Thesis for the degree of Doctor of Philosophy. Multipactor in Low Pressure Gas and in Nonuniform RF Field Structures.

Transcription

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

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

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

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

6 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 Multipactor Breakdown in RF Equipment, RVK 02, June 10-12, 2002, Stockholm, Sweden. [H] R. Udiljak, D. Anderson, U. Jostell, M. Lisak, J. Puech, V. E. Semenov, Detection of Multicarrier Multipaction using RF Power Modulation, 4th International Workshop on Multipactor, Corona and Passive Intermodulation in Space RF Hardware, 8-11 September, 2003, ESTEC, Noordwijk, The Netherlands. [I] J. Puech, L. Lapierre, J. Sombrin, V. Semenov, A. Sazontov, N. Vdovicheva, M. Buyanova, U. Jordan, R. Udiljak, D. Anderson, M. Lisak, Multipactor threshold in waveguides: theory and experiment, NATO Advanced Research Workshop on Quasi-Optical Control of Intense Microwave Transmission, February, 2004, Nizhny-Novgorod, Russian Federation. [J] R. Udiljak, D. Anderson, M. Lisak, J. Puech, V. E. Semenov, Microwave breakdown in the transition region between multipactor and corona discharge., RVK 05, June juni, Linköping [K] D. Anderson, M. Buyanova, D. Dorozhkina, U. Jordan, M. Lisak, I. Nefedov, T. Olsson, J. Puech, V. Semenov, I. Shereshevskii, R. Tomala, and R. Udiljak, Microwave breakdown in RF equipment., RVK 05, June juni, Linköping [L] V. E. Semenov, N. Zharova, R. Udiljak, D. Anderson, M. Lisak, J. Puech, and L. Lapierre, Multipactor inside a coaxial line, 5th International Workshop on Multipactor, Corona and Passive Intermodulation in Space RF Hardware, September, 2005, ESTEC, Noordwijk, The Netherlands. [M] R. Udiljak, D. Anderson, M. Lisak, V. E. Semenov, and J. Puech, Microwave breakdown in low pressure gas, 5th International Workshop on Multipactor, Corona and Passive Intermodulation in Space RF Hardware, September, 2005, ESTEC, Noordwijk, The Netherlands. [N] C. Armiens, B. Huang, R. Udiljak, D. Anderson, M. Lisak, U. Jostell, and P. Ingvarsson, Detection of Multipaction using AM signals, vi

7 5th International Workshop on Multipactor, Corona and Passive Intermodulation in Space RF Hardware, September, 2005, ESTEC, Noordwijk, The Netherlands. vii

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9 Contents Publications Acknowledgement Acronyms v xi xiii 1 Introduction 1 2 Multipactor in vacuum Single Carrier Basic theory Hybrid modes Factors effecting the threshold Methods of suppression Effect of random emission delays and initial velocity spread Multicarrier Design guidelines Single carrier Multicarrier Multipactor in low pressure gas Simple Model Model Multipactor boundaries Main results Advanced Model Model Analytical formulas for argon cross-sections Multipactor boundaries ix

10 3.2.4 Key findings Multipactor in irises Model and approximations Multipactor regions Comparison with experiments Main results Multipactor in coaxial lines Analytical study Model Multipactor resonance theory Main findings Particle-in-cell simulations Numerical implementation Simulations Comparison with experiments Main conclusions Detection of multipactor Common Methods of Detection Global methods Local methods Detection using RF Power Modulation Single carrier Multicarrier Main achievements Conclusions and outlook 111 References 115 Included papers A F 123 x

11 Acknowledgement I wish to thank Prof. Dan Anderson and Prof. Mietek Lisak for accepting me as a PhD student and for guidance and support in my daily work. I also want to thank Prof. Vladimir Semenov at the Institute of Applied Physics in Nizhny Novgorod, Russia, for fruitful discussions and for his patience with all my questions. Thank you Prof. Lars Eliasson, Director at the Institute of Space Research in Kiruna, for providing both financial and moral support making my PhD candidate appointment possible. A very warm thank you also to Jerome Puech for many interesting discussions about space related microwave problems and to his employer, Centre National d Études Spatiales, for financial support. Thanks to my friends in Toulouse and especially Dr. Omar Houbloss and Raquel Rodriguez. I thank my fellow members of the National Graduate School of Space Technology and my colleagues at Chalmers and especially Dr. Pontus Johannisson, Dr. Ulf Jordan and Dr. Lukasz Wolf for beneficial discussions and lots of support with Linux and LaTex. Many thanks also to our secretary Monica Hansen for guiding me through the administrative jungle. I want to thank my dear mother, Monica, for encouraging and supporting me and my family when 24 hours a day wasn t enough and my father, brother and sisters for believing in me. I am also grateful to my unofficial mentors: my father-in-law Lars-Göran Östling, my friends Jörgen Otbäck and Anders Wilhelmsson, and my brother-in-law Nicklas Östling. Most of all I thank my wife Malin and our daughters Janina and Lizette for encouragement and support during this time. xi

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

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

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

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

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

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

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

21 Figure 2.1: Initial stage of parallel plate multipactor, where a free electron is accelerated by the electric field and is forced into one of the plates, where it causes emission of secondary electrons. The motion of an electron in vacuum with an applied electric field can be studied by means of the equation of motion, mẍ = ee (2.1) where m ( kg) and e ( C) are the mass and charge of the electron, x the direction of motion, and E the electric field. Multipactor requires an alternating field and in the parallel-plate model a spatially uniform harmonic field E = E 0 sin ωt is assumed. Solving Eq. (2.1) with this field yields expressions for the velocity, ẋ, and the position, x, ẋ = ee 0 cos ωt + A (2.2) mω x = ee 0 sin ωt + At + B (2.3) mω2 where A and B are constants of integration, which will be determined by the initial conditions. By assuming that an electron is emitted at x = 0 with an initial velocity v 0 when t = α/ω, fully constrained expressions for the velocity and position are obtained, viz. ẋ = ee 0 mω (cos α cos ωt) + v 0 (2.4) x = ee 0 ( ) v 0 sinα sin ωt + (ωt α)cos α + mω 2 (ωt α) (2.5) ω For resonant multipactor to occur it is necessary for the electron to reach the other device wall (x = d) when ωt = Nπ + α, where N is an 7

22 odd positive integer (N = 1,3,5...). Applying this resonance condition to Eq. (2.5), the following expression is obtained for the amplitude of the harmonic electric field. E 0 = mω(ωd Nπv 0) e(nπ cos α + 2sin α) (2.6) An important quantity when studying multipactor is the impact velocity, since this determines the secondary electron yield. It can be found by inserting ωt = Nπ + α in Eq. (2.4), which yields Multipactor boundaries v impact = 2eE 0 mω cos α + v 0 (2.7) When constructing the multipactor boundaries, i.e. the boundaries of the regions in parameter space where multipactor can occur, an assumption will have to be made concerning the initial velocity. In reality, the initial velocity of the emitted electrons will follow some kind of distribution and a common choice is the Maxwellian distribution [21], f(v) exp ( (v v m) 2 ) 2v 2 T (2.8) where suitable parameters for the mean velocity, v m, and for the rmsvalue (or thermal spread), v T, have to be chosen. When performing particle-in-cell (PIC) simulations, such a distribution can be used to more accurately describe the initial velocity of the emitted electrons. However, for an analytical solution a simpler assumption will have to be made. There are two common approaches, one which assumes that the electrons are emitted with a constant initial velocity, v 0, regardless of the impact velocity. The other assumes that the ratio between the impact and initial velocities is equal to a constant, k = v impact /v 0. Both these approaches will be used and compared in this chapter, but in the following chapter, which deals with multipactor in low pressure gas, the constant k approach will be used only for the simple model while the constant initial velocity approach will be used for the more advanced model as that assumption is more physically correct. The reason why the constant k model has been used to such a great extent is the fact that it can successfully be fitted to experimental data. The cause of this success will be explained in the subsection on hybrid modes below. 8

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

24 2.5 Secondary electron yield 2 σ se [ ] σ se =1 0.5 W 1 W max W Primary electron energy [ev] Figure 2.2: Secondary electron yield as a function of the impact energy. Plotted using the formula for secondary electron yield presented in Ref. [22]. Parameters used are: W max = 400 ev, σ se,max = 2, and W zero = 10 ev. 10

25 solution or some kind of approximate solution will have to be used. In Ref. [21] an approximate formula for the non-returning electron limit is given and re-writing it for the constant v 0 approach yields, α min = 16v 0 5v 0 + 3v impact (2.11) Using Eqs. (2.6), (2.9), and (2.11) with v 1 equal to the velocity corresponding to the unity secondary electron yield, the lower multipactor threshold can be plotted. However, multipactor breakdown is possible also for impact velocities greater than v 1, in fact, for all impact velocities between the first and second (v 2 ) cross-over points, the phenomenon can occur, i.e. for v 1 < 2V ω cos α + v 0 < v 2, (2.12) where Eq. (2.7) has been re-written using the oscillatory velocity V ω = ee 0 /mω. Thus, in order to construct the complete multipactor boundaries, the thresholds for a number of different energies between these two points should be determined and then the envelope of all the thresholds will be the complete multipactor susceptibility zone (see Fig. 2.3). Furthermore, each order of resonance, N, will have its own zone and, as shown in Fig. 2.3, the zones become narrower with increasing N. This type of chart, based on the assumption of constant initial velocity, will be referred to as a Sombrin chart, since J. Sombrin is one of the major advocates of this assumption [29]. Using the other approach, with a constant ratio between the impact and initial velocities, k = v impact /v 0, 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 v 1 (k + 1)N π 2 ) (2.13) mω 2 d E 0 = e( k+1 k 1 Nπ cos α + 2sin α) (2.14) 16 α min = (2.15) 8 + 3(k 1) Using these formulas, multipactor charts similar to the one in Fig. 2.3 can be produced, cf. Fig When this is used, the charts are commonly referred to as Hatch and Williams charts, as they were the first 11

26 10 4 Voltage [V] N=3 N=5 N=9 N=7 N= Frequency Gap product [GHz mm] Figure 2.3: Multipactor susceptibility chart based on the constant initial velocity approach. Parameters used are: W 0 = 3.68 ev, W 1 = 23 ev, W 2 = 1000 ev, and N max = 9. 12

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

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

29 Phase-focusing In the previous subsection a mechanism called phase-focusing [1, 5, 32] was mentioned and in multipactor theory this is an important concept. In order for an electron to be a part of the discharge, it must have a phase close to the resonant phase as given by Eq. (2.9) or (2.13). Due to delays between the impact and emission of a new electron or a spread in the initial velocity, an electron will always acquire a small phase error. Inside the phase-focusing range, such an error will decrease as the electron traverses the electrode gap. In other words, the phases of the electrons will tend to converge towards the resonant phase, thus keeping all electrons close together. Outside the range of phase-focusing, the error will grow with each passage and after one or a few transits the electron will be lost. In order for a discharge to occur under such circumstances, the impact energy has to be large enough to produce a secondary yield sufficiently above unity to compensate for the incurred losses. To see in what range the phase focusing mechanism is active, a small phase error can be introduced in Eq. (2.5) while keeping the amplitude and phase constant and setting x = d. The ratio between the final and initial error is called the stability factor, G [33], and the condition for stable phase is: G < 1 (2.17) By setting G = 1, the phase range within which the phase is stable can be obtained. An interesting observation here is that even though the lower multipactor threshold for the constant k theory and the constant initial velocity model are identical, the range of stable phases varies substantially [34]. This can be seen clearly from the analytical expressions for the phase stability limits, which for constant k theory reads, φ R = arctan( 2 πn (v impact v 0 v impact + v 0 )) (2.18) φ L = arctan( 2 πn ) (2.19) and for the constant initial velocity approach, φ R = arctan( 2 πn ) (2.20) φ L = arctan( 2 πn (v impact + v 0 v impact v 0 )) (2.21) 15

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

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

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

33 and the field strength is increased at regular intervals. If no electron is in an advantageous position, i.e. has a suitable phase from a multipactor point of view, when the right amplitude is set, a discharge will not occur. As the amplitude is increased further, the impact velocity of any free electrons, also those that are not in a favourable position, will be high and the secondary yield will be an additional source of free electrons. Thus the chances of getting a breakdown increases until it eventually occurs. For experimental use, a hot filament or a radioactive source can be used to produce a sufficient amount of free electrons to achieve reliable measurement results [14]. Another factor that can have a significant effect on the threshold is contamination. Both the first cross-over point, W 1, and the maximum secondary yield, σ se,max, can be drastically affected. A lowered W 1 means that a discharge can occur at a lower voltage and an increased σ se,max can result in a faster growth of the total number of electrons. In Ref. [31] a detailed analysis of the impact of different types of contaminants was made. It was noted that the plastic bags, which were normally used to protect the microwave components from dirt, were the main source of contamination. A threshold reduction of up to 4 db was found. Also dust and fingerprints were a direct source of a lowered threshold. In the report [31] it was recommended that cleaned microwave parts for space use should be handled with cotton gloves and stored in hard plastic boxes. Microwave parts which have not been properly vented before power is applied can also have a threshold that is different from the expected multipactor threshold. If there is too much gas, corona breakdown may occur, and within a certain range of pressures, close to the minimum of the so called Paschen curve, the breakdown threshold can be significantly lower than in the multipactor case. In the pressure range corresponding to the transition region between corona and multipactor, a higher threshold can sometimes be expected. More details about this will be presented in chapter 3. Other factors that can have an indirect effect on the multipactor threshold are the voltage standing wave ratio, VSWR, and the temperature. If the VSWR is greater than what was intended with the design, the peak field strength in the system will also be greater than expected and thus a discharge may occur at a lower power level than assumed. An increased temperature can lead to increased outgassing from the device walls resulting in concerns similar to those of improper venting. 19

