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The German ~IE M TA JonCh pe ad

th emnIEECS'hpeMunich, )tbr2-519

92-0763

International Workshop on -Discrete Time Domain Modelling

of Electromagnetic Fields andNetworks

The German IEEE MTT/AP Joint Chapter andthe German IEEE CAS Chapter

MunichOctober 24-25 1991

* OIC 1Af, ~U. arm".0 tce d L

01t L bpn

LA0

nternational Workshop of the German IEEE MTT/AP Joint Chapterand the German IEEE CAS Chapter

Discrete Time Domain Modeling of ElectromagneticFields and Networks

October 24-25, 1991.

Technical University Munichr a H6rsaal 0606 (Theresianum)

The workshop will be organized by P. Ruser and J. Nossek (both Technical University Munich) undersponsorship of the European Research Office (ERO) of the U.S. Army.Program:

Thursday, October 24, 1991:

SESSION Al (Chairman: Prof. J.A. Nossek)

10 00 Opening Session K. Steinbach10:10 Overview of Discrete Time Domain Modelling of Electromagnetic Fields P. Russer1100 Radiation and Scattering of Transient Electromagnetic Fields L B. Felsen11:50 - Break -

SESSION A2 (Chairman: Prof. P. Russet)

13:20 Finite Difference Time Domain Modelling of Electromagnetic Fields I Wolff14:10 TLM Modelling of Electromagnetic Fields W.J.R. Hoefer15.00 - Break -15'30 Multi-Dimensional Wave Digital Filters A. Fettweis16:20 Recent Developments in Numerical Integration of Differential Equations W. Mathis

SOCIAL EVENT (at Seehaus/Kleinhesseloher See, Munich 40)

19 00 Song Recita with Piano and afterwards Dinner

Friday, October 25, 1991:

SESSION BI (Chairman: Prof. W.J.R. Hoefer)

8.30 Cellular Automata L. Thiele9.20 Cellular Automata G Wunsch100 - Break-

10:40 Analysis of Nonlinear Microwsve Circuits via the Time Domain T. ItohVoltage-Update Method

11:30 Nonlinear Time Domain Modelling of Networks M. Sobhy12.20 - Break -

SESSION B2 (Chairman: Prof. A. Fettwels)

Short contributions:13 10j Efficient AnalYticalNumcrical Modeling of Ultra-Wideband Pulsed L.B. Feisns, L. Carn

plane wave Scattering from a Large Strip Grating

13.25 Transient Currents and Fields of Wire Antenas with Diodes N. Sclieffer

13-40 Calculating Frequency Domain Data by Time Domain Methods M. Delder

11-55 - Break-14-15 Time Domain Analysis of Inkiomogeneouoly Loaded Structures Using Mr. Mrozowsi

Eigenfunction Expansion .lrupoP.use

14:30 The Hilbert Space Formulation of the TLM.MethodM.KupozP.Rse

14.45 Late Contributions

CONCLUDING SESSION

16 00 Open Forum,

Panel Discussion

Further short contributions will be accepted at the workshop.

LIST OF PARTICIPANTS

Invited speakers-

name

Prof. Dr. L.B. Felsen Polytechn Univ. Frmingdale New York~Prof. Dr. A. Fettweis Ruhr-Univ. Bochum

Prof. Dr. W.i.R. Hoefer Univ. of OttawaProf. Dr. T. Itoh Univ. California Los Angeles

Prof. Dr. W. Mathis GH WuppertaProf. Dr. P. Roser TV Miluchen

Prof. Dr. MlI. Sobhy Univ. of KentProf. Dr. L. Thsiele Univ. Saarbrslcken

Prof. Dr. I. Wolff Univ. DuisburgProf. Dr. G. Wunsch Univ. Radebeul

Further participants:

name

W. Anzill TV MtlnchenH. Bender TU MilochenProf. Hex FHi Aachen

Dr. E. Biebi TU MfinchenChr. Borokessel Univ. Ka~rlsruhe

C. Ciotti TH AachenProf. Dalichau, Univ. d. Bundeswehr MilochenDr. M. Dehier TH Darmstadt

Prof. Dr. Entenmann TU MtinchenProf. Dr.!1. Frost Univ. of York

T. Felgentreif TV MilnchenG. Gotthard Univ. Karlsruhe

3. Gras! Texas InstrumentsV. Gtsngerich TV Manchen

Prof. Dr. H.L. Hartnagel TH DarmostadtDr. Heidler Univ. d. Bundeswehr M(Inches

Prof Dr. E. Holzhauer Univ. StuttgartHlper TU MuIncher,

B. Isele TU MrunchenProf. Dr. R.H. Jansen Jansen Microwave

M. Kunig TU MtlnchenM. IKrumpholz TU Mtinchen

Dr. J.F. Luy Daimler Benr

name

Prof. Dr G. Mahler Univ. StuttgartIL. Meier TU Mtlnclien

H. Meinel DASAr Dr. M. Mrozowski TIJ GdanskDr. J.W. Mink ARO/US.Army

S. MfIller TU MilnelenG. Nitsche lRuhr.Uni. Bochium

Prof. Dr. J.A. Nossek TU MiluchenDr. G. Qibrich TU Mtlnchen

Hr. Paul TU MllnclenES. Panich Parzich GmnbH

G. Rohrbach Univ. StuttgartF. liostan Univ. Xaaruhe

Dr. N. Scheffer Tel-funken Systemlechnikr.M. Schneider MBB

Prof. Dr. R.Sorrentino Univ. PerugiaDr. K. Steinbach ERO/US.Army

RS. Stephan TH ElmenauW. Tewes, DLRl Oberpfaffenhofen

Prof. Tisim Univ. LinzW Thomann TU Miluchen

Dr. XC H. TflrknerProf. Uhlmnann TH lllmenau

Dr. RS. Weigel TU MilnehenDr. N. Zhu Univ. StuttgartTh. Zundi Untiv. d. Bundeswehr Mllnchen

2

Overview over Discrete Time Domain Methods inElectromagnetic Field Computation

Peter Russer I

Abstract

The modelling of fields in the time domain describes the evolution o physicalquantities in a natural way. Transient phenomena, nonlinear and dispersive beha-viour, the characteristics of systems with moving boundaries or with time dependentproperties are best described in the time domain. In this contribution different ap.proaches for time domain modelling of electromagnetic fields are compared.

1 Introduction

For electromagnetic field modelling numerous techniques have been developed [1,2,3). Themodelling of fields and networks in the time domain is highly attractive since it describes

the evolution of physical quantities in a natural way. Time domain modelling is especiallyadvantageous in the case of transient electromagnetic fields, fields in nonlinear, dispersiveor time-dependend media or in regions with moving boundaries. One of the main advan-

tages of time-domain modelling of electromagnetic fields is the local dependence of the

field variables on space as well as on time. Within discretized space and time the state

of the field in a given point and at a given time depends only on the field states of theneighbouring points at previous times This allows a highly parallel computation of the

time evolution of the discretized field.

In modelling of high frequency circuits we have to deal with the network as Aell with the

field concept (Table 1). Whereas the field has a spatial structure the network structureis topological. However if the field is described by a discrete set of base functions as it is

done for example in the method of moments [4,5] or if the field is discretized with respectto space we obtain topological relations between the state variables of the field This

allows to apply ne ssoik-tieor-tical methods to field problems.

'Lehrstuhl ffir llochfrequenztechink, Technische Universitgt Manchen, Arcisstrasse 21,

D-8000 Munich 2, Fed. Rep Germany

2

Table 1: Concepts of Field Theory and Network Theory

NETWORK FIELD

Topological structure Spatial structureTime, Amplitude Time, Amplitude, Space

Continuous

Analog Network * Electromagnetic Field

~Discrete

•a Digital network * Cellular Automata I

* Discrete modelling * Discrete Modellingof analog networks of fields

In the following we shall focus our attention on four approaches for time domain modellingof electromagnetic fields.

