protection device, the am-plifier may suffer permanent damage. Another concern is heating, whichis a result of the power dissipated to the device walls as the multipact-ing electrons strike the walls. Furthermore, the discharge can also causedirect physical damage to the component with the risk of permanentlychanging the electric properties of the device. However, the risk of suchdirect damage seems low, especially for metallic components. In caseswhere damage has been reported, it is not certain that it was caused bymultipactor [14,15]. Multipactor is known to be able to trigger ordinarygas discharges [16–18], either by increasing the outgassing from the com-ponent or just by starting the breakdown at a pressure and a voltagewhere corona is not expected, since gas breakdown can be sustained at amuch 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 thussuspect that the observed damage was due to a multipactor induced gasdischarge.

This chapter will present the basic theory of vacuum multipactorbetween two metallic parallel plates with an applied homogeneous, har-monic electric field. It is divided into two major parts, one describingthe single carrier case and another devoted to multicarrier multipactor.

2.1 Single Carrier

One of the first communication satellites, Telstar I, operated in singlecarrier mode. It had a capacity of 12 simultaneous telephone conversa-tions [19] and the solar panels provided a power of only 15 watts. Today,satellites operate in multicarrier mode with powers of several kilowattsand new satellites are being designed for tens of kilowatts. Thus, thesingle carrier mode is seldom found in real applications. Nevertheless,the single carrier case is important as it has been thoroughly studiedover the years and by making certain assumptions, the multicarrier casecan be approximated by the single carrier state and design and testingcan be done based on the simpler situation.

2.1.1 Basic theory

There are two main kinds of multipactor, the single-surface and thedouble-surface types. Single-surface multipactor can occur in structureswith nonuniform field or with crossed electric and magnetic fields [20],where the electron, accelerated by the electric field, returns to the orig-inal surface due to the circular motion caused by the magnetic field.This thesis, however, will focus on double-surface or parallel plate mul-tipactor, but some attention will be given to single-sided multipactor inthe case of a coaxial line.

A multipactor discharge starts when a free electron inside a mi-crowave device is accelerated by an electric field. In a strong field theelectron will quickly reach a high velocity and upon impact with one ofthe device walls, secondary electrons may be emitted from the wall. Ifthe field direction reverses at this moment, the newly emitted electronswill start accelerating towards the opposite wall and, when colliding withthis wall, knock out additional electrons. As this procedure is repeated,the electron density grows quickly and within fractions of a microseconda fully developed multipactor discharge is obtained (see Fig. 2.1).

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Figure 2.1: Initial stage of parallel plate multipactor, where a free electronis accelerated by the electric field and is forced into one of theplates, where it causes emission of secondary electrons.

The motion of an electron in vacuum with an applied electric fieldcan be studied by means of the equation of motion,

mx = eE (2.1)

where m (≈ 9.1× 10−31 kg) and e (≈ −1.6× 10−19 C) are the mass andcharge of the electron, x the direction of motion, and E the electric field.Multipactor requires an alternating field and in the parallel-plate modela spatially uniform harmonic field E = E0 sin ωt is assumed. SolvingEq. (2.1) with this field yields expressions for the velocity, x, and theposition, x,

x = −eE0

mωcos ωt + A (2.2)

x = − eE0

mω2sin ωt + At + B (2.3)

where A and B are constants of integration, which will be determined bythe initial conditions. By assuming that an electron is emitted at x = 0with an initial velocity v0 when t = α/ω, fully constrained expressionsfor the velocity and position are obtained, viz.

x =eE0

mω(cos α − cos ωt) + v0 (2.4)

x =eE0

mω2

(

sinα − sin ωt + (ωt − α) cos α)

+v0

ω(ωt − α) (2.5)

For resonant multipactor to occur it is necessary for the electron toreach the other device wall (x = d) when ωt = Nπ + α, where N is an

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odd positive integer (N = 1, 3, 5 . . .). Applying this resonance conditionto Eq. (2.5), the following expression is obtained for the amplitude ofthe harmonic electric field.

E0 =mω(ωd − Nπv0)

e(Nπ cos α + 2 sin α)(2.6)

An important quantity when studying multipactor is the impact ve-locity, since this determines the secondary electron yield. It can be foundby inserting ωt = Nπ + α in Eq. (2.4), which yields

vimpact =2eE0

mωcos α + v0 (2.7)

Multipactor boundaries

When constructing the multipactor boundaries, i.e. the boundaries ofthe regions in parameter space where multipactor can occur, an assump-tion will have to be made concerning the initial velocity. In reality, theinitial velocity of the emitted electrons will follow some kind of distri-bution and a common choice is the Maxwellian distribution [21],

f(v) ∝ exp

(

−(v − vm)2

2v2T

)

(2.8)

where suitable parameters for the mean velocity, vm, and for the rms-value (or thermal spread), vT , have to be chosen. When performingparticle-in-cell (PIC) simulations, such a distribution can be used tomore accurately describe the initial velocity of the emitted electrons.However, for an analytical solution a simpler assumption will have tobe made. There are two common approaches, one which assumes thatthe electrons are emitted with a constant initial velocity, v0, regardlessof the impact velocity. The other assumes that the ratio between theimpact and initial velocities is equal to a constant, k = vimpact/v0. Boththese approaches will be used and compared in this chapter, but in thefollowing chapter, which deals with multipactor in low pressure gas, theconstant k approach will be used only for the simple model while theconstant initial velocity approach will be used for the more advancedmodel as that assumption is more physically correct. The reason whythe constant k model has been used to such a great extent is the factthat it can successfully be fitted to experimental data. The cause of thissuccess will be explained in the subsection on hybrid modes below.

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In addition to fulfilling the resonance condition, which resulted inEqs. (2.6) and (2.7), the secondary electron yield, SEY or σse, must begreater than or equal to unity. For most materials the secondary yield asa 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 varyvery much between different materials. The impact energy where thesecondary yield first reaches unity is called the first cross-over point andis denoted W1, after that the yield increases and reaches a maximumat Wmax and the energy at which the yield drops below unity againis called the second cross-over point, W2. Below Wzero no secondaryyield is obtained [22, 23]. However, some researchers have publishedmeasurements of the SEY, which indicate that it is possible that theSEY does not drop to zero below a minimum impact velocity [24–28].On the contrary, it can increase after reaching a minimum yield and evenreach a yield close to unity for very low impact velocities. A yield closeto unity implies that the electron does not produce any secondaries, butrather that the electron bounces off the surface. This could have animportant effect on the multipactor threshold and development, but inthis thesis, the model by Vaughan [22] has been used unless otherwisespecified.

By setting the impact velocity, Eq. (2.7), equal to the first cross overpoint (converted to velocity, v1) and taking Eq. (2.6) into account, theresonant phase, α, can be found as a function of ωd,

tan α =1

2

(

2ωd − Nπ(v1 + v0)

v1 − v0

)

. (2.9)

Using this result, the amplitude can be plotted as a function of ωd orfd using Eq. (2.6) (or Eq. (2.7)).

One final thing that need to be confirmed before drawing the mul-tipactor charts is the non-returning electron limit. If the secondaryelectrons are emitted before the electric field reverses, the electrons willbe retarded by the field and if the velocity is low, they are likely toreturn to the wall of emission and thus being lost as their energy is toolow to produce new secondaries. The limit can be found by solving thefollowing system of equations:

x = 0

x = 0(2.10)

An analytical solution to this system of equations is not possible and inorder to establish the non-returning electron limit, either a numerical

9

0 500 1000 1500 2000 25000

0.5

1

1.5

2

2.5

σse

=1

W1

Wmax

W2

Secondary electron yield

Primary electron energy [eV]

σ se [−

]

Figure 2.2: Secondary electron yield as a function of the impact energy. Plot-ted using the formula for secondary electron yield presented inRef. [22]. Parameters used are: Wmax = 400 eV, σse,max = 2,and Wzero = 10 eV.

10

solution or some kind of approximate solution will have to be used. InRef. [21] an approximate formula for the non-returning electron limit isgiven and re-writing it for the constant v0 approach yields,

αmin = −√

16v0

5v0 + 3vimpact(2.11)

Using Eqs. (2.6), (2.9), and (2.11) with v1 equal to the velocity corre-sponding to the unity secondary electron yield, the lower multipactorthreshold can be plotted. However, multipactor breakdown is possiblealso for impact velocities greater than v1, in fact, for all impact velocitiesbetween the first and second (v2) cross-over points, the phenomenon canoccur, 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 bound-aries, the thresholds for a number of different energies between these twopoints should be determined and then the envelope of all the thresholdswill be the complete multipactor susceptibility zone (see Fig. 2.3). Fur-thermore, each order of resonance, N , will have its own zone and, asshown in Fig. 2.3, the zones become narrower with increasing N. Thistype of chart, based on the assumption of constant initial velocity, willbe referred to as a Sombrin chart, since J. Sombrin is one of the majoradvocates of this assumption [29].

Using the other approach, with a constant ratio between the impactand 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

(

kωd

v1− (k + 1)N

π

2

)

(2.13)

E0 =mω2d

e(k+1k−1Nπ cos α + 2 sin α)

(2.14)

αmin = −√

16

8 + 3(k − 1)(2.15)

Using these formulas, multipactor charts similar to the one in Fig. 2.3can be produced, cf. Fig. 2.4. When this is used, the charts are com-monly referred to as Hatch and Williams charts, as they were the first

11

100

101

102

103

104

Frequency − Gap product [GHz⋅mm]

Vol

tage

[V]

N=1

N=3

N=5

N=7N=9

Figure 2.3: Multipactor susceptibility chart based on the constant initial ve-locity 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 Hatchand Williams charts are that the multipactor zones are wider than theSombrin charts for increasing voltage. This occurs since a constantvimpact/v0 implies that v0 increases as the impact velocity increases.When e.g. Wimpact = 3000 eV, this means that for k = 2.5 the initialenergy W0 = 480 eV, clearly an unrealistic initial velocity.

100

101

102

103

104

Frequency − Gap product [GHz⋅mm]

Vol

tage

am

plitu

de [V

]

N=1

N=3

N=5N=7

N=9

Figure 2.4: Multipactor susceptibility chart produced with the constant kassumption. Parameters used are: k = 2.5 (corresponding toan 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 ma-terial as given by the parameters W1, W2, and W0 or k, multipactorcharts for that material can be designed. However, Woode and Pe-tit [31] performed a large series of multipactor experiments during the1980’s and used the Hatch and Williams charts to fit the experimentaldata. By tuning the k and W1 parameters for each zone, they were ableto produce multipactor charts that fit the experimental data quite well(cf. Fig. 2.5). The problem with this empirical approach is that it hasto assume different values for the first cross-over point for each zone inorder to obtain good fitting. This is clearly an unphysical approach andwill not contribute to an improved understanding of the phenomenon,

13

even though it may be sufficient from an engineering point of view. Onthe other hand, the higher order modes have a narrower phase-focusingrange (see below), which makes it difficult to compensate for e.g. initialvelocity spread, and thus a secondary yield of unity may not be suffi-cient to sustain a discharge. Consequently, an impact energy somewhathigher than the first cross-over point will be needed when constructingthe lower multipactor threshold for the higher order modes. This willbe discussed further in the subsection “Effect of random emission delaysand initial velocity spread.”

100

101

102

103

104

Frequency − Gap product [GHz⋅mm]

Vol

tage

am

plitu

de [V

]

Figure 2.5: Hatch and Williams charts for aluminium together with measure-ment data by Woode and Petit [31].

When the microwave engineer assesses the risk of having a multi-pactor discharge, it is usually not the boundary of the individual break-down region that is considered. Typically, the lower envelope of the allthe zones is taken as the “design threshold” (cf. Figs. 2.3 and 2.4). Bysetting the phase, α, in Eq. (2.7) to zero, the lowest field amplitude toachieve a certain vimpact is obtained. Thus the lower envelope, whichis the same for both the Sombrin Chart and the Hatch and Williamschart, is given by,

E0 =(v1 − v0)mω

2e. (2.16)

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 havea 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 ora spread in the initial velocity, an electron will always acquire a smallphase error. Inside the phase-focusing range, such an error will decreaseas the electron traverses the electrode gap. In other words, the phasesof the electrons will tend to converge towards the resonant phase, thuskeeping all electrons close together. Outside the range of phase-focusing,the error will grow with each passage and after one or a few transitsthe electron will be lost. In order for a discharge to occur under suchcircumstances, the impact energy has to be large enough to produce asecondary yield sufficiently above unity to compensate for the incurredlosses.

To see in what range the phase focusing mechanism is active, a smallphase error can be introduced in Eq. (2.5) while keeping the amplitudeand phase constant and setting x = d. The ratio between the final andinitial error is called the stability factor, G [33], and the condition forstable phase is:

|G| < 1 (2.17)

By setting |G| = 1, the phase range within which the phase is stable canbe obtained. An interesting observation here is that even though thelower multipactor threshold for the constant k theory and the constantinitial velocity model are identical, the range of stable phases varies sub-stantially [34]. This can be seen clearly from the analytical expressionsfor the phase stability limits, which for constant k theory reads,

φR = arctan(2

πN(vimpact − v0

vimpact + v0)) (2.18)

φL = − arctan(2

πN) (2.19)

and for the constant initial velocity approach,

φR = arctan(2

πN) (2.20)

φL = − arctan(2

πN(vimpact + v0

vimpact − v0)) (2.21)

15

where φL and φR are the left and right limits respectively. This differenceis illustrated graphically in Fig. 2.6. However, when v0 vimpact bothapproaches yield the same phase stability limits.

100

101

102

103

Frequency − Gap product [GHz⋅mm]

Vol

tage

[V]

Unstable phase range (constant v0)

Stable phase range (constant v0)

Unstable phase range (constant k)Stable phase range (constant k)

N=1

N=3

N=5

Figure 2.6: Lower multipactor thresholds in vacuum for the first 3 ordersof resonance (N = 1, 3, and 5). The curves for the constant kmodel 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 electronemission 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 witheach impact and if no saturation mechanism is considered the numberof electrons after a time t, if the field frequency is f , will be:

Ne(t) = Ne(0)(σse)2ftN (2.22)

The rapid growth of the number of electrons can be illustrated withan 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 initialelectron will be more than 1014.

In a very short time, the number of electrons will grow to very highvalues and it is clear that some kind of saturation mechanism will be-come active. Two main saturation processes have been described in theliterature. The first is the space charge effect [3], which is the mostobvious effect. Within the electron bunch the individual electrons willrepel each other causing a change in phase of those electrons and if thephase error is too large, the probability of losing electrons increases andeventually the effective secondary yield will be equal to unity and satu-ration has occurred. The second type of saturation process [35, 36] canset in if the discharge takes place inside a resonant cavity. Due to a highQ-value, the electric field strength is high and thus the risk for a dis-charge will increase. If a multipactor discharge is started, the electronstraversing the gap make up an alternating current, which loads the cav-ity. Loading the cavity means that the Q-value will decrease and thusalso the electric field strength. It is clear that this is a self-suppressingeffect. As the multipactor current increases, the field strength decreasesand with it the impact velocity of the electrons leading to a loweredsecondary emission yield. Eventually the secondary yield reaches unityand saturation has been reached.

2.1.2 Hybrid modes

It may seem somewhat contradictory to assert that the model based ona constant initial velocity is more physically correct than the constantk theory, when the latter approach can be better fitted to experimen-tal data. However, as briefly mentioned previously, the reason for thisparadox is the hybrid modes. Some of these modes were identified byRefs. [29,37,38] and a general treatment is given in [39]. The modes canbe found by allowing N in the resonance condition for Eq. (2.5) to bea sequence of odd half-cycles of the electric field, N1, N2, N3..., whereN1 = N and the remaining Nn ≥ N for the hybrid modes between theN th and (N + 2)th zones. Each such sequence will result in a narrowmultipactor zone located between the main multipactor areas. The low-est order hybrid mode in the parallel-plate case is the 1, 3 mode, whichmeans that the transit time in one direction takes 1/2 RF-cycle and thereturn transit takes 3/2 RF-cycles. This mode is then also associatedwith 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 differencesin the right boundaries of the main zones, between the constant initialvelocity and the constant k approaches, become negligible. This can beseen 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 hy-brid zones [40]. With the envelope of the hybrid zones includedthe resemblance between the Sombrin chart and the Hatch andWillams 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 sensitiveto an initial velocity spread. On the other hand, there are many hybridzones very close to one other and this spread will result in a mixing oroverlapping of the resonances [39].

