Rohde & Schwarz R&S ZNA Series - Page 153

Concepts and features
R&S
®
ZNA
153User Manual 1178.6462.02 ─ 29
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


















































nn
nn
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. Traveling 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
iiii
iiii
i
i
i
i
i
i
i
b
a
ZZZZ
ZZZZ
Z
Z
Z
Z
Z
b
a
0
0
1010
1010
0
1
1
0
0
1
1
)Re(
)Re(
2
1
The renormalized S-matrix S1 is calculated as:
  
PSESPS
1
00
1
1


 
with the unit matrix E and two additional matrices with the elements
ii
ii
ii
ZZ
ZZ
01
01


 
)Re(
)Re(
1
0
0
1
10
0
i
i
i
i
ii
i
ii
Z
Z
Z
Z
ZZ
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
iiii
iiii
ii
i
i
b
a
ZZZZ
ZZZZ
ZZ
b
a
0
0
1010
1010
10
1
1
)Re()Re(2
1
The renormalized S-matrix S1 is calculated as:
 
 
ASESAS
1
00
1
1



with the unit matrix E and two additional matrices with the elements
Measurement results