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Enhancement of Laguerre-FDTD with Initial Conditions for Fast Transient EM/Circuit Simulation

K. Srinivasan1, M. Swaminathan2 and E. Engin3School of Electrical and Computer Engineering

Georgia Institute of Technology777 Atlantic Drive NWAtlanta, Georgia 30332

{ krishna', madhavan.swaminathan2, engin3 } (ece.gatech.edu

AbstractIn this paper, a companion model for the 3D FDTD grid, toperform transient simulation using Laguerre-polynomials, hasbeen presented. The circuit model of the FDTD grid permitsthe problem to be solved in a modified nodal analysis (MNA)framework, making the matrix setup systematic and easier. Inaddition, companion models for an inductor, capacitor andmutual inductance have been derived. These models enablelinear transient circuit simulation with Laguerre-polynomialsusing MNA analysis. Prior work in this topic has thelimitation of being able to simulate only for a certain time-duration. An efficient solution has been proposed toovercome this limitation, so that simulation can be done forall time.

Discussion 1 IntroductionFDTD has been a ubiquitous time-domain simulation

method for electromagnetic analysis. Finite-differenceschemes, such as latency insertion method (LIM) [1], havebeen useful for transient circuit-simulation. One of thelimitations of time-domain finite-difference schemes is theCourant condition, that limits the size of the time-step that canbe used to obtain stable results. In EM analysis, the largesttime-step that can be used is a function of the smallest gridsize. Similarly, for transient circuit-analysis of componentsmade up of resistors, inductors and capacitors, the largesttime-step is a function of the smallest inductor and capacitorvalues. Solutions, such as ADI-FDTD [2], that areunconditionally stable, have been proposed to overcome thelimitation of the Courant condition to make FDTDsimulations run faster. This paper is an enhancement of anunconditionally stable FDTD scheme (referred to asLaguerre-MNA in this paper) proposed in [3]. The schemeproposed in [3] uses Laguerre polynomials as temporal basisfunctions to represent the EM fields/other unknownquantities. A system of linear equations is solved to obtain theunknown basis coefficients. Time-domain waveforms for thequantities of interest are generated using the solved basiscoefficients. The two limitations of Laguerre-FDTD methodproposed in [3] are (1) transient simulation can be carried outonly for certain duration of time (2) only structures wherefields eventually decay to zero with time can be accuratelysimulated. This paper proposes a solution that overcomes boththe limitations to enable accurate simulation of all types ofstructures for all time.

Discussion 2 Transient Simulation MethodologyTransient FDTD/circuit simulation using Laguerre

polynomials is presented in this section. Laguerre-MNA canbe used for (1) transient EM analysis and (2) transient

simulation in circuits composed of resistors, capacitors,inductors, mututal inductance, voltage/current sources. Theflowchart for simulation is shown in Fig. 1. The first step is torepresent the source waveforms in time-domain intoequivalent representations in the Laguerre-domain. The time-domain waveforms are represented as a sum of Laguerre-polynomials that are scaled by Laguerre basis coefficients.This representation is explained in Discussion 3 . The secondstep is to replace (1) the FDTD grid, or in the case of circuitsimulation, (2) capacitors, inductors, and mutual inductancewith their equivalent Laguerre-domain companion models.The circuit models are given in Discussion 5. The transientsources are replaced with DC sources. For each of the valuesin the Laguerre-domain that represents the time-domainsource waveform, a DC analysis is done once. The solution atthe end of each DC analysis is used to update the companionmodels, before the next DC analysis is performed. Afterupdating the companion models, a DC analysis is carried outusing the next value in the Laguerre-domain that representsthe source waveform. These series of steps are given by Steps3-5 in Fig. 1.

The final step (Step 6) is to construct the time-domainwaveform from the DC solution of the output of interest.

DIisuso3 Lagubasis Coefficitas ftis convert tavebrims

;Replace UIQgrid Or . -UC

wvPefrm itoLgurr-dmiAvesw tmlnedto wvfr

DCAiiiUlvl

1"d eC anionMOd e1s

Fig. 1: Flowchart for transient simulation using LaguerrepolynomialsDiscussion 3 Laguerre Basis Coefficients

The first step is to convert the time-domain sourcewaveforms into Laguerre-domain. A time-domain waveformcan be represented as a sum of Laguerre basis functions, witheach basis function scaled by its corresponding coefficient.

