IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 1, JANUARY 2003 137
III. NUMERICALEXPERIMENTS
In this section, we consider the plane wave scattering by a rectan- gular cavity, 1 m wide by 0.25 m deep, illuminated by a 300-MHz plane wave. Using the variational formulation described above, we compute the monostatic RCS of both the unfilled cavity ("r= 1) and the filled one ("r6= 1). At the same time, we compare the numerical results with those found by 2-D integral equation method (RAM2D) [3] and by the finite element-Green function method [2]. To verify the convergence rates predicted by our theorems, we also compute the error estimates.
• Test case 1. In this experiment, we assume that the cavity is un- filled, that is,"r = 1 (free-space) inside the cavity. Its RCS is computed for both TM and TE cases and compared with those found by RAM2D. The results agree very well in both polariza- tions (see Fig. 2).
• Test case 2. In this experiment, we assume that the cavity is filled with material that has"r= 4 0 i as its relative permittivity. The RCS for both fundamental polarizations are compared with those obtained by the finite element-Green function method [2] (see Fig. 3). Again, the results agree very well.
• Test case 3. The surface of the unfilled cavity is assumed to be coated with a 0.1-m-thick layer of material with"r = 3 0 3i.
The numerical results are compared with the finite element-Green function method [2] (see Fig. 4).
• Error rates. In this experiment, using the same cavity, we examine the convergence of the FEM. In each figure, we plot the relative errors inL2-norm andH1-norm (or energy norm) versush (the mesh size) to verify the estimates given in Theorems 2.1.6 and 2.2.4. We use the formulae
errorn= log2 u k0uu k ; for L2-norm log2 u k0uu k ; for H1-norm
wherehn= hn01=2. In Fig. 5, the error history of the total field uhfor both norms is plotted for the unfilled cavity and at normal incidence. In Fig. 6, we plot the error history for the filled cavity with the incident angle = 10. Fig. 7 is for = 1. In each figure, we observe that forh sufficiently small, the L2error rate approaches to 2, and theH1 error rate is between 1 and 2, as predicted by the theory.
IV. CONCLUSION
Using a Dirichlet–Neumann mapping and appropriate variational formulation, we reduce the problem of scattering by a 2-D cavity in an infinite ground plane to a problem in a bounded domain. The discretized variational problems for TM and TE polarizations are shown to have unique solutions. Optimal convergence results of the finite element approximation are established. Numerical experiments that confirm the optimal convergence rate predicted by the theory are presented. Moreover, our numerical results agree well with those by standard methods such as [2] and [3].
ACKNOWLEDGMENT
The authors thank Maj. W. D. Wood of the Air Force Research Lab for insightful discussions and helpful suggestions.
REFERENCES
[1] W. D. Wood, Jr. and A. W. Wood, “Development and numerical solution of integral equations for electromagnetic scattering from a trough in a ground plane,” IEEE Trans. Antennas Propagat., vol. 47, pp. 1318–1322, Aug. 1999.
[2] J. Jin, The Finite Element Method in Electromagnetics. New York:
Wiley, 1993.
[3] R. L. Clary, Ram2d: Two Dimensional Integral Equation Computer Code, Version 3.0. Los Angeles, CA: Northrop Grumman Corp., 1995.
[4] R. Kress, Linear Integral Equations, II ed. New York: Springer, 1999.
[5] D. Jerison and C. Kenig, “Unique continuation and absence of posi- tive eigenvalues for Schrödinger operators,” Ann. Math., vol. 121, pp.
463–388, 1985.
[6] G. Strang and G. J. Fix, An Analysis of the Finite Element Method. Wellesley, MA: Wellesley-Cambridge, 1988.
[7] G. Bao, “Numerical analysis of diffraction by periodic structures: TM polarization,” Numer. Math., vol. 75, pp. 1–16, 1995.
Dual-Band, Single CPW Port, Planar-Slot Antenna Daniel Llorens, Pablo Otero, and Carlos Camacho-Peñalosa
Abstract—A dual-band planar-slot antenna that operates through a single coplanar waveguide (CPW) port in two 10% wide-frequency bands, one octave apart is presented. An input T-match circuit is used to match the antenna in both bands, without increasing its size and keeping the CPW feed advantages. A case antenna covering the two global system for mobile communications bands (0.88–0.96 GHz and 1.71–1.88 GHz) and the digital enhanced cordless telecommunications band (1.88–1.9 GHz) has been designed, built, and measured, to demonstrate the technique. A proprietary program, based on the method of moments, has been used to analyze and optimize the antenna.
Index Terms—Coplanar waveguide (CPW), dual band, slot antenna.
