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Method of Moments computation technique for metal antennas.

The first step in the computational solution of electromagnetic problems is to discretize Maxwell's equations. The process results in this matrix-vector system:

$$V=ZI$$

*V*— Applied voltage vector. This signal can be voltage or power applied to the antenna or an incident signal falling on the antenna.*I*— Current vector that represents current on the antenna surface.*Z*— Interaction matrix or impedance matrix that relates*V*to*I*.

Antenna Toolbox™ uses method of moments (MoM) to calculate the interaction matrix and solve system equations.

The MoM formulation is split into three parts.

Discretization enables the formulation from the continuous
domain to the discrete domain. This step is called *meshing* in
antenna literature. In the MoM formulation, the metal surface of the
antenna is meshed into triangles.

To calculate the surface currents on the antenna structure, you first define basis functions. Antenna Toolbox uses Rao-Wilton-Glisson (RWG) [2] basis functions. The arrows show the direction of current flow.

The basis function includes a pair of adjacent (not necessarily coplanar) triangles and resembles a small spatial dipole with linear current distribution. Each triangle is associated with a positive or negative charge.

For any two triangle patches, $${t}_{n}^{+}$$ and $${t}_{n}^{-}$$, having areas $${A}_{n}^{+}$$ and $${A}_{n}^{-}$$, and sharing common edge $${l}_{n}$$, the basis function is

$$\begin{array}{l}{\overrightarrow{f}}_{n}(\overrightarrow{r})=\{\begin{array}{cc}\frac{{l}_{n}}{2{A}_{n}^{+}}{\overrightarrow{\rho}}_{n}^{+S},& \overrightarrow{r}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{t}_{n}^{+}\\ \frac{{l}_{n}}{2{A}_{n}^{-}}{\overrightarrow{\rho}}_{n}^{-S},& \overrightarrow{r}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{t}_{n}^{-}\end{array}\text{\hspace{0.05em}}\\ \end{array}$$

$${\overrightarrow{\rho}}_{n}^{+}=\overrightarrow{r}-{\overrightarrow{r}}_{n}^{+}$$ — Vector drawn from the free vertex of triangle $${t}_{n}^{+}$$ to observation point $$\overrightarrow{r}$$

$${\overrightarrow{\rho}}_{n}^{-}={\overrightarrow{r}}_{n}^{+}-\overrightarrow{r}$$ — Vector drawn from the observation point to the free vertex of the triangle $${t}_{n}^{-}$$

and

$$\nabla \cdot {\overrightarrow{f}}_{n}\left(\overrightarrow{r}\right)=\{\begin{array}{cc}\frac{{l}_{n}}{{A}_{n}^{+}},& \overrightarrow{r}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{t}_{n}^{+}\\ -\frac{{l}_{n}}{{A}_{n}^{-}},& \overrightarrow{r}\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}{t}_{n}^{-}\end{array}$$

The basis function is zero outside the two adjacent triangles $${t}_{n}^{+}$$ and $${t}_{n}^{-}$$. The RWG vector basis function is linear and has no flux (no normal component) through its boundary.

The interaction matrix is a complex dense symmetric matrix.
It is a square *N*-by-*N* matrix,
where *N* is the number of basis functions, that
is, the number of interior edges in the structure. A typical interaction
matrix for a structure with 256 basis functions is shown:

To fill out the interaction matrix, calculate the free-space Green's function between all basis functions on the antenna surface. The final interaction matrix equations are:

$${Z}_{mn}=\left(\frac{j\omega \mu}{4\pi}\right){\displaystyle \underset{S}{\int}{\displaystyle \underset{S}{\int}{\overrightarrow{f}}_{m}}}\left(\overrightarrow{r}\right).{\overrightarrow{f}}_{m}\left(\overrightarrow{{r}^{\prime}}\right)gd\overrightarrow{{r}^{\prime}}d\overrightarrow{r}-\left(\frac{j}{4\pi \omega \epsilon}\right){\displaystyle \underset{S}{\int}{\displaystyle \underset{S}{\int}\left(\nabla .{\overrightarrow{f}}_{m}\right)}}\left(\nabla .{\overrightarrow{f}}_{m}\right)gd\overrightarrow{{r}^{\prime}}d\overrightarrow{r}$$

where

$$g(\overrightarrow{r},\overrightarrow{{r}^{\prime}})=\frac{\mathrm{exp}(-jk\left|\overrightarrow{r}-\overrightarrow{{r}^{\prime}}\right|)}{\left|\overrightarrow{r}-\overrightarrow{{r}^{\prime}}\right|}$$ — Free-space Green's function

To calculate the interaction matrix, excite the antenna by a voltage of 1 V at the feeding edge. So the voltage vector has zero values everywhere except at the feeding edge. Solve the system of equations to calculate the unknown currents. Once you determine the unknown currents, you can calculate the field and surface properties of the antenna.

From the interaction matrix plot, you observe that the matrix
is diagonally dominant. As you move further away from the diagonal,
the magnitude of the terms decreases. This behavior is same as the
Green's function behavior. The Green's function decreases as the distance
between *r* and *r'* increases.
Therefore, it is important to calculate the region on the diagonal
and close to the diagonal accurately.

This region on and around the diagonal is called *neighbor
region*. The neighbor region is defined within a sphere
of radius *R*, where *R* is in terms
of triangle size. The size of a triangle is the maximum distance from
the center of the triangle to any of its vertices. By default, *R* is
twice the size of the triangle. For better accuracy, a higher-order
integration scheme is used to calculate the integrals.