* The finite-difference time-domain (FDTD) method

* The transmission line matrix (TLM) method

* The field modelling by cellular automata

* The field modelling by multi.dimensional wave digital filters (MDWDF)

2 The Finite-Difference Time-Domain Method

The finite-difference time domain (FDTD) method is the mathematical approach for the

solution of partial differential equations (6). The partial derivatives are simply replacedby finite differences. In 1966 Yee has first given a finite-difference time-domain scheme

for solution of the Maxwell equations 17,8,9]. In the FDTD method space and time are

discretized with increments At and At, respectively. The field component placement inthe FDTD unit ce;l is shown in Fig. 1. The side length of a unit cell in our notation is2AI

Space and time coordinates are given by x = I At, y = mAl, z = n At and t = k At. The

FDTD scheme for the solution of the Maxwell's equations is then given by

~ 3L1NODE(i,j, k)

H~ 1 ,nn+1 l'+lmnI I ,

H~(+lr+ n =Hiif(+,rn+l,n)Eklm~~)

+E E(I, + ,m + 1) - Ei (I,1+ + 1~~)+

H, ( 1E, (~ + , n + 1) + 2, + n)(3

+~2

ll n E, E ( + , m, n + 1)-E(,mn+)+

+Ii ' (+ , m,n - E,- (I +1, ,(n+ 2)~ ) J (4)

+~2Jinln + 1Q~, m)+2,n-E,(I+1 ,)+

+E (Is+i 1, ln )- ' ( , in + , n (3l )

41~

i4E+!

2(tm... 1) = E(I,m,n 1) + 1 -+

+H,'+'(l,m- l,n+l)-H,'+'(I,m+l,n+l)] (6)

with the stability factor

/-t "(7),

where c is the velocity of light, At is the time interval, Al is the length interval. The

condition for stability is given by

1(

The FDTD scheme gives the new field state at the time . Wt as a function of the field

states at (k - 1) At and (k - 2) At. Also the new spatial components at 1, m and n are

related to the spatial components at I 1, 1 ± 2, is ± 1, m ± 2, n ± land n ± 2. Howeverall electrical field components with even values of k, are only related to the magnetic fieldcomponents with odd values of k and vice versa.

Investigating planar circuits within the magnetic wall model a two-dimensional finitedifference scheme may be applied 1101. For the circuit plana. parallel to the z - y-plane

the electric field exhibits only the z-component and the magnetic field exhibits only thez- and y- components The surface current J flowing in the upper plane of the planarcircuit is given by

J = -e, x H (9)

where e, is the unit vector in z direction. The voltage V between the plates is given by

V = -dE, (10)

where d is the distance between the plates.

V V(x,y,t)= -jL, OJ(X-,Yt) (1)at

V J(X,Y,t) = -JC. OV(XYt) (12)atC, is the capacitance and L, is the inductance of an arbitrary square element of the

two-dimensional parallel plate line.

Jk+i' is ,) = S " ((m,n) +

+ -1VI(.+ l,n)-vb(,n+,n) (13)J +'(m,n) = -((n)

+At- [V'(m,n + 1)- V(m,n + 1)] (14)+',hv

Vk+

(m,n) = Vk' ((m,n)-

-ah [J * (- + 1, n) - J. (- - 1, n) +

~+j,*(m, + 1- jk(m,,n- 1)j (15)

Nonrectangular grids have been treated for the FD method [111. The analysis of radiatingstructures requires the termination of the grid with absorbing boundaries [11,12,13).

3 The TLM (Transmission Line Matrix) Method

The transmission line matrix (TLM) method was developed and first published in 1971by Johns and Beurle [14,15,16,171. In the TLM method the physical analogy and theisomorphism in the mathematical description between the electromagnetic field and amesh of transmission lines is used. The field evolution is modelled by voltage pulsespropagating on the mesh lines and being scattered in the mesh nodes. Field theoretically

the TLM method is based on the Huygens principle [18,19)

In the TLM model space and time are discretized in length intervals At and time intervalsAt. The intervals 61 and At correspond to the real space and time intervals withoutscaling if At = v2ceoAf//c- is chosen, where c, is the relative permittivity. Fig. (2)shows the port numbering of a two-dimensional TLM shunt node.

-4

-2

n

3 m

Figure 2: A two-dimensional TLM shunt node.

The scattering of impulses at a shunt nodeis described by the following equation:

v v2 (16)

[V4 1[V

6

Zo is the characteristic mesh line impedance, given by

Zo = q (17)

where no = 377fl, h is the height of the planar structure, and C, its permittivity.

The node scattering matrix for a shunt node is

Loony subregions can be msdelled in TLM by connecting a lumped resistor or an infinitelySlong transmission line stub across each mesh node (20]. Also the modelling of nonlinear

passive elements has already been treated [21,22,23]. Nonlinear active regions may be,modelled by connecting nonlinear active circuit elements to the r'.esh nodes (24,26]

The voltage wave pulbes V',, incident on a TLM mesh node depend on the voltagewave pulses .1 V,, emerging from the neighboring nodes as follows:

Eqs (16)-(18) and (19) describe the complete algorithm for the time discrete field evolis.

tion,

Fo the iwo-dimensional case Johns has shown the relation between the FDTD methodand the TLM method (28]

E. =pq + r (20)

with

qT - 11111) (21)p = (1 122)

r = -1 (23)

.,.= qC (,E. + ,v') (24)+E = g~p .E, qCrCp s..1E, + qCrCp ..1V' (25)

WithCrCr = al (26)

tF

2 2 2 2

7

10

87

2% z 6

24

9 x

Figure 3: Three-dimensional symmetric condensed TLM node.

where a is a constant we obtain

=+,E. = qCp sEqCrCp k.iE~a*.IE 4 (27)

\ , now have an algorithm relating k+,E, to hE, and tok.tE,. We have reduced the, ariables but increase the time depths of the algorithm

Different TLM schemes have been proposed for the three.dimensional case [16]. A sym.metrical three-dimensional condensed node has been introduced by Johns [26,27). Fig. 3shows the symmetric condensed TLM node In the three-dimensional case we haw tointroduce twelve wave amplitudes The voltage wave vector is given by

V, =(v;v;vvv;vv;ov;i v v., 1' (28)V, - IVIV2,VV4V$,V6,VV$,VgVoVIIVI,21 T (28)

the incident wave pulses kV' at f = kAt and the scttered wave pulses k+IV' at =

(k + I)At are related by

a+IV' SaV' (29)

8

the scattering matrix S given by [ 0 T T]S= TT 0 T (30)

T TT 0

with00 2

0 0 1 1 Iij1 0 - (31)10 0

2 2We introduce the field vector F given by

F =._ [E' ,I (32)I

The wave amplitude vector kVl,,,, is related to the field vector kFl,,,, by

F = QV' (33)

= jQTF (34)2

with0 00 0 10 01 1 1 10 01 1 0 0 0 0 0 0 0 0 1 1

Q 0 0 1 1 1 10 0 00 0 0 (35)0 0 0 0 -1 1 0 0 0 0 1 -1

0 0 1 -1 0 0 0 0 -1 1 0 0

-1 1 0 0 0 0 1 -1 0 0 0 0

Structures to be analyzed in TLM may be segmented into substructures. This methodfirst proposed by Kron is called diakoptics 129,30,311. Within diakoptics the scatteringof waves by boundaries is expressed via discrete Green's functions or so-called Johnsmatrices 1171. Discrete Green's functions may also be computed algebraically 132) The

TLM method in connedtion with absorbing boundary conditions has already been appliedto the analysis of a slot antenna 133]

4 Cellular Automata

John von Neumann's most extensive %%ork in the theory of automa~a was the investigation

of cellular automata [34]. The results of this work are contained ini the manuscript "The

L~-'w'

Theory of Automata- Construction, Reproduction, Homogeneity" [35]. John von Neu-

mann's cellular automata are homogeneous two-dimensional arrays of square cells, eachcontaining the same twentynine-state finite automaton. Any cell can assume at a giventime the unexcitable state, one of twenty quiescent states or one of eight sensitized states.

