HP 54753A - Page 184

TDR Fundamentals
Step Reflection Testing
9-10
Assuming Z
o
is real (approximately true for high quality commercial cable), it
is seen that resistive mismatches reflect a voltage of the same shape as the
driving voltage, with the magnitude and polarity of E
r
determined by the relative
values of Z
o
and R
L
.
Also of interest are the reflections produced by complex load impedances. Four
basic examples of these reflections are shown in Figure 9-7.
These waveforms could be verified by writing the expression for ρ(s) in terms
of the specific Z
L
for example:
multiplying ρ(s) by the transform of a step function of E
i
, and then
transforming this product back into the time domain to find an expression for
e
r
(t). This procedure is useful, but a simpler analysis is possible without
resorting to Laplace transforms. The more direct analysis involves evaluating
the reflected voltage at time t = 0 and at time t = ∞ and assuming any transition
between these two values to be exponential. (For simplicity, time is chosen to
be zero when the reflected wave arrives back at the monitoring point.) In the
case of the series R-L combination, for example, at t = 0 the reflected voltage
is +E
i
. This is because the inductor will not accept a sudden change in current;
it initially looks like an infinite impedance, and ρ = +1 at t = 0. Then current in
L builds up exponentially and its impedance drops toward zero. At t = ∞,
therefore e
r
(t) is determined only by the value of R.
The exponential transition of e
r
(t) has a time constant determined by the
effective resistance seen by the inductor. Since the output impedance of the
transmission line is Z
o
, the inductor sees Z
o
in series with R, and
Z
L
RsL
R
1 RCs+
--------------------
etc.,,+=
E
i
s
⁄
ρ
RZ
o
–
RZ
o
+
---------------
when
ι∞
==
ϒ
L
RZ
o
+
---------------=