Basics of computational electromagnetics

Updated February 11, 2012

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The basic principle of all computational techniques in electromagnetics can be summed up in one word: discretization. In the end, every CEM technique is a way to to solve Maxwell's equations. Maxwell's equations are continuous in the sense that the electric and magnetic fields are defined everywhere in space. A computer in general can only solve Maxwell's equations approximately at a finite set of points in space. A finite set of points in space is termed discrete, thus discretization is involved in any numerical solution to Maxwell's equations.

Finite element defined

The basic idea of finite element techniques is simple. Recall the definition of the derivative of a function f(x):

df/dx = limh->0{ [ f (x+h) - f(x) ]/ h }

(this is just the slope of the tangent line to the curve f(x), or the 'rise' over the 'run')

Finite element techniques generally replace all derivatives of the electric and magnetic field in Maxwell's equations with the formula above, perhaps applied multiple times to get second-order derivatives, except a finite but small value of h is used. The size of h defines a grid, or mesh. Moreover, Maxwell's equations can be rewritten as a set of linear algebraic equations in matrix form by taking h finite. Once the electric and magnetic fields have been discretized by using a finite value of h via the process above, there are a variety of methods which can be used to solve these equations, each yielding an approximate solution to Maxwell's equations at a finite set of points in space.

Expansion methods

Expansion methods take many forms. The basic idea is that one expands the electric and magnetic fields in Maxwells equations, using a series approximation to the functions, such as a Fourier Series, though in practice more complicated functions are used to increase the speed of these methods. Method of Moments (MoM) is an expansion method, as is the multi-level fast multipole (MLFM) method, as well as Mode-Matching. The basic idea is that you only need to keep a finite number of terms in the series. In an expansion method, one solves for the unknown coefficients of the series in which the solution is expanded. Generally, this is carried out by using an orthonormal function basis, like sines and cosines in a Fourier Series. (Click here for more information)

Approximate methods

Here, approximation is used to mean that one sort of approximation or another has been introduced to simplify Maxwell's equations. Approximation methods are common in reflector analysis. The four most common approximations are called

Geometrical theory of optics (GO)

Geometric theory of diffraction (GTD)

Physical optics (PO)

Physical theory of diffraction (PTD)

All four methods are generally recommended only for reflectors which are at least five wavelengths in diameter and have a fairly smooth shape. However, each method is more or less accurate for off-axis calculations of the far-field pattern. So GO might be fine if you only want the gain, but if you need to know the sidelobes accurately, PO is the method to use. It is difficult to provide cut and dry answers to accuracy questions related to these techniques, but the general guideline is 5 wavelengths for the size of the reflector, and if the reflector size is close to five wavelengths, use PO for off-axis calculations such as sidelobes. GTD should almost always be used in conjunction with GO, and PTD with PO . For more detailed discussion of these techniques and related accuracy issues, link here to our page on reflector analysis.