The unexcitable state represents the presence of no neuron. Quiescent cells can respondto stimuli from adjacent cells. The state of a cell at a given time is determined by a setof transition rides. The state after the transition is determined by the initial states ofthe cell and of its four inondiagonal ueighbours. John von Neumann has shown that thetwentymne states are sufficient to accomodate all logical and construction circumstances

that may arise and also to establish all transition rules for moving from one state to the

other. He demonstrated the logical universality of the cellular automata by showing howT ring's machine model can be reformulated in terms of cellular automata.

John von Neumann also has planned the continuous model as a further refinement. In 1969Konrad Zuse discussed the modelling of the Maxwell's equations by cellular automata [36],

Cellular automata now meet with growing interest for modelling of physical systems (38].

The discrete system may be described in state variable form [37]. An autonomous system

is described byz(r,t + 1) = A(rr)z(r',t) (36)

In vector notation this is given by

Z(t + 1) = Az(t) (37)

where z(t) is the state vector and A is called the transition matrix. The discrete time

variable is t, and r is the dscrete space variable.

As an example we consider the telegraphist's equation [37[.

Oi = R +n

-CL" = Gv+ ,± (38)

The corressponding difference equation is given by

-LAot = Ri + Av

-CAiv = Gv4+ Ai (39)

At and A, are given by

Aii(zt) = :(z',t +I) -:(z(,t)A~s(x,*) = i(z + 1,1) - s(x,t) (40) i

The state vector z and the transition matrix A in eq. (37) are given by

[ (X, t) (41)

andA = Ao + AIA,

(42)

with

Using the Z transformation with respect to x the variable {, given by

Z fz(x)) = E z(,)C - (45).. 0

and withZ {Az)} = (X-i - 1) Z (z(x)) (46)

we obtain

z( ,t + 1) -Az((,t) (47)

with

,A - (48)

This result may be transformed into a digital circuit Fig. 4. shows the corresponding

digital model of the lossy transmission line.

5 Field Modelling by Multi-Dimensional Wave Digital Filters

The numerical integration of partial differential equations using principles of multidi-

mensional wave digital filters (MDWDFs) has been proposed by Fettweis 143,441. Thecontinuous-domain physical system is simulated by means of a discrete passive dynami-cal system. In this method in a first step the partial differential equation is modelled

by a multidimensional analog circuit. This circuit is then transformed into a MDWDFequivalent circuit (45]. The advantage of this method is the robustness of the algorithm

even under conditions of of rounding and truncation operations.

L

- 1,1) ix,)( s(x + I,)

I -RIL I - RIL -

+ ++

11L I IC

+ + _

I/C!U(X- l,t) U(X,t) U(z+ 1,t)

Figure 4: Digital model of the lossy transmission line.

Let us consider as an example again the two--dimensional electromagnetic field problemassociated with a transverse electric field between two metal plates. We investigate theequations

+Ri +- = ut(t) (49)L4982 + t V

LT + = u2(t) (50)61 012

oil + O + O + G = U3(t) (51)

The variable 13 corresponds to time, whereas tl and f2 represent the spatial coordinates.The current density components in the upper metal plate in the directions tj and t2,respectively, are given by i and i2, and v is the voltage between the metal plates. Theinhomogeneous terms ul, U2 and U3 represent distributed impressed sources.

In the three-dimensional complex frequency domain we obtain the algebraic equations

(pL+R)Il+pIR313 = U, (52)

(psL+R)I2+p2Rals = U2 (53)

piRali+p2Rl3 2+(pC+G)Rfl1l = U3 (54)

where pl, pj and p3 are the complex frequencies The corresponding analog circuit repre-senting the partial differential equations is depicted in Fig. 5 Note that the this equivalentcircuit represents the complete field.

12

-P1 3 -pjr3 p3i + r

i Es" yPr &E

p3-c' + gr' -N2'3 R-Par3 psl+t r

11

Figure 5' Multidimensional analog circuit

Introducing the d;fferential operators

D,=! for v=1...3 (55)

yields

(D3L + R)I1 + DIR313 = U, (56)

(D3L + R)12 + D2R3) 3 = U2 (57)

D1 R311 + D2R3 12 + (D3C + G)R213 = U3 (56)

Using the trapezoidal rule for integration yields the replacement of the differential equa-tions by the following difference equations:

v(t) - v(t - T ) = R(t)i(t) - R(t - T )i(t - T ) (59)

i(t) - t(t - T) = G(t)v(t) - G(t - T )v(t - T,) (60)

where t is given byt = {ji02, 3 (61)

the vectors T. are given by

T 1=[T,,O,OT, T2 = [O,T2,O), Ta = [0,0,T3)T, (62)

Ts is an arbitrary positive time increment and T, and T2 are related to T3 via

T, = T, = r/2T3/V/'- (63)

and R(t) and G(t) are given by

R(t) = 2L(t)IT, G(t) = 2C(t)/T (64)

- -- tA

13

using the wave digital filter design rules the circuit according to Fig. 5 is converted into

the wave digital filter circuit shown in Fig. 6. The methods of MDWDFs ensure thatthe algorithm is recursible and that the full range of robustness properties of WDFs isconserved

- ~ ~~bs ! -- (

D rR4- 1 , - -1/2 -

D,,2' '2 0,T, 3 R

Figure 6: Wave digital filter

6 Conclusion

We have compared four different methods of discrete time domain analysis of electroma-gnetic fields. The methods originate from different theoretical frameworks. Whereas the

FDTD method is based on mathematical considerations, TLNI originates from a line mo.

del. The method of cellular automata is based on the theory of automata and systems andthe method of MDWDFs uses the analogy to multidimensional circuits and wave digitalfilters. There are interesting analogies between these methods. Each of these methodscontributes special advantages and interesting contributions to the solution of problems

also relevant in the other models

I.L 4

l 14

This work has been supported by the Deutsche Forschungsgemeinschaft.

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16

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lysis", Proc of the 20th European Microwave Conference, 1990, Vol.2, pp. 1495 - 1500.

[34] W Aspray, "John von Neumann and the Origins of Modern Computing", MIT Press, Cambrdge,

Masssacels, 1990

[35] J.v Neumann, "The Theory of Self-Reproducing Automata," in Burks "Essays cn Cellular Auto-mata," University of Illinois Press, Urbana 1970

[36] K Zuse, "Rechnender Raum," in. "Schriften zur Datenverarbeitung, Band I", Viereg, Braun.schwceig, 1969.

(37] G. Wunsch, "Zellulare Systeme - Mathematasche Theone kausaler Felder," Akdeeirc. Verlag Berlin,1977.

S[38] T. Toff'oh, 'Cellular Automata Machines",/,AiT Press, Canirndyc, Masisachusitls, 1987

39) R. Wait, A R Mitchell, "Finite Element Analysis and Applications", Chichester, 1985, John Wiley& Sons

[40] P.P Silvester, R.L Ferrari, 'Finite Elements for Electrical Engineers", Cambridge, 1983, 1990Cambridge Unversity Press

]41] A.C Cangellans, C -C. Lin, K K Mei, "Point-Matched Time-Domain Finite-Element Methods for

Electromagnetic Radiatior, and Scattering", IEEE Tans. Antennas Propagat., vol AP-35, pp. 1160-1173, Oct. 1987

[42] T. Toffoh, "Cellular Automata Mahies, MIT Press, Cambridge, Massachusetts, 1987.

