The Method of Moments

The Method of Moments (MoM) is the numerical technique used by AN-SOF to transform the Electric Field Integral Equation (EFIE) into a system of linear equations. Once digitized, these equations can be solved using standard algebraic methods to determine the current distribution on a structure.

From Calculus to Algebra

To solve the EFIE numerically, we represent the integral linear operator as $\mathcal{L}$. The fundamental equation is expressed as:

$\displaystyle \mathcal{L}(I) \,=\, E_{T} \qquad (1)$

Where $E_{T}$ is the tangential component of the incident electric field on the surface of the wire. Because the exact current distribution ($I$) is unknown, it is approximated as a finite sum of $N$ basis functions ($F_{n}$) with unknown amplitudes ($I_{n}$):

$\displaystyle I \,\approx\, \sum_{n} I_{n} \, F_{n} \qquad (2)$

By substituting this expansion into Eq. (1) and utilizing the linearity of the operator $\mathcal{L}$, we get:

$\displaystyle \sum_{n} I_{n} \, \mathcal{L}(F_{n}) \,=\, E_{T} \qquad (3)$

Weighting and Matrix Formation

To solve for the $N$ unknown amplitudes, Eq. (3) is weighted with a set of $N$ linearly independent testing functions ($T_m$), also known as weighting functions, as follows:

$\displaystyle \sum_n I_n \int T_m \, \mathcal{L}(F_n) \, du \,=\, \int T_m \, E_T \, du \qquad (4)$

where the integrals are evaluated over the wire surface, whose generally curvilinear axis is described parametrically by the real variable $u$. The testing process ensures that the number of independent equations equals the number of unknowns, resulting in the matrix equation:

$[\mathbf{Z}] [\mathbf{I}] \,=\, [\mathbf{V}] \qquad (5)$

Where:

• $[\mathbf{Z}]$ (Impedance Matrix): An $N \times N$ matrix where each element $Z_{mn} = \int T_{m} \, \mathcal{L}(F_{n}) \, du$.
• $[\mathbf{I}]$ (Current Matrix): An $N \times 1$ column vector containing the unknown current amplitudes $I_{n}$.
• $[\mathbf{V}]$ (Voltage Matrix): An $N \times 1$ column vector where each element $V_{m} = \int T_{m} \, E_{T} \, du$.

AN-SOF employs advanced linear algebra algorithms to solve this fully occupied system of equations.

Basis and Testing Functions in AN-SOF

The MoM process begins by dividing the wire structure into $N$ segments. AN-SOF specifically utilizes triangular basis functions and pulse testing functions (Fig. 1).

• Triangular Basis Functions: Used to approximate the current distribution smoothly across segments.
• Pulse Testing Functions: Used to satisfy the boundary conditions along the segments.

The Conformal Method of Moments (CMoM)

When a wire is curved and described parametrically, the support for the basis and testing functions follows that curve. This results in the Conformal Method of Moments (CMoM). Unlike legacy codes that force basis functions onto straight-wire approximations, CMoM ensures the mathematical model adheres strictly to the physical geometry.

Once the system is solved and the currents ($I_{n}$) are known, the engine can accurately compute secondary parameters such as input impedance, far-field radiation patterns, and near-field distributions.