A Microcontroller-based Liénard Oscillator

Yıl 2022, Cilt: 5 Sayı: 2, 80 - 85, 31.12.2022

Ersoy Mevsim Reşat Mutlu

https://doi.org/10.55581/ejeas.1194452

Cited By: 1

Öz

Van der Pol Equation is a special case of Liénard equations. Both oscillators have significant historical importance. There are lots of different circuit topologies for Van der Pol and Liénard oscillators. Such oscillators can be made using vacuum tubes, diodes, etc. Some oscillators are made using microcontrollers, which are cheap and easy-to-use devices. They provide accurate adjustability of the frequency and magnitude of the waveforms. Arduino Nano Klon V3.0 microcontroller is a commonly used microcontroller. In this study, to the best of our knowledge, for the first time in literature, a Liénard Oscillator has been made with the Direct digital synthesis (DDS) method using the Arduino Nano Klon V3.0 microcontroller and two DACs. The experimental results of the oscillator are given. The circuit is able to produce the state variables of the oscillator, the effect of quantization can be seen on the waveforms, and it is shown that it performs well. The two variable outputs of the system let its phase portrait be examined easily. Also, using a microcontroller helps to design the oscillator in mere a few days.

Anahtar Kelimeler

Microcontroller-based circuit design, Liénard Equation, Liénard Oscillator, Van der Pol oscillator, Digital Circuit Synthesis, Limit Cycle, Circuit Dynamics.

Mikrodenetleyici Tabanlı Bir Liénard Osilatörü

Öz

Van der Pol Denklemi, Liénard denklemlerinin özel bir halidir. Her iki osilatör de önemli bir tarihsel öneme sahiptir. Pek çok farklı Van der Pol ve Liénard osilatör devreleri mevcuttur. Bu tür osilatörler vakum tüpleri, diyotlar vb. kullanılarak yapılabilmektedir. Bazı osilatörler, ucuz ve kullanımı kolay cihazlar olan mikrodenetleyiciler kullanılarak yapılmaktadır. Mikrodenetleyiciler dalga biçimlerinin frekansının ve büyüklüğünün doğru şekilde ayarlanabilmesini sağlarlar. Arduino Nano Klon V3.0 mikrodenetleyici yaygın olarak kullanılan bir mikrodenetleyicidir. Bu çalışmada bildiğimiz kadarıyla literatürde ilk defa Arduino Nano Klon V3.0 mikrodenetleyici ve iki DAC denetleyici kullanılarak direct dijital sentezleme (DDS) yöntemi ile bir Liénard Osilatörü yapılmıştır. Çıkış akımını artırmak için bir tampon opamp kullanılmıştır. Osilatörün deneysel sonuçları verilmiştir. Devre, osilatörün durum değişkenlerini üretebilir, kuantalama hatasının dalga biçimleri üzerindeki etkisi görülebilmektedir ve bu devrenin iyi performans gösterdiği gösterilmiştir. Sistemin iki değişken çıkışı, faz portresinin kolayca incelenmesini sağlar. Ayrıca, bir mikrodenetleyici kullanmak, osilatörün sadece birkaç gün içinde tasarlanmasını sağlmaktadır.

Anahtar Kelimeler

Mikrodenetleyici tabanlı devre tasarımı, Liénard Denklemi, Liénard Osilatörü, Van der Pol osilatörü, Dijital Devre Sentezi, Limit Döngüsü, Devre Dinamiği

