14. 15. 16. 17. Large 2D Scatterers,’’ 11th Annual Re? iew of Progress in Applied Computational Electromagnetics, p. 606, 1995. S. Stein, ‘‘Addition Theorems for Spherical Wave Functions,’’ Q. Appl. Math. , Vol. 19, No. 1, 1961, pp. 15 24. W. C. Chew, Wa? es and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York, 1990, reprinted by IEEE Press, 1995. W. K. Tung, Group Theory in Physics, World Scienti? c Publishing Co. , Singapore, 1984. J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. this article, and characteristics of the antenna are experimentally investigated. 2.

DESIGN CONSIDERATIONS AND EXPERIMENTAL RESULTS 1997 John Wiley & Sons, Inc. CCC 0895-2477r97 A COMPACT MEANDERED CIRCULAR MICROSTRIP ANTENNA WITH A SHORTING PIN 1 Kin-Lu Wong,1 Chia-Luan Tang,1 and Hong-Twu Chen 2 Department of Electrical Engineering National Sun Yat-Sen University Kaohsiung, Taiwan 804, Republic of China 2 Department of Electrical Engineering Chinese Military Academy Fong-Shan, Taiwan 830, Republic of China Recei? ed 8 January 1997 ABSTRACT: By using a shorting pin and meandering the circular patch, a compact circular microstrip antenna with a patch size of less than 10% of the con? ntional circular patch antenna can be easily obtained. The design of such compact circular microstrip antennas is described, and experimental results are presented and discussed. 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 147 149, 1997 Key words: compact circular microstrip antenna; meandered circular patch; shorting pin 1. INTRODUCTION Figure 1 shows the con? guration of the short-circuited, meandered circular microstrip antenna. The circular patch is short-circuited at the edge with a shorting pin, and three narrow slots of the same length Z l . and width Z w . are cut in the patch.

The shorting pin makes the circular patch resonate at a much lower frequency w2x, as compared with a conventional circular patch of the same size. The narrow slots meander the patch and thus increase the effective electrical length of the patch. These two factors effectively reduce the required disk size for the antenna to be operated at a given frequency. Based on this design concept, the short-circuited circular microstrip antennas with different slot lengths were constructed. The circular patch has a radius Z d . of 7. 5 mm, and a shorting pin of radius Z r s . 0. 4 mm is placed near the patch edge at R s d s s 6. 5 mm.

The patch substrate has a dielectric constant Z r . of 4. 25 and a thickness Z h. of 1. 6 mm. Figure 2 shows the measured resonant frequency versus the slot length in the short-circuited circular patch. Results clearly indicate that, with increasing slot length, the resonant frequency of the meandered patch decreases. It is also found that the slot width has relatively little effect on the resonant frequency. From the results for the case of l s d, the circular patch of radius 7. 5 mm has a resonant frequency as low as 1. 652 GHz. As for a conventional circular patch antenna Zwithout a shorting pin and the slots in the patch. the radius of the circular patch to be operated at 1. 652 GHz needs to be about 25. 2 mm Zwith the same substrate material.. That is, the patch size is reduced to be about 9%, as compared with the conventional circular patch of the same operating frequency. When the slot length increases, the resonant frequency of the patch can be reduced further, which can reduce the patch size even further at a given frequency. The return loss for the case l s d is plotted in Figure 3. In order to obtain a good matching condition, the feed position is placed close to the shorting pin.

When the feed position is away from the shorting pin, the resonant input resistance is Because of the miniaturization of personal communication equipment, the demand for small antennas has increased. To meet such a requirement, several designs of the compact microstrip antennas have recently been proposed w1x. One of the effective ways to reduce the patch size of the microstrip antenna is to introduce a shorting pin at the edge of the patch w2x. By meandering the patch w3x, the electrical length of the patch can be increased, which makes the patch resonate at a much lower frequency.

