Digital Signal Processing Reference
In-Depth Information
a
1
a
2
b
1
b
1
Port 1
Two Port Network
Port 2
R
R
Figure 9-7
Two-port model used to define
S
-parameters.
losses. Similar to the derivation of the Poyting vector in Section 2.6, the power
balance equation can be expressed as
P
out
=
P
input
−
P
loss
−
P
radiated
(9-10)
where
P
input
is the total power injected into all ports,
P
out
is the total power
flowing out of all ports,
P
loss
is the power dissipated through ohmic losses (skin
effect, loss tangent, etc.), and
P
radiated
is the power radiated into free space.
The incident and reflected power waves are calculated from voltage and current
waves. The voltage waves are obtained from equation (6-49),
v(z)
=
v(z)
+
e
−
γz
+
v(z)
−
e
γz
and the current wave is derived by dividing the voltage wave by the characteristic
impedance of the structure,
v(z)
+
e
−
γz
Z
0
v(z)
−
e
γz
Z
0
i(z)
=
−
where
v(z)
+
+
z
-direction and
v(z)
−
is the voltage traveling in the
is the voltage
traveling in the
−
z
-direction. The power wave propagating on the network is
calculated by multiplying the current and voltage waves:
[
v(z)
+
e
−
γz
]
2
Z
0
[
v(z)
−
e
γz
]
2
Z
0
P(z)
=
+
(9-11)
If it is defined so that at port
j
,
z
=
0, the voltage at a port can be calculated where
v(
0
)
=
v
i
and
i(
0
)
=
i
i
, which are the incident voltage and current, respectively:
1
2
(v
i
+
Ri
i
)
v
+
=
(9-12)
1
2
(v
i
−
Ri
i
)
v
−
=
where
R
is the termination values at the ports of the network.
Since equation (9-10) says that the power must be balanced, the amount of
power delivered to the network or radiated is defined simply as the input power
minus the output power:
P
input
−
P
out
=
P
loss
+
P
radiated
Search WWH ::
Custom Search