Numerical Modeling of Ultrawide-Band Dielectric Horn ... - IEEE Xplore

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 52, NO. 5, MAY 2004

Numerical Modeling of Ultrawide-Band Dielectric Horn Antennas Using FDTD Neelakantam V. Venkatarayalu, Chi-Chih Chen, Member, IEEE, Fernando L. Teixeira, Member, IEEE, and Robert Lee

Abstract—A detailed characterization of the input impedance of ultrawide-band (UWB) dielectric horn antennas is presented using the finite-difference time-domain (FDTD) technique. The FDTD model is first validated by computing the characteristic impedance of two conical plate transmission lines (including planar bow-tie antennas) and comparing the results to analytical solutions. The FDTD model is next used to calculate the surge impedance of dielectric horn antennas using the conical plates as launchers. Design curves of the surge impedance for different choices of geometries and dielectric loadings are provided. The modeled antennas are particularly attractive for applications such as UWB ground penetrating radars (GPR) applications. Index Terms—Dielectric horn antenna, FDTD, impedance.

I. INTRODUCTION

D

IELECTRIC horn antennas are well known for having desirable properties such as increased directivity, reduced sidelobe level, wide bandwidth, low loss, and ease of fabrication [1]. These properties are particularly attractive for applications such as in ultrawide-band (UWB) ground penetrating radars (GPR). However, the characterization of such antennas with increasingly complex designs using analytical techniques is often not possible. On the other hand, a numerical model can provide a virtual test bench to explore different design possibilities before any costly prototyping. Although many numerical techniques can be used to model and study the characteristics of such antennas, the finite-difference time-domain (FDTD) technique [2] is perhaps the most robust and flexible method to solve UWB problems with increasingly complex geometries and constitutive properties. The use of FDTD method to study such class of antennas has already been considered before in some detail [3]–[6]. In particular, [6] computed the FDTD solution for the field radiated by a TEM horn, evidencing a very good agreement against the analytical solution. In this paper, we provide an in-depth analysis of the surge impedance of a special dielectric horn antenna design for UWB GPR applications employing two perfect electric conducting (PEC) plates as launcher. The complete antenna geometry is shown in Fig. 1. In practice the Manuscript received August 1, 2002; revised July 23, 2003. This work was supported in part by the U.S. Army Cold Regions Research Engineering Laboratory under Contract DACA89-99-K-0005 and in part by National Science Foundation under Grant ECS-0347502. N. V. Venkatarayalu was with the ElectroScience Laboratory and Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43212 USA. He is now with Temasek Laboratories, National University of Singapore, Singapore 119260, Singapore. C.-C. Chen, F. L. Teixeira, and R. Lee are with the ElectroScience Laboratory and Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43212 USA (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2004.827510

Fig. 1. Dielectric horn antenna geometry with the two-wire line feed and launcher plates.

PEC plates are terminated using resistive cards to control antenna ringing and hence increase the bandwidth. The study of the frequency behavior of the surge impedance of such a design is critical for their adequate performance for UWB applications. Being a time domain method, one single FDTD simulation is sufficient to extract the frequency behavior of the antenna over the entire frequency range with proper calibration. This paper is organized as follows. In Section II, the calibration procedure to extract the input impedance and the geometry of the feed structure (a balanced two wire transmission-line) are described. It should be noted that previously proposed antenna feed models [7] do not consider the presence of higher order modes (other than the TEM mode) which may influence the impedance at the driving point of the antenna. Section III includes validation results for two precursor geometries. First, the input impedance of a planar bow-tie antenna is obtained and compared against previous results [8]. Next, the characteristic impedance of a two conical plate transmission line in free space is computed for two different angle between the plates, (planar plates) and . The results are viz., then compared against analytical results given in [9]. In Section IV, the region between the plates is filled with the dielectric material (dielectric horn antenna) and the behavior of the surge impedance for various geometries is studied in detail. All the simulations were carried out with a source pulse having significant frequency content in a frequency range from 500 MHz to 2 GHz and a FDTD grid with uniform cell size

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VENKATARAYALU et al.: NUMERICAL MODELING OF UWB DIELECTRIC ANTENNAS USING FDTD

