twhertel august 2003

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 8, AUGUST 2003 1771

On the Convergence of Common FDTD Feed Modelsfor Antennas

Thorsten W. Hertel, Member, IEEE,and Glenn S. Smith, Fellow, IEEE

Abstract—The finite-difference time-domain (FDTD) methodis routinely used to calculate the input admittance/impedance ofsimple antennas. The value of the input admittance/impedancedepends on the level of discretization used in the method, andshould converge to a final value as the discretization becomes finer.In this paper, the level of discretization necessary for convergenceis studied using two common feed models: the hard-source feedand the transmission-line feed. First, the simplest and most naivemethods for introducing the voltage and the current in thesemodels are considered, and the results for the admittance areshown not to converge. Next, improved methods for introducingthe voltage and current in these models are constructed. Theresults for the admittance are then shown to converge, andguidelines are offered for the level of discretization needed forconvergence. In addition, two general problems associated withthe computation of the admittance are discussed: the agreementbetween admittances computed with different simple feed models,and the agreement between these admittances and measurements.

Index Terms—Convergence, discretization, finite-differencetime domain (FDTD), hard-source feed, transmission-line feed.

I. INTRODUCTION

T HE properties of antennas, such as input impedance andfield patterns, are now routinely determined using the

finite-difference time-domain (FDTD) method for solvingMaxwell’s equations. When the geometry of the antenna isaccurately represented in the FDTD model, extremely goodagreement can be obtained between the FDTD results andmeasurements. As an example, we present results for thecanonical problem shown in Fig. 1: a linear monopole antennafed by a coaxial line through an image plane. This is oneof the first antennas accurately analyzed using the FDTDmethod [1]. For this example, the ratio of monopole heightto radius is , with 5 cm, and the ratio of thecoaxial line outer radius to inner radius is with1.52 mm. The excitation is at a point in the transmission line(source plane) where the field is purely transverse electromag-netic (TEM). The rotational symmetry of the antenna and theexcitation allow an analysis in the two spatial coordinatesand. As shown schematically in Fig. 1, the computational domain

Manuscript received December 27, 2001; revised May 24, 2002. This workwas supported in part by the John Pippin Chair in Electromagnetics at theGeorgia Institute of Technology.

T. W. Hertel was with the School of Electrical and Computer Engineering,Georgia Institute of Technology, Atlanta, GA. He is now with Time DomainCorporation, Cummings Research Park, Huntsville, AL 35806 USA (email:[email protected]).

G. S. Smith is with the School of Electrical and Computer Engineering,Georgia Institute of Technology, Atlanta, GA 30332-0250 USA (email:[email protected]).

Digital Object Identifier 10.1109/TAP.2003.815414

Fig. 1. Model for the linear monopole antenna.

Fig. 2. A comparison of the FDTD results with measurements for the inputadmittance of the linear monopole.

is discretized using rectangular cells with the dimensionsand .

Fig. 2 is a comparison of theoretical (FDTD) and measuredinput admittances [2], [3]. For clarity, the conductanceand themagnitude of the susceptance are plotted on a logarithmicscale; the and signs next to the susceptance indicate where

is positive and negative, respectively. Theoretical results are

0018-926X/03$17.00 © 2003 IEEE

1772 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 8, AUGUST 2003

TABLE IPARAMETERS FOR THEDISCRETIZED LINEAR MONOPOLEANTENNA:

h=a = 32:9

shown for the three levels of discretization given in Table I:coarse, medium, and fine, where 1, 3, and 9 cells,respectively. Notice that the other numbers in this table, viz.,

and , do not scale with , because a subcellapproach was used to fit an integer number of cells within theinner and outer radii of the coaxial line. The number of cells perwavelength, , at the highest frequency used,4.5 GHz, is also given in the table. Clearly, the FDTD solutionconverges to the measured results with increasing level of dis-cretization over the nearly 300 : 1 range displayed forand

in the figure. Even the medium level of discretization, withonly three cells across the inner radius of the coaxial line, is inexcellent agreement with the measurements.

