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

48 4. CIRCUIT MODEL
For the latest term of the matrix, it is valid that
I1 = ( R
2
<strong>and</strong> again
V ′
2 = jωMI1 − jω L
2 I2 =
+ 1
jω2C
R
2
+ 1
jω2C
L I2
+ jω )
2 jωM → t22 = ( R 1
+
2 jω2C
I2
L 1
+ jω )
2 jωM
t22 = t11. (4.6)
Then we can write the whole transmission matrix
⎛
t11 (t
⎜
T = ⎝
2 ⎞
11 − 1)jωM
⎟
1
⎠ (4.7)
t11
jωM
<strong>and</strong> it is apparent that the matrix determinant is unitary, as expexted
for a reciprocal system.
In order to represent a chain of cavities, in the following sections
we use powers of the matrix T . In this sense a spectral decomposition
of matrix T would be useful, being
T = UΛU −1 → T N = UΛ N U −1 , (4.8)
where U is the eigenvectors matrix <strong>and</strong> Λ is the eigenvalues diagonal
matrix, namely
λ1
Λ =
0
0
λ2
(4.9)
where λ1 <strong>and</strong> λ2 are solutions of the characteristic polynomial
det(T − ΛI) = 0 → λ1,2 = t11 ±
<strong>and</strong> by introducing a new variable x, such that
t 2 11 − 1 (4.10)
t11 = cosh x, (4.11)
we obtain a rationalization of the expressions
cosh x + sinh x
Λ =
0
⎛
jωM sinh x
0
cosh x − sinh x
⎞
1
(4.12)
⎜
⎟
,U = ⎝
j
⎠
1
ωM sinh x
,U
(4.13)
−1 = 1
⎛
⎞
j
−
1
⎜
⎝
ωM sinh x
⎟
⎠ .
2
1 −jωM sinh x
(4.14)