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Fig. 2. Cross section of the FDTD grid with the magnetic current loop used to excite the two-wire transmission line.

of

where is the wavelength at the maximum frequency of 2 GHz in the excitation pulse. Complex anisotropic perfectly matched layer (PML) was used to terminate the FDTD domain. A quadratic profile for the electrical conductivity of the PML was chosen and the maximum conductivity was set . as 0.65

Fig. 3. Time domain and frequency content of the differentiated gaussian source pulse.

where

II. FDTD TRANSMISSION LINE MODEL AND CALIBRATION PROCEDURE

and

Various transmission line feed models to study antenna characteristics using FDTD have appeared in literature. In [7], a one-dimensional transmission line model was embedded inside the FDTD domain. The voltage and current in the transmission line were updated separately and the electric field (set at the driving point of the antenna) was obtained from the voltage at the end of the transmission line. The use of an embedded transmission line model in the FDTD grid avoids the need for very fine cell sizes to model details of the geometry of the feed. The disadvantage is that this simplified model does not account for the possible existence of higher order modes at the driving point which may affect the impedance value. In this paper, a balanced two wire transmission line is modeled directly in the FDTD grid with the cell size equal to . The transmission line was made of two one-cell thick perfect electrical conductors separated by a gap of 3 cells or . Note that the number of cells in the gap is related to number of higher order modes that can be incorporated in the model. In principle, the lines could have been modeled using a thin-wire approximation as well [10], but due to observed late-time instabilities, it was decided to introduce the conducting wires directly as one cell thick conductors instead. The transmission line is excited by a magnetic current loop around the two conductors with the polarity as shown in Fig. 2 to provide balanced excitation to the antenna. The broadband excitation pulse is a differentiated Gaussian pulse given by

(1)

(2)

The selection of as given by (2) gives the source pulse significant energy in the frequency band from to . The correspondingly normalized time domain waveform and the frequency domain spectrum content of the excitation pulse are shown in Fig. 3. The source excitation is coupled into the magnetic field update equations as

(3)

(4) (5) where

and are the time discretized excitation pulse given by (1) and and are the the direction is indicated in Fig. 2. Note that . components of the magnetic current density scaled by First, the characteristic impedance of the FDTD transmission-line model is determined numerically. This is done by

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extending the two one-cell thick conductors into the perfectly matched layer (PML) region [4] surrounding the FDTD domain so that any reflections are suppressed and the transmission line appears infinitely long. The characteristic impedance is then calculated by the voltage to current ratio at any point along the line. The voltage is obtained from integrating the electric field between the two conductors at a particular observation point and is given as (6)

(7) where , are reference cells indicated in Fig. 2. The current through the conductor is computed by integrating the magnetic field components around one of the conductors (due to the staggered nature of the FDTD grid this is done half cell away from the observation plane of the electric field), and applying Ampere’s law. Thus, the current at any time step is given as

Fig. 4. Input impedance versus normalized arm length of planar bowtie with 90 flare angle.

(8)

Fig. 5. Input impedance versus plate angle of infinitely long two conical plate transmission line with 180 and 60 angle between the plates.

(9) In the frequency domain, the characteristic impedance is given by the ratio of Fourier transforms of the computed voltage and current. This computed characteristic impedance gives , with zero imaginary component and virtually no dispersion over the entire frequency band, as desired. III. VALIDATION We designate the voltage response computed in this infinite length case as the incident voltage, . Once is determined, the impedance of the antenna can be computed from the reflection coefficient at its input terminals. To obtain the reflection coefficient, a calibration procedure is performed similar to that done in practical measurements. This requires the knowledge of the reflected voltage when the transmission line is terminated by a short, which can be obtained by an auxiliary . From FDTD simulation. This voltage is denoted as and , the reflection coefficient and hence the impedance of the antenna structure can be computed. The next step is to replace the \rm short in the transmission line with the antenna structure whose impedance needs to be

measured. The voltage observed in the transmission line with . The reflection coefficient of the antenna the antenna is is calculated as (10) From the reflection coefficient, the impedance of the antenna can be calculated as (11) where as determined before. To validate this method for measuring the antenna impedance, we apply it to calculate the input impedance of the planar bowtie antenna discussed in [8], with a plate flare angle of 90 . This antenna is a truncated bowtie of length . Reference [8] uses a monopole excitation and the antenna is fed through an image plane by a transmission line. Hence, the calculated impedance obtained here should be twice that of the impedance results in [8]. The input impedance versus normalized length is shown in Fig. 4, where the computed impedance is compared with twice the