34 2.1.4 Methods of suppression Many of the factors mentioned in the previous section that affect the multipactor threshold can also be utilised to suppress the discharge. The without doubt easiest method of avoiding a breakdown is to pressurise the component. The field strength required to achieve breakdown at atmospheric pressure is in general much higher than at low pressures or in vacuum. However, such a method is seldom feasible for components that will be used e.g. in space, where the external environment is a high vacuum. A small leakage can lead to slow venting of the component and thus risking severe corona discharge when the pressure reaches the range where the minimum breakdown field occurs. Another way of suppressing multipactor is to amplitude modulate the main carrier [21, 41]. If both signals are sinusoidal, the total field can be written: E tot = E 1 sinω 1 t + E 2 sin ω 2 t (2.23) This means that the envelope of the signal will vary according to (see also Fig. 2.8): E env = E1 2 + E E 1E 2 cos (ω 1 ω 2 )t (2.24) When the total field strength is well above the multipactor threshold (see Fig. 2.8), the secondary electron yield will increase quickly according to Eq. (2.22). However, as soon as the voltage drops below the threshold again, the electron loss will be large and according to Ref. [21] all electrons will be lost in just a few RF cycles. However, whether or not this is true also depends on the secondary yield properties of the electrode material. For materials with a very high maximum secondary yield, the number of electrons gained while above the threshold can be greater than the losses incurred while below. In such a case, no suppression is achieved and in some cases, the discharge may even become more powerful than before the modulation carrier was added [42] (cf. Fig. 2.9). Thus in order to successfully suppress a multipactor discharge using amplitude modulation, it is vital that the material has a low maximum secondary yield (preferably less than about 1.5). Due to the risk of contamination, which can greatly increase the maximum secondary yield, great care should be taken to assure a high level of cleanliness if this method of suppression is to be used. To AM-modulate the carrier is probably not feasible in most cases, as it would require extra hardware to produce the AM-signal. However, the 20

35 1.5 Amplitude Modulation, E0/E1=0.4 Multipactor threshold Sum signal Envelope Main signal Modulation signal 1 Amplitude Time Figure 2.8: Two signals and their sum signal (absolute values). The envelope varies and is partly above and partly below the multipactor threshold. typical bandpass filtering of a PSK (Phase-Shift Keying) signal causes modulations in the time-domain. The QPSK (Quadrature Phase-Shift Keying) signal in Fig has unity amplitude before filtering. Afterwards, the peaks are higher than the original amplitude, but the troughs can sometimes go down almost to zero amplitude. Comparing this with the AM-suppression, it is clear that an electron avalanche that is initiated during the peak periods, will be extinguished as the amplitude falls close to zero. However, for a typical PSK-signal, the duration between phase shifts (which normally coincides with the troughs) is several hundreds of RF cycles. Thus for most microwave systems, there will be ample time for a discharge to develop. But, when the amplitude drops below the threshold, the electron bunch will disappear and when the amplitude increases above the threshold again, there may not be any seed electrons present to restart the electron avalanche. Thus, the system will have sporadic discharges, which, if they do not occur too often, may not seriously degrade the signal. A very common way of suppressing a vacuum discharge is to apply a coating [26 28], a surface treatment [43], or a film [44] that has a high first cross-over point as well as a low maximum secondary electron 21

36 Amplitud Modulation, w2/w1=1.16) Noise (dbm) Power #1 (dbm) Match #1 (db) Power #2 (dbm) Time (ms) x 10 4 Figure 2.9: Multipactor experiment with two carriers with E 2 /E 1 = 0.36 and ω 2 /ω 1 = Due to a high maximum secondary yield, multipactor suppression is not possible (the material used in the experiment was plain aluminium, which can have a σ se,max 3). When the modulation signal is applied, the magnitude of the multipactor noise increases significantly. x 10 4 x 10 4 x 10 4 Square Root Raised Cosine filtered QPSK signal 1 Signal amplitude [ ] Time [µ s] Figure 2.10: Example of a QPSK modulated signal after bandpass filtering. 22

37 emission yield. So far, no practical coating with a σ se,max below unity has been found. However, alodine is a commonly used surface coating for space-bound microwave devices made of aluminium. It increases the first cross-over point to around 60 ev and reduces σ se,max to about 1.5, even though the actual values vary much between samples. The concern with a material with very good anti-multipactor properties is contamination. A few fingerprints or a very small layer of dust can drastically alter the properties of the material and make it prone to discharges. By applying a DC electric or magnetic field, the electron trajectory can be disturbed and the important resonance condition can be destroyed, thus making multipactor impossible. Simulations [45] have shown that an external DC magnetic field applied in the direction of wave propagation in a rectangular waveguide can efficiently suppress multipactor. A drawback with the method is the extra components required to produce the magnetic or electric field and thus the method may not be feasible for e.g. space applications, where extra weight is undesirable. The most efficient way of avoiding multipactor is to make a design where the mechanical dimensions are such that a power much higher than the nominal power is required to start a discharge. However, that may lead to large and heavy designs, which are to be avoided in space systems, and thus one may have to resort to one or several of the above mentioned methods of multipactor suppression Effect of random emission delays and initial velocity spread In the above analysis of multipacting electrons in a harmonic electric field, it was assumed that all secondary electrons were emitted with a fixed initial velocity, v 0. However, as briefly mentioned previously, the electrons are actually emitted with a distribution of velocities and the Maxwellian distribution is often used in simulations. Apart from the spread in initial velocities, there is also a finite time between impact of the primary electron and emission of the secondaries. Since this time in most cases is very small compared to the RF period, it was neglected in the previous analysis. Nevertheless, this time will cause a small phase error and if the resonant phase is close to the phase stability limits as given by Eqs. (2.18) - (2.21), the phase error may result in an increased electron loss. A detailed analysis of the effect of random secondary delay times and 23

