TY - JOUR T1 - Fraksiyonel dereceli osilatörlerin pratik gerçekleştirimi için bir yaklaşım TT - An approach for practical realization of fractional-order oscillators AU - Saçu, İbrahim Ethem PY - 2024 DA - October Y2 - 2024 DO - 10.28948/ngumuh.1336490 JF - Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi JO - NÖHÜ Müh. Bilim. Derg. PB - Niğde Ömer Halisdemir Üniversitesi WT - DergiPark SN - 2564-6605 SP - 1342 EP - 1346 VL - 13 IS - 4 LA - tr AB - Bu çalışmada, fraksiyonel dereceli osilatörlerin pratik gerçekleştirimi için bir yaklaşım önerilmektedir. Bu yaklaşımda, fraksiyonel integral operatörünü sağlamak için s-domeninde birinci dereceden bir yaklaşıklık fonksiyonu türetilmekte daha sonra bu fonksiyon ayrıklaştırma işlemine tabi tutularak z-domenine aktarılmaktadır. Böylece ilgili osilatörün sürekli zamanlı dinamik denklemleri fark denklemlerine dönüştürülerek dijital platformlarda gerçekleştirilebilme imkanı elde edilmektedir. Fark denklemlerinin avantajı, katsayılarının fraksiyonel derecenin değişimine bağlı olarak hızlıca güncellenebilmesidir. Önerilen yaklaşımın uygulanabilirliği Van der Pol (VdP) osilatüründe test edilmiştir. Önerilen yöntem kullanılarak fraksiyonel VdP osilatörünün fark eşitlikleri elde edilmiştir. Nümerik benzetimlerde bu fark denklemlerinden faydalanılmıştır. Ayrıca denklemler bir mikroişlemci uygulama kartında test edilerek önerilen yöntem deneysel olarak da doğrulanmıştır. KW - Fraksiyonel hesaplama KW - Dinamik sistem KW - Van der Pol osilatörü KW - Fraksiyonel türev KW - Kararlılık N2 - In this study, an approach for the practical implementation of fractional-order oscillators has been proposed. In this approach, a first-order approximation function is derived in the s-domain to satisfy the fractional integral operator, then this function is discretized and transferred to the z-domain. This allows the continuous-time dynamic equations of the oscillator to be transformed into discrete-time difference equations, making it feasible for realization on digital platforms. The advantage of difference equations lies in the ease with which their coefficients can be updated based on changes in the fractional order. The feasibility of the proposed approach is tested on the Van der Pol (VdP) oscillator. The difference equations of the fractional-order VdP oscillator have been obtained by using the proposed method. These difference equations are used in numerical simulations. Furthermore, the equations are tested on a microcontroller application board, and the proposed method is experimentally validated also. CR - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods. World Scientific, 2012. https://doi.org/10.1142/10044 CR - M. Dalir, and M. Bashour, Applications of fractional calculus. Applied Mathematical Sciences, 4(21), 1021–1032, 2010. CR - H. Sun, Y. Zhang, D. Baleanu, W. Chen, and Y. Chen, A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation, 64, 213–231, 2018. https://doi.org/10. 1016/j.cnsns.2018.04.019 CR - C. Li, and F. Zeng, Numerical Methods for Fractional calculus. CRC Press, 2015. CR - G. Carlson, and C. Halijak, Approximation of fractional capacitors (1/s)^(1/n) by a regular Newton process. IEEE Transactions on Circuit Theory, 11(2), 210–213, 1964. https://doi.org/10.1109/TCT.1964. 1082270 CR - M. S. Semary, M. E. Fouda, H. N. Hassan, and A. G. Radwan, Realization of fractional-order capacitor based on passive symmetric network. Journal of Advanced Research, 18, 147–159, 2019. https://doi. org/10.1016/j.jare.2019.02.004 CR - B. van der Pol, The nonlinear theory of electric oscillations. Proc. IRE 22, 1051–1086, 1934. CR - J. H. He, The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators. Journal of Low Frequency Noise, Vibration and Active Control, 38(3–4), 1252–1260, 2019. https://doi.org/10. 1177/1461348419844 CR - M. Li, Three classes of fractional oscillators. Symmetry, 10(2), 40, 2018. https://doi.org/10.3390/ sym10020040 CR - İ. E. Saçu, and M. Alçı, An electronically controllable fractional multivibrator. IETE Journal of Research, 67(3), 313–321, 2021. https://doi.org/10.1080/ 03772063.2018.1548909 CR - B. van der Pol, A theory of the amplitude of free and forced triode vibrations. Radio Review, 1, 701–710, 754–762, 1920. CR - R. S. Barbosa, J. A. T. Machado, B. M. Vinagre, and A. J. Calderón, Analysis of the Van der Pol oscillator containing derivatives of fractional order. Journal of Vibration and Control, 13(9–10), 1291–1301, 2007. https://doi.org/10.1177/1077546307077463 CR - M.S. Tavazoei and M. Haeri, A note on the stability of fractional order systems. Math. Comput. Simul., 79(5), 1566–1576, 2009. https://doi.org/10.1016/j.matcom. 2008.07.003 CR - B. T. Krishna, Studies on fractional order differentiators and integrators: a survey. Signal Processing, 91, 386–426, 2011. https://doi.org/10.1016 /j.sigpro.2010.06.022 CR - M. A. Al-Alaoui, Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems. Electr Eng, 90, 455–467, 2008. 10.1007/s00202-007-0092-0 UR - https://doi.org/10.28948/ngumuh.1336490 L1 - https://dergipark.org.tr/tr/download/article-file/3303055 ER -