Boundary Analysis for the Derivative of Driving Point Impedance Functions

Abstract

Four theorems are presented in this brief by performing boundary analysis of the derivative of driving point impedance functions evaluated at the origin. Also, the circuits corresponding to these driving point impedance functions, which are obtained as the natural results of presented theorems, are given. Driving point impedance functions are mainly used for synthesis of networks containing RL, RC, and RLC circuits. Considering that the driving point impedance function, Z(s), is an analytic function defined on the right half of the s-plane, we derive inequalities for the modulus of the derivative of driving point impedance function, vertical bar Z'(0)vertical bar, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis with Z(0) = 0. Finally, the sharpness of these inequalities are proved. Unique driving point impedance functions are obtained as intuitive results of presented theorems in the study and it is also shown that the extremal functions correspond to the driving-point impedances of simple LC circuits.