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Keywords:

  • polarizabilities;
  • inequality;
  • aperture

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Proof
  5. 3. Conclusion
  6. Acknowledgments
  7. References

[1] We prove the conjecture that αe,zz−1 ≥ αm,xx−1 + αm,yy−1 for a plane aperture of arbitrary shape, where αe and αm are the electric and magnetic polarizabilities.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Proof
  5. 3. Conclusion
  6. Acknowledgments
  7. References

[2] Aperture polarizabilities which describe low frequency coupling are useful in many EMI/EMC problems. In 1987 [Lee, 1987], it was conjectured that for an aperture of arbitrary shape in a perfectly conducting plane surface the inverse of the electric polarizability is greater than or equal to the sum of the inverses of magnetic polarizabilities along two principal axes. Since the magnetic polarizability tensor is a two by two real symmetric matrix, two orthogonal principal axes always exist. Let x and y be the principal axes. The conjecture can be stated as follows:

  • equation image

where

  • equation image
  • equation image
  • equation image

with p and m being the electric and magnetic dipole moments, and E0 and H0 the external fields.

[3] It is known that the equality sign of (1) holds for circular and elliptical apertures [Lee, 1987]. In 1990 [Gluckstern et al., 1990] it was shown that (1) is true for apertures symmetric with respect to the x and y axes, for example, a rectangle, a diamond, and a cross shaped aperture, for which numerical results exist in support of (1) [Gluckstern et al., 1990; Latham, 1972].

[4] In the following we will prove (1) for a plane aperture of any shape. The first step of the proof follows the idea suggested in the work of Gluckstern et al. [1990], i.e., to construct a stationary expression for each of the quantities in (1). We will then prove that these stationary expressions give minimum values when the trial functions are solutions of the associate boundary value problems. We will next manipulate the stationary expression for αe,zz−1 to a form similar to those for αm,xx−1 and αm,yy−1, which can be done for a plane aperture of any shape. On the other hand, the approach used in the work of Gluckstern et al. [1990] of manipulating the stationary expressions for αm,xx−1 or αm,yy−1 to a form similar to αe,zz−1 needed certain symmetry conditions imposed on the aperture.

2. Proof

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Proof
  5. 3. Conclusion
  6. Acknowledgments
  7. References

[5] The two principal magnetic polarizabilities are found from the solutions of the following magnetostatic boundary value problems formulated in terms of two integral equations [Latham, 1972; Gluckstern et al., 1990]:

  • equation image
  • equation image

where the constants cx and cy are determined by the constraints ∫AgdS = 0 and ∫AhdS = 0, equation image is a two-dimensional position vector, g and h are normalized vertical components of the magnetic fields in the aperture A, i.e., g = Hz/H0x, h = Hz/H0y. The magnetic polarizabilities are given by [Latham, 1972; Gluckstern et al., 1990; Lee, 1986]:

  • equation image
  • equation image

Multiplying (3) by g and h, respectively, integrating over A and noting that the terms corresponding to cx and cy drop out because of the constraints, we obtain the following stationary representations

  • equation image
  • equation image

where

  • equation image
  • equation image

which can be expanded in the following series [Magnus et al., 1966; Arfken, 1985]:

  • equation image
  • equation image

with

  • equation image
  • equation image

and ɛn is the Neumann number with ɛn = 1 for n = 0 and ɛn = 2 for n ≥ 1. Because K has the special form given by (7) we will be able to show that the stationary expressions (5) are minimums when g and h are solutions of the integral equations (3).

[6] Let X0 denote the true value of either of equations (5) when the solution g0 (or h0) of (3) is used as the trial function in (5), and X denote the value for any other trial function satisfying the constraint ∫AgdS = 0 (or ∫AhdS = 0). We will now prove XX0 following the same procedure given in the work of [Schwinger and Saxon, 1968; Borgnis and Papas, 1955]. Since

  • equation image
  • equation image

we have, after expanding the squares,

  • equation image

Multiplying (3) by the trial function g and integrating over A gives

  • equation image

recalling that g0 is the solution of (3). The sum in (8) is then simply X0, since

  • equation image

Thus (8) gives

  • equation image

or

  • equation image

i.e., any trial function other than g0 (or h0) in (5) will give a value greater than or equal to the true value.

