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© 2005 Wiley Periodicals, Inc.

ENHANCEMENTS TO THE FDTDMETHOD FOR STEADY-STATE POWER-DEPOSITION ANALYSIS INMICROWAVE-HEATING CAVITIES

Jacob George, Mark Muktoyuk, and Richard BergmanScience and Technology DivisionCorning IncorporatedCorning, NY 14831

Received 11 June 2005

ABSTRACT: We describe two enhancements to the finite-differencetime-domain (FDTD) method for the steady-state power-dissipationanalysis of microwave-heating cavities. The first enhancement accountsfor the accumulated numerical errors in the simulation due to nonphysi-cal reflection effects from the absorbing boundary condition (ABC). Thesecond enhancement is a new convergence metric for the determinationof steady state, using the variance of the power-balance error in thesystem over a number of excitation periods. © 2005 Wiley Periodicals,Inc. Microwave Opt Technol Lett 47: 530–534, 2005; Published onlinein Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21220

Key words: FDTD; microwave heating; power balance

1. INTRODUCTION

The FDTD method is a forward time-stepping algorithm that iswidely used for numerical solution of electromagnetic fields. Forheating problems [1–4], the method is typically terminated whensteady state is achieved. One method to determine steady state isto use a specific number of time cycles of the input wave form [1];however, this method has the drawback of being problem specific,requiring expertise in determining the number of time steps suffi-cient to achieve steady state. Alternatively, the relative error in thepower dissipation can be calculated periodically [2]. This approachis more general, but fails in the presence of an oscillatory transientresponse in the power dissipation in the material.

A third approach, described in [3], uses the instantaneouspower flow to the cavity to determine steady state. However, aswith the first two, this approach fails to identify simulation errorsdue to nonphysical reflections from nonperfect absorbing bound-ary conditions (ABCs) [5]. None of the above methods tracks thepower balance in the system by taking into account the power lostdue to both physical and numerical reflections.

In this paper, we present two enhancements to the FDTDmethod: first, we develop a power balance calculation whichaccounts for both physical reflections due to impedance mismatchand nonphysical (that is, numerical) reflections at an absorbingboundary. Second, we develop a convergence metric to determinesteady state for the FDTD method by tracking the incrementalchange in variance of the residual of the newly developed powerbalance calculation.

2. POWER BALANCE IN HEATING CAVITIES AND THECONVERGENCE METRIC

2.1 Power Balance to Account for Numerical ReflectionsFigure 1(a) shows the schematic of a cavity containing two lossymaterials: material A acts as a thermal insulator, while material Bis to be thermally processed. Typically, the forward power flowinside the feed waveguide is determined by the power dissipationin both lossy materials and on the cavity walls. However, weassume the cavity walls are perfect electric conductors (PECs),allowing us to ignore the power dissipated in the walls. We

530 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 6, December 20 2005

therefore attribute the time-averaged power flow into the cavity(across any transverse plane in the feed waveguide) to the powerdissipated in the lossy materials only. This is described using thereal part of the complex Poynting theorem:

Re���S

1

2E� � H� * � ds�� � ��0��A �

VA

�E� Arms�2dVA

� ��0��B �VB

�E� Brms�2dVB, (1)

where �0 is the permittivity of free space, � �A and � �B are thecomplex part of the dielectric permittivity of the lossy materials Aand B (respectively), � is the angular frequency of the inputsinusoid, VA and VB are the volume occupied by the materials Aand B (respectively), S is the area of the transverse excitationplane, E� is the electric field, H� * is the complex conjugate of themagnetic-field vector, and finally E� Arms and E� Brms are the root

mean square (RMS) values of the electric field at a point inmaterials A and B, respectively.

If EXrms, EYrms, and EZrms are the interpolated RMS fieldamplitudes at the center of a Yee cell [5] corresponding to the threespatial components (X, Y, and Z), then for �E� Arms�, material A isgiven by [4]:

�E� Arms�2 � EX rms2 � EY rms

2 � EZ rms2 . (2)

The RMS field corresponding to each of the twelve field compo-nents in the FDTD cell may be calculated as in [4]:

�Erms� � �2�¥n�1

N E�n�t�ej�n�t�N

, (3)

where N�t is the period of the input sinusoid, and N is the totalnumber of steps in a period.

