An asymptotic solution to nonlinear transmission lines. (English) Zbl 1136.35078
Summary: This paper explores an asymptotic approach to the solution of a nonlinear transmission line model. The model is based on a set of nonlinear partial differential equations without analytic solution. The perturbations method is used to reduce the system of nonlinear equations to a single nonlinear partial differential equation, namely to the modified Korteweg-de Vries equation (mKdV). By using the Laplace transform, the solution is represented in integral form in terms of Green’s functions. The solution for the nonlinear case is obtained by means of asymptotic methods. Thus, an approximate explicit analytic solution to the problem is obtained where the errors can be controlled. This allows us to analyze the nonlinear behavior of the solution. This kind of information is difficult to obtain by means of numerical methods due to the fact that for large periods of time greater computational resources are required and accumulated errors increase. For this reason, asymptotic methods have a great importance as a natural complement to numerical methods. Computer simulations support the presented developments.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35A35 | Theoretical approximation in context of PDEs |
| 35C15 | Integral representations of solutions to PDEs |
Keywords:
nonlinear transmission lines; asymptotic solution; Korteweg-de Vries equation (KdV); perturbations methodReferences:
| [1] | Ablowitz, M. J.; Segur, H., Solitons and Inverse Scattering Transform (1981), SIAM: SIAM Philadelphia · Zbl 0472.35002 |
| [2] | R.L. Burden, J.D. Faires, Análisis numérico, 3a. edición, Grupo Editorial Iberoamérica, S.A. de C.V., México, 1985, pp. 14-38.; R.L. Burden, J.D. Faires, Análisis numérico, 3a. edición, Grupo Editorial Iberoamérica, S.A. de C.V., México, 1985, pp. 14-38. |
| [3] | Calogero, F.; Degasperis, A., Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations (1982), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0501.35072 |
| [4] | Carneiro, S.; Martí, J. R., Evaluation of corona and line models in electromagnetic transient simulations, IEEE Trans. Power Delivery, 6, 334-342 (1991) |
| [5] | A. Castro Figueroa, Ecuaciones en derivadas parciales, Addison-Wesley Iberoamericana, S. A., Wilmington, Delaware, E.U.A., 1997.; A. Castro Figueroa, Ecuaciones en derivadas parciales, Addison-Wesley Iberoamericana, S. A., Wilmington, Delaware, E.U.A., 1997. |
| [6] | Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C., Solitons and Nonlinear Wave Equations (1988), Academic Press, Inc.: Academic Press, Inc. New York, USA · Zbl 0496.35001 |
| [7] | Gupta, R.; Kim, S. Y., Domain characterization of transmission line models an analysis, IEEE Trans. Computer-Aided Design Integrated Circuits Systems, 152, 184-193 (1996) |
| [8] | J.A. Gutierrez, J.L. Naredo, P. Moreno, J.L. Guardado, Transient analysis of nonuniform transmission line through the methods of characteristics, High voltage engineering, 1999, Eleventh International Symposium, vol. 1, pp. 339-342.; J.A. Gutierrez, J.L. Naredo, P. Moreno, J.L. Guardado, Transient analysis of nonuniform transmission line through the methods of characteristics, High voltage engineering, 1999, Eleventh International Symposium, vol. 1, pp. 339-342. |
| [9] | N. Hayashi, E.I. Kaikina, Non-linear Theory of Pseudodifferential Equations on Half-Line, Elsevier, Berlin, New York, 2004, 350pp.; N. Hayashi, E.I. Kaikina, Non-linear Theory of Pseudodifferential Equations on Half-Line, Elsevier, Berlin, New York, 2004, 350pp. · Zbl 1063.35154 |
| [10] | N. Hayashi, E.I. Kaikina, J.L. Guardado Zavala, On the boundary-value problem for the Korteweg-de Vries equation, in: Proceedings of the Royal Society of London: Series A, vol. 459, 2003, pp. 2861-2884.; N. Hayashi, E.I. Kaikina, J.L. Guardado Zavala, On the boundary-value problem for the Korteweg-de Vries equation, in: Proceedings of the Royal Society of London: Series A, vol. 459, 2003, pp. 2861-2884. · Zbl 1050.35099 |
| [11] | Hayashi, N.; Kaikina, E. I.; Ruiz Paredez, H. F., Korteweg-de Vries-Burgers equation on a half-line with large initial data, J. Evol. Equations, 2, 3, 319-347 (2002) · Zbl 1375.35445 |
| [12] | Kaikina, E. I.; Naumkin, P. I.; Shishmarev, I. A., Asymptotic behavior for large time of solutions to the nonlinear nonlocal Schrödinger equation on half-line, SUT J. Math. Japan, 35, 1, 37-79 (1999) · Zbl 0932.35188 |
| [13] | Olver, F. W.J., Asymptotics and Special Functions (1974), Academic Press: Academic Press New York, USA · Zbl 0303.41035 |
| [14] | B.N. Prasanna, Wave propagation on lossless nonlinear transmission lines exhibiting dispersion, Ph.D. Dissertation, Department of Electrical Engineering, University of Wisconsin, Madison, January 1966.; B.N. Prasanna, Wave propagation on lossless nonlinear transmission lines exhibiting dispersion, Ph.D. Dissertation, Department of Electrical Engineering, University of Wisconsin, Madison, January 1966. |
| [15] | Ramirez, A.; Naredo, J. L.; Moreno, P.; Guardado, J. L., Electromagnetic transients in overhead lines considering frequency dependence on corona effect via the method of characteristics, Int. J. Electric Power Energy Syst., 23, 3, 293-302 (2001) |
| [16] | Scott, A. C., Active and Nonlinear Wave Propagation in Electronics (1970), Wiley-Interscience: Wiley-Interscience New York, USA |
| [17] | Y. Tang, F. Dai, Transmission line model used in traveling wave studies, in: Proceedings of 1999 IEEE Power Engineering Society Transmission and Distribution Conference, vol. 2, pp. 797-803.; Y. Tang, F. Dai, Transmission line model used in traveling wave studies, in: Proceedings of 1999 IEEE Power Engineering Society Transmission and Distribution Conference, vol. 2, pp. 797-803. |
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