Transmission-line matrix method

A 2D series TLM node

If we consider an electromagnetic field distribution in which the only non-zero components are ${\displaystyle E_{x}}$, ${\displaystyle E_{y}}$ and ${\displaystyle H_{z}}$ (i.e. a TE-mode distribution), then Maxwell's equations in Cartesian coordinates reduce to

${\displaystyle {\frac {\partial {H_{z}}}{\partial {y}}}=\varepsilon {\frac {\partial {E_{x}}}{\partial {t}}}}$

${\displaystyle -{\frac {\partial {H_{z}}}{\partial {x}}}=\varepsilon {\frac {\partial {E_{y}}}{\partial {t}}}}$

${\displaystyle {\frac {\partial {E_{y}}}{\partial {x}}}-{\frac {\partial {E_{x}}}{\partial {y}}}=-\mu {\frac {\partial {H_{z}}}{\partial {t}}}}$

We can combine these equations to obtain

${\displaystyle {\frac {\partial ^{2}H_{z}}{\partial {x}^{2}}}+{\frac {\partial ^{2}{H_{z}}}{\partial {y}^{2}}}=\mu \varepsilon {\frac {\partial ^{2}{H_{z}}}{\partial {t}^{2}}}}$

The figure on the right presents a structure referred to as a series node. It describes a block of space dimensions ${\displaystyle \Delta x}$, ${\displaystyle \Delta y}$ and ${\displaystyle \Delta z}$ that consists of four ports. ${\displaystyle L'}$ and ${\displaystyle C'}$ 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 ${\displaystyle H_{z}}$, ${\displaystyle E_{x}}$ and ${\displaystyle E_{y}}$. If the voltages on the ports are considered, ${\displaystyle L_{x}=L_{y}}$, and the polarity from the above figure holds, then the following is valid

${\displaystyle -V_{1}+V_{2}+V_{3}-V_{4}=2L'\,\Delta l{\frac {\partial {I}}{\partial {t}}}}$

where ${\displaystyle \Delta x=\Delta y=\Delta l}$.

${\displaystyle \left(V_{3}-V_{1}\right)-\left(V_{4}-V_{2}\right)=2L'\,\Delta l{\frac {\partial I}{\partial t}}}$

${\displaystyle \left[E_{x}(y+\Delta y)-E_{x}(y)\right]\,\Delta x-[E_{y}(x+\Delta x)-E_{y}(x)]\Delta y=2L'\,\Delta l{\frac {\partial {I}}{\partial {t}}}}$

and dividing both sides by ${\displaystyle \Delta x\Delta y}$

${\displaystyle {\frac {E_{x}(y+\Delta y)-E_{x}(y)}{\Delta y}}-{\frac {E_{y}(x+\Delta x)-E_{y}(x)}{\Delta x}}=2L'\,\Delta l{\frac {\partial {I}}{\partial {t}}}{\frac {1}{\Delta x\,\Delta y}}}$

Since ${\displaystyle \Delta x=\Delta y=\Delta z=\Delta l}$ and substituting ${\displaystyle I=H_{z}\,\Delta z}$ gives

${\displaystyle {\frac {\Delta E_{x}}{\Delta y}}-{\frac {\Delta E_{y}}{\Delta x}}=2L'{\frac {\partial H_{z}}{\partial t}}}$

This reduces to Maxwell's equations when ${\displaystyle \Delta l\rightarrow 0}$.

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

${\displaystyle {\frac {\partial {H_{z}}}{\partial {y}}}=C'{\frac {\partial {E_{x}}}{\partial {t}}}}$

${\displaystyle -{\frac {\partial {H_{z}}}{\partial {x}}}=C'{\frac {\partial {E_{y}}}{\partial {t}}}}$

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 ${\displaystyle _{k}V_{1}^{i}}$. 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:

${\displaystyle _{k}V_{1}^{r}=0.5\left(_{k}V_{1}^{i}+_{k}V_{2}^{i}+_{k}V_{3}^{i}-_{k}V_{4}^{i}\right)}$

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

${\displaystyle _{k}\mathbf {V} ^{r}=\mathbf {S} _{k}\mathbf {V} ^{i}}$

where ${\displaystyle _{k}\mathbf {V} ^{i}}$ and ${\displaystyle _{k}\mathbf {V} ^{r}}$ are the incident and the reflected pulse amplitude vectors.

For a series node the scattering matrix S has the following form

${\displaystyle \mathbf {S} ={\frac {1}{2}}\left[{\begin{array}{cccc}1&1&1&-1\\1&1&-1&1\\1&-1&1&1\\-1&1&1&1\end{array}}\right]}$

A 2D series TLM node

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:

${\displaystyle _{k+1}V_{1}^{i}(x,y)=_{k+1}V_{3}^{r}(x,y-1)}$

${\displaystyle _{k+1}V_{2}^{i}(x,y)=_{k+1}V_{4}^{r}(x-1,y)}$

${\displaystyle _{k+1}V_{3}^{i}(x,y)=_{k+1}V_{1}^{r}(x,y+1)}$

${\displaystyle _{k+1}V_{4}^{i}(x,y)=_{k+1}V_{2}^{r}(x+1,y)}$

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

Apart from the series node, described above there is also the shunt TLM node, which represents a TM-mode field distribution. The only non-zero components of such wave are ${\displaystyle H_{x}}$, ${\displaystyle H_{y}}$, and ${\displaystyle E_{z}}$. With similar considerations as for the series node the scattering matrix of the shunt node can be derived.