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Fig. 6. FDTD grid showing the dielectric horn antenna, the balanced feed line, and the plate launcher geometry. Inset shows the feed point in detail.

impedance of the monopole configuration reported in [6]. A good agreement is observed. As a second validation study, the characteristic impedance of infinite conical plate transmission lines is calculated from the FDTD model and compared with the analytical results in [9], [11] obtained using stereographic projection and conformal mapping to reduce the conical plate geometry to an ideal parallel plate transmission line. To calculate the characteristic impedance of the infinite structure, the PEC plates are extended numerically into the PML region to eliminate reflections from the truncation. Furthermore, a time-domain gating is applied to keep only the response associated with the reflection occuring at the input terminal (surge impedance). The first case is a geometry with angle between the plates set as 180 , which is equivalent to a infinite planar bow-tie antenna. The input impedances of this planar antenna for various plate angles of 10 , 20 , 30 , 45 , 60 , and 90 are obtained from (10) and (11). The observed results are fairly constant and serve to verify the fact that such radiating structures are indeed frequency independent in the FDTD model. In Fig. 5, we compare the input impedance of the infinite bow-tie for various plate angles at the peak frequency of 1.2 GHz, with the results presented in [9]. A very good agreement between the analytical results and FDTD results is observed. The second case is a conical plate geometry with the angle between the plates set as 60 . This geometry is equivalent to the dielectric horn antenna shown in Fig. 1 with and horn angle, . Note that the properties of this geometry are different from the dielectric loaded horn antenna in that the radiation characteristics of a dielectric horn antenna is mainly determined by the dielectric body itself [1]. The conducting plates are merely part of the broadband wave launch mechanism. Several FDTD simulations have been carried out for this geometry for different plate angles. Also shown in Fig. 5 is the comparison of the theoretical results

presented in [9] and the obtained FDTD results. The results differ by a margin of 10–15 . This deviation can be attributed to the errors due to staircasing approximation introduced both by the discretization of the horn angle as well as the plate angle. The imaginary part of the computed impedance varies from 20 to 10 for different plate angles, which is less than 10% of the real part of the impedance. IV. FDTD MODELING OF DIELECTRIC HORN ANTENNAS The dielectric horn antenna is modeled by filling the space between the conical plates with a dielectric material. The complete discretized antenna geometry including the conical plate launchers and the two wire feeding transmission line is shown in Fig. 6. The inset of Fig. 6 shows the transition from the transmission line to the launcher plates. The tangential electric field components along this structure is set to zero. The antenna geometry in this case is of finite size, unlike the previous infinitely long conical plate launcher geometry. In order to study the surge reflection coefficient, a time domain gating is applied to isolate this reflection from residual reflections due to the truncation of the numerical model. We note that, in practice, the PEC launcher plates can be terminated using resistive cards to minimize reflections due to the finite length of the plates, and it is the surge impedance at the feed launcher that essentially governs the antenna bandwidth. In the numerical model, the horn geometry is long enough to separate out the surge reflection due to the driving point without introducing artificial errors due to gating. Utilizing a similar numerical calibration procedure as described in the previous section, the surge impedance of various dielectric horn antennas are computed. In Fig. 7 the surge impedance versus frequency for an antenna geometry with is shown for various plate angles of dielectric constant,

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Fig. 7. Real and imaginary part of input impedance versus frequency of dielectric horn antennas with horn angle = 60 and dielectric constant = 4 for various plate angles of the launcher.

