Variable-gain power amplifier for mobile WCDMA applications,IEEE Trans Microwave Theory Tech 12 (2001), 2464–2471.

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7. K. Nishinori, S. Watanabe, F. Sasaki, and K. Arai, A 15–50-GHz bandGaAs MMIC variable attenuator with 20-dB attenuation control,IEICE Trans Electron 10 (2001), 1543–1547.

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11. M. Inamori, K. Motoyoshi, T. Kitazawa, K. Tara, and M. Hagio, Anew GaAs variable-gain amplifier with a wide-dynamic-range andlow-voltage-operation linear attenuator circuit, IEEE Trans Micro-wave Theory Tech 2 (2000), 182–186.

© 2005 Wiley Periodicals, Inc.

SIMPLE AND ACCURATE FORMULASFOR THE PHYSICAL DIMENSIONS OFRECTANGULAR MICROSTRIPANTENNAS WITH THIN AND THICKSUBSTRATES

K. GuneyDepartment of Electronic EngineeringFaculty of EngineeringErciyes University38039, Kayseri, Turkey

Received 1 July 2004

ABSTRACT: New, very simple closed-form formulas for calculating thepatch length and width of electrically thin and thick rectangular microstripantennas (MSAs) are presented. They are obtained by means of a curve-fitting technique, and are useful for the computer-aided design of microstripantennas. The theoretical results obtained by using these new patch lengthand width formulas are in very good agreement with the experimental re-sults available in the literature. © 2005 Wiley Periodicals, Inc. MicrowaveOpt Technol Lett 44: 257–259, 2005; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.20603

Key words: microstrip antenna; patch length; patch width

1. INTRODUCTION

Microstrip antennas (MSAs) offer a number of unique advantagesover other types of antennas, such as low profile, light weight,conformal structure, low cost, and ease of integration with micro-wave integrated circuit or monolithic microwave integrated circuitcomponents [1–16]. The rectangular patch is the most commonlyused MSA, and is characterized by its length and width. Accuratedetermination of the patch length and width is important in thedesign of rectangular MSAs because the antenna-performancecharacteristics depend strongly on the physical dimensions of thepatch. Several formulas [1–23] varying in accuracy and computa-tional effort are available in the literature to determine the physicaldimensions of the patch. The theoretical values obtained by usingmost of these formulas are not in very good agreement with theexperimental results [17, 18].

In this paper, new, very simple and accurate patch length andwidth expressions based on experimental results are proposed for

rectangular MSAs with thin and thick substrates. After analyzingthe dependence of the patch length and width on antenna param-eters, a model for the patch length and width expressions is chosenfirst, then the unknown coefficient values of the model are obtainedby using a curve-fitting technique (CFT). The theoretical patchlength and width results obtained using the expressions derived inthis study agree well with the measured results [17, 18].

2. PATCH DIMENSIONS OF A RECTANGULAR MICROSTRIPANTENNA

Consider a rectangular patch of width W and length L over aground plane with a substrate of thickness h and relative dielectricconstant �r, as shown in Figure 1. The main objective of an MSAdesign is to achieve specific performance characteristics at a givenoperating frequency. The first design step is the selection of asuitable dielectric substrate material, then the patch dimensions Wand L are determined. It is difficult to accurately determine thesepatch dimensions because MSA is inhomogeneous and the radia-tion appears at the edges of the patch.

The patch width affects the resonant frequency, the efficiency,the bandwidth, the H-plane radiation pattern, and the cross polar-ization. In the selection of the patch width, it must be consideredthat, due to the excitation of surface waves, a small width leads toa large bandwidth and low antenna efficiency and gain. However,a large width leads to the excitation of higher-order modes whichmay distort the radiation pattern, decrease the bandwidth, andincrease the efficiency. To avoid exciting higher-order modes, thepatch width must be less than a wavelength in the substrate.

The patch length affects the resonant frequency, the resonantresistance, and the E-plane radiation pattern. The patch length isdetermined by the condition for resonance. It is approximatelyone-half of the wavelength in the substrate.

