Input Impedance - Special Line Lengths

Last week (or a few weeks ago) we talked about the concept of input impedance and how the impedance changes along the transmission line as a function of the line length, l. This week, we're going to show this change with some different segments of transmission line. No transmission line, a quarter wavelength transmission line, and a half wavelength transmission line.

We are going to assume a lossless transmission line for simplicity.

First, let's start with a line length of 0. With no transmission line, the input impedance equals the load impedance. This makes sense. There’s no transmission line. The load is connected directly to the generator/source.

Next, quarter wavelength transmission line. At a quarter wavelength line, the line inverts the load impedance relative to the square of the characteristic impedance. This inversion property is the basis of the quarter-wave transformer. Super useful when designing impedance matching networks.

Finally, half wavelength transmission line. With a half wavelength of transmission line, the input impedance is equal to the load impedance. For example, if the load impedance is a short, the source sees a short repeat at every half wavelength. If the load impedance is an inductor, the input impedance sees an inductor at every half wavelength.

Check out the attached document for full derivations. If you like what you read, there's way more in my book Understanding RF: From Theory to Practice. It's free. Go check it out.