## Transmission Lines and Time Domain Reflectometers

The behaviour of electromagnetic wave reflection occurs not only with electromagnetic waves propagating through the air, but also with signals propagating through electrical wires and cables. When electromagnetic waves in a wire reach a point where the properties of the wire change, a reflection can occur. Of course this reflection cannot just occur in any old direction, it must occur along the cable itself, so when a signal is reflected, it travels back along the cable in the opposite direction from which it came.

Reflections in a cable or wire occur when there is a discontinuity (a change in impedance) of some kind. This discontinuity may be one of the following:

- A connector connecting the cable to equipment or another cable.
- A change in the size of the cable
- The end of the cable (it may be left open, or it may be shorted out)
- A fault orbreak in the cable which may be a complete break or just a partial one.
- Any other case where there is a change in impedance along the path the signal is propagating

Depending on the type of impedance mismatch the signal encounters, the signal may be partially or fully reflected. If the signal reached the end of the cable, and there was no where else to go (because the end was open, or it was shorted to ground), all of the energy will be reflected back to the source. If the signal hits a discontinuity that is not the end of the cable (e.g., a connector or a partial break of the cable), then part of the signal will be reflected, and the rest will propagate forward (but with a loss of some power due to the reflection).

A time domain reflectometer (TDR) takes advantage of this behaviour to determine the location of faults and breaks in cables.

Wikipedia has a good description of how a TDR is used (from http://en.wikipedia.org/wiki/Time-domain_reflectometer):

Consider the case where the far end of the cable is shorted (that is, it is terminated into zero ohms impedance). When the rising edge of the pulse is launched down the cable, the voltage at the launching point “steps up” to a given value instantly and the pulse begins propagating down the cable towards the short. When the pulse hits the short, no energy is absorbed at the far end. Instead, an opposing pulse reflects back from the short towards the launching end. It is only when this opposing reflection finally reaches the launch point that the voltage at this launching point abruptly drops back to zero, signaling the fact that there is a short at the end of the cable. That is, the TDR had no indication that there is a short at the end of the cable until its emitted pulse can travel down the cable at roughly the speed of light and the echo can return back up the cable at the same speed. It is only after this round-trip delay that the short can be perceived by the TDR. Assuming that one knows the signal propagation speed in the particular cable-under-test, then in this way, the distance to the short can be measured.

A similar effect occurs if the far end of the cable is an open circuit (terminated into an infinite impedance). In this case, though, the reflection from the far end is polarized identically with the original pulse and adds to it rather than cancelling it out. So after a round-trip delay, the voltage at the TDR abruptly jumps to twice the originally-applied voltage.

Note that a theoretical perfect termination at the far end of the cable would entirely absorb the applied pulse without causing any reflection. In this case, it would be impossible to determine the actual length of the cable. Luckily, perfect terminations are very rare and some small reflection is nearly always cause

The magnitude of the reflection is referred to as the reflection coefficient or ρ. The coefficient ranges from 1 (open circuit) to -1 (short circuit). The value of zero means that there is no reflection. The reflection coefficient ( \rho) is calculated as follows:

\rho = \frac{Z_t - Z_0}{Z_t + Z_0}Where Z_{o} is defined as the characteristic impedance of the transmission medium and Z_{t} is the impedance of the termination at the far end of the transmission line.

Any discontinuity can be viewed as a termination impedance and substituted as Z_{t}. This includes abrupt changes in the characteristic impedance. As an example, a trace width on a printed circuit board doubled at its midsection would constitute a discontinuity. Some of the energy will be reflected back to the driving source; the remaining energy will be transmitted.

The Wikipedia entry has a few terms that need a little bit of explanation

**Impedance**is essentially the amount of opposition a cable or wire has to the current of a signal propagating down a cable. Impedance does not slow a signal down, but reduces the amount of current that can flow for a given voltage. Impedance can also cause a phase shift between voltage and current**Transmission Line**is simply two long conductors, electrically isolated from each other (except perhaps at the end) but somehow physically connected together. This “physical connection” may be that the two wires are twisted together – giving a twisted pair, it may consist of one wire conductor with the other conductor completely wrapped around it – giving a co-axial cable, or it may even consist of two traces on a printed circuit board. When we talk about “long” conductors, we are using the term “long” in a relative manner. It is long relative to the wavelength of the signals that are passing through it. If the conductors are longer than 1/100^{th}of the wavelength of the signal passing through it, then you have a transmission line.**Characteristic Impedance**is the impedance that a particular transmission line has.

Eric Bogatin uses an analogy for helping you to think about transmission lines. He says, you need to, in a zen-like manner, “be the signal”. Imagine you are a signal, just leaving the transmitter on your way down a transmission line. With your first step, you charge the line where you step to 1 volt. The transmission line in front of you is still at 0 volts, because you have not reached there yet. With every successive step, you are probing to see how much current it takes to charge the line up to 1 volt. Since impedance equals volt/current, as long as the amount of current it takes remains the same, the impedance that you see remains the same. As long as the impedance that you see remains the same, you can continue on forever.

You take a few more steps, pull more charge from the battery, and leave behind you a wake of charge keeping the line at 1 volt. The line in front of you is still at 0 volts. As long as the impedance to your movement remains the same, you can continue on forever. As soon as you hit somewhere where the impedance changes, you will reflect a little bit (or a lot) of energy back in the direction from which you came.

There are two important things to glean from this analogy. In a transmission line, it takes a finite length of time for a signal to propagate from one end to the other; if the line is long enough, there will be several cycles of a wave, or pulses of a signal on it at one time. Also, a more complete definition of **characteristic impedance **is that it is the instantaneous impedance that a signal sees as it propagates down the line. If the signal sees a change in the impedance for some reason, then some of the signal will be reflected back towards the source.

**Back to TDRs**

So you thought you were going to get a little description of what time domain reflectometry is, and what you ended up with was an in depth discussion of transmission lines. Now, let’s get back to TDRs. A TDR sends a pulse down a transmission line, and waits for any reflection that may come back. If there is some sort of impedance discontinuity, then there will be a reflection. The only time there will not be a reflection is if the signal path is terminated to the return path via an impedance that is equal to the characteristic impedance of the signal and return paths. In general, there will not be a perfect impedance match, so there will be some amount of reflection. When the TDR receives the reflection, it measures the time difference between sending the pulse out, and receiving the reflection back. Using that time, and the speed of light through the transmission line, you can calculate the distance to the impedance mismatch.

The speed of light through transmission lines is a noticeably smaller fraction of the speed of light through a vacuum (*c*). Typically it is between 0.4*c* and 0.7*c*. This fraction is called the *velocity factor*.

Another way to look at impedance mismatch is to consider the Voltage Standing Wave Ratio, or VSWR which is a way of identifying the wave resulting from the sum of the incident wave and the reflected wave. If you send a sine wave down a transmission line, and there are no mismatches, there is of course no reflection. If there is an impedance mismatch, some or all of the wave will be reflected. The incident wave, and the reflected wave will then interfere with each other creating a new wave equal to the sum of two waves. If you alter the frequency of the input sine wave, you will find that there is a maximum and a minimum peak to peak amplitude of the waves; the ratio of the maximum to the minimum is the VSWR.

VSWR = \frac{V_{p-p max}}{V_{p-p min}} = \frac{V_{incidentWave}+V_{reflectedWave}}{V_{incidentWave}-V_{reflectedWave}} = \frac{Z_L}{Z_0}Z_L is the impedance after the discontinuity

Z_0 is the original impedance