Applications of the Carathéodory’s Inequality for Driving Point Impedance Functions

Abstract

In this study, the Carathéodory’s Inequality, which is a highly popular topic of complex analysis theory, has been applied to electrical engineering to obtain novel driving point impedance functions. In electrical engineering, driving point impedance functions correspond to positive real functions and they are used for representation of the spectral characteristics of a particular circuit. Accordingly, boundary version of the Carathéodory’s inequality has been considered here assuming that the driving point empedance function, Z(s) has a fractional function structure with 0=0 and it is analytic in the right half plane. At the end of the analyses, new driving point impedance functions have been obtained and they have been presented with their spectral characteristics. According to simulation results, it is possible to say that the frequency responses of the obtained generic driving point impedance functions have spiky filter structures where the number of the spikes in the frequency response of these filters depend on a pre-defined parameter, n.

Keywords

• Driving point impedance function
• Carathéodory’s Inequality
• Circuit
• Filter

Süren Nokta Empedans Fonksiyonları için Carathéodory Eşitsizliği’nin Uygulamaları

Öz

Bu çalışmada, kompleks analiz teorisinde oldukça popular bir konu olan Carathéodory eşitsizliği, yeni süren nokta empedans fonksiyonları elde etmek için elektrik mühendisliğine uygulanmıştır. Elektrik mühendisliğinde süren nokta empedans fonksiyonları, pozitif reel fonksiyonlara karşılık gelmekte ve belli bir devrenin spektral özelliklerini temsil etmek için kullanılmaktadırlar. Buna göre, burada Carathéodory eşitsizliğinin bir sınır versiyonu, kesirli fonksiyon yapıdaki süren nokta empedans fonksiyonu Z(s) için, Z(s)’nin 0=0 olmak üzere sağ yarı düzlemde analitik olduğu varsayılarak değerlendirilmiştir. Analizler sonucunda, yeni süren nokta empedans fonksiyonları elde edilmiş ve bu fonksiyonlar spektral özellikleriyle birlikte sunulmuştur. Simülasyon sonuçlarına göre, elde edilen genel süren nokta empedans fonksiyonlarının frekans cevaplarının, sivri geçişli süzgeç yapısına sahip oldukları ve frekans cevabındaki bu sivri geçişlerin sayısının daha önce tanımlanmış olan bir n parametresine bağlı olduğunu söylemek mümkündür.

Anahtar Kelimeler

• Süren nokta empedans fonksiyonu
• Carathéodory eşitsizliği
• Devre
• Süzgeç

Kaynakça

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