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

39 10 4 Voltage [V] Frequency Gap product [GHz mm] Figure 2.11: Multipactor charts based on the same parameters as in Fig The solid line zones indicate the zones within which phasefocusing is active. The dashed line zone is produced by including also unstable phases until the non-returning electron limit. phase is very unstable, thus making a discharge impossible. A more correct boundary would be a rounded shape, which starts in the lower left corner of the solid line zone and smoothly joins the dashed left hand side [47]. This is confirmed by experiments [30,48], cf. Fig. 2.12, which shows measurement data from one of the early multipactor experiments by Hatch and Williams [48]. A similar rounded shape can also be seen in numerical simulations and examples of this is shown in chapter 5, which includes PIC simulations of multipactor in a coaxial line. 2.2 Multicarrier Modern satellites operate in multicarrier mode, i.e. several signals at different frequencies exist simultaneously in the microwave and electronic systems. An example of such a system is Sirius 3, which is one of the Nordic satellite [49]. It has 15 channels in the frequency range GHz and each channel has a bandwidth of 33 MHz. Assume that each channel has a power of 200 W. Then the maximum instantaneous power of the system, the peak power, is equal to 45 kw. The peak power increases with the square of the number of carriers. Such a high instan- 25

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

41 15 15 Amplitude [V] 10 5 Amplitude [V] Time [ns] Figure 2.13: In-phase multicarrier signal. The signal oscillates rapidly, which makes the signal envelope appear clearly in this time resolution Time [ns] Figure 2.14: Random phase multicarrier signal. taneous power is very unlikely in a real system since it will occur only when all the signals are in phase as illustrated by Fig The most likely scenario, if the carriers are not phase locked, is that the phase of each carrier is a random number and will result in a signal with much lower maximum instantaneous power as illustrated in Fig The signals in Figs 2.13 and 2.14 are characterised by all carriers having the same amplitude and a constant frequency. Consider a signal with N carriers, each carrier having the same amplitude E 0, but different phases φ n and with a frequency spacing f. The period of the envelope will then be T = 1/ f and the envelope is given by E env = E 0 ( N 1 2 cos (n2π ft + φ n )) + n=0 ( N 1 n=0 sin(n2π ft + φ n ) (2.25) A more realistic signal would have different amplitudes for each carrier and the frequency spacing would not be constant. The envelope of such a signal can be found from E env = ( N 1 2 E n cos (k n ω 0 t + φ n )) + n=0 ( N 1 n=0 E n sin(k n ω 0 t + φ n ) ) 2 ) 2 (2.26) 27

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

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

44 allows for another design margin, which is set with respect to the so called P 20 power level. The P 20 level corresponds to the peak power of the multicarrier waveform whose width at the single carrier multipaction threshold is equal to the time taken for the electrons to cross the multipacting region 20 times [50]. This level is illustrated in Fig Figure 2.15: An example where the in-phase peak power is above the single carrier threshold, while the P 20 level is more than 4 db below the same threshold. The peak voltage is V, the single carrier threshold is 91 V, and the P 20 voltage is 57 V. Signal data: 12 carriers, equally spaced, f min = GHz, f = 24 MHz and each carrier amplitude is 10.7 V. Material properties: W 1 = 23 ev and σ se,max = 3. In the case when a design is made with respect to the P 20 level, the design margins range from 4-6 db depending on the type of testing. A problem with the P 20 level is that it is not a trivial problem to find the peak power level for 20 electron gap crossings. This power level is usually referred to as the worst case scenario, even though it may not always be the worst case from a multipactor point of view. A number of different 30

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

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

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

48 carriers, because the NLSQ method requires a good seed phase in order to find a good local minimum. By combining these methods and using the result of the best random phase approach as a seed phase for the NLSQ optimisation, the best results are found. The problem with a random phase approach is that for a large number of carriers, the number of phase-sets needed to achieve a good result becomes discouragingly high [52]. By using a genetic algorithm, a great improvement in finding the worst case scenario in a short time has been achieved, especially for the non-equally spaced carrier case. This type of optimisation scheme has the advantage of being able to find not only a good local minimum as in the NLSQ case, but the actual global minimum can be found. The genetic algorithm was implemented by Merecki [40] and in addition to improving the optimisation part of the software, he also implemented the threshold of the hybrid modes (see Fig. 2.16) as well as many other useful functions. The main problem with multipactor in microwave systems is the electric noise that is generated, which degrades the signal to noise ratio. Thus the worst case may not always be the maximum power within the T 20 period. Depending on the material properties some other case may produce a substantially larger amount of energetic electrons. Therefore, WCAT also includes the possibility of finding the phase-set that will produce the largest amount of electrons. In addition to assessing the risk for a vacuum discharge, it can also be of value to investigate if a multipactor free design may have a risk of corona breakdown if the component is not thoroughly vented when it is brought into operation. In WCAT this is analysed using the mean carrier frequency and comparing the corona threshold with the in-phase peak voltage. If the minimum of the Paschen curve is greater than this voltage, then the corona margin is displayed in the output window of WCAT. If the opposite is true, the pressure range within which there is risk for corona discharge is presented (see Fig. 2.16). 34

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

50 multipactor discharge. Using a Monte Carlo algorithm, Gilardini [53] made quite a general study of the phenomenon and presented breakdown voltages normalised to the first cross-over point of the material for a wide range of dimensionless variables. He also paid special attention to a particular and realistic case, namely multipactor in low pressure argon [54]. This was done partly in an effort to compare the simulations with the experimental results of Höhn et al. [18]. In paper B of this thesis, low pressure multipactor was studied using an analytical model that takes into account only the friction force due to collisions between the electrons and the neutral gas particles. The main theory and results from this study will be presented in the first section below. In addition to the friction force, the collisions will also cause a random velocity spread of the electrons that results in a higher average impact energy. Furthermore, due to the long distance between molecules, the electrons are free to accelerate to very high velocities and upon impact with a gas molecule or atom the energy is sufficient to cause ionisation. In paper C of this thesis a more detailed analysis has been done, where all these effects have been considered and the used model as well some highlights from the results are presented in the section Advanced Model below. 3.1 Simple Model In a first attempt to understand the behaviour of multipactor in a low pressure gas, a simple analytical model was used, which takes only the friction force of the collisions with neutrals into account. By deriving explicit expressions for the multipactor threshold, qualitative comparison with experimental results [18] as well as results from computer simulations [53,54] could be made Model The differential equation governing the behaviour of the electrons in a low pressure gas is given by the equation of motion, Eq. (2.1), but augmented to include also the effects of collisions: mẍ = ee mν c ẋ (3.1) where ν c = σ c n 0 v is the collision frequency between the free electrons and the neutral particles. σ c is the collision cross-section, n 0 the neutral 36

