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

Boundary function Method
One of the best engineering methods for finding the worst case scenario
for equally spaced carriers, the boundary function method, was originally
designed by Wolk et al. [51]. Unfortunately the used function was
found empirically while studying the worst case scenario with an optimisation
tool, and is thus not physically founded. A consequence of this is
that under certain circumstances, the boundary function produces poor
results. It was also limited to work only for equally spaced carriers. However,
as part of the present thesis work, this method has been further
developed, and it has been found that the original boundary function
approximately describes a function that tries to squeeze all the energy
of the multicarrier signal during one envelope period into a specified,
shorter, time period. This works just as well in both the equally spaced
and the non-equally spaced carrier cases and can be summarized by the
following formulas:
⎧ �
FV (TX) =
⎪⎨
FV,max = N�
⎪⎩ FV,min =
TH
TX
Ei
i=1
� N�
E
i=1
2 i
N�
E
i=1
2 i
(2.29)
Here TX is the time period of interest, which is often set to T20, i.e.
the time it takes the electrons to traverse the gap 20 times. TH is the
period of the envelope and Ei is the voltage amplitude of each carrier.
FV (TX) is the design voltage and is shown as two symmetric curved lines
in Fig. 2.15. The design voltage can never exceed the in-phase voltage,
given by FV,max, and if all power is distributed evenly over the entire
envelope period the voltage amplitude will be FV,min, which is indicated
by a dashed line in Fig. 2.15.
The main advantage with the boundary function method is its simplicity.
It is also very reliable, although a little conservative and this is
especially true for non-equally spaced carriers, where the P20 level can
be much lower than FV . The method has been implemented as an auxiliary
method in WCAT, which is a software tool originally developed by
the present author and Genrong Li as part of a Master’s Thesis [52] at
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