3D TLM Models
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.
The topology of the SCN can not be analysed using Thevenin equivalent circuits. More general energy and charge conservation principles are to be used.
The electric and the magnetic fields on the sides of the SCN node number (l,m,n) at time instant k may be summarised in 12-dimensional vectors
They can be linked with the incident and scattered amplitude vectors via
where is the field impedance, is the vector of the amplitudes of the incident waves to the node, and is the vector of the scattered amplitudes. The relation between the incident and scattered waves is given with the matrix equation
The scattering matrix S may be calculated. For the symmetrical condensed node with ports defined as in the figure the following result is obtained
![\mathbf{S} = \left[
\begin{array}{ccc}
0& \mathbf{S}_0& \mathbf{S}^T_0\\
\mathbf{S}^T_0& 0& \mathbf{S}_0\\
\mathbf{S}_0& \mathbf{S}^T_0& 0
\end{array}\right]](http://upload.wikimedia.org/math/2/5/e/25e4bac3b24a634a779e3814cd3bf2e2.png)
where the following matrix was used
![\mathbf{S} = \frac{1}{2}\left[
\begin{array}{cccc}
0& 0& 1& -1\\
0& 0& -1& 1\\
1& 1& 0& 0\\
1& 1& 0& 0
\end{array}\right]](http://upload.wikimedia.org/math/b/3/2/b32d1b85abd895c2b0676587f6d94579.png)
The connection between different SCNs is done in the same manner as for the 2D nodes.
Read more about this topic: Transmission-line Matrix Method
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