- Electric field integral equation
The electric field integral equation is a relationship that allows one to calculate the
electric fieldintensity E generated by an electric currentdistribution J .
We consider all quantities in the frequency domain, and so assume a time-dependency that is suppressed throughout.
Following the third equation involving the
vector calculuswe can write any divergenceless vector as the curl of another vector, hence
where A is called the
magnetic vector potential. Substituting this into the above we get
and any curl-free vector can be written as the
gradientof a scalar, hence
where is the
electric scalar potential. These relationships now allow us to write
which can be rewritten by vector identity as
As we have only specified the curl of A, we are free to define the divergence, and choose the following:
which is called the
Lorenz gauge condition. The previous expression for A now reduces to
which is the vector
Helmholtz equation. The solution of this equation for A is
where is the three-dimensional homogeneous
Green's functiongiven by
We can now write what is called the electric field integral equation (EFIE), relating the electric field E to the vector potential A
We can further represent the EFIE in the dyadic form as
where here is the dyadic homogeneous Green's Function given by
The EFIE describes a radiated field E given a set of sources J, and as such it is the fundamental equation used in antenna analysis and design. It is a very general relationship that can be used to compute the radiated field of any sort of antenna once the current distribution on it is known. The most important aspect of the EFIE is that it allows us to solve the radiation/scattering problem in an
unboundedregion, or one whose boundary is located at infinity. For closed surfaces it is possible to use the Magnetic Field Integral Equationor the Combined Field Integral Equation, both of which result in a set of equations with improved condition number compared to the EFIE. However, the MFIE and CFIE can still contain resonances.
In scattering problems, it is desirable to determine an unknown scattered field that is due to a known incident field . Unfortunately, the EFIE relates the "scattered" field to J, not the incident field, so we do not know what J is. This sort of problem can be solved by imposing the
boundary conditionson the incident and scattered field, allowing one to write the EFIE in terms of and J alone. Once this has been done, the integral equation can then be solved by a numerical technique appropriate to integral equations such as the method of moments.
By the Helmholtz theorem a vector is described completely by its divergence and curl. As the divergence was not defined, we are justified by choosing the Lorenz Gauge condition above provided that we consistently use this definition of the divergence of A in all subsequent analysis.
This vector A should not be interpreted as a real physical quantity, it is just a mathematical tool to help us solve electromagnetic problems.
*Gibson, Walton C. "The Method of Moments in Electromagnetics". Chapman & Hall/CRC, 2008. ISBN 978-1-4200-6145-1
*Harrington, Roger F. "Time-Harmonic Electromagnetic Fields". McGraw-Hill, Inc., 1961. ISBN 0-07-026745-6.
*Balanis, Constantine A. "Advanced Engineering Electromagnetics". Wiley, 1989. ISBN 0-471-62194-3.
*Chew, Weng C. "Waves and Fields in Inhomogeneous Media". IEEE Press, 1995. ISBN 0-7803-4749-8.
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