APPLICATION OF HYBRID METHODS FOR NUMERICAL SOLVING OF ORDINARY DIFFERENTIAL EQUATIONS FOR ANALYSIS OF SELF-OSCILLATING CIRCUITS WITH VARIOUS DYNAMICS
DOI:
https://doi.org/10.18522/2311-3103-2026-1-%25pKeywords:
Numerical method, ordinary differential equation, autogenerator, harmonic oscillation, relaxation oscillatio, simulation errorAbstract
Ensuring the accuracy and stability of computer simulation of electronic devices is an important problem in their design. The greatest difficulties in simulation of electronic devices arise in the case of the analysis of self-oscillating circuits, since mathematical models of such circuits can be stiff and oscillating at the same time. The aim of this work is to develop an efficient numerical method for solving ordinary differential equations that provides higher accuracy of time domain analysis for various types of autogenerators compared to existing methods. The proposed method is a hybrid method and is based on the well-known Gear and trapezoidal methods used in simulators of electronic circuits. To evaluate the accuracy of the proposed method and known methods a generalized model of a self-oscillating circuit was used for which an analytical solution was determined in the steady-state operating mode. The accuracy of the numerical solution was determined based on the analysis of errors in estimating the main parameters of the oscillatory process – the amplitude and frequency of oscillations. A comparative analysis of errors in estimating the amplitude and frequency of oscillations in autogenerators demonstrates the high efficiency of the proposed hybrid method for analyzing both harmonic oscillators and relaxation oscillators. A further increase in the accuracy of the hybrid method is possible using implicit Runge-Kutta methods (Rado IIA and Lobatto IIIA subclasses), which have L- and P-stability, respectively. It should be noted that with an increase in the order of accuracy of implicit Runge-Kutta methods, the computational complexity of these methods increases, but for the Rado IIA and Lobatto IIIA subclasses the increase in computational complexity will be minimal.
References
1. Engel'khard M., Revenko A. SPICE Differentiation. Razlichiya v realizatsiyakh simulyatorov SPICE [SPICE Differentiation. Differences in SPICE Simulator Implementations], Komponenty i tekhnologii [Components and Technologies], 2015, No. 7 (168), pp. 82-88.
2. Prikota A. Programma SimOne. Sovremennye sredstva skhemotekhnicheskogo modelirovaniya [SimO-ne Program. Modern Means of Circuit Simulation], Elektronika NTB [Electronics Science Technology Business], 2011, No. 2, pp. 142-146.
3. Petzold L.R., Jay L.O., Yen J. Numerical solution of highly os-cillatory ordinary differential equations, Acta Numerica, 1997, pp. 437-483.
4. Guglielmi N., Hairer E. Applying Stiff Integrators for Ordinary Differential Equations and Delay Dif-ferential Equations to Problems with Distributed Delays, SIAM Journal on Scientific Computing, 2025, Vol. 47, No. 1. DOI: 10.1137/24M1632413.
5. Zhuk D.M., Kozhevnikov D.Yu., Manichev V.B. Problemy razrabotki matematicheskogo yadra dlya pro-gramm modelirovaniya dinamiki tekhnicheskikh sistem [Problems of developing a mathematical kernel for programs for modeling the dynamics of technical systems], Problemy razrabotki perspektivnykh mikro- i nanoelektronnykh sistem (MES) [Problems of developing promising micro- and nanoelectronic systems (MES)], 2020, No. 4, pp. 31-38. DOI: 10.31114/2078-7707-2020-4-31-38.
6. Bernardini A., Maffezzoni P., Sarti A. Linear Multistep Discretization Methods With Variable Step-Size in Nonlinear Wave Digital Structures for Virtual Analog Modeling, IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2019, Vol. 27, No. 11, pp. 1763-1776. DOI: 10.1109/TASLP.2019.2931759.
7. Pilipenko A.M., Biryukov V.N. Issledovanie effektivnosti sovremennykh chislennykh metodov pri analize avtokolebatel'nykh tsepey [Study of the efficiency of modern numerical methods in the analysis of self-oscillating circuits], Zhurnal radioelektroniki [Journal of Radio Electronics], 2013, No. 8. Available at: http://jre.cplire.ru/jre/aug13/9/text.html (accessed 24 August 2025).
