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How to find faults in coaxial cables using TDR TDR or time-domain reflectometry is a technique used to track down faults in cables – mainly coaxial cables but other types as well. But do you know how TDR actually works? This article is a primer on TDR. It’s a lot easier to understand than you may think and we also explain what the terms “velocity factor” and “characteristic impedance” mean. By JIM ROWE D ON’T BE PUT OFF by that complex sounding term “time-domain reflectometry” or its cryptic acronym “TDR”. They’re just techno jargon for a fault-finding technique that’s simpler than it sounds – at least in principle. First off, we need to explain that the main use for TDR is for finding faults and discontinuities in cables – primarily coaxial cables. These are the cables used to carry RF signals between antennas, receivers and transmitters and also to carry RF, video and high-speed digital signals between professional and domestic equipment. In essence, coaxial cables behave as transmission lines, in that when electrical energy is fed into one end of the cable, it takes a finite time for that energy to travel along the cable to HIGH SPEED OSCILLOSCOPE the other end. That’s because the distributed inductance and capacitance inside the cable force the energy to propagate along it in the form of an electromagnetic wave (a combination of electric and magnetic energy). This is very similar to light energy travelling along a fibre-optic cable – which is not surprising, because light is simply electromagnetic (EM) energy of a much higher frequency. When any kind of EM energy is propagating through empty space (ie, a vacuum), it does so at the speed of light, equal to 299,724,580 metres per second, or near enough to 300,000km/s. By the way, this equates to 300 metres per microsecond (300m/μs) and also to 300 millimetres per nano­ second (300mm/ns). Both of these figures are worth remembering. When EM energy is propagating through a more substantial medium like a coaxial cable (or a fibre-optic cable in the case of light), it moves at a slower speed; still very fast but not quite as fast as light in a vacuum. Velocity factor In the case of EM energy propagating along a coaxial cable or similar transmission line, its speed or velocity (Vp) is related to the speed of light in a vacuum by a factor known as the “velocity factor” (Vf) of the cable. In other words: Vp = Vf.c . . . (1) where c is the speed of light in a vacuum. Fig.1: the basic circuit for a Step TDR. It uses a step generator with a source resistance of Rsource, while the load at the end of the coaxial cable has a resistance of Rterm. A high-speed oscilloscope is used to monitor the voltage at the input end of the cable. INPUT Rsource STEP GENERATOR 76  Silicon Chip TRANSMISSION LINE (COAXIAL CABLE OR SIMILAR) CHARACTERISTIC IMPEDANCE = Zo LOAD (Rterm) siliconchip.com.au As you might expect, the value of Vf is closely related to the dielectric constant Er of the dielectric material used in the cable itself – between the centre conductor and the outer screen conductor. In fact: Vf = 1/√ √Er . . . (2) Most commonly available coaxial cables use a polyethylene (PE) dielectric in either solid or cellular foam form – or as small discs of solid PE spaced apart (“air-spaced” PE). A small number of cables for specialised applications use dielectric materials like fluorinated ethylene propylene (FEP), poly tetrafluoroethylene (PTFE) or polyvinyl chloride (PVC). Table 1 shows the dielectric constant Er, the velocity factor Vf and the propagation velocity Vp of some common types of coaxial cable dielectric, along with the figures for air or a vacuum for comparison. This information comes in handy when you’re using a TDR adaptor with a scope to locate the position of faults or discontinuities in cables. Characteristic impedance Now let’s consider another important aspect of coaxial cables and other transmission lines: their characteristic impedance. Just as the distributed capacitance and inductance of a cable forces EM energy to propagate along it at a specific velocity, they also force the energy to adopt a specific voltageto-current