1510 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,VOL. 50, NO. 6, JUNE 2002 Full-Wave Analysis of a Wide Class of Microstrip Resonators Fabricated on Magnetized Ferrites With Arbitrarily Oriented Bias Magnetic Field Germán León, Rafael R. Boix, Member, IEEE, and Francisco Medina, Senior Member, IEEE Abstract—A numerical code has been developed for the full-wave determination of the resonant frequencies and quality factors of microstrip patches with right-angle corners of arbitrary shape in the case in which the substrate of the patches is a mag- netized ferrite with arbitrarily oriented bias magnetic field. The code is based on the solution of an electric-field integral equation by means of Galerkin’s method in the spectral domain. The evaluation of the infinite integrals arising from the application of the numerical method is efficiently carried out by means of a technique based on the interpolation of the spectral dyadic Green’s function. The numerical results obtained indicate that microstrip patches fabricated on ferrite substrates present cutoff frequency regions in which resonances cannot occur owing to the excitation of magnetostatic modes. The limits of these cutoff regions are shown to be dependent on the orientation and the magnitude of the bias magnetic field, on the shape of the patches, and even on the nature of every particular resonant mode. The numerical results also show that the resonant frequencies of microstrip patches on magnetized ferrites can always be tuned over a wide frequency range provided the orientation of the bias magnetic field is suitably chosen. Index Terms—Magnetic tuning, magnetostatic modes, micro- strip resonators, microwave ferrites, spectral-domain approach. I. INTRODUCTION R ESONANT microstrip patches can be used either as an- tennas or as components of oscillators and filters in mi- crowave integrated circuits. Although the most conventional mi- crostrip patches are the rectangular and circular patches, other geometries such as the rectangular ring [1], the H-shaped patch [1], and the meander-shaped patch [2] have proven to be useful because of their size reduction capabilities (e.g., in the design of antenna arrays, antennas for personal communication systems, ). This means that the algorithms developed for the analysis of resonant microstrip patches should cover a spectrum of ge- ometries as wide as possible. Apart from their shape, the na- ture of the substrate of microstrip patches is another interesting degree of freedom for the designer of circuits and antennas. Although the most commonly used substrate materials are di- electrics, magnetized ferrites have proven to have potential ap- plication as substrates of microstrip patches. For instance, sev- Manuscript received January 23, 2001. This work was supported by the Comisión Interministerial de Ciencia y Tecnología, Spain, under Project TIC98-0630. The authors are with the Microwave Group, Department of Electronics and Electromagnetism, School of Physics, University of Seville, 41012 Seville, Spain (e-mail: boix@cica.es). Publisher Item Identifier S 0018-9480(02)05206-7. eral researchers have reported that resonant microstrip patches printed on ferrite substrates can be used in the fabrication of tunable band rejection filters [3] and tunable bandpass filters [4], [5]. Also, measurements have shown that the operating res- onant frequency of microstrip antennas printed on ferrite sub- strates can be varied over a wide frequency range by adjusting the bias magnetic field [6]. Apart from that, ferrite substrates can be used for reducing the radar cross section of microstrip antennas under certain conditions [7]–[9] and for achieving lin- early, as well as circularly polarized microstrip antennas with a single feed [10]–[12]. Finally, it should be pointed out that when ferrite materials are used as substrates of microstrip phased ar- rays, wide-angle impedance matching can be obtained by dy- namically adjusting the bias magnetic field with scan angle [11], [13]. In this paper, the authors present an algorithm for the deter- mination of the resonant frequencies and quality factors of mi- crostrip patches with right-angle corners of arbitrary shape in the case in which the patches are fabricated on magnetized fer- rites with arbitrarily oriented bias magnetic field. This algorithm is based on an efficient application of the spectral-domain ap- proach (SDA) [14], [15]. The study carried out in this paper is an ambitious generalization of that published in previous pa- pers [3], [16], [17] where the SDA was applied to the full-wave analysis of microstrip resonators of both circular shape [3], [17] and rectangular shape [16] fabricated on normally biased ferrite substrates. There is also an additional related paper in which rectangular microstrip resonators on in-plane biased ferrite sub- strates were analyzed by means of the cavity model [18]. How- ever, the drawback of the cavity model is that it fails to explain how the resonances of microstrip patches on ferrite substrates are affected by the excitation of magnetostatic modes along the substrate [17]. Fortunately, the SDA is capable of accounting for the excitation of magnetostatic modes since the information of these modes is included in the spectral dyadic Green’s function (in fact, the propagation constants of the magnetostatic modes are complex poles of the aforementioned spectral Green’s func- tion [9], [17]). Concerning the contents of this paper, Section II briefly describes the application of the SDA to the full-wave analysis of microstrip resonators fabricated on ferrite substrates. In this section, a powerful technique is explained, which makes possible the fast computation of infinite integrals arising from the application of the SDA. In Section III, numerical results are presented for the resonant frequencies and quality factors of microstrip patches of different shapes (rectangular, H-, and 0018-9480/02$17.00 © 2002 IEEE