1626 2007 Electronic Components and Technology Conference1-4244-0985-3/07/$25.00 02007 IEEE

The output of Step 1 is to calculate the set of Laguerre basiscoefficients for each of the time-domain source waveforms. Inthe methodology given in Fig. 1, a transient time-domainsource is replaced with a DC source, whose values are the setof Laguerre basis coefficients.A transient source waveform W(t) can be represented as a

sum ofN Laguerre basis functions pp (t), scaled by Laguerrebasis coefficients {Wp, as shown in Eq. 1 [3].

N-1

l) W(t) = ]Wp('p(t-)p=O

2) t s t

In Eq. 1-2, t is the real time t multiplied by a scaling factor s.The actual time scale at which the simulation is run is verysmall, typically picoseconds when rise/fall times are in theorder of picoseconds. To make the basis function work, thereal time is multiplied by s to scale the magnitude in the orderof seconds. The basis functions for orders p = 0-4 are plottedin Fig. 2 [3]. The basis functions span a time in the order ofseconds as shown by the x-axis in Fig. 2, hence the need forthe scale factor. The definition of the basis function is shownin Eq. 3.

3) (p, (t) = e-t 1L. (t)Laguerre polynomials are defined recursively as follows:4) Lo(t) 1

5) L1(t) I1- t

6) pLP (t) (2p -1 t)L (t) - (p - I)Lp-2 (t)

for p >2

The transient source waveforms are replaced by DC sources.The values of the DC sources are the set of Laguerre basiscoefficients tWp,}, which represent the transient sourcewaveform in the Laguerre-domain. tWp} is generated fromW(t) using Eq. 7.

7) WP= W(t)(p(tdi

The total simulation time is divided into different time-intervals. In Eq. 7, to and tf represent the start and final timeof an interval. The reason for dividing the simulation timeinto different intervals is explained in Discussion 4. Theoutput of Step 1 is to compute the set the Laguerre basiscoefficients tWJ,J for each of the transient source waveforms.

Discussion 4 Limitations in Prior Work & SolutionThis paper presents improvement over [3] with the

following changes/additions: (1) Methodology in [3] has thelimitation of being able to simulate only for certain time-duration. A memory and time-efficient solution has beenproposed by which simulation can be done accurately for alltime, and (2) A circuit representation of 3D FDTD grid inLaguerre-domain has been derived; converting from Theveninto Norton representation of some of the components, reducesthe number of unknowns by half, without the use of longcumbersome equations, and (3) transient simulationmethodology with linear circuit components, such as

voltage/current sources, resistors, mutual inductances,inductors and capacitors has been developed.

Changes/Additions (1)Laguerre basis functions approach 0, as time t tends to cc, asshown in Fig. 2. Therefore, any time-domain waveform that isspanned by these set of basis functions also goes to 0 as ttends to oc. Structures that are lossless or have a low losscannot be simulated accurately, because the fields can be non-zero for a long period of time.

p2p-

p=

P=

Time (s)

Fig. 2: Laguerre basis functions for orders p 0-4 [3]

Laguerre basis function is an exponentially decayingfunction multiplied by Laguerre polynomial. The exponentialfunction quickly decays to 0, and beyond a certain time-pointthe exponential function is treated exactly as 0. Laguerrepolynomials become very large with increasing time. Beyonda certain time, the numbers become very large to berepresented with the limitation of finite precision and isrepresented as Inf in the IEEE 754 floating point standard.Consequently, beyond a certain time-point, the basis functionis represented as 0 x Inf or NaN, not a number. A solution tofix the problem is to divide total simulation time into differentintervals; the final values at the end of an interval are used asinitial condition in the next time interval. Using the proposedsolution, simulation can be done for all time duration.

IntervalI Inter al 1_

Fig. 3: Total simulation-time of Laguerre-MNA is dividedinto different time-intervals

The algorithm given in Fig. 1 is applied in each of theintervals given in Fig. 2. The duration of an interval isselected such that simulation can be accurately done in thattime-interval. Differential equations for transient simulationhave been modified to explicitly include initial conditions, toenable restarting the simulation.

Changes/Additions (2)Transient simulation using Laguerre polynomials requiressolving a system of linear equations of the form Ax = b; thesolution is used to update the right-hand side b, before solvingthe matrix again [3]. The number of unknowns to be solvedare reduced in [3] by substituting one equation into another,

1627 2007 Electronic Components and Technology Conference

z

Il y

lf' 2

r

H1AHlxy~~ HN. H

E EyiX .. .. EZ wawXwww.................