2.1.3 Factors effecting the threshold

There are many different aspects that need to be considered when de-termining the multipactor threshold. The most important and most ob-vious ones are type of material, gap size, and amplitude and frequencyof the electric field. These are all part of the basic theory as describedabove. Apart from these there are other more or less important fac-tors. The supply of primary electrons does not effect the theoreticalthreshold, which can be determined with methods described earlier inthis thesis. Nevertheless, a weak source of seed electrons can result inan apparent higher threshold during testing. In a typical test setup fordetermination of the breakdown amplitude, an electric field is applied

18

and the field strength is increased at regular intervals. If no electronis in an advantageous position, i.e. has a suitable phase from a multi-pactor point of view, when the right amplitude is set, a discharge willnot occur. As the amplitude is increased further, the impact velocityof 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 freeelectrons. Thus the chances of getting a breakdown increases until iteventually occurs. For experimental use, a hot filament or a radioactivesource can be used to produce a sufficient amount of free electrons toachieve reliable measurement results [14].

Another factor that can have a significant effect on the thresholdis contamination. Both the first cross-over point, W1, and the max-imum secondary yield, σse,max, can be drastically affected. A loweredW1 means that a discharge can occur at a lower voltage and an increasedσse,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 con-taminants was made. It was noted that the plastic bags, which werenormally used to protect the microwave components from dirt, were themain source of contamination. A threshold reduction of up to 4 dBwas found. Also dust and fingerprints were a direct source of a low-ered threshold. In the report [31] it was recommended that cleanedmicrowave parts for space use should be handled with cotton gloves andstored in hard plastic boxes.

Microwave parts which have not been properly vented before poweris applied can also have a threshold that is different from the expectedmultipactor threshold. If there is too much gas, corona breakdown mayoccur, and within a certain range of pressures, close to the minimum ofthe so called Paschen curve, the breakdown threshold can be significantlylower than in the multipactor case. In the pressure range correspond-ing to the transition region between corona and multipactor, a higherthreshold can sometimes be expected. More details about this will bepresented in chapter 3.

Other factors that can have an indirect effect on the multipactorthreshold are the voltage standing wave ratio, VSWR, and the temper-ature. If the VSWR is greater than what was intended with the design,the peak field strength in the system will also be greater than expectedand thus a discharge may occur at a lower power level than assumed. Anincreased temperature can lead to increased outgassing from the devicewalls 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 themultipactor threshold can also be utilised to suppress the discharge. Thewithout doubt easiest method of avoiding a breakdown is to pressurisethe component. The field strength required to achieve breakdown atatmospheric pressure is in general much higher than at low pressures orin vacuum. However, such a method is seldom feasible for componentsthat will be used e.g. in space, where the external environment is a highvacuum. A small leakage can lead to slow venting of the component andthus risking severe corona discharge when the pressure reaches the rangewhere the minimum breakdown field occurs.

Another way of suppressing multipactor is to amplitude modulatethe main carrier [21, 41]. If both signals are sinusoidal, the total fieldcan be written:

Etot = E1 sinω1t + E2 sin ω2t (2.23)

This means that the envelope of the signal will vary according to (seealso Fig. 2.8):

Eenv =√

E21 + 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 accord-ing to Eq. (2.22). However, as soon as the voltage drops below thethreshold 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 ornot this is true also depends on the secondary yield properties of theelectrode material. For materials with a very high maximum secondaryyield, the number of electrons gained while above the threshold can begreater than the losses incurred while below. In such a case, no sup-pression is achieved and in some cases, the discharge may even becomemore powerful than before the modulation carrier was added [42] (cf.Fig. 2.9). Thus in order to successfully suppress a multipactor dischargeusing amplitude modulation, it is vital that the material has a low max-imum secondary yield (preferably less than about 1.5). Due to the riskof contamination, which can greatly increase the maximum secondaryyield, great care should be taken to assure a high level of cleanliness ifthis method of suppression is to be used.

To AM-modulate the carrier is probably not feasible in most cases, asit would require extra hardware to produce the AM-signal. However, the

20

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5Amplitude Modulation, E0/E1=0.4

Am

plitu

de

Time

Main signal

Sum signal

Envelope

Modulation signal

Multipactor threshold

Figure 2.8: Two signals and their sum signal (absolute values). The enve-lope varies and is partly above and partly below the multipactorthreshold.

typical bandpass filtering of a PSK (Phase-Shift Keying) signal causesmodulations in the time-domain. The QPSK (Quadrature Phase-ShiftKeying) signal in Fig. 2.10 has unity amplitude before filtering. After-wards, the peaks are higher than the original amplitude, but the troughscan sometimes go down almost to zero amplitude. Comparing this withthe AM-suppression, it is clear that an electron avalanche that is ini-tiated during the peak periods, will be extinguished as the amplitudefalls close to zero. However, for a typical PSK-signal, the duration be-tween phase shifts (which normally coincides with the troughs) is severalhundreds of RF cycles. Thus for most microwave systems, there will beample time for a discharge to develop. But, when the amplitude dropsbelow the threshold, the electron bunch will disappear and when theamplitude increases above the threshold again, there may not be anyseed electrons present to restart the electron avalanche. Thus, the sys-tem 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 applya coating [26–28], a surface treatment [43], or a film [44] that has ahigh first cross-over point as well as a low maximum secondary electron

21

0 1 2 3 4 5 6 7 8 9 10

x 104

−85−80−75−70−65

Amplitud Modulation, w2/w1=1.16)

Noi

se (

dBm

)

0 1 2 3 4 5 6 7 8 9 10

x 104

46.5

47

47.5

48

Pow

er #

1 (d

Bm

)

0 1 2 3 4 5 6 7 8 9 10

x 104

−20

0

20

40

Pow

er #

2 (d

Bm

)

0 1 2 3 4 5 6 7 8 9 10

x 104

−12

−11.5

−11

Mat

ch #

1 (d

B)

Time (ms)

Figure 2.9: Multipactor experiment with two carriers with E2/E1 = 0.36and ω2/ω1 = 1.16. Due to a high maximum secondary yield,multipactor suppression is not possible (the material used in theexperiment was plain aluminium, which can have a σse,max ≈ 3).When the modulation signal is applied, the magnitude of themultipactor noise increases significantly.

1 1.5 2 2.5 3 3.5

−1

−0.5

0

0.5

1

Time [µ s]

Sig

nal a

mpl

itude

[−]

Square Root Raised Cosine filtered QPSK signal

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 unityhas been found. However, alodine is a commonly used surface coating forspace-bound microwave devices made of aluminium. It increases the firstcross-over point to around 60 eV and reduces σse,max to about 1.5, eventhough the actual values vary much between samples. The concern witha material with very good anti-multipactor properties is contamination.A few fingerprints or a very small layer of dust can drastically alter theproperties of the material and make it prone to discharges.

By applying a DC electric or magnetic field, the electron trajec-tory can be disturbed and the important resonance condition can bedestroyed, thus making multipactor impossible. Simulations [45] haveshown that an external DC magnetic field applied in the direction ofwave propagation in a rectangular waveguide can efficiently suppressmultipactor. A drawback with the method is the extra components re-quired to produce the magnetic or electric field and thus the methodmay not be feasible for e.g. space applications, where extra weight isundesirable.

The most efficient way of avoiding multipactor is to make a designwhere the mechanical dimensions are such that a power much higherthan the nominal power is required to start a discharge. However, thatmay lead to large and heavy designs, which are to be avoided in spacesystems, and thus one may have to resort to one or several of the abovementioned 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 electricfield, it was assumed that all secondary electrons were emitted with afixed initial velocity, v0. However, as briefly mentioned previously, theelectrons are actually emitted with a distribution of velocities and theMaxwellian distribution is often used in simulations. Apart from thespread in initial velocities, there is also a finite time between impact ofthe primary electron and emission of the secondaries. Since this time inmost cases is very small compared to the RF period, it was neglected inthe previous analysis. Nevertheless, this time will cause a small phaseerror and if the resonant phase is close to the phase stability limits asgiven by Eqs. (2.18) - (2.21), the phase error may result in an increasedelectron loss.

A detailed analysis of the effect of random secondary delay times and

23

random 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 inthe range σse/2 < σ∗

se < σse. This means that the effective secondaryelectron yield will be a function not only of the impact velocity, but alsoof the resonant phase as well as the phase spread caused by the spreadin initial velocities and secondary delay times. Another study, whichsupports this result, investigated the effect on the different resonancezones for different values of the maximum SEY due to initial velocityspread [46]. It was found that, except for the first order mode, a real-istic thermal spread of the initial electrons raised the multipactor SEYrequirement from unity to above unity. For the higher order modes aSEY greater than approximately 1.5 was necessary to compensate forthe losses incurred. In addition, with increasing velocity spread, themultipactor zones started to overlap. The increased SEY requirementwill result in an increased threshold for the higher order modes and canexplain the success with increasing the first cross-over point in the Hatchand Williams charts when fitting experimental data (see Fig.2.5).

The importance of the spread in initial velocities can be seen whenconstructing multipactor charts for a constant initial velocity without al-lowing compensation for electron losses outside the phase stability range.In Fig. 2.11 zones bounded by solid lines indicate the region where mul-tipactor can take place under this assumption. The dashed lines makeup wider zones that encompass the other zones and are identical to thezones shown in Fig. 2.3.

By including a higher secondary electron yield and a spread in ini-tial velocities, the multipactor zones will become wider than the solidline zones. A σse greater than unity, which will be the case when theimpact velocity is greater than the first cross-over point, will compen-sate for some of the losses incurred due to phase instability. A spreadin 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 indicatedby the solid line multipactor zones in Fig. 2.11. This widening of themultipactor zones has been taken into account to a certain extent in thetraditional analytical approach, which is indicated by the wider dashedline zones in Fig. 2.11. However, the widening should not only be to-wards the left side, but also towards the right [39]. Furthermore, thesharp lower left corner of each dashed line zone is misleading, as thatindicate a point where the secondary electron emission is unity and the

24

100

101

102

103

104

Frequency − Gap product [GHz⋅mm]

Vol

tage

[V]

Figure 2.11: Multipactor charts based on the same parameters as in Fig. 2.3.The solid line zones indicate the zones within which phase-focusing is active. The dashed line zone is produced by includingalso unstable phases until the non-returning electron limit.

phase is very unstable, thus making a discharge impossible. A morecorrect boundary would be a rounded shape, which starts in the lowerleft corner of the solid line zone and smoothly joins the dashed left handside [47]. This is confirmed by experiments [30,48], cf. Fig. 2.12, whichshows measurement data from one of the early multipactor experimentsby Hatch and Williams [48]. A similar rounded shape can also be seen innumerical simulations and examples of this is shown in chapter 5, whichincludes PIC simulations of multipactor in a coaxial line.

2.2 Multicarrier

Modern satellites operate in multicarrier mode, i.e. several signals at dif-ferent frequencies exist simultaneously in the microwave and electronicsystems. An example of such a system is Sirius 3, which is one of theNordic 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 thateach channel has a power of 200 W. Then the maximum instantaneouspower of the system, the peak power, is equal to 45 kW. The peak powerincreases with the square of the number of carriers. Such a high instan-

25

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

26

0 5 10 15 200

5

10

15

Time [ns]

Am

plitu

de [V

]

Figure 2.13: In-phase multicarriersignal. The signal os-cillates rapidly, whichmakes the signal enve-lope appear clearly inthis time resolution.

0 5 10 15 200

5

10

15

Time [ns]

Am

plitu

de [V

]

Figure 2.14: Random phase multi-carrier signal.

taneous power is very unlikely in a real system since it will occur onlywhen all the signals are in phase as illustrated by Fig. 2.13. The mostlikely scenario, if the carriers are not phase locked, is that the phase ofeach carrier is a random number and will result in a signal with muchlower maximum instantaneous power as illustrated in Fig. 2.14.

The signals in Figs 2.13 and 2.14 are characterised by all carriershaving the same amplitude and a constant frequency. Consider a signalwith N carriers, each carrier having the same amplitude E0, but differentphases φn and with a frequency spacing ∆f . The period of the envelopewill then be T = 1/∆f and the envelope is given by

Eenv = E0

√

√

√

√

(

N−1∑

n=0

cos (n2π∆ft + φn)

)2

+

(

N−1∑

n=0

sin (n2π∆ft + φn)

)2

(2.25)A more realistic signal would have different amplitudes for each car-

rier and the frequency spacing would not be constant. The envelope ofsuch a signal can be found from

Eenv =

√

√

√

√

(

N−1∑

n=0

En cos (knω0t + φn)

)2

+

(

N−1∑

n=0

En sin (knω0t + φn)

)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 theworst case scenario from the multipactor point of view, it is importantto study a whole envelope period. For arbitrarily spaced frequencies, theenvelope period, T , can be found by solving the following Diophantinesystems of equations:

T =ni

∆fi, ni ∈ N i = 1, 2, ..., N − 1 (2.28)

where N is the number of carriers, f0 is the signal with the lowest fre-quency and ∆fi = fi − f0. The envelope period will be the solutionwith the smallest possible integers. For equally spaced carriers, the so-lution becomes n1 = 1, n2 = 2,...,nN−1 = N − 1, which is implies thatT = 1/∆f , like before.

When studying multicarrier multipactor it is common to make cer-tain simplifications that will allow using single carrier methodology toasses also the multicarrier case, e.g. the mean frequency of all the car-riers is used as the design frequency. Thus, most of what has been saidabout single carrier multipactor will then be valid also for the multiplesignals case.

2.3 Design guidelines

From an industrial point of view it is important not only to understandthe physics of multipactor, but also how the theoretical and experimen-tal results should be applied when making multipactor-free microwavehardware designs. In Europe, most space hardware designers follow thestandard issued by ESA [50]. This standard includes both the singleand the multicarrier cases, but for the latter it is stated that the de-sign guidelines are only recommendations. Most research support theserecommendations, but not enough tests have been performed to verifythe theoretical findings. When using the standard it is important to beaware of the fact that it is primarily based on the parallel-plate modelwith a uniform electric field. Design with respect to this approach forother geometries is normally a conservative and safe way. However, inmany common microwave structures, the geometry is such that lossesof electrons is much higher than in the parallel-plate case. Thus the

28

multipactor threshold in geometries such as coaxial lines, waveguidesand irises, can be higher or much higher than that obtained using theplane-parallel model.

2.3.1 Single carrier

In the ESA standard [50] components are divided into three categoriesor types. Type 1 is a well vented component where all RF paths aremetallic and the secondary electron emission properties are well known.This type of component has the lowest design margins with respect tomultipactor and, depending on the type of test, range from 3-8 dB.The second type of component may contain dielectrics with establishedmultipactor properties and the component should be well vented. Alsodepending on the type of test, the design margins range from 3-10 dB.All other components are categorised as type 3 and the design marginsrange from 4-12 dB.

