FAQs about Transmission Line Effects
A1: The general concept is that when the transition time of the signal is “short” compared to the interconnect propagation delay, then a “long line,” or transmission line, effect exists, and transmission line treatment of the circuit must be used.
For example, typical propagation delay of a 50 Ω cable, such as the RG58C/U, is 1.65 ns/ft and that of a 50 Ω microstrip line is 155 ps/in. The rise time of a CMOS device such as the 74C00 is about 10 ns and that of an ECL device in the 10EP/100EP series is in the range of 200-300 ps.
Therefore, a 74C00 device can safely drive a 3-ft RG58C/U cable without regard to line termination or reflections. An MC10EP16, a differential receiver, however, should not drive an unterminated line that is longer than ¼-in.
There is no hard and fast rule that separates a “short line” from a “long line,” although numbers from 2 to 5 for the ratio of the interconnect delay to the signal transition time have been suggested for the definition of a “long line.” More discussion on this subject can be found in Motorola MECL Systems Design Handbook and in a number of GaAs device data books. Rigorous treatments of the transmission theory can be found in the classical text book, Walter C. Johnson, “Transmission Lines and Networks,” McGraw-Hill, 1950, Part I, Chapters 1-9
The transmission line phenomenon can be intuitively explained by the existence of apparent signal delay, as we shall see in the following examples of a 6-ft “short line” and a 6-in. " long line".
A 6-FT “SHORT” COAXIAL LINE
Lets assume a pulse signal source with a rise time of 1000 ns is connected to a 6 ft long RG58/CU 50 Ω coaxial cable, which has a total propagation delay of approximately 10 ns. At t0, the signal is applied to the sending end of the line. 10 ns later, the signal arrives at the end of the line. Since the signal rise time is 1000 ns, the amplitude at the sending end has reached less than 1% of its final value, while the receiving end is just beginning to sense the incoming signal. So, between the two ends of the line, at any given time, the difference in signal amplitude is approximately 1%.
Therefore, as far as this signal is concerned, the entire line appears as an equipotential surface at any given instance. There is no apparent signal delay between the two ends of the cable, and, therefore, the cable appears as a lumped capacitive element, with a value of approximately 180 pf. This is an exaggerated case of a “short line”, where the ratio of the interconnect delay to the signal transition time is 0.01. For a “short line”, the signal generator sees the entire cable as a lumped capacitance only, and there is negligible transmission line phenomenon. Now lets look at a 6-in. “long line”.
A 6-INCH “LONG” COAXIAL LINE
Lets assume a pulse signal source with a rise time of 100 ps is connected to a 6-in. long RG58/CU cable, which has a total propagation delay of approximately 825 ps. Again, at t0, the signal is applied to the sending end of the line. After a little over 100 ps, the signal at the sending end will have reached its full amplitude, but the end of the line doesn’t even know a signal is due to arrive. Only after 825 ps later will have the signal arrived at the other end. So what happened between t0 and 824 ps?
The answer is that the signal is in transition; that is, the signal is traveling from one end of the line to the other end in some finite time. This case differs from the “short line” example described above in that there is apparent signal delay along the coaxial cable. Note that the ratio of the interconnect delay to the signal transition time is 8.3. Since the signal generator can not see the entire line at any given time, it appears as a “long line”.
The logical question to be asked is how does this apparent signal delay phenomenon affect the circuit operation? This leads to the discussion of Characteristic Impedance, the infinitely long line and reflections.
Q2: What is Characteristic Impedance, and how can it be measured?
A2: Before going further, we need to make some basic assumptions. First, we assume that the coaxial cable has a uniform cross section. Next, we assume that the cable can be represented by an equivalent lossless network of uniformly distributed, lumped L-C components, as shown in Fig. 9. Finally, we assume that the radiation effect can be neglected in that the electromagnetic field is confined entirely inside the cable. Now lets see what happens during the interval when the signal is traveling down the line, and our discussion will be restricted to the time interval before the signal reached the end of the line.
The application of a voltage step at the input end of the line begins to charge a small section of the line, say L1 and C1. Energy stored in the first section of these L-C elements is now transferred to the next section and then to the next section, and so on, as the rest of the L-C sections are being charged. When a given section is being charged, there must be an accompanying current flow. If the structure of the cable is uniform and if there is no IR or radiation loss, the voltage developed across each section and the resultant current flow in each section of the L-C network must be the same. Since the ratio of the charging voltage to the charging current has the dimension of an impedance, this equivalent impedance has a constant value, and it is given the name “Characteristic Impedance” of the line, Z0. Since we used a voltage step, a DC voltage, as the input, the resultant impedance is necessarily a resistance.
However, if the classical partial differential equations for the transmission line were solved using a sinusoidal input, Z0 will still be a real number or resistive. The value of Z0 is equal to √(L0/C0), where L0 is the inductance per unit length and C0 the capacitance per unit length of the line. In the ideal case, Z0 is purely resistive and is independent of frequencies.
The Infinitely Long Line.
It is interesting to see what would happen if the line were infinitely long. It was shown that the current flow in a uniformly distributed lossless line as a result of the application of a step voltage is constant before the signal reached the end of the line. As far as the generator is concerned, it sees an equivalent resistor Z0 instead of a cable, as long as the current stays constant. By assuming the line to be infinitely long, the current must, therefore, stay constant forever, as if the generator were connected to a resistor Z0. In another words, the generator can’t tell the difference whether its output is connected to an infinitely long line, to a lossless line with a resistor Z0 at the end, or simply to a resistor Z0.
The important concept here is that when a coaxial cable is terminated into its characteristic impedance Z0, the generator sees only the equivalent resistor Z0 and not the cable at all. That is, the generator does not see the total cable capacitance as so many pf/ft. This is a difficult concept to accept by many engineers who insist that the total cable capacitance must be included when driving a long cable. A simple argument should convince most disbelievers that the generator can’t possibly see the entire length of a “long line,” or the entire amount of the cable capacitance, at a given time when there is signal propagation involved.
An even more convincing proof can easily be made by observing a 1ns rise-time pulse signal at the end of several 50 Ω cables of different lengths, say 1 ft, 2 ft, 3 ft and 4 ft. One may be surprised to find that the rise time is more or less independent of the cable length. If the cable were to appear as a capacitance to the signal source, the rise time would double every time the cable length were doubled, say from 1 ft to 2 ft or from 2 ft to 4 ft, etc.
This example gives rise to the most important concept of matched impedance. When a generator is connected to a line terminated into Z0, the line becomes invisible, and the generator sees only an equivalent resistor Z0. Under this “matched impedance” condition, there will be no reflection.