That is, for a given operating frequency, the antenna size can be made much smaller. Such a design technique has been demonstrated for a rectangular microstrip patch antenna w3x, and signi? cant reduction in the antenna size has also been shown. In this article we present an experimental study, using a technique similar to that used for a circular microstrip antenna. The circular patch is shortcircuited at its edge with a shorting pin, and several slots are cut in the patch to force the excited patch surface current to travel a much longer path.

In this case the short-circuited, meandered circular patch will resonate at a much lower frequency, as compared with a conventional circular microstrip antenna of the same size. The design considerations of such compact circular microstrip antennas are described in Figure 1 The geometry of a meandered circular microstrip antenna with a shorting pin MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997 147 Figure 2 Resonant frequency versus slot length Z l . in the circular patch. d s 7. 5 mm, d s s 6. 5 mm, r s s 0. 4 mm, d p s 5. 2 mm, r p s 0. 65 mm, w s 1 mm, h s 1. mm, r s 4. 25 Figure 4 E- and H-plane radiation patterns at resonant frequency for the antenna with l s d. Other antenna parameters are given in Figure 2 Figure 3 Return loss versus frequency for the antenna with l s d. Other antenna parameters are given in Figure 2 quickly increased. For good matching the feed position is chosen at R s d p s 5. 2 mm, with the probe having a radius Z r p . of 0. 65 mm. The antenna bandwidth determined from 10-dB return loss is found to be about 1. 57%, which is less than that Z1. 9%. of a conventional circular microstrip antenna of the same operating frequency.

This reduction in the antenna bandwidth is expected w2x, due to the reduced antenna size. However, it is interesting to ? nd that, with slots in the patch, the reduction in the antenna bandwidth is smaller than the case of the short-circuited circular patch without slots w2x, where the antenna bandwidth is reduced by about 33% as compared with the conventional patch antenna. This is probably because the slots in the patch introduce an additional capacitive reactance that compensates the induc- Figure 5 E- and H-plane radiation patterns at resonant frequency for the antenna without slots Z l s 0..

Other antenna parameters are the same as in Figure 2 148 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997 tive reactance contributed from the shorting pin and the probe; thus a wider antenna bandwidth could be achieved w4x. The radiation patterns for the compact antenna with l s d are presented in Figure 4. It is seen that the radiation patterns remain broadside. Figure 5 shows the case for the antenna without slots Z l s 0. , where the patch resonates at about 1. 923 GHz. Similar broadside radiation is also observed.

However, because of the increase of the patch surface current component perpendicular to the main excitation direction w3x, the cross-polarization radiation in the H plane is increased Zcf. Figures 4 and 5.. On the other hand, the cross-polarization level in the E-plane is still in an acceptable level of less than y20 dB. Finally, it should also be noted that, due to the antenna size reduction, the antenna gain of a compact microstrip antenna will be lower than that of a conventional microstrip antenna operated at the same frequency w2x. 3. CONCLUSIONS INTRODUCTION

The design of a compact meandered circular microstrip antenna with a shorting pin has been described. Experimental results have been presented and discussed. Results indicate that, by combining short-circuiting and meandering of the circular patch, the antenna size can be reduced to be less than 10% that of a conventional circular microstrip antenna operated at the same frequency. This great reduction in antenna size makes it useful for applications where antenna size is a major concern. REFERENCES 1. K. Hiraswa and M. Haneishi, Analysis, Design, and Measurement of Small and Low-Pro? le Antennas, Artech House, London, 1992. 2. R.

Waterhouse, ‘‘Small Microstrip Patch Antenna,’’ Electron. Lett. , Vol. 31, April 13, 1995, pp. 604 605. 3. S. Dey and R. Mittra, ‘‘Compact Microstrip Patch Antenna,’’ Microwa? e Opt. Technol. Lett. , Vol. 13, Sept. 1996, pp. 12 14. 4. T. Huynh and K. F. Lee, ‘‘Single-Layer Single-Patch Wideband Microstrip Antenna,’’ Electron. Lett. , Vol. 31, Aug. 3, 1995, pp. 1310 1312. 1997 John Wiley & Sons, Inc. CCC 0895-2477r97 Planar waveguide structures containing a periodic corrugation along the waveguide have long been used for various applications, such as distributed feedback re? ectors w1x, optical ? lters, and leaky wave antennas w2x.

