Mathematical modelling of multi conductor cables

Previous Article
Modeling aspects to improve the solution of the inverse problem in scatterometry
DCDS-S Home
This Issue
Next Article
Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method

June 2015, 8(3): 521-546. doi: 10.3934/dcdss.2015.8.521

Mathematical modelling of multi conductor cables

Geoffrey Beck ^1, , Sebastien Imperiale ^2, and Patrick Joly ^1,

1.	POEMS, INRIA - CNRS:UMR7231 - ENSTA ParisTech, ENSTA ParisTech, 828 Boulevard des Maréchaux, 91762 Palaiseau, France, France
2.	M3DISIM, INRIA, Alan Turing, 1 rue Honoré d'Estienne d'Orves, 91120 Palaiseau, France

Received December 2013 Revised March 2014 Published October 2014

Abstract
Related Papers

This paper proposes a formal justification of simplified 1D models for the propagation of electromagnetic waves in thin non-homogeneous lossy conductor cables. Our approach consists in deriving these models from an asymptotic analysis of 3D Maxwell's equations. In essence, we extend and complete previous results to the multi-wires case.

Keywords: asymptotic analysis., Maxwell equations, Telegrapher models.

Mathematics Subject Classification: 35L05, 35A35, 73R05, 35A4.

Citation: Geoffrey Beck, Sebastien Imperiale, Patrick Joly. Mathematical modelling of multi conductor cables. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 521-546. doi: 10.3934/dcdss.2015.8.521

References:

[1]	I. Aganovic and Z. Tutek, A justification of the one-dimensional model of elastic beam,, Math. Methods in Applied Sci., 8 (1986), 1. doi: 10.1002/mma.1670080133.
[2]	C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains,, Mathematical Methods in the Applied Sciences, 21 (1998), 823. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.
[3]	A. Bermúdez, D. Gómez and P. Salgado, Mathematical Models and Numerical Simulation in Electromagnetism,, Springer, (2013). doi: 10.1007/978-3-319-02949-8.
[4]	G. Canadas, Speed of Propagation of Solutions of a Linear Integro-differential Equation with Nonconstant Coefficients,, SIAM Journal on Mathematical Analysis, 16 (1985). doi: 10.1137/0516009.
[5]	B. Cockburn and P. Joly, Maxwell equations in polarizable media,, SIAM Journal on Mathematical Analysis, 19 (1988), 1372. doi: 10.1137/0519101.
[6]	R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology,, 3, (1990).
[7]	M. Delfour and J. P. Zolésio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization,, Advances in Design and Control SIAM, (2001).
[8]	V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms,, Springer-Verlag 1986., (1986). doi: 10.1007/978-3-642-61623-5.
[9]	S. Imperiale and P. Joly, Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section,, Applied Num. Mathematics, (). doi: 10.1016/j.apnum.2013.03.011.
[10]	S. Imperiale and P. Joly, Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section,, Advances in Applied Mathematics and Mechanics, 4 (2012), 647.
[11]	P. Monk, Finite Element Methods for Maxwell's Equations,, Oxford science publications, (2003). doi: 10.1137/0729045.
[12]	C. R. Paul, Analysis of Multiconductor Transmission Lines,, 2nd. New York 2008., (2008). doi: 10.1109/9780470547212.
[13]	J. Stratton, Electromagnetic Theory,, second printing ed., (1941).
[14]	M. F. Veiga, Asymptotic method applied to a beam with a variable cross section,, in Asymptotic Methods for Elastic Structures (Lisbon, (1993), 237.
[15]	W. T. Weeks, Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface,, IEEE Trans. MTT, 18 (1970), 35. doi: 10.1109/TMTT.1970.1127130.
[16]	W. T. Weeks, Multiconductor transmission line theory in the TEM approximation,, IBM Journal of Research and Development, (1972), 604. doi: 10.1147/rd.166.0604.

show all references

References:

