Foster's Reactance Theorem - Poles and Zeroes

Poles and Zeroes

A consequence of Foster's theorem is that the poles and zeroes of any passive immittance function must alternate with increasing frequency. After passing through a pole the function will be negative and is obliged to pass through zero before reaching the next pole if it is to be monotonically increasing.

With the addition of a scaling factor, the poles and zeroes of an immittance function completely determine the frequency characteristics of a Foster network. Two Foster networks that have identical poles and zeroes will be equivalent circuits in the sense that their immittance functions will be identical.

Another consequence of Foster's theorem is that the plot of a Foster immittance function on a Smith chart must always travel around the chart in a clockwise direction with increasing frequency.