Analytical approximate solutions for a general nonlinear resistor–nonlinear capacitor circuit model

Abstract In this paper, the analytical approximate solutions of a general RC circuit comprised of a nonlinear resistor in series with a nonlinear capacitor are addressed. In the studied circuit, the capacitor is characterized by a quintic polynomial voltage–charge dependence and the resistor obeys a cubic polynomial voltage–current relation. An efficient and easy-to-implement algorithm based on a hybrid analytical–numerical mathematical technique, namely the multistage Adomian decomposition method (MADM) is applied for solving the nonlinear differential equation governing the circuit performance. It is shown that the classic Adomian decomposition method fails to provide accurate convergent solutions for the posed problem over the whole semi-infinite time domain; however, the MADM can easily achieve convenient solutions with any desired degree of accuracy for both the transient and steady state time zones by exploiting its two embedded precision adjustment parameters. For the sake of illustration, two relevant numerical examples are solved by the MADM and simulated by the MATLAB–Simulink, as well. The results by the MADM are evaluated as highly accurate, based on comparison. In addition to the circuit theory aspects, the present work might be of particular interest from a practical point of view as the quintic nonlinear capacitor typically represents the widely used ferroelectric ceramic capacitors.

[1]  Saeid Abbasbandy,et al.  Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method , 2003, Appl. Math. Comput..

[2]  Electrical properties and stability of Tb-doped zinc oxide-based nonlinear resistors , 2007 .

[3]  Hooman Fatoorehchi,et al.  Approximating the minimum reflux ratio of multicomponent distillation columns based on the Adomian decomposition method , 2014 .

[4]  Mehdi Dehghan,et al.  Application of the Adomian decomposition method for the Fokker-Planck equation , 2007, Math. Comput. Model..

[5]  Ji-Huan He.  A new approach to nonlinear partial differential equations , 1997 .

[6]  Abdul-Majid Wazwaz,et al.  A reliable modification of Adomian decomposition method , 1999, Appl. Math. Comput..

[7]  Jun-Sheng Duan,et al.  Recurrence triangle for Adomian polynomials , 2010, Appl. Math. Comput..

[8]  Randolph Rach,et al.  A new definition of the Adomian polynomials , 2008, Kybernetes.

[9]  Abdul-Majid Wazwaz,et al.  An analytic study of Fisher's equation by using Adomian decomposition method , 2004, Appl. Math. Comput..

[10]  Liang-ying Zhang,et al.  Sol-gel Glass-coated Zinc Oxide for Varistor Applications , 1998 .

[11]  G. Adomian,et al.  A modified decomposition , 1992 .

[12]  Randolph Rach,et al.  A bibliography of the theory and applications of the Adomian decomposition method, 1961-2011 , 2012 .

[13]  Peter A. Cook,et al.  Nonlinear dynamical systems , 1986 .

[14]  Abraham J. Arenas,et al.  Piecewise finite series solutions of seasonal diseases models using multistage Adomian method , 2009 .

[15]  H. Fatoorehchi,et al.  Adomian Decomposition Method to Study Mass Transfer within Wetted Wall Columns , 2011 .

[16]  R. Rach,et al.  An Efficient Numerical Scheme to Solve a Quintic Equation of State for Supercritical Fluids , 2015 .

[17]  Tamer A. Abassy.  Improved Adomian decomposition method (solving nonlinear non-homogenous initial value problem) , 2011, J. Frankl. Inst..

[18]  A. Rèpaci,et al.  Nonlinear dynamical systems: On the accuracy of adomian's decomposition method , 1990 .

[19]  Michael A. Gray.  Introduction to the Simulation of Dynamics Using Simulink , 2010 .

[20]  Muhammet Köksal,et al.  Analysis of nonlinear circuits by using differential Taylor transform , 2002, Comput. Electr. Eng..

[21]  Y. Cherruault,et al.  Convergence of Adomian's method applied to nonlinear equations , 1994 .

[22]  Saverio Morfu,et al.  Analog simulation of neural information propagation using an electrical FitzHugh–Nagumo lattice , 2004 .

[23]  Jafar Biazar,et al.  On the order of convergence of Adomian method , 2002, Appl. Math. Comput..

[24]  Gert Fregien,et al.  Non-linear capacitors in snubber circuits for GTO-thyristors , 1989 .

[25]  Songcheol Hong,et al.  A semi-empirical cad model of ferroelectric capacitor for circuit simulation , 1997 .

[26]  Abdul M. Siddiqui,et al.  Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid , 2010 .

[27]  Y. Cherruault,et al.  Adomian method for solving adaptive control problem , 2005 .

[28]  R. Rach,et al.  An accurate explicit form of the Hankinson–Thomas–Phillips correlation for prediction of the natural gas compressibility factor , 2014 .

[29]  Lucas Jódar,et al.  Piecewise finite series solution of nonlinear initial value differential problem , 2009, Appl. Math. Comput..

[30]  Dipankar Bhanja,et al.  An analytical prediction for performance and optimum design analysis of porous fins , 2011 .

[31]  H. Fatoorehchi,et al.  On Calculation of Adomian Polynomials by MATLAB , 2011 .

[32]  Jafar Biazar,et al.  An alternate algorithm for computing Adomian polynomials in special cases , 2003, Appl. Math. Comput..

[33]  G. Adomian.  A new approach to nonlinear partial differential equations , 1984 .

[34]  Abdul-Majid Wazwaz,et al.  A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions , 2013 .

[35]  Longtu Li,et al.  Improvement of electric fatigue in PLZT ferroelectric capacitors due to zirconia incorporation , 2001 .

[36]  H. Fatoorehchi,et al.  A more realistic approach toward the differential equation governing the glass transition phenomenon , 2013 .

[37]  Randolph Rach,et al.  Near-field and far-field approximations by the Adomian and asymptotic decomposition methods , 2011, Appl. Math. Comput..

[38]  Emanuel Gluskin,et al.  The use of non-linear capacitors , 1985 .

[39]  George Adomian,et al.  A review of the decomposition method and some recent results for nonlinear equations , 1990 .

[40]  Esmail Babolian,et al.  Restarted Adomian method for integral equations , 2004, Appl. Math. Comput..

[41]  Muhammet Köksal,et al.  A fast algorithm to compute the steady-state solution of nonlinear circuits by piecewise linearization , 2002, Comput. Electr. Eng..

[42]  D. Bedrosian,et al.  Time-domain analysis of networks with internally controlled switches , 1992 .

[43]  Hooman Fatoorehchi,et al.  Improving the differential transform method: A novel technique to obtain the differential transforms of nonlinearities by the Adomian polynomials , 2013 .

[45]  Emanuel Gluskin,et al.  A nonlinear resistor and nonlinear inductor using a nonlinear capacitor , 1999 .

[46]  Y. Cherruault,et al.  Decomposition methods: A new proof of convergence , 1993 .

[47]  Somchai Wongwises,et al.  A decomposition analysis on convecting-radiating rectangular plate fins for variable thermal conductivity and heat transfer coefficient , 2012, J. Frankl. Inst..

[48]  Jafar Biazar,et al.  Solution of the kinetic modeling of lactic acid fermentation using Adomian decomposition method , 2003, Appl. Math. Comput..

[49]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .