A SIMPLE AND ACCURATEEXPRESSION FOR THE BANDWIDTHOF ELECTRICALLY THICKRECTANGULAR MICROSTRIPANTENNAS

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

Received 30 July 2002

ABSTRACT: A new, very simple closed-form expression for computingthe bandwidth of rectangular microstrip antennas with thick substratesis presented. It is obtained by means of a curve-fitting technique, and isuseful for the computer-aided design of microstrip antennas. The theo-retical bandwidth results obtained by using this new bandwidth expres-sion are in very good agreement with the experimental results reportedelsewhere. © 2003 Wiley Periodicals, Inc. Microwave Opt TechnolLett 36: 225–228, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10727

Key words: microstrip antenna; bandwidth

1. INTRODUCTION

Microstrip antennas (MSAs), owing to their low profile, low cost,light weight, conformal structure, ease of fabrication, and integra-tion with solid-state devices, have lately become very popular, notonly with researchers but also in industrial applications [1–18].

An efficient use of MSAs requires knowledge of bandwidthbecause it is a critical parameter of a MSA. Several methods[1–34] varying in accuracy and computational effort are availablein the literature for computing the bandwidth of a rectangular patchantenna, as this is one of the most popular and convenient shapes.These methods can generally be divided into two groups: simpleanalytical methods and rigorous numerical methods. Simple ana-lytical methods can give a good intuitive explanation of antennaradiation properties. However, these methods do not rigorouslyconsider the effects of surface waves. Exact mathematical formu-lations in rigorous methods involve extensive numerical proce-dures, resulting in round-off errors, and may also need final ex-perimental adjustments to the theoretical results. They are alsotime consuming and not easily included in a computer-aideddesign (CAD) package. Furthermore, most of the previous theo-retical and experimental work has been carried out only withelectrically thin MSAs, normally of the order of h/�d � 0.14,where h is the thickness of the dielectric substrate and �d is thewavelength in the substrate. Interest in radiators etched on elec-trically thick substrates has developed recently, primarily for twomajor reasons. First, as these antennas are used for applicationswith increasingly higher operating frequencies, and consequentlyshorter wavelength, even antennas with physically thin substratesbecome thick when compared to a certain wavelength. Second, thebandwidth of the rectangular MSA is typically very small forlow-profile, electrically thin configurations. One of the techniquesto increase bandwidth is to increase the thickness proportionately.The design of MSA elements having wider bandwidth is an area ofmajor interest in MSA technology, particularly in the fields ofelectronic warfare, communication systems, and wideband radars.

In this study, a new very simple bandwidth expression based onexperimental results is proposed for MSAs with thick substrates.After analyzing the dependence of the bandwidth on substrate and

patch parameters, first, a model for the bandwidth expression ischosen, then the unknown coefficient values of the expression areobtained by a curve-fitting technique (CFT). The theoretical band-width results obtained using the bandwidth expression derived inthis study agree well with the measured results [34].

2. BANDWIDTH OF A RECTANGULAR MICROSTRIPANTENNA

Consider a rectangular patch of width W and length L over aground plane with a substrate of thickness h and a relative dielec-tric constant �r, as shown in Figure 1. The bandwidth of thisantenna can be expressed as [1]:

BW �s � 1

QT�s(1)

where s is voltage standing wave ratio (VSWR), QT is the totalquality factor. The total quality factor, QT, can be written as

1

QT� � 1

Qr�

1

Qc�

1

Qd�

1

Qs� (2)

where the four terms represent the radiation quality factor and thequality factors due to conductor loss, dielectric loss, and surfacewave.

Bandwidth was defined by Pozar [23] as the half-power widthof the equivalent circuit impedance response. For a series-typeresonance, this bandwidth is

BW �2R

wr

dX

dw�

wr

, (3)

where Z � R � jX is the input impedance at the radian resonantfrequency wr. For a parallel-type resonance, Eq. (3) is used with Rreplaced by G and X replaced by B, where Y � G � jB is theinput admittance at resonance. The derivative in Eq. (3) can beevaluated by calculating the input impedance at two frequenciesnear resonance and using a finite difference approximation. Theresonant resistance R is given by

R � Rr � Rd � Rc � Rs, (4)

Figure 1 Geometry of a rectangular microstrip antenna

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 3, February 5 2003 225

where the four terms represent the radiation resistance, the equiv-alent resistance of the dielectric loss, the equivalent resistance ofthe conductor loss, and surface wave radiation resistance.

