# Integrating the Currents

Summing or integrating the vector currents is the last step in the MoM procedure. This step leads to specific output parameters such as far fields and impedance.

## The Free Space Green Function

The free space Green's function is essential to the MoM to allow calculation of fields at arbitrary points in 3D space. Without going into the finer technical details of the equation, it can be stated that the Green function is contained inside the integral operator $ℒ$ operating on the surface currents,
(1) $ℒ{\left\{{J}_{scat}\right\}}_{\mathrm{tan}}=-{E}_{inc,\mathrm{tan}}$
Consider an infinitesimally small current element J in free space at a point r' radiating an electric field E and a magnetic field H.

The Green's function (Equation 2) gives the spatial response to a spatially impulsive current source. This means that for the current element (source) located at the point r', the Green's function gives the potential of this source at the point r, or any required point in 3D space.

(2) $G\left(r,r\text{'}\right)=\frac{{e}^{-j\beta |r-r\text{'}|}}{|r-r\text{'}|}$
with
(3) $|r-r\text{'}|=\sqrt{{\left(x-x\text{'}\right)}^{2}+{\left(y-y\text{'}\right)}^{2}+{\left(z-z\text{'}\right)}^{2}}$
the distance from the source to the field point. When there are multiple of these sources distributed in space, such as over the arbitrary PEC body, the response at the point r is given by summing all the sources (integration over all the sources).1

1 Computational Electromagnetics for RF and Microwave Engineering, Second Edition, David B. Davidson, p.265