Fig. 4: xSections of the Yee cell marked by the dotted lines in Fig. 5, parallel to the xz, yz and xy planes, respectively; dotsindicate direction of the fields pointing out of the page.

resulting in long cumbersome equations. In this paper, acompanion model of the FDTD grid in 3D, called as SLeEC(simulation using Laguerre equivalent circuit) model, hasbeen derived. Converting some of the components fromThevenin to Norton form, reduces the number of unknowns tobe solved by half, without the use of long cumbersomeequations. Solving the system of linear equations in [3] isequivalent to doing a DC analysis on the companion model ofthe FDTD grid.

Changes/Additions (3)Companion models of the capacitor, inductor and mutualinductance have been derived. Replacing these componentswith their respective companion models enables Spice-liketransient simulation using the Laguerre method.

Discussion 5 SLeEC ModelThe standard FDTD Yee cell is shown in Fig. 5 [4]. The

cross-sections of the FDTD cell at the locations marked bythe dotted lines in Fig. 5 are shown in Fig. 4. These representthe cross-sections as viewed by standing on the +oo of y, xand z axis and facing the Yee cell. A circuit model thatrepresents the Laguerre basis coefficients of the fields hasbeen derived, where nodal voltages represent the basiscoefficients of the Electric fields and branch currentsrepresent basis coefficients of the magnetic fields.

Eill5(t) H H9) at -E' 't =

The initial conditions are explicitly included in the differentialequations before converting them to Laguerre-domain, toenable restarting simulation beyond a certain time-duration, asexplained in Discussion 4. Using the procedure similar to [5],the time-domain differential equations Eq. 8-9 can beconverted to Laguerre-domain, given by Eq. 10-1 1.

10)

Hq li _,+ 2 Hini't -CH lij|(E - Eq + ) + CH

(E s2 2+ 22xl+2jk1i+lj+1+kkE °

k=O,q>OY, J+k)+ Cj2 { 2+1 k

2 liH i,+l,k + l+,k j,

k O,q 0k1+ ,+

CXE li,j,kE q li j+' k -2Ei li j+ k

(Hq li+ , j+1Jk -Hzq li_l I+,) C ij

( lj+2 ,+2 Hx +2 ,k 2)11)

-jyq ji,j+1 k -2 kEj i,j+I,kk=O,q>O

Eq. 10-11 can be represented in a circuit form as given in Fig.6. Fig. 6 represents the circuit model for the magnetic fieldHq 1,+ I j+ I k and the electric field Eq11j+ 'k at the location

marked by the solid edges and their intersection in Fig. 4.Only the partial 3D model is given in Fig. 6. The complete

model can be derived in a similar fashion. The branchcurrents represent the qth Laguerre basis coefficient of themagnetic fields and are given by,

Fig. 5: Yee cell

Consider the following two of the six Maxwell's differentialequations:

8) aH.at

Hn"itg(t)= K aExz

p y


H4

Hx

aEYax )

g

r

Hy

Hz

Ey

D

Ey

EzEz

Hx

EF

RIAl-6 X.J4 6 s .> RxJ-ii n. rq : < ->,I........

Fig. 6: Companion model of the 3D FDTD grid in Laguerre-domain

12) H i I'1IjI and1~Hq ji_ ,k12 H ,+1j+ 1,k = i+ l,j+ 1,k an ql ,j k =i- l,j+ 1 ,k

jk+ Ii,j+ 2 k+ and H i j+k 1k- 1

The nodal voltages represent the qth basis coefficient of theElectric fields:

iteration is used to update the companion model beforesolving for the next set of Laguerre basis coefficients.

The number of unknowns that needs to be solved in DCanalysis can be reduced by using the Norton equivalent formlooking into the circuit marked by the double arrow in Fig. 6.The values of the Norton equivalent circuit are given by,

Iva/,3RR+ Vsrc Jva/,4R3R R21) IN R +R RN = R2 + R3

IN has terms involving Ival43 and IvalJ4 and is therefore acurrent controlled current source. In MNA analysis, currentcontrolled current source terms in IN introduce additionalunknowns, besides the unknown nodal voltages [6]. However,IN can be implemented as voltage controlled current sourcesand current sources, by stamping the current in a branchdirectly and the additional unknowns can be eliminated.Voltage controlled current source does not introduceadditional unknowns [6]. The unknowns to be solved are onlythe electric field coefficients (nodal voltages) and the matrixdimension to be solved is in its optimal form.

With the values given by Eq. 15-21, it can be seen fromKCL and KVL equations that these satisfy Eq. 10-11. Thepartial model in Fig. 6 can be completed in a similar fashionand can satisfy the complete set of 3D Maxwell's differentialequations in the Laguerre-domain.

