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849E E E TRANSACTIONS O N MICROWAVE THEORY AND TEC IINIOUES, V o l . . 38, NO. 7, JU1.Y 1990

Application of the Three-DimensionalFinite-Difference Time-DomainMethod to the Analysis ofPlanar Microstrip Circuits

Abstract-A direct three-dimensional finite-difference time-domain(FDTD) method is applied to the full-wave analysis of various microstripstructures. The method is shown to be an efficient tool for modelingcomplicated microstrip circuit components as well as microstrip anten-nas. From the time-domain results, the input impedance of a line-fedrectangular patch antenna and the frequency-dependent scattering pa-rameters of a low-pass filter and a branch line coupler are calculated.These circuits are fabricated and the measurements are compared withthe FDTD results and shown to be in good agreement.

I . I N T R O D U C T I O NREQUENCY-domain analytical work with compli-F ated microstrip circuits has generally been doneusing planar circuit concepts in which the substrate isassumed to be th in enough that p ropagat ion can beconsidered in two dimensions by surrounding the mi-crostrip with magnetic walls [1]-[6]. Fringing fields areaccounted for by using either static or dynamic effectivedimensions and permittivities. Limitations of these m eth-ods are that fringing, coupling, and radiation must all behandled empirically since they are not allowed for in themodel. Also, the accuracy is questionable when the sub-strate becomes thick relative to the width of the mi-crostrip. T o fully account for these effects, i t is necessary

to use a full-wave solution.Full-wave frequency-domain methods have been usedto solve some of th e sim pler discontinuity problems [7],[SI. However, these methods are difficult to apply to atypical printed microstrip circuit.Modeling of microstrip circuits has also been per-formed using B ergeron's m ethod [9], [lo]. This method isa modification of the transmission l ine matrix (TLM)method, and has l imitations similar to the finite-dif-Manu script received Septem ber 29, 1989; revised March 21, 1990.This work was supported by NSF G r an t 8620029-ECS, the Joint ServicesElectronics Program (Contract DAAL03-89-C-0001), RADC ContractF19628-88-K-0013, AR O Contract DAAL03-88-J-0057, ON R ContractN00014-89-J-1019, and the D epartment of the Air Force.D. M. Sheen, S . M. Ali, and J. A. Kong are with the Department ofElectr ical Eng ineering and Comp uter Science and the Research Labora-tory of Electronics, Massachusetts Institute of Technology, Cambridge,MA 02139.M. D. Abouzahra is with MIT Lincoln Laboratory, Lexington, MA02173.IEEE Log Number 9036153.

ference t ime-domain (FDT D) method due to the d iscretemodeling of space and t ime [l l] 121. A unique problemwith this method is that the dielectric interface and theperfectly conducting strip are misaligned by half a spacest ep [121.The FDTD method has been used ex tensively fo r thesolution of two- and three-dimensional scattering prob-lems [13]-[17]. Recently, FDTD methods have been usedto effectively calculate the frequency-dependent charac-teristics of microstrip discontinuities [181-[21]. Analysis ofthe fundamental discontinuit ies is of great importancesince more complicated circuits can be realized by inter-connecting microstrip l ines with these discontinuit ies andusing transmission l ine and network theory. S ome circuits,however, such as patch antennas, may not be realized inthis way. Additionally, if the discontinuit ies are too closeto each othe r the use of network conc epts will not beaccurate due to the interaction of evanescent waves. Toaccurately analyze these types of structures it is necessaryto s imulate the en t i re s t ructure in one com putat ion. TheFDTD method shows great promise in i ts flexibil i ty inhandling a variety of circuit configurations. An additionalbenefit of the t ime-domain analysis is that a broad-bandpulse may be used as the excitation and the frequency-domain parameters may be calcu lated over the en t i refrequency range of interest by Fourier transform of thetransient results.In this paper, the frequency-dependent scattering pa-rameters have been calculated for several printed mi-crostrip circuits, specifically a line-fed rectangular patchantenna, a low-pass fi l ter, and a rectangular branch l inecoupler. These circuits represent resonant microstripstructures on an open substrate; hence, radiation effectscan be significant, especially for the microstrip antenna.Calculated results are presented and compared with ex-per imental measurements .The FDTD method has been chosen over the o therdiscrete methods (TLM or Bergeron's) because i t is ex-tremely efficient, i ts implementation is quite straightfor-ward, and it may be derived directly from Maxwell'sequations. Many of the techniques used to im plement this

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850 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 38, NO. 7, J U L Y 1990method have been demonstrated previously [18]-[20];however, simplification of the method has been achievedby using a simpler absorbing boundary condition [22].This simpler absorbing boundary condition yields goodresults for the broad class of microstrip circuits consid-ered by this paper. Additionally, the so urce treatmen t hasbeen enhanced to reduce the source effects documentedin [18]-[20].

