Digital Signal Processing Reference
In-Depth Information
permittivity is frequency dependent. Furthermore, the relationship between the
dielectric permittivity and the dielectric losses must be maintained; otherwise,
energy will propagate when it should be attenuated, inducting noncausal errors.
The dielectric models described in Chapter 6 will produce causal responses when
used to simulate transmission lines. Unfortunately, many commercially available
simulators do not account properly for the frequency dependence of the dielectric.
Consequently, the engineer should be very wary of prepackaged transmission-line
models to ensure that physical behavior is being observed in the simulations.
In reality, performing the Hilbert transform analytically is difficult, and numer-
ical methods have to be used. Also, with bandlimited frequency responses, the
Hilbert transform may not be a good check for causality, due to aberrations in
the time-domain waveform, which is discussed briefly in Example 9-8.
8.2.2 Passivity
A physical system is
passive
when it is unable to generate energy from within.
For example, an
n
-port network is said to be passive if
t
v
T
(τ )
·
≥
i
(τ ) dτ
0
(8-21)
−∞
where
v
T
(τ )
is the transpose of a matrix containing the port voltages and
i
(
τ
)is
a matrix containing the currents. The integral (8-21) represents the cumulative
net power absorbed by the system up to time
t
. In a passive system, this quantity
must be positive for all
t
.
A more useful approach for digital designers would be to test the passivity in
terms of the incident (
a
i
) and exiting (
b
i
) power waves at each port, as defined
in Section 9.3 and shown in Figure 8-19.
2
|
a
i
|
=
power incident to node
i
2
|
b
i
|
=
power flow out of node
i
This approach is particularly useful because in Section 9.3.1 it will be developed
into a practical passivity test using
S
-parameters.
Since power must be conserved, the power absorbed by the network (
P
a
)is
equal to the power driven into the network minus the power flowing out:
|
a
i
|
=
P
a
2
2
−|
b
i
|
(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.
When working in the frequency domain, the power waves
a
i
and
b
i
will be
complex. Since power is real, equation (8-22) must be implemented using the
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