the series of Bessel functions with 16 terms (Ns � 16) is sufficientfor limited input range 0.1 � � � 10. The asymptotic expression[12] presents better responses for large values of �. The DSFNNmodel presents mean error around 0.00005 dB.

5. CONCLUSIONS

In this article concise and easily computable directivity expres-sions of some circular-shape antennas were revisited and linkedthrough of the Q-integral notation. The simplest OTF-NN meth-odology was applied to found approximations of Q-integrals usingoptimal transfer functions. From several tested neural networks thebest one was selected. The OTF-NN named DSFNN that uses thedamped-sinusoid transfer functions was able to learn accuratesolutions of Q-integrals with small network complexities.

The computational cost of the DSFNN model was comparablewith approximate expressions and speedup (around 10�) the se-ries of Bessel functions approach. In addition, the accuracy ofDSFNN responses is in an intermediary range, between approxi-mate closed-form expressions and series of Bessel functions.

ACKNOWLEDGMENTS

We thank the Brazilian Research Agencies CNPq and CAPES forpartial financial support.

REFERENCES

1. S.V. Savov, An efficient solution of a class of integrals arising inantenna theory, IEEE Antennas Propag Mag 44 (2002), 98–101.

2. J.D. Mahony, Circular microstrip-patch directivity revisited: An easilycomputable exact expression, IEEE Antennas Propag Mag 45 (2003),120–122.

3. J.D. Mahony, A note on the directivity of a uniformly excited circularaperture in an infinite ground plane, IEEE Antennas Propag Mag 47(2005), 87–89.

4. J.D. Mahony, A comment on Q-type integrals and their use in expres-sions for radiated power, IEEE Antennas Propag Mag 45 (2003),127–128.

5. S.V. Savov, A comment on the radiation resistance, IEEE AntennasPropag Mag 45 (2003), 129.

6. M. Abramowitz and I.A. Stegun, Handbook of mathematical func-tions, Dover, New York, 1970.

7. I. Gradshteyn and I. Ryzhik, Tables of integrals, series and products,Academic Press, New York, 1965.

8. J.D. Mahony, Approximations to the radiation resistance and directiv-ity of circular-loop antennas, IEEE Antennas Propag Mag 36 (1994),52–55.

9. J.D. Mahony, Approximate expressions for the directivity of a circularmicrostrip-patch antenna, IEEE Antennas Propag Mag 43 (2001),88–90.

10. K. Verna and Nasimuddin, Simple expressions for the directivity of acircular microstrip antenna, IEEE Antennas Propag Mag 44 (2002),91–95.

11. K. Verna and Nasimuddin, Simple and accurate expression for direc-tivity of circular microstrip antenna, J Microwaves Optoelectron 2(2002), 70–74.

12. C.A.Balanis, Antenna theory analysis and design, Wiley-Interscience,New Jersey, 2005.

13. M. Riedmiller and H. Braun, A direct adaptive method for fasterbackpropagation learning: The rprop algorithm, In: Proceedings of theIEEE International Conference on Neural Networks, San Francisco,EUA, 1993, pp. 586–591.

14. W. Duch and N. Jankowski, Optimal transfer function neural net-works, in Proceedings of the European Symposium on Artificial Neu-ral Networks, Bruges, Belgium, 2001, pp. 101–106.

© 2007 Wiley Periodicals, Inc.

PROPERTIES OF MULTIPLE GAPSMICROSTRIP FILTER WITH FRACTALMETALLIC PATTERNS

Sheng Wang,1,2,3,4 Ming Sen Guo,1,2,3,4 H. L. W. Chan,3,4 andXing-Zhong Zhao1,2

1 Department of Physics, Wuhan University, Wuhan, 430072, China;Corresponding author: xzzhao@whu.edu.cn2 Center of Nanoscience and Nanotechnology, Wuhan University,Wuhan, 430072, China3 Department of Applied Physics, Hong Kong Polytechnic University,Hung Hom, Kowloon, Hong Kong4 Materials Research Center, Hong Kong Polytechnic University,Hung Hom, Kowloon, Hong Kong

Received 29 March 2007

ABSTRACT: In this study, we demonstrate a multiple-gaps resonatorfilter, consisting of a multilayer structure with a fractal metallic pattern,that can tune the resonant frequencies of the multiple gaps. It is foundthat the resonant frequency of the gaps shifts to low frequencies with theincrease of fractal patterns dimension. The experimental observationsare in good agreement with the simulated results using the finite differ-ence time domain method. © 2007 Wiley Periodicals, Inc. MicrowaveOpt Technol Lett 49: 2726–2728, 2007; Published online in Wiley In-terScience (www.interscience.wiley.com). DOI 10.1002/mop.22882

