A Bound for the Derivative of Positive Real Functions and Corresponding Circuits

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In this paper, driving point impedance functions, Z(s)=A+c_1*(s-b)+c_2*(s-b)^2+..., which are frequently used in electrical engineering, have been considered for boundary analysis of the Schwarz lemma. Accordingly, considering the s_1, s_2, s_3…, s_n points in the right half plane which are different than s=b, Schwarz lemma has been obtained for positive real functions. In addition, a result of the Rogisinski’s lemma has been used to prove the new inequalities and the derivative of the driving point impedance function has been evaluated from below by considering Taylor expansion coefficients c_1 and c_2. For all presented inequalities, sharpness analysis has been carried out and extremal functions corresponding to different driving point impedance functions have been obtained. It is possible to say that simple circuits can be synthesized using the obtained transfer functions.