[1]	I. Aganovic and Z. Tutek, A justification of the one-dimensional model of elastic beam,, Math. Methods in Applied Sci., 8 (1986), 1. doi: 10.1002/mma.1670080133.
[2]	C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains,, Mathematical Methods in the Applied Sciences, 21 (1998), 823. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.
[3]	A. Bermúdez, D. Gómez and P. Salgado, Mathematical Models and Numerical Simulation in Electromagnetism,, Springer, (2013). doi: 10.1007/978-3-319-02949-8.
[4]	G. Canadas, Speed of Propagation of Solutions of a Linear Integro-differential Equation with Nonconstant Coefficients,, SIAM Journal on Mathematical Analysis, 16 (1985). doi: 10.1137/0516009.
[5]	B. Cockburn and P. Joly, Maxwell equations in polarizable media,, SIAM Journal on Mathematical Analysis, 19 (1988), 1372. doi: 10.1137/0519101.
[6]	R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology,, 3, (1990).
[7]	M. Delfour and J. P. Zolésio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization,, Advances in Design and Control SIAM, (2001).
[8]	V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms,, Springer-Verlag 1986., (1986). doi: 10.1007/978-3-642-61623-5.
[9]	S. Imperiale and P. Joly, Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section,, Applied Num. Mathematics, (). doi: 10.1016/j.apnum.2013.03.011.
[10]	S. Imperiale and P. Joly, Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section,, Advances in Applied Mathematics and Mechanics, 4 (2012), 647.
[11]	P. Monk, Finite Element Methods for Maxwell's Equations,, Oxford science publications, (2003). doi: 10.1137/0729045.
[12]	C. R. Paul, Analysis of Multiconductor Transmission Lines,, 2nd. New York 2008., (2008). doi: 10.1109/9780470547212.
[13]	J. Stratton, Electromagnetic Theory,, second printing ed., (1941).
[14]	M. F. Veiga, Asymptotic method applied to a beam with a variable cross section,, in Asymptotic Methods for Elastic Structures (Lisbon, (1993), 237.
[15]	W. T. Weeks, Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface,, IEEE Trans. MTT, 18 (1970), 35. doi: 10.1109/TMTT.1970.1127130.
[16]	W. T. Weeks, Multiconductor transmission line theory in the TEM approximation,, IBM Journal of Research and Development, (1972), 604. doi: 10.1147/rd.166.0604.

[1]	Laurent Boudin, Bérénice Grec, Francesco Salvarani. A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1427-1440. doi: 10.3934/dcdsb.2012.17.1427
[2]	Johannes Eilinghoff, Roland Schnaubelt. Error analysis of an ADI splitting scheme for the inhomogeneous Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5685-5709. doi: 10.3934/dcds.2018248
[3]	Domenica Borra, Tommaso Lorenzi. Asymptotic analysis of continuous opinion dynamics models under bounded confidence. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1487-1499. doi: 10.3934/cpaa.2013.12.1487
[4]	Kun Wang, Yangping Lin, Yinnian He. Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 657-677. doi: 10.3934/dcds.2012.32.657
[5]	Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415
[6]	Dina Kalinichenko, Volker Reitmann, Sergey Skopinov. Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion. Conference Publications, 2013, 2013 (special) : 407-414. doi: 10.3934/proc.2013.2013.407
[7]	Xueke Pu, Min Li. Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-33. doi: 10.3934/dcdsb.2019055
[8]	Makram Hamouda, Chang-Yeol Jung, Roger Temam. Asymptotic analysis for the 3D primitive equations in a channel. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 401-422. doi: 10.3934/dcdss.2013.6.401
[9]	Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048
[10]	Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3185-3213. doi: 10.3934/dcdsb.2015.20.3185
[11]	Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305
[12]	José A. Carrillo, Stéphane Cordier, Giuseppe Toscani. Over-populated tails for conservative-in-the-mean inelastic Maxwell models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 59-81. doi: 10.3934/dcds.2009.24.59
[13]	W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431
[14]	Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097
[15]	Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631
[16]	Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257
[17]	Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271
[18]	Matthias Eller. Stability of the anisotropic Maxwell equations with a conductivity term. Evolution Equations & Control Theory, 2019, 8 (2) : 343-357. doi: 10.3934/eect.2019018
[19]	Chiun-Chang Lee. Asymptotic analysis of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3251-3276. doi: 10.3934/dcds.2016.36.3251
[20]	Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences