Transmission Lines

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Impedance of Transmission Lines

Transmission line.png

A transmission line with characteristic impedance , driven by a source with impedance , and terminated with a load impedance of

Transmission lines are a bit different than the normal wires we’re used to dealing with. For example, if you measured the resistance of a 10m piece of wire, and found it to be 0.01, then you might reasonably expect that you’d measure the impedance of a 20m piece of wire to be 0.02. However, when we say that a coaxial cable (SMA, BNC, or otherwise) has an impedance of , there is no mention of a length. 50 coaxial cable is 50 whether it is 1m or 100m long. How can this be?

Transmission line rlgc.png

A per-length transmission line model consisting of a (small) series resistance , a series inductance , a (small) parallel resistance caused by dielectric conduction, and a parallel capacitance .

It turns out that the impedance of a transmission line, although it is real-valued (i.e. resistive), is not caused by the resistance of the wire (which is typically quite small, and results in signal loss along the wire). Rather, for a lossless transmission line, capacitance and inductance are what give rise to the characteristic impedance. If you’ve ever cut a cable in half and seen the dielectric that sits between the conducting wire and the exterior sheath, you are probably not surprised that capacitance plays a role. The other key to understanding transmission lines is to recognize that they are for carrying signals. You have to launch a signal down a transmission line, so we should really be thinking about the relationship between the voltage and current of the signal that is transmitted.

Transmission line deriv.png

Adding a differential piece of transmission line to an infinite line.

Here is a cute pedagogical derivation of how this works. Supposing a lossless transmission line, we add a differential piece of line (with two half-inductances and a capacitor, as shown above), and argue that this shouldn’t change the overall impedance. In this configuration, the overall impedance of the line is given by

Now if we define to be an inductance per unit length, and to be a capacitance per unit length, we have and , where is some small unit of length. In this case:

Notice how the differential length and frequency dependence (and, even the definition of and as being per unit length) fall out of the left-hand term under the root. And, of course, as , we are left with: