American Institute of Mathematical Sciences

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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

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), 1-14. doi: 10.1002/mma.1670080133.  Google Scholar

[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), 823-864. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.  Google Scholar

[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.  Google Scholar

[4]

G. Canadas, Speed of Propagation of Solutions of a Linear Integro-differential Equation with Nonconstant Coefficients, SIAM Journal on Mathematical Analysis, 16 (1985), 143. doi: 10.1137/0516009.  Google Scholar

[5]

B. Cockburn and P. Joly, Maxwell equations in polarizable media, SIAM Journal on Mathematical Analysis, 19 (1988), 1372-1390. doi: 10.1137/0519101.  Google Scholar

[6]

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 3, Springer-Verlag, 1990.  Google Scholar

[7]

M. Delfour and J. P. Zolésio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization, Advances in Design and Control SIAM, Philadelphia, PA, 2001.  Google Scholar

[8]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[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.  Google Scholar

[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, (AAMM), 4 (2012), 647-664.  Google Scholar

[11]

P. Monk, Finite Element Methods for Maxwell's Equations, Oxford science publications, 2003. doi: 10.1137/0729045.  Google Scholar

[12]

C. R. Paul, Analysis of Multiconductor Transmission Lines, 2nd. New York 2008. doi: 10.1109/9780470547212.  Google Scholar

[13]

J. Stratton, Electromagnetic Theory, second printing ed., Mcgraw Hill, 1941. Google Scholar

[14]

M. F. Veiga, Asymptotic method applied to a beam with a variable cross section, in Asymptotic Methods for Elastic Structures (Lisbon, 1993), de Gruyter, Berlin, 1995, 237-254.  Google Scholar

[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), 35-43. doi: 10.1109/TMTT.1970.1127130.  Google Scholar

[16]

W. T. Weeks, Multiconductor transmission line theory in the TEM approximation, IBM Journal of Research and Development, (1972), 604-611. doi: 10.1147/rd.166.0604.  Google Scholar

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), 1-14. doi: 10.1002/mma.1670080133.  Google Scholar

[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), 823-864. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B.  Google Scholar

[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.  Google Scholar

[4]

G. Canadas, Speed of Propagation of Solutions of a Linear Integro-differential Equation with Nonconstant Coefficients, SIAM Journal on Mathematical Analysis, 16 (1985), 143. doi: 10.1137/0516009.  Google Scholar

[5]

B. Cockburn and P. Joly, Maxwell equations in polarizable media, SIAM Journal on Mathematical Analysis, 19 (1988), 1372-1390. doi: 10.1137/0519101.  Google Scholar

[6]

R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, 3, Springer-Verlag, 1990.  Google Scholar

[7]

M. Delfour and J. P. Zolésio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization, Advances in Design and Control SIAM, Philadelphia, PA, 2001.  Google Scholar

[8]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer-Verlag 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar

[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.  Google Scholar

[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, (AAMM), 4 (2012), 647-664.  Google Scholar

[11]

P. Monk, Finite Element Methods for Maxwell's Equations, Oxford science publications, 2003. doi: 10.1137/0729045.  Google Scholar

[12]

C. R. Paul, Analysis of Multiconductor Transmission Lines, 2nd. New York 2008. doi: 10.1109/9780470547212.  Google Scholar

[13]

J. Stratton, Electromagnetic Theory, second printing ed., Mcgraw Hill, 1941. Google Scholar

[14]

M. F. Veiga, Asymptotic method applied to a beam with a variable cross section, in Asymptotic Methods for Elastic Structures (Lisbon, 1993), de Gruyter, Berlin, 1995, 237-254.  Google Scholar

[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), 35-43. doi: 10.1109/TMTT.1970.1127130.  Google Scholar

[16]

W. T. Weeks, Multiconductor transmission line theory in the TEM approximation, IBM Journal of Research and Development, (1972), 604-611. doi: 10.1147/rd.166.0604.  Google Scholar

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