Surface equivalence principle

Illustration of the equivalence principle for an imaginary closed surface with impressed electric <math>(J_s)</math> and magnetic current <math>(M_s)</math> sources: original (a) and equivalent (b) problems over the imaginary surface, <math>S</math>. <math>J_1</math> and <math>M_1</math> represent the original source distributions inside the surface. The equivalent formulation yields the same external electric and magnetic fields distribution as in the original problem. The internal fields and surface currents are chosen to enforce the boundary conditions.

In electromagnetism, surface equivalence principle or surface equivalence theorem relates an arbitrary current distribution within an imaginary closed surface with an equivalent source on the surface. It is also known as field equivalence principle,<ref name="Rengarajan00"/> Huygens' equivalence principle<ref name="chew93"/> or simply as the equivalence principle.<ref name="FOOTNOTEHarrington2001106-109">Harrington 2001, p. 106-109.</ref> Being a more rigorous reformulation of the Huygens–Fresnel principle, it is often used to simplify the analysis of radiating structures such as antennas.

Certain formulations of the principle are also known as Love equivalence principle and Schelkunoff equivalence principle, after Augustus Edward Hough Love and Sergei Alexander Schelkunoff, respectively.

Physical meaning

General formulation

The principle yields an equivalent problem for a radiation problem by introducing an imaginary closed surface and fictitious surface current densities. It is an extension of Huygens–Fresnel principle, which describes each point on a wavefront as a spherical wave source. The equivalence of the imaginary surface currents are enforced by the uniqueness theorem in electromagnetism, which dictates that a unique solution can be determined by fixing a boundary condition on a system. With the appropriate choice of the imaginary current densities, the fields inside the surface or outside the surface can be deduced from the imaginary currents.<ref name="FOOTNOTEBalanis2012328-331">Balanis 2012, p. 328-331.</ref> In a radiation problem with given current density sources, electric current density <math>J_1</math> and magnetic current density <math>M_1</math>, the tangential field boundary conditions necessitate that

<math>J_s = \hat{n} \times (H_1 - H)</math>

<math>M_s = -\hat{n} \times (E_1 - E)</math>

where <math>J_s</math> and <math>M_s</math> correspond to the imaginary current sources that are impressed on the closed surface. <math>E</math> and <math>H</math> represent the electric and magnetic fields inside the surface, respectively, while <math>E_1</math> and <math>H_1</math> are the fields outside of the surface. Both the original and imaginary currents should produce the same external field distributions.<ref name="FOOTNOTEBalanis2012328-331">Balanis 2012, p. 328-331.</ref>

Love and Schelkunoff equivalence principles

Illustration of Love and Schelkunoff equivalence principles: a) The original problem, b) Love equivalent problem, c) Schelkunoff equivalent problem with perfect electric conductor, d) Schelkunoff equivalent problem with perfect magnetic conductor. All problems have the same external fields.

Per the boundary conditions, the fields inside the surface and the current densities can be arbitrarily chosen as long as they produce the same external fields.<ref name="FOOTNOTEHarrington2001106-109">Harrington 2001, p. 106-109.</ref> Love's equivalence principle, introduced in 1901 by Augustus Edward Hough Love,<ref name="Love01"/> takes the internal fields as zero:

<math>J_s = \hat{n} \times H_1</math>

<math>M_s = -\hat{n} \times E_1</math>

The fields inside the surface are referred as null fields. Thus, the surface currents are chosen as to sustain the external fields in the original problem. Alternatively, Love equivalent problem for field distributions inside the surface can be formulated: this requires the negative of surface currents for the external radiation case. Thus, the surface currents will radiate the fields in the original problem in the inside of the surface; nevertheless, they will produce null external fields.<ref name="Rengarajan00"/>

Schelkunoff equivalence principle, introduced by Sergei Alexander Schelkunoff,<ref name="Schelkunoff38"/><ref name="Schelkunoff39"/><ref name="Schelkunoff51"/> substitutes the closed surface with a perfectly conducting material body. In the case of a perfect electrical conductor, the electric currents that are impressed on the surface won't radiate due to Lorentz reciprocity. Thus, the original currents can be substituted with surface magnetic currents only. A similar formulation for a perfect magnetic conductor would use impressed electric currents.<ref name="Rengarajan00"/>

The equivalence principles can also be applied to conductive half-spaces with the aid of method of image charges.<ref name="Rengarajan00"/><ref name="FOOTNOTEBalanis2012328-331">Balanis 2012, p. 328-331.</ref>

Applications

The surface equivalence principle is heavily used in the analysis of antenna problems to simplify the problem: in many of the applications, the close surface is chosen as so to encompass the conductive elements to alleviate the limits of integration.<ref name="FOOTNOTEBalanis2012328-331">Balanis 2012, p. 328-331.</ref> Selected uses in antenna theory include the analysis of aperture antennas<ref name="FOOTNOTEBalanis2005653-660">Balanis 2005, p. 653-660.</ref> and the cavity model approach for microstrip patch antennas.<ref name="FOOTNOTEKinaymanAksun2005300">Kinayman & Aksun 2005, p. 300.</ref> It has also been used as a domain decomposition method for method of moments analysis of complex antenna structures.<ref name="Li07"/> Schelkunoff's formulation is employed particularly for scattering problems.<ref name="chew93"/><ref name="FOOTNOTEBalanis2012333">Balanis 2012, p. 333.</ref><ref name="barber-osa"/><ref name="fem-mom"/>

The principle has also been used in the analysis design of metamaterials such as Huygens’ metasurfaces<ref name="Huygens-metasurface"/><ref name="active-cloak"/> and plasmonic scatterers.<ref name="array-plasmonic"/>

See also

• Babinet's principle
• Electromagnetism uniqueness theorem
• Huygens–Fresnel principle
• Reciprocity (electromagnetism)

References

Bibliography

• Balanis, Constantine A. (2005). Antenna Theory: Analysis and Design (3rd ed.). John Wiley and Sons. ISBN 047166782X.
• Balanis, Constantine A. (2012). Advanced Engineering Electromagnetics. John Wiley & Sons. ISBN 978-0-470-58948-9.
• Harrington, Roger F. (2001). Time-Harmonic Electromagnetic Fields. McGraw-Hill. ISBN 9780070267459.
• Kinayman, Noyan; Aksun, M. I. (2005). Modern Microwave Circuits. Norwood: Artech House. ISBN 9781844073832.