11. PROBLEM FORMULATIONThe FDTD method is formulated by discretizingMaxwell's curl equations over a finite volume and approx-imating the derivatives with cente red difference approxi-mations. Conducting surfaces are treated by setting tan-gential electric field components to 0. The walls of themesh, however, require special treatment to prevent re-flections from the mesh termination.A. GoLeming Equations

Formula t ion of the FDTD method begins by consider-ing the differential form of Maxwell's two curl equationswhich govern the propagation of fields in the structures.For simplicity, the media are assumed to be piecewiseuniform, isotropic, and homogeneous. The structure isassumed to be lossless (i.e., no volume currents or finiteconductivity). With these assumptions, Maxwell's curlequations may be written as

dHp = - V X EatdEatE - E V Y H .

In order to find an approximate solution to this set ofequations, the problem is discretized over a finite three-dimensional computational domain with appropriateboundary conditions enforced on the source, conductors,and mesh walls .B. Finite-Difference Equations

To obtain discrete approximations to these continuouspartial differential equations the centered difference ap-proximation is used on both the time and space first-orderpartial d ifferentiations. For convenience, the six fieldlocations are considered to be interleaved in space asshown in Fig. 1, which is a drawing of the F DT D unit cell[131. The entire computational domain is obtained bystacking these rectangular cubes into a larger rectangularvolume. The P, j , and 2 dimensions of the unit cell areA x , A y , and A z , respectively. The advantages of this fieldarrangement are that centered differences are realized inthe calculation of each field component and that continu-ity of tangential field components is automatically satis-fied. Because there are only six unique field componentswithin the unit cell, the six field components touching theshaded upper eighth of the unit cell in Fig. 1are cons id-ered to be a unit node with subscript indices i , j , and kcorresponding to the node numbers in the 2 9, and 2

NODE Li , k )

YJ/XFig. 1. Field compon ent placement in the F DTD uni t ce ll .

directions. This notation implicitly assumes the 1/2space indices and thus simplifies the notation, renderingthe formulas directly implementable on the computer.Th e time steps are indicated with the sup erscript n . Usingthis field component arrangement, the above notation,and the centered difference approximation, the explicitfinite differen ce approximations to 1) and (2) a re

3 )


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The half t ime steps indicate that E and H are alter-nately calculated in order to achieve centere d differencesfor the t ime derivatives. In these equa tions, the permitt iv-i ty and the permeabili ty are set to the appropriate valuesdepending on the location of each field component. Forthe electric field components on the dielectric-air inter-face the averagc of the two permittivities, E , ) + ~ , ) / 2 ,sused. The validity of this treatme nt is explained in [20].Du e to the use of centered differences in these approx-imations, the error is second order in both the space andtime steps; i.e., if A x , A y , A z , an d A t are proportional toAl, then the global error is O(hl*) . he maximum timestep that may be used is limited by the stability restrictionof the finite difference equations,

A t < - ,+-+- ( 9)I ; , ; , , A x A y A zzwhere L ,,,,,s the m aximum velocity of light in the compu-tational volum e. Typically, im nill be the velocity of lightin free space unless the entire volume is fi l led withdielectric. The se equ ations will al low the approximatesolution of E r , r ) an d H r , t ) in the volume of thecomputational domain or mesh; however, special consid-eration is required for the source, the conductors, and themesh walls.C. Source Considerations

