Transmission Lines

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Impedance of Transmission Lines

A transmission line with characteristic impedance Z0, driven by a source with impedance Zs, and terminated with a load impedance of ZL

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 Z0 is given by

\begin{align}  Z_0 & = \frac12 Z_ L + \frac12 Z_ C\parallel (\frac12 Z_ L+Z_0)\\ Z_0^2 & = \frac14 Z_ L^2 + Z_ L Z_ C\\ Z_0 & = \sqrt {Z_ L Z_ C + \left(\frac12 Z_ L\right)^2}\\ \end{align}\,\!

Now if we define L to be an inductance per unit length, and C to be a capacitance per unit length, we have Z_ L=j\omega L \Delta \ell and Z_ C=1/j\omega C \Delta \ell , where \Delta \ell is some small unit of length. In this case:

  Z_0=\sqrt {\frac{L}{C} - \left(\frac12\omega L\Delta \ell \right)^2}. \,\!

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 \Delta \ell \to 0, we are left with:

  Z_0=\sqrt {\frac{L}{C}} \,\!
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