HFSS15: Integral Equation Method Used in HFSS-IE

The most commonly used electromagnetic integral equation is the Electric Field Integral Equation, or more commonly referred to by its acronym, EFIE. Consider a perfect conductor that is illuminated by a plane wave or any other source. On the surface of a perfect conductor the total tangential electric field must be zero. The total electric field comprises the incident electric field and the scattered electric field; hence, on the conducting surface.

The above relation only specifies the scattered field on the surface of the conductor. To fully understand the system, we need to be able to calculate the scattered field everywhere; one way to do this is to calculate the induced current which generates the scattered electric field.

The field generated by a current is

where G is the Green's function defined as

and J is the surface current density. Combining the relationship between current and scattered field with the boundary condition yields the EFIE

The unknown current density is within the integral operator; these types of equations are known as integral equations. Also note the integral is a convolution. Electromagnetic integral equations are convolutions, and this allows us to make an analogy to system theory. The Green's function is the transfer function, or the impulse response of the system, and the incident electric field is the output. Our task is to find the input, or the current density, from the output knowing the transfer function.

Solving the EFIE is typically done through a finite element technique. We divide the surface of the conductor into many, many triangles, and on each triangle we assume a current distribution. This converts EFIE into a matrix equation, and in the electromagnetics literature this technique is called the method of moments (MoM).

The EFIE is just one of many different integral equations used in electromagnetics. We can vary the boundary conditions we enforce, or the method we use to specify the field everywhere. For example, if on the surface of the conductor, we enforce the continuity of the magnetic field, following a similar procedure yields the magnetic field integral equation (HFIE). All of them are similar in form to the EFIE and are solved in a similar manner.

References:

1. Roger Harrington, Time Harmonic Electromagnetic Fields, McGraw-Hill, New York, NY 1961.

2. Andrew Peterson, Scott Ray, Raj Mittra, Computational Methods for Electromagnetics, IEEE Press, New York, NY 1998.