8. Pilipenko A.M., Biryukov V.N. Otsenka tochnosti chislennogo analiza relaksatsionnogo generatora [Esti-mation of the accuracy of numerical analysis of a relaxation oscillator], Zhurnal radioelektroniki [Zhur-nal Radioelektroniki], 2013, No. 11, pp. 7. Available at: http://jre.cplire.ru/jre/nov13/6/text.html (ac-cessed 24 August 2025).
9. Pilipenko A.M., Agabekyan A.V. Model' avtokolebatel'noy tsepi dlya testirovaniya metodov chislennogo analiza perekhodnykh protsessov v SPICE-simulyatorakh [Model of a self-oscillating circuit for testing methods of numerical analysis of transient processes in SPICE simulators], Izvestiya YuFU. Tekhniches-kie nauki [Izvestiya SFedU. Engineering Sciences], 2022, No. 3 (227), pp. 243-254. DOI: 10.18522/2311-3103-2022-3-243-254.
10. Biryukov V.N. Rukovodstvo k laboratornym rabotam po kursu «Osnovy avtomatizirovannogo analiza tsepey» [Laboratory Manual for the Course "Fundamentals of automated circuit analysis"], Issue 3. Ta-ganrog: Izd-vo TRTU, 2001, 27 p.
11. Panteleev A.V., YAkimova A.S., Bosov A.V. Obyknovennye differentsial'nye uravneniya: osnovy teorii i algoritmy resheniy: ucheb. posobie [Ordinary differential equations: fundamentals of theory and solution algorithms: textbook]. Moscow: Vuzovskaya kniga, 2012, 188 p.
12. Gear C.W. Algorithm 407: DIFSUB for Solution of Ordinary Differential Equation, Communications of the ACM, 1971, Vol. 14, No. 4, p. 185-190.
13. Chua L.O. Pen-Min L. Mashinnyy analiz elektronnykh skhem: algoritmy i vychislitel'nye metody [Ma-chine analysis of electronic circuits: algorithms and computational methods]: transl. from Engl. by E.S. Vilenkina i dr., ed. by V.N. Il'ina. Moscow: Energiya, 1980, 640 p.
14. Pilipenko A.M., Biryukov V.N. Development and testing of high accuracy hybrid methods for time-domain simulation of electronic circuits and systems, Proceedings of 2017 IEEE East-West Design and Test Symposium, EWDTS, 2017, pp. 8110124. DOI 10.1109/EWDTS.2017.8110124.
15. Biryukov V.N., Pilipenko A.M. Chislennyy analiz zhestkikh uzkopolosnykh sistem [Numerical analysis of rigid narrowband systems], Radiosistemy [Radio Systems], 2002, No. 2 (62), pp. 36-38.
16. Pilipenko A.M. Biryukov V.N. Gibridnye metody resheniya obyknovennykh differentsial'nykh uravneniy zhestkikh i/ili kolebatel'nykh tsepey [Hybrid methods for solving ordinary differential equations of rigid and/or oscillatory circuits], Radiotekhnika [Radio Engineering], 2011, No. 1, pp. 11-15.
17. Pilipenko A.M. Gibridnye metody vysokogo poryadka tochnosti dlya chislennogo analiza vo vremennoy oblasti zhestkikh i kolebatel'nykh tsepey [Hybrid methods of high order of accuracy for numerical analy-sis in the time domain of rigid and oscillatory circuits], Modelirovanie, optimizatsiya i informatsionnye tekhnologii [Modeling, Optimization and Information Technology], 2017, No. 3 (18). Available at: https://moitvivt.ru/ru/journal/pdf?id=367 (accessed 24 August 2025).
18. Khayrer E. Vanner G. Reshenie obyknovennykh differentsial'nykh uravneniy: Zhestkie i differentsi-al'no-algebraicheskie zadachi [Solution of ordinary differential equations: Stiff and differential-algebraic problems], transl. from the second Engl. ed. by E.L. Starostina i dr., ed. by S.S. Filippova. Moscow: Mir, 1999, 685 p.
19. Rakitskiy Yu.V., Ustinov S.M., Chernorutskiy I.G. Chislennye metody resheniya zhestkikh system [Nu-merical methods for solving stiff systems]. Moscow: Nauka, 1979, 208 p.
20. Maffezzoni P. Codecasa L., D’Amore D. Time-domain simulation of nonlinear circuits through implicit Runge-Kutta methods, IEEE Transactions. Circuits and Systems, 2007, Vol. 54, No. 2, pp. 391-400.