Fig. 8. Time snap shots of E component in the xz plane for the antenna geometry with parameters, = 9:0, = 60 .

the launcher. These results are computed from the time-domain gated voltage response observed in the line feeding the antenna structure. Thus, the geometry appears to be of infinite length and, as a result, the surge impedance is fairly constant in the range of 500 MHz to 2 GHz. In Fig. 8 the time domain snap component of the near field radiated by the shots of the , the horn angle, antenna with parameters, and the launcher plate angle is is shown. The component is normalized to its peak value in the entire time period of the simulation. In Fig. 8(a), the excitation of the TEM mode in the two-wire transmission line is seen. In Fig. 8(b), the excited wave reaches the first discontinuity near the transmission line feed and the antenna apex. The voltage reflected back at this point is used to find the surge impedance of the antenna structure. In Fig. 8(d), the discontinuity at the antenna arm ends is reached. The secondary reflection from this end, typically minimized in practice due to resistive cards, is gated out using

Fig. 9. Real part of input impedance of 60 pyramidal dielectric horn antenna versus plate angle of the conical plate launcher for various dielectric constants of the horn.

Fig. 10. Real part of input impedance of dielectric horns with horn angle 30 , 60 , and 90 for various plate angles of the launcher.

the rectangular window in the numerical model so that is does not contributes to the surge impedance computation. The surge impedance calculated for various cases with dielec2, 4, 9, and 16 and plate angles from 10 to tric constants, 60 is shown in Fig. 9. It is interesting to see that, for lower permittivities, the impedance is dependent on the plate angle of the PEC plate launcher. However, for higher dielectric constants of the horn, typical in GPR applications, the impedance is less significantly affected by the PEC conical plate angle. The dependence of the antenna surge impedance with respect to the dielectric horn angle is investigated next. For a dielectric , three different horn angles of 30 , 60 , constant of and 90 have been considered. For each horn angle, different FDTD simulations have been carried out for each PEC conical plate angle. The results are shown in Fig. 10. It is observed that changes in the horn angles seem to cause the impedance to shift without affecting the impedance behavior as a function of the launcher plate angle.


V. CONCLUSION The FDTD method has been used for a detailed study of the surge impedance of UWB dielectric horn antennas with a two conical plate transmission line as launcher. The FDTD model has been first validated by comparing the computed impedances against theoretical results for two-conical plates transmission lines with cone angle of 180 (planar bowtie antenna) and cone angle of 60 . Various plate angles have been considered, ranging from 10 to 90 . The case with cone angle of 60 was then studied with a dielectric loading in the space between the plates (dielectric horn antenna). The impact of variations of the dielectric constant ( 2, 4, 9, and 16) and the cone angle (30 , 60 , and 90 ) on the antenna impedance has been studied, and design curves have been provided. The results show that the FDTD model introduced here is sufficiently robust for the optimization of the electrical properties of such UWB antennas before any actual, costly prototyping.

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Chi-Chih Chen (S’92–M’97) was born in Taiwan, R.O.C., in 1966. He received the B.S.E.E. degree from the National Taiwan University, Taiwan, R.O.C., in 1988 and the M.S.E.E. and Ph.D. degrees from the Ohio State University, Columbus, in 1993 and 1997, respectively. He joined the Ohio State University ElectroScience Laboratory as a Postdoctoral Researcher in 1997 and became a Senior Research Associate in 1999. His main research interests include the ground penetrating radar, UWB antenna designs, radar target detection and classification methods, automobile radar systems. In recent years, his research activities have been focused on the detection and classification of buried landmines, unexploded ordnance and underground pipes. Dr. Chen is a Member of Sigma Xi and Phi Kappa Phi.

