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SHARPENED FORMS FOR DRIVING POINT IMPEDANCE FUNCTIONS AT BOUNDARY OF RIGHT HALF PLANE

Year 2021, Volume: 9 Issue: 4, 1093 - 1105, 20.12.2021
https://doi.org/10.21923/jesd.945359

Abstract

Driving point impedance functions (DPIFs) are frequently used in electrical engineering, and they represent characteristic properties of various types of circuits such as RL, RC, LC and RLC networks. In this paper, boundary analysis of driving point impedance functions are investigated using Schwarz lemma. Assuming that the driving point impedance function, Z(s), is given as Z(s)=A/2+c_p (s-1)^p+c_(p+1) (s-1)^(p+1)+... and it is analytic in the right half of the s-plane, novel boundaries are obtained for |Z^' (0)|. Accordingly, it is aimed to obtain novel inequalities which presents higher boundaries for |Z'(0)| and derive novel generic driving point impedace functions by performing extremal analysis of these obtained inequalities. It is also aimed to investigate how |Z'(s)| can be interpreted when it is considered at the boundary. According to simulation results, frequency characteristics of obtained driving point impedance functions can be used to design of multi-notch filters which are localized at certain frequency values.

References

  • Boas, H. P., 2010. Julius and Julia: Mastering the Art of the Schwarz lemma. The American Mathematical Monthly, 117 (9), 770-785.
  • Dineen, S., 2016. The Schwarz Lemma. Courier Dover Publications, USA.
  • Dubinin, V. N., 2004. The Schwarz inequality on the boundary for functions regular in the disk. Journal of Mathematical Sciences, 122 (6), 3623-3629.
  • Hazony, D., 1963. Elements of network synthesis. Reinhold Publishing Corporation, New York, USA.
  • Kresin, G., Maz'ja, V. G., 2007. Sharp real-part theorems. Berlin: Springer.
  • Krueger, R. J., Brown, D. P., 1969. Positive real derivatives of driving point functions. Journal of the Franklin Institute, 287 (1), 51-60.
  • Mercer, P. R., 1997. Sharpened versions of the Schwarz lemma. Journal of Mathematical Analysis and Applications, 205 (2), 508-511.
  • Mercer, P. R., 2018a. Boundary Schwarz inequalities arising from Rogosinski’s lemma. Journal of Classical Analysis, 12, 93-97.
  • Mercer, P. R., 2018b. An improved Schwarz Lemma at the boundary. Open Mathematics, 16 (1), 1140-1144.
  • Osserman, R., 2000. A sharp Schwarz inequality on the boundary. Proceedings of the American Mathematical Society, 128 (12), 3513-3517.
  • 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.
  • Örnek, B. N., Düzenli, T., 2019. Schwarz lemma for driving point impedance functions and its circuit applications. International Journal of Circuit Theory and Applications, 47 (6), 813-824.
  • Örnek, B. N. (2015). Caratheodory's inequality on the boundary. The Pure and Applied Mathematics, 22 (2), 169-178.
  • Reza, F. M., 1962. A bound for the derivative of positive real functions. SIAM Review, 4 (1), 40-42.
  • Van Der Pol, B., 1937. A new theorem on electrical networks. Physica, 4 (7), 585-589.

SAĞ YARI DÜZLEMİN SINIRINDAKİ SÜREN NOKTA EMPEDANS FONKSİYONLARI İÇİN KESKİNLEŞTİRİLMİŞ FORMLAR

Year 2021, Volume: 9 Issue: 4, 1093 - 1105, 20.12.2021
https://doi.org/10.21923/jesd.945359

Abstract

Süren nokta empedans fonksiyonları (SNEF), elektrik mühendisliğinde sıklıkla kullanılmaktadır ve RL, RC, LC, ve RLC ağları gibi farklı tipteki devrelerin karakteristik özelliklerini temsil etmektedirler. Bu çalışmada, süren nokta empedans fonksiyonlarının sınır analizi, Schwarz lemması kullanılarak araştırılmaktadır. Z(s) süren nokta empedans fonksiyonunun Z(s)=A/2+c_p (s-1)^p+c_(p+1) (s-1)^(p+1)+... yapısında olduğu ve sağ yarı s-düzleminde analitik olduğu varsayılarak, |Z'(0)| için yeni sınırlar belirlenmektedir. Buna göre, |Z'(0)| için yeni üst sınırlar temsil eden eşitsizlikler türetilmesi ve bu eşitsizliklerin ekstremal analizi ile yeni genel süren nokta empedans fonksiyonları elde edilmesi amaçlanmaktadır. Ayrıca, sınırda olduğu düşünüldüğü takdirde, |Z^' (s)|’nin nasıl yorumlanacağı meselesinin çözülmesi de hedeflenmektedir. Benzetim sonuçlarına göre, elde edilen süren nokta empedans fonksiyonlarının frekans karakteristikleri, belli frekanslarda konumlanmış çok çentikli süzgeçlerin tasarlanması için kullanılabilmektedir.

References

  • Boas, H. P., 2010. Julius and Julia: Mastering the Art of the Schwarz lemma. The American Mathematical Monthly, 117 (9), 770-785.
  • Dineen, S., 2016. The Schwarz Lemma. Courier Dover Publications, USA.
  • Dubinin, V. N., 2004. The Schwarz inequality on the boundary for functions regular in the disk. Journal of Mathematical Sciences, 122 (6), 3623-3629.
  • Hazony, D., 1963. Elements of network synthesis. Reinhold Publishing Corporation, New York, USA.
  • Kresin, G., Maz'ja, V. G., 2007. Sharp real-part theorems. Berlin: Springer.
  • Krueger, R. J., Brown, D. P., 1969. Positive real derivatives of driving point functions. Journal of the Franklin Institute, 287 (1), 51-60.
  • Mercer, P. R., 1997. Sharpened versions of the Schwarz lemma. Journal of Mathematical Analysis and Applications, 205 (2), 508-511.
  • Mercer, P. R., 2018a. Boundary Schwarz inequalities arising from Rogosinski’s lemma. Journal of Classical Analysis, 12, 93-97.
  • Mercer, P. R., 2018b. An improved Schwarz Lemma at the boundary. Open Mathematics, 16 (1), 1140-1144.
  • Osserman, R., 2000. A sharp Schwarz inequality on the boundary. Proceedings of the American Mathematical Society, 128 (12), 3513-3517.
  • 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.
  • Örnek, B. N., Düzenli, T., 2019. Schwarz lemma for driving point impedance functions and its circuit applications. International Journal of Circuit Theory and Applications, 47 (6), 813-824.
  • Örnek, B. N. (2015). Caratheodory's inequality on the boundary. The Pure and Applied Mathematics, 22 (2), 169-178.
  • Reza, F. M., 1962. A bound for the derivative of positive real functions. SIAM Review, 4 (1), 40-42.
  • Van Der Pol, B., 1937. A new theorem on electrical networks. Physica, 4 (7), 585-589.
There are 15 citations in total.

Details

Primary Language English
Subjects Electrical Engineering
Journal Section Research Articles
Authors

Bülent Nafi Örnek 0000-0001-7109-230X

Timur Düzenli 0000-0003-0210-5626

Publication Date December 20, 2021
Submission Date May 30, 2021
Acceptance Date July 5, 2021
Published in Issue Year 2021 Volume: 9 Issue: 4

Cite

APA 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