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

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the assumption of a constant ratio k = vimpact/v0 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 cvimpact(Φ − k) ξω(Φ + 1) (3.7) where Φ = exp(−νcNπ/ω), β = kdνc + (k + 1)vimpact, γ = kdνc − vimpact(k + 1), and ξ = [kdνc + (k − 1)vimpact]νc. Equation (3.7) can be used together with Eq. (3.5) to plot the 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
Voltage (V) 10 3 10 2 10 1 Multipaction threshold curves in low pressure air (d=0.1 m) p=0.1 Pa p=1 Pa p=10 Pa Vacuum 10 −3 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 W1 = 23 eV. Phase stability and the non-returning electron limit are not considered, only resonance. Parameters used are: σc = 6.9 × 10 −20 m 2 , k = 2.5, N = 1 and d = 0.1 m. 10 −2 39
Page 1 and 2: Thesis for the degree of Doctor of
Page 3: Multipactor in Low Pressure Gas and
Page 6 and 7: Conference contributions by the aut
Page 8 and 9: viii
Page 10 and 11: 3.2.4 Key findings . . . . . . . .
Page 12 and 13: xii
Page 14 and 15: xiv
Page 16 and 17: addition, it is difficult to make c
Page 18 and 19: numerical study of the phenomenon i
Page 20 and 21: to physically damage microwave comp
Page 22 and 23: odd positive integer (N = 1,3,5...)
Page 24 and 25: σ se [−] 2.5 2 1.5 1 0.5 W 1 W m
Page 26 and 27: Voltage [V] 10 4 10 3 10 2 10 0 N=1
Page 28 and 29: even though it may be sufficient fr
Page 30 and 31: where φL and φR are the left and
Page 32 and 33: (cf. Fig. 2.7). When taking the env
Page 34 and 35: 2.1.4 Methods of suppression Many o
Page 36 and 37: Noise (dBm) Power #1 (dBm) Match #1
Page 38 and 39: andom spread in emission velocities
Page 40 and 41: 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 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 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.
Page 145:
Paper D R. Udiljak, D. Anderson, M.
Page 149:
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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