feed-gap height (g) suggests that the feed-gap height of 1 mm can yield the widest BW. It is shown that the proposed antenna can operate in lower band from 1525 to 2440 MHz continuously, which may find applications in many wireless communication systems such as GPS, DCS, PCS, UMTS, and IMT-2000. ACKNOWLEDGMENTS This work was supported by the National Science Council of the Republic of China under grant NSC 94 –2213-E-006 – 045. REFERENCES 1. S. Dey, K.A. Jose, C.K. Aanandan, P. Mohanan, and K.G. Nair, Wideband printed dipole antenna, Microwave Opt Technol Lett 4 (1991), 417– 419. 2. G. Kumar and K.P. Ray, Broadband microstrip antennas, Artech House, Norwood, 2003. 3. K.L. Wong, Compact and broadband microstrip antennas, Wiley, New York, 2002. 4. H.M. Chen, Y.F. Lin, C.C. Kuo, and K.C. Huang, A compact dual- band microstrip-fed monopole antenna, IEEE Antennas Propag Soc Int Symp 2 (2001), 124 –127. 5. Y.F. Lin, H.D. Chen, and H.M. 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Parini, Analysis and design of UWB disc monopole antennas, IEE Seminar Ultra Wideband Commun Technol Syst Des (2004), 103–106. © 2007 Wiley Periodicals, Inc. SUPER-HYPER SINGULARITY TREATMENT FOR SOLVING 3D ELECTRIC FIELD INTEGRAL EQUATIONS Mei Song Tong and Weng Cho Chew Center for Computational Electromagnetics and Electromagnetics, Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 Received 20 October 2006 ABSTRACT: The super-hyper singularity treatment is developed for solving three-dimensional (3D) electric field integral equations (EFIE). EFIE usually takes two forms: one of which includes a super-hyper sin- gular kernel generated by the double gradient of the Green’s function. So far, there is no way to evaluate these super-hyper singular integrals needed for constructing the pertinent matrix equation. We apply the se- ries expansion of the Green’s function to the super-hyper singular ker- nel and derive closed-form expressions for their evaluations in the Cauchy principal-value sense. The derivation is based on the constant current approximation over a flat triangle patch, but it can be extended easily to higher-order approximations of the current. The scheme can be used to accurately calculate the near and self interaction terms in the impedance matrix for the method of moments (MoM), Nystro ¨m method or boundary element method (BEM). © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1383–1388, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop. 22443 Key words: singularity treatment; electric field integral equation; Cauchy principal value 1. INTRODUCTION The widely-used electric field integral equation (EFIE) in electro- magnetic (EM) scattering and radiation problems assumes two forms with different kernels: one includes the gradient of the Green’s function times the divergence of current (first form); the other includes the double gradient of the Green’s function times the current (second form). These two forms are equivalent to each other, demonstrable using integration by parts. The first form has a 1/R 2 singularity in the kernel for self-terms, where R is the distance between the source point and the obser- vation point. This singularity, together with 1/R singularity in the Green’s function in the kernel, can be efficiently handled by singularity subtraction technique and Duffy’s method [1–7]. Therefore, the first form is more widely used, and because of the divergence of current in the kernel, a linear approximation for the current expansion is often used. The second form is free of the divergence of current. This form allows constant current approximation over a discretized patch with great simplicity. It also provides a simple interface with fast algorithms, such as the fast multipole algorithm (FMA) [8], be- cause it associates an unknown with a point in space, where this locality improves FMAs efficiency. However, this form includes a 1/R 3 super-hyper singularity in the kernel generated by the double gradient of the Green’s function. Due to the higher degree of the singularity, there is no direct method to handle such a kernel. One usually changes the kernel back to the first form to evaluate the self-terms and uses the second form just for far-interaction terms in the Nystro ¨ m method [9 –11]. This treatment is inefficient and is not convenient when working with FMA. Under some circumstances, it is obviously very desirable to apply the second form without changing the kernel, e.g., in de- signing a localized Nystro ¨ m method. The problem, however, is the Figure 8 Measured peak gain of the planar monopole antenna: (a) 1.5–2.7 GHz; (b) 4 –5.2 GHz; W = 23 mm, D = 5 mm, S = 0 mm, g = 1 mm DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 6, June 2007 1383