Transmission-line matrix method: Difference between revisions

Revision as of 08:46, 23 April 2012

Revision as of 03:55, 11 May 2012

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== Basic principle ==

[[ImageFile:SingleNode2DTLM.png|thumb|500px|right|2D TLM example: an incident voltage pulse in two consecutive scattering events.]] The TLM method is based on [[Huygens Principle|Huygens' model of wave propagation]] and scattering and the analogy between field propagation and transmission lines. Therefore it considers the computational ___domain as a mesh of transmission lines, interconnected at nodes. In the figure on the right is considered a simple example of a 2D TLM mesh with a voltage pulse of amplitude 1 V incident on the central node. This pulse will be partially reflected and transmitted according to the transmission-line theory. If we assume that each line has a characteristic impedance <math>Z</math>, then the incident pulse sees effectively three transmission lines in parallel with a total impedance of <math>Z/3</math>. The reflection coefficient and the transmission coefficient are given by

: <math>R = \frac{Z/3-Z}{Z/3+Z} = -0.5</math>

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=== The scattering matrix of an 2D TLM node ===

[[ImageFile:SeriesTlmNode.png|thumb|400px|right|A 2D series TLM node]]

If we consider an electromagnetic field distribution, in which the only non-zero components are <math>E_x</math>, <math>E_y</math> and <math>H_z</math> (i.e. a TE-mode distribution), the Maxwell's equations in [[Cartesian coordinates]] reduce to

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=== Connection between TLM nodes ===

[[ImageFile:2DTLMmes.png|272px|thumb|right|A 2D series TLM node]]

In order to describe the connection between adjacent nodes the mesh of series nodes look at the figure on the right. As the incident pulse in timestep ''k+1'' on a node is the scattered pulse from an adjacent node in timestep ''k'' the following connection equations are derived:

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== 3D TLM models ==

[[ImageFile:SymmetricCondensedNode.png|thumb|320px|right|A 3D symmetric condensed node]]

Most problems in electromagnetics require a three-dimensional computing. As we have structures, that describe TE and TM-field distributions, intuitively it seem possible to provide a combination of shunt and series nodes, which will provide a full description of the electromagnetic field. Such attempts have been made, but they proved not very useful because of the complexity of the resulting structures. Using the normal analogy, presented above, leads to calculation of the different field components at physically separated points. This causes difficulties in simple and efficient boundary definition. A solution to these problems was provided by Johns in 1987, when he proposed the structure, known as the '''symmetrical condensed node''' (SCN), presented in the figure. It consists of 12 ports, because two field polarisations are to be assigned to each of the 6 sides of a mesh cell.

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== References ==

<references/>

* C. Christopoulos, ''The Transmission Line Modeling Method: TLM'', Piscataway, NY, IEEE Press, 1995. ISBN 978-01985653380-19-856533-8

* Russer, P., Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering, Second edition, Artec House, Boston, 2006, ISBN 978-15805390741-58053-907-4

* J. L. Herring, Developments in the Transmission-Line Modelling Method for Electromagnetic Compatibility Studies, [http://www.nottingham.ac.uk/ggiemr/publications/JLH_thesis.htm PhD thesis], University of Nottingham, 1993.

[[Category:Numerical differential equations]]