Before the actual Method of Moments can be applied, the problem type has to be defined. This determines the surface integral equation that has to be used for calculating the impedance matrix. For structures which only consist of perfect electric conductors, the electric field integral equation (EFIE) is usually applied, which is derived from Maxwell’s equations. Combining this integral equation with the boundary condition of the tangential electric field on a perfectly electrically conducting (PEC) surface yields the corresponding impedance operator \(Z\), which links the electric field \({{\vec{E}}_{inc}}\) incident on the structure with the surface current density \({\vec{J}}\) on the structure: \[{{\overrightarrow{E}}_{inc}}(\overrightarrow{r})=j\omega \mu \iint\limits_{S}{\overrightarrow{J}(\overrightarrow{r}')\frac{{{e}^{-jk\left| \overrightarrow{r}-\overrightarrow{r}' \right|}}}{4\pi \left| \overrightarrow{r}-\overrightarrow{r}' \right|}}dS'+\overrightarrow{\nabla }\left( \frac{-1}{j\omega \varepsilon }\iint\limits_{S}{\overrightarrow{\nabla }\cdot \overrightarrow{J}(\overrightarrow{r}')\frac{{{e}^{-jk\left| \overrightarrow{r}-\overrightarrow{r}' \right|}}}{4\pi \left| \overrightarrow{r}-\overrightarrow{r}' \right|}}dS' \right)=\overrightarrow{L}\left( \overrightarrow{J}(\overrightarrow{r}) \right)\]
with the observation vector \(\vec{r}\), the source vector \(\vec{r}^{'}\), the structure’s surface \(S\), the permeability \(\mu\), the permittivity \(\epsilon\), the angular frequency \(\omega\) and the angular wavenumber \(k\).
By inverting the impedance operator, the surface current density on a PEC structure can be computed if the incident electric field is known. This is usually done numerically using of the Method of Moments by converting the operator equation into a matrix equation. For the computation of characteristic modes, the impedance matrix resulting from this conversion is of utmost interest. Basically, there are three steps to obtain this impedance matrix:
1. Discretization of the structure (Meshing),
2. Expansion of the unknown current densities with basis functions,
3. Testing in order to transfer the vector integral equation into a matrix equation.
Typically, a triangular mesh is used in the Method of Moments as it is capable of modelling arbitrarily shaped three-dimensional surfaces. After a mesh has been defined, the unknown surface current densities are expanded into a series of known basis functions \(\vec{f}_m\) with unknown coefficients \(I_m\): \[\overrightarrow J (\overrightarrow r ) = \sum\limits_{m = 1}^M {{I_m}{{\overrightarrow f }_m}(\overrightarrow r )}\] where \(M\) is the total number of MoM-unknowns. At the Institute of Microwave and Wireless Systems, the popular Rao-Wilton-Glisson (RWG) basis function [3] is employed. It is defined over the common edge of two triangles where one triangle is named the plus-triangle \(T_{m}^{+}\) and the other one the minus-triangle \(T_{m}^{-}\) (see Fig. 2): \[\overrightarrow f_m{\rm{(}}\overrightarrow r {\rm{) = }}\left\{ {\begin{array}{*{20}{c}} {\frac{{{l_m}}}{{2A_m^ + }}\overrightarrow \rho _m^ + ,{\rm{ }}\overrightarrow r \in T_m^ + }\\ {\frac{{{l_m}}}{{2A_m^ - }}\overrightarrow \rho _m^ - ,{\rm{ }}\overrightarrow r \in T_m^ - }\\ {0,{\rm{ otherwise}}} \end{array}} \right.\] with the length of the common edge \(l_m\), the triangle areas \(A_{m}^{\pm }\) and the position vectors \(p_{m}^{\pm }\) with respect to the free vertices. The RWG basis function can be considered as a small dipole oriented from the plus-triangle to the minus-triangle.
