Rohde & Schwarz ZNB20, User Manual

Page: 141 / 1332

Concepts and Features
R&S
®
ZNB/ZNBT
126 User Manual 1173.9163.02 ─ 38
Renormalization can be based on two alternative waveguide circuit theories whose
conversion formulas may yield different results if the reference impedance of at least
one test port has a non-zero imaginary part.
Conversion formula for wave quantities and S-parameters
Renormalization transforms the "raw" S-matrix S
0
for the default reference impedances
Z
0i
(with physical port number index i = 1,2,...,n) into a "renormalized" S-matrix S
1
for
the modified reference impedances Z
1i
. In terms of raw and renormalized wave quanti-
ties a
0i
, b
0i
and a
1i
, b
1i
, S
0
and S
1
are defined as follows:
.
... ...
;
... ...
1
12
11
1
1
12
11
0
02
01
0
0
02
01












 
























 












n n n n
a
a
a
S
b
b
b
a
a
a
S
b
b
b
The renormalized wave quantities (a
1
and b
1
) and the S-matrix S
1
can be calculated
from S
0
and the reference impedances Z
0i
, Z
1i
according to two alternative waveguide
circuit theories.
1. Travelling waves
In the model of Marks and Williams ("A General Waveguide Circuit Theory"), the
wave quantities a and b are transformed as follows:













 
 
 






i
i
i i i i
i i i i
i
i
i
i
i i
i
b
a
Z Z Z Z
Z Z Z Z
Z
Z
Z
Z
Z b
a
0
0
1 0 1 0
1 0 1 0
0
1
1
0
0 1
1
) Re(
) Re(
2
1
The renormalized S-matrix S1 is calculated as
  
P S E S P S
1
0 0
1
1

     
with the unit matrix E and two additional matrices with the elements
i i
i i
ii
Z Z
Z Z
0 1
0 1




) Re(
) Re( 2
1
0
0
1
1 0
0
i
i
i
i
i i
i
ii
Z
Z
Z
Z
Z Z
Z
P


2. Power waves
In the model of Kurokawa ("Power Waves and the Scattering Matrix"), the wave
quantities a and b are transformed as follows:













 
 
 






i
i
i i i i
i i i i
i i
i
i
b
a
Z Z Z Z
Z Z Z Z
Z Z
b
a
0
0
1 0 1 0
1 0 1 0
1 0
1
1
) Re( ) Re( 2
1
Measurement Results