On boundary analysis for derivative of driving point impedance functions and its circuit applications

Abstract

In this study, a boundary analysis is carried out for the derivative of driving point impedance (DPI) functions, which is mainly used for the synthesis of networks containing resistor-inductor, resistor-capacitor and resistor-inductor-capacitor circuits. It is known that DPI function, Z(s), is an analytic function defined on the right half of the s-plane. In this study, the authors present four theorems using the modulus of the derivative of DPI function, |Z′(0)|, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis and finally, the sharpness of the inequalities obtained in the presented theorems are proved. It is also shown that simple inductor-capacitor tank circuits and higher-order filters are synthesised using the unique DPI functions obtained in each theorem.

A new method to realize nonrational driving‐point functions

“…An extension of the realizability conditions, as given in Örnek and Düzenli, to the nonrational functions requires that each and every approximant is convergent and realizable (as stated above). Therefore, for the function N ( s ) to be realizable, P n ( s ) must be a polynomial with positive coefficients for all n ; then, using , we can prove the theorem that, if the coefficients of P n ( s ) are positive for all n and is positive for x ∈ ζ , then Q n ( s ) is a polynomial with positive coefficients. The degree of Q n ( s ) is one less than the degree of P n ( s ). …”

“…Finite subset of an RL ladder circuit for the Lambert W function An extension of the realizability conditions, as given in Örnek and Düzenli, 23 to the nonrational functions requires that each and every approximant is convergent and realizable (as stated above). Therefore, for the function N(s) to be realizable, P n (s) must be a polynomial with positive coefficients for all n; then, using (8), we can prove the theorem that, if the coefficients of P n (s) are positive for all n and [N(s)](x) is positive for x ∈ , then 1.…”

Rogosinski Lemması ile ilgili Süren Nokta Empedans Fonksiyonları için Carathéodory Eşitsizliği

“…( ) süren nokta empedans fonksiyonunun türevinin modülünün sıfır noktasındaki değeri, yani | ′(0)| göz önüne alınarak üç teorem içerisinde ′( ) için üç alt sınır sunulmaktadır. [9] ve [10]…”

Upper bound of Hankel determinant for a class of analytic functions

“…Usage of positive real function and boundary analysis of these functions for circuit synthesis can be given as an exemplary application of the Schwarz lemma in electrical engineering. Furthermore, it is also used for analysis of transfer functions in control engineering and multi-notch filter design in signal processing [14][15][16].…”