Th e volume in which the microstrip circuit simulation isto be performed is shown schematically in Fig. 2.A t t = 0the fields are assumed to be identically 0 throughout thecomputational domain. A Gaussian pulse is desirable asthe excitation because i ts frequency spectrum is alsoGaussian and will therefore provide frequency-domaininformation from dc to the desired cutoff frequency byadjusting the w idth of the pulse.In order to simulate a voltage source excitation it isnecessary to impose the vertical electric field, E,, in arectangular region underneath port 1 as shown in Fig. 2.The remaining electric field components on the sourceplane must be specified or calculated. In [181-[201 a nelectric wall source is used; i.e., the remaining electricfield components on the source wall of the mesh are sett o 0. An unwanted side effect of this type of excitation isthat a sharp magnetic field is induced tangential to thesource wall. This results in som e distortion of th e laun chedpulse. Specifically, the pulse is reduced in magnitude dueto the energy stored in the induced magnetic field and anegative tail to the pulse is immediately evident. Analternative excitation scheme is to simulate a magneticwall a t the source plane. T he source plane consists only ofE , and E , compon ents, with the tangential magnetic fieldcomponents offset f A y / 2 . I f the magnetic wall is en-forced by sett ing the tangential magnetic field compo-nents to z ero just behind the source plane , then signifi-cant distortion of the pulse still occurs. If the magneticwall is enforced directly on the source plane by usingimage thcory (i.e., HtZlnutside the magnetic wall is equal

Fig. 2. Computational domain.

to HI,, inside the magnetic wall) , then the remainingelectric field components on the source p lane may bereadily calculated using the finite-difference equations.Using this excitation, only a minimal amount of sourcedistortion is apparent. The launched wave has nearly unitampl i tude a n d is Gaussian in t ime and in the j direction:

It is assumed that excitation specified in this way willresult in the fundamental mode only propagating downthe m icrostrip in the frequency range of interest.The finite-difference formulas are not perfect in theirrepresen tat ion of the propagation of electromagneticwaves. O ne effect of this is numerical dispersion; i.e., thevelocity of propagation is slightly frequency dependenteven for uniform plane waves. In order to minimize theeffects of numerical dispersion and truncation errors, thewidth of the Gaussian pulse is chosen for at least 20points per wavelength at the highest frequency repre-sented significantly in the pulse.D. Conductor Treatment

The circuits considered in this pape r have a conductingground pla ne an d a single dielectric substr ate with metal-lization on top of this substrate in the ordinary m icrostripconfiguration. These electric conductors are assumed tobe perfectly conducting and have zero thickness and aretreated by sett ing the electric field com ponents tha t l ie onthe conductors to zero . The edge of the conductor shouldbe modeled with electric field components tangential tothe edge lying exactly on the edge of the microstrip asshown in Fig. 2.E. Absorbing Boundary Treatment

Due to the finite capabili t ies of the computers used toimplement the finite-difference equations, the mesh mustbe limited in the i an d i directions. The differenceequat ions cannot be used to evaluate the field compo-nents tangential t o the ou ter bound aries since they wouldrequire the values of field components outside of themesh. One of the six mesh boundaries is a ground planea n d i ts tangential electric field values are forced to be 0.


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852 1T.T.T TRANSACTIONS O N MIC ROWAVF THEORY ANI Tf( lINIOIJT.S, VOI . 38, O . 7, JlJ1.Y 1990The tangential electric field components on the othe r fivemesh walls must be specified in such a way that outgoingwaves are not reflected using the absorbing boundarycondition [22], [23]. For t he structu res considered in thispaper, the pulses on the microstrip lines will be normallyincident to the mesh walls. This leads to a simple approxi-mate continuous absorbing boundary condition, which isthat the tangential fields on the outer boundaries willobey the one-dimensional wave equation in th e directionnormal to the mesh wall. For the 9 normal wall theone-dimensional wave equation may be w ritten

I d E, , , = 0 .

