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Time Domain Reflectometry
Theory
Application Note 1304-2
For Use with
HP 54750A and
HP 83480A Mainframes
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Introduction The most general aproach to evaluating the time domain response
of any electromagnetic system is to solve Maxwell's equations in the
time domain. Such a procedure would take into account all the
effects of the system geometry and electrical properties, including
transmission line effects. However, this would be rather involved
for even a simple connector and even more complicated for a
structure such as a multilayer high speed backplane. For this
reason, various test and measurement methods have been used to
assist the electrical engineer in analyzing signal integrity.
The most common method for evaluating a transmission line and
its load has traditionally involved applying a sine wave to a system
and measuring waves resulting from discontinuities on the line.
From these measurements, the standing wave ratio () is
calculated and used as a figure of merit for the transmission
system. When the system includes several discontinuities, however,
the standing wave ratio (SWR) measurement fails to isolate them.
In addition, when the broadband quality of a transmission system
is to be determined, SWR measurements must be made at many
frequencies. This method soon becomes very time consuming and
tedious.
Another common instrument for evaluating a transmission line is
the network analyzer. In this case, a signal generator produces a
sinusoid whose frequency is swept to stimulate the device under
test (DUT). The network analyzer measures the reflected and
transmitted signals from the DUT. The reflected waveform can be
displayed in various formats, including SWR and reflection
coefficient. An equivalent TDR format can be displayed only if the
network analyzer is equipped with the proper software to perform
an Inverse Fast Fourier Transform (IFFT). This method works well
if the user is comfortable working with s-parameters in the
frequency domain. However, if the user is not familiar with these
microwave-oriented tools, the learning curve is quite steep.
Furthermore, most digital designers prefer working in the time
domain with logic analyzers and high speed oscilloscopes.
When compared to other measurement techniques, time domain
reflectometry provides a more intuitive and direct look at the DUT's
characteristics. Using a step generator and an oscilloscope, a fast
edge is launched into the transmission line under investigation.
The incident and reflected voltage waves are monitored by the
oscilloscope at a particular point on the line.
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This echo technique (see Figure 1) reveals at a glance the
characteristic impedance of the line, and it shows both the position
and the nature (resistive, inductive, or capacitive) of each
discontinuity along the line. TDR also demonstrates whether losses
in a transmission system are series losses or shunt losses. All of
this information is immediately available from the oscilloscope's
display. TDR also gives more meaningful information concerning
the broadband response of a transmission system than any other
measuring technique.
Since the basic principles of time domain reflectometry are easily
grasped, even those with limited experience in high frequency
measurements can quickly master this technique. This application
note attempts a concise presentation of the fundamentals of TDR
and then relates these fundamentals to the parameters that can be
measured in actual test situations. Before discussing these
principles further we will briefly review transmission line theory.
X
e x (t)
Ei ex Zo ZL
Ei +Er
Ei
Zo Z L
t
Transmission Line Load
Figure 1. Voltage vs time at a particular point on a mismatched
transmission line driven with a step of height Ei
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Propagation on a The classical transmission line is assumed to consist of a
Transmission Line continuous structure of R's, L's and C's, as shown in Figure 2. By
studying this equivalent circuit, several characteristics of the
transmission line can be determined.
If the line is infinitely long and R, L, G, and C are defined per unit
length, then
R + j L
Zin = Zo ----------
G + jC
where Zo is the characteristic impedance of the line. A voltage
introduced at the generator will require a finite time to travel down
the line to a point x. The phase of the voltage moving down the line
will lag behind the voltage introduced at the generator by an
amount per unit length. Furthermore, the voltage will be
attenuated by an amount per unit length by the series resistance
and shunt conductance of the line. The phase shift and attenuation
are defined by the propagation constant , where
= + j = (R + jL) (G + jC)
and = attenuation in nepers per unit length
= phase shift in radians per unit length
ZS L R L R
ES C G C G ZL
Figure 2. The classical model for a transmission line.
The velocity at which the voltage travels down the line can be
defined in terms of :
Where = -- Unit Length per Second
The velocity of propagation approaches the speed of light, c, for
transmission lines with air dielectric. For the general case, where
er is the dielectric constant:
c
= ----
er
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The propagation constant can be used to define the voltage and the
current at any distance x down an infinitely long line by the relations
Ex = Eine