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

52 the assumption of a constant ratio k = v impact /v 0 between impact and initial velocities. This is true also in the presence of collisions, when the above simple model is used and thus the constant k approach will be used in the following expressions. Combining Eqs. (3.5) and 3.6 under this assumption yields an expression for the resonant phase, viz. tan α = ω2 [βφ + γ] + 2ν 2 cv impact (Φ k) ξω(φ + 1) (3.7) where Φ = exp( ν c Nπ/ω), β = kdν c + (k + 1)v impact, γ = kdν c v impact (k + 1), and ξ = [kdν c + (k 1)v impact ]ν c. Equation (3.7) can be used together with Eq. (3.5) to plot the multipactor threshold in a low pressure gas as a function of gap size, pressure, or frequency. However, by multiplying the expression for the amplitude of the electric field with the gap size, d, an expression for the voltage as a function of the frequency-gap size and the pressure-gap size products can be obtained. This approach will be used in a subsequent section, where a more advanced model is used to analyse the phenomenon. Figure 3.1 shows the lower multipactor threshold in low pressure air for different pressures. The graphs are based on Eqs. (3.5) and (3.7) only and do not consider the non-returning electron limit nor the phase stability limits. From Fig. 3.1 it is clear that the multipactor threshold increases with increasing pressure. By sweeping the pressure instead of the frequency and comparing the changing threshold with the corona breakdown threshold, an understanding can be obtained of how the transition between these two types of discharges can occur. Fig. 3.2 shows that the multipactor threshold first increases until a certain point, where it intersects the curve corresponding to the corona threshold. It will then follow this curve towards the minimum of the Paschen curve. This is just a qualitative picture and the sharp intersection would in reality be a smooth transition. Figure 3.1 does not consider the non-returning electron limit, nor the phase stability limits. As explained in the previous chapter, phase focusing is needed to maintain the generated electrons in a close bunch, since an electron with a too large phase error will be lost. By introducing a small phase error in Eq. (3.2) and keeping the amplitude and phase constant while setting x = d, the error after the passage can be found. The ratio between the final and initial error is the stability factor, G, and when the absolute value of this factor is less than one, the phase focusing effect is active. With the present model, the expression for the 38

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

54 10 3 Threshold curves in low pressure air (fd=1ghz mm) Multipactor in air Multipactor in vacuum Corona in air Voltage (V) Pressure (Pa) Figure 3.2: Thresholds for multipactor and corona discharges in air as functions of pressure together with the multipactor vacuum threshold as a reference level. Parameters used are: σ c = m 2, W 1 = 23 ev, k = 2.5, N = 1, f d =1 GHz mm and d = 0.1 m. 40

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

56 3.1.3 Main results The main result found in paper B is that a higher microwave power is required to initiate breakdown in a low pressure gas, since the collisions tend to slow down the electrons. By combining the low pressure multipactor graph with the corona threshold curve, it was concluded that with increasing pressure, the required threshold will first increase and, after reaching a plateau, it will make a smooth transition to the low pressure branch of the Paschen curve. This behaviour is confirmed by the investigations made by Gilardini [53] for materials with a low first cross-over point, close to the ionisation energy of the gas, and for N = 1, i.e. for the first order of multipactor. For materials with a higher first cross-over energy and for higher order multipactor, Gilardini found no initial increase in the multipactor threshold, instead a monotonically decreasing breakdown voltage was seen. A possible explanation for the differences between the result obtained by the simple analytical model and the result of Gilardini is also presented in paper B and it is suggested that the reason is that for materials with a higher W 1 and for N > 1 the contribution of electrons from impact ionisation decreased the required W 1 (see Fig. 3.4). However, as will be seen in the next section, electron contribution from impact ionisation is not the only reason for this behaviour. The collisions will also cause an electron velocity spread, which will result in a larger total impact velocity and thus a lower voltage is needed to achieve the necessary first cross-over energy. A comparison was made with experiments by Höhn et al. [18] in low pressure argon as well as with PIC-simulations by Gilardini [54] in the same gas (see Fig. 7 in paper B). However, the fd-product chosen by these authors was located in the middle of the right boundary of the first multipactor zone, an area dominated by the hybrid modes [39]. The simple model used in the presented analytical approach is not applicable to these modes and consequently the behaviour found in the experiments and simulations could not be confirmed. Furthermore, the impact energy of the electrons at this f d-product is several times higher than the ionisation energy of argon and thus a significant contribution of electrons from collisional ionisation would be expected. In addition, the required W 1 would be reduced due to the electron velocity spread and thus a behaviour similar to curves (b) or (c) in Fig. 3.4 should be expected and it is also what is found. To further analyse multipactor in a low pressure gas, a better ana- 42

57 Threshold curves in a low pressure gas a Voltage (V) b c Multipactor in a low pressure gas without ionisation Multipactor in vacuum Corona Pressure (Pa) Figure 3.4: Qualitative form of the dependence of breakdown threshold with pressure in the region between which multipactor and corona, respectively, dominate the breakdown process: (a) the friction force due to collisions with neutrals dominates, (b) electron velocity spread reduces the required W 1 and collisional ionisation contributes significantly to the total number of electrons, (c) intermediate situation. 43

58 lytical model is obviously needed. The model must, in addition to the friction force, be able to take collisional ionisation into account as well as collision induced velocity spread of the electrons. In the next section a more advanced model that includes all these effects will be presented. 3.2 Advanced Model The simple model used in the previous section provided important qualitative understanding of the multipactor threshold behaviour in a low pressure gas. However, due to the inherent limitations of the model, some of the results found by other researchers could not be confirmed. This section will present an improved model for multipactor in a low pressure gas and it is based on paper C of this thesis. As a representative gas, the noble gas argon will be used in the included examples Model Just as in the simple model, the basic geometric configuration is electron motion between two parallel plates perpendicular to the x-direction. During the passage, no electron loss, only generation through collisional ionisation, will occur. Using the differential equations for the total electron momentum and for the change in the number of electrons, one can derive the following equation for the electron drift acceleration: du dt = ee m u(ν c + ν iz ). (3.9) where u is the drift velocity and ν iz the ionisation frequency. In general, the collision and ionisation frequencies are functions of the electron velocity. However, by assuming that ν c and ν iz are constants, Eq. (3.9) becomes a first order linear differential equation. Multipactor requires an alternating driving electric field and as in the previous model a harmonic field E = ˆxE 0 sin ωt is used, where ˆx is the unit vector, ω the angular frequency, and t the time. Assuming the electric field to be homogeneous, the drift velocity will be parallel to the field, u = ˆxu, and the vector notation for E and u can be dropped in the following analysis. By setting ν = ν c + ν iz, u = ẋ, and du/dt = ẍ, Eq. (3.9) can be written, ẍ = ee m ẋν. (3.10) 44

