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

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G 100 90 80 70 60 50 40 30 20 10 0 0 0.5 1 1.5 2 Z/Z (Z =50 Ω) 0 0 Figure 5.7: The normalised parameter G vs. normalised characteristic impedance Z. Each mark represents a stable phase solution and an effort has been made to suppress polyphase modes in order to clearly show the behaviour of the main resonance modes. The chart was obtained by numerically solving the equation of motion. Stars mark (blue): double-sided multipactor, dots (red): single-sided multipactor with 0 < αR < 20 o , and crosses (green): single-sided multipactor with αR > 20 o . The dashed line indicates Ri,min = Ro/ √ 3 and the dash-dot line Ri,min = Ro/ √ 2. 80
lines indicates that there is no dependence on Z, which implies that a simple scaling law exists in the single-sided case, viz. P ∝ (ωRo) 4 Z. (5.19) In Fig. 5.6 this law has been used to normalise the axes, but voltage is used on the ordinate instead of power. The chosen normalisation of the axes in Fig. 5.6 is general and using the analytical solution of Eq. (5.2) presented above, it can be shown that this normalisation is valid also for non-zero initial velocity. It is important, however, to be careful when scaling to a different radii ratio, since for smaller values of the characteristic impedance, the single-sided multipactor zones may not exist at all. In the double-sided case, it is evident that G is a function of Z. Consequently a more complicated scaling law should be expected. For small values of Z, however, the coaxial case becomes similar to the parallel-plate geometry, where the resonance voltage can be written as function of the frequency-gap-size product. For the coaxial case, this scaling law becomes P ∝ (ω(Ro − Ri)) 4 1 . (5.20) Z and for the first order resonance this scaling law is quite accurate (cf. Fig. 5.3), but for the higher order modes it quickly loses its validity with increasing Z. 5.1.3 Main findings A qualitative comparison with experiments [63] shows good agreement with the present analysis. The experimental data shows an increase in the multipactor threshold for increasing radii-ratio Ro/Ri. It was also found that the first multipactor zone became narrower for increasing Ro/Ri. These features are in agreement with the results of this study as shown in Figs. 5.3 and 5.7. By mapping the data of Fig. 5.3 into the voltage vs. frequency-gap-size space, used in the experiments, a clear threshold increase compared with the parallel-plate case can be seen as well. Even though the experiments used quite large values of Ro/Ri, no case of single-sided multipactor was observed. This can be explained by the fact that a material with a low first cross-over point was used, where the initial velocity will play an important role when the applied voltage is not high enough. More importantly, only the first order mode was studied and in this case, as shown in Fig. 5.3, multipactor will be 81
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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 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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Multipactor in Low Pressure Gas and in ... - of Richard Udiljak

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