Transmission-line matrix method: Difference between revisions

Revision as of 09:02, 25 September 2015

Revision as of 05:06, 15 August 2016

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== Basic principle ==

[[File: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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: <math>E_S = \left[0.5^2+0.5^2+0.5^2+(-0.5)^2\right](\Delta t/Z) = \Delta t/Z</math>

Therefore, the [[energy conservation law]] is fulfilled by the model.

The next scattering event excites the neighbouring nodes according to the principle described above. It can be seen that every node turns into a secondary source of spherical wave. These waves combine to form the overall waveform. This is in accordance with Huygens principle of light propagation.

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Since <math>\Delta x = \Delta y = \Delta z = \Delta l</math> and substituting <math>I = H_z \,\Delta z</math> gives

: <math>\frac{\Delta E_x}{\Delta y} - \frac{\Delta E_y}{\Delta x} = 2L'\frac{\partial H_z}{\partial t}</math>

This reduces to Maxwell's equations when <math>\Delta l \rightarrow 0</math>.

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* 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.

* Mansour Ahmadian, Transmission Line Matrix (TLM) modelling of medical ultrasound [https://www.era.lib.ed.ac.uk/handle/1842/427 PhD thesis], University of Edinburgh 2001

[[Category:Numerical differential equations]]

[[Category:Computational science]]