When designing with respect to multipactor a complete electric fieldanalysis is performed and regions with high voltages and critical gapsizes are identified. Using the frequency-gap size product, the multi-pactor threshold can be found in a susceptibility chart for the materialin question. A susceptibility chart in the ESA standard is basically anenvelope of the multipactor zones as shown in e.g. Fig. 2.5. If a mar-gin larger than the largest design margin, 12 dB, is found, no testing isrequired. However, in most cases, the component will have to be testedand 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 therecommendations given in the ESA standard. Type 2 and type 3 com-ponents will require further research before they can be included in thestandard. In the single carrier case, the level that is compared with themultipactor threshold in the susceptibility chart is the amplitude of thesignal and no ambiguities exist. For multicarrier designs, the traditionalway of designing was to set the design margin with respect to the peakpower of in-phase carriers, shown in Fig. 2.13. This design method isstill allowed by the ESA standard and for type 1 components the designmargins range from 0-6 dB depending on the type of testing that willbe performed. However, as previously mentioned, in-phase carriers fornon-phase locked signals is extremely unlikely and thus the standard

29

allows for another design margin, which is set with respect to the socalled P20 power level. The P20 level corresponds to the “peak power ofthe multicarrier waveform whose width at the single carrier multipactionthreshold is equal to the time taken for the electrons to cross the mul-tipacting 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 singlecarrier threshold, while the P20 level is more than 4 dB below thesame threshold. The peak voltage is 128.4 V, the single carrierthreshold is 91 V, and the P20 voltage is 57 V. Signal data: 12carriers, equally spaced, fmin = 1.545 GHz, ∆f = 24 MHz andeach 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, thedesign margins range from 4-6 dB depending on the type of testing. Aproblem with the P20 level is that it is not a trivial problem to find thepeak power level for 20 electron gap crossings. This power level is usuallyreferred to as the worst case scenario, even though it may not always bethe 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. usingparabolic or triangular phase distribution in the equally spaced carrierscase (cf. Ref. [51]). Some of the better methods for finding the worstcase scenario will be described in the following subsections after a briefdiscussion about the 20 gap crossings rule (TGR).

Twenty gap crossings rule

The TGR was proposed in Ref. [14] in 1997 and in its original versionit reads:

“As long as the duration of the multicarrier peak and the mode orderof the gap are such that no more than twenty gap-crossings can occurduring the multicarrier peak, then multipaction-generated noise shouldremain well below thermal noise (in a 30 MHz band).” [14]

The rule is a result of an analysis of simulated multi-carrier mul-tipactor discharges. Comparison with experiments showed great devi-ations, where the simulated noise could be as much as 75 dB greaterthan the measured noise level. In the experiments, a minimum of 99gap-crossings were required before the produced noise was detectableabove the noise floor of -70 dBm. Of course, there may be bit errorseven at lower noise levels, but as the number of electrons grows expo-nentially with the number of gap crossings (see Eq. (2.22), there is ahuge difference between 20 and 99 gap-crossings.

However, the TGR is certainly a good first attempt to lower therequirements for multi-carrier multipactor. It is a fairly conservativemethod and thus the risk of applying it should be quite limited. How-ever, more appropriate guidelines should be based on an unambiguoustheoretical 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 materialproperties of the test samples are well known and that these proper-ties are also being used in the simulations. Due to the large differencein secondary emission properties between different materials, it wouldseem reasonable that for a material with a low σse,max one would allowmore gap crossings than, in the opposite case, for a material with a highσse,max.

31

Boundary function Method

One of the best engineering methods for finding the worst case scenariofor equally spaced carriers, the boundary function method, was origi-nally designed by Wolk et al. [51]. Unfortunately the used function wasfound empirically while studying the worst case scenario with an optimi-sation tool, and is thus not physically founded. A consequence of this isthat under certain circumstances, the boundary function produces poorresults. It was also limited to work only for equally spaced carriers. How-ever, as part of the present thesis work, this method has been furtherdeveloped, and it has been found that the original boundary functionapproximately describes a function that tries to squeeze all the energyof the multicarrier signal during one envelope period into a specified,shorter, time period. This works just as well in both the equally spacedand the non-equally spaced carrier cases and can be summarized by thefollowing formulas:

FV (TX) =

√

TH

TX

N∑

i=1E2

i

FV,max =N∑

i=1Ei

FV,min =

√

N∑

i=1E2

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 theperiod 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 linesin 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 entireenvelope period the voltage amplitude will be FV,min, which is indicatedby a dashed line in Fig. 2.15.

The main advantage with the boundary function method is its sim-plicity. It is also very reliable, although a little conservative and this isespecially true for non-equally spaced carriers, where the P20 level canbe much lower than FV . The method has been implemented as an auxil-iary method in WCAT, which is a software tool originally developed bythe 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 func-tionality by the present author as well as by Mariusz Merecki as part ofa Master’s Thesis [40] at Centre National d’Etudes Spatiales, Toulouse,France. Fig. 2.16 shows the graphical user interface of the present versionof WCAT and an example when the worst case of non-equally spacedcarriers have been assessed using the built in genetic algorithm.

0 5 10 15 20 25 30 35 400

50

100

150

200

← Fv =129

Threshold =192

Env Threshold =139

Hyb. Threshold =158

← Fv =129

Threshold =192

Env Threshold =139

Hyb. Threshold =158

0 5 10 15 20 25 30 35 4010

0

101

102

103

0 5 10 15 20 25 30 35 4010

0

101

102

103

Boundary FunctionFVminIn−Phase EnvelopeEnvelope

Figure 2.16: Assessment of worst case scenario for multicarrier multipactorusing WCAT, Worst Case Assessment Tool. The example showsa case with 10 non-equally spaced carriers with varying ampli-tude.

Optimisation methods

A more direct method of finding the worst case scenario is to use somekind of optimisation tool. In WCAT several different methods of findingthe worst case scenario are implemented. One uses the non-linear leastsquare (NLSQ) functionality of Matlab to find the set of phases that willfit as much energy as possible inside a time period TX ≤ TH , where TH

is the envelope period. Another method generates a variable number ofsets of random phases and the corresponding envelopes are compared tofind the worst case. This method is better than the NLSQ optimisationmethod when it comes to finding the worst case for non-equally spaced

33

carriers, because the NLSQ method requires a good seed phase in orderto find a good local minimum. By combining these methods and usingthe result of the best random phase approach as a seed phase for theNLSQ optimisation, the best results are found. The problem with arandom phase approach is that for a large number of carriers, the numberof phase-sets needed to achieve a good result becomes discouraginglyhigh [52].

By using a genetic algorithm, a great improvement in finding theworst case scenario in a short time has been achieved, especially for thenon-equally spaced carrier case. This type of optimisation scheme hasthe advantage of being able to find not only a good local minimum asin the NLSQ case, but the actual global minimum can be found. Thegenetic algorithm was implemented by Merecki [40] and in addition toimproving the optimisation part of the software, he also implementedthe threshold of the hybrid modes (see Fig. 2.16) as well as many otheruseful functions.

The main problem with multipactor in microwave systems is theelectric noise that is generated, which degrades the signal to noise ratio.Thus the worst case may not always be the maximum power within theT20 period. Depending on the material properties some other case mayproduce a substantially larger amount of energetic electrons. Therefore,WCAT also includes the possibility of finding the phase-set that willproduce the largest amount of electrons.

In addition to assessing the risk for a vacuum discharge, it can alsobe of value to investigate if a multipactor free design may have a riskof corona breakdown if the component is not thoroughly vented whenit is brought into operation. In WCAT this is analysed using the meancarrier frequency and comparing the corona threshold with the in-phasepeak voltage. If the minimum of the Paschen curve is greater than thisvoltage, then the corona margin is displayed in the output window ofWCAT. If the opposite is true, the pressure range within which there isrisk 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, thephenomenon is usually called multipaction or multipactor and the theoryfor such vacuum microwave breakdown was presented in the previouschapter. In a gas it is normally called corona discharge or gas breakdownand it can occur when the electron mean free path between collisionswith molecules is smaller than the characteristic dimensions of the vessel.An applied microwave electric field can widen the velocity distributionof the free electrons and thus make more electrons energetic enough toionise the gas. If the production of electrons exceeds the loss throughdiffusion, attachment, and recombination, the electron density will growexponentially and microwave gas breakdown will occur.

When the mean free path between collisions is of the same order asthe device dimensions, classical theory for microwave gas and vacuumdischarge can not be used. Diffusion loss can no longer be assumed,like in the gas breakdown case, since that requires a mean free pathseveral times shorter than the characteristic length of the component.Nevertheless, the electrons will meet a resistance due to collisions withthe neutral gas molecules and thus pure vacuum can not be assumedeither.

Low pressure multipactor has received comparatively little attention.However, a few studies, theoretical as well as experimental, have revealedsome parts of the complicated picture. Vender et al. [17] performedPIC-simulations to study the electron density development and showedthat 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 breakdownvoltages normalised to the first cross-over point of the material for awide range of dimensionless variables. He also paid special attentionto a particular and realistic case, namely multipactor in low pressureargon [54]. This was done partly in an effort to compare the simulationswith the experimental results of Hohn et al. [18].

In paper B of this thesis, low pressure multipactor was studied usingan analytical model that takes into account only the friction force dueto collisions between the electrons and the neutral gas particles. Themain theory and results from this study will be presented in the firstsection below. In addition to the friction force, the collisions will alsocause a random velocity spread of the electrons that results in a higheraverage impact energy. Furthermore, due to the long distance betweenmolecules, the electrons are free to accelerate to very high velocities andupon impact with a gas molecule or atom the energy is sufficient to causeionisation. In paper C of this thesis a more detailed analysis has beendone, where all these effects have been considered and the used modelas 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 lowpressure gas, a simple analytical model was used, which takes only thefriction force of the collisions with neutrals into account. By deriving ex-plicit expressions for the multipactor threshold, qualitative comparisonwith experimental results [18] as well as results from computer simula-tions [53,54] could be made.

3.1.1 Model

The differential equation governing the behaviour of the electrons ina low pressure gas is given by the equation of motion, Eq. (2.1), butaugmented to include also the effects of collisions:

mx = eE − mνcx (3.1)

where νc = σcn0v is the collision frequency between the free electronsand the neutral particles. σc is the collision cross-section, n0 the neutral

36

gas density and v the electron velocity. The collision cross-section isgenerally a function of the electron velocity, but in order to avoid anon-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 conditionsas for the vacuum case, i.e. that an electron is emitted from x = 0 withan initial velocity v0 when t = α/ω, the position and velocity of theelectron can be found:

x =1

νc(1 − eνc(

αω−t))(v0 + Λ[ω cos α − νc sin α])

+Λ

ω[ω(sin α − sin(ωt)) + νc(cos α − cos(ωt))] (3.2)

x = v0eνc(

αω−t) + Λ[eνc(

αω−t)(ω cos α − νc sin α)

− ω cos(ωt) + νc sin(ωt)] (3.3)

where

Λ =eE0

m(ω2 + ν2c )

(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. Applyingthis condition to Eqs. (3.2) and (3.3) yields expressions for the requiredelectric field and the impact velocity:

E0 =me (ω2 + ν2

c )(d + v0

νc(e−

Nπνcω − 1))

(1 + e−Nπνc

ω ) sin α + ((1 − e−Nπνc

ω ) ωνc

+ 2νc

ω ) cos α(3.5)

vimpact = v0e−

Nπνcω + Λ(1 + e−

Nπνcω )(ω cos α − νc sin α) (3.6)

In order to draw the multipactor boundaries, an expression for thenon-returning electron limit is needed. Like in the vacuum case no ex-plicit analytical expression for this can be found and thus the limit willbe 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 un-der the assumption of a constant initial velocity is the same as with

37

the assumption of a constant ratio k = vimpact/v0 between impact andinitial velocities. This is true also in the presence of collisions, when theabove simple model is used and thus the constant k approach will beused in the following expressions. Combining Eqs. (3.5) and 3.6 underthis assumption yields an expression for the resonant phase, viz.

tan α =ω2[βΦ + γ] + 2ν2

c vimpact(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 multi-pactor threshold in a low pressure gas as a function of gap size, pressure,or frequency. However, by multiplying the expression for the amplitudeof the electric field with the gap size, d, an expression for the voltage as afunction of the frequency-gap size and the pressure-gap size products canbe obtained. This approach will be used in a subsequent section, wherea more advanced model is used to analyse the phenomenon. Figure 3.1shows the lower multipactor threshold in low pressure air for differentpressures. The graphs are based on Eqs. (3.5) and (3.7) only and do notconsider the non-returning electron limit nor the phase stability limits.

From Fig. 3.1 it is clear that the multipactor threshold increaseswith increasing pressure. By sweeping the pressure instead of the fre-quency and comparing the changing threshold with the corona break-down threshold, an understanding can be obtained of how the transitionbetween these two types of discharges can occur. Fig. 3.2 shows thatthe multipactor threshold first increases until a certain point, where itintersects the curve corresponding to the corona threshold. It will thenfollow this curve towards the minimum of the Paschen curve. This isjust a qualitative picture and the sharp intersection would in reality bea smooth transition.

Figure 3.1 does not consider the non-returning electron limit, northe phase stability limits. As explained in the previous chapter, phasefocusing is needed to maintain the generated electrons in a close bunch,since an electron with a too large phase error will be lost. By introducinga small phase error in Eq. (3.2) and keeping the amplitude and phaseconstant 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 phasefocusing effect is active. With the present model, the expression for the

38

10−3

10−2

101

102

103

Multipaction threshold curves in low pressure air (d=0.1 m)

Vol

tage

(V

)

Frequency (GHz)

p=0.1 Pap=1 Pap=10 PaVacuum

Figure 3.1: Multipactor chart showing the lower multipactor threshold atdifferent pressures. i.e. each curve is based on an impact ve-locity corresponding to W1 = 23 eV. Phase stability and thenon-returning electron limit are not considered, only resonance.Parameters used are: σc = 6.9 × 10−20 m2, k = 2.5, N = 1 andd = 0.1 m.

39

10−2

10−1

100

101

102

101

102

103

Threshold curves in low pressure air (fd=1GHz ⋅ mm)

Vol

tage

(V

)

Pressure (Pa)

Multipactor in airMultipactor in vacuumCorona in air

Figure 3.2: Thresholds for multipactor and corona discharges in air as func-tions of pressure together with the multipactor vacuum thresholdas a reference level. Parameters used are: σc = 6.9 × 10−20 m2,W1 = 23 eV, k = 2.5, N = 1, f × d =1 GHz·mm and d = 0.1 m.

40

stability factor becomes:

G =(νc

ω (CΦ − 1) + C ωνc

(Φ − 1)) tan α + 1 − Cνc

ω (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. Boththe positive and negative phase error limits tend to decrease with increas-ing pressure. However, the limit for non-returning electrons increases,which is a more important limit when the electron impact energy exceedsthe first cross-over energy, thus reducing the width of the multipactorzone.

10−3

10−2

10−1

100

101

−100

−80

−60

−40

−20

0

20

40Phase Limits

Pha

se li

mit

(deg

rees

)

Pressure (Pa)

Postive phase errorNegative phase errorNon−ret. el. in vacuum (Semenov et. al.)Numerical non−returning electron limit

Figure 3.3: Phase limits in a low pressure gas based on the simple analyticalmodel. The dashed line and the solid line (dashed at the end)show the upper and lower phase limits beyond which a phaseerror will start growing. The dash-dot line is the phase belowwhich emitted electrons will not be able to escape from the wallof emission. The dotted line is the phase limit obtained usingEq. (2.15) and is an approximation of the dash-dot line in thevacuum case. Parameters used are: σc = 6.9× 10−19 m2, N = 1,k = 7.6, d = 0.1 m, and W1 = 23 eV.

41

3.1.3 Main results

The main result found in paper B is that a higher microwave power isrequired to initiate breakdown in a low pressure gas, since the collisionstend to slow down the electrons. By combining the low pressure mul-tipactor graph with the corona threshold curve, it was concluded thatwith increasing pressure, the required threshold will first increase and,after reaching a plateau, it will make a smooth transition to the lowpressure branch of the Paschen curve. This behaviour is confirmed bythe investigations made by Gilardini [53] for materials with a low firstcross-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 higherorder multipactor, Gilardini found no initial increase in the multipactorthreshold, instead a monotonically decreasing breakdown voltage wasseen. A possible explanation for the differences between the result ob-tained by the simple analytical model and the result of Gilardini is alsopresented in paper B and it is suggested that the reason is that for mate-rials with a higher W1 and for N > 1 the contribution of electrons fromimpact ionisation decreased the required W1 (see Fig. 3.4). However,as will be seen in the next section, electron contribution from impactionisation is not the only reason for this behaviour. The collisions willalso cause an electron velocity spread, which will result in a larger to-tal impact velocity and thus a lower voltage is needed to achieve thenecessary first cross-over energy.

