# Defining electric current source excitations for surface integral equation formulations

In a finite difference (FD) based electromagnetic formulation based on a Yee cell grid, one can define electric current source excitations ($J$) on the $E$ field grid points. At a distance, the fields generated from these excitations (in free-space) closely match with those of an analytical Hertzian dipole (which is infinitesimally thin). A Hertzian dipole has a constant current $I$ over its small length $l$, but since it is infinitesimally thin, its effective current area density $J = \frac{I}{dxdy} = \infty$, so the fields are singular at the origin and do not match up with those of the "real" current in a FD formulation (which is kind of like a cube of current rather than a line and thus has finite $J$ and finite $E$ everywhere including the near-field) close to the source. If I have multiple of these sources in my system at different locations, I can compute the power radiated from each individual one by integrating the quantity $J\cdot E$ over a small volume enclosing each source.

In an integral equation (IE) formulation, instead of directly specifying current excitations, one just specifies incident fields and the corresponding currents over dielectric/conducting surfaces and scattered fields are computed. How could I achieve the same thing as a current excitation with finite dimensions with an IE formulation (like with a FD method)?

I guess I could use the closed-form fields from an analytical Hertzian dipole as an incident wave source and calculate scattered fields, however, this would not have the same result as a finite current "cube" in an FD formulation. Is there any straightforward way to define electric current excitations with finite current density J (or at least surface current density) in a surface integral equation formulation? Further, how would you go about calculating the incident power from each source when using an IE method?

-
Does no one here much experience with surface integral equation methods or is my question confusing? –  Costis Jun 16 '12 at 0:45

## 1 Answer

I think your question is fine. Depends upon what exactly you're trying to do - seems like it could be 1 of 2 scenarios:

1. If you are interested in illuminating an arbitrary meshed object by a nearby dipole source with electrically small but nonzero length, I think you could safely assume some current distribution over the length of the dipole (e.g. half sinusoid or maybe hat function) and then impose its incident field through quadrature. That is, when integrating E-inc over a surface mesh element, evaluate it by perfoming an auxiliary integral over the length of the driven wire, looking up the current density at that position on the wire, then push that through your analytical Hertzian dipole formula.

2. If you are interested in driving your pure MoM model with an electrically small ideal current source (e.g. modeling a bowtie antenna), you should look into/google "delta gap" feed models. Unfortunately I have never implemented this and don't know what the best original reference is for it. All the antenna codes I've worked were either FEM or FEM-BEM hybrids and were fed by either an FEM lumped element model or an FEM waveguide boundary condition. I would bet that "The Method of Moments In Electromagnetics" by Walton C. Gibson would contain some discussion - at least there's a picture of it on the cover. (on a bowtie, no less)

-