ratio. This V/I ratio is called the cable’s characteristic impedance, and is usually represented as “Zo”. The value of Zo for any particular cable depends mainly on the ratio of the outer conductor’s inside diameter (D) to the inner conductor’s outside diameter (d), together with the dielectric constant of the insulating material between them (Er). In fact, if you neglect the series resistance of the inner and outer conductors per unit length, Table 1: Common Coaxial Cable Dielectrics DIELECTRIC MATERIAL (DIELECTRIC Er CONSTANT) AIR (OR VACUUM) 1.00 1.00 300mm/ns SOLID PE 2.3 0.66 198mm/ns CELLULAR FOAM PE 1.4 – 2.1 ~0.87 261mm/ns Vf (VELOCITY FACTOR) Vp (VELOCITY OF PROPAGATION) AIR SPACED PE ~1.1 0.95 285mm/ns SOLID PTFE 2.1 0.69 207mm/ns CELLULAR FOAM PTFE 1.4 0.84 252mm/ns SOLID FEP 2.1 0.69 207mm/ns CELLULAR FOAM FEP 1.5 0.82 246mm/ns -9 Note: velocities shown in millimetres per nanosecond (10 s) the Zo of a coaxial cable can be found from this simple formula: Zo = (138Ω/√ √Er) x log(D/d)        . . . (3) which can be simplified to: Zo = 138Ω x Vf x log(D/d) . . . (4) Although you can calculate the Zo of any particular cable with this formula, it’s generally not necessary because cable manufacturers usually provide this information. Table 2 shows the relevant details for some common coaxial cables. All of them have a Zo of either 50Ω or 75Ω. Knowing the Zo of a cable is important because when the cable is used to transfer electrical energy from a source or “generator” to a load, you only get maximum power transfer when the generator’s source resistance/ impedance and the load’s resistance/ impedance are both matched to the Zo of the cable. If the load resistance is not matched to the cable impedance (Zo), some of the energy reaching the load end of the cable is reflected back along the cable to the generator (with a polarity which may be the opposite of the ‘incident’ energy). If the source resistance of the generator is not matched to the cable Zo either, some of this returned energy is reflected back towards the load again. The net result is that some of the energy bounces back and forth along the cable and is wasted as heat. So a cable’s Zo or characteristic impedance is most important in that it allows you to match the resistance of the load and generator to it, in order to achieve the most efficient transfer of energy/power (and preserve signal integrity). TDR basics Having explained coaxial cable operation and the significance of Vp (velocity of propagation) and Zo (characteristic impedance), we are primed to understand the basics of TDR. First, there are actually three versions of TDR, known as “Step TDR”, “Pulse TDR” and “Spread-Spectrum TDR”. We’re going to be dealing mainly with Step TDR because it’s the version most commonly used nowadays and it’s the easiest to understand. Now take a look at the simple circuit in Fig.1. It shows a length of coaxial cable connected between a voltage step generator and a load resistor. The step generator has a source resist- Table 2: Typical Characteristics Of Some Common Coaxial Cables (VELOCITY OF PROPAGATION) D/d (mm/mm) Zo (CHIMAPREADCATNERCISET) IC Loss (dB/m <at> F) COMMON USES 4.7/1.0 75W 0.2 <at> 1GHz CABLE TV & SATELLITE ANTENNA INSTALLATIONS 198mm/ns 2.9/0.81 50W 1.056 <at> 2.4GHz THIN ETHERNET, RF & INSTRUMENTATION SOLID PE 198mm/ns 3.7/0.64 75W 0.39 <at> 1GHz BASEBAND VIDEO, DOMESTIC TV ANTENNAS RG-174/U SOLID PE 198mm/ns 1.5/(7 x 0.16) 50W 2.46 <at> 2.4GHz WI-FI PIGTAILS, GPS & INSTRUMENTATION RG-213/U SOLID PE 198mm/ns 7.2/(7 x 0.75) 50W 0.27 <at> 1GHz LOW LOSS UHF ANTENNA CABLES Vp CABLE TYPE DIELECTRIC RG-6/U CELLULAR FOAM PE 261mm/ns RG-58/U SOLID PE RG-59/U Note: RG-XX/U type numbers are a carr y-over from US militar y specs during WW2. They are nowadays used mainly to identify matching connectors. siliconchip.com.au November 2014  77 Table 3: Scope Displays With Step Generator CONDITIONS RESULT Zo = Rsource Rterm = Zo (CORRECT MATCHING) ALL ENERGY IS CARRIED TO THE LOAD, WITH NO REFLECTED ENERGY Ei Zo = Rsource Rterm = ZERO (SHORT CIRCUIT AT LOAD END) ALL ENERGY REFLECTED BACK TO THE GENERATOR, BUT WITH REVERSED POLARITY (Tr = 2 x cable transit time) Ei Zo = Rsource Rterm = INFINITY (OPEN CIRCUIT AT LOAD END) ALL ENERGY REFLECTED BACK TO THE GENERATOR, BUT WITH THE SAME POLARITY (Tr = 2 x cable transit time) OSCILLOSCOPE DISPLAY Er (= –Ei) Tr Er (= +Ei) Ei Tr Zo = Rsource Rterm = 2 x Zo (TWICE Zo & Rsource) ENERGY REFLECTED BACK TO THE GENERATOR, BUT WITH THE SAME POLARITY (Tr = 2 x cable transit time) Ttransit = L/Vp Ei Er (= +Ei/3) Tr Zo = Rsource Rterm = Zo/2 (HALF Zo & Rsource) ENERGY REFLECTED BACK TO THE GENERATOR, BUT WITH REVERSED POLARITY (Tr = 2 x cable transit time) Er (= –Ei/3) Ei Tr ance of Rsource, while the load has a resistance of Rterm (shortened from Rterminating). The only other item in the circuit is a high-speed oscilloscope with its input being used to monitor the voltage at the input end of the cable. To begin, consider the situation where both Rterm and Rsource are correctly matched to the Zo of the cable. What would you expect to see on the scope? In this case, you would see a single voltage step as shown in the top trace of Table 3. The step would have an amplitude of Ei volts and would continue at that level indefinitely – or at least until the generator output falls again. Note that the value of Ei will be equal to half of the generator’s open-circuit output voltage, because the effective input resistance of the cable will be equal to Rsource and together they will constitute a 2:1 voltage divider. Shorting the cable Now consider what the scope would show if the load resistor Rterm were replaced with a short circuit – in effect, a resistor of zero ohms. This would be an extreme mismatch at the load end of the cable and as a result all of the voltage step energy would be reflected Scheduled for publication in the December issue, this do-it-yourself adaptor lets you use any reasonably “fast” scope to perform step TDR on your own cables. 78  Silicon Chip back towards the generator as another voltage step Er – with the same amplitude as Ei but of opposite polarity. So the scope display would look like the second trace in Table 3, with the voltage at the cable input dropping to zero as soon as the reflected energy arrived back. Note the significance of Tr. It is the time between the start of the voltage step and its sudden end. It represents the time taken for the incident step to travel to the end of the cable, plus the time taken for the reflected step to travel back to the start. In other words, it will be equal to twice the cable transit time. And we can work out the transit time. It’s equal to: . . . (5) where Vp is the velocity of propagation in the cable (as before, measured in mm/ns), while L is the cable length in millimetres. So Tr will equal twice this value and if we measure Tr using the scope we can calculate the effective length of the cable using this rearranged equation: L = (Tr x Vp)/2                    . . . (6) where L is the cable length in millimetres, Tr is the step “length” in nanoseconds and Vp is the velocity of propagation in mm/ns. So by measuring Tr, we can quite easily work out the cable length – or more precisely the distance to the short circuit. Disconnecting the cable Next consider what would happen if we removed the short circuit from the load end of the cable and left it without any termination at all – an open-circuit. This will again represent an extreme mismatch but of the opposite kind to a short circuit. All of the voltage step energy will again be reflected back to the generator as a voltage step Er, but this time with the same polarity as Ei. When the reflected step reaches the start of the cable, the scope will show the voltage suddenly rising to twice its initial value, as shown in the third trace of Table 3. The Tr time will still have the same significance as before, in this case allowing us to work out the cable length to the open circuit. Get the idea? Now let’s consider what would happen if we don’t connect a short circuit or an open circuit to the end of the cable but instead connect a load siliconchip.com.au Voltage reflection coefficient Perhaps you’re wondering why the value of Er is only equal to a third of Ei, when the load resistance is twice the value of Zo? That’s because Er and Ei are related by a factor called the voltage reflection coefficient (Rho), which has a value given by this expression: Er/Ei = Rho = (Zload - Zo)/(Zload + Zo) ...