## 1. Introduction

[2] Early applications of one dimensional periodicity in substrate and substrate-superstrate geometries for microstrip elements produced very useful results for microstrip antenna gain enhancement, radiation pattern shaping, as well as antenna efficiency and bandwidth improvement [*Alexopoulos and Jackson*, 1984; *Alexopoulos et al.*, 1985; *Yang and Alexopoulos*, 1987]. More recently, progress has been obtained with the introduction of periodic arrays of material blocks [*Yang et al.*, 1997a; *Yang*, 1999] or metallic implants [*Yang et al.*, 1997b; *Sievenpiper et al.*, 1999; *Kyriazidou*, 1999; *Contopanagos et al.*, 1999; *Kyriazidou et al.*, 2000, 2001] within multilayered dielectric structures, these structures being referred to as photonic band gap (PBG) or electromagnetic band gap (EBG) materials. These materials have enabled some key and novel contributions such as the realization of high impedance surfaces or magnetic walls [*Yang et al.*, 1997b; *Sievenpiper et al.*, 1999] and a variety of other applications [*Contopanagos et al.*, 1999; *Kyriazidou et al.*, 2000, 2001].

[3] Numerical methods and commercial software have provided the backbone for the modeling and characterization of these complex geometries, such as the method of moments [*Tsao and Mittra*, 1987], finite elements [*Zhang et al.*, 1999; *Coccioli et al.*, 1997], the combined FEM/MOM approach [*Gedney et al.*, 1992], the plane wave expansion techniques [*Leung and Liu*, 1990] and the mode-matching technique [*Chen*, 1970; *Huang et al.*, 1994]. A distinct semianalytical methodology was developed however which not only provides a better insight on the electromagnetic characteristics of these structures but it also results in a 2 orders of magnitude faster computational methodology [*Kyriazidou*, 1999; *Kyriazidou et al.*, 2000]. This is a significant approach for the characterization of the electromagnetic properties of general media with ordered metallic inclusions well beyond the use of effective medium theory. This method creates the fundamental analogies between natural and artificial (PBG) crystals through the development of the Lorentzian generator function *p*, henceforth referred to as the Kyriazidou-Cantopanagos-Alexopoulos (KCA) Lorentzian generator [*Alexopoulos et al.*, 2004]. Furthermore, it satisfies the Kramers-Kronig relations and Causality and it provides therefore a correct mechanism for the derivation of the effective wave impedance, index of refraction, and permittivity and permeability of the equivalent homogeneous bulk materials. In fact, this method provided the first analytical means to prove that a purely dielectric-metal structure yields a composite with magnetic as well as dielectric properties [*Contopanagos et al.*, 1999; *Kyriazidou et al.*, 2000]. The first step of this semianalytical procedure is the derivation of the polarizability of the implant elements. This was first demonstrated for metallic disk implants [*Contopanagos et al.*, 1999; *Kyriazidou et al.*, 2000].

[4] In this paper we consider a generalization of this method to metallic ring elements and the time dependance *e*^{jωt} is adopted. Metallic ring implants provide a variety of flexibilities, including metallic disks as a limiting case, and moreover the possibility of a gap variable impedance load in each ring implant which will result in reconfigurable materials, filters etc [*Alexopoulos et al.*, 2004] as well as in metamorphic structures [*Liu and Alexopoulos*, 2006; *Kyriazidou et al.*, 2006, 2007]. The latter initiated the concept of artificial magnetic conductors [*Alexopoulos et al.*, 1974a, 1974b; *Alexopoulos and Katehi*, 1979] or high impedance walls [*Sievenpiper et al.*, 1999]. A negative index of refraction has been reported in an ordered variable impedance loaded PEC ring media [*Liu and Alexopoulos*, 2006]. We will discuss the validity of deriving the effective parameters of layered artificial materials from the reflection and transmission coefficients. We will start from a general case and a more universal conclusion will be established for symmetric and nonsymmetric structures. We will begin the derivation of the electric and magnetic polarizabilities for a narrow planar annular ring which requires knowledge of the current distribution on the ring. For a narrow annular ring, the current distribution is obtained from that of a wire ring with a wire radius being a fourth of the width of the annular ring [*Balanis*, 1982, and references therein]. With this background we derive closed form expressions for the electric and magnetic polarizabilities and the corresponding shunt susceptances for different incident wave polarizations. We also obtain the composite medium's effective parameters such as effective wave impedance, refractive index, permittivity and permeability. Furthermore, we discuss the general ring band gap characteristics and we provide comparisons to metallic disk implants. A comparison of this methodology to the use of a commercial code such as Ansoft Designer is provided. Further we show that the effective parameters of the metamaterial composite can be obtained from *S*_{11} and *S*_{21} for centrosymmetric structures. Otherwise, it is possible to derive effective parameters for noncentrosymmetric composites only if the structure is semi-infinite.