[43] A Fettweis, G. Nitsche, "Numerical Integration of Partial Differential Equations Using Principlesof Multidimensional Wave Digital Filters," "Proc IEEE Int Symp. Circuits and Systems, New

Orleans, May 1990, pp. 954-957

[44] A Fettweis, G. Nitsche, "Numerical Int-gration of Partial Differential Equations Using Principles ofMultidimensional Wave Digital Filters," Joural of VLSI Signal Processing, vol. 3, pp 7-24,1991

[45] A Fettweis, "Wave Digital Filters Theory and Practice", Proc IEEE, vol 74, pp. 270-327, Feb1988

RADIATION AND SCATIERING OF TRANSIENT ELECTROMAGNETIC FIELDS

by

Leopold B. Felseni Dept of Electrical Engineering/Weber Re!.arch Institute

Polytechnic UniversityFarmlngdale, NY 11735, USA

"Ultrawideband" (UWB) and "Very Short Pulse' (SP) provide alternative designationsfor the same class of transient electromagnetic wave phenomena. However, UWB relatesthese phenomena to the frequency domain whereas SP relates the same phenomena to thetime domain (TI). By generating SP.TD data through UWB frequency synthesis, theevolution of the TD signal with increasing bandwidth can be tracked systematically to itshighly localized UWB form. Localized pulse-like featurt: (observables) In data suggest thatmodeling and iaterpretation in terms of a TD "observable-based r!arametrizaton" (OBP) isphysically more appropriate. Implementing a TD.OBP requires new thinking and conceptsdirectly in the time domain. A systematic OBP strategy for learning to "think TD" viaidentification of TI wave objects is proposed and illustrated by examples Involving SPexcitation of layered media, strip gratings and aperture-coupled cavities. Of special interestis the TD evolution of the strongly dispersive leaky modes, Floquet modes and cavity modes,and their role in synthesizing the SP response.

1i7

PROCEEDINGSOF THE SECOND

INTERNATIONALCONFERENCE

ON

ELECTROMAGNETICSIN

AEROSPACEAPPLICATIONS

SWIEMMBR 17-20, 1991 - POLTECNICO DI TORINO. ITALY

-I!

hi TIMdEDOMM ESERVABLEEASED . 17mXAMON

Deporme ofRteduR hof ds ere nstitute,

and St~~~~ycO~~rOS , boat O OSf pfisstl I

dtrsct ~ref the~nb~too bais to 4 ecF 7U pro$n o gdirtrn

aedsstiehiif cottttteti "bo4 naliciss beor.Is&smn

eussed aynms vns re brnirsi auete rhhe h eg utlu. to tef O N funst tisboanaysi an sytheis f tansentsouca~~dtd wves SM ott~nsof an n t h dor asnreiopre

sa it ntretu O heewve it tts~enttl ti s o efalo d r 'D L.oltssi =

(they, bred ponao) tieherbyredni n 'Ueperti aut efre a l tit dsdamet"A wae

at baexpfor bertz t has beent~ rCoemed the ptIt4 ', WA N, terati trcrA l

a srodtie dar n &A mrefr yc so Irorim n lotetrogatlots 1 er

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saris avefes hasso behoseden Inrleal =esttctv obes e t t V predsa sho resLas

had, detrbeiv esitorforonc beweW Xsene so WslOJs ossedsslli Msrelw

ysle st wipth a -dthe S orteay ati sa eteels n doan reeoftefrsl pil'aveme

Contrast to ftees. ste saal - SI e~saitb (epenn NO see a:It w .mm~T~e~h ~ a~~

Il one ) m illesteat s etcpre

Drec W ID tilse n cgersis sues Inhi sie la O erbhad u ds af efgat b ~ ~ ede

(Isshedel S pesoas ofebas~. pstao no pyh.I ~

L3s.Feltent. Very Short Poise SctteorgTune Domtain Parametrirestion

p&adon nwayftom the cotromcaj condodns. and tesseloas, which thereby playsa fundamtental role sn fieldthereby aceomodtc a clar; of ovnrnvooical syntheod Webh: and1, - (rxy) deeminbg the coordirntesproblems instead of a single prototype. It Is also pexpendicolse aad parallel to 1Te laI Inefaces.

poteally more efficiet or cotmpuststo si re=retm , the ptentialutiones my Cosructe Ioprpate parameter regonts. F~alOil forms by-e (Forer decomposition into the tratwves

dosewyome hqecyUoe:a os hdibest forpe'.see Waare of= doei h ie aeobrdwes, k ) sod the temporal

dependetl preblem i the (t ,w) domeainby tnvolsssg theThe analytical tools employed in the implementation of hidsy condittlons amaoss layer Interfaces snd the fetlil011 Include (1.133. coditions at the source loeatlen T' . (x,y'.z'). and

thereafter performing spectal synthesi Thfis leads to

3. The spectral theory of Itasleots (lIn wlorl the following generic form for the Potential functiontI at

dsocrly insA iime domran by&sp"se spectral-

* ~superpeadonoOf f. iebs. fle . f~~;'t)IId ddl F(t'tt.3 ) II

2L The hybrd moaqreol-rdsoraace olalson.m which where the wove object F, defined in the (spac~le- .constructs time-dependent fields scattered by tergets (wayotsomher-retisuncy) Phase spore has thOr litherumeos media in tern' of sticmstwet conPiplto

cehae f 1, e lossht t ( lese'rol (wed

sysoeotsedPthe totieal distinction between aod the deonminalor D is written emoleotIy Astheg early tife end Ile time reapot, and hasthereby clarified the Iittatio assncioted with the D(w,(, . R.i 1 R.wQ (lb)siogularity aeansiort method Here, R. and Rt. ore spectral domain raflocllon

3 phseopee hats lgoithmforfficientsiro jae ' o oing ulong the (+n) an (z3.M ph s odsa bse ue agrtcm ino i o fI Firethens respectively, from the soercot Plante n. -'.

decomposition n ott c eobntinO O h most versatile treatment of the, spectral Integralstimc.harmoolt or transient fields into windowed at (I) illis usefut to separate (I/I)) telf-eonsistenly into

waveismnder phose spae atice or in a phase apacecontinuum. The bea-type fields are *good yyt*piggatius* and Are tracked coovrulently throughcomp esp pogato andosttering enCOoftrtera JRR 1 . (2)

4. The complex $0=#c poit (C21') method dot Obtained by partial power soures exansion. etch tetm inmodeling of poisd beam ts t puls. and the th seisprinrtrsos oittmea reflected waves

aratectgofthee pise bems y env~~aiteas. hic ca beatttiluised ntogenetaizred roy fieldss mterfgo teft ptue drbe ab ve has beers after synthesis, whereas the remainder,1 with its resonant

The 01? mdlr srtdecribdaoehsbe der~iiotOf, c&a he ord ed ino gided modeapplied to a varnc - propaqotion and seeitleong fiels. Note that the amplittoda of the Modal termenironments in X.ctolmagneties and utnderwater dsossaldoeltsl nt e daes Ns And N1 Of theacooslims (12.29). These, sppilina have beersyrovrtml dped soctrd ras it Ne on 0. 3 Ilsmsnapr as the frequency doanpoari rod~etsoy d serier. and intact retention of Dr furishn apure

utreas. Transient SP scattering byh o synthesis, No mod ft el siaiin.Jseiuemt direct iD-OPB. is Presently bei Formod eafield s exi yi ub eutino huiplareeted for various enoiroomeolol conditions. IS o reta lsiiiyi so qt Ca. tnoo h

plano wave scattering ifro a alit eoupled esitY has formal spectral iieisl in.l). wih(~ It Is useful tnalreadly been implemented 130). Other exarmplea iclude trat the real potential fa an atulylio goral f. definedSP excitation of la the lower half of the COMPtotC t plsna. single sad molttpleStiy targetsb. periodic sad quas periodic arays Of 'ttip$ without fye t).6xfr1 'I 1 e0 lt0 (i

We with a reflectlog plate backing.e.plainar and eyllndricslly layered dielectrics r(..t.m) *Jf(..t) ell di: f(. t) - Re f.,(-t) (3a)