The problem in the literature is that an expression as simple aspossible for the bandwidth should be obtained, but the theoreticalbandwidth results obtained by using the expression must be ingood agreement with the experimental results. In this work, thisproblem is solved efficiently by using the CFT. First, the antennaparameters related to the bandwidth are determined by using theresults available in the literature. After this determination, a modelfor the bandwidth expression depending on these parameters ischosen, then the unknown coefficients of the model are determinedby the CFT.

The feeding method or position are not considered in calculat-ing the bandwidth because the feeding method or position does noteffect the intrinsic patch bandwidth. The bandwidth of a patch issignificantly greater than that of a printed dipole, at least over therange for which the patch actually resonates (h � 0.12�o, where�o is the free space wavelength at the resonant frequency fr). Thisfact is consistent with the antenna gain/bandwidth relation toantenna size. Therefore, the effect of the patch width W on thebandwidth of rectangular microstrip antennas must be taken intoconsideration in the bandwidth calculation of these antennas. Fromthe results of the methods available in the literature [1–34], we seethat for a given frequency, larger bandwidth is possible by choos-ing a thicker substrate and a wider patch. The results also indicatethat a lower value of �r results in a larger bandwidth. It is clearfrom all of the methods and formulas proposed in the literature[1–34] that only four parameters, h, W, �r, and �o are needed todescribe the bandwidth of electrically thick rectangular MSAs.

To find the proper model for the bandwidth, many experimentswere carried out in this work. After many trials, the followingmodel (depending on h, W, �r, and �o), which produces goodresults, was chosen:

BW � %��1� hW

�r�o2��2

� �3� h

�0��, (5)

where the unknown coefficients �1, �2, and �3 are determined bythe CFT. Based on the experimental bandwidth results reportedelsewhere [34], the coefficient values found by the CFT are

�1 � 89, �2 � 0.45, �3 � 91. (6)

The following bandwidth expression is then obtained by substitut-ing the coefficient values given by Eq. (6) into Eq. (5):

BW � %�89� hW

�r�o2�0.45

� 91� h

�o�� (7)

Eq. (7) has been found to fit closely the experimental resultsavailable in the literature [34].

3. RESULTS AND CONCLUSIONS

In order to determine the most appropriate suggestion given in theliterature, we compared our computed values of the bandwidth fordifferent electrically thick rectangular patch antennas with thetheoretical [1, 21, 31, 32, 34] and experimental [34] results re-ported by other scientists, which are all given in Table 1. The sumof the absolute errors between the theoretical and experimentalresults in Table 1 for every suggestion are also listed in Table 2.The antennas given in Table 1 vary in electrical thickness, definedas h/�d, from 0.1405 to 0.2284, and in physical thickness from3.30 to 12.81 mm, and operate over the frequency range 2.980–8.000 GHz.

In Table 1, the results of Carver and Mink [21] were obtainedby using a program called MSAnt which has been written by Pozar

TABLE 1 Comparison of Measured and Calculated Bandwidths of Rectangular Microstrip Antenna Elementswith Thick Substrates

L(mm)

W(mm)

h(mm)

fr

(GHz) h/�d*

BW (%)

Measured[34]

PresentMethod [21] [1] [31] [34] [32]