59 Since the equation has the same form as Eq. (3.1), it will also have the same solutions and as the initial conditions are identical, the formulas for the resonant field amplitude and the impact velocity will be identical. However, it should be noted that instead of ν c one will have ν and v 0 should be replaced by u 0. An important difference is that the velocity in the previous model was only directed in the x-direction, but now there is also a thermal velocity component, v t, i.e. the total velocity is v = u + v t. With these new designations, the expressions for the resonant field amplitude and the impact velocity become: E 0 = m e (ω2 + ν 2 )(d + u 0 ν (Φ 1)) (1 + Φ)sin α + ((1 Φ) ω ν + 2ν ω )cos α (3.11) u impact = u 0 Φ + Λ(1 + Φ)(ω cos α ν sinα) (3.12) where Φ = exp ( Nπν/ω) has been introduced for simplicity. Λ is given by Eq. (3.4) as before, but ν c should be replaced by ν. In order to construct the multipactor boundaries, the same approach as in the simple model case is taken and an expression for the resonant phase is obtained by combining Eqs. (3.11) and (3.12), which yields, tan α = ω2 [ρφ + χ] + 2ν 2 (Φu 0 u impact ) (dν + u impact u 0 )νω(1 + Φ) (3.13) where ρ = dν + u impact + u 0 and χ = dν u impact u 0 have been used for convenience. The reason why the expression looks somewhat different from Eq. (3.7) is that the constant initial velocity approach has been used instead of the assumption of a constant ratio between impact and initial velocities. This will also affect the expression for the phase stability factor, which in this case becomes G = (Φ 1)(ν2 + ω 2 )sin αλ Φνu 0 ν((1 + Φ)(ν sin α ω cos α)λ Φu 0 ) (3.14) So far, the differences between the simple and the more advanced model are fairly trivial. However, the parameters used (ν c and ν iz ) are not constants, they depend to a great extent on the total electron velocity, which is the vector sum of the drift and thermal velocities. The thermal velocity will have a random direction and therefore the average total velocity will be equal to the drift velocity. However, the total (average) energy, ɛ, will still depend on both velocities and it becomes, ɛ = mv2 2 = m 2 (u2 + v t 2 ) (3.15) 45

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

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

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

63 accurately describes the cross-section for the entire measurement range can be given by (cf. Fig. 3.6) ln ɛ/ɛ i σ iz = q 1 [m 2 ] (3.23) ɛ/ɛi + 0.1(ɛ/ɛ i ) 2 where ɛ i is the ionisation threshold of argon and q 1 = m Ionisation cross section for Argon Ionisation Cross Section [m 2 ] S.C.Brown H.C.Straub Analytical approximation Electron energy [ev] Figure 3.6: Ionisation cross-section for electron-argon collisions. The circles and stars indicate measurement data by S. C. Brown [58] and Straub et al. [59] respectively and the solid line is an analytical approximation given by Eq. (3.23) Multipactor boundaries In the following section the above model will be used to determine the multipactor boundaries. The best accuracy is attained when solving the basic differential equations numerically while using good approximate formulas for the different parameters. However, such computation takes very long time, since both the initial and the final multipactor conditions have to be fulfilled. Faster computation can be achieved by using different approximations, e.g. constant parameters as shown in Eqs. (3.10)-(3.12), Eq. (3.13), and Eq. (3.14). Two different implementations are used in paper C, one purely numerical and one semi- 49

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

65 Numeric stable phase Numeric unstable phase Analytic stable phase Analytic unstable phase Voltage [V] 10 2 pd=7 Pa mm pd=5 Pa mm pd=15 Pa mm 10 0 Frequency Gap product [GHz mm] Figure 3.7: Lower multipactor thresholds in low pressure argon for three different fixed pd-values, one for each zone. The dotted lines represent the multipactor zones for vacuum multipactor. Parameters used are: W 1 = 23 ev, W 0 = 3.68 ev, σ se,max (0) = 3, ɛ 0 = 0. 51

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

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

68 N=1 Normalised Voltage N=5 N= Pressure gapsize [Pa mm] Figure 3.10: Normalised multipactor thresholds for varying pd. The thresholds are normalised with respect to the vacuum threshold. Curves for the three first orders of resonance are shown for a material with a first cross-over point more than 7 times greater than in Fig Parameters used are: W 1 = 170 ev, W 0 = 4 ev, σ se,max (0) = 1.3, ɛ 0 = 0, fd N=1 = 1 GHz mm, fd N=3 = 3.2 GHz mm, and fd N=5 = 5 GHz mm. 54

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

70 10 3 µ=0 µ=0.75 Voltage [V] µ=0 Vacuum multipactor µ= Frequency Gap product [GHz mm] Figure 3.12: Multipactor thresholds in low pressure argon for the two lowest order modes (N = 1 and N = 3) with µ = 0.75 and µ = 0 respectively. In this case, the second cross-over energy is about 7 times greater than in Fig Parameters used are: W 1 = 170 ev, W 0 = 4 ev, σ se,max (0) = 1.3, ɛ 0 = 0, pd = 25 Pa mm for N = 1, and pd = 15 Pa mm for N = 3. 56

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

72 58

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

74 the increased threshold is losses of electrons out of the iris region [60]. In this analysis, we show that one of the contributing factors to this electron loss is a random drift due to the axial component of the initial velocity of the secondary emitted electrons. Other loss mechanisms, which are due to the inhomogeneity of the field, tend to further enhance the losses and these effects will be more pronounced for small gap lengths. This means that by taking only losses due to the random drift into account, a conservative increase of the breakdown threshold should be obtained. 4.1 Model and approximations The geometry used in the model is the 2-dimensional structure shown in Fig The iris has a gap height h in the y-direction, a length l in the z-direction and is assumed to be fitted into a waveguide with a height that is much greater than h. The harmonic electric field E is assumed uniform in the gap, as a simple approximation of the actual field. There are two main reasons for choosing a uniform field. Firstly, the deterministic model developed for the parallel plate case, which is described in chapter 2, can be used to describe the basic behaviour of an electron trajectory inside the gap. Secondly, the effect of the initial velocity spread of the secondary electrons along the z-axis on the multipactor threshold can be analysed separately from the drift force due to inhomogeneities in the electric field. In addition, it gives a convenient base for comparing the results with those of the parallel plate model. By assuming a uniform E-field in the y-direction, the electron motion along the z-direction is not affected by the field. The motion in this direction, the drift motion, will occur with a fixed velocity v z between the impacts. Lets assume that a seed electron is emitted inside the gap at the coordinate z 0, l/2 < z 0 < l/2, at one of the walls. As the electron traverses the gap and hits the opposite side of the iris, it has become displaced a distance z in the z-direction. This drift is determined by the velocity in the z-direction, v z, together with the transit time, t g, and is given by z = v z t g. For a fixed mode order, N, and frequency, f, of the field, each transit time is the same and is given by, t g = Nπ ω, (4.1) where ω = 2πf. The electron trajectory in the z-direction will perform a random walk with a change of velocity, v z, after each impact. When the impact 60