The antenna designer needs practical and accurate formulas forcomputing the patch dimensions. The problem in the literature isthat formulas as simple as possible for the patch dimensions shouldbe obtained, but the theoretical patch-dimension results obtainedby using the formulas must be in good agreement with the exper-imental results. In this paper, this problem is solved efficiently byusing the CFT. First, the antenna parameters related to the patchdimensions are determined by using the results available in theliterature. After this determination, a model for the patch dimen-sions is chosen according to these parameters, then the unknowncoefficients of the model are determined by the CFT.

It is apparent from all of the formulas presented in the literature[1–23] that only three parameters, h, �r, and resonant frequency fr,

Figure 1 Geometry of the rectangular microstrip antenna

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 3, February 5 2005 257

are needed to describe the patch dimensions of electrically thin andthick rectangular MSAs.

To find the proper model for the patch dimensions, manyexperiments were carried out in this work. After many trials, thefollowing model, which produces very good results depending onh, �r, and fr, was chosen:

patch dimension � �1�d�h/�d��2 � �3h�h/�d�

�4

� �5�o for W or for L (1a)

with

�d ��o

��r

�c

fr��r

, (1b)

where �d is the wavelength in the dielectric substrate, �o is thefree-space wavelength, and c is the velocity of electromagneticwaves in free space. The unknown coefficients �1, �2, �3, �4, and�5 in Eq. (1a) are determined by the CFT for electrically thin andthick rectangular MSAs. Based on the experimental patch-dimen-sion results available in the literature [17, 18], the coefficientvalues were optimally found by the CFT. The following patch-dimension formulas are then obtained by substituting these opti-mum coefficient values into Eq. (1a) as follows:

L � 0.51�d�h/�d�0.0112 � 0.565h�h/�d�

�0.117 � 0.014�o

for h/�d � 0.13, (2a)

L � 383.0445�d�h/�d�0.00052 � 0.36h�h/�d�

0.014 � 239.37�o

for h/�d 0.13, (2b)

W � 9.8636�d�h/�d�0.4561 � 10.9h�h/�d�

�0.446 � 0.00204�o

for h/�d � 0.13, (2c)

W � 382.466�d�h/�d��0.000907 � 0.0003h�h/�d�

�0.04 � 239.73�o

for h/�d 0.13. (2d)

Eqs. (2a)–(2d) have been found to fit closely the experimentalresults reported elsewhere [17, 18]. For simplicity, we choose themodel for patch width to be the same as that for patch length. Thisis clearly illustrated in Eqs. (1a) and (2).

3. RESULTS AND CONCLUSION

In order to determine the most appropriate suggestion given in theliterature, we compared our computed values of the patch lengthand width for different electrically thin and thick rectangularMSAs with the theoretical [1, 9, 12, 17–23] and experimental [17,

TABLE 1 Comparison of Measured and Computed Patch Dimensions of Electrically Thin and Thick Rectangular MicrostripAntennas

PatchNo. h (cm) (h/�d)

fr

(MHz)

W (cm) L (cm)

Measured[17, 18]

PresentMethod [19] [20] [1, 21, 22] [17, 18]

Measured[17, 18]

PresentMethod [23] [1, 9, 22] [12] [17, 18]

1 0.0170 0.006535 7740 0.850 0.848 1.163 2.341 1.527 0.848 1.290 1.291 1.275 1.299 1.300 1.2982 0.0170 0.007134 8450 0.790 0.794 1.065 2.145 1.399 0.794 1.185 1.182 1.168 1.189 1.190 1.1893 0.0790 0.015577 3970 2.000 2.000 2.267 4.565 2.978 2.001 2.500 2.502 2.485 2.508 2.512 2.5084 0.0790 0.032505 7730 1.063 1.062 1.164 2.187 1.457 1.063 1.183 1.181 1.191 1.183 1.184 1.1835 0.1270 0.062193 4600 0.910 0.910 1.957 1.838 1.378 0.905 1.000 1.002 1.001 1.010 0.987 1.0096 0.1570 0.040421 5060 1.720 1.722 1.779 3.496 2.297 1.725 1.860 1.865 1.903 1.864 1.867 1.8647 0.1570 0.038384 4805 1.810 1.808 1.873 3.681 2.419 1.811 1.960 1.969 2.004 1.968 1.971 1.9688 0.1630 0.056917 6560 1.270 1.273 1.372 2.577 1.716 1.275 1.350 1.350 1.403 1.351 1.349 1.3519 0.1630 0.048587 5600 1.500 1.495 1.607 3.019 2.010 1.497 1.621 1.598 1.644 1.600 1.599 1.600

