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

<strong>and</strong> then
1. TRANSMISSION MATRIX REPRESENTATION 47
V1 =
R
2
+ jω L
2
1 1
+
jω2C jωM V2
<strong>and</strong>, finally, the term t11 is
R L
t11 = + jω 1 −
2 2
1
ω2LC = 1
ω0
K jωQ + 1 − ω2 0
ω2
.
1
jωM
(4.1)
Note that ω0 = 1
√ LC is the resonant frequency of an half cavity, <strong>and</strong> of
a cell as well, Q is the quality factor, Q = ω0L
<strong>and</strong> we use the relation
R
K = 2M/L, leading to
t11 = 1
ω0
k jωQ + 1 − ω2 0
ω2
, (4.2)
Then, let us approach to the term t12:
<strong>and</strong>
<strong>and</strong> so
V ′
2 = jωMI1 − jω L
2 I2 =
R
2
I1 = I2
R 1
+
jωM 2 jω2C
1
+ I2
jω2C
L
+ jω
2
V1 = ( R 1 L
+ + jω
2 jω2C 2 )I1 − jωMI2
= ( R
1 L 1
+ + jω )2 − jωM I2
2 jω2C 2 jωM
= ( R 1 L
+ + jω
2 jω2C 2 )2
1
− 1 jωMI2
(jωM) 2
<strong>and</strong> by using the equation (4.2), one obtains
Next, we approach to the term t21
V1 =
(4.3)
t12 = jωM(t 2 11 − 1). (4.4)
R
2
+ 1
jω2C
<strong>and</strong> imposing again I2 = 0, one obtains
<strong>and</strong> finally
V2 = V ′
2 = jωMI1 → I1
t21 = 1
jωM
L
+ jω I1,
2
V2
= 1
jωM ,
(4.5)