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

46 4. CIRCUIT MODEL
where the matrix elements are defined as follows
t11 = V1
, t12 = V1
V2
I2=0
t21 = I1
V2
I2=0
, t22 = I1
I2
V2=0
I2
V2=0
The great advantage of such a representation is that a chain of dispositives
is simply represented by the product of the respective transmission
matrices, while others representation, such as resistances <strong>and</strong>
conductances matrices, do not allow this operation.
1.1. Transmission Matrix for a single cavity. As stated in
the previous chapter, a resonant cavity mode can be represented by the
behavior of an equivalent lumped circuit. In this paragraph we deduce
the transmission matrix of such a circuit. It should be better that the
transmission matrix represent two half cavities magnetically coupled;
this fact simplifies very much the building of the model, because it
well represents the tiles composing a LIBO tank, <strong>and</strong> therefore, in the
following, we call this circuit cell. In other words, a cell represents a
tile with two half cavities magnetically coupled.
Figure 4.2. The equivalent circuit for a single cell representing
two magnetically coupled half cavities.
Let us start from the calculation of the parameter t11; from the
figure 4.2 it is easy to write
V ′
1 = jω L
2 I1 − jωMI2, V ′
2 = jωMI1 − jω L
V1 =
R
2
+ 1
jω2C
L
+ jω I1 − jωMI2,
2
where V ′
1 <strong>and</strong> V ′
2 are the voltages on the inductances. When I2 = 0 it
is valid that
V2 = V ′
2 = jωMI1 → I1 = V2
jωM ,
,
.
2 I2