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

120
100
80
60
40
20
0
(Vci − 0.999) · 10 5
77
0
5. CONCLUSION 65
100
0
82
0
114
1 2 3 4 5 6 7 # cell
Figure 4.5. The relative level of field in the cavities
for a generic disposition of the cavities, N = 8,
δω0/ω0 = 0.001 <strong>and</strong> K = 0.04 <strong>and</strong> σrms = 8.81 · 10 −8 .
The computed values are normalized to the unity, but are
shown using the formula (Vci − 0.999) · 10 5 for a better
visualization.
Even though, it is worth remember that these results were obtained
under the hypothesis of negligible non-adjacent cavities coupling <strong>and</strong>
of infinite quality factors for all the cavities.
The second point, which should be true also without the previous
hypothesis, leads to very interesting tuning procedure of such a
structure. One could think to act on all the accelerating cavities frequencies
of the same quantity, in order to change only the mean value.
This point simplifies the tuning procedure <strong>and</strong> is discussed in the next
chapter.
Finally, let us give some comments on the third point: the δω0p is
due to the machining tolerances <strong>and</strong> one can assume that quantity as
a r<strong>and</strong>om gaussian variable with a mean equal to zero <strong>and</strong> a variance
equal to σ 2 . It follows that the error ∆ω is still a gaussian variable
whose variance is σ 2 ∆ω = σ2 /N.
About the flatness of the axial field, starting from the expressions
(4.66) <strong>and</strong> (4.67), we have already identified the parameter (4.68) as
a good figure of merit. One could use a numerical program making
all the permutation <strong>and</strong> using σrms as optimization parameter. For
example, in the best disposition of figure 4.6, it is σrms = 4.54 · 10 −10 ,
where in the generic disposition of figure 4.5, it is σrms = 8.81 · 10 −8 .
Concerning this last point, it is worth noting that the errors in the cavities
are experimentally measured <strong>and</strong>, as we state in the next chapter,