Digital Signal Processing Reference
In-Depth Information
line terminated with a capacitor beginning at time
t
=
τ
d
,
=
v
ss
1
−
e
−
(t
−
τ
d
)/τ
]
v
capacitor
t>τ
d
(3-108)
where
τ
=
CZ
0
is the time constant,
τ
d
the time delay of the transmission line
as given by (3-107), and
v
ss
the steady-state voltage determined by the voltage
source
v
s
and the voltage divider between the source resistance
R
s
and the ter-
mination resistance
R
t
. Note that (3-108) is an approximation because it assumes
a step function with an infinite edge rate (i.e ., the rise time is infinitely fast).
Figure 3-39 shows the response of a line terminated with a capacitive load. The
waveform shape at node B follows equation (3-108). Notice that the waveform
at the source (node A) dips toward zero at
t
=
500 ps (which is 2
τ
d
)
because the
capacitor initially looks like a short circuit, so the reflection coefficient is
1.
Note that the shape of this wave at node A is also dictated by (3-108) when the
exponent term is
e
−
[
(t
−
2
τ
d
)/τ
]
, which simply shifts the time. The voltage reflected
back toward the source is initially
v
i
, where
v
i
is the initial voltage launched
onto the transmission line. After the capacitor is fully charged, it will look like
an open and have a reflection coefficient of
−
1. Consequently, the reflected wave-
form at the receiver will double. As seen in Figure 3-39b, the waveform at the
receiver (B) reaches a steady-state value of 2 V after about three time constants
[3
τ
=
+
3
(
50
)(
2pF
=
300 ps] after arriving at the receiver, just as circuit theory
predicts.
If the line is terminated with a parallel resistor and capacitor, as depicted in
Figure 3-40, the voltage at the capacitor will be dependent on the time constant
t
d
=
250
ps
v
s
R
s
=
50
Ω
B
A
Z
0
=
50
Ω
C
L
=
2pF
0-2
V
(a)
2.5
2
B
1.5
1
A
0.5
0
0
0.25
0.5
0.75
1
Time, ns
(b)
Figure 3-39
(a) Transmission line terminated with a capacitive load; (b) step response
showing reflections from the capacitor.
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