10 0.2000 0.066004 6200 1.337 1.335 1.452 2.727 1.816 1.337 1.412 1.411 1.485 1.412 1.407 1.41211 0.2420 0.090814 7050 1.120 1.120 1.277 2.398 1.597 1.123 1.200 1.202 1.306 1.200 1.189 1.20012 0.2520 0.077800 5800 1.403 1.400 1.552 2.915 1.941 1.404 1.485 1.486 1.587 1.485 1.477 1.48513 0.3000 0.083326 5270 1.530 1.539 1.708 3.240 2.152 1.543 1.630 1.639 1.764 1.636 1.625 1.63614 0.3000 0.126333 7990 0.905 0.897 1.126 2.137 1.419 0.902 1.018 1.020 1.164 1.017 0.992 1.01615 0.3000 0.103881 6570 1.170 1.172 1.370 2.599 1.726 1.177 1.280 1.279 1.415 1.275 1.258 1.27516 0.4760 0.129219 5100 1.375 1.377 1.765 3.315 2.208 1.385 1.580 1.577 1.805 1.577 1.534 1.57717 0.3300 0.140525 8000 0.776 0.773 1.125 2.114 1.407 0.777 1.080 1.080 1.151 0.997 0.960 1.08518 0.4000 0.151895 7134 0.790 0.796 1.262 2.370 1.578 0.798 1.255 1.262 1.290 1.109 1.054 1.26419 0.4500 0.145395 6070 0.987 0.982 1.483 2.786 1.855 0.987 1.450 1.449 1.517 1.309 1.254 1.45420 0.4760 0.147462 5820 1.000 1.009 1.546 2.905 1.935 1.013 1.520 1.523 1.582 1.365 1.303 1.52721 0.4760 0.16165 6380 0.814 0.826 1.411 2.650 1.765 0.828 1.440 1.458 1.443 1.232 1.158 1.45922 0.5500 0.175363 5990 0.790 0.791 1.503 2.823 1.880 0.793 1.620 1.619 1.537 1.300 1.199 1.61823 0.6260 0.155278 4660 1.200 1.187 1.931 3.628 2.416 1.192 1.970 1.955 1.975 1.691 1.603 1.95724 0.8540 0.209105 4600 0.783 0.780 1.957 3.676 2.448 0.783 2.300 2.301 2.001 1.672 1.452 2.30125 0.9520 0.181413 3580 1.256 1.262 2.514 4.723 3.145 1.265 2.756 2.756 2.571 2.169 1.982 2.75426 0.9520 0.201683 3980 0.974 0.961 2.261 4.248 2.829 0.964 2.620 2.613 2.313 1.933 1.706 2.61227 0.9520 0.197629 3900 1.020 1.015 2.308 4.335 2.887 1.018 2.640 2.640 2.360 1.976 1.757 2.63828 1.0000 0.211852 3980 0.883 0.881 2.261 4.248 2.829 0.884 2.676 2.676 2.313 1.932 1.668 2.67729 1.1000 0.228353 3900 0.777 0.773 2.308 4.335 2.887 0.777 2.835 2.831 2.360 1.970 1.637 2.83630 1.2000 0.221646 3470 0.920 0.925 2.594 4.873 3.245 0.929 3.130 3.137 2.653 2.217 1.871 3.14031 1.2810 0.218197 3200 1.030 1.035 2.813 5.284 3.518 1.039 3.380 3.376 2.877 2.404 2.045 3.37932 1.2810 0.203196 2980 1.265 1.267 3.020 5.674 3.778 1.271 3.500 3.502 3.089 2.585 2.272 3.50133 1.2810 0.214787 3150 1.080 1.084 2.857 5.368 3.574 1.088 3.400 3.404 2.922 2.442 2.094 3.405

258 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 3, February 5 2005

18] results reported by other scientists, as given in Table 1. Thesum of the absolute errors between the theoretical and experimen-tal results in Table 1 for every suggestion is also listed in Table 2.The antennas given in Table 1 vary in electrical thickness (definedas h/�d) from 0.006535 to 0.228353 and in physical thicknessfrom 0.17 to 12.81 mm, and operate over the frequency range2.980–8.450 GHz.