In the FDTD analysis of an antenna, a simple model is oftenused for the feed region of the antenna, i.e., a model that doesnot correspond exactly to the geometry of the feed region forthe actual antenna. There are several reasons for using a simplemodel, for example, the level of discretization required to accu-rately model the actual geometry in the feed region may be im-practical, then the simple model is used to save computationalresources. Or, the geometry of the actual feed region may be un-known, such as when a balanced antenna is fed by a circuit thatincludes a balun, then the simple model is used in an attempt toseparate the analysis of the antenna from that of the balun.

There are two important points we would like to make aboutthese simple feed models: The first point: The input admittanceof the antenna computed using a simple feed model, in gen-eral, will not be in excellent agreement with the measured inputadmittance, viz., not in as good agreement as shown in Fig. 2.This is a result of the geometries of the theoretical and experi-mental models being different in the feed region. This problemis not new or unique to numerical methods such as FDTD; ithas been a concern in antenna analysis for at least the last 70years, and it is discussed in some detail in the treatise by King[4]. The second point: The admittances computed using two dif-ferent simple feed models will not necessarily agree preciselywith each other. This is a result of the two simple feed modelsnot having exactly the same geometry in the feed region. Wewill illustrate these two points later in this paper. However, theobjective of this paper is not to rehash these well-known prob-lems. Rather, it is to show, once a simple feed model is adopted,what steps must be taken to ensure that the results for the inputadmittance/impedance converge, and to offer some estimate forthe level of discretization required for convergence.

In Section II of this paper, a brief overview is given of severalsimple feed models, and this is followed by a detailed descrip-tion of the two simple feed models analyzed in this paper: the

(a) (b)

(c) (d)

Fig. 3. Feed models for a simple wire antenna: (a) gap feed, (b) frill feed,(c) monopole transmission-line feed, and (d) infinitesimal-gap feed.

hard-source feed and the transmission-line feed. The remainderof this paper is concerned with a convergence study for thesetwo feed models. In Section III, the simple dipole antenna to beanalyzed and the discretizations for the convergence study areintroduced. The results obtained, when the elements of the feedmodels are arranged in the simplest and most naive way, are pre-sented in Section IV. Section V describes the critical factors thatmust be considered to obtain convergence, presents the resultsof the convergence study, and provides guidelines for the dis-cretization necessary for convergence. In Section VI, the resultsfor the two feed models are compared with measurements andwith each other.

II. A N OVERVIEW OF SIMPLE FDTD FEED MODELS

The purpose of this section is to review previous work onFDTD feed models. Section II-A gives a brief overview of sev-eral feed models, and Section II-B describes in more detail thetwo feed models that will be studied in this paper.

A. Background

Several different feed models have been proposed for systemssuch as microstrip circuits, printed antennas, and linear wire an-tennas. A feature many of these models have in common is asource gap, i.e., the metallic structure of the antenna/circuit isbroken at a point, and a source is placed in the gap. This is il-lustrated in Fig. 3(a) for the monopole above a ground planewhere the gap is located at the antenna/ground–plane interface.The tangential electric field components within the gap are thenrelated to an impressed voltage. In the most common configura-tion, often referred to as the hard-source feed [5]–[8], these field

HERTEL AND SMITH: ON THE CONVERGENCE OF COMMON FDTD FEED MODELS FOR ANTENNAS 1773

components force the total voltage to follow a time-domain ex-pression, usually a finite-duration signal, e.g., a differentiatedGaussian pulse in time. When the damping due to radiation issmall, the settling time for the current on the antenna can belong, and thus require excessively long computer run time forthe hard-source feed. This run time can be reduced significantlyby including a resistance in the source which absorbs energy [9].

In another feed model, the impressed voltage is introducedin a one-dimensional transmission line that virtually attaches atthe gap in the antenna [10]. This transmission-line feed is es-pecially advantageous for time-domain simulations because thereflected voltage in the transmission line is obtained directly.The computer run time for the transmission-line feed is signifi-cantly less than that for the hard source, because the resistanceof the matched transmission line absorbs energy.

Despite the popularity of the gap-feed model, these modelshave an inherent flaw: the drive-point gap introduces errors inthe susceptance due to its own susceptance [11]. These errorscan be avoided if feed models are implemented that do not phys-ically break the antenna geometry at the drive point. One suchmodel uses an equivalent frill generator [5], [6]. In this model,the magnetic or electric fields surrounding the conductor areused to impress a current on the antenna, see Fig. 3(b).

Other gapless feed models were developed for just thin-wireantennas in which the contours that are used to update the mag-netic field components next to the drive point are slightly mod-ified. In the transmission-line feed for monopole antennas, thecoaxial line, see Section I, is replaced by a simple one-dimen-sional transmission line [12] that virtually attaches at the an-tenna/ground–plane interface, as shown in Fig. 3(c). Here, thereflected voltage of the line enters the contourin the seg-ment along the ground plane. A similar hard-source approachimpresses a total voltage in an “infinitesimal gap” [13]. Here,the total voltage enters the contour in the segment along the an-tenna conductor, as shown in Fig. 3(d).

B. Hard-Source and Transmission-Line Feed

Fig. 4 presents schematic drawings for the two feed models tobe analyzed in this paper: Fig. 4(a) is for the hard-source feed,and Fig. 4(b) is for the transmission-line feed. For both models,the perfect conductor of the antenna is broken at the drive pointby a gap of width . The electric field in this gap is related tothe terminal voltage between the drive-point terminals by

(1)

In the FDTD method, this integral is converted into a summa-tion that easily can be rewritten to obtain the modified updateequations for the electric field in the gap as a function of theterminal voltage, e.g.,

(2)

where the gap extends in thedirection, as in Fig. 4, andis the number of cells within the gap, i.e., . In bothfeed models, the currentat the drive point is determined using

(a)

(b)

Fig. 4. Schematic drawings showing (a) the hard-source feed and (b) thetransmission-line feed.

the magnetic field surrounding the conductor of the antenna withthe relation

(3)

The contour is shifted away from the center of the gap in thedirection by the distance , see Fig. 4(a).In the hard-source feed, Fig. 4(a), the voltage in (1) is the

total impressed voltage , and it is specified. The trans-mission-line feed, Fig. 4(b), contains a one-dimensional, virtualtransmission line that is attached at the drive-point gap. The vari-ables for this transmission line are the currentand the voltage

; as shown in Fig. 5, they are staggered in space and time asin the conventional FDTD approach. The excitation is producedby a “one-way injector” located back in the transmission line atthe source plane [12], [14]. It is referred to as one-way injector,because the incident signal is launched at the source plane inonly one direction—toward the drive point. This nonphysicalsource implementation makes it possible to study the reflectedvoltage from the antenna at the observation plane located belowthe source plane without the superposition of the incident signal.The incident voltage is specified in this model. A simpleabsorbing boundary condition is placed at the bottom end of thetransmission line. Thecoupling of the transmission line to theantennais established by updating the electric field in the gapusing the last voltage in the line, i.e., (2) with .The voltage coupled to the line is marked with an open dot inFigs. 4(b) and 5. On the other hand, thecoupling of the antenna


Fig. 5. The one-dimensional transmission line used in the transmission-linefeed.

to the transmission lineis established by relating the last currentin the line to the magnetic field near the drivepoint of the antenna using (3).

For the examples to be discussed, the excitation for bothmodels will be the differentiated Gaussian voltage pulse

(4)

where is the characteristic time of the pulse. Note thatfor the hard-source feed, and

for the transmission-line feed. This pulse has the advantagethat it has a well-defined spectral peak, and it does not containany dc component, which can cause long settling times in thenumerical simulation. Frequency-domain quantities, such asthe input admittance, are obtained using the Fourier transformof the time-domain results. In the hard-source feed, the inputadmittance is determined using the current into the antenna arm,(3), and the impressed voltage, (4). In the transmission-linefeed, the input admittance is determined using both the incidentand reflected voltages within the transmission line.

III. PROBLEM GEOMETRY AND FDTD DISCRETIZATION

The convergence study is based on the input admittance andis performed for the perfectly conducting, linear dipole antennaof length with square cross section of width, shown inFig. 6(a). The dipole’s length to width ratio is held fixed at

with 10 cm. Notice that the half-length of thisdipole matches the heightof the monopole discussed inSection I, and the width of the square conductor has been setto , so that it is approximately equivalent to the roundconductor (radius) of the monopole [15]. This structure is veryeasy to discretize with the FDTD method using cubic Yee cells.For simplicity, at each step in the convergence study, the nextfiner discretization is achieved by decreasing the size of the cells

(a)

(b)

Fig. 6. (a) Model of the linear dipole with square cross section, (b) coarsestdiscretization for the model (w=�x = 2, L=�x = 77).

