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 ==

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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.

In order to show the TLM schema we will use time and space discretisation. The time-step will be denoted with <math>\Delta t</math> and the space discretisation intervals with <math>\Delta x</math>, <math>\Delta y</math> and <math>\Delta z</math>. The absolute time and space will therefore be <math>t = k\,\Delta t</math>, <math>x = l\,\Delta x</math>, <math>y = m\,\Delta y</math>, <math>z = n\,\Delta z</math>, where <math>k=0,1,2,\ldots</math> is the time instant and <math>m,n,l</math> are the cell coordinates. In case <math>\Delta x = \Delta y = \Delta z</math> the value <math>\Delta l</math> will be used, which is the [[lattice constant]]. In this case the following holds:

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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:NumericalComputational differential equationselectromagnetics]]

[[Category:Computational science]]

[[Category:Computational physics]]

[[Category:Electromagnetism]]

[[Category:Electrodynamics]]