This equation is Mur's first approximate absorbingboundary condition and it may be easily discretized usingonly field components on or just inside the mesh wall,yielding an explicit finite difference equation [22],

where E,, represents the tangential electric field compo-nents on the mesh wall and E , represents the tangentialelectric field components one node inside of the meshwall. Similar expressions are immediately obtained for theother absorbing boundaries by using the correspondingnormal directions for each wall. It should be noted thatthe normal incidence assumption is not valid for thefringing fields which are propagating tangential to thewalls; therefore the sidewalls should be far enough awaythat the fringing fields are negligible at the walls. Addi-tionally, radiation will not be exactly normal to the meshwalls . Second-order absorbing boundary conditions [22],[231 which acco unt for obliqu e in cidence will no t work onthe mesh walls where the microstrip is incident becausethese absorbing boundary conditions are derived in uni-form space.The results presented show that the first-order absorb-ing bound ary trea tmen t is sufficiently accur ate and partic-ularly well suited to th e micro strip geometry. It is possibleto obtain more accurate normal incidence absorbingboundary conditions [241, which have been used in theFD TD calcula tion of microstrip discontinuities [20]. Du eto the more dynamic, resonant behavior of the circuitsconsidered, Mur's first approximate absorbing boundarycondition is used and this allows for accurate simulationof the various microstrip circuits.F. Time Marching Solution

Th e finite difference equation s, (3)-(8), are used withthe above boundary and source conditions to simulate thepropagation of a broad-band Gaussian pulse on the mi-crostrip structure. The essential aspects of the time-

domain algorithm are as follows:Initially (at t = n = 0) all fields are 0.Th e following are repea ted until the response is =: 0:Gaussian excitation is imposed on port 1.is calculated from FD equations.f I + I /2is calculated from FD equations.Tangential E is set to 0 on conductors.Save desired field quantities.n + n + l .domain results.

E I I I

Compute scattering matrix coefficients from time-One additional consideration is that the reflectionsfrom the circuit will be reflected again by the source wall.To eliminate this, the circuit is placed a sufficient dis-tance from the source, and after the Gaussian pulse hasbeen fully launched, the absorbing boundary condition isswitched on at the source wall.

G. Frequency-Dependent ParametersIn addition to the transient results obtained naturallyby the FDTD method, the frequency-dependent scatter-ing matrix coefficients are easily calculated.

where [VI ' and [VI' a re the reflected and incident voltagevectors, respectively, and [S ] is the scattering matrix. Toaccomplish this, the vertical electric field underneath thecenter of each microstrip port is recorded at every timestep. As in [20], it is assumed that this field value isproportional to the voltage (which could be easily ob-tained by numerically in tegrating the vertical electric field)when considering propagation of the fundamental mode.To obta in the sca t te r ing parameter S , w ) , the incidentand reflected waveforms must be known. The FDTDsimulation calculates the sum of incident and reflectedwaveforms. To o btain the incident waveform, the calcula-tion is performed using only the port 1 microstrip line,which will now be of infinite extent (i.e., from source tofar absorbing wall), and the incide nt waveform is recor ded.This incident waveform may now be subtracted from theincident plus reflected waveform to yield the reflectedwaveform for port 1. The other ports will register onlytransmitted waveforms and will not need this computa-tion. The scattering parameters, S,k , may then be ob-tained by simple Fourier transform of these transientwaveforms as

Note that the reference planes are chosen with enoughdistance from the circuit discontinuities to eliminateevanescent waves. These distances are included in thedefinition of the circuit so that no phase correction isperfo rme d for the scatterin g coefficients. F or all of thecircuits considered, the only unique coefficients are in


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200 Time Steps 400 ime Stepsa

T0 794 r r600 Tune Steps 800Tune Steps2.46mrn

Fig. 3. Line-fed rectangular microstrip antenna detail.

the f i rs t co lumn of the scat ter ing matr ix ( i .e . ,S , , w ) ,S 2, w ) ,S J w ) , 1.

111 N U M E R I C A LESULTSNumerical results have been computed for three con-figurations, a line-fed rectangular patch antenna, a low-pass filter, and a branch line coupler. These circuits havedimensions on the ord er of 1 cm, and the frequency rangeof interest is from dc to 20 GHz. T he operating regions ofall of these circuits are less than 10 GHz; however, theaccuracy of the computed results at higher frequencies isexamined. These circuits were constructed on Duroidsubstrates with E = 2.2 and thickness of 1/32 inch (0.794

mm). Sca ttering matrix coefficients were measu red usingan HP 8510 network analyzer, which is calibrated to 18GHz, but provides measurement to 20 GHz.A. Line-Fed Rectangular M icrostrip Antenna

