Digital Signal Processing Reference
In-Depth Information
are wrong. A practical test for the passivity of
S
-parameters can be developed
directly from equations (8-22) and (8-23)[Ling, 2007].
Since power must be conserved, the power absorbed by the network (
P
a
)is
equal to the power driven into the network less power flowing out:
(
|
a
i
|
2
2
−|
b
i
|
)
=
P
a
(8-22)
where
P
a
≥
0 for a passive network. If
P
a
<
0, the network is generating power
and the system would be considered nonpassive. This allows equation (8-22) to
be written in terms of the power wave matrices that will produce a real value for
the power absorbed by the network. A system is passive if
a
H
a
b
H
b
−
≥
0
(8-23)
where
a
is a matrix that contains all the power waves incident to each port,
b
contains the power waves coming out of each port, and
x
H
indicates the
conjugate
transpose
(sometimes called the
Hermitian transpose
), which is calculated by
taking the transpose of
x
and then taking the complex conjugate of each entry.
Using the definition of
S
-parameters from equation (9-18),
the passivity
requirement of (8-23) can be rewritten as
Sa
,
b
H
S
H
a
H
b
=
=
a
H
a
S
H
a
H
Sa
−
≥
0
(9-68)
a
H
(
U
S
H
S
)
a
−
≥
0
where
U
is the unity (identity) matrix. Equation (9-68) leads to the general
requirement for passivity:
S
H
S
U
−
≥
0
(9-69)
If
S
H
S
is greater than 1, the requirement of (8-22) is violated and the system is
not passive.
A quick test to ensure passivity can be derived based on the eigenvalues of
S
H
S
. The eigenvectors
ζ
and eigenvalues
λ
are determined from the solution of
the equation
S
H
S
ξ
=
λξ
Techniques for calculating the eigenvectors and eigenvalues are detailed by
O'Neil [1991]; however, there are many software packages that can be used,
such as Mathematica or Matlab.
The eigenvectors can be formed into an
N
×
N
matrix,
ζ
1
1
ζ
2
1
···
ζ
N
1
ζ
1
2
ζ
2
2
···
ζ
N
2
V
=
[
ξ
1
ξ
2
···
ξ
N
]
=
.
.
.
.
···
ζ
1
N
ζ
2
N
···
ζ
N
N
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