Slot Resonator with PBG Layers Sérgio Pinheiro dos Santos and Humberto César Chaves Fernandes Abstract  This paper shows the study and results of the three layers slot resonator with Photonic Band Gap (PBG) substrate. The full wave Transverse Transmission Line (TTL) method is used to calculate the complex resonant frequency, using double spectral variables. This method presents efficient and concise calculus in the determination of the structure parameters. Numerical results for the resonant frequency of three Layer’s substrate slot antenna, for s and p polarization of PBG material, for different thickness substrate and resonator length are presents. Key Worlds  TTL Method, PBG-Photonic Band Gap, Multilayer, Slot antenna. I. INTRODUCTION Multilayer microwave resonator shows complexity when more accuracy methods are applied [1]–[2]. In this paper the concise full wave Transverse Transmission Line (TTL) method is used to calculate the complex resonant frequency of three Layer’s substrate slot antenna with Photonic Band Gap (PBG) substrate. Photonic band gap material exhibits energy band gap. In a photonic crystal if a photon has energy in the band gap it not propagates through of the material in the light direction. This PBG has periodic array of cylinder air with diameters and spacing less than one light wavelength [3]- [6]. This substrate can improve the bandwidth and eliminate propagation of undesirable modes and increases the antenna efficiency. The choice of the structure has objective expand application the method in multilayer resonators because the increase of bandwidth for 4%[7] if maintain a fine substrate (h/λ 0 <0.01) and low permittivity. This paper presents the comportment of the microwave resonator in function aperture of slot. Fig. 1 shows one three Layer’s substrate slot resonator. The thickness of the patch resonator and the ground plane are perfect conductors __________________________________ H. C. C. Fernandes humbeccf@ct.ufrn.br and S.P. Santos, are with Department of Electrical Engineering - Federal University of Rio Grande do Norte, P.O.Box: 1583, 59078-970-Natal RN, Brazil Fax/ Phone: +55 84 2153732. This work was supported by CNPQ. (a) (b) Fig. 1 – (a) Cross section of the three Layer’s substrate slot resonator, (b) Cut of slot resonator region. II. THEORY II.1 - TTL METHOD The general equations of the fields in the TTL method are obtained after using the Maxwell’s equations,as: (1) were the index ‘T’ shows the transversal components directions (x, z): z E x E E zi xi Ti ˆ ˆ G G G + = (2.1) z H x H H zi xi Ti ˆ ˆ G G G + = (2.2) z z x x z x z x T ˆ ˆ ˆ ˆ ∂ ∂ + ∂ ∂ = ∇ + ∇ = ∇ (2.3) The equations are used for the analysis in the spectral domain, in the " x " and " z "directions. Therefore it should be applied to the field equations of double Fourier transform defined as: dz dx e e z y x f y f z j x j k n k n ∫∫ ∞ ∞ - ∞ ∞ - ⋅ ⋅ = β α β α ) , , ( ) , , ( ~ (3) The electromagnetic fields are obtained:       + - + = yi k yi n i i xi H E y j k E ~ ~ 1 ~ 2 2 ωμβ ∂ ∂ α γ (4.1)       - - + = yi n yi k i i zi H E y j k E ~ ~ 1 ~ 2 2 ωμα ∂ ∂ β γ (4.2)                   ∇ ∂ ∂ +           - × ∇ + =           yi yi T yi yi T i i Ti Ti H E y E H j k H E G G G G G G 1 2 2 1 ε μ ω γ