Build Your Own Time-Domain Reflectometer

What's a TDR?

Time Domain Reflectometry is a powerful technique in which a pulse is generated to propagate down a cable, after which the reflected signal returns to the generator and is then interpreted based on its shape, phase, and delay. The results can be used to determine the length of a cable, if and where there is an open circuit, what kind of load a cable is terminated with, and even the relative permittivity and permeability of a dielectric. TDRs are used in a wide range of applications including aviation and naval craft troubleshooting where there are often miles of cable and a technician can accurately pinpoint a malfunction. They are also used in devices measuring soil moisture content in which a cable is placed in the soil and the relative permittivity (dielectric constant) of the soil can be calculated. Because water has a very high relative permittivity and dry soil does not, the permittivity that is detected will correlate to the amount of moisture present. TDRs are used in this same way as a level indicator for liquids, where a thin probe is submerged and detects lower permittivity as the level drops. Most circuit board verification jigs will implement this technique as a method of determining if a chip on a ball-grid-array solder pad has any open pins.

How It Works

The physics behind this method can be pretty complex (single equations that take up half a page), but we will work with a few assumptions that will make life easier and still get us a very close answer; in engineering it is important to not let perfection be the enemy of good-enough. Firstly, we will assume we are working with a lossless dielectric, meaning the vector describing this dielectric contains no real component ‘R’ and the loss tangent is 0°. Secondly, we will assume that our cable material (copper) is a perfect conductor and the vector describing it contains no imaginary component “ωj”. The basic principal is that we want to send a short pulse into a cable and use a visualization tool to examine the reflection. When a signal is incident to a cable, the signal does not travel to the end of the cable instantaneously. For starters, the finite speed of light (299,792,458 m/s) is the upper boundary, but as we will soon see, we will not even achieve these speeds; the reason is that the constants of relative permittivity and permeability for our cable's dielectric will slow down transmission speed or phase velocity, Vp. The signal will still be traveling at a significant fraction of the speed of light but keeping this value precise is key to the math coming out right. Phase velocity with our assumptions is defined as

Thus the time it will take for our incident signal to travel to the load can be calculated as

We will then expect to see a delay in our reflected signal corresponding to the results of these equations. We will get into more detailed calculations in a moment, for now let’s get building

Implementation

TDRs are expensive, but fortunately a rudimentary version can be easily assembled using a signal generator and an oscilloscope. In addition to these items you will need a set of BNC connectors and of course, a cable (can be to test it; you may also use a set of precision matched loads as seen in the image below, or you can use a potentiometer connected to the end of the cable, though this method introduces a bit of noise. The matched set I will be using consists of a 50, 75, and 93 Ohm loads.

 

One of the most readily available cable available is a piece of coaxial cable. Coaxial cable has been around for a very long time, but it has improved in construction significant over the years. Any coaxial cable will have a characteristic impedance partly determined by the diameter of the core cable that corresponds to its intended usage; they will range between 30 Ohms to 93 Ohms, with 30 ohms having the highest power handling capacity, and 77 ohms having the least signal attenuation. 50 Ohms is a common coaxial cable impedance as it is a compromise between power handling and signal loss. For this test I will use a long piece of coax cable, but I won’t say just yet how long it is or what its characteristic impedance is yet- that’s what we are here to find out! I will tell you that the manufacturer lists its relative permittivity as 1.2. For all coaxial dielectrics, the relative permeability is so close to 1 that we can just assume it is for our purposes ( accurate to within 0.00000001%). Substituting these values into our first equation we get
Vp = 273,671,819.7 m/s
or 10.7745 in/nS
which means our signal will propagate at about 91.29% of the speed of light, this percentage is known as the velocity factor or VF. Now we will hook up our gear and see what we get. Connect the signal generator to your scope with a T connector and the shortest cable you can find, this will help minimize error and the T will give you a place to connect the coaxial mystery cable- seen in the image as a white cable.

Open Circuit Configuration

After connecting one end of the coaxial cable to the T connector, leave the other end open; this first test will be an open circuit test. The open circuit test will give a very good representation of delay time and reflection coefficient. The frequency that you the the signal generator to is arbitrary, but it works best if you have a really fast rise time. In my case I am connecting to the 'Pulse Out' terminal that can be seen in the image below.
With the signal generator hooked up, trigger the scope and you should see an image like the one below:

Here we see our incident wave as the bottom step with the reflected wave as the top step. Because there is a finite transit time (calculated earlier) in the line, its input impedance will look like the characteristic impedance of the line for   so the line appears to be infinitely long until the pulse has had a chance to traverse the cable and reflect back to the input. the pulse begins as V0/2 because it is expecting the input impedance and load to be matched, creating a voltage divider. In the open circuit case the entire signal is reflected back towards the generator as can be seen using the equation for the reflection coefficient

giving us a coefficient of 1 for the load being infinite. After reflecting back, the signal jumps back up to V0, in this case about 4.5-5 volts. Let's take a closer look at what is going on: