Transmission-line matrix method: Difference between revisions

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{{short description|Method of computing electromagnetic fields}}

The '''transmission-line matrix (TLM) method''' is a space and time discretising method for computation of [[electromagnetic fields]]. It is based on the analogy between the electromagnetic field and a mesh of [[transmission line]]s. The TLM method allows the computation of complex three-dimensional electromagnetic structures and has proven to be one of the most powerful time-___domain methods along with the finite difference time ___domain ([[FDTD]]) method.

The '''transmission-line matrix''' ('''TLM''') '''method''' is a space and time discretising method for computation of [[electromagnetic fields]]. It is based on the [[analogy]] between the electromagnetic field and a mesh of [[transmission line]]s. The TLM method allows the computation of complex three-dimensional electromagnetic structures and has proven to be one of the most powerful [[Time ___domain|time-___domain]] methods along with the [[finite difference time ___domain]] (FDTD) method. The TLM was first explored by British electrical engineer [[Raymond Beurle]] while working at [[English Electric Valve Company]] in [[Chelmsford]]. After he had been appointed professor of [[electrical engineering]] at the [[University of Nottingham]] in 1963 he jointly authored an article, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix", with [[Peter B. Johns]] in 1971.<ref name="de Cogan TLM">{{cite book |last1=de Cogan |first1=Donard |title=Transmission Line Matrix (TLM) Techniques for Diffusion Applications |date=12 December 2018 |publisher=Routledge |isbn=978-1-351-40712-0 |url=https://books.google.com/books?id=1lEPEAAAQBAJ |language=en}}</ref>

== 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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: <math>_kV^r_1 = 0.5\left(_kV^i_1 + _kV^i_2 + _kV^i_3 - _kV^i_4\right)</math>

If all incident waves as well as all reflected waves are summarizedcollected in one vector, then this equation may be written down for all ports in matrix form:

: <math>_k\mathbf{V}^r=\mathbf{S}_k\mathbf{V}^i</math>

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In order to describe the connection between adjacent nodes by a 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:

: <math>_{k+1}V^i_1(x,y)=_kV_{k+1}V^r_3(x,y-1)</math>

: <math>_{k+1}V^i_2(x,y)=_kV_{k+1}V^r_4(x-1,y)</math>

: <math>_{k+1}V^i_3(x,y)=_kV_{k+1}V^r_1(x,y+1)</math>

: <math>_{k+1}V^i_4(x,y)=_kV_{k+1}V^r_2(x+1,y)</math>

By modifying the scattering matrix <math>\textbf{S}</math> inhomogeneous and lossy materials can be modelled. By adjusting the connection equations it is possible to simulate different boundaries.

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The connection between different SCNs is done in the same manner as for the 2D nodes.

== Open-sourced code implementation of 3D-TLM ==

The [[George Green (mathematician)|George Green]] Institute for Electromagnetics Research (GGIEMR) has open-sourced an efficient implementation of 3D-TLM, capable of [[Parallel computing|parallel computation]] by means of [[Message Passing Interface|MPI]] named GGITLM and available online. <ref>{{cite web|title=George Green Institute for Electromagnetics Research - TLM time ___domain simulation code|url=https://www.nottingham.ac.uk/research/groups/ggiemr/our-research/large-scale-electromagnetic-modelling/large-scale-electromagnetic-modelling.aspx|website=University of Nottingham - George Green Institute for Electromagnetics Research|publisher=University of Nottingham|accessdate=23 March 2017}}</ref>

== References ==

<references/>

* C. Christopoulos, ''The Transmission Line Modeling Method: TLM'', Piscataway, NY, IEEE Press, 1995. {{ISBN |978-0-19-856533-8}}

* Russer, P., Electromagnetics, Microwave Circuit and Antenna Design for Communications Engineering, Second edition, Artec House, Boston, 2006, {{ISBN |978-1-58053-907-4}}

* P. B. Johns and M.O'Brien. "Use of the transmission line modelling (t.l.m) method to solve nonlinear lumped networks", The Radio Electron and Engineer. 1980.

* 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 scienceelectromagnetics]]

[[Category:Computational physics]]

[[Category:Electromagnetism]]

[[Category:Electrodynamics]]