Published in Symposium and Workshop on Time Domain Reflectometry in Environmental, Infrastructure, and Mining Applications held at
Northwestern University, Evanston, Illinois, September 17-19, 1994 (Washington, DC: U.S. Bureau of Mines, 1994), pp. 4-13.
USBM special publication SP 19-94
by James R. Andrews, Ph.D., IEEE Fellow
President, Picosecond Pulse Labs, Inc.
P.O. Box 44, Boulder, CO 80306
(303) 443-1249
Time Domain Reflectometry, or TDR, is an electrical measurement technique that has been used for many years to determine the spatial location and nature of various objects. An early form of TDR was radar. This paper discusses the principles of TDR with an emphasis on TDR within coaxial transmission lines. A review will be made of commercially available, coaxial, TDR instruments, including a state-of-the-art, 10 ps resolution TDR.
Time Domain Reflectometry, or TDR, is a remote sensing electrical measurement technique that has been used for many years to determine the spatial location and nature of various objects. An early form of TDR, dating from the 1930s, that most people are familiar with is RADAR. Radar consists of: a radio transmitter which emits a short pulse of microwave energy, a directional antenna, and a sensitive radio receiver. After the transmitter has radiated the pulse, the receiver then listens for an echo to return from a distant object, such as an airplane or ship. By measuring the time from the transmitted pulse until the echo returns and knowing the speed of light, the distance to the reflecting object may be easily calculated. Detailed analysis of the echo can reveal additional details of the reflecting object which helps in identifying it. The same principles hold for radar, lidar, coax TDR, optical fiber OTDR and broadband impulse radars.
This paper discusses TDR as applied to coaxial transmission lines. Coaxial TDR is essentially a "closed circuit radar". It involves sending an electrical pulse along a coaxial cable and using an oscilloscope to observe the echos returning back to the input. This technique was reported in the literature in the 1930's and 40's for testing telephone coaxial cables (17, 7). Numerous TDR articles and books have been written on the subject since. A few of these are listed in the references ( 1, 8, 11, 12, 13, 15). Several references are good application notes (2, 3, 4, 9, 10) which are available from TDR manufacturers.
A coaxial TDR is usually configured as shown in FIGURE 1. The pulse generator is
represented by its Thevenin equivalent circuit of an open circuit voltage source,
, and its
source impedance, 
. It generates a fast rising step function, pulse waveform. This pulse is
launched into the coaxial cable. A high impedance oscilloscope is bridged across the input
to the coaxial cable. If a sampling oscilloscope is used, then this is usually a feed-thru
sampling head. There are also other alternate arrangements such as a terminated sampler and
feed-thru pulser, or combining a pulse generator and sampler with a matched resistive tee. A
coaxial cable designated as the "Reference Cable" is connected to the TDR output port. If the
ref. cable impedance, Ro, is known, then quantitative results may be obtained from TDR
measurements. Usually the reference cable impedance is matched to the generator impedance,
i.e. 
. This is done to prevent reflections which come back into the TDR from being
rereflected and corrupting the test results. At the far end of the reference cable is the
"Reference Plane". This is the port at which the unknown impedance, 
,
is attached.
In coaxial TDR the pulse generator sends a pulse through the sampler into the reference
coax. The pulse propagates through the coax at a velocity,
, and arrives at the far end after
a time TD.
Where c is the speed of light 
and k is the relative dielectric constant of the
coaxial transmission line. If the load impedance matches the coax impedance,
,
then the TDR pulse is perfectly absorbed. However if is not
equal to 
, some of the
incident pulse energy will be reflected back to the left towards the generator. This reflected
pulse will arrive back at the TDR output port at t = 2TD. The feed-thru oscilloscope allows
us to visually see the total waveform, 
, at the cable
input. is the algebraic sum
of the pulse generator outgoing step pulse and any returning echoes. An examination of the
time delay and wave shape of the echoes present on allows
us to determine the location
and nature of discontinuities within the coax and/or mismatched terminations to the coaxial
transmission line.
