Representation of Impedance Functions in Terms of Resonant Frequencies

The conventional extension of Foster's reactance theorem to electric circuits with an infinite number of degrees of freedom (sections of transmission lines and cavity resonators) leads to series which converge so slowly that often seemingly natural approximations make the series actually divergent. There exist, however, modified expansions which are suitable for numerical calculations and which admit of an attractive physical interpretation. Similar expansions can also be obtained for the transfer impedance. The method of approach is function-theoretic and is based on the assumption that the driving-point impedance and the transfer impedance are analytic functions of the oscillation constant. When these functions are single-valued, they may be represented as certain series of partial fractions or series of functions analogous to partial fractions. In the case of pure reactances each term of these series corresponds to a resonant circuit coupled to the input element or a resonant transducer. The results are approximately true for slightly dissipative systems.