The actual dimensions of the microstrip antenna ana-lyzed are shown in Fig. 3. The operat ing resonance ap-proximately corresponds to the frequency where L ,=1.245 mm = A /2. Simulation of this circuit involves thestraightforward application of the finite-difference equa-tions, source, and boundary conditions. To model thethickness of the substrate correctly, A Z is chosen so thatthree nodes exactly match the thickness. An additional 13nodes in the 2 direction are used to model the free spaceabove the substrate. In order to correctly model thedimensions of the antenn a, Ax and Ay have been chosenso that an integral number of nodes will exactly fit therectangular patch. Unfortunately, this means the portwidth and placement will be off by a fraction of the spacestep. The sizes of the spac e steps are carefully chosen tominimize the effect of this error.The space s teps used are Ax = 0.389 mm, A y = 0.400mm, and Az = 0.265 mm, and the total mesh dimensionsa re 60 X 100X 16 in the f 8, and directions respectively.The rectangular ante nna patch is thus 32 Ax x 40 A y . Th elength of th e microstrip line from the source plane to th eedge of the ante nna is 50Ay, and the reference plane forport 1 is 10Ay f rom the edge of the patch. Th e microstripline width is modeled as 6Ax.The time step used is A t = 0.441 ps. The Gaussianhalf-width is T = 15 ps, and the time delay t , , is set to be3 T so the Gaussian will start at approximately 0. Th esimulation is performed for 8000 t ime steps, somewhatlonger than for other circuits, due to the highly resonant

Fig. 4 . Rectangular microstrip antenna distribution of E , x , y , l ustunder neath the dielectric interface at 200, 400, 600, and 800 t ime steps.

m 25303540 _ - - - MEASUREMENTALCULATION45500 2 4 6 8 10 12 14 16 18 20

Frequency ( CH z)Fig. 5. Return loss of the rectangular antenna.

behavior of the antenna. The computation t ime for thiscircuit is approxim ately 12 h on a V AX station 3500 work-station.The spatial distribution of EZ(x, , t ) jus t beneath themicrostrip at 200, 400, 600, and 800 time steps is shown inFig. 4, where the source Gaussian pulse and subsequentpropagation on the antenna are observed. Notice that theabsorbing boundary condition for the source has beenimplemented several nodes from the source plane toeliminate any undesirable effects of switching from sourceto absorbing boundary conditions. Three-dimensionalproperties of the propagation are observed including en-hancement of the field near the edges of the microstrip.The scattering coefficient results, shown in Fig. 5, showgood agreement with the measured data. The operatingresonance at 7.5 GHz is almost exactly shown by boththeory and measurement. This result is a significant ad-vancement over planar circuit techniques, which withoutempirical treatment will allow only \S,,I= 1.0 becausethey do not allow for radiation. Also, the resonancefrequency calculated using planar circuit concepts will besensitive to errors in the effective dimensions of thepatch. Additional resonances are also in good ageementwith experiment, except for the highest resonance ne ar 18GH z, which is somewhat shifted.


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0 -N 10

-20-30

- - --. _ _ _ _ - ---- __ Real

Imaginary4050 ' ' . ' . ~ . ~ ~ ~ .7.0 7.1 7.2 7.3 7.4 7.5 7 6 7.7 7.8 7.9 8 0Frequency (CHz )

Fig. 6. Input impedance of the rectangular ante nna near the operatingresonance of 7.5 GHz as calculated from the FDTD results.

2 4 1 3 m mFlg 7. Low-pa\s filter detail.

Input impedance fo r the an tenna may be calcu latedf ro m t h e S, w ) calculation by transforming the referencep lane to the edge of the microstr ip an tenna,

where k is the wavenumber on the microstrip, L is thelength from the edge of t h e an t en n a t o t h e r e f e r en cep lane 10A y), and Z,, is the characterist ic impedance ofthe microstrip line. For simplicity of the Z,, calculation,the microstrip is assumed to have a c onstan t characterist icimpedance of 50 C l and an effective permittivity of 1.9 isused to calculate the wavenumber. Results for the inputimpedance calculation near the operating resonance of7.5 GHz ar e shown in Fig. 6.B. Microstrip Low-Pass Filter

