Boundary Analysis for the Derivative of Driving Point Impedance Functions

• Published in: IEEE transactions on circuits and systems. II, Express briefs Vol. 65; no. 9; pp. 1149 - 1153
• Main Authors: Ornek, Bulent Nafi, Duzenli, Timur
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.09.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: analytic function; Analytic functions; Driving point impedance functions; Impedance; Inequalities; Integrated circuit modeling; LC circuits; Mathematical analysis; RLC circuits; Schwarz lemma; Sharpness; Theorems
• ISSN: 1549-7747; 1558-3791
• DOI: 10.1109/TCSII.2018.2809539

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, <inline-formula> <tex-math notation="LaTeX">{Z(s)} </tex-math></inline-formula>, 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, <inline-formula> <tex-math notation="LaTeX">{|}{Z}^{\prime } {(0)|} </tex-math></inline-formula>, by assuming the <inline-formula> <tex-math notation="LaTeX">{Z(s)} </tex-math></inline-formula> function is also analytic at the boundary point <inline-formula> <tex-math notation="LaTeX">{s=}{0} </tex-math></inline-formula> on the imaginary axis with <inline-formula> <tex-math notation="LaTeX">{Z(0)=}{0} </tex-math></inline-formula>. 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.