The analytical expression of power flow across a transverseplane for the TE10 mode in a uniform waveguide is given by [6]:

Figure 1 Large-cavity details and results: (a) cavity details (the cavity has a front door with 76.2-mm-thick material A block, as well); (b) feed detailsshowing the TE10 mode excitation plane and scattered power estimation plane (feed is located centrally at the back of the cavity); (c) power balance; (d)convergence metric (steady state when tolerance is less than 0.005)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 6, December 20 2005 531

P10 � �ab

4 �� �

��� �E0�2, (4)

where a and b are the broadside width and height of thewaveguide, respectively, � is the phase-shift constant, and E0 isthe maximum electric-field amplitude in the transverse plane cor-responding to the TE10 mode. The left-hand side of Eq. (1)accounts for the time-averaged power flow into the cavity acrossthe transverse plane S [see Fig. 1(b)]. This can be obtained bytaking the difference of the Eq. (4) and the numerically computedscattered power flow across a transverse plane [plane B in Fig.1(b)] in the waveguide. However, this approach does not take intoaccount the forward-propagating waves in this region due to thenumerical reflections at the absorbing boundary behind thewaveguide feed. Indeed, these reflections produce standing wavesbetween the ABC and the source plane, thus decreasing the pre-cision to which power balance is calculated, in that (i) the steady-state power estimate will depend upon the location at which it iscalculated, and (ii) forward-propagating power will include non-physical reflections and therefore be inaccurate.

In our approach, we correct for numerical reflections by calcu-lating the average (Es

av) of the maximum and minimum of thestanding-wave pattern between the ABC and the source plane. Thisaverage is the power of the scattered fields in the absence ofnumerical reflections from the ABC, and is used for two purposes:(i) to calculate the true scattered power flow across a transverseplane in the waveguide using Eq. (4) and (ii) to obtain the correc-tion factor, defined as

E0c � Es

source Esav, (5)

where Esav is the ABC reflection-free scattered field in the

waveguide and Essource is the maximum electric-field value on the

plane just one cell behind the source plane (the source plane is notused because the source plane uses total field formulation). Wethen calculate the corrected source power P10

c in order to eliminatethe effect of the reflections from the ABC, using

P10c � �ab

4 �� �

��� �E0 � E0c�2. (6)

2.2 Convergence MetricIn order to develop the convergence metric to identify steady state,we define the normalized steady state error ess using

ess � 1 �P̃d � P̃r�, (7)

where P̃d and P̃r are the normalized power dissipation and powerreflection, respectively, calculated by the method described above.For instance, for the cavity shown in Figure 1(a), the normalizeddissipated power in the two materials is given by

P̃d �PA � PB

P10c , (8)

where PA and PB are the time-averaged power dissipation inmaterials A and B, respectively. The convergence metric is basedupon calculating the maximum incremental change in normalizedvariance of ess over a running-time window, using

vardiff�P� �

maxi�P��,P

�var��i� var��i 1��

�var��� � 1��, (9)

where P is the period number, � is the number of periods in therunning window, and

var��i� � var�ess�i ��, ess�i � � 1�, . . . , ess�i 1�, ess�i��.

(10)

Here, � � 10 was chosen to ensure monotonicity in theconvergence metric, where is approximately the number ofwavelengths in the longest dimension of the cavity. When vard-

iff(P) 0.005 (which is sufficient in the authors’ experience)over the time window �, steady state is achieved and we terminatethe simulation. With this metric we are able to correctly identify anumerical steady state (beyond which the simulation fails to in-crease accuracy), even in the presence of transient oscillatorypower dissipation in the material.

3. SIMULATION DETAILS, NUMERICAL RESULTS, ANDDISCUSSIONS

The two enhancements presented in the previous sections areapplied and demonstrated for two cavities here. The steady-stateerror in both cases is estimated in terms of the deviation from unityfor the sum of all the dissipations plus reflections normalized withrespect to the input power. This estimation is performed at thepoint in time where steady state is identified by the metric pro-posed in this paper.

