Transmission-line matrix method: Difference between revisions

Revision as of 07:57, 25 September 2015

Revision as of 08:15, 25 September 2015

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[[File:SeriesTlmNode.png|thumb|400px|right|A 2D series TLM node]]

If we consider an electromagnetic field distribution, in which the only non-zero components are <math>E_x</math>, <math>E_y</math> and <math>H_z</math> (i.e. a TE-mode distribution), thethen Maxwell's equations in [[Cartesian coordinates]] reduce to

: <math>\frac{\partial{H_z}}{\partial{y}} = \varepsilon\frac{\partial{E_x}}{\partial{t}}</math>

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: <math>\frac{\partial^2H_z}{\partial{x}^2}+\frac{\partial^2{H_z}}{\partial{y}^2} = \mu\varepsilon\frac{\partial^2{H_z}}{\partial{t}^2}</math>

The figure on the right presents a structure, referred to as a ''series node''. It describes a block of space dimensions <math>\Delta x</math>, <math>\Delta y</math> and <math>\Delta z</math> andthat consists of four ports. <math>L'</math> and <math>C'</math> are the distributed inductance and capacitance of the transmission lines. It is possible to show that a series node is equivalent to a TE-wave, more precisely the mesh current ''I'', the ''x''-direction voltages (ports 1 and 3) and the ''y''-direction voltages (ports 2 and 4) may be related to the field components <math>H_z</math>, <math>E_x</math> and <math>E_y</math>. If the voltages on the ports are considered, <math>L_x = L_y</math>, and the polarity from the above figure holds, then the following is valid

: <math>-V_1+V_2+V_3-V_4 = 2L'\,\Delta l\frac{\partial{I}}{\partial{t}}</math>

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

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

Similarly, using the conditions across the capacitorcapacitors on ports 1 and 4, it can be shown that the corresponding to thetwo other two Maxwell equations are the following:

: <math>\frac{\partial{H_z}}{\partial{y}} = C'\frac{\partial{E_x}}{\partial{t}}</math>

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: <math>-\frac{\partial{H_z}}{\partial{x}} = C'\frac{\partial{E_y}}{\partial{t}}</math>

Having these results, it is possible to compute the scattering matrix of a shunt node. The incident voltage pulse on port 1 at time-step ''k'' is denoted as <math>_kV^i_1</math>. Replacing the four line segments from the above figure with their [[Thevenin equivalent]] it is possible to show that the following equation for the reflected voltage pulse holds:

: <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 are summarised in one vector as well as all reflected waves are summarized 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>