Th e low-pass fi l ter analyzed is one designed accordingto the c riteria es tablished by [2] an d is shown in Fig. 7. Asin the rectangular antenna, Ax, Ay, and Az are carefullychosen to fi t the dimensions of the circuit . The spacesteps Ax and A y are chosen to exactly match the dimen-sions of the rectangular patch; however, the locations andwidths of the ports will be modeled with some error.Th e sp ace s tep s u sed a r e Ax = 0.4064 mm, Ay = 0.4233mm, and Az = 0.265 mm, and the total mesh dimensionsar e SOX l O O X 16 in the 2, 9 an d 2 directions respectively.Th e long rectangular patch is thus 5 0 A x X6 Ay . Th edis tance f rom the source p lane to the edge of the longpatch is 50 A y , and the reference p lanes fo r por ts 1 and 2

200 Time Steps 400 Time Steps

' 1

600 Time Steps 800 Time Steps

- - - . /Fig. 8. Low-pass filter distribution of E ; x , y . r ) ust underneath thedielectric interface at 200, 400. 600. and 800 time steps.

0 2 4 6 8 10 12 14 16 18 2050 ' 'Frequency ( C H z )Fig. 9. Return 1 0 s of the low-pass filter

are 1OAy from the edges of the patc h. The strip widths ofpor ts 1 and 2 are modeled as 6A x.The t ime step used is At = 0.441 ps. The Gaussianhalf-width is T = 15 ps and the t ime delay t , , is set to be3 T . Th e s imulat ion is performed for 4000 time s tep s toallow the response on both ports to become nearly 0. T h ecomputa tion t ime for this circuit is approximately 8 h on aVAX station 3500 workstation.The spatial distribution of E,(x, y , t ) ust beneath themicrostrip at 200, 400, 600, and 800 time steps is shown inFig. 8. Th e scattering coefficient results, shown in Figs. 9and 10, again show good agreement in the location of theresponse nulls. The desired low-pass fi l ter performance isseen in the sha rp S,, roll-off beginning at approximately 5GHz. There is again some shift near the high end of thefrequency range. In the S,, results, the stopband for thecalculated curve is somewhat narrower than th e measuredresults. Some experimentation with planar circuit tech-niques has led to the conclusion that this narrowing iscaused predominantly by the slight misplacement of theports inherent in the choice of Ax and Ay.C. Microstrij2 Brunch Lin e CoiiplerT he branch line coupler, shown in Fig. 1 1 is used todivide power equally between ports 3 and 4 from ports 1o r 2. This occurs a t the f requency where the cen ter- to -center distance between the four l ines is a quarter wave-


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I O ' " " ' '

m -25-30

35-40-45

MEASUREMENT__ CALCULATION

2 4 6 8 10 12 14 16 16 20Frequency GHz)Fig. 10. Insertion loss of the low-pass filter.

2.413 mmFig. 11. Branch line coupler detail.

length. Also, the phase difference between ports 3 and 4is 90 at this frequency. To model this circuit, Ax, A y ,and Az are chosen to match the dimensions of the circuitas effectively as possible. Th e sp ace ste p, A z, is chosen tomatch the substrate thickness exactly. The space steps,A x and Ay, are chosen to match the cen ter- to -cen terdistance (9.75 mm) exactly. Again, small errors in theo ther an d j dimensions occur.Th e sp ace s t eps used a r e Ax = O.4 0 6 mm, A y =O.4 0 6mm, and Az = 0.265 mm and the total mesh dimensionsa re 6 0 X l OOX 16 in the 2 j an d 2 directions, respec-tively. Th e cen ter-to-ce nter distances are 24 Ax and 24 A y .The d is tance f rom the source p lane to the edge of thecoupler is SOAy, and the reference planes for ports Ithrough 4 are lOAy from the edges of the coupler. Thestrip widths of por ts 1 through 4 a re mo d e l ed as 6 Ax .The wide strips in the coupler are modeled as lOAxwide.The t ime step used is At = 0.441 ps. The Gaussianhalf-width is T = 15 ps and the t ime delay t, , is set to be3 T . The simulation is performed for 4000 t ime s teps toallow the response on all four ports to become nearlyzero. The computation t ime for this circuit is approxi-mately 6 h on a VAXstation 3500 workstation.The spatial distribution of E z x ,y , ) just beneath themicrostrip at 200 400, 600, an d 800 time steps is shown inFig. 12 . The scattering coefficient results, shown in Fig.13, again show good agreement in the location of theresponse nulls and crossover point. The desired branchline coupler performance is seen in the sha rp S I , n d S,,nulls which occur at approximately the same point (6.5

200 Time Steps 400 Time Step?

- .600 Time Steps 8 Time Step,

Fig. 12. Branch line coupler distribution o f E , x . y . 1 ) ust underneaththe d ielectric interface at 200, 400. 600, and 800 time steps.