[7] The electrostatic boundary value problem can be formulated in terms of the differential-integral equation [Latham, 1972; Gluckstern et al., 1990]

  • equation image

or

  • equation image

where K is given by (6) and f is the electric potential in the aperture divided by the external field E0z. To ensure convergence one may replace K(equation image, equation image′) by K(equation image, equation image′, z) = [(xx′)2 + (yy′)2 + z2]−1/2/π, and let z [RIGHTWARDS ARROW] 0 after integration by parts. The electric polarizability is given by [Latham, 1972; Gluckstern et al., 1990; Lee, 1986]

  • equation image

The stationary representation for αe,zz can be found from (11) and (12) as

  • equation image

Integrating by parts and noticing that f vanishes at the boundary of the aperture, we have

  • equation image
  • equation image

which has been cast in the form of (5) if we use equation image for g and equation image for h and realize that

  • equation image
  • equation image

Notice that ∫Aequation imagedS = 0, ∫Aequation imagedS = 0 satisfy the constraints ∫AgdS = 0, ∫AhdS = 0. With (5), (10), (14), and (15) the proof for the conjecture (1) is completed.

3. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Proof
  5. 3. Conclusion
  6. Acknowledgments
  7. References

[8] Using the tangential aperture electric field (Ex, Ey) as the trial function for the normal component of the aperture magnetic field in the stationary minimum representation of the inverse of the magnetic polarizabilities (i.e., equation image for g and equation image for h) we have thus proved the conjecture expressed in (1) with the help of (10). When these electric trial functions are proportional to the actual solutions of the magnetic integral equations (3) the equality sign of (1) holds. For the simple case of a circular aperture g (or h) is indeed proportional to equation image (or equation image).

[9] As a corollary for a plane perfectly conducting disk of any shape we have

  • equation image

where the z axis is perpendicular to the plane of the disk; Mzz and Pxx, Pyy are the magnetic and electric polarizabilities (Mzz is negative for the perfectly conducting disk).

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Proof
  5. 3. Conclusion
  6. Acknowledgments
  7. References

[10] Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Proof
  5. 3. Conclusion
  6. Acknowledgments
  7. References
  • Arfken, G. (1985), Mathematical Methods for Physicists, 3rd ed., 921 pp., Academic, San Diego, Calif.
  • Borgnis, F. E., and C. H. Papas (1955), Randwertprobleme Der Mikrowellenphysik, pp. 122124, Springer-Verlag, New York.
  • Gluckstern, R. L., R. Li, and R. K. Cooper (1990), Electric polarizability and magnetic susceptibility of small holes in a thin screen, IEEE Trans. Microwave Theory Tech., 38(2), 186192. (Correction, IEEE Trans. Microwave Theory Tech., 38(10), 1529–1530, 1990.).
  • Latham, R. W. (1972), Small holes in cable shields, Interact. Note 118, Air Force Res. Lab., Albuquerque, N. M.
  • Lee, K. S. H. (Ed.) (1986), EMP Interaction: Principles, Techniques, and Reference Data, Taylor and Francis, Philadelphia, Pa.
  • Lee, K. S. H. (1987), Relations between electric and magnetic polarizabilities and other related quantities, Radio Sci., 22(7), 12351238.
  • Magnus, W., F. Oberhettinger, and R. P. Soni (1966), Formulas and Theorems for the Special Functions of Mathematical Physics, 487 pp., Springer-Verlag, New York.
  • Schwinger, J., and D. S. Saxon (1968), Discontinuities in Waveguides, pp. 6668, Gordon and Breach, Newark, N. J.