CAD OF RECTANGULAR MICROSTRIP ANTENNAS David R. Jackson, Stuart A. Long, Jeffery T. Williams, and Vickie B. Davis Department of Electrical and Computer Engineering University of Houston Houston, TX 77204-4005 Publication information: Chapter 5 of Advances in Microstrip and Printed Antennas, Kai Fong Lee and Wei Chen, Eds., John Wiley, 1997.15.1 INTRODUCTION This chapter develops simple CAD formulas for the rectangular microstrip patch antenna. The CAD formulas are closed-form approximate expressions that describe the basic properties of the patch antenna. CAD formulas are presented for the resonance frequency, input resistance at resonance, radiation efficiency, bandwidth, and directivity. With the exception of the formulas for resonance frequency, all of the CAD formulas are derived from accurate analytical approximations of exact formulas, and are therefore not simply empirical in nature. The CAD formulas account for radiation into space, surface-wave radiation, dielectric loss, and conductor loss. The formulas in all cases become more accurate as the substrate thickness decreases. With the exception of input resistance, the CAD formulas are independent of the specific feeding mechanism. The formula for resonant input resistance applies for the specific case of a probe feed. A CAD formula for the probe reactance is also given. Although derived for the rectangular patch, the CAD formulas for bandwidth, radiation efficiency, and directivity may be used in an approximate fashion for the circular patch as well, by applying the formulas to an equivalent square patch having the same area as the original circular patch. In addition, a simple CAD model of the patch antenna is introduced for the calculation of input impedance. This model consists of a parallel RLC circuit (modeling the patch resonance) in series with a reactance (the probe reactance). This simple circuit, which follows directly from cavity model theory, can be used to calculate the input impedance at any frequency once the resonant input resistance, resonance frequency, and bandwidth of the patch are known. Hence, this simple circuit can be used directly with the above mentioned CAD formulas. Alternatively, the CAD formulas for radiated and dissipated power can be used to calculate an effective loss tangent of the substrate, which can be used directly in a cavity-model analysis of the patch. Results from the CAD formulas are compared with rigorous results from a spectral-domain analysis. Results show that the formulas for bandwidth, radiation efficiency, and directivity are very accurate, and even suffice for final design equations provided the substrate thickness is small enough so that ελrh /0≤ 0.10. The CAD formula for resonant input resistance loses accuracy sooner as the substrate thickness increases, but is accurate for substrate thicknesses in the range ελrh /0≤ 0.03. This chapter also discusses CAD formulas for the far-field radiation pattern of the rectangular patch on an infinite substrate. A comparison is made between formulas obtained from an electric current model and a magnetic current model. Both models are derived from different applications of the equivalence principle. In the electric current model the horizontal patch current is integrated to find the far-field pattern. In the magnetic current model the equivalent magnetic current at the boundary of the patch is used as the radiating source. It is demonstrated that both models yield the same result provided that the electric and magnetic currents in the two models correspond to the same cavity mode, and that the frequency of radiation is the resonance frequency of the cavity mode. Formulas obtained from both models are also presented for the far-field radiation pattern of a patch with a substrate that is truncated at the edges, and results are presented to show how these patterns differ from those for an infinite substrate.2 5.2 CAD MODEL FOR RECTANGULAR PATCH ANTENNA A probe-fed rectangular patch is shown in Fig. 1. In the cavity model [1]-[2], a perfect magnetic-wall boundary is placed on the edges of the patch to form an ideal closed cavity. In order to account for fringing, the effective length of the patch is taken as Le = L + 2 ΔL, where ΔL is an edge extension that is chosen to produce the correct resonance frequency for the dominant cavity mode of the patch. CAD formulas for ΔL are discussed in the next section. The resonance frequency of the dominant cavity mode, f0 , is related to the effective patch length by fcLrre02=εμ, (5.1) where c is the speed of light, 2.9979254 × 108 m/s, and εr and μr are the relative permittivity and permeability of the substrate, respectively. The effective width of the patch, We, is chosen as We = W + 2 ΔW, where the fringing width is chosen as [3] ΔWh≈FHGIKJln4π. (5.2) The fringing width is much less important than the fringing length, since it is the fringing length that determines the resonant frequency of the patch. The ideal magnetic wall allows for a simple modal expansion of the fields in terms of an eigenfunction expansion. The electric field Ez inside the cavity, as well as the eigenfunctions, are independent of z, provided the assumption is made that the probe current Jzis constant (this is one restriction that limits the validity of the model to substrates that are thin compared to a wavelength). The eigenfunctions ϕmnxy( , ) satisfy the eigenvalue equation ∇+ =220ϕϕmn mn mnxy k xy( , ) ( , ) . (5.3) The eigenfunctions are cavity modes that can exist inside the magnetic-wall cavity, and the eigenvalues kmn are the corresponding wavenumbers of the resonant cavity modes. Because of the ideal cavity approximation, the eigenfunctions are complete and orthogonal, and the total field excited by the feed may be expanded in terms of these functions. Furthermore, the eigenvalues are all real numbers, independent of the substrate loss tangent (they are actually the cutoff wavenumbers of a corresponding rectangular waveguide with magnetic walls). The eigenfunction expansion of the electric field inside the cavity is [1] Exy A xyzmnnmmn(,) (,)==∞=∞∑∑00ϕ, (5.4)3 where AjJkkmnzmnmn mn e mn=−ωμϕϕϕ(, )(,)122. (5.5) The inner product notation (f, g) denotes integration of the product f•g