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

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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. 3.2.1 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(νc + νiz). (3.9) where u is the drift velocity and νiz the ionisation frequency. In general, the collision and ionisation frequencies are functions of the electron velocity. However, by assuming that νc and νiz 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 = ˆxE0 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 = νc + νiz, u = ˙x, and du/dt = ¨x, Eq. (3.9) can be written, ¨x = eE − ˙xν. (3.10) m 44
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 νc one will have ν and v0 should be replaced by u0. 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, vt, i.e. the total velocity is v = u + vt. With these new designations, the expressions for the resonant field amplitude and the impact velocity become: E0 = m e (ω2 + ν2 )(d + u0 ν (Φ − 1)) (1 + Φ)sin α + ((1 − Φ) ω ν 2ν + ω )cos α (3.11) uimpact = u0Φ + 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 νc 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 (Φu0 − uimpact) (dν + uimpact − u0)νω(1 + Φ) (3.13) where ρ = dν + uimpact + u0 and χ = dν − uimpact − u0 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 − Φνu0 ν((1 + Φ)(ν sin α − ω cos α)Λ − Φu0) (3.14) So far, the differences between the simple and the more advanced model are fairly trivial. However, the parameters used (νc and νiz) 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 + 〈vt 2 〉) (3.15) 45
Page 1 and 2:
Thesis for the degree of Doctor of
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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 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 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
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Amplitude of acceleration [Gm/s 2 ]
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tipactor events that are short-live
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software and the time evolution of
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ditional test sample, a resonant ca
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the FFT. It was concluded that ther
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6.2.1 Single carrier When using the
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Reference signal generator Phase co
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Amplitude [A.U.] 2.5 2 1.5 1 0.5 De
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Chapter 7 Conclusions and outlook T
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tioned there were multipactor in no
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References [1] W. Henneberg, R. Ort
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[27] A. Ruiz et al, “Ti, V, and C
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[51] D. Wolk, D. Schmitt, and T. Sc
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[75] R. Udiljak, Multipactor in low
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Included papers A-F 123
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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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