A comparison was made with experiments by Hohn et al. [18] in lowpressure argon as well as with PIC-simulations by Gilardini [54] in thesame gas (see Fig. 7 in paper B). However, the fd-product chosen bythese authors was located in the middle of the right boundary of thefirst multipactor zone, an area dominated by the hybrid modes [39].The simple model used in the presented analytical approach is not ap-plicable to these modes and consequently the behaviour found in theexperiments and simulations could not be confirmed. Furthermore, theimpact energy of the electrons at this fd-product is several times higherthan the ionisation energy of argon and thus a significant contributionof electrons from collisional ionisation would be expected. In addition,the required W1 would be reduced due to the electron velocity spreadand thus a behaviour similar to curves (b) or (c) in Fig. 3.4 should beexpected and it is also what is found.

To further analyse multipactor in a low pressure gas, a better ana-

42

Threshold curves in a low pressure gas

Vol

tage

(V

)

Pressure (Pa)

a

c

b

Multipactor in a low pressure gas without ionisationMultipactor in vacuumCorona

Figure 3.4: Qualitative form of the dependence of breakdown threshold withpressure in the region between which multipactor and corona,respectively, dominate the breakdown process: (a) the frictionforce due to collisions with neutrals dominates, (b) electron ve-locity spread reduces the required W1 and collisional ionisationcontributes significantly to the total number of electrons, (c) in-termediate situation.

43

lytical model is obviously needed. The model must, in addition to thefriction force, be able to take collisional ionisation into account as wellas collision induced velocity spread of the electrons. In the next sectiona 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 qual-itative understanding of the multipactor threshold behaviour in a lowpressure 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 lowpressure gas and it is based on paper C of this thesis. As a representa-tive 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 elec-tron motion between two parallel plates perpendicular to the x-direction.During the passage, no electron loss, only generation through collisionalionisation, will occur. Using the differential equations for the total elec-tron momentum and for the change in the number of electrons, one canderive 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 ve-locity. However, by assuming that νc and νiz are constants, Eq. (3.9)becomes a first order linear differential equation. Multipactor requiresan alternating driving electric field and as in the previous model a har-monic field E = xE0 sin ωt is used, where x is the unit vector, ω theangular frequency, and t the time. Assuming the electric field to behomogeneous, the drift velocity will be parallel to the field, u = xu,and the vector notation for E and u can be dropped in the followinganalysis. By setting ν = νc + νiz, u = x, and du/dt = x, Eq. (3.9) canbe written,

x =eE

m− xν. (3.10)

44

Since the equation has the same form as Eq. (3.1), it will also have thesame solutions and as the initial conditions are identical, the formulasfor 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 velocityin the previous model was only directed in the x-direction, but nowthere is also a thermal velocity component, vt, i.e. the total velocityis v = u + vt. With these new designations, the expressions for theresonant field amplitude and the impact velocity become:

E0 =me (ω2 + ν2)(d + u0

ν (Φ − 1))

(1 + Φ) sin α + ((1 − Φ)ων + 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 givenby Eq. (3.4) as before, but νc should be replaced by ν.

In order to construct the multipactor boundaries, the same approachas in the simple model case is taken and an expression for the resonantphase is obtained by combining Eqs. (3.11) and (3.12), which yields,

tan α =ω2[ρΦ + χ] + 2ν2(Φu0 − uimpact)

(dν + uimpact − u0)νω(1 + Φ)(3.13)

where ρ = dν + uimpact + u0 and χ = dν − uimpact − u0 have beenused for convenience. The reason why the expression looks somewhatdifferent from Eq. (3.7) is that the constant initial velocity approach hasbeen used instead of the assumption of a constant ratio between impactand initial velocities. This will also affect the expression for the phasestability factor, which in this case becomes

G =(Φ − 1)(ν2 + ω2) 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 advancedmodel are fairly trivial. However, the parameters used (νc and νiz)are not constants, they depend to a great extent on the total electronvelocity, which is the vector sum of the drift and thermal velocities. Thethermal velocity will have a random direction and therefore the averagetotal velocity will be equal to the drift velocity. However, the total(average) energy, ε, will still depend on both velocities and it becomes,

ε =mv2

2=

m

2(u2 + 〈vt

2〉) (3.15)

45

where 〈vt2〉 represents the average of the square of the magnitude of

the thermal velocity. In paper C, a differential equation for the thermalvelocity is derived, viz.

d〈vt2〉

dt+ (νcδ + νiz)〈vt

2〉 = u2(νc(2 − δ) + νiz) (3.16)

where δ is the energy loss coefficient. By assuming that νc, νiz and δ areconstants, like before, Eq. (3.16) can be solved explicitly and with theinitial condition 〈vt

2(t = α/ω)〉 = 0, the thermal impact velocity, whenωt = Nπ + α, can be found and thus the total impact velocity can bedetermined. However, the expression is very complicated and will notbe reproduced here.

The total impact velocity will determine the secondary electron emis-sion yield. For vacuum multipactor as well as in the previous simplemodel for low pressure multipactor, the impact velocity was perpendic-ular to the electrodes. In such a case, the secondary yield depends onlyon the impact velocity. However, for angular incidence, which will bethe case now with the random three dimensional thermal velocity com-ponent, the yield will be a function not only of the impact energy butalso of the angle of incidence. To account for the angular incidence theexpressions given in Ref. [22] have been used and for ease of referencethey are reproduced here,

εmax(θ) = εmax(0)(1 + θ2/π) (3.17)

σse,max(θ) = σse,max(0)(1 + θ2/2π) (3.18)

η =εimpact − ε0

εmax(θ) − ε0(3.19)

σse = σse,max(θ)(η exp 1 − η)k (3.20)

where θ is the impact angle with respect to the surface normal. εmax isthe impact energy when the secondary emission reaches its maximum,σse,max. εimpact is the total impact energy and ε0 is the energy limitfor 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 < 1and k = 0.25 for η > 1.

In vacuum multipactor, the only source of new electrons is secondaryyield from each impact. When the phenomenon takes place in a gas,another potential source of new electrons is impact ionisation of thegas 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 ofmost materials and thus when the electron energy is sufficient to initiatemultipactor, it is also enough to ionise the gas molecules. In this model,the contribution from impact ionisation is included by modifying thebreakdown condition from σse = 1 to

σse +〈νiz〉Nπ

ωµ = 1 (3.21)

where 〈νiz〉 is the average ionisation frequency and µ is an ionisationfactor, which ranges from 0 − 1 and indicates the fraction of the elec-trons from ionisation that is able to become a part of the multipactingbunch. Determination of the correct value of µ is not a trivial problemand for simplicity a constant µ = 0.75 is used. This is quite a roughapproximation, but for materials with a low first cross-over point, it willbe shown that the ionisation contribution is fairly small and the exactvalue for µ is not so important. On the other hand, for materials witha high first cross-over energy, the importance of µ can not be neglectedand thus a detailed investigation of µ should be performed, but due tothe complexity, it will be left as future work.

Apart from µ, there are other parameters, which need to be deter-mined with good accuracy in order to obtain useful quantitative results.The energy loss coefficient, δ, which is used in Eq. (3.16), is a smallquantity 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, thevalue is about 10 to 100 times larger than the elastic value [56]. How-ever, the value is still quite small and will not have major effect on thelow pressure multipactor threshold and for simplicity a value 10−3 willbe used, which is about 37 times greater than the elastic value. Theremaining parameters, νc and νiz, are very important for the thresh-old and a detailed description of these values will be given in the nextsection.

3.2.2 Analytical formulas for argon cross-sections

For most materials of interest, the first cross-over energy is in the range20-70 eV and it is this value that determines the lower multipactorthreshold. Thus it would be of value to have expressions for the col-lision 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 electron-argon collision cross section, the data given in Ref. [57] is used and itcovers the range up to 20 eV. An analytical formula has been devised,which approximates the given data quite well in the measurement re-gion (cf. Fig. 3.5). Outside the measurement points, the cross-sectionfor very low energy electrons has been set to converge towards the geo-metrical cross-section of the argon atom. For high energy electrons, thecross-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 + (8ε)3+

ε

1 + (0.07ε)2)1.5 × 1.15 × 10−20 [m2] (3.22)

where ε is the total electron energy, given by Eq. (3.15).

10−3

10−2

10−1

100

101

102

10−21

10−20

10−19

10−18

Electron energy [eV]

Tot

al c

ollis

ion

cros

s se

ctio

n [m

2 ]

Buckman and LohmanAnalytical approximation

Figure 3.5: Absolute total collision cross-section for electrons scattered fromargon. The stars indicate measurement data by Buckman andLohmann [57] and the solid line is the analytical approximationgiven by Eq. (3.22).

The ionisation cross-section increases rapidly for electron energiesslightly above the ionisation threshold and thus it is of importance tohave an accurate description in this range, especially since this is closeto the first cross-over energy of many materials. A simple function that

48

accurately describes the cross-section for the entire measurement rangecan be given by (cf. Fig. 3.6)

σiz = q1ln ε/εi

√

ε/εi + 0.1(ε/εi)2[m2] (3.23)

where εi is the ionisation threshold of argon and q1 = 4.8 × 10−20 m2.

102

103

10−22

10−21

10−20

10−19

Electron energy [eV]

Ioni

satio

n C

ross

Sec

tion

[m2 ]

Ionisation cross section for Argon

S.C.BrownH.C.StraubAnalytical approximation

Figure 3.6: Ionisation cross-section for electron-argon collisions. The circlesand stars indicate measurement data by S. C. Brown [58] andStraub et al. [59] respectively and the solid line is an analyticalapproximation given by Eq. (3.23).

3.2.3 Multipactor boundaries

In the following section the above model will be used to determine themultipactor boundaries. The best accuracy is attained when solvingthe basic differential equations numerically while using good approxi-mate formulas for the different parameters. However, such computationtakes very long time, since both the initial and the final multipactorconditions have to be fulfilled. Faster computation can be achievedby using different approximations, e.g. constant parameters as shownin Eqs. (3.10)-(3.12), Eq. (3.13), and Eq. (3.14). Two different imple-mentations are used in paper C, one purely numerical and one semi-

49

analytical. Attempts were made to find a purely analytical implementa-tion as well, but due to the strong non-linearities in the functions for thecross-sections, no accurate such implementation could be found. Detailsconcerning the two implementations are presented in paper C and willnot be reproduced here.

As mentioned in chapter 2, when constructing the complete multi-pactor zones, the multipactor thresholds corresponding to impact ve-locities between the first and second cross-over points are determinedfor a specific order of resonance within the phase range from the non-returning electron limit to the upper phase stability limit. The zonefor 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 phe-nomenon, it is sufficient to study the threshold corresponding to unitySEY. Thus, in most of the following charts, only the lower multipactorthreshold will be considered. However, in keeping with the multipactortradition, the complete zones will be presented as well.

One concern that appears when making low pressure multipactorcharts is the parameters which should be used on the chart axes. Clas-sical vacuum multipactor charts use engineering units with voltage as afunction of the frequency-gap size product, like in Fig. 2.3. By multi-plying Eq. (3.11) by the gap size, d, to get the voltage and rearrangingEqs. (3.12), (3.13), and (3.14), these expressions can all be written asfunctions of two natural parameters, viz. fd and pd, i.e. the frequency-gap size and the pressure-gap size products. Thus, for a given pd themultipactor zones can be constructed in the classical engineering unitsas shown in Fig. 3.7. Note that three different pd-values are used, onefor each zone. The chosen values are close to the limit of stability of thenumerical implementation for each zone.

In Fig. 3.7 both the analytical (semi-analytical) and the numericalimplementations are used to plot the thresholds. Very good agreementbetween the two implementations is found and therefore the faster an-alytical version is used to produce all other figures. The most strikingfirst impression of the graphs in Fig. 3.7 is the difference in behaviourbetween the first and the higher order modes. The first order modeshows an increased threshold, which is a consequence of the frictionforce experienced by the electrons due to collisions with neutrals. Thisis in agreement with the model presented in paper B, which only con-sidered the friction force. However, for higher order modes, the resultis the opposite. Instead of an increased threshold as in the friction

50

100

102

Frequency − Gap product [GHz⋅mm]

Vol

tage

[V]

pd=15 Pa mm

pd=7 Pa mm

pd=5 Pa mm

Numeric stable phaseNumeric unstable phaseAnalytic stable phaseAnalytic unstable phase

Figure 3.7: Lower multipactor thresholds in low pressure argon for three dif-ferent fixed pd-values, one for each zone. The dotted lines repre-sent the multipactor zones for vacuum multipactor. Parametersused are: W1 = 23 eV , W0 = 3.68 eV, σse,max(0) = 3, ε0 = 0.

51

only model, the inclusion of ionisation and thermal spread leads to adecreased threshold with increasing pressure.

Thresholds as in Fig. 3.7 can be found for all impact velocities be-tween W1 and W2 and by constructing the envelope of all these curvesfor each order of resonance, the complete, classical multipactor zones canbe 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 draw-back with this chart is that the model does not account for the hybridzones and thus the right boundary of each zone will not accurately re-flect the true multipactor threshold for those fd-values. The model can,however, be extended to include also the hybrid modes, but due to theincreased complexity, this is left as future work

100

101

102

103

Frequency−Gap product [GHz⋅mm]

Vol

tage

[V]

pd=10 Pa mm

pd=4 Pa 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. Pa-rameters used are: W1 = 23 eV, W2 = 1000 eV, W0 = 3.68 eV,σse,max(0) = 3 and ε0 = 0.

3.2.4 Key findings

Among the main results is that the friction force dominates the lowpressure multipactor threshold for materials with a low first cross-over

52

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 orderto see what happens when the gas density increases, the threshold can beplotted as a function of the pd-product. This has been done in Fig. 3.9for a material with a low first cross-over energy and, as expected, thethreshold increases with increasing pressure for the lowest order mode,N = 1, and after reaching a maximum, the threshold starts to decreaseagain. For higher order modes, the threshold decreases monotonicallyas the gas becomes dense enough to affect the multipacting electrons.This behaviour is identical to that found by Gilardini [53] in his MonteCarlo simulation of low pressure multipactor. He also observed that formaterials with a higher first cross-over point, the threshold does notincrease with increasing pd, instead it falls off monotonically, which isthe behaviour shown in Fig. 3.10. The main reason for this differencein behaviour is the contribution of electrons from collisional ionisation,which increases drastically when the electron energy is well above theionisation threshold.

10−2

10−1

100

101

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Nor

mal

ised

Vol

tage

Pressure × gapsize [Pa⋅mm]

N=1

N=3

N=5

Figure 3.9: Normalised multipactor thresholds for varying pd. The thresholdsare normalised with respect to the vacuum threshold. Curves forthe three first orders of resonance are shown. Parameters usedare: W1 = 23 eV, W0 = 3.68 eV, σse,max(0) = 3, ε0 = 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 differencein the basic behaviour regardless of material. The threshold falls off

53

10−2

10−1

100

101

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Nor

mal

ised

Vol

tage

Pressure × gapsize [Pa⋅mm]

N=1

N=3

N=5

Figure 3.10: Normalised multipactor thresholds for varying pd. The thresh-olds are normalised with respect to the vacuum threshold.Curves for the three first orders of resonance are shown fora material with a first cross-over point more than 7 timesgreater than in Fig. 3.9. Parameters used are: W1 = 170 eV,W0 = 4 eV, σse,max(0) = 1.3, ε0 = 0, fdN=1 = 1 GHz·mm,fdN=3 = 3.2 GHz·mm, and fdN=5 = 5 GHz·mm.