TDR waveforms observed for various resistive terminations, 
, are shown in
FIGURE 2. For
this and the rest of the TDR discussion we will assume that the generator
produces a step
function pulse. If Rt = Ro, then no reflection occurs and the TDR display
on the scope is a flat
line. If is greater than
, then a positive step is observed. For
, a negative step
is observed. The actual value of
may be calculated from
the size of the incident step 
,
and the reflected pulse, 
. The reflection coefficient,
rho, is defined as:
Note that may have either a positive or negative value and likewise for rho. Transmission
line analysis shows rho is also given by
Rearranging terms in equation (4) allows us to solve for .
For the matched case, 
and rho = 0. For an open circuit,
rho = +1. For a short circuit,
and rho = -1. Equations (3)-(5) hold for both pure resistive terminations and
connections to other transmission lines of different characteristic impedances.
Reactive components can also be measured using TDR. FIGURE 3 shows the TDR waveforms obtained for terminations of simple inductors or capacitors. The inductor initially appears as an open circuit to the fast rising edge of the TDR step pulse. (i.e. the high frequencies). Thus it initially gives a rho of +1. Later in time, the inductor appears as a short circuit to the flat top of the TDR step pulse (i.e. the DC portion). Thus the final TDR value is a rho of -1. The capacitor performs exactly opposite. The L or C value may be determined by measuring the exponential time constant, tau, of the TDR response.
FIGURE 4 shows a very common situation encountered when dealing with coaxial cables. These are the cases of either a series inductance, L, or a shunt capacitance, C, within the coax. This could be caused by a poor connector for example. Again by measuring the time constant, tau, L or C may be determined.
The TDR waveforms shown in
FIGURES 3 and 4 were for the case when the TDR step waveform is ideal with a risetime of zero
picoseconds. With a finite risetime pulse generator and oscilloscope, these waveforms will
no longer have sharp corners, but will have smooth rounded corners. For "Large" inductors
or capacitors, their time constants will be much greater than the system risetime,
, and the
leading edge will still go to a reflection coefficient of +1 or -1 respectively. In these cases, the
TDR displays will still be as shown in FIGURES 3
and 4. No visible reflections will occur for
"Tiny" inductors or capacitors with time constants much less than the TDR system risetime,
. Thus they can not be measured.
FIGURE 5 shows the TDR displays that will occur for "Small" inductors and capacitors. This
is the situation when the time constant associated with the reactive component is of the same
order of magnitude as the TDR system risetime, . For these "Small" reflections the max.
or min. reflection coefficient is always less than +1 or -1. These "Small" inductors or
capacitors can be measured even though their TDR responses are not the same as
FIGURES 3 and 4. Measure ,
, and 
as shown in
FIGURE 5. For reflections of less than 10%, equations
(10) and (11) are good approximate formulas to calculate L and C (5).
For impedance discontinuities located
very close to the TDR output port, the risetime of the displayed output step,
FIGURE 5, may be
used as . However, if a long cable is present between the TDR output and the reference
plane at which the measurements are to be made, then the effective risetime will be slower
than the input risetime due to the cable pulse response. In this case the system should first be
calibrated by attaching a short circuit at the reference plane and measuring the falltime,
, of
the reflection from the short, FIGURE 6. Use the short circuit
falltime in place of in equations
(10) and (11).
FIGURE 7 illustrates the TDR display for two closely spaced discontinuities. The minimum
temporal resolution is the system risetime, . The minimum spatial resolution,
, is given by:
As an example, with a 10 ns risetime TDR system and ordinary RG-58 coax (k = 2.2), the resolution would be 1 meter. For a typical ultra-fast TDR with a 30 ps generator and a 17 ps scope, the TDR system risetime would be 35 ps. The resultant minimum spatial resolution would be 3.5 mm (for k=2.2).
If a computer is available, considerably more information can be extracted from TDR waveforms (1). The frequency domain, scattering parameter S11(f) can be computed using Fast Fourier Transforms (FFT).
is the measured reflection waveform from a short circuit,
FIGURE 6. 
is the
measured TDR waveform of the device under test.
