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

64 4. CIRCUIT MODEL
is unitary, one obtains
Vc1 = 1
jωC L ⎧
⎨ N
1 − ε
1 ⎩
p=1
∗ 2 N
p +
N
− ε
p=1
∗ N
p ε
q=1
−
2q − N − 1
q
N
⎧
∼ 1
⎨
N
1
= 1 − ε
jωC ⎩ 2
∗ 2 p + 1
N
2
− 1
2
N
N
ε
p=1
∗ p ε
q=1
− q
p=1
2q − N − 1
N
N
ε
p=1
∗ p ε
q=1
+ q
+ 1
jωC R N
N
ε
p=1
∗ p ε
q=1
+ q
− K
4
N − (−1) (1−q)
ε L 1 + ε R N
2
N
−
(1−q)
N − (−1)
N
(4.67)
Each voltage is characterized by an imperturbed term, a perturbed
term that depends on the errors of all the cells <strong>and</strong> a local term that
depends on the capacitances of the cavity we are considering.
It is apparent that the field is flat, if the voltages are near the mean
value Vcm, evaluated on the odd cavities. Therefore, the expression of
the mean square error σrms is defined as
σrms =
oddi
(Vcm − Vci) 2
(4.68)
<strong>and</strong> this value is an index for the flatness of the voltages in the equivalent
model, <strong>and</strong> therefore of the field in the cavities.
One can use the value (4.68) in the research of the best disposition of
the cavities after all the possible permutation. For example in figure 4.5
is shown the voltages on the capacitance of a generic disposition, where
the figure 4.6 shows the best disposition with the same errors. The
numeric values were obtained with the help of a MATLAB program.
5. Conclusion
In this short section we resume the results obtained with the perturbation
approach. Let us start from the resonant frequency of the
whole structure.
• The π/2 mode frequency error depends only on the accelerating
cavities errors <strong>and</strong> does not depend on the position of
cavities.
• Moreover, the frequency is the arithmetic mean of the accelerating
cavities frequencies.
• The variance of ∆ω is N times smaller than the one of the
r<strong>and</strong>om variable δ a ω0p.