(7) where Zload is the load impedance, which in this case is equal to 2Zo. Rearranging this and substituting for Zload, we find that the value of Er becomes: Er = Ei(2Zo - Zo)/(2Zo + Zo)    = Ei.Zo/3Zo = Ei/3          2.00 1 .00 0.80 0.60 0.40 0.30 RG-174/U 0.20 RG-213/U 0.10 0.08 0.06 RG-58/U 0.05 0.04 0.03 0.02 0.01     . . . (8) Next consider what will happen if we again connect a load resistor to the end of the cable but this time with a value which is HALF the value of Zo and Rsource. Again this is a mismatch, so some of the voltage step energy will be reflected back towards the generator as before. This time though, the reflected voltage Er will be reversed in polarity compared with Ei, because the load resistance is lower than Zo. You can see the resulting downward step in the fifth (lowest) trace in Table 3. You’ll also see that the value of Er is again equal to one third of Ei, which is confirmed thus: Er = Ei(Zo/2 - Zo)/(Zo/2 + Zo)    = Ei(-Zo/2)/(3Zo/2) = -Ei/3          . . . (9) From these five examples you’ll be starting to see how TDR works and how it allows us to calculate some important details about the operation of a cable and transmission line: (1) Whether the cable is correctly terminated in a matched load, which means no reflected energy. This is shown by the voltage step extending indefinitely. (2) If there is a further step in the scope display, indicating some kind of mismatch, then the Tr time between the siliconchip.com.au 3.00 CABLE LOSS IN DECIBELS PER METRE (dB/m) resistor with a value Rterm which is TWICE the value of Zo and Rsource. This is again a mismatch, although not as severe as a short or open circuit. Some of the voltage step energy will be reflected back towards the generator but not as much as before – and with the same polarity as Ei. So the scope will show an upward step after time Tr, with a step height Er in this case equal to Ei/3 as shown in the fourth trace of Table 3. 0.006 1 2 4 6 8 10 20 40 60 100 200 400 600 1GHz 2 3 FREQUENCY IN MEGAHERTZ & GIGAHERTZ Fig.2: these curves show the losses in three common types of 50Ω coaxial cable as a function of frequency. RG-174/U cable ranges from 0.06dB/m at 1MHz up to almost 2.5dB/m at 2.4GHz, while RG-213/U cable ranges from just 0.006dB/m at 1MHz up to 0.49dB/m at 2.4GHz. RG-58/U cable is midway between these two. initial step and the “reflection” step can be used to work out the length of cable L between the generator end and the mismatch. (3) The amplitude and polarity of the reflected voltage step Er can be used to work out the effective resistance of the mismatched load. Cable losses There’s one complication we need to consider before moving on: the effect of cable losses. In the discussion so far, we’ve made the assumption that the cables being tested are “perfect”, in the sense that when the generator and load resistances are properly matched to the cable’s Zo, all of the energy fed into the cable at one end emerges from the other end and passes into the load. In other words, we’ve assumed that the cables are lossless. But in the real world, nothing is perfect. As shown in column six in Table 2, all cables have a loss which is usually listed in terms of decibels per metre (dB/m), or decibels per 100 feet (dB/100ft) in countries like the USA which still use the Imperial system. Because cable losses rise with increasing frequency, the loss figure is usually qualified with a frequency figure, as shown. To put things into perspective, look at the curves in Fig.2. These show the losses in three common types of 50Ω coaxial cable, all plotted against frequency. As you can see, the small diameter RG-174/U cable has a loss figure ranging from 0.06dB/m at 1MHz up to almost 2.5dB/m at 2.4GHz, while RG-213/U cable with its much larger diameter has a loss figure ranging from only 0.006dB/m at 1MHz up to 0.49dB/m at 2.4GHz. The common RG-58/U cable is midway between the other two in terms of its loss factor – ranging from November 2014  79 Fig.3: a Pulse TDR is almost idential to a Step TDR, the