54

100

102

Frequency − Gap product [GHz⋅mm]

Vol

tage

[V]

µ=0

µ=0.75

µ=0

µ=0.75

Vacuum multipactor

Figure 3.11: Multipactor thresholds in low pressure argon for the two lowestorder modes (N = 1 and N = 3) with µ = 0.75 and µ = 0respectively. Parameters used are: W1 = 23 eV, W0 = 3.68 eV,σse,max(0) = 3, ε0 = 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 andthe 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, amonotonically decreasing threshold is obtained. The cause of this dis-tinct lowered threshold is the partial thermalisation of the electrons dueto the collisions. The velocity spread results in a total impact veloc-ity, which is greater than the drift velocity alone and thus for the samesecondary electron emission, a lowered impact drift velocity is possible.Even though the friction force requires a higher voltage to achieve thesame impact drift velocity, the thermalisation effect dominates, with alowered threshold as a result. This becomes clear in Fig. 3.11, where it isapparent that it is not the electrons from ionisation that constitute themain reason for a decreased threshold, rather it is a consequence of thepartial thermalisation. In the case with a high W1, the thermalisationeffect is also important, but without the contribution from collisionalionisation, the behaviour would not be the same, which can be seen inFig. 3.12.

To summarise the key findings from the more advanced model, it

55

100

103

Frequency − Gap product [GHz⋅mm]

Vol

tage

[V]

Vacuum multipactor µ=0

µ=0.75

µ=0.75

µ=0

Figure 3.12: Multipactor thresholds in low pressure argon for the two lowestorder modes (N = 1 and N = 3) with µ = 0.75 and µ = 0respectively. In this case, the second cross-over energy is about7 times greater than in Fig. 3.11. Parameters used are: W1 =170 eV , W0 = 4 eV, σse,max(0) = 1.3, ε0 = 0, pd = 25 Pa·mmfor N = 1, and pd = 15 Pa·mm for N = 3.

56

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

57

58

Chapter 4

Multipactor in irises

A common microwave component is the waveguide iris, which is oftenused as a shunt susceptance for the purpose of matching a load to thewaveguide. There are many different types of irises, but a typical config-uration consists of a step-like, short length, reduction of the waveguideheight. Similar structures also appear in other configurations, e.g. asapertures in array antennas, as coupling slots in directional couplers,and as irises in waveguide filters. As the field strength in the iris canbe very high and the gap height is small, there is a pronounced risk ofhaving a multipactor discharge.

So far in this thesis, all the models considered have been based onthe plane-parallel model with a spatially uniform harmonic electric field.In general, most theoretical studies of the multipactor phenomenon havebeen limited to this or similar approximations. However, many RF de-vices involve more complicated electric field structures where predictionsbased on the parallel-plate model are not applicable. This is true fore.g. the waveguide iris, where the electric field will be a combinationof several different electromagnetic modes, most of which typically areevanescent. However, due to the short length of the iris, these modeswill be of importance. Nevertheless, in this analysis, which is describedin more detail in paper D, it is only the importance of the random driftdue to initial velocity spread of the secondary electrons that has beenconsidered. Thus, as far as the field is concerned, a spatially uniformharmonic field based on the parallel-plate model is used.

Experiments [60, 61] as well as numerical studies [61, 62] of multi-pactor in an iris have shown that the discharge threshold increases withdecreasing 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]. Inthis analysis, we show that one of the contributing factors to this electronloss is a random drift due to the axial component of the initial velocityof the secondary emitted electrons. Other loss mechanisms, which aredue to the inhomogeneity of the field, tend to further enhance the lossesand these effects will be more pronounced for small gap lengths. Thismeans 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 shownin Fig. 4.1. The iris has a gap height h in the y-direction, a length lin the z-direction and is assumed to be fitted into a waveguide with aheight that is much greater than h. The harmonic electric field E isassumed uniform in the gap, as a simple approximation of the actualfield. There are two main reasons for choosing a uniform field. Firstly,the deterministic model developed for the parallel plate case, which isdescribed in chapter 2, can be used to describe the basic behaviourof an electron trajectory inside the gap. Secondly, the effect of theinitial velocity spread of the secondary electrons along the z-axis on themultipactor threshold can be analysed separately from the drift force dueto inhomogeneities in the electric field. In addition, it gives a convenientbase for comparing the results with those of the parallel plate model.

By assuming a uniform E-field in the y-direction, the electron motionalong the z-direction is not affected by the field. The motion in thisdirection, the drift motion, will occur with a fixed velocity vz betweenthe impacts. Lets assume that a seed electron is emitted inside the gap atthe coordinate z0, −l/2 < z0 < l/2, at one of the walls. As the electrontraverses the gap and hits the opposite side of the iris, it has becomedisplaced a distance ∆z in the z-direction. This drift is determined bythe 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

ω, (4.1)

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

z

y

h

l

E

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 edgeshas been passed, the electron trajectory is lost. The probability of sur-vival, p(k) (see Fig. 4.2), for the electron trajectory decreases with thenumber of transits, k, and for a general one-dimensional random walkproblem, with the jump size governed by a continuous distribution func-tion, Φk(z), an explicit solution for this, the first passage time problem,is not always possible. However, in paper D it is explained that theasymptotic behaviour of p(k) is determined by the largest eigenvalue γ0

of the expansion of Φk(z), i.e.

p(k) ∝ γk0 . (4.2)

A detailed description of how to determine p(k) is given in paper Dtogether with approximate solutions for γ0 when the normalised irislength, η = l/(vT tg), is either very small or very large. This summary,however, will focus on the effect the random electron drift has on themultipactor susceptibility zones.

Each seed electron inside the iris gap will start to multiply with thesuccessive wall collisions. Due to the stochastic losses, the number ofelectrons will sometimes become large and sometimes small. However, ifon average the generation of electrons due to wall collisions is larger thanthe loss over the gap edges, there is a finite probability that a sufficiently

61

50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

Number of collisions

Pro

babi

lity

of s

urvi

val

l=2 mm

l=8 mm

l=16 mm

Figure 4.2: The probability of survival, p(k), for an electron emitted in thecenter of the iris gap, z = 0, for three different iris lengths. Pa-rameters used: f = 1 GHz, N = 1, and WT = 2 eV (corre-sponding to vT , the rms-velocity of the Maxwellian distributionof initial velocity in the z-direction).

strong discharge will appear. The generated number of electrons overthe initial number of electrons after k collisions is given by,

Ne

N0≡ g(k) = p(k)σk

se. (4.3)

Depending on the start position of the seed electrons, the initial be-haviour 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 largeenough, it will start increasing again. But if the start position is inthe 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 asymptoticbehaviour of p(k) that will determine whether or not there will be adischarge. Thus from Eq. (4.2) and Eq. (4.3) one can conclude that theasymptotic change in the electron number is given by,

g(k) ∝ (σseγ0)k. (4.4)

Thus the average number of electrons will grow if

σseγ0 > 1 (4.5)

62

0 50 100 150 2000.5

1

1.5

2

2.5

3

σ = 1.151

σ = 1.159

Number of collisions

Ne/N

0 [−]

Multipactor in iris

Figure 4.3: The growth in electron number as a function of the number ofgap crossings for two different SEY-coefficients. Parameters used:f = 1 GHz, N = 1, l = 2 mm, and WT = 2 eV.

or equivalentlyσse > 1/γ0 > 1. (4.6)

This implies that the secondary electron yield must be greater than avalue that is larger than unity (1/γ0 > 1) to have growth of the numberof electrons. This modified breakdown criterion is the only differencebetween the model considered here and the conventional resonance the-ory of multipactor inside a plane-parallel gap (where σse > 1 is usedwhen determining the threshold). The condition for σse, Eq. (4.6), canbe 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, re-spectively, in the parallel-plate model, multipactor regions that accountfor the electron losses due random drift can be obtained.

63

4.2 Multipactor regions

Using the above described model and by employing a natural scalingparameter of η, viz. l · f , multipactor charts in the traditional engi-neering units (voltage vs. frequency-gap-size product) can be devisedfor a specific value of the ratio of gap height and iris length, h/l. Fig-ure 4.4 shows an example of this, where the multipactor regions havebeen constructed for 5 different h/l-ratios.

100

101

102

103

104

Frequency gap height product [GHz ⋅ mm]

Pea

k vo

ltage

[V]

Multipactor regions for different height/length ratios

0.001135.26.3

Figure 4.4: The first four multipactor susceptibility zones for a microwaveiris 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 toEq. (4.1) and thus the distance traversed in each step becomes smaller,which implies that the probability of surviving k steps increases. Thiscauses the multipactor zones to shrink towards higher frequencies withincreasing h/l as is evident in Fig. 4.4. The transit time is also a functionof the mode order, which increases the transit time for higher ordermodes. This counteracts the decrease due to increasing frequency andconsequently a behaviour similar to that of the first resonance zone canbe 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 zoneswill disappear for relatively small values of h/l. However, in the oppositecase for a material with a large SEY, like e.g. aluminium, multipactorwill be possible also for relatively thin irises.

4.3 Comparison with experiments

By comparing the current model with experimental data [61], good qual-itative agreement can be observed (see Fig. 4.5). As the h/l-ratio in-creases, the threshold increases until, beyond a certain limit, no multi-pactor is possible. Since only the electron losses due to the random driftare accounted for, the model predicts the existence of a discharge beyondthe limit found in the experiments. Consequently, from an engineeringpoint of view, this is a conservative measure of the increased thresholdand thus it should be safe to apply it when designing multipactor freemicrowave hardware.

0 1 2 3 4 5 6 7 8500

1000

1500

2000

2500

3000

3500

Height/Width [−]

Pea

k V

olta

ge [V

]

Multipactor in iris

Current model a)Current model b)Current model c)Current model d)Measur. −presentationMeasur. −proceedings

Figure 4.5: The multipactor threshold as a function of h/l for different pa-rameters of the current model. For comparison, measurementdata 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, andσse,max = 2.22, (i.e. W1 = Wf2 and σse,max = αmax in table A-6for 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 factthat several multipactor zones are involved (f · h ≈ 11.5 GHz·mm).Starting with mode number N = 7 for ’current model a)’, the lowerthreshold for the parallel-plate case is found and as the h/l-ratio in-creases, the zone corresponding to N = 7 shrinks until, with an almostsudden voltage step, the next threshold, being determined by the N = 5zone, is reached. Finally the last N = 3 zone determines the thresholdbefore it also vanishes.

In addition to the experimental comparison, Fig. 4.5 brings forwardthe importance of different parameters of the current model as well as ofthe used model for SEY [22]. By increasing the initial velocity (’currentmodel b)’), the overall threshold decreases as a lower field strength willbe sufficient to reach the same impact velocity (cf. Eq. (2.7)). The effectof an increased thermal spread, WT , is that the electron losses increasesand the threshold starts to increase for lower h/l-values (also shownin ’current model b)’). By lowering the first cross-over point (’currentmodel c)’), the parallel-plate threshold decreases, since an additionalzone, N = 9, comes into play. However, as it shrinks away, the thresholdincreases in a sudden step to the same level as in case ’a)’ and then itfollows ’a)’ except that the steps occur at higher h/l-values. An increasedfirst 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 zonethat determines this threshold.

In the current model a uniform electric field has been used. Due tothe geometry of the iris, the actual electric field will tend to be curvedoutwards 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 theedges, the Miller force [10], which is proportional to the negative gradi-ent of the square of the electric field amplitude, will tend to push theelectrons out of the iris. This effect is most important for the higherorder resonances, where several RF-cycles are required to cross the gap.In addition to the Miller force, the curved electric field will have a com-ponent in the z-direction, which, in particular for the first order mode,will drive the electrons toward the iris edges. This means that the elec-tron losses will be greater than in the case of a uniform field, whichwill lead to an even further increase of the multipactor threshold. Thiseffect should be more pronounced for thin irises and could explain whythe current model predicts the existence of a discharge beyond a certainh/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 irislength due to the initial velocity of the secondary electrons tends tosignificantly increase the multipactor threshold in a waveguide iris ascompared to predictions based on the classical two parallel-plate model.Inherent in the presented model is the scaling parameter h/l, whichmakes it possible to produce useful multipactor susceptibility charts inthe traditional engineering units. An increase in the h/l-ratio results ina shrinkage of each multipactor resonance zone. For each zone, the zonereduction effect is more pronounced for lower frequencies. A consequenceof this is that the multipactor free region in the parallel plate model atlow frequency-gap-height products grows with increasing h/l-ratio.

67

68

Chapter 5

Multipactor in coaxial lines

The coaxial line is a very common and important component in mi-crowave systems. It is a transmission line that consists of an innercylindrical conductor and a coaxial outer conductor. A constant cross-section is maintained by means of a dielectric medium, which is con-tained between the conductors. In some space applications, as well as inother systems, the dielectric medium has been partly omitted in order tosave weight or to reduce the dielectric losses. In a vacuum environment,the line may become evacuated, which makes it exposed to the risk of amultipactor discharge.

The electric field in a coaxial line is nonuniform, which makes analyt-ical analysis difficult, since the the equation of motion for an electron be-comes a non-linear differential equation. However, multipactor in a coax-ial line has been studied experimentally [63] and numerically [64–67]. Inthese studies it was found that two different types of resonant multi-pactor can occur, namely a two-sided discharge between the outer andthe inner conductors and the one-sided analogue on the outer conduc-tor. In an attempt to understand the effect of varying the relative innerradius, i.e. varying the characteristic impedance of the coaxial line,different scaling laws were suggested in these studies. Another study fo-cused on the current due to the multipacting electrons and treated thisas a radially oriented Hertzian dipole in order to determine the electricfield generated by the multipactor discharge [68].

In paper E resonant multipactor in a coaxial line is analysed bymeans of an approximate analytical solution of the non-linear differen-tial equation of motion, which in a large range of microwave frequen-cies and amplitudes agrees very well with the numerical solution. As

69

support for the qualitative analytical results, paper F presents PIC-simulations of the phenomenon in the same geometry. The advantageof PIC-simulations is the ability to include aspects, which are stochasticin nature, like e.g. the initial velocity spread of the secondary elec-trons. This summary will focus on the results of the study and for amore detailed description of the model and approximations used, seepapers E and F.

5.1 Analytical study

By finding an analytical solution of the equation of motion, general prop-erties of the multipactor can be found, which may be difficult to identifywhen numerically studying the phenomenon. In addition, the time ofcomputation can be radically reduced when using explicit analytical ex-pressions instead of a numerical scheme. In this section an approximateanalytical solution of the non-linear differential equation that governsthe electron motion in the nonuniform field between the inner and outerconductor of a coaxial line is found. Using these expressions, the effecton the multipactor resonances and thresholds is studied. The validityof the expressions is then confirmed by solving the differential equationnumerically.

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, whichmeans that the electric field, E, is radially directed and the amplitudewill be inversely proportional to the distance from the centre of the line.There will be no dependence on the angle around the coaxial axis, whichmeans that the problem can be studied as a one dimensional problem,provided that the effect of the magnetic field is neglected and only across-section of the coaxial line is considered. In vacuum, the equationof 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 instan-taneous strength of the electric field. The radial position of the electronis 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 thenotation Λ = qEoRo/m, Eq. (5.1) can be written:

r′′ = −Λ

rsin ωt (5.2)

The relation between the field amplitude and the voltage amplitude isgiven by Uc = EoRo ln (Ro/Ri). Since the field is inhomogeneous andstronger near the centre conductor, there will be a net average force thatslowly, compared to the fast harmonic oscillations, pushes the electrontowards the outer conductor. This force is called the ponderomotive orMiller force [10] and it tends to push the electrons away from regions withhigh amplitudes of the RF electric field. By separating r(t) accordingto 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 approximatesolution of Eq. (5.2) can be derived (see paper E) where the positionand velocity of the electron are given by:

r(t) ≈ Λ

ω2

sin (ωt)√

C1(t − C2)2 + Λ2

2ω2C1

+

√

C1(t − C2)2 +Λ2

2ω2C1(5.3)

and

r′(t) ≈ 1

R(t)

(

C1(t − C2) +Λ

ωcos (ωt)

)

, (5.4)

71

where R(t) is the average position,

R(t) =

√

C1(t − C2)2 +Λ2

2ω2C1. (5.5)

The constants of integration, C1 and C2, are determined by the initialconditions, which for an electron starting at the outer conductor arer(t = t0) = Ro and r′(t = t0) = −v0. Using Eq. (5.3), the position of anelectron emitted from the outer conductor with no initial velocity hasbeen plotted in Fig. 5.2. The accuracy of the expression is evident fromthe comparison with the numerical solution.