REFERENCES [1] M. Chatterji, Dielectric and Dielectric Loaded Antennas, U.K.: Research Studies Press Ltd., 1985. [2] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302–307, May 1966. [3] J. G. Maloney, G. S. Smith, and W. R. Scott, “Accurate computation of the radiation from simple antennas using the finite-difference time-domain method,” IEEE Trans. Antennas Propagat., vol. 38, pp. 1059–1068, July 1990. [4] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 1995, ch. 14, pp. 477–520. [5] A. Taflove and S. Hagness, Computational Electrodynamics: The FiniteDifference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000. [6] K. L. Shlager, G. S. Smith, and J. G. Maloney, “Accurate analysis of TEM horn antennas for pulse radiation,” IEEE Trans. Electromagn. Compat., vol. 38, pp. 414–423, Aug. 1996. [7] J. G. Maloney, K. L. Shlager, and G. S. Smith, “A simple FDTD model for transient excitation of antennas by transmission lines,” IEEE Trans. Antennas Propagat., vol. 42, pp. 289–292, Feb. 1994. [8] K. L. Shlager, G. S. Smith, and J. G. Maloney, “Optimization of bow-tie antennas for pulse radiation,” IEEE Trans. Antennas Propagat., vol. 42, pp. 975–982, July 1994. [9] F. C. Yang and K. S. H. Lee, Impedance of a two-conical-plate transmission line, in Sensor and Simulation Notes, Nov. 1976. [10] K. R. Umashankar, A. Taflove, and B. Becker, “Calculation and experimental validation of induced currents on coupled wires in an arbitrary shaped cavity,” IEEE Trans. Antennas Propagat., vol. AP-35, pp. 1248–1257, Nov. 1987. [11] R. L. Carrel, “The characteristic impedance of two infinite cones of arbitrary cross section,” IRE Trans. Antennas Propagat., vol. AP-6, pp. 197–201, Apr. 1958.

Neelakantam V. Venkatarayalu was born in Chennai, India, in 1979. He received the B.E. degree in electronics and communication engineering from Anna Unversity, India, in 2000 and the M.S degree in electrical engineering from The Ohio State University, Columbus, OH, in 2002. During his graduate study, he was with the ElectroScience Laboratory, Department of Electrical Engineering, Ohio State University, as a Graduate Research Associate. Currently he is with Temasek Laboratories, National University of Singapore, as an Associate Scientist where he is involved with the Electromagnetics group. His current research interests include time domain numerical methods for solving Maxwell’s equations and their application for ultrawide-band antenna design.

Fernando L. Teixeira (S’89–M’93) received the B.S. and M.S. degrees in electrical engineering from the Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Brazil, in 1991 and 1995, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1999. From 1999 to 2000, he was a Postdoctoral Research Associate with the Research Laboratory of Electronics at the Massachusetts Institute of Technology (MIT), Cambridge, MA. Since 2000, he has been an Assistant Professor at the ElectroScience Laboratory (ESL) and the Department of Electrical Engineering, at The Ohio State University, Columbus. His current research interests include analytical and numerical techniques for wave propagation and scattering problems in communication, sensing, and devices applications. He was the Editor of Geometric Methods for Computational Electromagnetics (PIER 32, EMW: Cambridge, MA, 2001), and has published over 40 journal articles and book chapters and 50 conference papers in those areas. Dr. Teixeira is a Member of Phi Kappa Phi and Sigma Xi. He was awarded the Raj Mittra Outstanding Research Award from the University of Illinois, and a 1998 MTT-S Graduate Fellowship Award. He received paper awards at the 1999 USNC/URSI National Radio Science Meeting, Boulder, CO and at the 1999 IEEE AP-S International Symposium, Orlando, FL. He received a Young Scientist Award at the 2002 URSI General Assembly and the NSF CAREER Award in 2004. He was the Technical Program Coordinator of the Progress in Electromagnetics Research Symposium (PIERS), Cambridge, MA, in 2000. He is currently the Vice-Chairman of the Columbus IEEE MTT/AP Joint Chapter.

Robert Lee received the B.S.E.E. degree in 1983 from Lehigh University, Bethlehem, PA and the M.S.E.E. and Ph.D. degrees from the University of Arizona, Tucson, in 1988 and 1990, respectively. From 1983 to 1984, he worked for Microwave Semiconductor Corporation in Somerset, NJ, as a Microwave Engineer. From 1984 to 1986, he was a Member of the Technical Staff at Hughes Aircraft Company in Tucson, AZ. From 1986 to 1990, he was a Research Assistant at the University of Arizona. In addition, during the summers of 1987 through 1989, he worked at Sandia National Laboratories, Albuquerque, NM. Since 1990, he has been at The Ohio State University where he is currently a Professor. His major research interests are in the development and application of numerical methods for electromagnetics.