# Transmission Lines

### From AstroBaki

### [edit] Short Topical Videos

- Tektronix - Transmission Lines (amusing. bear through the "this is a transmission line" intro)

### [edit] Reference Material

## Contents |

## Impedance of Transmission Lines

*A transmission line with characteristic impedance Z_{0}, driven by a source with impedance Z_{s}, and terminated with a load impedance of Z_{L}*

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 0.01Ω, then you might reasonably expect that you’d measure the impedance of a 20m piece of wire to be 0.02Ω. However, when we say that a coaxial cable (SMA, BNC, or otherwise) has an impedance of 50Ω, there is no mention of a length. 50Ω coaxial cable is 50Ω whether it is 1m or 100m long. How can this be?

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

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.

*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 *Z*_{0} is given by

Now if we define *L* to be an inductance per unit length, and *C* 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 *L* and *C* as being per unit length) fall out of the left-hand term under the root. And, of course, as , we are left with: