# Properties of Purely Reactive Foster and non-Foster Passive Networks

In the year 2012 while completing his doctoral studies, Dr. Andrei Müller introduced the world's first publically available software for portraying the classical 2-dimensional Smith Chart in a 3-dimentional format. A brief report on the 3D Smith Chart software was posted here on RF Cafe, and an article entitled "The 3D Smith Chart and Its Practical Applications" was published in Microwave Journal magazine. Since that time, Dr. Müller has authored many scholarly articles on circuit theory dealing largely with complex impedances. Recently and in collaboration with Dr. Stepan Lucyszyn, the paper "Properties of Purely Reactive Foster and non-Foster Passive Networks" dealt with Foster and non-Foster passive networks with 'a strongly real function of real frequency of positive type' (IET Electronic Letters publication). It can be read in its entirety on the IET Digital Library website for a modest fee. An Abstract is reproduced with permission below to whet your appetite for the information contained therein.

Visit the 3D Smith Chart website and/or on Facebook.

Application of the 3D Smith Chart software for depicting group delay.

Properties of Purely Reactive Foster and non-Foster Passive Networks

By Andrei A. Müller and Stepan Lucyszyn  -  ABSTRACT:

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.

Access the full paper on the IET Digital Library website.

Posted October 23, 2015