ADMITTING YOUR SUSCEPTANCE TO MY RESISTANCE TO IMPEDANCE
If I thought about that title for just a little while longer, I might be able to come up with a ribald limerick about impedance and reactance, susceptance and admittance. But that may not be as great an idea as it seemed at first, so instead let's carry on!
I think most of us deal with impedance all the time, maybe to the point that we've stopped thinking about some of the basic questions. A few columns ago, I pointed out that our so-called 600-ohm balanced audio standard apparently originated when pole-mounted telegraph wires were re-used for telephone transmission. An old story tells us that fifty and seventy-five ohm RF transmission lines came along because that's what you got when you used standard sizes of copper tubing to make coaxial cables. So we owe our selection of 50- and 75-ohm cables at least partly to the plumbing industry? More on that later. Who amongst us remembers the 230-ohm balanced "open-wire" transmission lines that were used before high-power co-ax became available?
And what do we mean when we say that a chunk of co-ax is 50 ohms? Some smart apple is going to reply that means that's the cable's characteristic impedance. But what exactly does that mean? If you measure between the centre conductor and the shield of that co-ax with an ohmmeter, it will read open circuit, and it will measure close to that at audio frequencies. I daresay if you measured its impedance at a few GigaHertz with a bridge, you might find that the cable's impedance was close to a short circuit.
Well, the reactive components of a coaxial cable are the series inductance (L) of the inner conductor, and shunt capacitance (C) between the inner conductor and the shield. Then there's series resistance (R) of the inner conductor, and susceptance (G)(very high shunt resistance of the insulation between the inner and outer conductor). So if we look at the whole spectrum of RF frequencies, there is a broad range where the characteristic impedance holds true. And I guess that's why it's called the "characteristic" impedance. Ignoring the two resistive components, the simplified formula for calculating the characteristic impedance is the square root of L/C. And there are formulas to calculate impedance based on the ratio of the diameters of the inner and outer conductor. Here's where it gets interesting: in actual practice, we find that cable attenuation increases faster with increasing frequency than the simple L/C formula would lead us to expect. This turns out to be because of skin effect, which causes R to increase with the square of frequency, until it can't be ignored with our simplified formula. The obvious way to reduce skin effect (and that attenuation) is to increase the surface of the inner conductor, by increasing its diameter. But this will cause the characteristic impedance of the cable to drop, so that to pass a certain power of signal, greater current will be required, which increases losses due to resistance, and eventually we reach a point where we're not improving anything this way. By continued experimentation, we find that there is an optimum ratio of inner and outer conductor to minimize cable attenuation, and it's about 1:3, which gives us an impedance of… 75 ohms. If instead you try to optimize the amount of RF power a given size of cable can safely carry, you end up at about… 50 ohms. So there you have it: where signal losses must be minimized, 75 ohms is your best bet. In transmission, where we're more concerned about maximizing the power we can crank out of our lines, 50 ohms turns out to be the wise choice.
Sometimes it's reassuring to find out that some standard is what it is for good scientific reasons, and not due to the whims of someone trying to figure out what size of pipe to connect to your bathroom.