Circuit Applications of Schwarz-Pick Lemma

Abstract

In this brief, the relation between Schwarz-Pick lemma and driving point impedance function is investigated. Schwarz-Pick lemma is given as the generalized version of Schwarz lemma. Here, a generic driving point impedance function is obtained by applying Schwarz-Pick lemma to positive real functions. At first, a novel inequality is obtained for the modulus of the derivative of the driving point impedance function and in the next step, an extremal function is found by applying sharpness analysis to this inequality. At the end, obtained extremal function is considered as a generic driving point impedance function and corresponding circuit schematics are presented with their spectral analyses. Obtained driving point impedance function is not an arbitrary function but it is the intuitive result of the considered problem in this study. Accordingly, it is guaranteed to obtain a practically realizable driving point impedance function. Also, novel upper bounds are determined for the magnitude of driving point impedance function and for its derivative.