Digital Signal Processing Reference
In-Depth Information
The frequency-dependent delay can be calculated from the phase of the inser-
tion loss,
θ
S
21
360
◦
f
τ
p
=
(9-22)
where
arctan
Im
(S
21
)
Re
(S
21
)
θ
S
21
=
(9-23)
Note that equation (9-22), called the
phase delay
, has a slightly different meaning
than that of the propagation delay of a digital pulse on a transmission line.
Since a digital pulse is composed of numerous harmonics, and realistic dielectrics
have values that vary with frequency, the propagation delay of a time-domain
pulse will have numerous frequency components, each propagating with a unique
phase velocity. The propagation delay of a digital pulse can be thought of as a
group of harmonics propagating simultaneously and is sometimes called the
group
delay
. The phase delay, as described with (9-22), is associated with only a single
frequency.
Figure 9-12a shows an example of the phase of
S
21
for a 1-in. transmission-line
model constructed with a realistic frequency-dependent dielectric, as described in
Chapter 6. To calculate the phase delay, the phase must be unwrapped, as shown
in the figure. Figure 9-12b is the phase delay calculated from the unwrapped
phase using equation (9-22). Note the frequency-dependent nature of the delay,
which is required for a causal transmission-line model.
For a loss-free network, the total power exiting the
N
ports must equal the
total incident power. This means that for a two-port loss-free network, the power
transmitted from port 1 to port 2 plus the power reflected from port 1 must be
conserved:
P
reflected,port1
P
incident
P
transmitted,port1
−
port2
P
incident
+
=
1
(9-24)
Essentially, equation (9-24) says that if the power is not transmitted from port 1
to port 2, it must be reflected. This allows us to write an equation that relates the
168
50
0
166
−
50
−
100
164
−
150
−
200
162
−
250
0.0
1.0
2.0
3.0
4.0
5.0
0.0
1.0
2.0
3.0
4.0
5.0
Frequency, GHz
Frequency, GHz
(a)
(b)
Figure 9-12
(a) Phase of the insertion loss (
S
21
) for a 1-in. causal transmission line
model; (b) phase delay.
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