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

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possible for quite large Ro/Ri-values. In the simulations by Sakamoto et al [64], the experimental results were confirmed. In addition, single-sided multipactor was observed, which confirms the result of this theoretical study. Among the more important results of this part of the study are the following. The analytical approximate solution of the nonlinear differential equation of motion for an electron in a coaxial line. The dual regions of stable phase, Eqs. (5.13) - (5.17), which explain why single-sided multipactor will be possible also for smaller values of Ro/Ri. The scaling law for single-sided multipactor, Eq. (5.19), simplifies the presentation of multipactor prone regions of the single-sided case. The limit formula for the transition from double- to single-sided multipactor, Eq. (5.7), is an interesting feature for future experiments to confirm. The reduced threshold for the first order zone of single-sided multipactor, which must be taken into account when constructing the lower boundary of all the zones. Finally, this analysis shows that the behaviour of resonant multipactor is significantly affected by the nonuniform field and it shows the benefits of analytically studying different geometries to understand the basic behaviours before performing numerical simulations. 5.2 Particle-in-cell simulations In this part of the coaxial study, which is based on paper F, extensive PIC-simulations have been performed in order to verify the analytical results as well as to investigate the importance of initial velocity spread and different maximum secondary emission. One of the advantages with PIC-simulations is the ability to include parameters that are stochastic in nature. The stochastic properties of some of the parameters as well as the actual value of the maximum secondary emission coefficient may have significant effects on the multipactor threshold as well as on the existence of a discharge. This has previously been shown in the case of plane-parallel geometry [46] and it is demonstrated that the effect is similar also in coaxial geometry. 5.2.1 Numerical implementation The geometry and field is described in the analytical section. The code uses normalised parameters such as G and λ and the SEY follows the model by Vaughan [22]. The initial velocities of the secondary electrons 82
are assumed to have a Maxwellian distribution, i.e. f(vx,vy,vz) ∝ vn v exp � − 1 v ( ) 2 vT 2 � (5.21) where v is the absolute value of the initial velocity, vn its normal component with respect to the surface of emission, and vT is the thermal initial velocity spread. Another of the used parameters related to this is the normalised spread of initial electron velocity defined as vT/vmax, where vmax is the impact velocity for maximum SEY. Calculations were performed for 2-D arrays of different sets of the normalised parameters (e.g. ρ = Vω/vmax vs. λ with the other parameters fixed) and each run corresponds to one particular point in one of these arrays. Each run was primed with 200 seed electrons, uniformly distributed over initial phase, and the run was terminated when either the number of particles exceeded 4500 or when 200 RF-periods had elapsed. The run was also terminated in case the number of electrons dropped below 10 before 200 RF-cycles had passed. At the end of each run, the following parameters were recorded: • Number of RF-periods needed to exceed 4500 particles. If 4500 particles were not attained within 200 RF-cycles, this parameter was set to 200. • Number of electrons at the end of each run. • Heating asymmetry, i.e. the ratio between the average power deposited on the inner conductor and the average power deposited on the outer conductor. • Average electron growth rate (over the 10 last RF-periods), normalised with respect to the RF-period. 5.2.2 Simulations To facilitate comparison between the theoretical result presented in Fig. 5.7 a simulation was made in the same parameter space. Figure 5.8 shows the number of electrons obtained after 200 RF-cycles. As expected, the lower order resonances (i.e. at lower and leftmost G-values) indicate high electron numbers, since more impacts with the conductors will occur during the same number of RF-periods. Due to this fact, a parameter space was chosen, which did not include any points in the 83
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Thesis for the degree of Doctor of
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Multipactor in Low Pressure Gas and
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Conference contributions by the aut
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viii
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3.2.4 Key findings . . . . . . . .
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xii
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xiv
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addition, it is difficult to make c
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numerical study of the phenomenon i
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to physically damage microwave comp
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odd positive integer (N = 1,3,5...)
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σ se [−] 2.5 2 1.5 1 0.5 W 1 W m
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Voltage [V] 10 4 10 3 10 2 10 0 N=1
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even though it may be sufficient fr
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where φL and φR are the left and
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(cf. Fig. 2.7). When taking the env
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2.1.4 Methods of suppression Many o
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Noise (dBm) Power #1 (dBm) Match #1
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andom spread in emission velocities
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Figure 2.12: Multipactor experiment
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where kn is a factor determining th
Page 44 and 45:
allows for another design margin, w
Page 46 and 47: Boundary function Method One of the
Page 48 and 49: carriers, because the NLSQ method r
Page 50 and 51: multipactor discharge. Using a Mont
Page 52 and 53: the assumption of a constant ratio
Page 54 and 55: Voltage (V) 10 3 10 2 10 1 10 −2
Page 56 and 57: 3.1.3 Main results The main result
Page 58 and 59: lytical model is obviously needed.
Page 60 and 61: where 〈vt 2 〉 represents the av
Page 62 and 63: these quantities in the range from
Page 64 and 65: analytical. Attempts were made to f
Page 66 and 67: only model, the inclusion of ionisa
Page 68 and 69: Normalised Voltage 1.1 1.05 1 0.95
Page 70 and 71: Voltage [V] 10 3 10 0 µ=0 µ=0.75
Page 73 and 74: Chapter 4 Multipactor in irises A c
Page 75 and 76: h l Figure 4.1: The geometry used i
Page 77 and 78: N e /N 0 [−] 3 2.5 2 1.5 1 Multip
Page 79 and 80: to compensate for electron losses i
Page 81: 4.4 Main results This analysis has
Page 84 and 85: support for the qualitative analyti
Page 86 and 87: where R(t) is the average position,
Page 88 and 89: where d stands for the gap width, V
Page 90 and 91: there are different oscillatory vel
Page 92 and 93: Phase [degrees] 60 50 40 30 20 10
Page 94 and 95: G 100 90 80 70 60 50 40 30 20 10 0
Page 98 and 99: upper right region of the figures,
Page 100 and 101: σse,max the zones become wider and
Page 102 and 103: G 50 45 40 35 30 25 20 15 10 5 0.2
Page 104 and 105: creasing ratio Ri/Ro was observed.
Page 107 and 108: Chapter 6 Detection of multipactor
Page 109 and 110: Amplitude of acceleration [Gm/s 2 ]
Page 111 and 112: tipactor events that are short-live
Page 113 and 114: software and the time evolution of
Page 115 and 116: ditional test sample, a resonant ca
Page 117 and 118: the FFT. It was concluded that ther
Page 119 and 120: 6.2.1 Single carrier When using the
Page 121 and 122: Reference signal generator Phase co
Page 123 and 124: Amplitude [A.U.] 2.5 2 1.5 1 0.5 De
Page 125 and 126: Chapter 7 Conclusions and outlook T
Page 127 and 128: tioned there were multipactor in no
Page 129 and 130: References [1] W. Henneberg, R. Ort
Page 131 and 132: [27] A. Ruiz et al, “Ti, V, and C
Page 133 and 134: [51] D. Wolk, D. Schmitt, and T. Sc
Page 135 and 136: [75] R. Udiljak, Multipactor in low
Page 137: Included papers A-F 123
Page 141: Paper B R. Udiljak, D. Anderson, M.
Page 145: Paper D R. Udiljak, D. Anderson, M.
Page 149:
Paper F V. E. Semenov, N. Zharova,
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