MoM and electric intregrodifferential equations

Modern computers have all the prerequisites for developing numerical resolution methods for a large class of physical problems that have been left without an analytical solution for years. There are several numerical solution methods known in literature and among them the method of moments (MoM) is a generable method applicable to a wide class of electromagnetic problems. It allows to easily deal with open structures in which the equivalence theorem is applicable and electric and magnetic currents are unknowns.

MoM

In a differential method like FEM or finite difference type, both the antenna and the environment are considered and everything is discretized into elements. The unknown, typically the electric field, is calculated in each element and therefore the field will be obtained over the entire calculation volume. These methods are solved using Faraday’s formulation which says how things in one element are related to neighboring elements. It is an interaction that expands from one point and widens, so the relations that must be solved are written in the form of partial differential equations.

So it is clear that if we want to calculate the field in a very big volume, the computation domain increases. However, there are also integral methods where only the current over the elements will be considered and only the objects will be discretized and so the unknown will then be the current over the single element considered. Each stream will interact with the others. So remote interaction is implemented.

EFIE

Considering a metallic object (but the discussion is easily extensible also to dielectrics ones) and the sources they radiate, we will have an incident field and a scattered field. The total field will be the sum of the two fields and the field produced by the interaction is what interests us precisely . To analyze this situation, we remove the metal object, by bringing it back to an equivalent source.

Once the current densities induced by the source are known, we should lead back to the equivalent current produced by the metal object, then we would be able to lead back to the equivalent field through the potential vectors.

We are in free space for which we have that the vector potential is expressed by:

\underline{A}(r)=\mu_{0} \iint_{S} J_{e}\left(\underline{r}^{\prime}\right) \frac{e^{-j k_{0}\left|r-r^{\prime}\right|}}{4 \pi\left|r-r^{\prime}\right|} d r^{\prime}

Furthermore, considering the boundary condition such that the cross product between the electric field and the normal is zero, we could lead to the expression of the scattered field starting from the induced one:

\underbrace{\hat{n} \times \underline{E}^{i}}_{\text {noto }}=-\hat{n} \times \underline{E}^{s}=j \omega \hat{n}+\left[\underline{A}+\frac{1}{k_{0}^{2}} \nabla \nabla \cdot \underline{A}\right]\\
\hat{\underline{n}} \times \underline{E}^{i}=\underline{\hat{n}} \times\left[j \omega \mu \iint_{S}\left(1+\frac{1}{k_{0}^{2}} \nabla \nabla \cdot\right) \underline{J}_{e}\left(r^{\prime}\right) G_{0}\left(r, r^{\prime}\right) d_{r}^{\prime}\right]

This is an integrodifferential expression of electrical type, since the known term expressed as an electric field is referred to as an electrical equation. Known in the literature as EFIE, electric field integral equation. Analyzing these equations they all have the same formula, known term outside (function representing the incident field) and the unknown inside the equation trapped by derivatives or functions which are linear operators (integral / derivative of the sum is the sum of integral / derivative ). Although the form is complicated, from a mathematical point of view, this is written as a linear operator applied to the unknown (function, generally vector) equal to a known term. All EFIEs are presentable as in the following equation and to free the current we have to construct the inverse so as to be able to calculate the scattered field.

\underline{L} \circ I(\underline{r})=f(\underline{r}) \Longrightarrow I(\underline{r})=\underline{L}^{-1} \circ f(\underline{r})

Numerical method

This is typically impossible to do by hand and is done using the computer. In order to be processed by the computer, it must be traced back to a matrix calculation, which can be easily implemented. It is therefore necessary to discretize it and do it by looking for the unknown function on a basis of known functions and writing the solution as their combination.

L \circ \sum_{n=1}^{N} I_{n} i_{n}(\underline{r}) \longrightarrow \sum_{n=1}^{N} I L \circ i_{n} \approx \underline{f}

To find the solution to the N equations in N unknowns, we select another set of equations such that they are an orthonormal basis. These are the weight functions and their name derives from the fact that they are used to test the unknown function:

\forall m \quad\left\langle\underline{w}_{m}, \sum_{n=1}^{N} I_{n} L \circ i_{n}\right\rangle=\left\langle\underline{w}_{m}, \underline{f}\right\rangle

We are therefore brought back to an algebraic system where the matrix of the system takes the name of generalized impedance matrix.

{\left[\begin{array}{c}
Z_{m n}
\end{array}\right]}_{N \times N} \underbrace{\left[\begin{array}{c}
I_{1} \\
\vdots \\
I_{n}
\end{array}\right]}_{N \times 1}=\underbrace{\left[\begin{array}{c}
V_{1} \\
\vdots \\
V_{n}
\end{array}\right]}_{N \times 1}\longrightarrow [I]={\color{red}{[Z]^{-1}}}[V]

Matlab

Using the Antenna Toolbox in Matlab it is possible to design, analyze and visualize antennas and antenna arrays. This toolbox for Matlab uses electromagnetic solvers including MoM to calculate impedance, current distribution, efficiency, radiation pattern both in the near and far field.

The toolbox has a wide catalog of antennas and arrays but also allows you to build particular antennas and to import STL structures. Planar antennas are easily designed, even with complex geometries. Furthermore, it is possible to improve the antenna bandwidth, build networks and optimize the performance by exploiting Machine Learning techniques integrated in the tool.

Applications

References

• Wireless Electromagnetic Technologies – Medical Engineering – Tor Vergata
• Antenna Design – Balanis
• Method of moments – Wikipedia
• Antenna ToolBox – MathWorks
• A review on Computational Electromagnetics Methods – Sumithra