14) EYq li,j+ k = Vi,j+2,kThe branch current circuitry represents Eq. 10 and thecircuitry connected to the node with voltage Vi+'k

represents Eq. 11. The values of the branch current circuitryare,

q-1-2 EHk li+I,j+2,kk=O,q>O

i+ ,j,k); R1 1x

The circuitry connected to the node with voltage Vi1+'k.have the values,17) 'val/3 I,j+',k+ i,j+',k-1J2 2+ 2 2

19) Rval,4 =Ii_ ,j+2,k Ci+.j+Ck19) R2 = CX; R3=C

src li j+±k = 2E" li,j+1,k20)

2 Jq li,j+ 2,kY

q-12 E li,j+ 1,k °

k=O,q>O

The circuit given in Fig. 6 can be stamped in an MNA matrixand solved to find the unknown Laguerre basis coefficients ofthe electric and magnetic fields. The qth value of the basis-coefficient is used for the DC source, in the qth iteration. Thesolution at the end of the qth iteration in the flowchart given inFig. 1 represents qth Laguerre basis coefficient of the electricand magnetic fields. The DC solution at the end of the qh

15) IVa = 2Hinit li+ l j+ J k

1) IVal/2 = CY V+ l,j+l,k

Discussion 6 Companion model of mutual inductanceIn this section, companion model for inductors with

mutual coupling is derived. Consider two inductors L1 and L2with coupling M, shown in Fig. 7.

i1

Vi

26

_L2+ V2 -

Fig. 7: Two inductors with mutual coupling

The voltages across the inductors are V1 and V2 whosepolarity is as shown in Fig. 7. The direction of the currentthrough the inductors is given in Fig. 7. Using the dotconvention, the voltages across the inductors including initialconditions are given by,

22) V1 L1 di +M d2 L1iil (0)(t) - Mi2 (0),(t)dt dt

23) V2 M di +L2 d2 _ Mil(0),(t) - L2i2(0)'(t)dt dt

In Eq. 22-23, il(O) and i2(0) represent initial current throughthe inductors at time t=0. The Laguerre-domain equations Eq.24-25, corresponding to the time-domain differentialequations given by Eq. 22-23, can be obtained similar to theprocedure given in [5].


V1P = SLL 0.5ilP + k - sL1i1(0) +

L k=O,p>O

2i2 =L V2P - 2 ik + 212(0)

2S k=O,p>O24) 35)

Ms!0.5i P + i Msi2(°)k=O,p>O

Similarly, Eq. 23 can be represented in the Laguerre-domainas,

V2P = sL2 0.5i P + i2j sL2i2(0)+25) k=O,p>O j25)~~~~~~~~-

Ms 0.5i1P + pi0k Msi1(0)_ k=O,p>O_

Eq. 24-25 can be represented using a Thevenin form circuitmodel shown in Fig. 8.

VACCVS VAit VAH

2M M+ i1(°) --ig

L2 L22M p>°

L2k=O,p>0Eq. 34-35 can be represented in a Norton equivalentcompanion model shown in Fig. 9.

cccsA ,--

ipI1 ip2

IHA

IinitB

IHB

Fig. 9: Norton equivalent companion model for mutualinductance

init

2

Fig. 8: Thevenin equivalent companion model for mutualinductance.

The values of the resistors are given by,26) RA = 0.5L1s

27) RB = 0.5L2sThe current-controlled voltage sources are,

28) VCCVS = 0.5Msi P

29) VCCVS = 0.5Msi P

Independent voltages sources which represent the initialconditions are,

30) V"it =L si1(0)Msi2(0)

31) Vf'B L2 si2 (0) -Msil (0)Independent voltage sources which represent the history termsare,

p-1 p-1

32) VA=H l Eik+ Ms Eikk=O,p>O k=O,p>O

p-1 p-1

33) H L2s i + Ms 1kk=O,p>O k=O,p>O

Solving for if, iP in Eq. 24-25,

2ip = VP l2 k + 2i(0)

1s k=O,p>O34)

2M.+ L2(i )

M .