0 1 2 3 4 5 6 7 8

5

6

7

8

9

10

Time [ns]

Pos

ition

[mm

]

Multipactor in coax

Figure 5.2: Motion of an electron emitted from the outer conductor of a coax-ial line. The solid line corresponds to the analytical expressionEq. (5.3), the dotted line is a numerical solution of the differen-tial equation Eq. (5.2) (almost covered by the solid line) and thedashed line is the average motion according to Eq. (5.5). Param-eters used: Vc = 1200 V, f = 3 GHz, W0 = 0 eV (the initialelectron energy), Ro = 10 mm, and Ri = 5 mm.

An important result can be obtained by only looking at the averageposition, Eq. (5.5). The minimum of this equation, Rmin, is the small-

72

est achievable radial position for an electron emitted from the outerconductor, provided that the oscillations are not too large. The expres-sion 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(Λ2 + 2Λ2 cos α2 + 4Λcos αv0ωRo + 2v2

0ω2R2

o)−1/2 (5.6)

However, for v0 = 0 a much more compact expression, which is indepen-dent 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 than58% of the outer radius, Ro, then two sided multipactor is not possiblewhen 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 theouter conductor) are possible. First, double-sided discharge will be con-sidered and typical for this is that the one way transit time correspondsto an odd integer of half RF field periods. However, in a coaxial line,the transit time is normally longer for electrons emitted from the outerconductor than for electrons emitted from the inner conductor. Thus,the sum of two transits must be considered and the condition for this isthat it should be an integer number of RF periods. This is the resonancecriterion and in addition to this the phase-focusing effect should be ac-tive, which for the parallel-plate case is given by Eqs. (2.18)- (2.21). Itis instructive to compare the coaxial case with the parallel-plate case,since in the limit when the Ri ≈ Ro the coaxial and parallel-plate mod-els should give the same results. For the parallel-plate case, when theinitial velocity is neglected (v0 = 0), the phase stability range is givenby the following inequalities [39],

πk < λ <√

(πk)2 + 4, (5.8)

where k is an odd positive integer. The normalised gap width, λ, isdefined by

λ = ωd/Vω = m(ωd)2/eU, (5.9)

73

where d stands for the gap width, Vω = eEω/mω is the amplitude ofthe electron velocity oscillations in the spatially uniform RF field, Eω isthe RF field amplitude inside the gap, and U is the voltage between theconductors. In addition, for an electron avalanche to start, the impactvelocity 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 analyticalanalysis that is feasible in the parallel plate case is not applicable forthe coaxial line, despite the fact that the approximate electron positionand velocity are known explicitly (Eqs. (5.3) and (5.4)). One way offinding the resonance zones in the parameter space is to compute a seriesof successive electron trajectories, searching for the conditions when itconverges to a periodically repeated sequence [69]. In Fig. 5.3 one cansee the result of such a method, where stable resonance zones have beenfound numerically. To allow simple comparison with the parallel-platecase, the normalised gap size has been used and in terms of the coaxialline it becomes,

λ =m(ωd)2

eUc=

G(Ro − Ri)2

R2o ln (Ro/Ri)

, (5.11)

which coincides with Eq. (5.9) when Ro/Ri is close to unity. The con-venient parameter, G = ωRo/Vω,o, has been introduced as representinga normalised outer radius and Vω,o = qEo/mω.

When Ro/Ri is close to unity the parallel-plate and coaxial modelsgive similar results. When the ratio becomes larger, the zones deviatefrom the straight lines predicted by Eq. (5.8) and the deviation is morepronounced for the higher modes. When the ratio becomes too large,all two sided resonances disappear. This is a consequence of the Millerforce and for the higher order modes, where the approximate analyticalsolution is very accurate, the prediction (Eq. (5.7)) is that the doublesided resonances should disappear roughly at Ro/Ri =

√3 ≈ 1.73. Fig-

ure 5.3 shows that this is indeed true. The first order mode, however, isnot accurately described by the analytical expression and the numericalcalculations show that this mode can exist for values of Ro/Ri as highas 4.

In Fig. 5.4 the double-sided discharge regions have been computednumerically for Ro/Ri = 1.4. The classical multipactor zones have upperand lower thresholds that satisfy Eq. (5.10) in the following sense: since

74

1 1.5 2 2.5 3 3.5 4

4

6

8

10

12

14

16

18

20

22

Ro/R

i [−]

λ

Double sided multipactor, numerical data

1c2h

3 3c

4

5

5c

6

7 8

Figure 5.3: Normalised gap width according to Eq. (5.11) vs. Ro/Ri. Thesolid straight lines form the classical zones according to Eq. (5.8).The dots (blue) indicate stable resonances where the sum of twotransits equal an odd number of RF-cycles. The crosses (red) arefor sums equal to an even number of RF-cycles. The sum of thetransit time in RF-cycles for some distinct zones are indicated.The lowest order hybrid modes are marked with an ’h’ and theclassical resonances with a ’c’. The dashed vertical line indicatesRo/Ri =

√3 ≈ 1.73.

75

there are different oscillatory velocities depending on whether the elec-tron starts on the outer or the inner conductor, an average value mustbe computed. By setting Vω,o = qEo/mω = v1/2 the corresponding volt-age, VRo, can be derived. Similarly, by setting Vω,i = qEi/mω = v1/2the voltage VRi can be obtained. The lower threshold voltage is thenapproximately equal to:

Uth ≈ VRo + VRi

2. (5.12)

The upper threshold is computed in a similar manner, only with v2

instead of v1. The hybrid modes, which in general have a lower averageimpact velocity will require a stronger electric field to reach an energyequal to the first cross-over point. Consequently, a threshold higher thanthe lower envelope is obtained for these zones (cf. Fig. 5.4).

100

101

102

103

104

Am

plitu

de o

f Con

duct

or V

olta

ge [V

]

Frequency [GHz]

Double sided multipactor: Zc = 20Ω

1c

2h

3c

4

6

7

Figure 5.4: Numerically obtained double sided multipactor chart. Thedashed lines are the approximate lower and upper envelopes givenby Eq. (5.12). The sum of the transit time in RF-cycles for eachzone 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 be-comes the dominating scenario. Single-sided multipactor is less compli-cated than its double-sided counterpart, as the complicated asymmetrydoes not appear in this case. This allows an analytical analysis based

76

on the approximate solution for the electron trajectory, Eq. (5.3). Theanalytical expression is accurate when the oscillations are small and aconsequence of this is that accurate stable phase multipactor regions areonly found for N ≥ 2, i.e. when the duration of the trajectory is at least2 RF-periods.

By analysing the resonance and stability conditions, one can showthat single-sided breakdown will have not only one region of stable reso-nant phase, but rather two stable regions can be found. One somewhatwider region with resonance close to zero and another, which is reso-nant close to π/4. It can be shown that these regions, in the case whenv0 = 0, are approximately given by:

0 < αR < α1 (5.13)

andα2 < αR < α3 (5.14)

where

α1 ≈ 4

Nπ(5.15)

α2 ≈ π

4− 1

Nπ(5.16)

α3 ≈ π

4+

1

Nπ(5.17)

For increasing N , the second region converges to αR = π/4, which ac-cording 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 . Exceptfor the lowest order resonance (N = 1), the regions of phase stability forthe numerically and analytically obtained phases agree very well.

To obtain the multipactor threshold, it is necessary to know theimpact 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 maximumimpact velocity for each mode, i.e. vimpact = 2Vω,o when v0 = 0. In thesame figure one can also identify a second set of regions with a higherbreakdown threshold. These zones correspond to the second stable phaseregion, 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

0 5 10 15 200

10

20

30

40

50

60

α1

α2

α3

N (rf−cycles)

Pha

se [d

egre

es]

Figure 5.5: Stable resonant phase for single-sided multipactor. Analyticallyobtained stable phases are shown as diamonds (red) and the onesobtained 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), whichwill yield a higher impact velocity and consequently a lower threshold.Furthermore, in the numerical solution of Eq. (5.2) for the first ordermode, 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 maximumimpact velocity of the second zone is slightly lower than 2Vω,o, resultingin a somewhat higher threshold. The following higher order modes thenquickly converge to the analytical limit (cf. Fig. 5.6).

When the initial velocity is zero, the only parameters left to vary areG and the ratio Ro/Ri. Since the characteristic impedance in ohms ofa coaxial line in vacuum is given by Z ≈ 60 ln (Ro/Ri), it follows thatonly two parameters remain to be varied, viz. G and Z. By followingtrajectories for different values of G and Z, stable phase points werefound in this parameter space and the result is plotted in Fig. 5.7, whichwas produced using a numerical solution of the equation of motion (aversion of this figure using the analytical expressions can be found inpaper E).

The straight lines in Fig. 5.7 on the right hand side are regions ofstable single-sided resonances. The fact that these appear as straight

78

100

101

102

102

103

104

105

Frequency x Ro [GHz ⋅ mm]

Am

plitu

de o

f Con

duct

or V

olta

ge/ln

(R0/R

i) [V

]

Single sided Multipactor: Zc=100 Ω

Figure 5.6: Single-sided multipactor breakdown regions based on the numer-ical solution of Eq. (5.2). Regions corresponding to N = 1 − 22RF-periods are shown. The regions with an initial phase, α, closeto zero are produced using dots (blue) and the regions with α closeto π/4 are indicated by crosses (red). The dash-dot line is theapproximate lower envelope given by Vω,o = v1/2 and the dashedline is given by Vω,o = v1/

√2. Parameters used: W1 = 23 eV,

W2 = 2100 eV, W0 = 0 eV, and Z = 100 Ω (corresponding toe.g. Ro = 10 mm and Ri = 1.88 mm.)

79

0 0.5 1 1.5 20

10

20

30

40

50

60

70

80

90

100

Z/Z0 (Z

0=50 Ω)

G

Figure 5.7: The normalised parameter G vs. normalised characteristicimpedance Z. Each mark represents a stable phase solution andan effort has been made to suppress polyphase modes in orderto clearly show the behaviour of the main resonance modes. Thechart was obtained by numerically solving the equation of mo-tion. Stars mark (blue): double-sided multipactor, dots (red):single-sided multipactor with 0 < αR < 20o, and crosses (green):single-sided multipactor with αR > 20o. The dashed line indi-cates 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 asimple scaling law exists in the single-sided case, viz.

P ∝ (ωRo)4Z. (5.19)

In Fig. 5.6 this law has been used to normalise the axes, but voltageis used on the ordinate instead of power. The chosen normalisationof the axes in Fig. 5.6 is general and using the analytical solution ofEq. (5.2) presented above, it can be shown that this normalisation isvalid also for non-zero initial velocity. It is important, however, to becareful when scaling to a different radii ratio, since for smaller valuesof the characteristic impedance, the single-sided multipactor zones maynot 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. Forsmall values of Z, however, the coaxial case becomes similar to theparallel-plate geometry, where the resonance voltage can be written asfunction of the frequency-gap-size product. For the coaxial case, thisscaling law becomes

P ∝ (ω(Ro − Ri))4 1

Z. (5.20)

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 withincreasing Z.

5.1.3 Main findings

A qualitative comparison with experiments [63] shows good agreementwith the present analysis. The experimental data shows an increase inthe multipactor threshold for increasing radii-ratio Ro/Ri. It was alsofound that the first multipactor zone became narrower for increasingRo/Ri. These features are in agreement with the results of this studyas shown in Figs. 5.3 and 5.7. By mapping the data of Fig. 5.3 into thevoltage vs. frequency-gap-size space, used in the experiments, a clearthreshold increase compared with the parallel-plate case can be seen aswell. Even though the experiments used quite large values of Ro/Ri,no case of single-sided multipactor was observed. This can be explainedby 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 appliedvoltage is not high enough. More importantly, only the first order modewas 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 etal [64], the experimental results were confirmed. In addition, single-sidedmultipactor was observed, which confirms the result of this theoreticalstudy.

Among the more important results of this part of the study are thefollowing. The analytical approximate solution of the nonlinear differen-tial equation of motion for an electron in a coaxial line. The dual regionsof stable phase, Eqs. (5.13) - (5.17), which explain why single-sided mul-tipactor will be possible also for smaller values of Ro/Ri. The scalinglaw for single-sided multipactor, Eq. (5.19), simplifies the presentationof multipactor prone regions of the single-sided case. The limit formulafor the transition from double- to single-sided multipactor, Eq. (5.7), isan interesting feature for future experiments to confirm. The reducedthreshold for the first order zone of single-sided multipactor, which mustbe taken into account when constructing the lower boundary of all thezones. Finally, this analysis shows that the behaviour of resonant multi-pactor is significantly affected by the nonuniform field and it shows thebenefits of analytically studying different geometries to understand thebasic behaviours before performing numerical simulations.

5.2 Particle-in-cell simulations

In this part of the coaxial study, which is based on paper F, extensivePIC-simulations have been performed in order to verify the analyticalresults as well as to investigate the importance of initial velocity spreadand different maximum secondary emission. One of the advantages withPIC-simulations is the ability to include parameters that are stochasticin nature. The stochastic properties of some of the parameters as wellas the actual value of the maximum secondary emission coefficient mayhave significant effects on the multipactor threshold as well as on theexistence of a discharge. This has previously been shown in the caseof plane-parallel geometry [46] and it is demonstrated that the effect issimilar also in coaxial geometry.

5.2.1 Numerical implementation

The geometry and field is described in the analytical section. The codeuses normalised parameters such as G and λ and the SEY follows themodel 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

vexp

(

−1

2(

v

vT)2)

(5.21)

where v is the absolute value of the initial velocity, vn its normal com-ponent with respect to the surface of emission, and vT is the thermalinitial velocity spread. Another of the used parameters related to thisis 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 thenormalised parameters (e.g. ρ = Vω/vmax vs. λ with the other param-eters fixed) and each run corresponds to one particular point in one ofthese arrays. Each run was primed with 200 seed electrons, uniformlydistributed over initial phase, and the run was terminated when eitherthe number of particles exceeded 4500 or when 200 RF-periods hadelapsed. The run was also terminated in case the number of electronsdropped below 10 before 200 RF-cycles had passed. At the end of eachrun, the following parameters were recorded:

• Number of RF-periods needed to exceed 4500 particles. If 4500particles were not attained within 200 RF-cycles, this parameterwas set to 200.

• Number of electrons at the end of each run.

• Heating asymmetry, i.e. the ratio between the average power de-posited on the inner conductor and the average power depositedon the outer conductor.

• Average electron growth rate (over the 10 last RF-periods), nor-malised with respect to the RF-period.

