Image Impedance - Derivation

Derivation

As an example, the derivation of the image impedances of a simple 'L' network is given below. The L network consists of a series impedance, Z, and a shunt admittance, Y.

The difficulty here is that in order to find Z_{i 1} it is first necessary to terminate port 2 with Z_{i 2}. However, Z_{i 2} is also an unknown at this stage. The problem is solved by terminating port 2 with an identical network: port 2 of the second network is connected to port 2 of the first network and port 1 of the second network is terminated with Z_{i 1}. The second network is terminating the first network in Z_{i 2} as required. Mathematically, this is equivalent to eliminating one variable from a set of simultaneous equations. The network can now be solved for Z_{i 1}. Writing out the expression for input impedance gives;

and solving for Z_{i 1},

Z_{i 2} is found by a similar process, but it is simpler to work in terms of the reciprocal, that is image admittance Y_{i 2},

Also, it can be seen from these expressions that the two image impedances are related to each other by;