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

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Phase [degrees] 60 50 40 30 20 10 α 3 0 0 5 10 N (rf−cycles) 15 20 α 2 α 1 Figure 5.5: Stable resonant phase for single-sided multipactor. Analytically obtained stable phases are shown as diamonds (red) and the ones obtained numerically are indicated with dots (blue). Eqs. (5.15) - (5.17) are shown as solid lines. The dashed line indicates α = π/4 . zones as there are phases that are smaller than π/4 (cf. Fig. 5.5), which will yield a higher impact velocity and consequently a lower threshold. Furthermore, in the numerical solution of Eq. (5.2) for the first order mode, an impact velocity of as much as four times Vω,o can be observed. This results in a threshold much lower than the envelope. The maximum impact velocity of the second zone is slightly lower than 2Vω,o, resulting in a somewhat higher threshold. The following higher order modes then quickly converge to the analytical limit (cf. Fig. 5.6). When the initial velocity is zero, the only parameters left to vary are G and the ratio Ro/Ri. Since the characteristic impedance in ohms of a coaxial line in vacuum is given by Z ≈ 60ln (Ro/Ri), it follows that only two parameters remain to be varied, viz. G and Z. By following trajectories for different values of G and Z, stable phase points were found in this parameter space and the result is plotted in Fig. 5.7, which was produced using a numerical solution of the equation of motion (a version of this figure using the analytical expressions can be found in paper E). The straight lines in Fig. 5.7 on the right hand side are regions of stable single-sided resonances. The fact that these appear as straight 78
Amplitude of Conductor Voltage/ln(R 0 /R i ) [V] 10 5 10 4 10 3 10 2 10 0 Single sided Multipactor: Z c =100 Ω 10 1 Frequency x R o [GHz ⋅ mm] Figure 5.6: Single-sided multipactor breakdown regions based on the numerical solution of Eq. (5.2). Regions corresponding to N = 1 − 22 RF-periods are shown. The regions with an initial phase, α, close to zero are produced using dots (blue) and the regions with α close to π/4 are indicated by crosses (red). The dash-dot line is the approximate lower envelope given by Vω,o = v1/2 and the dashed line is given by Vω,o = v1/ √ 2. Parameters used: W1 = 23 eV, W2 = 2100 eV, W0 = 0 eV, and Z = 100 Ω (corresponding to e.g. Ro = 10 mm and Ri = 1.88 mm.) 10 2 79
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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
Page 42 and 43: 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 94 and 95: G 100 90 80 70 60 50 40 30 20 10 0
Page 96 and 97: possible for quite large Ro/Ri-valu
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.
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Paper D R. Udiljak, D. Anderson, M.
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Paper F V. E. Semenov, N. Zharova,
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