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

where R(t) is the average position,
�
R(t) =
C1(t − C2) 2 + Λ2
2ω 2 C1
. (5.5)
The constants of integration, C1 and C2, are determined by the initial
conditions, which for an electron starting at the outer conductor are
r(t = t0) = Ro and r ′ (t = t0) = −v0. Using Eq. (5.3), the position of an
electron emitted from the outer conductor with no initial velocity has
been plotted in Fig. 5.2. The accuracy of the expression is evident from
the comparison with the numerical solution.
Position [mm]
10
9
8
7
6
5
Multipactor in coax
0 1 2 3 4
Time [ns]
5 6 7 8
Figure 5.2: Motion of an electron emitted from the outer conductor of a coaxial
line. The solid line corresponds to the analytical expression
Eq. (5.3), the dotted line is a numerical solution of the differential
equation Eq. (5.2) (almost covered by the solid line) and the
dashed line is the average motion according to Eq. (5.5). Parameters
used: Vc = 1200 V, f = 3 GHz, W0 = 0 eV (the initial
electron energy), Ro = 10 mm, and Ri = 5 mm.
An important result can be obtained by only looking at the average
position, Eq. (5.5). The minimum of this equation, Rmin, is the small-
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