Method of Moments (MoM) — An Integral Equation-Based Electromagnetic Field Analysis Technique
Theory and Physics
Overview — What is MoM?
What's the difference between the Method of Moments and FEM or FDTD? They're all methods for solving electromagnetic fields, right?
Good question. MoM is integral equation-based and treats only the current distribution on the conductor surface as the unknown. It doesn't require a volume mesh and is overwhelmingly strong for open-region problems. For wire antennas or RCS (Radar Cross Section) calculations, it's orders of magnitude more efficient than FEM.
No volume mesh needed? That sounds incredibly convenient!
Yes, that's MoM's greatest strength. With FEM, you have to create a 3D volume mesh including the air around the structure. Moreover, handling open space requires absorbing boundary conditions (like PML). With MoM, you only need to discretize the conductor surface, and the Green's function automatically satisfies the radiation condition at infinity. NEC2 is a free MoM code widely used even by amateur radio operators.
NEC2, that's the one amateur radio people often use, right? Can it be used in professional settings too?
NEC2 is a thin-wire approximation MoM code specialized for wire antennas. It's still the first choice for designing dipole or Yagi-Uda antennas. However, it cannot handle surface elements, so for complex shapes including patch antennas or enclosures, you'd use Altair Feko or the Ansys HFSS MoM solver. In this article, let's systematically look at MoM from its theoretical foundations to implementation.
Fundamental Differences from FEM/FDTD
There are three main approaches to numerical electromagnetic field analysis. Understanding the formulation and strengths of each is key to selecting the appropriate method.
| Method | Formulation | Discretization Target | Matrix Property | Strengths |
|---|---|---|---|---|
| MoM | Integral Equation | Conductor Surface Only | Dense Matrix (Small Scale) | Antenna Radiation, RCS, Open Regions |
| FEM | Differential Equation (Weak Form) | Entire Volume | Sparse Matrix (Large Scale) | Complex Shapes, Inhomogeneous Media, Resonators |
| FDTD | Differential Equation (Finite Difference) | Entire Volume (Orthogonal Grid) | No Matrix Needed (Explicit Method) | Wideband Transient Response, Nonlinear |
Governing Equation — EFIE (Electric Field Integral Equation)
I'd like to see the mathematical formula for MoM. What kind of equation is it based on?
The fundamental equation is the EFIE (Electric Field Integral Equation). It is derived from the boundary condition that the tangential component of the total electric field on a perfect electric conductor (PEC) surface is zero.
On a perfect conductor surface $S$, the relationship between the incident electric field $\mathbf{E}^{inc}$ and the surface current $\mathbf{J}(\mathbf{r}')$ is described by the following EFIE:
Here, the Green's function $G(\mathbf{r}, \mathbf{r}')$ represents the point source response in free space:
Wow, that's pretty complex... What does it mean in simple terms?
In simple terms, it's an equation that says "currents at all points on the conductor electromagnetically influence each other through the Green's function, which acts as a 'propagation weight'." The left side is the incident wave (the electromagnetic wave coming from outside), and the right side is the scattered field generated by that current. MoM's job is to find the current distribution $\mathbf{J}$ that balances these on the conductor.
Physical Meaning of Each Term
- $\hat{n} \times \mathbf{E}^{inc}$: Tangential component of the incident electric field. Equivalent to the "source" where the external electromagnetic wave incident on the conductor induces surface currents. For example, the incident field when a radar wave hits a stealth aircraft.
- $j\omega \mu \int G \cdot \mathbf{J} \, dS'$: Term originating from the vector potential $\mathbf{A}$. Represents the magnetic field component generated by the time variation of current, directly linked to antenna radiation.
- $\frac{1}{j\omega\varepsilon} \nabla \int G \, \nabla' \cdot \mathbf{J} \, dS'$: Term originating from the scalar potential $\phi$. Represents the electric field component generated by charge accumulation ($\nabla' \cdot \mathbf{J}$). This term becomes dominant at low frequencies.
- $G(\mathbf{r}, \mathbf{r}')$ (Green's function): Kernel function representing "the electric field created at $\mathbf{r}$ by a point current source at $\mathbf{r}'$". The form $e^{-jkR}/4\pi R$ is the propagation of a spherical wave itself. It decays inversely with distance $R$ and is delayed by phase $kR$.
Assumptions and Applicability Limits
- Perfect Electric Conductor (PEC) Assumption: Ignores finite conductivity of the surface. Almost accurate for metal conductors in the microwave band, but extension to impedance boundary conditions is needed at optical frequencies or for thin films.
- Free Space Green's Function: Assumes a homogeneous infinite space. For ground reflections, modification via Sommerfeld integral or image theory is needed.
- Surface Equivalence Theorem: Replacing volume currents in a structure with surface currents is only possible for PEC or homogeneous dielectrics.
- Frequency Domain: Basic MoM analyzes at a single frequency. Wideband characteristics are obtained via frequency sweep or AWE (Asymptotic Waveform Evaluation).
Dimensional Analysis and Unit System
| Variable | SI Unit | Notes |
|---|---|---|
| Surface Current Density $\mathbf{J}$ | A/m | Distinguish from volume current density [A/m²] |
| Green's Function $G$ | 1/m | Dimension of $e^{-jkR}/(4\pi R)$ |
| Wavenumber $k$ | rad/m | $k = 2\pi/\lambda = \omega\sqrt{\mu\varepsilon}$ |
| Impedance Matrix Element $Z_{mn}$ | Ω (Ohm) | Obtained as the integral result of basis functions |
| Electric Field $\mathbf{E}$ | V/m | Electric field strength of the incident wave |
Discretization via Galerkin Method
How do you solve this integral equation on a computer?
You approximate the unknown surface current $\mathbf{J}$ as a linear combination of basis functions $\mathbf{f}_n$. This is the core of MoM.
Expand the unknown current using $N$ basis functions:
$I_n$ is the coefficient (unknown) for each basis function. Substitute this expansion into the EFIE and take the inner product (testing) with a testing function $\mathbf{f}_m$. In the Galerkin method, the same function is used for both the testing function and the basis function:
Here, $\mathcal{L}$ is the integral operator of the EFIE, and $\langle \cdot, \cdot \rangle$ denotes the inner product.
I see, it's the same concept as the Galerkin method in FEM!
Yes, in the sense of being a weighted residual method, it's the same framework as FEM. The difference is that the operator is integral, not differential. That's why MoM is also called "Method of Weighted Residuals applied to Integral Equations". It was systematized by Harrington in his 1968 book.
Impedance Matrix [Z][I]=[V]
Testing via the Galerkin method transforms the continuous integral equation into the following matrix equation:
Each component is defined as follows:
So $[Z]$ is the impedance matrix, $[I]$ is the current coefficient vector, and $[V]$ is the excitation vector. The structure is similar to FEM's $[K]\{u\} = \{F\}$!
Exactly. However, there is one critical difference. The FEM stiffness matrix $[K]$ is a sparse matrix (only adjacent elements are coupled), but the MoM impedance matrix $[Z]$ is a dense (full) matrix. That's because all basis functions are coupled to each other through the Green's function.
So that means if there are $N$ unknowns, you need $O(N^2)$ memory?
Sharp observation. Solving with a direct method requires $O(N^2)$ memory and $O(N^3)$ computation time. That's why MoM has long been used as a "small-scale but accurate" method. To break through this limitation and...
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