Z0 is the characteristic impedance of a transmission line. I’ve understood the notion of the impedance of a transmission line, but never internalized it.

This morning on the bus I was reading The Circuit Designer’s Companionon the bus to work and I finally grokked the notion of the impedance of a transmission line.

Now, this will likely either sound stupid because it’s so obvious, or, alternately, stupid because well, why the fuck are you even thinking about this? No matter. I’ve wondered about this for quite a while now and have never gotten a good explanation for why this matters.

So, what the hell am I talking about?

Let’s take the simpler example: coaxial cable. Coax is all the hell over the place these days; though typically it’s quite well hidden. (Pro tip: you very likely have some in your pocket or purse right now — connecting your cell phone’s circuit board to its antenna) The one that most people can see is the typical 75-ohm cable from the shockingly appropriately named cable company. (Most coax is actually 50-ohm… there are some reasons for that I’m not going to get into right now)

What does 75-ohm mean?

If you measure any aspect of a normal length of cable with a typical multimeter you will find no way to measure 75 ohms.

Now, on the other hand, you measure an infinite length of cable with your multimeter you’d measure 75 ohms from the center conductor to the ground shield. But they are not connected… true… but there exists inductance and capacitance on that infinite cable that would get you to 75 ohms.

The problem is that short cable in your hand isn’t infinite and it doesn’t measure 75 ohms, right?

Well, that’s where it gets interesting.

When you’re dealing with short timeframes, nanosecond short, things start to matter. Take a cable around a meter long and short the center conductor to the ground. If you measure the resistance at the non-shorted end you’d read just around 0 ohms as you would expect — a shorted circuit.

But, let’s put a square pulse on the line.

For the first 10 nanoseconds you’d be feeding what looks like a 75-ohms resistance before the voltage across the conductors of the line drops to 0. Until the signal travels down (and back) the cable, the only thing providing anything you can measure is the cable itself. 10 nanoseconds is around how long it takes the electromagnetic signal to travel down and back the cable. Similarly if the far end is open, you’d measure the 75 ohms for 10-ns before you read the infinite ohms of an open circuit. Now, on the other hand, if the far end had the same impedance as the cable, you wouldn’t be able to tell the cable from the load because they are matched.

The notion of “reflected wave” isn’t a reflection as much as the signal coming back as the voltage difference propagating back and forth on the cable until it equalizes and finds an equilibrium.

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I love to learn. I really do. Even if it’s rather mundane, I absolutely love the feeling of finally understanding something. It’s a bit of magic. In many ways we are all so fortunate that there is so much that we can learn that we’ll never, ever, run out of things.