TABLE IIPARAMETERS FOR THEDISCRETIZED LINEAR DIPOLE ANTENNA OF

SQUARE CROSSSECTION

by a factor of two in each direction, i.e., doubling the number ofcells in each direction. Fig. 6(b) shows the coarsest discretiza-tion used for the antenna. The dipole has 77 cells along its totallength, 2 cells along its width, and 1 cell across the gap. Theshaded areas in this picture are the faces of the FDTD cells alongwhich the tangential electric field components are set to zero, sothey delineate the boundaries of the perfectly conducting struc-ture. The excitation is the differentiated Gaussian voltage pulse(4), with a ratio of characteristic times for the pulse and theantenna of .

Generally, four different discretizations were used for theconvergence study: coarse, medium, fine, and superfine. Shownin Table II are the number of cells per antenna width ,the number of cells per antenna length , and the numberof cells per wavelength at the highest frequencyused, 7.0 GHz. The number of cells per wavelength


(a) (b) (c)

Fig. 7. Basic feeding scheme shown for the first three discretizations.

is often used to determine the numerical dispersion for themethod, i.e., the deviation of the phase velocityof a wavein the FDTD grid from the speed of light. A straightforwardanalysis was used to determine the ratio at the shortestwavelength , assuming the wave propagated along onlyone direction in the three-dimensional grid [16]. Theseresults are presented in the last row of the Table II. Notice thatalready for the coarsest discretization, with approximately 30cells per wavelength, there is very little dispersion.

As a point of reference, we note that the “rule of thumb” oftenused for FDTD analysis is that at least 10 cells per shortest wave-length should be used to reduce numerical dispersion to a rea-sonable level. For this level of discretization, .Of course, numerical dispersion is not the only factor that deter-mines the accuracy of the FDTD computation. In regions wherethe details of the local geometry are important, such as at edgesand corners, a much finer discretization may be required to ob-tain acceptable accuracy. The results presented later will makethis point evident.

IV. M OTIVATION

In this section, the motivation for this work is described bypresenting results for the admittance when thebasic feedingschemeis employed for both the hard-source feed and the trans-mission-line feed. The basic feeding scheme is the simplestand most naive approach for relatingand to the electro-magnetic field components. The geometry of the feed regionis shown in Fig. 7 for the first three discretizations (coarse,medium, and fine). Fig. 7(a) is a “bird’s eye” view of the feedregion, Fig. 7(b) is a top view, and Fig. 7(c) is a cross-sectionalview. In each discretization, a one-cell gap is employed, i.e.,the physical length of the gap changes with the discretization,see Fig. 7(b), while the total length of the antenna remainsthe same. Here, just a single electric field in the center of the

(a)

(b)

Fig. 8. Input admittance obtained for the dipole with the basic feeding scheme:(a) hard-source feed and (b) transmission-line feed.

Fig. 9. Schematic illustration of the change of the capacitance with varyinglength of the gap.

gap is updated with the modified update equation based on(1). The current is determined by integrating directly aroundthe drive-point gap, i.e., the contour is not shifted [in Fig. 4(a)] and has the smallest cross-sectional area; it justencloses the surface of the conductor, see Fig. 7(c). The mag-netic field components and that mark the rectangularcontour are seen to move closer to the antenna conductorthe finer the discretization. The distance between the surface


(a) (b) (c)

Fig. 10. Improved feeding scheme shown for the first three discretizations.

of the conductor and the magnetic field component is exactlya half spatial step of the corresponding grid.

The results for the input admittance (, ) from this studyare shown in Fig. 8 for the first three discretizations. Fig. 8(a) isfor the hard-source feed and Fig. 8(b) is for the transmission-linefeed. Again, the admittance is shown on a logarithmic scale forclarity. The results appear to converge with finer discretizationat the lower frequencies, but they clearly do not converge at thehigher frequencies.