It can be clearly seen from Tables 1 and 2 that most of theprevious formulas give comparable results—some cases are ingood agreement with the measurements, and others are far frombeing in agreement. The results calculated by using the formulasproposed in this paper are better than those predicted in otherstudies. The very good agreement between the measured valuesand our computed patch-dimension values supports the validity ofthe formulas obtained in this work.

In this paper, the patch-dimension expression models, whichare simpler and more complicated than the model given by Eq.(1a), were also tried. It was observed that the results of the simplermodels are not in good agreement with the experimental results,and that the more complicated models provide only a little im-provement in the results, at the expense of the simplicity of theformula. The advantages of the formulas given here are theirsimplicity and accuracy.

As a consequence, new closed-form formulas for calculatingthe patch length and width of rectangular MSAs with substratessatisfying 0.006535 � h/�d � 0.228353 and 0.17 mm � h �12.81 mm have been developed with the use of the measured data.It was shown that the theoretical results obtained by using theseformulas are in very good agreement with the measurements.Better accuracy with respect to the previous formulas was ob-tained. The approach presented here allows the designers to use asingle model for both patch width and length, thus simplifying thedesign.

REFERENCES

1. I.J. Bahl and P. Bhartia, Microstrip antennas, Artech House, Dedham,MA, 1980.

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8. J.R. James and P.S. Hall, Handbook of microstrip antennas, IEEelectromagnetic wave series no. 28, vols. 1 and 2, Peter PeregrinusLtd., London, 1989.

9. Y.T. Lo, S.M. Wright, and M. Davidovitz, Microstrip antennas, K.Chang (Ed.), Handbook of microwave and optical components, vol. 1,Wiley, New York, 1989, pp. 764–889.

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microstrip and printed circuit antennas, Artech House, Canton, MA,1991.

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13. J.F. Zurcher and F.E. Gardiol, Broadband patch antennas, ArtechHouse, Norwood, MA, 1995.

14. R.A. Sainati, CAD of microstrip antennas for wireless applications,Artech House, Norwood, MA, 1996.

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17. M. Kara, Formulas for the computation of the physical properties ofrectangular microstrip antenna elements with various substrate thick-nesses, Microwave Opt Technol Lett 12 (1996), 234–239.

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© 2005 Wiley Periodicals, Inc.

A NOVEL VARIABLE OPTICALATTENUATOR AND TEMPERATURESENSOR BASED ON HIGH-BIREFRINGENCE FIBER-LOOP MIRROR

Y. Yan, Q. Zhao, S. Chen, L. Liu, H. Zhang, and X. DongInstitute of Modern OpticsNankai UniversityTianjin 300071, P. R. China

Received 19 June 2004

ABSTRACT: A novel variable optical attenuator (VOA) based onhigh-birefringence (HiBi) fiber-loop mirror is proposed. The HiBifiber-loop mirror is put into the temperature-controlling box. As thetemperature increases, the transmission spectrum shifts with nearlyunaltered shape. The attenuation at 1558 nm is more than 30 dBwith about 2.5-dB insertion loss. Meanwhile, the wavelength shiftwith the temperature is vigorously linear. In this scheme, we proposea temperature sensor with fairly good linearity of 0.9997. Moreover,using theoretical analysis and numerical simulation, we explicitly

TABLE 2 Total Absolute Errors Between the Measured and Computed Patch Dimensions

W (cm) L (cm)

PresentMethod [19] [20] [1, 21, 22] [17, 18]

PresentMethod [23] [1, 9, 22] [12] [17, 18]

Errors (cm) 0.143 22.553 74.122 36.918 0.164 0.140 5.244 9.349 12.762 0.170

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 44, No. 3, February 5 2005 259