Along the diagonal, *r* and *r'* are
equal and the defines Green's function becomes singular. To remove
the singularity, extraction is performed on these terms.

$$\begin{array}{l}{\displaystyle \underset{{t}_{p}}{\int}{\displaystyle \underset{{t}_{q}}{\int}\left({\overrightarrow{\rho}}_{i}.{{\overrightarrow{\rho}}^{\prime}}_{j}\right)}}g(\overrightarrow{r},\overrightarrow{{r}^{\prime}})ds\text{'}ds={\displaystyle \underset{{t}_{p}}{\int}{\displaystyle \underset{{t}_{q}}{\int}\frac{\left({\overrightarrow{\rho}}_{i}.{{\overrightarrow{\rho}}^{\prime}}_{j}\right)}{\left|\overrightarrow{r}-\overrightarrow{{r}^{\prime}}\right|}}}ds\text{'}ds+{\displaystyle \underset{{t}_{p}}{\int}{\displaystyle \underset{{t}_{q}}{\int}\frac{\left(\mathrm{exp}\left(-jk\left|\overrightarrow{r}-\overrightarrow{{r}^{\prime}}\right|\right)-1\right)\left({\overrightarrow{\rho}}_{i}.{{\overrightarrow{\rho}}^{\prime}}_{j}\right)}{\left|\overrightarrow{r}-\overrightarrow{{r}^{\prime}}\right|}ds\text{'}ds}}\\ {\displaystyle \underset{{t}_{p}}{\int}{\displaystyle \underset{{t}_{q}}{\int}g(\overrightarrow{r},\overrightarrow{{r}^{\prime}})ds\text{'}ds}}={\displaystyle \underset{{t}_{p}}{\int}{\displaystyle \underset{{t}_{q}}{\int}\frac{1}{\left|\overrightarrow{r}-\overrightarrow{{r}^{\prime}}\right|}}}ds\text{'}ds+{\displaystyle \underset{{t}_{p}}{\int}{\displaystyle \underset{{t}_{q}}{\int}\frac{\left(\mathrm{exp}\left(-jk\left|\overrightarrow{r}-\overrightarrow{{r}^{\prime}}\right|\right)-1\right)}{\left|\overrightarrow{r}-\overrightarrow{{r}^{\prime}}\right|}ds\text{'}ds}}\end{array}$$

The two integrals on the right-hand side of the equations, called potential or static integrals are found using analytical results [3].

The MoM formulation for finite arrays is the same as for a single antenna element. The main difference is the number of excitations (feeds). For finite arrays, the voltage vector is now a voltage matrix. The number of columns are equal to the number of elements in the array.

For example, the voltage vector matrix for a `2x2`

array
of rectangular patch antenna has four columns as each antenna can
be excited separately.

To model an infinite array, you change the MoM to account for the infinite behavior. To do so you replace the free-space Green's functions with periodic Green's functions. The periodic Green's function is an infinite double summation.

Green's Function | Periodic Green's Function |
---|---|

$$\begin{array}{l}g=\frac{{e}^{-jkR}}{R}\\ R=\left|\overrightarrow{r}-{\overrightarrow{r}}^{\prime}\right|\end{array}$$ | $$\begin{array}{l}{g}_{\text{periodic}}={\displaystyle \sum _{m=-\infty}^{\infty}{\displaystyle \sum _{n=-\infty}^{\infty}{e}^{j{\varphi}_{mn}}\frac{{e}^{-jk{R}_{mn}}}{{R}_{mn}}}}\\ {R}_{mn}=\sqrt{{\left(x-{x}^{\prime}-{x}_{m}\right)}^{2}+{\left(y-{y}^{\prime}-{y}_{n}\right)}^{2}+{\left(z-{z}^{\prime}\right)}^{2}}\\ {\varphi}_{mn}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}=k\left({x}_{m}\mathrm{sin}\theta \mathrm{cos}\phi +{y}_{n}\mathrm{sin}\theta \mathrm{cos}\phi \right)\\ {x}_{m}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}=m\cdot {d}_{x},\text{}{y}_{n}=n\cdot {d}_{y}\end{array}$$ |

*d*_{x} and *d*_{y} are
the ground plane dimensions that define the *x* and *y* dimensions
of the unit cell. *θ* and *Φ* are
the scan angles.

Comparing the two Green's functions, you observe an additional
exponential term that is added to the infinite sum. The *Φ _{mn}* accounts
for the scanning of the infinite array. The periodic Green's function
also accounts for the effect of mutual coupling.

For more information see, Infinite Arrays.

[1] Harringhton, R. F. *Field Computation by Moment
Methods*. New York: Macmillan, 1968.

[2] Rao, S. M., D. R. Wilton, and A. W. Glisson. “Electromagnetic
scattering by surfaces of arbitary shape.” *IEEE.
Trans. Antennas and Propagation*, Vol. AP-30, No. 3, May
1982, pp. 409–418.

[3] Wilton, D. R., S. M. Rao, A. W. Glisson, D. H. Schaubert,
O. M. Al-Bundak. and C. M. Butler. “Potential Integrals for
uniform and linear source distribution on polygonal and polyhedral
domains.” *IEEE. Trans. Antennas and Propagation*.
Vol. AP-30, No. 3, May 1984, pp. 276–281.

[4] Balanis, C.A. *Antenna Theory. Analysis and
Design*. 3rd Ed. New York: John Wiley & Sons, 2005.

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