# Conformal Moment Method

## Conformal Method of Moments with Exact Kernel

In the traditional Method of Moments (MoM) the structures to be modeled are divided into straight wire segments. Straight segments fit well the shape of linear antennas like dipoles and arrays constructed using dipoles. However, there are many antennas and structures that have curved shapes. In these cases, a curved wire is approximated using a string of straight-line segments, Fig. 1(a). **Sharp junctions** between adjacent wires introduce a **modeling error** at the very beginning of the simulation that can never be fixed. Poor results for curved antennas like loops, helices and spirals are often obtained when the linear approximation is applied, especially large errors in the feed point impedances.

Furthermore, in the traditional MoM other problems arise due to the use of the so-called **thin-wire Kernel**. The Kernel is the heart of the integral equation to be solved by the MoM, and the thin-wire approximation, which considers that the currents are concentrated as a filament along the wire axes, produces large errors in the results. For the math involved, refer to Section “18. Background Theory”. One of the problems that appears due to the thin-wire Kernel is erratic numerical oscillations when there are **wires bent at right angles** or for angles less than 30 degrees between adjacent segments, Fig. 1(b).

Another problem that should be pointed out is about the **spacing between parallel wires.** Segments cannot be very close to each other since misleading results are obtained when the spacing between them is less than a quarter of a segment length, Fig. 1(c).

**The segment length itself has a limitation**, it must be greater than 0.001 of a wavelength, and consequently the traditional MoM cannot be applied at very low frequencies, Fig. 1(d). For example, consider an electric circuit around 1 meter in size operating at 60 Hz. The free space wavelength can be calculated as (300/60) x 1,000,000 = 5,000,000 meters. Thus, the size of the circuit measured in wavelengths is 1/5,000,000 = 0.0000002, so segments shorter than 0.0000002 of a wavelength are needed to model the circuit. This segment length is at least 5,000 times shorter than the minimum segment length supported by the MoM. Therefore, an electric circuit at low frequencies cannot be modeled using the traditional implementation of the MoM for wire antennas.

The limitations of the traditional MoM have been removed in its improved version: the** Conformal Method of Moments (CMoM) with Exact Kernel**. In the CMoM, **conformal segments** are used that exactly follow the contour of the structure, so an exact description of geometry details is achieved, Fig. 2. A conformal segment is a curved cylindrical tube that correctly fit the shape of curved wires. The limitations regarding bent wires, small spacings between wires, and segment length have been removed in AN-SOF by using the **exact Kernel** instead of the thin-wire approximation, which allows us to perform calculations with much higher accuracy than the traditional method.

Therefore, with the CMoM with exact Kernel we remove the limitations of the old MoM and obtain the following advantages:

- Decreased number of calculations and increased accuracy of results.
- Decreased simulation time and computer memory usage, allowing us to model larger and more complex designs.
- Ability to simulate from extremely low frequencies (circuits at 60 Hz) to very high ones (microwave antennas).