VNA Concepts and Features
R&S
®
ZNL/ZNLE
229
User Manual 1178.5966.02 ─ 07
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
The renormalized S-matrix S1 is calculated as:
A
S
E
S
A
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
ii
ii
ii
ii
ii
A
1
1
1
8.3.7
Stability Factors
The stability factors K, μ1 and μ2 are real functions of the (complex) S-parameters,
defined as follows:
|
|
2
|
|
|
|
|
|
1
:
21
12
2
21
12
22
11
2
22
2
11
S
S
S
S
S
S
S
S
K
|
|
|
)
(
|
|
|
1
:
21
12
21
12
22
11
11
22
2
11
1
S
S
S
S
S
S
S
S
S
|
|
|
)
(
|
|
|
1
:
21
12
21
12
22
11
22
11
2
22
2
S
S
S
S
S
S
S
S
S
where denotes the complex conjugate of S.
Stability factors are calculated as functions of the frequency or another stimulus
parameter. They provide criteria for linear stability of two-ports such as amplifiers. A lin-
ear circuit is said to be unconditionally stable if no combination of passive source or
load can cause the circuit to oscillate.
●
The K-factor provides a necessary condition for unconditional stability: A circuit is
unconditionally stable if K>1 and an additional condition is met. The additional con-
dition can be tested with the stability factors μ
1
and μ
2
.
●
The μ
1
and μ
2
factors both provide a necessary and sufficient condition for uncondi-
tional stability: The conditions μ
1
>1 or μ
2
>1 are both equivalent to unconditional
Measurement Results
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