V. IMPROVED FEEDING SCHEME AND DISCRETIZATION

NECESSARY FORCONVERGENCE

To obtain convergence, it is crucial to properly position andscale the elements in the feed model. Fig. 9 illustrates how dif-ferent geometrical lengths of the gap introduce a change in thesusceptance. As shown schematically, the decrease in length in-creases the capacitance at the drive point. To avoid this sourceof error, the length of the drive point-gap must be the same foreach discretization.

The above criteria are met in theimproved feeding scheme.Schematic drawings of the drive-point region for this schemeare shown in Fig. 10. Here, the drive-point gap properly scalesfor each discretization by keeping the length of the gap constantat . For discretizations finer than the coarsest one, thisrequires a multiple-cell gap, see Fig. 10(b). In order to apply thevoltage in the gap uniformly, all tangential electric field com-ponents within the gap are updated based on (1). The currentcontour in this new model is shifted to to avoid thefringing of the electric field in the gap.1 Furthermore, the con-tour is chosen to be the same for every discretization. Noticethat in order to properly scale and position the current contour,

1The motivation of an offset contour [s > l =2 in Fig. 4(a)] is to enclose thetotal current into the antenna conductor, i.e., the conduction current, instead ofa portion of the displacement current that is obtained when the contour enclosesthe gap.

Fig. 11. Proper scaling of the reference planes in the transmission line.

averages in the transverse and longitudinal directions are nec-essary for the discretizations finer than the coarsest one. Thiscan clearly be seen in the schematic drawings in Fig. 10(b) (topview) and in Fig. 10(c) (cross-sectional view).

For the transmission-line model, it is crucial to properly po-sition the reference planes for the voltages and currents in thetransmission line. In the present approach, the reference planefor the current in the last cell is chosen to be the same for alldiscretizations.2 This plane is shown as a solid line in Fig. 11.The reference plane for the voltage that couples the line to theantenna must be placed at the same location for all discretiza-tions.3 This plane is set by the position of the voltage in the lastcell of the coarsest grid and is shown as a dashed line in Fig. 11.Notice that for discretizations finer than the coarsest one, av-erages are necessary to determine the corresponding voltage atthis plane.

The input admittance is shown as a function of frequency inFig. 12 for the first three discretizations. The results are seento converge for the hard-source feed in Fig. 12(a) and for thetransmission-line feed in Fig. 12(b). To better examine the con-vergence of the results, the admittances are shown for the im-proved feeding scheme in Fig. 13 over the limited frequency

2Recall thatI(l ) couples the antenna in the three-dimensional FDTD gridto the one-dimensional transmission line.

3Recall that in the basic feeding scheme,V(l ) is used for the coupling.


(a)

(b)

Fig. 12. Input admittance obtained for the dipole with the improved feedingscheme: (a) hard-source and (b) transmission-line feeds.

range 3.9 GHz 4.1 GHz. Clearly, the results for theconductance and the susceptance converge for both thehard-source and transmission-line feed.

To analyze the results for convergence quantitatively, thefrequency and the conductance at antiresonance(the point where is zero for the second time, , orequivalently ) are given for the hard-source feed inTable III. Clearly, and are seen not to converge for thebasic feeding scheme in Table III (top); whereas, they convergefor the improved feeding scheme in Table III (bottom). Noticethat the number of cells required for convergence is fairly large:Even for the coarse discretization, with about 100 cells perwavelength, the conductance at antiresonance differs from thefinal value (superfine discretization) by about 3%. Thus, wesee that the rule of thumb of 10 cells per wavelength neededto control numerical dispersion is rather meaningless for thisexample. A much finer discretization is needed to model the

Fig. 13. Comparison of the results for the admittance: improved feedingscheme.

TABLE IIIFREQUENCY ANDCONDUCTANCE AT ANTIRESONANCE FOR THEHARD SOURCE:(TOP) BASIC FEEDING SCHEME: (BOTTOM) THE IMPROVEDFEEDING SCHEME

details in the feed region of this antenna and to determine anaccurate value for the input admittance.

VI. DISCUSSION

To complete our discussion, we will return to the two pointsmade in the Introduction to the paper: The first point is thatthe input admittance of the antenna computed using a simplefeed model, in general, will not be in very good agreement withthe measured input admittance. And the second point is that theadmittances computed using two different simple feed modelswill not necessarily agree with each other.

