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

10 2. LINAC AND SCL ACCELERATORS
where φ is the phase advance for one period L. From the previous
(2.1), one obtains N + 1 solutions for φ
φ (n) = nπ
, n = 0, 1, . . . , N (2.2)
N
Therefore the phase advance assumes only a discrete set of values.
There are N + 1 solutions, each of them is called resonant mode of the
structure <strong>and</strong> it is characterized by its phase advance. The Brillouin
diagram is sampled in these cases, as it is shown in figure 2.3. It is
apparent that for N → ∞ the curve becomes continuous again.
f [GHz]
3.06
3.04
3.02
3
2.98
2.96
0 0.2 0.4 0.6 0.8 1
φ/π
Figure 2.3. N = 7 coupled cavities. The Brillouin diagram
is sampled in 7 frequencies. We can recognize
modes 0, π/2 <strong>and</strong> π.
Let us give some definitions on single cavities. There are several
figure of merit that are used to characterize the single cavity. The
first one, which is also used as a parameter in the equivalent lumped
circuits, is the quality factor, defined as
Q = ω W
P
, (2.3)
where ω is the resonant pulsation, W is the stored energy in the cavity
<strong>and</strong> P is the average power loss in the cavity. Another important
parameter is the shunt impedance defined as
Rsh =
0
L
2
Ez(z) dz
2P
, (2.4)