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Time Domain Reflectometry Theory
For Use with Keysight 86100 Infiniium DCA
Application Note
Introduction
The most general approach 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 (s) 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.
Introduction (continued)
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
04 | Keysight | Time Domain Reflectometry Theory - Application Note
Propagation on a Transmission Line
The classical transmission line is assumed to consist of a 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 wL
Zin = Zo ----------
G + jwC
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 b per unit length. Furthermore,
the voltage will be attenuated by an amount a per unit length by the series
resistance and shunt conductance of the line. The phase shift and attenuation
are defined by the propagation constant g, where
g = a + jb = (R + jwL) (G + jwC)
and a = attenuation in nepers per unit length
b = 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 b:
w
Where nr= -- Unit Length per Second
b
The velocity of propagation approaches the speed of light, nc, for transmission
lines with air dielectric. For the general case, where er is the dielectric constant:
nc
nr= ------
er
05 | Keysight | Time Domain Reflectometry Theory - Application Note
The propagation constant g can be used to define the voltage and the current at
any distance x down an infinitely long line by the relations
Ex = Eine