I. ANTENNACONFIGURATION ANDOPERATION
There is growing research activity on dual and multiband antennas for new wireless communications applications. In addition, a great in- terest in coplanar waveguide (CPW) fed antennas has been present in literature, so far due to the advantages of this type of circuit, e.g., they are completely uniplanar and are easily integrated with active devices or MMICs to provide active integrated antennas [1].
In this letter, we present a CPW-fed planar-slot antenna that operates through a single port in two-frequency bands up to one octave apart.
The antenna consists of a rectangular slot loop, which presents nearly omnidirectional radiation patterns.
The antenna configuration is shown in Fig. 1. The operation band- width and the symmetry of the patterns depend on the aspect ratio of the loop = wl=hl. A square loop ( = 1) shows better symmetry of the patterns, but has a narrower band in terms of matching than a loop having a smaller . This behavior can be deduced, via the complemen- tarity principle from a wire-loop antenna [2]. Here, an aspect ratio of = 0:5 has been chosen to favor operation bandwidth. Moreover, when the slots are made wider the operation bandwidth also increases,
Manuscript received September 6, 2000; revised January 29, 2002.
The authors are with the Department of Communications Engineering, Universidad de Málaga (DIC-UMA), E-29071 Málaga, Spain (e-mail:
otero@ic.uma.es).
Digital Object Identifier 10.1109/TAP.2003.809105 0018-926X/03$17.00 © 2003 IEEE
138 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 1, JANUARY 2003
Fig. 1. Antenna geometry.
Fig. 2. T-match circuit.
but then the positive input reactance of the antenna rises at frequencies close to the resonances.
The dual-band operation is based on an open-circuited stub located at the mid point of the loop side opposite to the input CPW feed. It is used to obtain the appropriate equivalent magnetic current distribution in the radiating slots, while still keeping a compact size. Its length is nearly a quarter-guided wavelength in the upper band. The open-circuited stub plays another significant role: the frequency difference between the two bands can be adjusted by modifying its characteristic impedance. An increasing factor of two inZc gives a decrease of about 10% in the frequency difference.
II. THET-MATCHCIRCUIT
When loaded with the open-circuited stub, the slot loop needs to be matched in the desired bands. A matching network or feeding structure was needed which would not sacrifice ease of fabrication and moderate size. Eventually, a T-match circuit was devised. This circuit is based on the T-match connection used with wire dipoles [2].
Its interest lies on the fact that the input impedance of the original loop is decreased by a factor that depends on the ratios of the geomet- rical parameters of the T-match circuit,st1=st2andwa=wt(see Fig. 2.
Increasing the ratiost1=st2increases the input impedance. That scale factor is almost constant with frequency.
The T-match circuit also contributes to the total input impedance with the reactance of the odd mode in a series open-circuited stub, made with an asymmetric CPW (st16= st2). That reactance depends on the stub length and on the characteristic impedance of the stub, which can be adjusted by means of parameterswa,wt,st1, andst2. This behavior can again be explained via the complementarity principle and the be- havior of the original wire T-match. In the lower frequency band, the reactance of the open-circuited stub cancels out that of the loop at fre- quencies below the first resonance, which means a small additional size
Fig. 3. Input impedance.a: without T-match; b: with T-match. Dashed line, imaginary part; continuous line, real part (computed values).
TABLE I
DIMENSIONS FOR THECASEANTENNA(IN MILLIMETERS)
Fig. 4. Measured and computedjS j (including T-match).
reduction. Matching at the upper band is achieved where the input resis- tance of the slot loop reaches a second maximum. However, the loop’s reactance does not come to a null at this point. The T-match circuit itself is made to resonate at a higher frequency, therefore providing a negative reactance that adds to the positive reactance of the loop. Fig. 3 explains this behavior.
III. RESULTS
Arlon-512 substrate (r= 3:52, h = 1:02 mm) has been used for a case antenna in L-band. The antenna dimensions are shown in Table I.
A characteristic impedance of 100 has been chosen for reference, for the CPW port to have practical dimensions. Using Booker’s formula with the input resistance of a wire loop one wavelength long, relatively high input impedance values ( 300 ) are expected for the slot loop.
Therefore, the T-match circuit is used to decrease the input impedance of the single slot loop if it is to be matched to a 100 line. In order to analyze and optimize the structure, a proprietary program, based on
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 1, JANUARY 2003 139
Fig. 5. Measured radiation patterns. a) Lower band (0.92 GHz), E plane. b) Upper band (1.805 GHz), E plane. c) Lower band, H plane. d) Upper band, H plane. Crosspolar component shown in dashed line. is measured from the axis perpendicular to the antenna plane.
the integral equation and the method of moments, has been used [3].
