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

62 4. CIRCUIT MODEL
the bars is proportional to the amplitude of the field <strong>and</strong> all the bars
are normalized to one. The graph is in a percent scale. Note that the
coupling cavities, which are represented by the even bars, are nominally
uncharged.
100
90
80
70
60
50
40
30
20
10
0
%
50
0
100
0
100
0
100
1 2 3 4 5 6 7 8 9 # cell
Figure 4.4. The relative level of field in the cavities in
the imperturbed case.
Therefore, the lateral cavities produce an half voltage <strong>and</strong> we can
drop this dissymmetry by considering only a contribution i = 1 which
is the sum of the two half voltages. In such a way we do not directly
consider the cavity i = N +1 <strong>and</strong> we deal with N/2 accelerating cavities
<strong>and</strong> index i goes from 1 to N. Then, the value of Vci for the generic
cavity is
Vci(ε1, . . . , εN) =
+
−
1
jωCeqi
N
ε
p=1
∗ p ε
q=i
+ q
N
⎧
⎨
N
(N − i + 1)2
1 −
⎩
N
N
ε
p=1
∗ p ε
q=i
− q
N 2
0
p=1
ε ∗ p
50
2
(i−q)
N − i + 1 − (−1)
N
2q − N − i
for i = 2s + 1 with s = 1, 2, . . . , N/2 − 1
N
+
(4.63)
where Ceqi is the capacitance of the cavity with number i <strong>and</strong> it is the
series of the capacitances of the two half cavities with indexes i−1 <strong>and</strong>