XIX Sympozjum Srodowiskowe PTZE - materialy.pdf

XIX Sympozjum PTZE, Worliny 2009
Fig.2. Scattering parameters of plastic walls (Eik) and substrate under test (Sik)
Taking into account the reciprocal and source-free networks the scattering matrix satisfies:
Sik=Ski. In our case for symmetrical two-port networks: S11=S22, S12=S21 and also E11=E22,
E12=E21. For each network it is possible to formulate the following scattering equations
⎡b1
⎤ ⎡S
⎢ ⎥ = ⎢
⎣b2
⎦ ⎣S
11
21
S
S
12
22
⎤ ⎡a1
⎤
⎥ ⎢ ⎥
⎦ ⎣a2
⎦
Scattering equations (1) refer to single networks and for sample composed of 3 layers such
equations become a bit complicated. To receive mathematical formulas which allow to extract
scattering parameters of networks representing the powder substrate the so called graph
method of calculation has been introduce. The graph method has been derived from the theory
of power flow in the networks branches. As a example how to use graph method in practice
the reflection coefficient in the gate A-A (Fig.2) has been presented in Fig. 3.
A
E21
E22 S22
E22
E11 Eik
S11 Sik
E11 Eik
S22
ΓA S11 Sik E11 Eik
A
S21
S12
A
84
Fig.3. The graph method used for exampled
analyzing of the reflection coefficient in the gate
A-A.
The final formulas of SWik of whole sample (substrate under test and 2 plastic walls) are as
follow:
2
( 1−
E11S11)
S11
+ E11S21
SW11
= E11
+ E21
(2)
2 2 2
( 1 − E S ) − E S
E
2
S
S21
11
21 21
SW21
= (3)
2 ( 1− E11S11)
− E11S
21
where:
SWik – scattering parameters of whole sample
Values of unknown parameters of ε’, ε”, µ’ µ” can be obtained by solving the above complex
equations (2 and 3).
Receive data of complex relative permittivity εr and permeability µr of ferrite powders in
function of frequency will be presented.
11
11
21
E21
E12 S12
E21
A
(1)