Theory, Design and Tests on a Prototype Module of a Compact ...

1. TRANSMISSION MATRIX REPRESENTATION 51
<strong>and</strong> if it is short-circuited, then
VN
0 (TN)
= TN =
12
IN
1 (TN) 22
<strong>and</strong> in an analogous way, for the other cavities, it is
VN−1
VN
0
= TN−1 = TN−1 · TN =
IN−1
IN
1
VN−2
0
= TN−2 · TN−1 · TN =
1
IN−2
(TN−1 · TN)12
(TN−1 · TN)22
(TN−2 · TN−1 · TN)12
(TN−2 · TN−1 · TN)22
(4.22)
Last, we can deduce the rule for the generic capacitance
N
Vci ∝ Ii =
for i = 1, 2, . . . , N (4.23)
l=i
Tl
22
<strong>and</strong>, of course, VcN+1 ∝ 1.
In the expression (4.23) we need the product of N−i+1 transmission
matrix, but remembering the matrix (4.17), we can obtain in a similar
way
⎛
⎞
N ⎜
Tl = ⎝
l=i
<strong>and</strong> therefore
cosh (N − i + 1)x jωM sinh x sinh (N − i + 1)x
−j sinh (N − i + 1)x
ωM sinh x
cosh (N − i + 1)x
Vci ∝ cosh (N − i + 1)x for i = 1, 2, . . . , N
VcN+1 ∝ 1
⎟
⎠
(4.24)
(4.25)
1.5. Asymptotic expression of the ratio between the voltages
on the capacitances of two adjacent cavities in an infinite
structure. In this short subsection we consider an interesting property
of an infinite chain of cavities, that is given from the ratio of the
voltages of two adjacent cavities. Let us start from the ratio
Vci
Vci+1
= cosh (N − i + 1)x
cosh (N − i)x
<strong>and</strong> imposing that N − i + 1 = i ′ , one obtains
Vci
Vci+1
=
=
cosh i ′ x
cosh (i ′ − 1)x =
1
cosh x + sinh x tanh i ′ x
cosh i ′ x
cosh i ′ x cosh x + sinh i ′ x sinh x
(4.26)
(4.27)