Schwarz lemma for driving point impedance functions and its circuit applications

Abstract

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 + c(p)(s - 1)(p) + c(p + 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.