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

76 1 Probability of survival l=16 mm 0.2 l=2 mm l=8 mm Number of collisions Figure 4.2: The probability of survival, p(k), for an electron emitted in the center of the iris gap, z = 0, for three different iris lengths. Parameters used: f = 1 GHz, N = 1, and W T = 2 ev (corresponding to v T, the rms-velocity of the Maxwellian distribution of initial velocity in the z-direction). strong discharge will appear. The generated number of electrons over the initial number of electrons after k collisions is given by, N e N 0 g(k) = p(k)σ k se. (4.3) Depending on the start position of the seed electrons, the initial behaviour of N e can vary. If the start position is close to the iris edge, the average electron number will first decrease and then if σ se is large enough, it will start increasing again. But if the start position is in the center, it may first start to increase, but after a number of transits, it will start decreasing (cf. Fig. 4.3). Eventually, it is the asymptotic behaviour of p(k) that will determine whether or not there will be a discharge. Thus from Eq. (4.2) and Eq. (4.3) one can conclude that the asymptotic change in the electron number is given by, g(k) (σ se γ 0 ) k. (4.4) Thus the average number of electrons will grow if σ se γ 0 > 1 (4.5) 62

77 3 Multipactor in iris 2.5 σ = N e /N 0 [ ] σ = Number of collisions Figure 4.3: The growth in electron number as a function of the number of gap crossings for two different SEY-coefficients. Parameters used: f = 1 GHz, N = 1, l = 2 mm, and W T = 2 ev. or equivalently σ se > 1/γ 0 > 1. (4.6) This implies that the secondary electron yield must be greater than a value that is larger than unity (1/γ 0 > 1) to have growth of the number of electrons. This modified breakdown criterion is the only difference between the model considered here and the conventional resonance theory of multipactor inside a plane-parallel gap (where σ se > 1 is used when determining the threshold). The condition for σ se, Eq. (4.6), can be converted into a range of impact energies, W 1 < W min < W impact < W max < W 2, (4.7) where the impact energies W min and W max are determined by σ se = 1/γ 0. Consequently, using W min and W max instead of W 1 and W 2, respectively, in the parallel-plate model, multipactor regions that account for the electron losses due random drift can be obtained. 63

78 4.2 Multipactor regions Using the above described model and by employing a natural scaling parameter of η, viz. l f, multipactor charts in the traditional engineering units (voltage vs. frequency-gap-size product) can be devised for a specific value of the ratio of gap height and iris length, h/l. Figure 4.4 shows an example of this, where the multipactor regions have been constructed for 5 different h/l-ratios. Multipactor regions for different height/length ratios Peak voltage [V] Frequency gap height product [GHz mm] Figure 4.4: The first four multipactor susceptibility zones for a microwave iris with five different height/length-ratios. Parameters used are: W 0 = 2 ev (y-direction), W T = 2 ev (z-direction), W 1 = 85.6 ev, and σ se,max = 1.83 (material properties for alodine [50]). The transit time decreases with increasing frequency according to Eq. (4.1) and thus the distance traversed in each step becomes smaller, which implies that the probability of surviving k steps increases. This causes the multipactor zones to shrink towards higher frequencies with increasing h/l as is evident in Fig The transit time is also a function of the mode order, which increases the transit time for higher order modes. This counteracts the decrease due to increasing frequency and consequently a behaviour similar to that of the first resonance zone can be observed also for the higher order modes. For materials with a low maximum SEY, like in Fig. 4.4, the ability 64

79 to compensate for electron losses is not very good and thus the zones will disappear for relatively small values of h/l. However, in the opposite case for a material with a large SEY, like e.g. aluminium, multipactor will be possible also for relatively thin irises. 4.3 Comparison with experiments By comparing the current model with experimental data [61], good qualitative agreement can be observed (see Fig. 4.5). As the h/l-ratio increases, the threshold increases until, beyond a certain limit, no multipactor is possible. Since only the electron losses due to the random drift are accounted for, the model predicts the existence of a discharge beyond the limit found in the experiments. Consequently, from an engineering point of view, this is a conservative measure of the increased threshold and thus it should be safe to apply it when designing multipactor free microwave hardware Multipactor in iris 3000 Peak Voltage [V] Current model a) Current model b) 1000 Current model c) Current model d) Measur. presentation Measur. proceedings Height/Width [ ] Figure 4.5: The multipactor threshold as a function of h/l for different parameters of the current model. For comparison, measurement data from [51] is included. Parameters used in a): h = 1.2 mm, f = 9.56 GHz, W T = 2 ev, W 0 = 2 ev, W 1 = 59.1 ev, and σ se,max = 2.22, (i.e. W 1 = W f2 and σ se,max = α max in table A-6 for silver in [50])). Modified parameters in: b) W 0 = W T = 4 ev, c) W 1 = 40 ev, and d) W 1 = 80 ev. 65

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

81 4.4 Main results This analysis has shown that the random electron drift along the iris length due to the initial velocity of the secondary electrons tends to significantly increase the multipactor threshold in a waveguide iris as compared to predictions based on the classical two parallel-plate model. Inherent in the presented model is the scaling parameter h/l, which makes it possible to produce useful multipactor susceptibility charts in the traditional engineering units. An increase in the h/l-ratio results in a shrinkage of each multipactor resonance zone. For each zone, the zone reduction effect is more pronounced for lower frequencies. A consequence of this is that the multipactor free region in the parallel plate model at low frequency-gap-height products grows with increasing h/l-ratio. 67

82 68

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

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

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

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

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

88 where d stands for the gap width, V ω = ee ω /mω is the amplitude of the electron velocity oscillations in the spatially uniform RF field, E ω is the RF field amplitude inside the gap, and U is the voltage between the conductors. In addition, for an electron avalanche to start, the impact velocity should be between the first and the second cross-over points (cf. Eq. (2.12)), which for zero initial velocity is given by, v 1 < 2V ω < v 2. (5.10) Due to the asymmetry in the electron motion, the simple analytical analysis that is feasible in the parallel plate case is not applicable for the coaxial line, despite the fact that the approximate electron position and velocity are known explicitly (Eqs. (5.3) and (5.4)). One way of finding the resonance zones in the parameter space is to compute a series of successive electron trajectories, searching for the conditions when it converges to a periodically repeated sequence [69]. In Fig. 5.3 one can see the result of such a method, where stable resonance zones have been found numerically. To allow simple comparison with the parallel-plate case, the normalised gap size has been used and in terms of the coaxial line it becomes, λ = m(ωd)2 = G(R o R i ) 2 eu c Ro 2 ln (R o /R i ), (5.11) which coincides with Eq. (5.9) when R o /R i is close to unity. The convenient parameter, G = ωr o /V ω,o, has been introduced as representing a normalised outer radius and V ω,o = qe o /mω. When R o /R i is close to unity the parallel-plate and coaxial models give similar results. When the ratio becomes larger, the zones deviate from the straight lines predicted by Eq. (5.8) and the deviation is more pronounced for the higher modes. When the ratio becomes too large, all two sided resonances disappear. This is a consequence of the Miller force and for the higher order modes, where the approximate analytical solution is very accurate, the prediction (Eq. (5.7)) is that the double sided resonances should disappear roughly at R o /R i = Figure 5.3 shows that this is indeed true. The first order mode, however, is not accurately described by the analytical expression and the numerical calculations show that this mode can exist for values of R o /R i as high as 4. In Fig. 5.4 the double-sided discharge regions have been computed numerically for R o /R i = 1.4. The classical multipactor zones have upper and lower thresholds that satisfy Eq. (5.10) in the following sense: since 74