Thes letter esasepts permit investigation of the TD in the redoiotr. the moat $1llflsa opliOs arebuldup of mtipli e suors and of diaperolee edecls ased" with whther the aeMSWav IS. performedasociated wsth ridic structures sod guided modes; qie, or be" the V Inversion. Te former is the

slin 5tde fV ' aepee oh coivenationali route, gan fro~m the foil soluitton in theueful foes" Wi studie aScatteringby clatter fi rcloan th tydmiwieteatr

adby targets in clatter. arlnrayrah r r~esydmi otetm oa.wietelteI kaCs o a n ui tlestinatiaortl5~

presented here. to the point source moheass. Ercsing tse2. Symelsa eptioas for TO dyadle Geeea'a Funtios alternative together with the C" deopaion In (2). andc

performuing con=ee defortoatios la th eOmple spectralA. ?aa~ayeed ielctrc mdiapoin s tee t paths (SDp) trough saddle

. P le a n e l a y r e e c t ri m oda e r alO O o l e d 0 m a t e x a h y b r id

Dyad i rea't tsnctON for platte stratifiled I gnrlie a fed)cliekentri =wbith may tend 10 perfect conductors In the

naavo ee wel eplredin (fdifed rode felds),

msyoCdrs from sortabty, choen aSWAr pesnoal

La. claer. Very Short Pulse Scattering. rime DomainParameraation

+ (Ra mode reruwde)ou ( M Ti relation aon hq sehematlued bj the space-timey ~ ~ ~ (4 dependentslW onuto ray-mode trajectories in Fig. 2,BAch n-tcrm at the first ssc comis an SOP Into wtttv = = etIng themdal p sed. The modalwith a'ra Iths aosymptotic eviluation thee di~e P evAed a(,su. be es,itmto yd pc-ieryfedwivefroat).pr m ailescn series arusfroma residue at 414~ t '). .0 ,.tz) l4 4 1

oefororation, while tlast term n s aWrspetaintefral rumaitder alcmjot N, s peav where v /; steydlhs peiW .t the seeso d terms in (4) are derived from ~ Nanns~ss

the sCoWd term In (21 Derails of thes reductions may NmseJrslsfrteeoirtusI l.be fnin In3323 bep peraed for an input pulse fq) deried from the

The conventional sod nonicornventional routes in = ciyi _delt functiont i. (r I (sr)y, r - t.1r1, Istr<0pefr~gthe spectral synthests lead to the following Th oeta field at (r. IsaW been syrstesized a) viaalerIV treatments of the dipersin qa t io that dirct summation of s reflected wayvefros (firstdefuss the cma contruibuiorts In (4): tarm =t with N, T -0J 3 N) and b) vist leaksy mode

m)~ 4.e~~hD s..s~(s (5) saero~ end em n 4wilh Use fell spectraler.q-p Identifies the conventional frequency withicnus the tpereqe W fte wapule sru

depeden sptialwevoumer plesIs~(w).wheets a) are nottdlsperlve, they cart he evaluated A1 simpleidentifies the noesnrerions waversunt form; the wavefou series is tItitiated d~rly by the a*9;.enet frequency poles c. (Is5). The corresponding delayed arrival limes that fit into the ime i Inta Ofmodl1lsases Wsigird may be Inferred fro th observation, and Irjtreoy the dscresse at ach err.

respective 31.41 0 ar h in igs. 3. sard ar " ffloed ins the figurewith (i).)') esar e aluated e cl ter sts l Ithe Usyohas of the,

ode a~mpiodesthe trro dispersiv 1ab mode& Thillusttates theThe spectral Inregrils which remain in the camle role of dispersion under SFconditionts and the virtue of

saM the comaplex Is piano far the options in (Uanrr OBPhInthe flly SYiithealued dta(Sb). reapecively, can be evaluated asymrptotically by the The above analyic sursuary and nurerical samplessaddlec point method. The respective saddle point have beer extracted from a full manuscript beingconditions wt - A. and Is5 . , are specified by Prepredlor poblicatio 132).a) d*,(-)/OdI . - 0,. b) VX, 0.(.,) Ic k. 0 (6) Acknowledgentwhere V denotes the graient operator hr the T he longtlerm resach sarmmsrid ad relerstiod

sp ct i. Thi Imleht'% ia this paper has been supported by the OWks of Navalspectral~~ ~ ~ d$PAts Thi mplatacadtos5)nd() eeA.t Amy Pissese offte, ead the jointmast be Satisfied soltnoste sa jtednthe, fitr a(ace 5Rs cr sPormwI=$ modal spectral wavunumbers .,l.t *, t*) ad elolsrgat

Is0(r~sg %V); thatfs/d values are Idependenr of the Referencesmousie (conoetheral or aionconentianal) by Which they 111 11 HqeM1an d LL. Fallea. 'Creno w ansdwere derived. The aUtrulraneou requirements above canretecsi tailn citro ysotbe schesudued by graphical construction whidi locates convanebes I E t iniss $atLt pg Pon t four-dimuensional dispersion surface Dcla.Is).' 3 1.(1913).pp.4 .those points (at,,kr ) that how a surface normal It parallel 323 The Nease. N4the- an. SeisE Apldto the faijr-dsncnslonwspaetm 2 ThieeHer. N terans Sui. (198 pp. ie2Uo.1' 5.11'). with the orientationsaof the % e.r~ne area N.8 (94 M292lanthe spectal Ind conlraltloal domains arange 1s33 1 Heymn ari La. fele, *A wavefontm Fill 131333.) rh . 9 15 h) iste hclterprtialo of she sirgolarlty expsansion

ametitielWaVVCl~.'method,' IEEE The~t u ML Ppegau, AP*33diusesiosl wavvuelr.%Wiu.e are geaerally (I19 06)p,74718.cemyes good appromimattions for the Phases of weshly (43 1. Heyman adL.tlt.?odaesvcampen imodel can be obtained by platting the real parts ado La, re fr tsnaus((4. I~,)ats~t&1. eepagto and scattleelag of ray fMle' Wave

COoSlderallon from here on will be restricted soa Mofw.Vol. 7, (19M) pp. M3-31sl dselectic layer on a perfectl condlucting &round (5) R. Heymian vAd LIL Feisee. -Wu*h daperiveplans. wilth she *eee pomw (p ',s) (%Dz'.8) insole, spectral them7 Of lrssacal: WIMfaem lae andand the oheervation point Irjt) loated ouside. the layer oiteureslb E fa.MLoq ,APMWrelevaet mode fields are sow the leaky .w(987), pp. 8O.8

W oa3catuc aba rapped mouh hove exteral 363 X. Heymsan and LB. Felsee. 'We&iydsesv

evaasen fels.Th sdde oin ~t oniva m (6), spectrmal t01e" Of tisolalvl aas U-YI MOf she*whc yield Idearical reuts fo mode q Op orM. are Spectal bitav1 Y" Tsei ML r~eytqe.