Key words: multiple gaps; fractal patterns; tunable ability

1. INTRODUCTION

The development of tunable microstrip filter is an important area inmicrowave electronics. The tunability of microstrip resonant filtershas been reported by many researchers [1], who show that theresonant frequency of the filter can be tuned by adding a ferritesubstrate to the microstrip configuration and exciting a variablecurrent bias in a coil that surrounds this modified geometry [2, 3].In the last decade, high temperature superconductor (HTS) hadalso been applied to the microstrip resonant filter [4]. By thismethod, the resonant frequency of the filter can be shifted towardlower values as the additional DC bias decreased. This approachcan be used to control tuning in any microstrip filter if the initialdesign is based on a resonant frequency, which is somewhat higherthat the desired resonant frequency. Both methods share somecommon features (e.g., tunability of resonant frequency by chang-ing the dielectric constant of substrate) but they use differentmethods to tune the resonant frequency and have some shortcom-ing in the application. In the first method, the tunability is only 4%and the variable current bias must be larger than 100 V. In thesecond method, although the tunability of the filter is higher thanthe first method, the operating temperature must be lower thanroom temperature, which makes it hard to use in practical appli-cations. Recently, a type of structure [5-7], consisting of a metallicplanar fractal and a metal sheet separated by a thin dielectric layer,has been fabricated, which can exhibit multiple stop and passbands for electromagnetic (EM) waves. In this article, the fractalmetal patterns are inserted into the two layers of high-frequencysubstrates-FR4 (�r � 4.5) using printed circuit board technology.From the experiments and finite difference time domain (FDTD)[8] simulations, we found that the band gaps of microstrip resonantfilter in the transmission characteristics are downshifted to lowerfrequency when the fractal patterns dimension is increased.

2. THE MICROSTRIP FILTER DESIGN

Figure 1 shows the cross section of a microstrip resonator filterwith a fractal metallic pattern.

2726 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 11, November 2007 DOI 10.1002/mop

Using printed circuit board technology, the resonator filter isconstructed by end coupling a metallic fractal pattern layer that isinserted between two similar dielectric substrate layers (�r � 4.5,see Fig. 2)

The fractal metallic pattern [7] layer is generated from a me-tallic line length, defined as the first level of the fractal structure,and placed parallel to the x axis in the xy plane. The (k � 1)th levelstructure contains 2k lines, with the midpoint of each perpendicu-larly connected to the ends of the kth level lines, is scaled from thatof the kth level line by a factor of 2 if k is an even (odd) number.

3. RESULTS AND DISCUSSION

The measurements are performed with the Anritsu 37269C VectorNetwork Analyzer together with soldered SMA contact on theresonator filter. Before the measurements are conducted, the equip-ment must be calibrated using a coaxial calibration method. Thesolid circles in Figure 3 denote the measured transmission char-acteristic when the resonator filter measured by the Vector Net-work Analyzer and the open triangles in Figure 3 denote thesimulation results by the FDTD.

The analysis of the results shows that the EM field excitescurrent in the metallic lines of the fractal; with the current ampli-tude reaching maxima at those strong stop band frequency so that

the effective dielectric constant is also varied along with theincident wavelength following Eq. (1) [5].

�eff� f � � �0 � �l

�l

fl2 � f2 (1)

where f denotes operating frequency in giga hertz, the index l runsover all resonance, and �0, fl, and �l are parameters obtainable fromFDTD calculated spectra [9]. Equation (1) shows that in varying fromone resonance frequency fl to the next fl�1, it is necessarily that thereis a point at which �eff � 1, hence the existence of the pass band.Because there are different resonant frequencies at which �eff � 1 inthe fractal pattern, the microstrip resonant filter shows multiple bandgap transmission characteristics. Figure 3 shows the transmissioncharacteristic of the microstrip resonant filter with the fractal patternand we find that the microstrip filter has four resonant frequenciescentered at 1.25, 4.1, 6.5, and 8.2 GHz. The main reason for thedifference between the measurement and simulation is measurementerror; the simulation is idealization and does not include loss.

An interesting phenomenon occurs if the dimensions of thefractal patterns are increased. We measured four transmissionspectra of the different dimensions of fractal patterns when thefractal metallic patterns layer is inserted between the two sub-strates of microstrip resonant filter, as shown in Figure 4.

As the dimensions of the fractal patterns increases, the resonantfrequency of the microstrip resonant filter shifts to lower frequency.From Figure 4, we find that the frequency shift of the filter is higherthan 0.5 GHz. We simulated the different dimensions of fractalpatterns by FDTD to explain the experimental observations. Simula-tions results agree quite well with the measured results, which man-ifest that the stop bands are shifted to lower frequency with theincreased dimension of the fractal patterns. We attribute this to thefact that stop bands are caused by resonances intrinsic to the fractal [5,9] and that the attachment of a dielectric slab would lower theresonance frequency due to the increased capacitance and inductancein the microstrip circuit [10]. Thus, the stop bands shift downwardafter the dimension of the fractal patterns are increased.