In Fig. 14, we compare the input admittances for the dipoleantenna computed using the two simple feed models (hard-source and transmission-line feeds) with the measured resultsfor the monopole antenna described in Section I. The measuredresults for the monopole shown in Fig. 2 have been dividedby a factor of two for this comparison. Both of the computed


Fig. 14. Comparison of the numerical results for the input admittance with themeasurement.

results are for the fine discretization, so, as shown in Fig. 12,the admittances have converged. The two theoretical results forthe input conductance are in very good agreement with themeasurements. However, neither of the computed results forthe input susceptance is in very good agreement with themeasurements, although the results for the transmission-linefeed are significantly closer to the measurements than thosefor the hard-source feed. Clearly, to obtain excellent agreementbetween theory and experiment, as demonstrated in Fig. 2, allof the details of the experimental model must be included inthe theoretical model.

In Fig. 14, the two theoretical results for the input susceptanceare significantly different. This difference is caused by the

differences in the geometry in the drive-point region for the twotheoretical models. When the two susceptances are subtractedand a capacitance computed, i.e., , theresult shown in Fig. 15(a) is obtained. Notice that the capac-itance is approximately 0.04 pF over the range of frequenciesof interest. As shown in Fig. 15(b), when this capacitance isadded to the admittance calculated with the hard-source feed,it is in excellent agreement with the admittance calculated withthe transmission-line feed. So the difference in the two simplefeed models can be attributed to a local capacitance at the drivepoint. Clearly, to obtain excellent agreement between results fortwo theoretical models, all of the details for the two models, suchas those in the feed region, must be the same.

From the results for the susceptance in Fig. 14, onemight conclude that the transmission-line feed is a better modelthan the hard-source feed because it is in better agreement withthe measurements. However, this conclusion would not be true.The susceptance for the hard-source feed can be put into betteragreement with the measurements by adjusting the size of thegap and the position of the contour. This points out a weaknessof simple feed models. They have free parameters, and there areno unique values for these parameters that will produce equallygood agreement with measurements for a variety of antennas.

(a)

(b)

Fig. 15. (a) Plot of the additional local capacitance at the drive point.(b) Comparison of the input susceptance for the transmission-line feed andthe hard-source feed when the capacitanceC = 0.04 pF is added in thehard-source feed.

VII. CONCLUSION

Two antenna feed models (hard-source and transmission-linefeeds) commonly used in the FDTD method were shown to con-verge only when the elements of the feed are properly scaledand positioned. These elements are the length of the drive-pointgap and the size and position of the current contour. For thetransmission-line feed, this also includes the proper positioningof the reference plane for the voltage in the transmission linethat couples the line to the antenna. The discretization necessaryfor convergence was shown to be much finer than one typicallyexpects.

REFERENCES

[1] J. G. Maloney, G. S. Smith, and W. R. Scott Jr., “Accurate computa-tion of the radiation from simple antennas using the finite-differencetime-domain method,”IEEE Trans. Antennas Propagat., vol. 38, pp.1059–1068, July 1990.


[2] W. R. Scott Jr., “Dielectric spectroscopy using shielded open-circuitedcoaxial lines and monopole antennas of general length,” Ph.D. disserta-tion, Georgia Inst. Technol., Atlanta, GA, 1985.

[3] T. W. Hertel and G. S. Smith, “The insulated linear antenna—revisited,”IEEE Trans. Antennas Propagat., vol. 48, pp. 914–920, June 2000.

[4] R. W. P. King,The Theory of Linear Antennas. Cambridge, MA: Har-vard Univ. Press, 1956.

[5] R. Luebbers, L. Chen, T. Uno, and S. Adachi, “FDTD calculation of ra-diation patterns, impedance, and gain for a monopole antenna on a con-ducting box,”IEEE Trans. Antennas Propagat., vol. 40, pp. 1577–1583,Dec. 1992.

[6] P. A. Tirkas and C. A. Balanis, “Finite-difference time-domain methodfor antenna radiation,”IEEE Trans. Antennas Propagat., vol. 40, pp.334–340, Mar. 1992.