1 .n~ 141 tL0 (7 .Hemnsid eLlses. ?ropastio paie1: V9 C bea soutl~ota oplex source parameter

(1 101 p.12.1043

ra moe a

Lt.Flen Very lSod PalmSocateftin: e Doin PaCAMetisZation

321 Li Fese an 2 Flyran.'Dac end beam 1243) .lt LiB. Poison ad D&M ulster,method For .tseHasssofrome dassnbused beras due so bea-s-mode ccovsion ofA~tt*Proceedingspf she SPIt Symposdsto, a hlfib-reqtiu y gaossisn p-wave isnputs na

Powa w rck Beam Sotares end aluminm losnt va=etmfA Acosee Soc Amn

(93 HeMyns and L.D. Pison, 'Complex souree [3 L. Po. Flset Ho snd L. Is, Iee-

(10 = Steinberi, F. Heymn aMd LB. Poison, so pcu~ o. AcamSm~ 80

bwi sypasce ofm sumanae oncitod psopoptlcoeradsd1tim from lut s. nortm IAORLso SeAnd U XH a B ese,1ocneto n

-A2 (1191). vs. 7559. Twsl tic ed utw rvng W e

11 B. Siverg, . e an d L B, l'lsn.M en, ucSWWstl t to n-Kdre place beamlloss summaltion ra .i meld (1n0 M 23is9-241o4. E

radation fro (198 e p a 8.1.'a cteigfo[13 L. eram~~vestsr a ki epbNgayt4 . J7)L PesnT rvb.~c .. e voerior

OS A m A218p. Wae bjct-spcta3A.W eptssslos so be

yet'w -OZ(I pb p. 7S-9. (2$) Soc. Fete and0.0]c 'We th r ing photowvslt-oule2cli1rca cylinderwibooked by a

11 iTK. A od. els. an L s, 'Mode s o loybrid 1-obeato lb led t

rs snt So A. 7 (5, pp

[173~ ~ ~ ~ ~~~~(9 H.LsBadL.Fles.Wvfstad [1 4. rolo, and . lezopuuloe sctand from

rSuc a tLy Ie) op. 9ca4l963.b a di n-eo antoennasoitob(1)co.L nducting elser its , m Et unode and 019I.JU .l nAan

[18 Lelest resss~l soercn (32] LB Poison it' .D. elos cr ~rpreprslonwav1161 eb~ LT -1. (187 pnd LI3D.2

31*11,Mo ea ybi 11.. m w er c e y ,a

t19 S ourcan L tPto aato iRns s an modeson infoetitl aeMo .H

conlas, ' 1 'nnet Sto 9m , (198) pp. Bro n &Flit(d.,Peu rm

3203 T. sihra And LB Felses, iWvotyb nd a4 rmilu O.Aeop m en O

bpskaeeptsu)ss ys of p perfsectynjoi 'aito rpriso lotiocean~es. JEWsti Tsrdam ArdennesnentsJ.oVOL

Sn.am g 83.(lV2.,N.8)ppp..985ONv. 97PJ213 L.. Meser a rge 0. rlen so e L ecen 1Poisontnad .M ,aucrp n rprain

(19) LH. osodn sd 1,. ritsm 'Rys n rods&se oavsisl a se l rec nd dslte

'I'-V.Ls i Felo endo M IRc 'PIPsemode T o.veraln t an lssinl'Hybsidat aseltraso u ato n &AOB opl oa s slS .

SeA~ma1,(14s a9Z ioA'tog. 12 (90121 Hyaz 0 riieuan LBJ4lsnOf=t eonne1fi

LSE Ftise VocySlson e Scattering: IimeDmanPrmetation

F*j Harfzmn diapla-eneted diemctis aestsrelatieamtSt. .nted F=4ie emabtt

ptu Mt - asnoi (PW C)&

wIt)defined in (6). The subsciopt s bee thate t esals fttm- ~ = _

U..0.r.. i=iijf feld o th bserme aus phase

purameterdterent theeo oosfia

It 2X-M tirat ra =I%$ 9eat(ut 0tMA

A,. t

I10.2 w NY 1

a ;,SPp~ an t~rs lace l o ceT P

tttteaye . Numriafl esult for ~T'aE cwbto oF fodom omlzdtecerodby it pulse a nt.asitd E.. Physita onst rursud asin IS vor teuc prbslm lwo~ra~

paraeter raye tona each ealsrllcte 0 StFC

Direct ~~ S&rn aclt.

DOW xftL10 d, P'.L4

LL Fell=o VesyShort Poise Scattesini Time Dettss Peosmethtion

3,d mode 3model. from, M02 to mc4

-00 .00.

1 ~ ' L OZ/ ) me: t~o (Bem:

70 oe.fo5=1t e oe, rmm lt w

000

-0,

.101

cesttc~o od esrotie oteleeoe. Mod sm -041 Co mpltie [trfel

4 it.. I'l4

- -Finite-Differenoe Time Domain Modelillin

of Electromagnetio Fielcd.

Inao Wolff. Duliburo Unvereitv

For the design of planar microwave integrated circuits up to 1985 mainly analysistechniques in the frequency domain have been used With the requirements fornew and flexible tools in the design of planar circuits eg with closely coupled

S pi mpntq and three-dimensional dlscontiuitloc lihko airbridgoo, alternative tachniquet

must be studied. One of these techniqres is the finite-difference time domain

analysis (FOrD) which In principle is known since 30 years. Yee already in 1966proposed this technique for the analysis of electromagnetic boundary value problems[I)

ItDuring a long time the FDTD technique only was used to quantitatively demonstrate

tlectromaognnfic, finld snhilinn in fhp time domain Only the Introduotion of oooorbingwalls made this technique to a powerful tool for the solution of real problems.

The FDT is a numerical method for the solution of electromagnetic field problemswhich has a large numerical, but a low analytical expense. Despite the large

numerical expense i is believed to be on,) of the most efficient techniques, becausebasically it stores only the field distribution at one moment in memory instead of

working with a large equation system matrix, The field solution for each other

time then is determined from Maxwell's equations and is calculated using a timestepping procedure based on the finite-difference method in time domain. Available

algorithm, called the leapfrog algorithm fits very well on modern computer archi-

tectures, so that the data required to describe a three-dimensional field distributioncan be handled in a reasonable time. Therefore It can efficiently be implementedon vector or parallel computers as well Sufficiently accurate results can be received

by using a single precision floating point expression requiring only four bytes. It isa further advantage that the transient analysis delivers a broad band frequency

response in one single computation run.

In this talk it shall be demonstrated that the FDT technique can be applied to

real microwave circuit design problems It will be shown that this technique enablesto model arltrarfly shaQped planar structures with multiple coupled discontinuitiesand planar lines and three-dimensional circuit structures Several applications to

realistic problems of modern monolithic Integrated microwave circuit design problems

will be demonstrated In the future the application of FDTD method surely Is a

powerful analysis technique for nonlinear microwave integrated circuit design by

combining physical semiconductor models which work in the time domain with the

FDT description of the passive crcuit elements.

(I] KS Ye, 'Numerical solution of initial boundary value problems involving

Mexwell's equations in isotropic meda', JEEE Trans Antennas Propagoat, Vol

14, 1966. pp. 302-307

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4.

1

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E,

V ii

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E2 020 3000 600 tAt 9M0~~ t4

-Voltage history at port I (left port) and port 2(right port) of the spiral inductor. A1203,

6r =9.8,height .635 rom, aidth = .625 n,

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100

TLM Modelling of Electromagnetic Fields

Wolfgang J. R. Hoefer

Address: Laboratory for Electromagnetics and Microwaves,Dept of Electrical Engineering, Univ. of Ottawa,Ottawa, Ontario, Canada, KIN 6N5.

ABSTRACT

In this workshop paper the principal features of TLM analysis of electromagneticfields will be summarized, and research trends in this area will be discussed. Time domainmodeling in general, and TLM modeling in particular, is focusing on the realization ofa new generation of time domain simulation tools which link geometry, layout, physicaland processing parameters of a microwave or high speed digital circuit with its systemspecifications and the desired time and frequency performance, including electromagneticsusceptibility and emissions. Such CAD systems will most likely employ dedicated parallelprocessors configured in a 3D array. Ftirthermore, the specific nature of discrete timedomain algorithms affords optimization and synthesis procedures which differ radicallyfrom those employed in traditional frequency domain CAD tools.

1 PROPERTIES OF TIME DOMAIN FIELD MODELS

1.1 Time-Stepping Algorithms

Most time domain field models describe only the local properties of the propagationspace. The most current forms are based either on a discretization of Maxwell's Equations(Finite Difference - Time Domain or FD-TD formulation) or on the description of space bya discrete spatial network (Transmission Line Matrix or TLM formulation). Fig. 1 showsthe basic 2D TLM impulse scattering process which can be considered as a computerimplementation of Huygens' principle. Finite Element formulations in the time domainare also possible but have not been used extensively so far.