This program assumes an infinite ground plane.
Computed and measured (with a HP-8753C network analyzer)jS11j parameter of the prototype antenna are presented in Fig. 4. Agreement is satisfactory. As shown, the measured return loss is larger than 10 dB in both GSM and digital enhanced cordless telecommunications (DECT) bands. Measured radiation diagrams for E and H planes are shown in Fig. 5. Crosspolarization level in the H plane is high at the upper band; this is due to the fact that, to achieve good dual matching, the impedance levels for both resonances can not be too different. When increasing the length of the inset stubleto obtain this effect, the current nulls in the second band move from the horizontal (shorter) sides of the loop to the backside. As a consequence, the polarization linearity is degraded in the H plane in the upper band.
IV. CONCLUSION
A planar, single-port, CPW-fed antenna showing dual-band opera- tion has been presented. The input T-match circuit for the CPW-fed slot loop has been demonstrated as a useful, compact–matching mech- anism. An additional stub is used to determine the appropriate current distribution on the loop, and to set the frequency difference between the operation bands. A prototype antenna working in L-band at two frequencies separated by one octave, has been built and measured. Two 10% wide bands are obtained with a compact size.
ACKNOWLEDGMENT
The authors would like to thank J.-F. Zürcher (LEMA-EPFL) and Prof. I. Molina, Prof. E. Márquez, and Ing. S. Ortigosa (DIC-UMA) for their valuable help.
REFERENCES
[1] K. H. Y. Ip, T. M. Y. Kan, and G. V. Eleftheriades, “A single-layer cpw-fed active patch antenna,” IEEE Microwave Guided Wave Lett., vol.
10, pp. 64–66, Feb. 2000.
[2] C. A. Balanis, Antenna Theory and Design. New York: Wiley, 1982.
[3] P. Otero, G. V. Eleftheriades, and J. R. Mosig, “Modeling the coplanar transmission line excitation of planar antennas in the method of mo- ments,” Microwave Opt. Technol. Lett., vol. 16, no. 4, pp. 219–225, Nov.
1997.
Hybrid-Coupled Broad-Band Triangular Microstrip Antennas
K. P. Ray, Girish Kumar, and Harish C. Lodwal
Abstract—Two broadband configurations consisting of three hy- brid-coupled equilateral and isosceles triangular microstrip antennas have been proposed. Both configurations yield more than four times bandwidth as compared with the corresponding single triangular microstrip antenna.
The radiation patterns of a hybrid-coupled isosceles triangular microstrip antenna is in the broadside direction with very small variation over the entire bandwidth. In addition, this antenna has wide half power beamwidth, making it suitable as an element for the large scan broadband antenna array.
Index Terms—Broad-Band microstrip antennas, hybrid-coupled microstrip antennas.
I. INTRODUCTION
The bandwidth of microstrip antennas (MSAs) in planar configura- tion has been increased by gap or direct coupling of multiple resonant patches, such as rectangular [1]–[3], circular [4], and triangular MSA [5]. A hybrid-coupling method (which uses both gap and direct cou- pling) for some of these configurations has been reported to control the coupling to get the desired response. The four hybrid-coupled equilat- eral triangular MSA (ETMSA) [5] is unsuitable as an array element because of its large size. In this paper, experimental investigations on two broadband configurations using three hybrid-coupled ETMSA and isosceles triangular MSA (ITMSA) are presented. The hybrid-coupled ITMSA radiates in the broadside direction over the entire bandwidth and is compact enough to be used as an array element.
II. HYBRID-COUPLEDETMSA
A single ETMSA of side lengtha = 5:8 cm was fabricated on glass epoxy substrate with following parameters:"r= 4:3, h = 0:159 cm, andtan = 0:02. The feed point location to match with 50 co-axial feed was computed to be 3.9 cm from the vertex using improved linear transmission line model [6]. The measured bandwidth was 40 MHz at 1.650 GHz against the theoretical value of 38 MHz at 1.655 GHz. To achieve wider bandwidth, similar to the case of gap coupled RMSA and CMSA [1]–[4], parasitic elements of slightly different side lengths were chosen. Initially, only the gap coupled configuration without the
Manuscript received December 20, 1999; revised December 17, 2001.
K. P. Ray is with the SAMEER, I.I.T. Campus, Powai, Mumbai-400076, India.
G. Kumar and H. C. Lodwal are with the Electrical Engineering Department, I.I.T. Bombay, Powai, Mumbai-400076, India.
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