CRITERIOS PARA EL ANÁLISIS DE LA INFORMACIÓN

Throughout this chapter we have shown the foundations of an IE-MoM technique for the EM modeling of printed circuits in bounded layered media.

The basis of this approach consists of the formulation of a canonical problem consisting of the radiation of an infinitesimal current inside bounded layered media. This defines the GF. On the other hand, using the equivalence principle, the printed circuits can be replaced by equivalent surface currents. In this way, the initial structure has been transformed into a problem of equivalent surface currents radiating in a bounded layered media. Using a IE technique in combination with the MoM these currents are obtained for the BCs imposed in the original problem. With this result any other field or derived magnitude can be then calculated, e.g. a multiport network characterization of the structure.

Due to the symmetry presented by a bounded layered media, the GF’s problem has been reduced into a two-dimensional transverse boundary problem, in coordinates u1, u2 and a one-dimensional transmission line problem in z direction. The former consists in deriving the eigensolutions or modes satisfying Helmholtz equation in the transverse contour with certain BCs, while the latter describes how these modes propagate in the layered media.

The formulation of the overall problem is thus based on two main ingredients which can be treated independently. On one side the resolution of a transverse boundary problem, meaning the derivation of modes and the solution to operators associated with the IE for a giving transverse contour. On the other side, an efficient resolution of the equivalent transmission line problem along the normal coordinate for a generic mode. These two auxiliary problems will be the object of, respectively, the two next Chapter 3 and Chapter 4 in the thesis.

References 21

References

[1] J. R. Mosig, “Integral-equation technique,” in Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, T. Itoh, Ed. New York: Wiley, 1989, ch. 3, pp. 133–213. [2] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [3] ——, Field Computation by Moment Methods. Florida: Krieger Publishing Company, 1968. [4] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. New

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[5] J. R. Mosig, R. C. Hall, and F. E. Gardiol, “Numerical analysis of microstrip patch antennas,” in Handbook of microstrip antennas. I, James and Hall, Eds. London: IEE electromagnetic waves series, 1989, ch. 8, pp. 393–453.

[6] I. Stevanovi´c, P. Crespo-Valero, and J. R. Mosig, “An integral-equation technique for solving thick irises in rectangular waveguides,” IEEE Trans. Microwave Theory Tech., vol. 54, no. 1, pp. 189–197, Jan. 2006.

[7] E. Miller, “A selective survey of computational electromagnetics,” IEEE Trans. Antennas Prop- agat., vol. 36, no. 9, pp. 1281–1305, Sept. 1988.

[8] R. E. Collin, Field Theory of Guided Waves, 2nd ed. Wiley-IEEE Press, 1990.

[9] N. Marcuvitz, Waveguide Handbook, ser. Radiation Laboratory. New York: McGraw Hill, 1941, vol. 10.

[10] V. Rumsey, “Reaction concept in electromagnetic theory,” Phys. Rev., vol. 94, pp. 1483–1491, June 1954.

[11] L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice Hall, 1973.

[12] D. Llorens del R´ıo, “Electromagnetic analysis of 2.5d structures in open layered media,” Ph.D. dissertation, Ecole Polytecnique F´ed´eral de Lausanne, Lausanne, 2005, th`ese No 3183 (2005). [Online]. Available: http://library.epfl.ch/theses/?nr=3183

[13] G. Eleftheriades and J. Mosig, “On the network characterization of planar passive circuits using the method of moments,” IEEE Trans. Microwave Theory Tech., vol. 44, no. 3, pp. 438–445, March 1996.

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3 Transverse Boundary Problem

3.1 Introduction

In this chapter we present the details of the IE-MoM formulation for the EM modeling of printed circuits in laterally bounded media. This scheme encompasses a wide range of prac- tical structures such as discontinuities in waveguides, planar circuits embedded in shielded multilayered media or even 2-D printed periodic structures. Each of the problems enumerated above has already been solved by other authors using IE-MoM techniques. Some examples can be found in the work of [1, 2] applied to waveguide iris filters, or the techniques for boxed printed circuits proposed in [3–5] or even the studies on infinite planar arrays or FSS in [6–8]. Nevertheless, our formulation introduces a unified approach applicable to all the aforemen- tioned problems. Our method is based on a modal representation of the transverse boundary problem and on the expansion of the equivalent surface currents by divergence-conforming BFs. It starts with general and simplified forms of the integral operators. Then it establishes a systematic procedure that particularizes the solution to a specific combination of transverse BC and type of BFs.

This chapter is organized as follows. The first section commences with a review of the IE- MoM formulation in connection with the techniques presented in §§2.3–2.5. The purpose is to give a precise picture of the type of integral expressions to be solved and the role they play within the overall IE-MoM technique. The next section develops the approach followed to achieve a generic solution for these integrals. The formulation is then applied to rectangular and circular perfect electric conductors BCs (i.e. waveguides) and periodic BC in order to prove the generality, simplicity and elegance of this technique. Finally, the validity of the approach is demonstrated confronting the results obtained using the presented technique and other references from the literature.