3.1 Large CavityA cavity with lossy materials A (insulation, with �r � 1.7 �j0.001) and B (laminated fiberglass [7], with �r � 4.4 � j0.128)is depicted in Figure 1(a). The TE10 mode in the WR-340 feedwaveguide is excited with a sinusoid of frequency 2.45 GHz in thesource plane S [Fig. 1(b)]. The cell size used is 5.83 mm in allthree directions and the time step was chosen to satisfy the Courantstability condition [5]. Mur’s 2nd-order ABC [8] is used to absorbthe scattered fields at the back side of the feed. Figure 1(c) showsthe dissipated and reflected powers (corrected using the proposedmethod) as a function of time.

In this case, our method improves the steady state error to2%, whereas a method similar to the one described in [3] isfound to introduce more than 5% error due to numerical reflec-tions. Also, a steady-state identification scheme based on power-dissipation information from single cycles of input excitation [2]might easily get caught near 80 ns [Fig. 1(c) for material B] of thepower-dissipation response, and result in a wrong prediction inabsolute-power dissipation. From Figure 1(d), we see that theconvergence metric is monotonically decreasing, resulting in con-vergence near 150 ns. Note that since the variance defined by Eq.(10) requires � excitation periods, its first value is calculated at(� � 1)th period. Further, since the vardiff metric defined by Eq.(9) identifies the maximum change in consecutive variances overthe previous � variances, the first of the convergence-metric dataoccurs at the (2� � 1)th period of excitation.

3.2 Small CavityThe example considered here is adopted from [2]: a 17.2 � 17.2 �14.907 cm cavity with a 14.32-cm-long WR-340 waveguide feed.The uniform cell size used here is 5.73 mm, the excitation fre-quency is 2.45 GHz, and the time step is chosen to meet theCourant condition [5]. A lossy material (�r � 2 � j0.5) is placed4.587 cm from the top of the cavity, as shown in Figure 2(a). Eq.(1), suitably modified for a single material case, is implementedhere for 50 time periods of the input waveform and the normalizedpowers are plotted in Figure 2(c).

532 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 6, December 20 2005

Comparing Figures 2(b) and 2(c), we see that the lack of acorrection scheme for the numerical reflections is prone to makinga total steady-state error as high as 10%, whereas the methodproposed here reduces this error to 1%. From Figure 2(d), we seethat the convergence metric is monotonically decreasing, thusresulting in convergence near 26 ns.

The convergence history of the two problems presented above[Figs. 1(d) and 2(d)] demonstrates the dependency of convergencespeed to the geometry, size, and type of materials considered in themodel. However, the results indicate that the new generalizedconvergence metric proposed is well suited for a wide range ofcavity parameters.

4. CONCLUSIONS

Two enhancements to the FDTD method have been demonstrated.We have defined a correction to the source field that eliminates theeffects of numerical reflections from the ABCs to the power-

balance calculation. In addition, we develop an algorithm to de-termine convergence, using the residual of the normalized totalestimated power from the power-balance calculations. Thismethod is sufficiently robust to correctly identify convergence,despite (i) the presence of plateaus in the power balance, whichmay falsely indicate steady state, and (ii) the presence of ripples atsteady state, which may falsely indicate the transient regime.

ACKNOWLEDGMENTS

The authors wish to acknowledge the fruitful discussions withZagorka Gaeta and Leslie J. Button.

REFERENCES

1. M. Sundberg, P.O. Risman, P.-S. Kildal, and T. Ohlsson, Analysis anddesign of industrial microwave ovens using the finite difference timedomain method, J Microwave Power Electromagn Energy 31 (1996),142–157.

Figure 2 Small-cavity details and results: (a) cavity schematic; (b) power balance with no correction; (c) power balance with correction; (d) convergencemetric (steady state when tolerance is less than 0.005)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 6, December 20 2005 533

2. F. Liu, I. Turner, and M. Bialkowski, A finite-difference time-domainsimulation of power density distribution in a dielectric loaded micro-wave cavity, J Microwave Power Electromagn Energy 29 (1994), 138–149.

3. S. Hanafusa, T. Iwasaki, and N. Nishimura, Electromagnetic fieldanalysis of a microwave oven by the FD-TD method: A considerationon steady state analysis, IEEE Antennas Propagat Soc Int Symp, Seattle,WA, 1994, vol. 3, pp 1806–1809.