FD-TD and TLM methods employ similar but different formulations. While FD-TDis expressed in terms of total electric and magnetic field components, TLM uses incidentand reflected wave quantities in a spatial network. As a general rule, both formulationsare equivalent; for each TLM scheme there exists an equivalent FD-TD formulation. Fig.2 shows two such pairs. Figs. 2a and b show Johns' distributed TLM node 11] and Yee'sunit FD-TD cell. [2]. Figs. 2c and d compare Johns' condensed TLM node [3) andthe equivalent FD-TD scheme derived by Chen et al. [4). Fig. 3 shows the dispersion

i12

characteristics of the discretization schemes in Fig. 2 as derived by Nielsen and Hoefer [5].For low frequencies the dispersion surfaces form a unit sphere in all cases. However, athigher frequencies the dispersion characteristics of the condensed TLM node and Chen'sFD-TD scheme (Fig. 3b) are far superior to the other two (Fig. 3a).

Clearly, the equivalent TLM and FD-TD schemes possess identical dispersion and er-ror characteristics They can also be derived formally one from the other. Furthermore,optimized codes for equivalent schemes require a similar computer memory and executiontime. Nevertheless, they have their respective advantages and disadvantages when im-plementing boundaries, dispersive constitutive parameters, and nonlinear devices. In thefinal analysis, the choice between TLM or FD-TD is based mere on personal preferencesand familiarity with one or the other method rather than on objective criteria. In thefollowing, the salient features of time domain simulators will thus be described in terms ofTLM formalism with the understanding that there exists, or could be found, an equivalentFD-TD formulation unless indicated otherwise.

1.2 Requirements for Time Domain Field A nalysis

The principal advantages of modeling electromagnetic fields in the time domain arewell known. However, in order to exploit them in a practical application, dispersive andnonlinear properties, moving boundaries, and sophisticated signal processing proceduresmust be implemented, which include forward and inverse Fourier transform, convolution,and absorbing boundaries. Another important requirement for practical applicabihty is agraphic user interface for 3D geometry editing, parameter extraction and display, as wellas dynamic visualization of fields, charges and currents.

The feasibility of these features has been demonstrated :ctli in TLM and FD-TD en-vironments [6]-[S]. However, the computational requirements for wrodeling complex struc-tures with such methods are still extremely severe. Therefore, research efforts are beingfocused on the development of accelerating techniques, the most important of which willbe discussed below.

2 ACCELERATING TECHNIQUES IN TLM MODELLING

In the following, the most important accelerating techniques will be briefly described.The first exploits the localised nature of the time domain algorithms through parallelprocessing, the second is based on the numerical processing of the time domain outputsignal using the Prony-Pisarenko Method, and the third involves the reduction of the

so-called coarseness error by improving the properties of the discrete TLM mesh in thevicinity of sharp corners and edges.

2

2.1 Parallel Processing

The principle of causality ensures that any change in the state of a TLM node affects

only its immediate neighbours at the next computational step. This allows the implemen-tation of TLM in a form quite dIfferent from the program on a serial machine. Since ina parallel computer each processor has its own memory, it is practical to assign to eachof them an impulse scattering matrix and a set of boundary conditions. The impulsescattering matrix incorporates the local properties of the computational space such aspermittivity, permeability, conductivity, and mesh size in the three co-ordinate directions.The boundary conditions specify whether there are boundaries between a node and itsneighbours, or whether the nodes are connected together. This parallel implementationgreatly facilitates variable mesh grading, conformal boundary modeling, and the simulation

of highly inhomogeneous materials and complicated geometries.Fig. 4 compares, on a logarithmic scale, the improvements made over the last year in

computing speed using various programming techniques [9] and parallelisation. The orig-inal matriz formulation by Johns [3] requires 144 multiplications and 126 additions andsubtractiors per scattering per node. Through manipulation of the highly symmetricalimpulse rcattering matrix, Tong and Fujino (9] have reduced the scattering to six multi-plications, S6 additions/subtractions and 12 divisions by four, increasing computing speedover six times. Programming in Assembler rather than C++ accelerates the process againfour times. Finally, parallel processing increases speed by more than two orders of mag-nitude over the fastest serial version implemented on a 386 computer in C++ language.The combined rotasures effectively reduce computation times from hours to seconds.

T:Ls comparison strongly suggests that future implementations of time domain simu-lators fmi, CAD purposes will be based on dedicated parallel processors or supercomputersthat emulate parallel processing.

2.2 Signal Processing

'fht fast Fourier Transform (FFT) is the most frequently used signal processingmeth ! .or extracting the spectral characteristics of a structure from a time domain simu-ltioL. For efBcient computations it is of prime importance to reduce the number of timesamples requi.ed to extract a meaningful frequency response. To achieve this goal, process-ing of the time response using the Prony-Pisarenko method has been applied successfully.[10].

in this appruach the discrete time domain output signal is treated as a deterministicsignal drowned in noise. (Fig. 5). The signal is then approximated by a superpositionof diumped exponential Junctions (Prony's method), and the noise is minimized using Pis-arenko's model. This signal processing technique reduces the number of required timesamples by typically one order of magnitude.

3

2.3 Reduction of Coarseness Error

One of the principal sources of error in the TLM analysis of structures with sharp

edges and comers is the so-called coarseness error. It is due to the insufficient resolutionof the edge field by the discrete TLM network. The error is particularly severe when

boundaries and their comers are placed halfway between nodes as shown in Fig. 6. Itis clearly seen that the nodes situated diagonally in front of an edge are not interacting

* directly with the boundary but receive information about its presence only across theirneighbours who have one branch connected to it. The network is thus not sufficiently"stiff" at the edge, and results obtained are always shifted towards lower frequencies. Theclassical remedy for this problem is to use a finer mesh in the vicinity of the edge, butthis introduces additional complications and computational requirements. On the other

hand, the dispersion characteristics of the condensed 3D TLM node (see Fig. 3b) are sogood that the velocity error is practically negligible even for rather coarse meshes. A muchbetter and more efficient way is thus to modify the corner node such that it can interact

directly with the corner through an additional stub as shown in Fig. 7 for the 2D case.Since this stub is longer than the other branches by a factor V2 it is simply assumed tohave a correspondingly larger propagation velocity. In the 3D case up to three stubs mustbe added depending on the nature of the corner. The effect of this corner correction isdemonstrated in Fig. 8 which shows typical results for the first resonant frequency of acavity containing a sharp edge as a function of the mesh parameter At. The parameterp is proportional to the fraction of power carried by the fifth branch of the comer node

and is equal to half the characteristic admittance of the comer branch when normalizedto the link line admittance (see Fig 7). For p = 0 (no comer correction) the coarsenesserror increases almost linearly with increasing Al, while for p = 0.1 the frequency remainsaccurate even for a very coarse mesh.

3 BOUNDARIES IN ARBITRARY POSITIONS

3.1 Accurate Dimensioning and Curved Boundaries

The accurate modeling of waveguide components, discontinuities and junctions re.quires a precision in the positioning of boundaries that is identical to, or better than themanufacturing tolerances. If boundaries can only be introduced either across nodes orhalfway between nodes, then the mesh parameter A would have to be very small indeed,leading to unacceptable computational requirements. Similar considerations apply whencurved boundaries with very small radii of curvature must be modeled. It is thereforeimportant to provide for arbitrary positiomng of boundaries. The basis for this featurehas been described already in 1973 by Johns 111).

4

i Fig. 9 shows the concept of arbitrary wall positioning in two-dimensional TLM. Theboundary branch which has a length different from A1/2 is simply replaced by an equivalentbranch of length A1/2 having the same input admittance. This ensures synchronism,but requires a different characteristic admittance for the boundary branch and hence, a

modification of the impulse scattering matrix of the boundary node. (see [11]). Theeffect of such boundary tuning is shown in Fig. 10 which indicates that the length of theboundary branch can be continuously tuned over a range of more than one mesh parameterlength At without appreciable error. This important technique removes the restriction thatdimensions of TLM models can only be integer multiples of the mesh parameter.