_ - - _ MEASUREMENT-40-45-50

__ CALCULATION0 1 2 3 4 5 6 7 8 9 I OFrequency GHz)

Fig. 13. Scattering parame ters of the branch line coupler.

GHz) as the crossover in S,, an d S,,. At this crossoverpoint S,, an d S,, are both approximately -3 dB, indicat-ing that the power is being evenly divided between p orts 3an d 4. The nulls in S I , an d S,, at the operating pointindicate that l i t t le power is being transm itted by ports 1an d 2. The phase d i f ference between S,, and S, , isverified to be approximately Yo at the operating point( =.6.5 GHz) in both the calculated and measured coeffi-cients, as shown in Fig. 14. Some shift in the location ofthe nulls is again observed and is likely due to the sl ightmodeling errors in the widths of the lines.D. Error Discussion

In general , agreement between measured and calcu-lated results has been good; however, there are severalreasons to explain the small discrepancies in the results.The modeling error occurs primarily in the inabil i ty tomatch all of the circuit dimensions. The space steps, A x,A y , and A z , may be freely chosen; this allows exactmatching of only one circuit dimension (or two edges) ineach of the 2, j an d 2 directions. Another small sourceof error is the exclusion of dielectric and conductor loss inthe FDTD calculation. This causes the calculated S pa-rameters to be shifted up in amplitude from the measureddata at the higher frequencies. Measurement errors occurbecause of the microstrip-to-coaxial transitions, which are


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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,OL. 8, NO.7, JULY 1490856

180160

140

120M

al2 100

60

g 60dr:L 40

20

.-a

MEASUREMENTALCULATION1

O l A 0 1 2 3 4 5 6 7 6 9Frequency (GHz)

Fig. 14. Phase difference of port 3 relative to port 4 of the branch linecoupler .

not de-embedded in the measurement , and because thenetwork analyzer and i t s connectors are rated on ly to 18GHz.IV. CONCLUSIONS

The f in i te-d i f ference t ime-domain method has beenused to perform time-domain simulations of pulse propa-gation in several printed microstrip circuits. In ad dition tothe t ransien t resu lt s , f requency-dependent scat ter ing pa-rameters a nd th e inpu t impeda nce of a l ine-fed rectangu-lar patch an tenna have been calcu lated by Four ier t rans-form of the t ime-domain results. These results have beenverified by comparison with me asure d da ta. Th e versati l-ity of the FD TD method al lows easy calcu lation of manycomplicated microstrip structures. With the computa-tional power of computers increasing rapidly, this methodis very promising for the computer-aided design of manytypes of microstrip circuit components.REFERENCES[ l ] G. DInzeo, F. Giannini, C. Sodi, and R. Sorrentino, Method ofanalysis and f ilter ing properties of microwave planar networks,IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 462-471,July 1978.[2] G. DInzeo, F. Giannini, and R. Sorrentin o, Novel microwaveintegrated low-pass filters, Electron. Lett., vol. 15, no. 9, pp.258-260, Apr. 26, 1979.[3] K. C. Gu pta, R. Garg, and R. Chadha , Computer-Aided Design ofMicrowaue Circuits.[4] T. Okoshi, Planar Circuits or M icrowaves and Lighhvaues. Berlin:Springer-Verlag, 1985.[5] W. K. Gwarek, Analysis of arbitrarily shaped two-dimensionalmicrowave circuits by finite difference time domain method,IEEE Trans. Microwaoe Theory Tech., vol. 36, pp. 738-744, Apr.1988.[6] K. C. Gup ta and M . D. Abouzahra, Pla nar circuit analysis, inNumerical Techniques for Microwave and Millimeter-Wave PassiveStructures, T. I toh, Ed.

