The Exact Kernel

The kernel is the mathematical core of the Electric Field Integral Equation (EFIE) solved by AN-SOF. It determines how the current at one point on a wire influences the field at another. While several approximations exist to simplify this calculation, AN-SOF utilizes an Exact Kernel to maintain high numerical precision where other solvers fail.

Defining the Kernel

The kernel is obtained by averaging the free-space Green’s function $G(\mathbf{r}, \mathbf{r}’)$ over the circumference of the wire’s cross-section. Mathematically, it is expressed as:

$\displaystyle K(s,s’) \,=\, \frac{1}{4 \pi^2} \int_0^{2\pi} \int_0^{2\pi} G(\mathbf{r},\mathbf{r}’) \, d\phi’ d\phi \qquad (1)$

In this equation:

• $s$ and $s’$ are coordinates along the wire axis (observation and source points).
• $\phi$ and $\phi’$ are the angular positions on the wire’s surface.
• The result represents the interaction between a source point $\mathbf{r} = (s, \phi)$ and an observation point $\mathbf{r}’ = (s’, \phi’)$ on the actual tubular surface of the wire.

The Thin-Wire Approximation

Because the exact integral is difficult to solve analytically, many legacy Method of Moments (MoM) codes use the thin-wire approximation:

$\displaystyle K(s,s’) \approx \frac{e^{-j k R}}{4 \pi R}, \qquad R = \sqrt{|\mathbf{r}(s) \ – \ \mathbf{r}(s’)|^2 + a^2} \qquad (2)$

where $a$ is the wire radius.

This approximation assumes that the current is concentrated on the wire axis and the observation point is on the surface, or vice versa. While efficient for very thin wires, this simplification breaks down as the wire becomes thicker relative to the segment length.

Consequences of Approximation

When a wire is divided into segments where the diameter is larger than the segment length (a ratio $> 1$), the thin-wire approximation introduces significant numerical errors:

1. Current Oscillations: The current distribution begins to exhibit non-physical “ripples” or oscillations, particularly near discrete sources (feed points) and at the wire ends.
2. Impedance Divergence: As the mesh is refined (more segments added), the input impedance does not settle on a single value. Instead, it diverges, making it impossible to obtain a reliable result.

Figure 1 shows the current distribution in amplitude along a center-fed half-wave dipole obtained using both the thin-wire approximation and the exact kernel. The antenna has been divided into segments with a diameter three times greater than their lengths, resulting in very thick wire segments.

The thin-wire kernel exhibits the well-known oscillatory effect on the current distribution near the position of discrete sources and at wire ends when the segment diameter-to-length ratio is greater than 1. As shown, these oscillations disappear when the exact kernel is used instead of the thin-wire approximation.

The AN-SOF Advantage: Stability and Convergence

By using the Exact Kernel, AN-SOF eliminates these numerical artifacts. The current distribution remains smooth and physically accurate, even for segments with a high diameter-to-length ratio.

As shown in comparisons between AN-SOF and NEC-2 (which relies on the thin-wire kernel), AN-SOF demonstrates monotonic convergence (see Fig. 2). Whether using a finite-gap or delta-gap source, the resistance and reactance values stabilize as the number of segments ($N$) increases, providing a trustworthy solution for complex, high-precision antenna designs.