Schwarz lemma for driving point impedance functions and its circuit applications

Abstract

Summary In this paper, a boundary version of the Schwarz lemma is investigated for driving point impedance functions and its circuit applications. It is known that driving point impedance function, Z(s) = 1 + cp(s − 1)p + cp + 1(s − 1)p + 1 + ..., p > 1, is an analytic function defined on the right half of the s‐plane. Two theorems are presented using the modulus of the derivative of driving point impedance function, |Z′(0)|, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis with Z()0=0. In the obtained inequalities, the value of the function at s = 1 and the derivatives with different orders have been used. Finally, the sharpness of the inequalities obtained in the presented theorems is proved. Simple LC circuits are obtained using the obtained driving point impedance functions.

Some Results Concerned with Hankel Determinant

“…As exemplary applications, the use of positive real functions and boundary analysis of these functions for circuit synthesis can be given. Moreover, it is also possible to utilize Schwarz lemma for the analysis of transfer functions in control engineering and to design multi-notch filter structures in signal processing [14,15].…”

Boundary Analysis of Control Systems using Schwarz Lemma

“…Positive real functions (PRFs) play an important role in electrical engineering. Although they are mainly used in network synthesis as driving point impedance functions (DPIFs) [1,2], it is also possible to encounter PRFs in signal processing [3], control systems [4], and even in electromagnetic and microwave engineering [5]. Positive realness for the systems is frequently investigated in control theory literature [6,7].…”

Sharpened Forms for Driving Point Impedance Functions at Boundary of Right Half Plane

“…Theoretical analysis of positive real derivatives of DPIFs is given by (Krueger and Brown, 1969) where it's proved that the derivative of an RC driving point admittance is positive real under certain coefficient conditions. There are also other studies on boundary analysis of DPIFs using Schwarz lemma in the literature (Örnek and Düzenli, 2018;2019).…”