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

References

1. Akkaya, R., Endiz, M. S. (2020). Yarı empedans kaynaklı i̇nverter Devresinin Performans analizi. European Journal of Science and Technology (EJOSAT), Special Issue, 13–20. https://doi.org/10.31590/ejosat.801852
2. Örnek, B. N., Düzenli, T. (2018). On boundary analysis for derivative of driving point impedance functions and its circuit applications. IET Circuits, Devices & Systems, 13(2), 145–152. https://doi.org/10.1049/iet-cds.2018.5123
3. Örnek, B. N., Düzenli, T. (2018). Boundary Analysis for the derivative of driving point impedance functions. IEEE Transactions on Circuits and Systems II: Express Briefs, 65(9), 1149–1153. https://doi.org/10.1109/tcsii.2018.2809539
4. Tavazoei, M. S. (2018). Passively realisable impedance functions by using two fractional elements and some resistors. IET Circuits, Devices & Systems, 12(3), 280–285. https://doi.org/10.1049/iet-cds.2017.0342
5. Mukhtar, F., Kuznetsov, Y., Russer, P. (2011). Network modelling with Brune's Synthesis. Advances in Radio Science, 9, 91–94. https://doi.org/10.5194/ars-9-91-2011
6. Wunsch, A. D., Sheng-Pin Hu. (1996). A closed-form expression for the driving-point impedance of the small inverted L Antenna. IEEE Transactions on Antennas and Propagation, 44(2), 236–242. https://doi.org/10.1109/8.481653
7. Reza, F. M. (1962). A bound for the derivative of positive real functions. SIAM Review, 4(1), 40–42. https://doi.org/10.1137/1004005
8. Richards, P. I. (1947). A special class of functions with positive real part in a half-plane. Duke Mathematical Journal, 14(3), 777–789. https://doi.org/10.1215/s0012-7094-47-01461-0

9. Dineen, S. (2016). Schwarz lemma. Dover Publications Inc.
10. Kresin, G., Maz'Ya, V., Shaposhnikova, T. (2007). Sharp real-part theorems: A unified approach. Springer.
11. Örnek, B. N. (2015). Carathéodory's inequality on the boundary. The Pure and Applied Mathematics, 22(2), 169–178. https://doi.org/10.7468/jksmeb.2015.22.2.169 Ornek, B. N. (2016). The caratheodory inequality on the boundary for holomorphic functions in the unit disc. Journal of Mathematical Physics, Analysis, Geometry, 12(4), 287–301. https://doi.org/10.15407/mag12.04.287
12. Osserman, R. (2000). A Sharp schwarz inequality on the boundary. Proceedings of the American Mathematical Society, 128(12), 3513–3517. https://doi.org/10.1090/s0002-9939-00-05463-0
13. Mercer, P. R. (1997). Sharpened versions of the Schwarz lemma. Journal of Mathematical Analysis and Applications, 205(2), 508–511. https://doi.org/10.1006/jmaa.1997.5217
14. Mercer, P. R. (2018). Boundary Schwarz inequalities arising from Rogosinski's lemma. Journal of Classical Analysis, (2), 93–97. https://doi.org/10.7153/jca-2018-12-08
15. Mercer, P. R. (2018). An improved Schwarz lemma at the boundary. Open Mathematics, 16(1), 1140–1144. https://doi.org/10.1515/math-2018-0096
16. Dubinin, V. N. (2004). The Schwarz inequality on the boundary for functions regular in the disk. Journal of Mathematical Sciences, 122(6), 3623–3629. https://doi.org/10.1023/b:joth.0000035237.43977.39
17. Mateljevic, M. (2018). ‘Rigidity of holomorphic mappings & Schwarz and Jack lemma’, https://doi.org/10.13140/RG.2.2.34140.90249.
18. Azeroǧlu, T. A., Örnek, B. N. (2012). A refined Schwarz inequality on the boundary. Complex Variables and Elliptic Equations, 58(4), 571–577. https://doi.org/10.1080/17476933.2012.718338
19. Harold P. Boas. (2010). Julius and julia: Mastering the art of the schwarz lemma. The American Mathematical Monthly, 117(9), 770-785. https://doi.org/10.4169/000298910x521643
20. Örnek, B. N., Düzenli, T. (2021). Rogosinski Lemması ile ilgili Süren Nokta Empedans Fonksiyonları için Carathéodory Eşitsizliği. Dicle Üniversitesi Mühendislik Fakültesi (DÜMF) Mühendislik Dergisi, 12(1), 61-68. https://doi.org/10.24012/dumf.860229
21. Örnek, B. N., Düzenli, T. (2021). Sharpened forms for driving point impedance functions at boundary of right half plane. Mühendislik Bilimleri ve Tasarım Dergisi, 9(4), 1093-1105. https://doi.org/10.21923/jesd.945359