Excitation of the Structure

In the Method of Moments (MoM), the excitation represents the right-hand side of the matrix equation $[\mathbf{Z}] [\mathbf{I}] = [\mathbf{V}]$. This “Voltage Vector” $[\mathbf{V}]$ defines the external electromagnetic energy applied to the structure, which can take the form of discrete voltage sources or incident external fields.

Discrete Voltage Sources

When a voltage generator is placed on a specific wire segment (the $i$-th segment), the corresponding element in the voltage column vector is set to the voltage $V_i$ of that generator. This results in a vector where all elements are zero except for the driven segment:

$\displaystyle [\mathbf{V}] \,=\, \begin{bmatrix} 0 \\ \vdots \\ V_i \\ \vdots \\ 0 \end{bmatrix}$

The Delta-Gap Source Model

In a delta-gap model, the source is assumed to exist at a single point (an infinitesimal gap). In a numerical simulation, this source occupies the entire length of the segment where it is placed. As you increase the number of segments ($N$) to refine the mesh, the length of that segment diminishes, approaching zero. While mathematically convenient, this can lead to convergence issues.

The Finite-Gap Source Model

To model a finite-gap source, which more accurately represents physical connectors, a specific, short wire segment is defined to house the excitation. Unlike the delta-gap model, when the overall antenna mesh is refined, the length of this specific source segment remains fixed. This approach ensures the convergence of the input impedance, which is calculated by dividing the source voltage by the computed current at that specific segment.

Incident Plane Wave Excitation

When the structure is excited by an external source, such as a distant transmitting antenna, it is modeled as an incident plane wave. In this scenario, every segment of the wire is excited simultaneously by the incoming field.

The incident field $\mathbf{E}_i$ at any point $\mathbf{r}$ is defined as:

$\displaystyle \mathbf{E}_i(\mathbf{r}) \,=\, \mathbf{E}_0 \; e^{-j \mathbf{k} \cdot \mathbf{r}}$

Where:

• $\mathbf{k}$: The wavevector, indicating the direction of propagation.
• $k = |\mathbf{k}|$: The wavenumber.
• $\mathbf{r}$: The evaluation point on the structure.

To fill the voltage vector $[\mathbf{V}]$, the solver calculates the contribution for each segment $m$ by integrating the incident field along the segment’s length:

$\displaystyle V_m \,=\, \int_{s_m} \mathbf{E}_i(\mathbf{r}(s)) \cdot \hat{\mathbf{s}} \; ds$

In this integral, $\mathbf{r}(s)$ represents the parametric description of the curved wire axis, and $\hat{\mathbf{s}}$ is the unit vector tangent to the wire at that point. This ensures that only the tangential component of the electric field, the part responsible for inducing current, is accounted for in the calculation (Fig. 1).