Error Free Butcher Algorithms for Linear Electrical Circuits

ETRI Journal

关键词: single term Walsh series (STWS) and RK-Butcher algorithms;    Singular systems;   

Others  :  1185294
DOI  :  10.4218/etrij.05.0104.0055

【 摘 要 】

In this paper, an error-free Butcher algorithm is introduced to study the singular system of a linear electrical circuit for time invariant and time varying cases. The discrete solutions obtained using Runge-Kutta (RK)-Butcher algorithms are compared with the exact solutions of the electrical circuit problem and are found to be very accurate.Stability regions for the single term Walsh series (STWS) method and the RK-Butcher algorithm are presented. Error graphs for inductor currents and capacitor voltages are presented in a graphical form to show the efficiency of the RK-Butcher algorithm.This RK-Butcher algorithm can be easily implemented in a digital computer for any singular system of electrical circuits.

   

【 参考文献 】

• [1]R.K. Alexander and J.J. Coyle, "Runge-Kutta Methods for Differential-Algebric Systems," SIAM J. of Numerical Analysis, vol. 27, no. 3, 1990, pp. 736-752.
• [2]D.J. Evans, "A New 4th Order Runge-Kutta Method for Initial Value Problems with Error Control," Int.l J. Computer Mathematics, vol. 139, 1991, pp. 217-227.
• [3]D.J. Evans and A.R. Yaakub, "A New Fifth Order Weighted Runge-Kutta Formula," Int. J. Computer Mathematics, vol. 59, 1996, pp. 227-243.
• [4]D.J. Evans and A.R. Yaakub, "Weighted Fifth Order Runge-Kutta Formulas for Second Order Differential Equations," Int. J. Computer Mathematics, vol. 70, 1998, pp. 233-239.
• [5]J.D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, Chichester, UK, 1991.
• [6]K. Murugesan, D. Paul Dhayabaran, and D.J. Evans, "Analysis of Different Second Order Systems via Runge-Kutta Method," Int. J. Computer Mathematics, vol. 70, 1999, pp. 477-493.
• [7]K. Murugesan, D. Paul Dhayabaran, and D.J. Evans, "Analysis of Different Second Order Multivariable Linear System Using Single Term Walsh Series Technique and Runge-Kutta Method," Int. J. of Computer Mathematics, vol. 72, 1999, pp. 367-374.
• [8]K. Murugesan, D. Paul Dhayabaran, and D.J. Evans, "Analysis of Non-Linear Singular System from Fluid Dynamics Using Extended Runge-Kutta Methods," Int. J. Computer Mathematics, vol. 76, 2000, pp. 239-266.
• [9]K. Murugesan, D. Paul Dhayabaran, E.C. Henry Amirtharaj, and D.J. Evans, "A Comparison of Extended Runge-Kutta Formulae Based on Variety of Means to Solve System of IVPs," Int.J. of Computer Mathematics, vol. 78, 2001, pp. 225-252.
• [10]K. Murugesan, D. Paul Dhayabaran, E.C. Henry Amirtharaj and D.J. Evans, "A Fourth Order Embedded Runge-Kutta RKACeM(4,4) Method Based on Arithmetic and Centrodial Means with Error Control," Int. J. Computer Mathematics, vol. 79, no. 2, 2002, pp. 247-269.
• [11]J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, John Wiley & Sons, UK, 1987.
• [12]J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, UK, 2003.
• [13]L.F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, USA, 1994.
• [14]E. Fehlberg, "Klassische Runge-Kutta-Formeln Vierter und Niedriger Ordnung Mit Schrittweitenkontolle und Ihre Anwendung auf Warmeleitungsprobleme," Computing, vol. 6, 1970, pp. 61-71.
• [15]G.E. Forsythe, M.A. Malcolm, and C.D. Moler, Computer Methods for Mathematical Computations, Englewood Cliffs. NJ: Prentice-Hall, New Jersey, p.135. 1977.
• [16]M. Bader, "A Comparative Study of New Truncation Error Estimates and Intrinsic Accuracies of Some Higher Order Runge-Kutta Algorithms," Computational Chemistry, vol. 11, 1987, pp. 121- 124.
• [17]M. Bader, "A New Technique for the Early Detection of Stiffness in Coupled Differential Equations and Application to Standard Runge-Kutta Algorithms," Theoretical Chemistry Accounts, vol. 99, 1998, pp. 215-219.
• [18]K. Murugesan, S. Sekar, V. Murugesh, and J.Y. Park, "Numerical Solution of an Industrial Robot Arm Control Problem Using the RK-Butcher Algorithm," Int. J. Computer Applications in Technology, vol. 19, no. 2, 2004, pp. 132-138.
• [19]J.Y. Park, David J. Evans, K. Murugesan, S. Sekar, and V. Murugesh, "Optimal Control of Singular Systems Using the RK-Butcher Algorithm," Int. J. Computer Mathematics, vol. 81, no. 2, 2004, pp. 239-249.
• [20]L.F. Shampine and M.K. Gordon, Computer Solution of Ordinary Differential Equations-The Initial Value Problem, W.H. Freeman & Co, San Francisco, USA, 1975.
• [21]L.O. Chua and P.M. Lin, Computer-Aided Analysis of Electronic Circuits, Prentice-Hall, New Jersey, USA, 1975.
• [22]K. Balachandran and K. Murugesan, "Analysis of Electronic Circuits Using the Single-Term Walsh Series Approach," Int. J. of Electronics, vol. 69, no. 3, 1990, pp. 327-332.
• [23]H.C. Yan, Investigation of Generalized State-Space Systems, Ph.D. thesis, Clarkson University, Potsdam, USA, 1986.