difference being that the stimulus generator delivers a narrow voltage pulse rather than a DC voltage step. HIGH SPEED OSCILLOSCOPE INPUT TRANSMISSION LINE (COAXIAL CABLE OR SIMILAR) Rsource NARROW PULSE GENERATOR CHARACTERISTIC IMPEDANCE = Zo 0.013dB/m at 1MHz up to just over 1.00dB/m at 2.4GHz. So real cables do lose some of the input EM energy (as heat), even when the generator and load are correctly matched to their Zo. But what effect does this have when you are checking a cable using Step TDR? This depends on things like the cable loss factor and the cable’s length. These are not likely to have much effect on a fairly short cable but when you’re checking a fairly long run of a relatively lossy cable, the cable loss will tend to attenuate the indicated level of reflected step Er. So any mismatch will appear to be less serious than it should. Pulse TDR Remember that the version of TDR we’ve been discussing so far is Step TDR – the name referring to the way it uses a voltage step waveform as the incident “stimulus” being fed into the cable to be tested. But we’re now going to look briefly at the other basic version: Pulse TDR, where a short voltage pulse is used as the stimulus rather than a step. Fig.3 shows the basic circuit for a Pulse TDR. It’s almost identical to the Step TDR circuit of Fig.1, the only difference being that the stimulus generator is now labelled ‘Narrow Pulse Generator’; it generates a narrow voltage pulse rather than a step. In effect, Pulse TDR works in much the same way as Step TDR. If you compare the traces shown in Table 4 with those for Step TDR in Table 3, you’ll see that the only differences are that Table 4: Scope Displays With Pulse Generator CONDITIONS RESULT Zo = Rsource Rterm = Zo (CORRECT MATCHING) ALL ENERGY IS CARRIED TO THE LOAD, WITH NO REFLECTED ENERGY Zo = Rsource Rterm = ZERO (SHORT CIRCUIT AT LOAD END) ALL ENERGY REFLECTED BACK TO THE GENERATOR, BUT WITH REVERSED POLARITY (Tr = 2 x cable transit time) OSCILLOSCOPE DISPLAY Ei Ei Er (= –Ei) Tr Zo = Rsource Rterm = INFINITY (OPEN CIRCUIT AT LOAD END) ALL ENERGY REFLECTED BACK TO THE GENERATOR, BUT WITH THE SAME POLARITY (Tr = 2 x cable transit time) Ei Er (= +Ei) ENERGY REFLECTED BACK TO THE GENERATOR, BUT WITH THE SAME POLARITY (Tr = 2 x cable transit time) Er (= +Ei/3) Ei Tr Zo = Rsource Rterm = Zo/2 (HALF Zo & Rsource) ENERGY REFLECTED BACK TO THE GENERATOR, BUT WITH REVERSED POLARITY (Tr = 2 x cable transit time) Ei Tr 80  Silicon Chip each voltage step of Table 3 is now replaced with a voltage pulse. The basic behaviour is unchanged, because we’re still looking at the effects caused by the interaction between cable parameters Vp and Zo and changes in load resistance. Step TDR is more popular But if there’s so little difference between the two, why is Step TDR more popular than Pulse TDR? For a couple of reasons, one being that during the stimulus pulse in Pulse TDR, the scope can’t be allowed to monitor the cable input voltage Ei because it would be overloaded. So with this approach, the stimulus pulse creates a ‘dead zone’, during which the scope can’t look for reflections. But when the pulse width is made very narrow to reduce the dead zone, this also reduces the TDR’s range. So Pulse TDRs generally need to provide a number of different pulse widths, to achieve different trade-offs between dead zone and range. Another allied problem with Pulse TDRs is that because a pulse stimulus carries much less energy than the step stimulus, the technique is not capable of delivering the same signal to noise ratio. So with real-world (read “lossy”) cables, Pulse TDR can’t reveal cable faults or discontinuities as clearly as Step TDR. Summarising Tr Zo = Rsource Rterm = 2 x Zo (TWICE Zo & Rsource) LOAD (Rterm) Er (= –Ei/3) You should now have a reasonable understanding of what TDR is, how it works and how it’s used for checking coaxial cables in particular. Think of it as “echo location for cable faults” if you like. And as you may have guessed, this article is a prelude for a planned lowcost adaptor which lets you use any reasonably “fast” scope to perform Step TDR on your own cables. Look SC for it in the December issue. siliconchip.com.au