One advantage of the present design, which is very importantfor practical application, is the tunability according to the dimen-sion of fractal patterns. The resonance frequency of microstripresonant filter can be tuned by various dimensions of fractal

Figure 1 The cross section of microstrip resonant filter with fractalmetallic patterns. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]

Figure 2 The microstrip resonant filter with the fractal metallic patterns.[Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]

Figure 3 The comparison between simulation and measured results (thefirst line length of the fractal pattern is 8 mm). [Color figure can be viewedin the online issue, which is available at www.interscience.wiley.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 11, November 2007 2727

patterns without the additional of DC bias, and it works at the roomtemperature. The unique property of the tunability can overcomethe drawback of the current tunable microstrip resonant filters andimprove the tunability performance of microstrip resonant filters.

5. CONCLUSION

In conclusion, through FDTD simulations and experiments, wehave shown that microstrip resonant filters with fractal metallicpatterns exhibit multiple band gaps. The resonant frequencies ofthe gaps can be tuned by changing the dimensions of the fractalpatterns without additional electric field.

REFERENCES

1. J.S.A. Hong and M.J. Lancaster, Microstrip filters for RF/microwaveapplications, Wiley, New York, NY 2001.

2. G. Subramanyam, F.W. Van Keuls, and F.A. Miranda, A K-bandtunable microstrip bandpass filter using a thin film conductor/ferro-electric/dielectric multilayer configuration, IEEE Microwave GuidedWave Lett 8 (1998), 1011–1014.

3. F.A. Miranda, G. Subramanyam, F.W. Van Keuls, R.R. Romanofsky,J.D. Warner, and C.H. Mueller, Design and development of ferroelec-tric tunable microwave components for Ku- and K-band satellitecommunication systems, IEEE Trans MTT 48 (2000), 1181–1189.

4. S. Pal, C. Stevens, and D. Edwards, Tunable HTS microstrip filters formicrowave electronics, Electron Lett 41 (2005), 286–288.

5. W.J. Wen, L. Zhou, J.S. Li, W.K. Ge, C.T. Chan, and P. Sheng,Subwavelength photonic band gaps from planar fractals, Phys Rev Lett89 (2002), 223901–223904.

6. B. Hou, G. Xu, and W.J. Wen, Tunable band gap properties of plarnatmetallic fractals, J Appl Phys 95 (2004), 3231–3233.

7. L. Zhou, W.J. Wen, C.T. Chan, and P. Sheng, Reflectivity of planarmetallic fractal patterns, Appl Phys Lett 82 (2003), 1012–1014.

8. K.S. Yee, Numerical solutions of initial boundary value problemsinvolving Maxwell’s equations in isotropic media, IEEE Trans Anten-nas Propag 14 (1966), 302–307.

9. L. Zhou, C.T. Chan, and P. Sheng, Theoretical studies on the trans-mission and reflection properties of metallic planar fractals, J Phys D:Appl Phys 37 (2004), 368–373.

10. B.A. Munk, Frequency selective surfaces, theory and design, Wiley,New York, 2000.

© 2007 Wiley Periodicals, Inc.

A CIRCULAR CPW-FED SLOTANTENNA FOR BROADBANDCIRCULARLY POLARIZED RADIATION

I-Chung Deng,1,2 Jin-Bo Chen,1,2 Qing-Xiang Ke,1,2

Jun-Rong Chang,1,2 Woon-Fa Chang,3 and Yueh-Tsu King3

1 Department of Electronics Engineering, Technology and ScienceInstitute of Northern Taiwan, Taipei, Taiwan 112, Republic of China;Corresponding author: icdeng@seed.net.tw2 Institute of Mechatronic Engineering, Technology and ScienceInstitute of Northern Taiwan, Taipei, Taiwan 112, Republic of China3 Animal Technology Institute Taiwan, Miaoli, Taiwan 350, Republic ofChina

Received 30 March 2007

ABSTRACT: The design of a circular slot antenna for obtainingbroadband circularly polarized (CP) radiation using a coplanarwaveguide (CPW) feed is presented. The operation of the proposedantenna is achieved by protruding metallic mono-strip from the cir-cular ground plane towards the slot center at � � 0° and feedingantenna using a 50-� CPW with a protruded single strip at � �270°. In this design, the impedance bandwidth and 3 dB axial-ratiobandwidth can reach as large as 50 and 36%, respectively. The peakof antenna gain is about 3.9 dBi, with gain variations �1 dBi for thefrequencies within CP bandwidth. © 2007 Wiley Periodicals, Inc.Microwave Opt Technol Lett 49: 2728 –2733, 2007; Published onlinein Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22881

Key words: circular polarization; CPW-fed; circular slot antenna

1. INTRODUCTION

CPW-fed antennas are popular for many communication applica-tions because of being compatible with the monolithic microwaveintegrated circuits (MMIC) [1-4] and they own relatively much

Figure 4 The transmission characteristics of microstrip resonators withdifferent dimensions of fractal patterns. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.com]

Figure 1 Geometry of the proposed CPW-fed circularly polarized slotantenna

2728 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 11, November 2007 DOI 10.1002/mop