89 Double sided multipactor, numerical data λ c c 2 h 1 c R o /R i [ ] Figure 5.3: Normalised gap width according to Eq. (5.11) vs. R o /R i. The solid straight lines form the classical zones according to Eq. (5.8). The dots (blue) indicate stable resonances where the sum of two transits equal an odd number of RF-cycles. The crosses (red) are for sums equal to an even number of RF-cycles. The sum of the transit time in RF-cycles for some distinct zones are indicated. The lowest order hybrid modes are marked with an h and the classical resonances with a c. The dashed vertical line indicates R o /R i =

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

91 on the approximate solution for the electron trajectory, Eq. (5.3). The analytical expression is accurate when the oscillations are small and a consequence of this is that accurate stable phase multipactor regions are only found for N 2, i.e. when the duration of the trajectory is at least 2 RF-periods. By analysing the resonance and stability conditions, one can show that single-sided breakdown will have not only one region of stable resonant phase, but rather two stable regions can be found. One somewhat wider region with resonance close to zero and another, which is resonant close to π/4. It can be shown that these regions, in the case when v 0 = 0, are approximately given by: and 0 < α R < α 1 (5.13) α 2 < α R < α 3 (5.14) where α 1 4 (5.15) Nπ α 2 π 4 1 (5.16) Nπ α 3 π (5.17) Nπ For increasing N, the second region converges to α R = π/4, which according to Eq. (5.7) corresponds to R min R o / 2. In Fig. 5.5 the resonant stable phase, α R, has been plotted as a function of N. Except for the lowest order resonance (N = 1), the regions of phase stability for the numerically and analytically obtained phases agree very well. To obtain the multipactor threshold, it is necessary to know the impact velocity, which is given by v impact 2V ω,o cos α + v 0 (5.18) The lower boundary shown in Fig. 5.6 is obtained for the maximum impact velocity for each mode, i.e. v impact = 2V ω,o when v 0 = 0. In the same figure one can also identify a second set of regions with a higher breakdown threshold. These zones correspond to the second stable phase region, Eq. (5.14). Since the phases in this region are close to π/4, the impact velocity is v impact 2V ω,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

92 60 50 α 3 Phase [degrees] α 2 10 α N (rf cycles) Figure 5.5: Stable resonant phase for single-sided multipactor. Analytically obtained stable phases are shown as diamonds (red) and the ones obtained numerically are indicated with dots (blue). Eqs. (5.15) - (5.17) are shown as solid lines. The dashed line indicates α = π/4. zones as there are phases that are smaller than π/4 (cf. Fig. 5.5), which will yield a higher impact velocity and consequently a lower threshold. Furthermore, in the numerical solution of Eq. (5.2) for the first order mode, an impact velocity of as much as four times V ω,o can be observed. This results in a threshold much lower than the envelope. The maximum impact velocity of the second zone is slightly lower than 2V ω,o, resulting in a somewhat higher threshold. The following higher order modes then quickly converge to the analytical limit (cf. Fig. 5.6). When the initial velocity is zero, the only parameters left to vary are G and the ratio R o /R i. Since the characteristic impedance in ohms of a coaxial line in vacuum is given by Z 60ln (R o /R i ), it follows that only two parameters remain to be varied, viz. G and Z. By following trajectories for different values of G and Z, stable phase points were found in this parameter space and the result is plotted in Fig. 5.7, which was produced using a numerical solution of the equation of motion (a version of this figure using the analytical expressions can be found in paper E). The straight lines in Fig. 5.7 on the right hand side are regions of stable single-sided resonances. The fact that these appear as straight 78

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

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

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

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

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

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

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

100 σ se,max the zones become wider and zones previously suppressed by the losses can appear (cf. Fig. 5.11). This behavior is very similar to that noted for the parallel plate geometry [46]. The lower (left) envelope is G V /V ω max Figure 5.10: Number of electrons after 200 rf-cycles. The vertical straight lines indicate the lower and upper theoretical envelopes according to Eq. (5.12). Parameters used: R i /R o = 0.7, f = 1.5 GHz, σ se,max = 1.3, v T /v max = 0.01, and W 1 = 50 ev. obeyed, but the upper can be exceeded since a non-zero initial phase will yield a lower impact velocity and a concomitant higher upper threshold. In Figs and 5.11 the velocity spread is quite low, v T /v max = When increasing this ratio, a greater portion of the electrons can have a large negative initial phase, which leads to overlapping of the multipactor regions when σ se,max is large (see Fig. 5.12). On the other hand, the increased velocity spread also increases the losses, which especially affects the higher order resonances, since the phase-focusing effect gets weaker with increasing mode order. If σ se,max is low, the losses are not sufficiently compensated for and this leads to suppression of the higher order modes (cf. Fig. 5.13). This is a result similar to that found in the parallel plate case [46]. The corresponding behaviour is seen also for single-sided multipactor (see paper F). For single-sided multipactor it was noted that for the first order mode, the envelope of the breakdown zones V ω,o = v 1 /2 was not obeyed. This is confirmed by the PIC-simulations, where the first order mode 86

101 50 G V /V ω max Figure 5.11: Same as Fig only with σ se,max = G V /V ω max Figure 5.12: Number of electrons after 200 rf-cycles. The vertical straight lines indicate the lower and upper theoretical envelopes according to Eq. (5.12). Parameters used: R i /R o = 0.7, f = 1.5 GHz, σ se,max = 2.0, v T /v max = 0.1, and W 1 = 50 ev. 87

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

103 50 G V /V ω max 0 Figure 5.14: Number of electrons after 200 rf-cycles. The vertical straight lines indicate V ω,o = v 1 /2 and V ω,o = v 2 /2. Parameters used: R i /R o = 0.1, f = 1.5 GHz, σ se,max = 2.0, v T /v max = 0.01, and W 1 = 50 ev log 10 (Amplitude) [V] log (Frequency) [GHz] 10 Figure 5.15: Multipactor breakdown regions for copper electrodes. The region confined by circles (or white crosses, when inside a dark region) is from Ref. [63]. The dark regions are obtained by the PIC-code with: R o /R i = 2.3 (Z = 50 Ω), σ se,max = 2.25 (from table A-6 in [50], W 1 = 27 ev, and v T = 3 ev (v T /v max = 0.111). 89