[7] M. R. Zunoubi, N. H. Younan, J. H. Beggs, and C. D. Taylor, “FDTDanalysis of linear antennas driven from a discrete impulse excitation,”IEEE Trans. Electromagn. Compat., vol. 39, pp. 247–250, Aug. 1997.

[8] C. E. Brench and O. M. Ramahi, “Source selection criteria for FDTDmodels,” inProc. IEEE Electromagnetic Compatibility Symp., vol. 1,1998, pp. 491–494.

[9] R. J. Luebbers and H. S. Langdon, “A simple feed model that reducestime steps needed for FDTD antenna and microstrip calculations,”IEEETrans. Antennas Propagat., vol. 44, pp. 1000–1005, July 1996.

[10] J. M. Bourgeois and G. S. Smith, “A fully three-dimensional simula-tion of a ground-penetrating radar: FDTD theory compared with exper-iment,” IEEE Trans. Geosci. Remote Sensing, vol. 34, pp. 36–44, Jan.1996.

[11] E. Semouchkina, W. Cao, and R. Mittra, “Source excitation methodsfor the finite-difference time-domain modeling of circuits and devices,”Microwave Opt. Technol. Lett., vol. 21, pp. 93–100, 1999.

[12] J. G. Maloney, K. L. Shlager, and G. S. Smith, “A simple FDTD modelfor transient excitation of antennas by transmission lines,”IEEE Trans.Antennas Propagat., vol. 42, pp. 289–292, Feb. 1994.

[13] S. Watanabe and M. Taki, “An improved FDTD model for the feedinggap of a thin-wire antenna,”IEEE Microwave Guided Wave Lett., vol.8, pp. 152–154, Apr. 1998.

[14] J. G. Maloney and G. S. Smith, “Modeling of antennas,” inAdvancesin Computational Electrodynamics: The Finite-Difference Time-DomainMethod, A. Taflove, Ed. Norwood, MA: Artech House, 1998.

[15] C. A. Balanis,Antenna Theory Analysis and Design. New York: Wiley,1997.

[16] A. Taflove, Computational Electrodynamics: The Finite-DifferenceTime-Domain Method. Norwood, MA: Artech House, 2000.

Thorsten W. Hertel (S’96–M’02) was born in Holz-minden, Germany, on April 19, 1974. He receivedthe Vordiplom degree in electrical engineeringfrom the Technische Universität Braunschweig,Braunschweig, Germany, in 1995, and the M.S.and Ph.D. degrees from the Georgia Institute ofTechnology (Georgia Tech), Atlanta, in 1998 and2001, respectively, both in electrical and computerengineering.

In 2002, for six months, he continued to work atGeorgia Tech as Postdoctoral Fellow, working on

pulsed and broad-band antennas. Since July of 2002, he has been an AntennaDesign Engineer with Time Domain Corporation, Huntsville, AL, working onultra-wide-band antennas. His special interests include numerical modelingwith the finite-difference time-domain (FDTD) method and antenna analysis.

Glenn S. Smith(S’65–M’72–SM’80–F’86) receivedthe B.S.E.E. degree from Tufts University, Medford,MA, in 1967 and the S.M. and Ph.D. degrees in ap-plied physics from Harvard University, Cambridge,MA, in 1968 and 1972, respectively.

From 1972 to 1975, he served as a PostdoctoralResearch Fellow at Harvard University and also as apart-time Research Associate and Instructor at North-eastern University, Boston, MA. In 1975, he joinedthe faculty of the School of Electrical and ComputerEngineering at the Georgia Institute of Technology,

Atlanta, where he is currently Regents’ Professor and John Pippin Chair in Elec-tromagnetics. His technical interests include: basic electromagnetic theory andmeasurements, antennas and wave propagation in materials, and the radiationand reception of pulses by antennas. He is the author of the bookAn Introduc-tion to Classical Electromagnetic Radiation(Cambridge Univ. Press, 1997) andcoauthor of the bookAntennas in Matter: Fundamentals, Theory and Applica-tions (MIT Press, 1981). He also authored the chapter ”Loop Antennas” in theMcGraw-Hill Antenna Engineering Handbook, 1993.

Prof. Smith is a Member of Tau Beta Pi, Eta Kappa Nu, and Sigma Xi and amember of URSI Commissions A and B.

twhertel august 2003

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