5.2.2 Simulations

To facilitate comparison between the theoretical result presented inFig. 5.7 a simulation was made in the same parameter space. Figure 5.8shows the number of electrons obtained after 200 RF-cycles. As ex-pected, the lower order resonances (i.e. at lower and leftmost G-values)indicate high electron numbers, since more impacts with the conductorswill occur during the same number of RF-periods. Due to this fact, aparameter 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 transitionfrom double-sided to single-sided multipactor occurs at more or less thesame impedances for the different zones in both the PIC-simulation andthe theoretical data, since the non-zero initial velocity in the PIC-datais small relative to the oscillatory velocity.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

10

20

30

40

50

60

70

80

90

100

Z/Z0 (Z

0=50 Ω)

G

500

1000

1500

2000

2500

3000

3500

4000

4500

Figure 5.8: Number of electrons after 200 RF-cycles. The white dots arepoints of 2-sided multipactor, the green and yellow dots 1-sidedmultipactor, 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 thisshould be single-sided multipactor. This is confirmed by looking at theratio of power deposited on the inner and the outer conductors, whichdirectly identifies the type of discharge. In Fig. 5.9, the dark blue areasare regions where most or all power is deposited on the outer conductor,which implies single-sided multipactor on this conductor. The orange

84

and yellow areas indicate that a similar amount of power is depositedon both conductors and thus the double-sided scenario dominates.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

10

20

30

40

50

60

70

80

90

100

Z/Z0 (Z

0=50 Ω)

G

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Figure 5.9: Ratio of power deposited on the inner and outer conductors dueto 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 PIC-simulations, this confirms the validity of the scaling laws, Eqs. (5.19) and(5.20). The two different types of modes for single-sided discharge, onewith a phase α ≈ 0 and the other with α ≈ π/4 can also be seen. Theformer type of mode is discontinued before reaching the first dashed linefrom the right hand side and the latter before the other dashed line inFig. 5.7. This is also evident in the PIC-data, especially for values of Gbetween 30 and 40 in Fig. 5.9 where they are fairly well separated andonly the lower of the paired bands extend into the region between thedashed 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-valueswhile 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 thezones are well defined and fairly narrow (cf. Fig. 5.10). With increasing

85

σse,max the zones become wider and zones previously suppressed by thelosses can appear (cf. Fig. 5.11). This behavior is very similar to thatnoted for the parallel plate geometry [46]. The lower (left) envelope is

500

1000

1500

2000

2500

3000

3500

4000

4500

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.85

10

15

20

25

30

35

40

45

50

Vω/Vmax

G

Figure 5.10: Number of electrons after 200 rf-cycles. The vertical straightlines indicate the lower and upper theoretical envelopes accord-ing 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 willyield 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 electronscan have a large negative initial phase, which leads to overlapping ofthe multipactor regions when σse,max is large (see Fig. 5.12). On theother hand, the increased velocity spread also increases the losses, whichespecially affects the higher order resonances, since the phase-focusingeffect gets weaker with increasing mode order. If σse,max is low, thelosses are not sufficiently compensated for and this leads to suppressionof the higher order modes (cf. Fig. 5.13). This is a result similar tothat found in the parallel plate case [46]. The corresponding behaviouris seen also for single-sided multipactor (see paper F).

For single-sided multipactor it was noted that for the first ordermode, 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

500

1000

1500

2000

2500

3000

3500

4000

4500

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.85

10

15

20

25

30

35

40

45

50

Vω/Vmax

G

Figure 5.11: Same as Fig. 5.10 only with σse,max = 2.0.

500

1000

1500

2000

2500

3000

3500

4000

4500

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.85

10

15

20

25

30

35

40

45

50

Vω/Vmax

G

Figure 5.12: Number of electrons after 200 rf-cycles. The vertical straightlines indicate the lower and upper theoretical envelopes accord-ing 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.

87

500

1000

1500

2000

2500

3000

3500

4000

4500

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.85

10

15

20

25

30

35

40

45

50

Vω/Vmax

G

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 caninclude aspects of the multipactor, which are difficult to analyse theoreti-cally. As a more realistic description is possible, quantitative comparisonwith experiments becomes feasible.

In the experimental study by Woo [63] coaxial lines made of copperwere used. When taking values for the secondary electron emission prop-erties for copper from the ESA standard [50], the lower thresholds of thePIC-simulations are not in very good agreement with the experimentaldata. However, the secondary emission properties can vary a great dealbetween different samples of the same material and contamination canreduce the first cross-over point and increase the maximum SEY. Thusby slightly lowering the first cross-over point in the PIC-simulations,very good agreement is obtained (see Fig. 5.15). In paper F an alter-native method of obtaining the experimental threshold when using thesecondary emission properties given in the ESA standard [50] is pre-sented, 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

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.85

10

15

20

25

30

35

40

45

50

Vω/Vmax

G

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Figure 5.14: Number of electrons after 200 rf-cycles. The vertical straightlines 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, andW1 = 50 eV.

log 10

(Am

plitu

de)

[V]

log10

(Frequency) [GHz]−1.6 −1.4 −1.2 −1 −0.8 −0.6

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

500

1000

1500

2000

2500

3000

3500

4000

4500

Figure 5.15: Multipactor breakdown regions for copper electrodes. The re-gion confined by circles (or white crosses, when inside a darkregion) is from Ref. [63]. The dark regions are obtained by thePIC-code with: Ro/Ri = 2.3 (Z = 50 Ω), σse,max = 2.25 (fromtable A-6 in [50], W1 = 27 eV, and vT = 3 eV (vT /vmax =0.111).

89

creasing ratio Ri/Ro was observed. It was also found that the first mul-tipactor zone became narrower for decreasing Ri/Ro. This behaviourwas seen in the analytical analysis and it is evident also in the PIC-simulations. Figure 5.16 shows the result of a simulation for Z = 174 Ω(Ro/Ri = 18.26). The agreement between simulations and experimentsis good. The multipactor zone becomes narrower and the threshold in-creases. The position of the zone is slightly shifted compared with theexperimental data and the reason for this deviation may partly be ex-plained by an inaccuracy of the dimensions of the copper electrodes.The main reason, however, seems to be related to the SEY-model andthe description of the initial velocity of the secondary electrons, but amore detailed study will be necessary to settle this matter.

log 10

(Am

plitu

de)

[V]

log10

(Frequency) [GHz]−1.6 −1.4 −1.2 −1 −0.8 −0.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Figure 5.16: Ratio of power deposited on the inner and outer conductorsdue to electron impacts (log-scale) for the same parameters andSEY-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 ex-periments. This is in agreement with the PIC-simulations and showsthat no measurements were made at high enough voltage and frequency,where single sided multipactor would be the main phenomenon (the darkblue regions in Fig. 5.16).

90

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 differ-ent maximum secondary electron emission have been highlighted. Theresults have shown that the behaviour with respect to these conditions,in both the single-sided and the double-sided cases, is qualitatively thesame as for the parallel plate multipactor. The results are in good agree-ment with available experimental data, but the commonly used modelfor SEY seems to be inadequate for large Ro/Ri-ratios, since a small,but not negligible, deviation from the experiments was noticed.

91

92

Chapter 6

Detection of multipactor

In the space industry, where multipactor is a well-known problem, greatcare is taken to avoid the phenomenon. According to the ESA stan-dard on multipactor design and test [50] only single carrier parts with amargin of 8-12 dB (depending on the type of component) in the analysisstage are exempt from testing. The recommended corresponding marginfor multicarrier parts is 6 dB between the peak power of the multicarriersignal and the single carrier threshold. By performing unit acceptancetests, the margins can be reduced to 3-4 dB in the single carrier caseand to 0 dB in the multicarrier case. In order to take advantage of thesmaller margins, accurate and reliable testing is required to verify thatthe prescribed margins are actually fulfilled and successful testing relieson accurate and unambiguous methods of detection.

This chapter starts by briefly reviewing some of the most commonmethods of multipactor detection. Then a method of detection is pre-sented, which can be used in combination with other methods to achieveaccurate and unambiguous test results. Beginning with the single carriercase, a theory for the underlying mechanism making the method possi-ble is given together with some supporting data. This is followed by adiscussion outlining how the method can be applied also in the multicar-rier case. Detection of multipactor using RF power modulation, whichis 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 bedivided into two fairly distinct categories, viz. global and local methods

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of detection. The global methods are characterised by being able todiscern whether or not microwave breakdown is taking place somewherewithin the tested system. However, it can not pinpoint the location ofthe discharge. For flight hardware it is important to completely avoidmultipactor in the entire microwave system and consequently this typeof testing is preferred. During the product development stage, it maybe of interest to know not only that a discharge is taking place, but alsoits exact position within the system. In such a case, a local method canbe 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 ofdetection are normally used and there are several reasons for this. Themethods can usually be applied without modifying the component, whichis advantageous as modifications can affect the electromagnetic proper-ties and give inadequate measurement results. In cases where the dis-charge is weak, many local methods are unable to detect the phenomenonand thus the often more sensitive global ones are a better choice. Fur-thermore, it is a requirement of the ESA standard [50] that two methodsof detection should be used and at least one of them should be global.By using two methods of detection, the risk for misinterpretations is re-duced. The most common global methods of detection will be describedin the following subsections.

Close-to-carrier noise

Multipacting electrons will be accelerated to high velocities by the elec-tric field and at regular intervals, 2/N times per field cycle, the electronswill hit an electrode or device wall and experience a sudden deceleration.The radiated power is a function of the electron acceleration and it isdescribed by Larmor’s formula,

P =2

3

e2

4πε0

v2

c3(6.1)

where v is the acceleration, ε0 the dielectric constant of vacuum, and cthe speed of light. For an applied sinusoidal electric field, the electronswill experience a sinusoidal acceleration, except for the sudden inter-ruptions when colliding with the electrodes. Fig. 6.1 shows what theacceleration may look like for first order multipactor (N = 1), where theelectrons hit the electrodes once every half cycle of the electric field.

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1.76 1.78 1.8 1.82 1.84 1.86

x 10−8

−1.5

−1

−0.5

0

0.5

1

1.5x 10

6 C2C−Noise Timedomain

Am

plitu

de o

f acc

eler

atio

n [G

m/s

2]

Time (s)

Figure 6.1: A qualitative plot of the electron acceleration for multipactingelectrons with N = 1 (average value for a large number of elec-trons).

The regular deviation from a pure sinusoid will generate harmonics,but these will be discussed in the next subsection. Due to variations inthe time between impact and emission of new electrons, in the time ittakes to decelerate the electrons, and also in the number of electrons,close-to-carrier noise will be generated. In fact, most noise is generatedat the carrier frequency, but this can not be seen in a measured spectrum,as the signal amplitude masks the noise. By performing a Fast FourierTransform (FFT) of a sequence like the one shown in Fig. 6.1, the noisegenerated close to the carrier can be seen (see Fig. 6.2). Very close tothe carrier, the noise level is even higher than what can be discerned inFig. 6.1 and by using a bandpass filter with high rejection at the carrierfrequency, the noise increase close to the carrier can be detected with aspectrum analyser. In combination with a low-noise amplifier, this canbe a very sensitive method of detection.

This method of detection can be used for both single and multicarriersignals. Care should be taken, however, when using a pulsed signal,since such a signal will generate harmonics and if the pulse length andtype is not chosen properly, the harmonics may be generated in themeasurement band [31]. A risk with this method is that other sources of

95

2 4 6 8 10 12 14 16

−100

−90

−80

−70

−60

−50

−40

−30

−20

Multipactor Noise Power

(dB

)

Frequency (GHz)

Figure 6.2: Noise spectrum of a multipactor simulation like the one inFig. 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 f0/3are clearly visible.

noise may be misinterpreted as multipactor noise. Nevertheless, by usingtwo different methods of detection as required by the ESA standard, thisrisk is greatly reduced.

Third harmonic

The resonant behaviour of the multipactor discharge and the repetitiveacceleration and sudden deceleration of the electrons will generate noise,which will have harmonics at the basic frequency, f0, and at odd mul-tiples of this frequency (cf. Fig. 6.2). Higher order multipactor willhave harmonics not only at odd multiples of the basic harmonic, but ateach odd multiple of f0/N [14] (cf. Fig. 6.2). In such cases there aremany different frequencies available for detection. However, since thethird harmonic usually is the most powerful harmonic and it is alwayspresent, regardless of the order of resonance, it is the best choice fordetection.

Third harmonic detection is a very reliable and fast method of de-tection. According to Ref. [14], it gives the fastest indication of mul-tipactor and that makes it the method of choice when studying mul-

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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 asclose-to-carrier noise and, thus, it is not recommended to use only thirdharmonic and close-to-carrier noise detection in a test setup.

Reflected power

Mismatches in the transitions between microwave components implythat some of the input power will be reflected. However, in a well-designed system, very little power is reflected. Components containinghigh Q-value parts, like e.g. cavity resonators, are only well matchedat certain frequencies and minor changes in the component propertiescan lead to detuning of the part. Multipactor is known to be able todetune high Q-value components [70]. Thus the reflected power froma component can be used as an indication of a multipactor event. Tostudy the absolute value of the reflected power is usually not a good wayof detection, since the input power can vary during a multipactor testand consequently the reflected power will vary as well, as the reflectedpower is a fixed fraction of the input power. This fraction is called thereturn loss and is commonly measured in decibel. The return loss will bea stable value, insensitive to power fluctuations, until the component isdetuned. Figure 6.3 shows an example where both close-to-carrier noiseand the return loss are monitored simultaneously during a multipactortest. Both methods indicate a change around t = 50 s and thus it canbe determined with great confidence that a multipactor discharge wasinitiated at that time.

The main advantage with this method is that it is quite reliable andthere is little risk that other phenomena will cause a mismatch that canbe confused with multipactor. However, for low Q-value components orbadly matched systems, the sensitivity of the method is low.

Electron monitoring

A new global method of detection was presented at the 4th Interna-tional Workshop on Multipactor, Corona and Passive Intermodulationin Space RF Hardware [71]. It is called the Electron Density Detec-tion Method, abbreviated EDDM, and uses a set of tri-axial cables asa probe and electrons picked up by the probe are then monitored witha high precision electron meter. The data is collected using a computer

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0 10 20 30 40 50 60 70 80 90 100

−90

−80

−70

−60

Noi

se (

dBm

)

Time (s)

0 10 20 30 40 50 60 70 80 90 100

42

44

46

Inpu

t Pow

er (

dBm

)

Time (s)

0 10 20 30 40 50 60 70 80 90 100−16

−14

−12

−10

−8

Ret

urn

Loss

(dB

)

Time (s)

Figure 6.3: Multipactor monitored by studying close-to-carrier noise as wellas the return loss. At t ≈ 50 s the component starts to experiencemultipactor discharge and becomes increasingly mismatched. Ex-cellent agreement with the second detection methods, close-to-carrier noise, can be seen.

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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 wherethe discharge will take place. Even though it is advantageous for highersensitivity to have the probe pointing at the critical gap it is also reli-able when located outside the device under test and thus it can be usedfor waveguides and coaxial transmission lines. It is not completely clearfrom the paper, however, how the electrons escape from a completelyconfined device like a typical waveguide or coaxial cable and one willhave to assume that there must be some small opening somewhere inthe system, where electrons can leak out.

Furthermore, the method can be used to quantitatively measure theamount of generated electrons. however, this requires some kind of cali-bration of each test setup and that may prove to be problematic. Whenused 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 ithas become a local method. Among the main advantages of the methodis the low cost involved, since no expensive microwave instruments areneeded.

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 bombard-ment during a multipactor event. The gas molecules consist of resid-uals of water, air and contaminants, and using a mass spectrometer,the different molecules can be identified. It has been noted [31] that adetectable increase in the water spectrum can be seen during a multi-pactor discharge. The major drawback of this method of detection isits inability to detect fast multipactor transients (not enough moleculesare released from the walls) and thus it is not a suitable method formulticarrier multipactor studies. Another disadvantage is that there isa certain delay between onset of the discharge and indication in the in-strumentation. However, it can be useful as a diagnostic tool togetherwith 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 adischarge in the system, but also to determine the exact position, localmethods of detection will have to be used. The two most common local

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methods are charge detection using a probe and optical monitoring andthese two methods will be described below.

Charge probe

A special case of the method of electron monitoring, which was describedin a previous subsection, is the use of a probe that monitors only acertain position within the microwave device. A very common approachused when detecting electrons inside a waveguide is to flush mount anSMA connector and apply a positive potential to the centre pin. Thenegatively charged electrons are attracted to the pin and causes a small,but detectable, current to flow, which can serve as an indication of theelectron density.

The method is easy to implement and therefore it has been quitecommon in many test setups. Unfortunately, it is also quite slow dueto the circuit used to amplify the weak current [14] and thus it is notfeasible for measuring fast multipactor events. In addition, it requiresmodification of the component, which makes it useless when testing flighthard ware. In such a case the EDDM is a better choice.