The testing is now done in order to transform the vector terms into scalar values which are required to set up the impedance matrix. The scalar product with a testing function is applied to the impedance operator and the incident field. Here, the RWG basis function is used as testing function as well (Galerkin method) since this results in a symmetric impedance matrix which is essential for the characteristic mode computation. The resulting system of linear equations is calculated as follows: \[\small\left( {\begin{array}{*{20}{c}}{\left\langle {{{\overrightarrow f }_1}(\overrightarrow r ),{{\overrightarrow E }_{inc}}(\overrightarrow r )} \right\rangle }\\{\left\langle {{{\overrightarrow f }_2}(\overrightarrow r ),{{\overrightarrow E }_{inc}}(\overrightarrow r )} \right\rangle }\\ \vdots \\ {\left\langle {{{\overrightarrow f }_M}(\overrightarrow r ),{{\overrightarrow E }_{inc}}(\overrightarrow r )} \right\rangle } \end{array}} \right)=\small\left( {\begin{array}{*{20}{c}} {\left\langle {{{\overrightarrow f }_1}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_1}(\overrightarrow r )} \right)} \right\rangle }&{\left\langle {{{\overrightarrow f }_1}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_2}(\overrightarrow r )} \right)} \right\rangle }& \ldots &{\left\langle {{{\overrightarrow f }_1}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_M}(\overrightarrow r )} \right)} \right\rangle }\\ {\left\langle {{{\overrightarrow f }_2}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_1}(\overrightarrow r )} \right)} \right\rangle }&{\left\langle {{{\overrightarrow f }_2}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_2}(\overrightarrow r )} \right)} \right\rangle }& \ldots &{\left\langle {{{\overrightarrow f }_2}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_M}(\overrightarrow r )} \right)} \right\rangle }\\ \vdots & \vdots & \ddots & \vdots \\ {\left\langle {{{\overrightarrow f }_M}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_1}(\overrightarrow r )} \right)} \right\rangle }&{\left\langle {{{\overrightarrow f }_M}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_2}(\overrightarrow r )} \right)} \right\rangle }& \ldots &{\left\langle {{{\overrightarrow f }_M}(\overrightarrow r ),\overrightarrow L \left( {{{\overrightarrow f }_M}(\overrightarrow r )} \right)} \right\rangle } \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{I_1}}\\ {{I_2}}\\\vdots \\ {{I_M}}\end{array}} \right)\] with the scalar products defined as \[\left\langle {{\overrightarrow{f}}_{l}}(\overrightarrow{r}),\overrightarrow{L}\left( {{\overrightarrow{f}}_{m}}(\overrightarrow{r}) \right) \right\rangle =\iint\limits_{S}{{{\overrightarrow{f}}_{l}}(\overrightarrow{r})\cdot \overrightarrow{L}\left( {{\overrightarrow{f}}_{m}}(\overrightarrow{r}) \right)}dS={{Z}_{l,m}}\] and \[\left\langle {{\overrightarrow{f}}_{l}}(\overrightarrow{r}),{{\overrightarrow{E}}_{inc}}(\overrightarrow{r}) \right\rangle =\iint\limits_{S}{{{\overrightarrow{f}}_{l}}(\overrightarrow{r})\cdot {{\overrightarrow{E}}_{inc}}(\overrightarrow{r})}dS={{V}_{l}}.\]This is usually abbreviated as a matrix equation: \[V=ZI,\] with \(Z=R+jX\). Now, the characteristic modes of the structure can be calculated by solving the following generalized eigenvalue equation: \[XI_{n} = \lambda_{n}RI_{n} \]
[2] Yikai Chen and Chao-Fu Wang, Characteristic Modes – Theory and Applications in Antenna Engineering, John Wiley & Sons, Inc., Hoboken, New Jersey, USA, 2015.
[3] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic Scattering by Surfaces of Arbitrary Shape,” Antennas and Propagation, IEEE Transaction on, Volume 30, Issue 3, pp. 409-418, May 1982.
