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

52 4. CIRCUIT MODEL
<strong>and</strong> if N → ∞, namely the chain is infinite, i ′ also goes to the infinity,
tanh i ′ x = 1 <strong>and</strong> therefore
Vci
Vci+1
=
1
cosh x + sinh x
= 1
λ1
= λ2
(4.28)
Therefore, we obtain that the ratio is equal to the eigenvalue λ2 of the
transmission matrix T . It is worth noting that this could be a way to
obtain the dispersion diagram for an infinite chain <strong>and</strong> it is interesting
that this concept is related to the eigenvalues of the transmission
matrix.
2. The perturbed transmission matrix
This section deals with the case of a chain of equivalent circuits
with different lumped parameters, i.e. the inductances <strong>and</strong> capacitances
depend from the cavities.
This situation represents the realistic case of finite machining tolerance
<strong>and</strong> errors for the copper tiles containing two coupled half cells.
Such errors lead to a different resonant frequency <strong>and</strong> a different shunt
impedance for each half cell <strong>and</strong> could be interesting to find a connection
between the machining tolerances <strong>and</strong> the reachable relevant parameters
of the whole chain that are the π/2-mode resonant frequency
<strong>and</strong> the axial field.
In the equivalent model these errors are schematized by a shift in
the resonant frequency ω0 becoming ω0 + δω0i for each cavity.
2.1. The perturbed transmission matrix for one cell. Let
us consider the single two-ports shown in figure 4.3, which is composed
by two half cavities <strong>and</strong> where i is the two-ports index.
Figure 4.3. The two-ports representing two coupled
half cavities with different parameters.
The machining tolerances are represented by errors in the cavity
inductance <strong>and</strong> capacitance which are different for the left <strong>and</strong> right