Properties of purely reactive Foster and non-Foster passive networks

The mathematical concept of strongly real functions of positive and negative types is introduced to network theory for the first time. The driving-point reactance/susceptance of a pure Foster network, made up of only ideal positive inductance and capacitance elements, is a strongly real function of real frequency of positive type. As a corollary, for a pure non-Foster network made up of only ideal negative inductance and capacitance elements, the driving-point reactance/susceptance is a strongly real function of real frequency of negative type. It is shown that a condition for a purely reactive passive network to exhibit a positive or negative reactance/susceptance-frequency gradient is that the driving-point immittance should have alternating poles and zeroes lying on the real frequency axis. Finally, it is shown that either purely Foster or non-Foster networks can be constructed by combining ideal Foster and non-Foster reactive elements.