P. B. Katehi and N. C. Alexopoulos, Frequency dependent char-acteristics of microstrip discontinuities in millimeter wave inte-grated circuits, IEEE Trans. Microwave Theory Tech., vol. MTT-[8] R. W. Jackson, Full-wave, finite element analysis of irregularmicrostrip discontinuities, IEEE Trans. Microwave Theory Tech.,vol. 37, pp. 81-89, Jan . 1989.[9] S . Koike, N. Yoshida, and 1. Fukai, Transient analysis of mi-

Dedham, MA: Artech House, 1981.

New York: Wiley, 1989, pp. 214-333.[7]

33, pp. 1029-1035, Oct. 1985.

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David M. Sheen was born in Richland, WA, onMarch 6, 1964. He received the B.S. degree inelectr ical engineering from Washington StateUniversity in December 1985 and the M.S. de-gree in electr ical engineering from the Mas-sachusetts Institute of Technology, Cambridge,MA, in September 1988. He is currently study-ing for the Ph.D. degree in the Department ofElectr ical Engineering and C omputer Science atM.I.T. His research interests are in the are as ofmicrowave engineering and numerical solutionof electromagnetic wave propagation problems.


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SHEEN er al.: APPLICATION OF THE THREE-DIMENSIONAL FINITE-DIFFERENCE TIME-DOMAIN MFTHOD 85 7Sami M. Ali (M79-SM86) was born in Egypton December 7, 1938. He received the B.S.degree from the Military Technical College,Cairo, Egypt, in 1965, and the Ph.D. degreefrom the Technical University of Prague, Prague,Czechoslovakia, in 1975, both in electrical engi-neering.H e joined the Electr ical Engineering Dep art-men t, Military Technical College, Cairo, in 1975.In 1985, he became a Professor and head of theBasic Electrical Engineering Department there.From 1981 to 1982 he was a visiting scientist at the Research L aboratory

of Electronics, Massachusetts Institute of Technology, Cambridge, MA.Since 1987, he has again been a visiting scientist there. His currentresearch interests deal with microwave integrated circuits and microstripanten na applications.

Lincoln Laboratory, where he has worked on microstrip line discontinu-ities, wide-band dielectric directional couplers, and dielectric imagelines. H e is currently interested in the com puter-aided design of mi-crowave and millimeter-wave wide-band pla nar circuits and in the devel-opment of wide-band automated measurement schemes for characteriz-ing composite materials.Dr. Abouzahra h as published over 30 research papers a nd holds onepatent in the microwave area. H e also coauthored a chapter in Numen-cal Techniques for Microwave and Millimeter Wav e Passive Structures(John Wiley). Dr. Abouzahra is on the editorial board of the IEEETRANSACTIONSN MICROWAVEHEORYND TECHNIQUES.

Jin Au Kong (S6S-M69-SM74-F85) is Pro-fessor of Electrical Engineering and Chairmanof Area IV and Energy and ElectromagneticSystems in the Department of Electrical Engi-neering and Computer Science at the Mas-sachusetts Institute of Technology, Cambridge,MA. From 1977 to 1980 he served the UnitedMohamed D. Abouzahra (S79-M8S-SM87) Nations as a High-Level Consultant to the Un-was born in Beirut, Lebanon, on Ju ne 15, 1953. der-Secretary-General on Science and Technol-H e received t he B.S. degree (with distinction) in ogy. and as an Interregional Advisor on remoteelectronics and communications from Cairo sensing technology for the Department of Tech-University, Cairo, Egypt, in 1976 and th e M.S. nical Coo peratio n for Developm ent. His researc h interest is in the areaand Ph.D. degrees in electr ical engineering from of electromagnetic wave theory and applications He has published S I Xthe University of Colo rado, Boulder, in 1978 books and mor e than 300 refereed journal and conference papers andand 1984, respectively. has supervised more than 100 thesesFrom 1979 to 1984 he worked as a research Dr. Kong IS the Editor for the Wiley series on remote sensing, theand teaching assistant at the University of Col- Editor-in-Chief of the Journal of Electromagnetic Waies and A cationsorado, Boulder. Since 1984 he has been a mem- J E WA ) , and th e Chief Ed itor for the Elsevier book series on Progressber of the technical staff at the Massachusetts Institute of Technology, in Electromagnetics Research (PIER)

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