Basic Principle
The TLM method is based on 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, then the incident pulse sees effectively three transmission lines in parallel with a total impedance of . The reflection coefficient and the transmission coefficient are given by
The energy injected into the node by the incident pulse and the total energy of the scattered pulses are correspondingly
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 and the space discretisation intervals with, and . The absolute time and space will therefore be, where is the time instant and are the cell coordinates. In case the value will be used, which is the lattice constant. In this case the following holds:
where is the free space speed of light.
Read more about this topic: Transmission-line Matrix Method
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