L 2L1

2M p°

LIk=Op>O

The values of the conductances are given by,

36) GA

37) GB

2

Lls2

L2s

The current controlled current sources are,cccs -M38) IA i2

LIcccs -M39) IB iP

L2Independent current sources which represent the initialconditions are,

40) IAit= 2i1(0) + 2M(0)

41) IBit= 212(0) + i2(0)L2Independent current sources which represent the history termsare,

p-l

42) IAH = 2 1kk=O,p>O

2M p- 1

L O1pI k=O,p>°

43) IH = -2 k 2M k

k=O,p>O 2 k=O,p>O

It can be seen from KCL and KVL equations that Eq. 26-43,can be represented using the circuit models given in Fig. 8and Fig. 9, and these satisfy Eq. 24-25. In a similar fashion,companion models for inductors and capacitors can bederived. Simulation result from such a test-case is given in[7].Discussion 7 Time-domain Waveform

The final step (Step 6) is to obtain the time-domainwaveform, from the DC solution of the output quantity. The


B!

time-domain waveform can be obtained by using Eq. 1. Thepth basis coefficient is multiplied by the pth basis function;assuming there are N basis coefficients that represent theoutput waveform, the N waveforms are added to obtain theoutput transient waveform.

Transient simulation using Laguerre polynomials is a verymemory efficient method. At the end of each DC solution,only the basis coefficient of the output quantity (branchcurrent or nodal voltage) needs to be saved, in order tocompute the final time-domain waveform of the output. Oncethe companion model has been updated with the current DCsolution to make it ready for the next iteration, there is noneed to save these values.

The final values of the nodal voltages and branch currentsat the end of an interval need to be calculated as well, toenable restarting the simulation, as explained in Discussion 4.The memory required to save these values, to enablerestarting the simulation is O(M+N), where M, N are thenumber of branches, nodes in the circuit. None of the DCvalues in the solution need to be saved in order to calculatethe final values. More information on an efficient way tocalculate the final time-domain values at the end of an intervalis given in [7].

Discussion 8 A 2D EM Test-caseConsider TE, field in a 2D metal box shown in Fig. 10.

The dimension of the box is 0.im x 0.im with free-spacematerial properties inside the box. The grid size is 0.mOl xO.Olm. The vertical line in Fig. 10 represents the location ofthe Jy current source. The same source waveforms as [3] areused for this simulation.

PEC

yi

PEgFig. 10: A 2D Example

Ey(t) at the location marked probe in Fig. 10 for Interval I isshown in Fig. 11; red waveform is by using Laguerre-MNAand the blue dots is from conventional FDTD scheme. Thetime-scale factor used in Interval I is s = 7.0 x 1010. Thefinal values of the fields at the end of Interval I are used asinitial conditions for simulation in Interval II. The length ofInterval II is 1.5ns; simulation results are shown in Fig. 12.The value of the time-scale factor used for Interval II iss = 7.56 x 1011. The number of basis coefficients used is400, for all the simulations.

ConclusionsCompanion model of a 3D FDTD grid for simulation usingLaguerre polynomials was presented. Mutual inductancemodels were also derived. In a similar fashion, companion

Simulation results were presented for an EM test-case,showing good correlation with conventional FDTD scheme.

FD'TD Va Laquef,@e~DTD4 1

3 t n 0 1A+

Fig. 11: Simulation results from Ons to 5ns; solid red:Laguerre-MNA, dotted blue: FDTD

Lauerre-MNA dottedblue:FDTDW L

w4

Trhce x)S104

Fig. 12: Simulation results fromOns to 65ns; solid red:Laguerre-MNA, dotted blue: FDTD

References1. J. E. Schutt-Aine, "Latency Insertion Method (LIM) for

the Fast Transient Simulation of Large Networks," IEEETrans. Circuits and Systems-I: Fundamental Theory andApplications, Vol. 48, pp. 8 1-89, Jan. 2001.

2. T. Namiki, "A New FDTD Algorithm Based onAlternating-Direction Implicit Method," IEEE Trans.MTT, Vol. 47, No. 10 Oct. 1999.

3. Y. Chung, T. K. Sarkar, B. Jung, M. Salazar-Palma, "AnUnconditionally Stable Scheme for the Finite-differenceTime-domain Method," IEEE Trans. MTT, Vol. 51, No. 3,Mar. 2003.

4. 5. M. Rao, Time-domain Electromagnetics, AcademicPress, 2006.

5. K. Srinivasan, M. Swaminathan, and E. Engin,"Overcoming Limitations of Laguerre-FDTD for FastTime-domain EM Simulation," IEEE MTT-S InternationalMicrowave Symposium June, 2007.

6. L. T. Pillage, R. A. Rohrer, C. Visweswariah, ElectronicCircuit and System Simulation Methods, McGraw-Hill,1994.

7. K. Srinivasan, E. Engin, and M. Swaminathan, "FastFDTD Simulation Using Laguerre Polynomials in MNAFramework,"1 IEEE Electromagnetic CompatibilitySymposium, July 2007.

models for inductors and capacitors can also be obtained.


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