10.80 7.76 3.30 8.000 0.1405 17.50 17.64 15.29 18.48 8.39 17.00 11.2812.55 7.90 4.00 7.134 0.1519 18.20 18.20 13.62 20.09 8.15 17.77 12.1814.50 9.87 4.50 6.070 0.1454 17.90 17.91 14.54 19.17 8.31 17.34 11.7015.20 10.00 4.76 5.820 0.1475 18.00 17.96 14.08 19.46 8.19 17.47 11.8014.40 8.14 4.76 6.380 0.1617 19.00 18.68 12.45 21.47 7.95 18.42 12.9316.20 7.90 5.50 5.990 0.1754 20.00 19.41 10.73 23.41 7.63 19.29 14.1019.70 12.00 6.26 4.660 0.1553 18.70 18.46 13.01 20.55 8.10 18.01 12.5723.00 7.83 8.54 4.600 0.2091 20.90 20.93 7.85 28.24 6.76 21.26 16.4927.56 12.56 9.52 3.580 0.1814 20.00 19.68 10.10 24.27 7.46 19.66 14.5426.20 9.74 9.52 3.980 0.2017 20.60 20.66 8.45 27.17 7.02 20.85 16.1026.40 10.20 9.52 3.900 0.1976 20.30 20.45 8.76 26.59 7.10 20.61 15.7626.76 8.83 10.00 3.980 0.2119 20.90 21.04 7.63 28.64 6.67 21.40 16.6528.35 7.77 11.00 3.900 0.2284 21.96 21.69 6.50 31.03 6.14 22.26 17.5631.30 9.20 12.00 3.470 0.2216 21.50 21.39 6.92 30.06 6.32 21.91 17.1333.80 10.30 12.81 3.200 0.2182 21.60 21.26 7.11 29.56 6.41 21.73 16.9535.00 12.65 12.81 2.980 0.2032 20.40 20.66 8.26 27.39 6.90 20.92 16.0734.00 10.80 12.81 3.150 0.2148 21.20 21.13 7.39 29.07 6.54 21.55 16.77

* �d � �o/��r � c/( fr��r).

TABLE 2 Total Absolute Errors Between the Measured andCalculated Bandwidths

PresentMethod [21] [1] [31] [34] [32]

Errors (%) 3.09 165.97 85.99 214.62 7.47 88.08

226 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 3, February 5 2003

[6]. The MSAnt program fails to accurately predict the bandwidthbecause it is only valid for electrically thin configurations. Themethod of [1] is found to yield bandwidths of MSAs with sub-strates thinner than approximately 0.17�d with reasonable accu-racy; however, it becomes increasingly inaccurate as the substratethickness increased. The results of an approximate bandwidthformula [31] based on a rigorous Sommerfeld solution are also notin good agreement with the measured results. The closed-formexpression, based on the modified cavity model and the exactGreen’s function for a grounded dielectric slab, was presented byKara [34] for calculation of the bandwidth. To closely match themeasured bandwidth results, the correction factor derived bymeans of a CFT was also included in the closed-form expression[34]. The results of this closed-form expression are close to theexperimental results, as shown in Table 1. However, the formulagiven in this paper is simpler than the formula given by Kara [34].The results of Guney [32] were obtained from the curve-fittingexpression based on the results of the Green’s function methods. Itwas shown by Guney [32] that the results of the curve-fittingexpression are in good agreement with the results of the method ofmoments approach [23] and electric surface current model [26].However, it is apparent from Table 1 that the results of thecurve-fitting expression are not in good agreement with the mea-sured results.

We observe that our results, calculated by using the bandwidthexpression given by Eq. (7), are better than those predicted byother scientists. This is clear from Tables 1–2. The very goodagreement between the measured values and our computed band-width values supports the validity of the bandwidth expressionobtained in this work.

It is evident from Tables 1 and 2 that the theoretical bandwidthresults calculated from the theories available in the literature arenot in very good agreement with the experimental results. For thisreason, the data sets obtained from the existing theories are notused in this work. Only the measured data set is used in thedetermination of the coefficient values of the bandwidth expres-sion, given by Eq. (5).

In this work, the bandwidth expression models, which aresimpler and more complicated than the model given by Eq. (5),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 formula given here are simplicityand accuracy.

As a result, a new closed-form expression for computing thebandwidth of rectangular MSAs with substrates satisfying h/�d �0.14 and 3.3 mm � h � 12.81 mm has been developed with theuse of the measured bandwidth data. It was shown that the theo-retical results obtained by using this expression are in good agree-ment with the measurements. Better accuracy with respect to theprevious methods was obtained.

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© 2003 Wiley Periodicals, Inc.

