Environmental Engineering Reference
In-Depth Information
at the
i
th
port, when a matched generator is connected to the
i
th
port and all the other
ports are terminated with a matched load (i.e.,
a
j
=
0
,∀
i
=
j
)[30]:
b
i
a
i
|
a
j
=
0
,∀
j
=
i
.
S
ii
=
(2.24)
The terms
S
ij
are referred to as transmission scattering parameters. They are mea-
sured as the ratio of the outgoing wave at the
i
th
port (
b
i
) and the incoming wave at
the
j
th
port (
a
j
) when a matched generator is connected at port
j
and the other ports
are connected to a matched load:
b
i
a
j
|
a
i
=
0
,∀
i
=
j
.
S
ij
=
(2.25)
Often, the matrix notation is used
⎡
⎤
⎡
⎤
⎡
⎤
b
1
b
2
...
...
b
N
S
11
S
12
... ...
S
1
N
a
1
a
2
...
...
a
N
⎣
⎦
⎣
⎦
⎣
⎦
S
21
... ... ...
...
=
...
... ... ...
...
×
(2.26)
...
... ... ...
...
S
N
1
... ... ...
S
NN
=
For a reciprocal network, the scattering matrix is symmetrical, i.e.
S
mn
S
nm
,
=
,
, ...,
(
m
N
). As can be seen from the previous equations, to measure the
generic scattering parameter
S
ij
, which involves ports
i
and
j
, the remaining ports
have to be terminated with matched loads
1
(rather than with short or open circuits, as
must be done with other network parameters). This is great advantage of character-
izing electrical networks through
S-parameters
; in fact, at microwave frequencies,
open and short circuits are more difficult to implement and may lead to instability
(severe distortion or oscillation may occur) [29].
n
1
2
Fig. 2.7
Two-port network
scheme
Fig. 2.7 shows the schematic of a two-port network:
I
1
and
I
2
are the currents
at the terminals;
V
1
and
V
2
are the voltages at the terminals; and
a
and
b
are the
traveling waves. The square of the coefficient represents the power:
2
can be considered as the power entering
port 1
and exiting
port 1
, respectively. For
two-port networks, the system can be modeled by two equations:
2
|
a
1
|
and
|
b
1
|
1
Clearly, when performing reflection measurements (i.e.,
S
ii
), all ports except the
i
th
port
should be terminated with a matched impedance.
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