Kaynakça

• [1] B. van der Pol, “A theory of the amplitude of free and forced triode vibrations”, Radio Review, 1, pp. 701–710, 754–762, 1920.
• [2] J. M. Ginoux, C. Letellier, “Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(2), 023120, 2012.
• [3] B. Van der Pol, LXXXVIII. On “relaxation oscillations”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11), 978- 992, 1926.
• [4] B. van der Pol, “The nonlinear theory of electric oscillations”, Proc. IRE, 22, pp. 1051–1086, 1934.
• [5] M. L. Cartwright, “I. Van der Pol’s Equation for Relaxation Oscillations”, In Contributions to the Theory of Nonlinear Oscillations (AM-29), Volume II, Princeton University Press., (pp. 1-18), 2016.
• [6] T. Marios, “Theoretical and Numerical Study of the Van der Pol equation”, Dissertation, 2006.
• [7] A. Liénard, "Etude des oscillations entretenues," Revue générale de l'électricité, 23, pp. 901–912 and 946–954, 1928.
• [8] J. Gleick, M. Berry, “Chaos-making a new science”, Nature, 330, 293, 1987.
• [9] S. Ahmad, “Study of Non-linear Oscillations Using Tunnel Diode”, Doctoral dissertation, 1962.
• [10] J. Brechtl, X. Xie, P. K. Liaw, “Investigation of chaos and memory effects in the Bonhoeffer-van der Pol oscillator with a non-ideal capacitor”, Communications in Nonlinear Science and Numerical Simulation, 73, 195-216, 2019.
• [11] T. J. Slight, B. Romeira, L. Wang, J. M. Figueiredo, E. Wasige, C. N. Ironside, “A Liénard oscillator resonant tunnelling diode-laser diode hybrid integrated circuit: model and experiment”, IEEE journal of quantum electronics, 44(12), 1158-1163, 2008.
• [12] T. J. Slight, B. Romeira, L. Wang, J. M. Figueiredo, E. Wasige, C. N. Ironside, “A Liénard oscillator resonant tunnelling diode-laser diode hybrid integrated circuit: model and experiment”, IEEE journal of quantum electronics, 44(12), 1158-1163, 2008.
• [13] Çakır, K., Mutlu, R., & Karakulak, E. Ters-Paralel Bağlı Schottky Diyot Dizisi Tabanlı Van der Pol Osilatörü Devresinin Modellenmesi ve LTspice ve Simulink Kullanarak Analizi. EMO Bilimsel Dergi, 11(21), 81-91.
• [14] Çakır, K., Mutlu, R., Modeling and Analysis of Schottky diode bridge and JFET based Liénard oscillator circuit, DOI: 10.14744/sigma.2022.00082.
• [15] M. Dursun, E. Kaşifoğlu, “Design and implementation of the FPGA-based chaotic van der pol oscillator”, International Advanced Researches and Engineering Journal, 2(3), 309-314, 2018.
• [16] Bilgin, S., Üser, Y., & Oktay, M. (2016). Low cost laboratory type signal generator using DDS method. International Journal of Engineering and Applied Sciences, 8(4), 59-65.
• [17] Abdullah, A. I., Mohammed, I. A., & AL-Helali, R. A. (2008). Microcontroller-Based Function Generator. Al-Khwarizmi Engineering Journal, 4(1).
• [18] Yener, S. C., Barbaros, C., Mutlu, R., & Karakulak, E. (2017). Implementation of Microcontroller-Based Memristive Chaotic Circuit. Acta Physica Polonica A, 132(3), 1058-1061
• [19] Yener, Suayb Cagri, and Resat Mutlu. "A microcontroller-based ECG signal generator design utilizing microcontroller PWM output and experimental ECG data." 2018 Electric Electronics, Computer Science, Biomedical Engineerings' Meeting (EBBT). IEEE, 2018.
• [20] Yener, Ş. Ç., & Mutlu, R. (2019, November). A Microcontroller Implementation of Hindmarsh-Rose Neuron Model-Based Biological Central Pattern Generator. In 2019 1st International Informatics and Software Engineering Conference (UBMYK) (pp. 1-4). IEEE.
• [21] Yener, Ş. Ç., Mutlu. R., & Karakulak. E. (2020). Implementation of a Microcontroller-Based Chaotic Circuit of Lorenz Equations. Balkan Journal of Electrical and Computer Engineering, 8(4), 355-360.
• [22] Karthikeyan, R., Çiçek, S., Pham, V. T., Akgul, A., & Duraisamy, P. (2020). A class of unexcited hyperjerk systems with megastability and its analog and microcontroller-based embedded system design. Physica Scripta, 95(5), 055214.
• [23] Karakulak, E., Tan, R. K., & Mutlu, R. (2021). STM32F429 Discovery Board-Based Emulator for Lotka-Volterra Equations. Journal of the Institute of Science and Technology, 11(3), 1887-1895.
• [24] Usta, B. N., Tepeyurt, B., & Karakulak, E. (2021, October). Simple Synthetic ECG Generation via PWM Output of Microcontroller. In 2021 5th International Symposium on Multidisciplinary Studies and Innovative Technologies (ISMSIT) (pp. 27-30). IEEE.
• [25] Karakulak, E., ARM MCU-Based Experimental EEG Signal Generator Using Internal DAC and PWM Outputs. Gazi University Journal of Science, 1-1.