ELECTROMAGNETIC SCATTERINGFROM A THICK CIRCULAR APERTURE

Haeng S. Lee1 and Hyo J. Eom2

1 LG Electronics Institute of Technology16, Woomyeon DongSeocho Gu, Seoul, Korea2 Department of Electrical EngineeringKorea Advanced Institute of Science and Technology373-1, Kusong Dong, Yusung GuTaejon, Korea

Received 1 August 2002

ABSTRACT: Electromagnetic scattering from a circular aperture in athick conducting plane is analyzed. Numerical computations are per-formed to evaluate the near and far zone radiation fields in terms of theaperture geometry and incident polarization state. A salient difference inthe near-zone field behavior between TM and TE wave incidences isdiscussed. © 2003 Wiley Periodicals, Inc. Microwave Opt TechnolLett 36: 228 –231, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10728

Key words: electromagnetic scattering; circular aperture; near fields

1. INTRODUCTION

Electromagnetic scattering from a circular aperture in a conductingplane is a canonical problem in diffraction theory. Electromagneticfield penetration into a circular aperture is also of practical interestin electromagnetic compatibility-related problems [1]. Varioustechnical approaches have been utilized to rigorously solve thiscanonical problem of circular-aperture scattering. While most re-searchers have dealt with scattering from a circular aperture in athin conducting plane [2, 3], Roberts [4] considered scatteringfrom a circular aperture in a thick conducting plane. Extensivenumerical computations were performed in [4], using a plane-waveexpansion technique, to illustrate the behavior of far-zone scatter-ing from a circular aperture in a thick plane. Recently, simple yetrigorous series solutions have been formulated for electromagneticscattering from a thick circular aperture [5] and a thick annularaperture [6], based on a transform approach. It is possible toperform circular-aperture scattering computation using final theo-retical formulations [5, 6]. The purpose of this paper is to presentthe behavior of near- and far-zone transmitted fields based on thepreviously-derived theoretical formulation. In the following sec-tions we summarize the field representations and present the nu-merical results for transmitted fields.

2. FIELD REPRESENTATIONS

Assume that an electromagnetic wave E� i impinges on a circularaperture with the radius a and thickness d in a thick perfectly-

conducting plane, as shown in Figure 1. The normalized wave-numbers of regions I, II, and III are kj � �nj, ( j � 1, 2, 3),respectively, and nj is the refractive index of the respective re-gions. To represent the scattered fields, it is convenient to intro-duce the fields that are transverse to the z-direction using circularcylindrical coordinates (r, , z). The transverse E� t and H� t fieldscan be expressed in terms of two eigenvectors e� jm and e� je ( j � 1,2, 3), where e� jm and e� je are related to the transverse magnetic(TM) and the transverse electric (TE) waves in the z-direction,respectively. The explicit expressions for e� jm and e� je are availablein [5]. The transverse transmitted field in region I ( z � d/ 2) isrepresented in terms of the continuous modes as

E� t1 � ����

� �0

�

�̃m��e� 1m � ̃e��e� 1e�ei�1 z�d/ 2�d�, (1)

H� t1 � ����

� �0

�

�̃m��h� 1m � ̃e��h� 1e�ei�1 z�d/ 2�d�, (2)

where �1 � ��2n12 � �2.

The transverse field in region II (�d/ 2 � z � d/ 2, r � a)consists of the discrete eigenfunctions

E� t2 � ����

� �j

� Ajei� j z � Bje

�i� j z�e� 2m

� Cjei� j z � Dje

�i� j z�e� 2e�, (3)

H� t2 � ����

� �j

� Ajei� j z � Bje

�i� j z�h� 2m

� Cjei� j z � Dje

�i� j z�h� 2e�, (4)

where �j � ��2n22 � ( j/a)2, � j � ��2n2

2 � ( j/a)2, j

is the jth root of J( x) � 0, and j is the jth root of J ( x) � 0.Note that J( x) is the Bessel function of the first kind and J ( x)denotes its differentiation.

The total field in region III ( z � �d/ 2) is composed of theincident, reflected, and scattered fields. For TE wave incidence, theincident electric field is

Figure 1 Problem geometry ( x � r cos , y � r sin ). Region I: halfspace above conducting plane; Region II: circular aperture; Region III: halfspace below conducting plane

228 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 36, No. 3, February 5 2003