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

2. THE PERTURBED TRANSMISSION MATRIX 53
sides of the two-ports. In this case, the transmission matrix becomes
⎛
⎞
Ti =
where we have defined
t L 11i = 1
1 −
Ki
t R 11i = 1
Ki
1 −
⎜
⎝
ω L 0i
ω
t L 11i
1
jωM
2
L L i
L R i
R 2
ω0i L
ω
R i
LL i
(t L 11it R 11i − 1)jωM
t R 11i
∼ = 1
Ki
∼ = 1
Ki
1 −
1 −
⎟
⎠ (4.29)
L 2
ω0i ω
R 2
ω0i ω
(4.30)
Therefore, starting from the scheme of figure 4.3 we introduce the following
notation
C L i = 2C + δC L i
L L i = L
2 + δLL i
C R i = 2C + δC R i
L R i = L
2 + δLR i
(4.31)
where the apexes L <strong>and</strong> R indicate respectively the left <strong>and</strong> the right
side half cell parameters. If ω L 0i <strong>and</strong> ω R 0i are the resonant frequencies
then
ω L 0i = ω0 + δω L 0i =
ω R 0i = ω0 + δω R 0i =
1
L Li CL i
1
R Li CR i
(4.32)
Let us find now the frequencies perturbations δω L 0i e δω R 0i. The perturbation
on the resonant frequency can be related to the inductance <strong>and</strong>
capacitance in the following way
ω0 = 1
√ LC → δω0
ω0
then the perturbations for both sides are
δωL 0i
= −
ω0
1
L δLi 2 LL i
δωR R
0i δLi ω0
= − 1
2
= − 1
δL δC
+ , (4.33)
2 L C
L R i
+ δCL i
C L i
+ δCR i
C R i
(4.34)
The terms in parenthesis can be assumed of the same order of magnitude,
because they are consequence of the same machining tolerances.