4. L. Ma, D. Paul, N. Pothecary, C. Railton, J. Bows, L. Barratt, J. Mullin,and D. Simons, Experimental validation of a combined electromagneticand thermal FDTD model of a microwave heating process, IEEE TransMicrowave Theory Tech 43 (1995), 2565–2572.

5. A. Taflove and S.C. Hagness, Computational electrodynamics—thefinite-difference time-domain method, 2nd ed., Artech House, Norwood,MA, 2000.

6. R.E. Collin, Foundations for microwave engineering, McGraw Hill,New York, 1966.

7. A.C. Metaxas and R.J. Meredith, Industrial microwave heating, IEEPower Engineering Series, Peter Peregrinus Ltd., London, 1993.

8. G. Mur, Absorbing boundary conditions for the finite-difference ap-proximations of the time domain electromagnetic field equations, IEEETrans Electromagn Compat 23 (1981), 377–382.

© 2005 Wiley Periodicals, Inc.

A BROADBAND PRINTED BOW-TIEANTENNA WITH A SIMPLIFIEDBALANCED FEED

Guiping Zheng, Ahmed A. Kishk, Allen W. Glisson, andAlexander B. YakovlevDepartment of Electrical EngineeringCenter for Applied Electromagnetic Systems Research (CAESR)University of MississippiUniversity, MS 38677-1848

Received 5 June 2005

ABSTRACT: A broadband printed bow-tie antenna with a simplifiedbalanced feeding network and modified tapering is presented. Mi-crostrip and parallel-strip transmission lines printed on the substratewith high dielectric permittivity realize the proposed feeding net-work. The ground-plane transition between the microstrip line andthe parallel-strip line is exponentially tapered so as to reduce thereflection losses and produce a balanced feed for the antenna. Thisprinted bow-tie antenna achieves a 68% measured bandwidth and astable radiation pattern within the X-band. Commercial FEM soft-ware is used for optimization of the bow-tie antenna and the simula-tion results agree very well with the experiment. © 2005 Wiley Peri-odicals, Inc. Microwave Opt Technol Lett 47: 534 –536, 2005;Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.21221

Key words: broadband antenna; bow-tie antenna; feeding network; mi-crostrip line

1. INTRODUCTION

New configurations of a simplified feeding network for a printeddipole antenna have been recently presented in [1, 2]. In theproposed designs, the feeding network is realized by a microstripline (implemented on a grounded dielectric slab) coupled to aparallel-strip transmission line. The upper part of the feed/antennastructure consists of a microstrip line connected to one of theparallel lines and one arm of the printed dipole antenna. The lowerpart consists of a truncated ground plane directly connected to theother parallel line, which is transitioned to the second arm of the

dipole antenna (printed on the same surface of substrate). Thetruncated ground plane acts as a reflector, resulting in a highfront-to-back ratio radiation pattern. In [3], a similar feeding-network configuration was proposed for a printed bow-tie antenna.However, in this design the truncated ground plane has a differentrole and does not act as a reflector, as in the designs presented in[1, 2]. The feeding networks presented in [4–10] achieve a bal-anced antenna current in a very narrow band. In addition, thesefeeding networks are structurally complicated and include addi-tional elements, such as a T-junction, power divider, microstrip-line discontinuity taper, and microstrip-to-coplanar stripline (CPS)balun. In the proposed simplified feeding network used in themodified printed Yagi antenna designs [1, 2], reflection losses arereduced significantly and the efficiency of the network is greatlyimproved. However, these designs have a relatively narrow band-width compared to those in [4, 5] because the two arms of thedipole antenna reside on the opposite sides of substrate with highdielectric permittivity.

In this paper, we propose a new design based on the simplifiedfeeding network in conjunction with a broadband printed bow-tieantenna [11]. The new design features broadband operation and astable broad main-beam radiation pattern in the X-band.

Figure 1 Geometry of printed bow-tie antenna with a simplified feedingnetwork: (a) 3D schematic view; (b) top layer; (c) bottom layer; and (d)photograph of the fabricated antenna

534 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 47, No. 6, December 20 2005