transmission line

A 50-ohm cable?

a length of coaxial cable< i> with the label “50 ohms” printed along its outer sheath. (Figure ) Now, coaxial cable is a two-conductor cable made of a single conductor surrounded by a braided wire jacket, with a plastic insulating material separating the two. As such, the outer (braided) conductor completely surrounds the inner (single wire) conductor, the two conductors insulated from each other for the entire length of the cable. This type of cabling is often used to conduct weak (low-amplitude) voltage signals, due to its excellent ability to shield such signals from external interference.

Coaxial cable contruction.
I was mystified by the “50 ohms” label on this coaxial cable. How could two conductors, insulated from each other by a relatively thick layer of plastic, have 50 ohms of resistance between them? Measuring resistance between the outer and inner conductors with my ohmmeter, I found it to be infinite (open-circuit), just as I would have expected from two insulated conductors. Measuring each of the two conductors’ resistances from one end of the cable to the other indicated nearly zero ohms of resistance: again, exactly what I would have expected from continuous, unbroken lengths of wire. Nowhere was I able to measure 50 Ω of resistance on this cable, regardless of which points I connected my ohmmeter between.
What I didn’t understand at the time was the cable’s response to short-duration voltage “pulses” and high-frequency AC signals. Continuous direct current (DC) — such as that used by my ohmmeter to check the cable’s resistance — shows the two conductors to be completely insulated from each other, with nearly infinite resistance between the two. However, due to the effects of capacitance and inductance distributed along the length of the cable, the cable’s response to rapidly-changing voltages is such that it acts as a finite impedance, drawing current proportional to an applied voltage. What we would normally dismiss as being just a pair of wires becomes an important circuit element in the presence of transient and high-frequency AC signals, with characteristic properties all its own. When expressing such properties, we refer to the wire pair as a transmission line.
This chapter explores transmission line behavior. Many transmission line effects do not appear in significant measure in AC circuits of powerline frequency (50 or 60 Hz), or in continuous DC circuits, and so we haven’t had to concern ourselves with them in our study of electric circuits thus far. However, in circuits involving high frequencies and/or extremely long cable lengths, the effects are very significant. Practical applications of transmission line effects abound in radio-frequency (“RF”) communication circuitry, including computer networks, and in low-frequency circuits subject to voltage transients (“surges”) such as lightning strikes on power lines.

Circuits and the speed of light

Suppose we had a simple one-battery, one-lamp circuit controlled by a switch. When the switch is closed, the lamp immediately lights. When the switch is opened, the lamp immediately darkens: (Figure

Lamp appears to immediately respond to switch.
Actually, an incandescent lamp takes a short time for its filament to warm up and emit light after receiving an electric current of sufficient magnitude to power it, so the effect is not instant. However, what I’d like to focus on is the immediacy of the electric current itself, not the response time of the lamp filament. For all practical purposes, the effect of switch action is instant at the lamp’s location. Although electrons move through wires very slowly, the overall effect of electrons pushing against each other happens at the speed of light (approximately 186,000 miles per second!).
What would happen, though, if the wires carrying power to the lamp were186,000 miles long? Since we know the effects of electricity do have a finite speed (albeit very fast), a set of very long wires should introduce a time delay into the circuit, delaying the switch’s action on the lamp: (Figure

At the speed of light, lamp responds after 1 second.
Assuming no warm-up time for the lamp filament, and no resistance along the 372,000 mile length of both wires, the lamp would light up approximately one second after the switch closure. Although the construction and operation of superconducting wires 372,000 miles in length would pose enormous practical problems, it is theoretically possible, and so this “thought experiment” is valid. When the switch is opened again, the lamp will continue to receive power for one second of time after the switch opens, then it will de-energize.
One way of envisioning this is to imagine the electrons within a conductor as rail cars in a train: linked together with a small amount of “slack” or “play” in the couplings. When one rail car (electron) begins to move, it pushes on the one ahead of it and pulls on the one behind it, but not before the slack is relieved from the couplings. Thus, motion is transferred from car to car (from electron to electron) at a maximum velocity limited by the coupling slack, resulting in a much faster transfer of motion from the left end of the train (circuit) to the right end than the actual speed of the cars (electrons): (Figure

Motion is transmitted sucessively from one car to next.
Another analogy, perhaps more fitting for the subject of transmission lines, is that of waves in water. Suppose a flat, wall-shaped object is suddenly moved horizontally along the surface of water, so as to produce a wave ahead of it. The wave will travel as water molecules bump into each other, transferring wave motion along the water’s surface far faster than the water molecules themselves are actually traveling: (Figure

Wave motion in water.
Likewise, electron motion “coupling” travels approximately at the speed of light, although the electrons themselves don’t move that quickly. In a very long circuit, this “coupling” speed would become noticeable to a human observer in the form of a short time delay between switch action and lamp action.