The electromagnetic problem of a straight uniform periodic structure has been analyzed by various methods, such as the coupled mode theory w1x, the ? nite element method w3x, the method of lines w4x and the rigorous mode matching method w2x. With these basic analyses, the properties of Bragg waveguide gratings can be considered to be well understood. In recent years, many authors have conducted work on different types of guided-wave gratings, such as sinusoidal Zor triangular . pro? le gratings and tapered period interval Zor pro? le depth, index contrast, and so on. gratings to improve and tailor the performance of active or passive devices.

Thus, it is mandatory to analyze these periodic and aperiodic structures with ef? cient and accurate numerical methods. The combination of the mode-matching method with multimode network theory has been shown to be very ef? cient for analyzing the nonuniform dielectric waveguides for integrated optics applications w5x. In this Letter, we will utilize this combined method, which can accurately include both fundamental modes and higher-order modes, to analyze the scattering characteristics of nonuniform waveguide gratings for a millimeter-wave system or for optical integrated circuit applications.

Numerical results presented for different types of nonuniform Bragg gratings, such as concave waveguide gratings and bent waveguide gratings, provide guidelines for optimally designing or improving the performance of optoelectronic devices. METHOD OF ANALYSIS AN EQUIVALENT NETWORK METHOD FOR THE ANALYSIS OF NONUNIFORM PERIODIC STRUCTURES Jin Jei Wu Institute of Electro-Optical Engineering National Chiao Tung University Hsinchu, Taiwan, Republic of China Recei? ed 5 No? ember 1996; re? ised 10 February 1997 ABSTRACT: The characteristics of guided wa? s scattered by nonuniform wa? eguide gratings are systematically in? estigated with the use of an equi? alent network method. This procedure is based on a combination of the multimode network theory and the rigorous mode-matching method. Conca? e Bragg gratings and bent wa? eguide gratings are taken as examples to demonstrate the present approach, and numerical results are gi? en to illustrate their potential for millimeter-wa? e and optical integrated circuit applications. 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 149 153, 1997.

Key words: conca? e Bragg grating; staircase approximation As an illustration of this method, consider a two-dimensional structure as depicted in Figure 1Za. , which shows a parallelplate waveguide ? lled with a concave grating. The scattering of guided waves by nonuniform waveguide gratings Zsuch as concave waveguide gratings and bent waveguide gratings. cannot be analyzed exactly, even though the geometric pro? le is simple. Thus a useful approximation is needed. We utilize the staircase approximation of the continuous nonuniform pro? e of the waveguide gratings; this is a discretization in geometry. Figure 1Zb. shows the staircase approximation of a concave Bragg grating in the neighborhood of x i . Therefore, the whole structure can be approximated by a sequence of basic units, each consisting of a waveguide discontinuity and a uniform partially ? lled parallel-plate waveguide. According to the literature w5x, the general ? eld solution in each uniform waveguide region can be expressed in terms of the complete set of mode functions A n4, which are the solutions of a Sturm Liouville eigenvalue problem.

For the TE fundamental-mode incident case, the tangential ? eld in each uniform waveguide region can be represented by Ez Z x, y . s H y Z x, y . s Y VnZ x . n n Z y. , Z y. , Z1. Z2. Y In Z x . n n where Vn and In are the equivalent voltage and current of the nth mode, respectively. By matching the tangential ? eld components at the ith discontinuity, the scattering of modes by a waveguide discontinuity can be quanti? ed by the analysis of MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 3, June 20 1997 149