Optical detection

Optical detection is possible since the electrons that make up the multi-pactor discharge can excite or ionise either the remaining gas moleculeswithin the device or the molecules in the device wall. It can be dividedinto two groups, viz. photon detection via optical probe and photondetection via photographs or video camera [72, 73]. Both methods arecommon, but the former seems to be more frequently used.

The main advantage with these methods is that they can be used topin-point the location of the discharge inside the device. A major dis-advantage is that they may be impossible to use for studying real parts,as the devices may not have any suitable openings. This is especiallytrue for the methods based on photographic techniques.

6.2 Detection using RF Power Modulation

During verification of a test setup for multipactor at Saab EricssonSpace, Goteborg, Sweden, an odd, spike-like phenomenon was foundin the noise generated by a discharge in a coaxial test sample. An ad-

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ditional test sample, a resonant cavity, was manufactured and the sametype of spikes were noted again (cf. Fig. 6.4).

92.82 92.84 92.86 92.88 92.9 92.92 92.94 92.96 92.98 93

−90

−85

−80

−75

−70

−65

−60

−55

Noise Power (92.8 − 93.0 s)

(dB

m)

Time (s)

Figure 6.4: Periodic spikes, which appeared during a multipactor experiment.The main periodicity is 100 Hz and emanates from the powersupply of the high power amplifier.

A fast Fourier transform revealed that the spikes were periodic andit was noted that the same periodicity could be found also in the inputsignal after it had been amplified by the TWTA (travelling wave tubeamplifier). Due to interference from the main power supply, the signalwas amplitude modulated with a main modulation frequency of 100 Hzand with harmonics at multiples of this frequency. The interference wasvery weak and would in most cases be disregarded. In order to see ifthe periodicity was present only in conjunction with multipactor events,a large number of test runs were performed. The results were consis-tent - 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-periodicbefore onset of multipactor but periodic directly afterwards.

The AM (amplitude modulation) that was present in the input signalwas very weak and not deliberately added. A stronger AM was added tothe signal before the high power amplifier, resulting in a more distinctmodulation. This made it possible to study if there was any correla-tion between the modulation strength and the corresponding peak in

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34 36 38 40 42 44 46 48 50 52

−90

−85

−80

−75

−70

−65

−60

−55Noise Power (32 − 52 s)

(dB

m)

Time (s)

0 50 100 150 200 250 300 350 400 450 500−160

−140

−120

−100

−80Fourier transform (32 − 52 s)

(dB

m)

frequency (Hz)

Figure 6.5: The beginning of a multipactor test sequence showing the timebefore onset of the discharge. The FFT (fast Fourier transform)gives no indication of dominant frequency components.

54 56 58 60 62 64 66 68 70 72

−90

−85

−80

−75

−70

−65

−60

−55Noise Power (52 − 72 s)

(dB

m)

Time (s)

0 50 100 150 200 250 300 350 400 450 500−160

−140

−120

−100

−80Fourier transform (52 − 72 s)

(dB

m)

frequency (Hz)

Figure 6.6: The end of the same test sequence as in Fig. 6.5. The sudden in-crease in the noise floor indicates onset of multipactor. The FFTshows dominant frequency components at 100 Hz and multiplesthereof.

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the FFT. It was concluded that there is a positive correlation betweenthe modulation depth and the strength of the detected signal at thecorresponding frequency. By taking the time average of one of the testsequences 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 betweenthe 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.

50 55 60 65 70 75 80 85 90 95 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

Linear Plot of Noisepower vs Input Signal Power

Noi

se P

ower

[nW

]

Time (s)

26 28 30 32 34 36 38 40 42 44

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Input Signal Power [W]

Figure 6.7: Time average of the test sequence shown in Fig. 6.3 with a linearscale on both axes. The straight line has been added to show theclose to linear relationship between input power and noise power.Note: The input power is increased every 4 seconds, which is thereason for the step like behaviour.

The reason why the small modulation was so noticeable in the mul-tipactor noise (see Fig. 6.4) is that the noise signal is a function of thedifference between the input power and the multipactor threshold, i.e.no discharge noise is generated until the multipactor threshold has beenreached. Furthermore, since the decibel scale is a relative scale, the smallincrease in absolute numbers becomes very noticeable in relation to theexisting noise floor. As a comparison, the first small steps in Fig. 6.7,

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correspond to huge increases in the decibel scale, which can be seen inFig. 6.8.

0 10 20 30 40 50 60 70 80 90 100

−85

−80

−75

−70

−65

−60

Noise Power (100 samples noise average)

(dB

m)

Time (s)

Figure 6.8: The same sequence as in Fig. 6.7 (except that in this case theentire sequence is shown). The small initial steps of Fig. 6.7become huge steps on the logarithmic scale.

The above examples, which used the new detection method, reliedon close-to-carrier noise measurement data. However, the mechanismwhich is utilised requires primarily that the input signal is amplitudemodulated, that the detected signal is proportional to the differencebetween the input power and the multipactor threshold and that thedetected signal responds quickly to changes in the multipactor event.The two first conditions are likely to be fulfilled by all detection methodsfor multipactor, but the last will have to be verified for each method.Results from measurements presented in [14] show that third harmonicdetection is faster than close-to-carrier noise detection, which shouldmake it excellent for AM detection. In general, probably any methodcan be used provided that a suitable AM frequency is chosen and thatthe instrument used for detection has a sampling frequency that is morethan two times larger than the modulation frequency in order to fulfilthe Nyquist criterion.

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6.2.1 Single carrier

When using the AM detection method in the single carrier case, thetest setup implementation is quite straight forward. In practice, theonly difference between a standard multipactor test setup and one thatuses the AM detection method is that the signal data must be sent toa computer or some other instrument that can perform a FFT. It isalso important that the signal generator is capable of producing an AMsignal, but that is a standard feature of most signal generators.

The spectrum analyser, which receives the signal, should be set to asampling rate that is at least two times larger than the AM signal, i.e. ifthe AM signal has a frequency of 1000 Hz, then a minimum sampling rateof 2000 samples/second should be used. However, in order to study theshape of the modulation signal, it is better to use a sampling frequencymore than twenty times greater than the frequency of the AM. The signaldata can be stored for future processing, or if a powerful computer isused, real time FFT can be performed, thus allowing the operator to getan immediate indication when a discharge takes place.

6.2.2 Multicarrier

AM detection in the multicarrier case [74] is somewhat more difficult andthe method has not yet been experimentally confirmed. When perform-ing a multicarrier multipactor test, the phase of each carrier will have tobe adjustable in order to enable the test engineer to produce the wantedshape of the signal envelope. The aim is to produce a signal envelopecorresponding to the assumed worst case. When the phases have beenset to their predefined values, the envelope will be periodic. If the enve-lope exceeds the multipactor threshold for a time long enough to allow asufficient number of gap crossings, a discharge will occur, provided thata suitable seed electron is available. When the envelope drops belowthe threshold again, the electrons will disappear quickly [21] and therewill normally be no electrons left from the discharge the next time theenvelope exceeds the threshold. If no specific source of seed electrons isavailable, a multipactor discharge may not occur every envelope period.However, if an efficient electron seeding source is used, there should bean ample amount of electrons available to initiate a discharge each timethe envelope exceeds the threshold for a sufficiently long time.

The use of AM detection of multipactor requires that the dischargeevent is continuous or that it occurs regularly. Single or very sporadi-

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cally occurring discharges will be difficult to identify in a FFT plot. Byusing a source of free electrons in the test setup, e.g. a hot filamentor a UV light [14], seed electrons will be abundant and a multipactorevent is likely to occur each time the envelope exceeds the threshold fora long enough time. If the envelope is amplitude modulated, the dis-charge events will vary in strength with the AM frequency. This periodicvariation will appear as a peak in the FFT plot of the detected signal atthe same frequency. By applying a weak, 1-5% depth, synchronised AMto the input signals, the multipactor threshold can be determined withhigh accuracy and without risking ambiguous test results. A 1% AMcorresponds to less than ±0.1 dB variation in the input signal and willhave no significant effect on the measured threshold.

The detected signal must then be processed by a computer or a sim-ilar 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 toachieve a good AM in the multicarrier case, all the signals should bemodulated using the same reference signal of modulation. Many signalgenerators have a signal reference input, thus allowing the user to syn-chronise several signal generators. To perform a successful multicarrierexperiment, the phases have to be stable in relation to each other andthus a common reference signal will have to be used in any case. An-other way of achieving synchronised modulation could be to modulatethe gain adjustment of the high power amplifier.

In Fig. 6.9 it is suggested that the third harmonic should be moni-tored and that is probably the best choice when studying multicarriermultipactor, since third harmonic detection is fast and sensitive. Close-to-carrier noise detection is a possible alternative, but it may not be assensitive and thus weak multipactor events may be overlooked.

6.2.3 Main achievements

A method of multipactor detection has been devised, which can be usedto obtain accurate and unambiguous measurement results for both singleand multicarrier multipactor. The method does not aim to replace any ofthe existing methods of detection, rather it can serve as a complementto the other methods to improve accuracy and confidence in the testresults.

Close-to-carrier noise and third harmonic detection are two fast andsensitive methods of multipactor detection. Both methods rely on noisegeneration, which makes them prone to non-multipactor generated noise.

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HPA

VacuumChamber

DUTSpect.Analys.LNA

Third harmonic detection

PowerMeter

PowerMeter

Reflected power

dBdB

Forward power

Instrument control&

Data storage and processing

∆φ

∆φ

MUX

Phasecontrol

Reference signal generator

dB

Spect.Analys.

Wave form monitor

Figure 6.9: Test setup for multicarrier multipactor measurements using RFpower modulation. Each input signal is amplitude modulated andby phase locking the signals using a signal reference, the entiresignal envelope will be modulated. A computer can be used tocontrol the instruments and collect the output data, which canthen be real-time Fourier transformed to reveal any periodicityin the detected signal.

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In a typical test setup there can be many sources of noise, which couldresult in ambiguous test results. On the one hand, the multipactorthreshold could be established at a too low value if non-multipactorgenerated noise is misinterpreted as the result of a discharge. On theother hand, a short-lived multipactor event could be disregarded andlead to determination of a too high threshold based on a more distinctindication. The AM detection method resolves this concern by onlysignalling for true multipactor events.

Another advantage of the AM detection method is the fact that itis particularly sensitive close to the multipactor threshold, since it onlyresponds to the signal difference between the input signal and the thresh-old, 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 themultipactor threshold, will produce a very distinct modulated outputsignal as indicated in Fig. 6.11. Even though the signal is very noisy,the periodicity is very distinct. Not always will the periodicity be asnoticeable as in Fig. 6.11, but a FFT will reveal any periodicity in themeasured signal.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120Input signal envelope, single carrier, 1%depth AM

Am

plitu

de [A

.U.]

Time

Input signalMultipactor threshold

Figure 6.10: Example of a single carrier input signal envelope with a 1%depth AM. The signal has barely exceeded the multipactorthreshold.

One of the shortcomings of the AM detection method is that it re-

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

Detected signal

Am

plitu

de [A

.U.]

Time

Figure 6.11: Qualitative form of the detected multipactor signal when theinput signal to the DUT is as shown in Fig. 6.10. The same typeof signal can be seen in Fig. 6.4, but in this case, the samplingfrequency 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 minimumof 5-10 times the period of the AM is needed for a reliable FFT. If thefrequency of AM is 1 kHz, then the time needed for the measurementwould be 5-10 ms. In most cases this would be acceptable, but if singleevents of multipactor are to be detected, the method will not work.

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Chapter 7

Conclusions and outlook

This thesis has presented basic theory as well as new developments con-cerning the phenomenon called multipactor. Its deleterious effects onmicrowave systems operating in a vacuum environment have been em-phasised. For satellites, the discharge can be catastrophic as basically nomeans of repair or modification is available for parts in orbit. To avoidthe risks associated with vacuum discharges, lots of effort has been putinto studying the phenomenon and today well established engineeringmethods exist, which are used by the microwave engineer when designinghardware bound for space. However, the available methods are basedon the simple parallel-plate model, which in many instances is the worstcase. Thus, when designing important microwave features, in particularsuch involving a nonuniform electromagnetic field, like e.g. irises andcoaxial lines, the parallel-plate model may not be applicable and thereis a risk of unnecessarily conservative designs. Such designs are oftenlarge and heavy, which is a great disadvantage when it comes to devicesto be used in space. As a first attempt to establish new methods ofassessing the risk for having a discharge in such structures, a large partof 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 sec-ondary electrons in an iris, the threshold can be significantly increasedcompared with the pure parallel-plate case. Many interesting aspects ofthe phenomenon in a coaxial line have been found, e.g. the dual sta-ble regions of single-sided resonance and its unexpectedly low thresholdof the first order mode. In addition, the increased threshold for highimpedance coaxial lines that was found in experiments, was also foundin this study, both in the theoretical analysis and in the PIC-simulations.

111

At altitudes where most satellites operate, the pressure is very lowand for most purposes it can be approximated as a perfect vacuum.However, during the launch phase of a satellite and during its first daysof operation as well as at times when the satellite fires its attitude andaltitude control engines, the microwave parts may not be completelyvented and thus it is important to understand what happens with themultipactor threshold with increasing pressure. This has been one of themain topics of this thesis. It was found that for materials with a low firstcross-over energy, for the lowest order resonance, the threshold increaseswith increasing pressure until reaching a maximum, after which it startsto decline. The reason for the increase is the friction force experiencedby the electrons when colliding with the neutral gas particles. In allother cases, for materials with a high first cross-over energy and in gen-eral for the higher order modes, a monotonically decreasing thresholdis noted. This behaviour can be explained by the thermalisation of theelectrons, which leads to a higher total impact energy, as well as thecontribution of electrons from collisional ionisation, which reduces thenecessary secondary emission yield and consequently also the requiredimpact velocity. Improved quantitative results require a more detailedinvestigation of the fraction of the electrons from impact ionisation thatactually contribute to the multipactor bunch, but that is left for a futurestudy. Furthermore, in order to make the model useful for all frequency-gap size products, an extension of the model to include also the hybridmodes is necessary. Due to the complexity of such a study, it was notincluded in this first analysis. However, this may be an interesting topicfor a future investigation.

In addition to designing with respect to the proper thresholds, mosthardware will require some type of testing to ensure compliance withthe standard. Such tests must be of good quality to avoid ambiguities,which could disqualify a component that is multipactor free. In thisthesis a method that gives unambiguous and highly reliable test resultswas presented. It is a method based on a weak amplitude modulation ofthe input signal, which becomes very distinct in the multipactor signal,since the generated noise power is a function of the difference betweeninput power and the multipactor threshold. By using this auxiliarymethod 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 workin my licentiate thesis [75] and realizing that the two main topics men-

112

tioned there were multipactor in nonuniform fields and irises, which werethen successfully studied during the second part of my PhD work. How-ever, there are still interesting projects to consider, e.g. multipactor inirises where not only the effect of the random walk is considered, but alsothe effect of the actual nonuniform field in the structure. There are manyother important structures in microwave systems, which have not beenstudied with respect to multipactor discharge. Among these are crossedirises, septum polarisers and ridged waveguides. The 20 gap-crossingsrule, which is part of the ESA standard [50], should be re-investigatedboth theoretically and experimentally to make sure that the rule hasa sound theoretical base with good agreement between simulations andexperiment. As a continuation of the coaxial study, the axial dimensioncould be included by considering not only a travelling wave signal butalso a standing wave. Since electrons affected by the Miller force willdrift towards positions with low field amplitude, it may not be feasibleto directly apply the peak amplitude of the standing wave in the coaxialmodel presented in this thesis.

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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 forDetection 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, “Mul-tipactor 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, “Im-proved 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, “Mul-tipactor 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, “Mul-tipactor in a coaxial transmission line, part I: analytical study”, acceptedfor publication in Phys. Plasmas

Paper F

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