A complete Time Domain Network Analyzer (TDNA) can be built to measure all "S" parameters if one includes an additional sampling head to measure Time Domain Transmission (TDT) waveforms in addition to TDR waveforms (1). Recent work by various university researchers (6, 16) has resulted in calibration schemes for TDNAs which allow them to now approch the performance of frequency domain network analyzers.
If one has additional knowledge of the termination, such as the geometry, then it is possible using modeling to determine other parameters such as dielectric and permability functions (5). Picosecond Pulse Labs uses the computer program PSPICE to simulate the response of its electronic circuits to various transient signals such as TDR pulses. Comparing the computer simulation with actual measured TDR and TDT responses allows us to better understand our circuits and improve our designs.
Using deconvolution techniques (1, 14) and digital filtering (10) it is also possible with a computer to correct a smeared out TDR waveform, such as FIGURE 5, to give a sharper TDR waveform that would result if a near perfect step function had been used as the test signal. All TDRs show rather "dirty" TDR reference baselines that are corrupted by small bumps and wiggles that are present on the step generator pulse and also minor reflections within the sampler. Deconvolution effectively removes most of these effects. It can also give an effective increase in bandwidth and reduction in the TDR system risetime and hence its spatial resolution. Software signal processing can also slow down the TDR's system risetime if desired to show what reflections might occur in a transmission system when slower risetime pulses are used.
The above examples all used a "step" pulse as the TDR test signal. The "step" is very useful because the long step plateau conveys "DC" information about the reflecting object, while the very fast risetime of a step pulse contains very high frequencies and thus gives "HF" information and good spatial resolution. Other pulse types are also used, depending upon the application. See FIGURE 8. The radar pulse discussed earlier was a short pulse of a sine wave which was turned on briefly. This signal is ideal for testing narrow-band systems, such as waveguides. For systems that do not support DC, a narrow impulse is often used. For systems where the test TDR pulse is radiated from broadband antennas, either the impulse or monocyle pulse is used.
REVIEW OF COMMERCIALLY AVAILABLE COAXIALTDRs
An inexpensive TDR can be assembled for under $1,500 using a conventional 100 MHz oscilloscope and a general purpose pulse generator. Inexpensive pulse generators with < 5 ns risetimes are available. Or, one could build a simple pulse generator circuit with fast TTL ICs. A 20 dB BNC attenuator should be attached to the pulse generator output to assure a good 50 Ohm generator source impedance. A BNC tee adapter is connected directly to the scope 1 MOhm input. Attach the pulse generator and attenuator to one arm of the tee. The other arm is the TDR output port. Do not use any coax cable between the BNC tee and the scope input as this would cause undesirable multiple reflections. A 5 ns risetime pulser and a 3.5 ns risetime, 100 MHz scope will give a 6 ns TDR with spatial resolution of 60 cm in ordinary coax cable.
Since the 1960's, the major oscilloscope companies have all built dedicated TDR instruments or oscilloscopes that include TDR capabilities. By the late 60's, TDRs were available which had 35 ps risetime capabilities. Many of these older TDRs with risetimes from 1 ns to 35 ps have been discontinued, but are still available, relatively inexpensively, on the used equipment market. The TDRs built today by the oscilloscope companies are completely digital, programmable instruments with many features. There are also several smaller companies making specialized TDRs. TABLE 1 lists the very fast (< 200 ps) coaxial TDRs which are now available. The HyperLabs instrument is unique in that it is a plug-in card for a PC computer with a small remote TDR pulser/sampling head.
The commercial, "state-of-the-art" in coaxial cable TDRs is a 10 ps risetime instument made by Hewlett-Packard and Picosecond Pulse Labs (PSPL) (3). This consists of the HP-54124A, 50 GHz, 9.4 ps risetime, digital sampling oscilloscope and the PSPL 4015B, 9 Volt, 15 ps pulse generator. The basic system risetime is 18 ps. However, the HP-54120B oscilloscope includes built-in firmware programs which allow one to make "normalized" measurements which include deconvolution and digital filtering as mentioned earlier. The result of combining this very fast sampler and pulser plus the microprocessor is a powerful TDR instrument with 10 ps risetime. This is equivalent to a spatial resolution of 1.0 mm in coaxial cable (k=2.2).
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