An alternative method %hich avoids the modification of the S-Matrix of the boundarynodes is to replace the extension of a boundary beyond its standard position by an equiv-alent reactance. The differential equation of that reactance is discretized, resulting in arecursive formula for the impulse reflected by the boundary. This method is preferrable fora serial type computer implementation while the former is more appropriate for a parallelversion.

3.2 Moving Boundaries, ard Time Domain Optimization

Since it usually takes considerable time to build up a quasi-stationnary field in astructure of high Q-factor, optimization based on a new complete analysis after everymodification is extremely time consuming. Instead, techniques for continuously varyingthe boundary position and other characteristics of a structure during a TLM simulationwill be developed. Two different methods will be investigated. One is to modify the

scattering matrix of nodes situated close to a boundary, the other is to generate theimpulses reflected by moving boundaries using recursive algorithms. In order to implementautomatic optimal tuning these measures will be coupled with appropriate optimizationstrategies. Furthermore, if optimization criteria are to be formulated in the frequencydomain, a sliding Fourier transform window will be introduced as well in order to extractthe time-varying frequency domain characteristics from the evolving time domain response.

4 NUMERICAL SYNTHESIS BY REVERSE TLM

It has been shown recently by Sorrentino et a. [12] that the TLM process can bereversed without modification of the algorithm, yielding the source distribution from theresulting field by going backwards in time. Direct numerical electromagnetic synthesis iscompletely unchartet territory as yet, and the exact procedure and its implementation arenot very dear. The desired characteristic of a structure or component is usually given fora limited frequency range and for the dominant mode of propagation. This informationis insufficient to synthesise the exact topology of the structure. Therefore, the missing

- --

information must be generated and added by the designer. Implementation will mostlikely be an alternate sequence of analyses and syntheses which will converge much fasterthan repeated analysis and optimization in traditional CAD.

t 5 CONCLUSION

~Computer time and memory required to model realistic electromagnetic structures

' are still obstales when it comes to practical applications of time domain modeling tech-I ' niques. Therefore, considerable research efforts are concentrating on ways to reduce thecomputation count significantly. In this workshop paper we describe three different ways toachieve this, namely parallel processing, Prony-Pisarenko signal processing, and coarenesserror compensation at sharp comers and edges. All these methods can be combined toaccelerate TLM simulations by several orders of magnitude. Since the computation countfor TLM analyses increases faster than the fourth power of the linear mesh density, theseaccelerating features enhance our ability to model complex structures to a much greaterextend than the mere memory size and speed of the computer. Procedures for fine tuningof wall pceitiouns have also been described.

Future time domain CAD systems will most likely employ dedicated parallel proces-sors configured in a 3D array. Furthermore, the specific nature of discrete time domainalgorithms requsres optimization and synthesis procedures different from those employedin traditional frequency domain CAD tools. These include the implementation of mov-ing boundaries for geometrical tuning during a simulation as well as numerical synthesisthrough reversal of the TLM process in time. It is conceivable that at the present rate of

progress in time domain modeling these procedures will equal or surpass the capabilitiesof frequency domain CAD tools in the next decade.

REFERENCES

[1) S. Akhtarzad and P.B. Johns, "Solution of Maxwell's Equations in Three Space Di-mensions and Time by the T.L.M. Method of Analysis," Proc. IEE, vol. 122, no. 12,pp. 1344-1348, Dec. 1975.

[2] K.S. Yee, "Numerical Solution of Initial Boundary Value Problems involving Maxwell'sEquations in Isotropic Media," IEEE Trans. Antennas Propagation, vol. AP-14, no.5, pp. 302-307, May 1966.

[3] P.B. Johns, "A Symmetrical Condensed Node for the TLM Method," IEEE Trans.Microwave Theory Tech., vol. MTT-35, no. 4, pp. 370-377, April 1987.

[4] Z. Chen, W.J.R. Hoefer and M. Ney, "A new Finite-Difference Time-Domain For-mulation equivalent to the TLM Symmetrical Condensed Node," in 1991 IEEE Intl.Microwave Symp. Dig., pp. 361-364, Boston, Mass., June 11-13, 1991.

67

[5] J. Nielsen, W.J.R. Hoder, "A Complete Dispersion Analysis of the Condensed NodeTLM Mesh," in 4th Biennial IEEE Conference on Electromagnetic Field Computation

* , Dig., Toronto, Ont., Oct. 22-24, 1990.

[6] P.P.M. So, W J.R. Hoefer, "3D-TLM Time Domain Electromagnetic Wave Simulatorfor Microwave Circuit Modeling," in 1991 IEEE Intl. Microwave Symp. Dig., pp.631-634, Boston, Mass., June 11-13, 1991.

[7] P.P.M. So, Eswarappa, W.J.R. Hoefer, "A Two-dimensional TLM Microwave Field

Simulator using New Concepts and Procedures," IEEE Trans. Microwave TheoryTeclhiCU.3, vol. MTT-37, no. 12, pp. 1877-1884, Dec. 1989.

[8] M.A. Morgan, Editor, 'Finite Element and Finite Difference Methods in Electromagnetic Scattering," PIER 2 Progress in Electromagnetics Research, Elsevier, 1990.

[9) C.E. Tong, Y. Fujino, "An Efficient Algorithm for Transmission Line Matrix Analysisof Electromagnetic Problems using the Syonietrical Condensed Node," IEEE Trans.Mir:owave Theory Techniques, vol. MTT-39, no. 8, pp. 1420-1424, Au.. 1991.

[10] J.L. Dubard, D. Pompei, J. Le Roux, A. Papiernik, "Characterization of MicrostripAntennas using the TLM Simulation Associated with a Prony-Pisarenko Method,"

Intl. Journal of Numerical Modelling, vol. 3, no. 4, pp. 269-285, Dec. 1990.

111] P.B. Johns, "Transient Analysis of Waveguides with Curved Boundaries," ElectronicsLetters, vol. 9, no. 21, 18th Oct. 1973.

[12 R. Sorrentino, P.P.M. So, W.J.R. Hoofer, "Numerical Microwave Synthesis by Inver-

sion of the TLM Process", in 21st European Microwave Conference Dig., Stuttgart,Germany, 9 - 12 Sept. 1991.

t7

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Recent Developments of in Numerical Integrationof Differential Equations

Wolfgang MathisUniversity of Wuppertal

AbstractNumerical Integration of differential equation is a standard discipline in numer-ical mathematics and basically for simulating dynamical systems in all areas ofengineering. In dependence of the kind of modelling dynamical systems will bedescribed by ordinary or partial differenti % equations. In this paper we restrict usmainly to the former case. The most general type of ordinary differential equationhas the implict form

F~x,:,) = O, ()and is called differentil.algebraic equation (DAE). By means of a suitable trans-formation the explicit dependence of t can be dropped If F is solvable for iglobally we obtain

- fx); (2)we refer this type as ODE Because these equations possess a manifold of solutionsrather than a unique solution additional conditions must be prescribed to x. If xis determined at one time s this situation is called initial valued problem (IVP)whereas conditions in different time points are called boundary valued problem(BVP).In order to simulate a dynamical system we start with a system description oftype (1) or (2). In dependence of our interests we formulate a IVP or a BVP. Forcalculating the solution x(t) we associate a convenient difference equation to (1) or(2). It is obviously to replace the derivative i by a difference approximation witha step-size h and to construct a discrete sequence of x(,.). The quality of suchapproximations will be characterized by consistency, convergency and stability.The theory of numerical integration of ODE (1) and the art of its implementationare available. In this paper we are interested mainly but not only in multistepmethods. In most applications, e g. mechanics and electrical engineering, mostdynamical systems are represented by DAE's in a natural manner. For this reasonwe discuss the main aspects of to the theory and implementations of numericalintegration methods for DAE's and remark that the essential results are developedduring the last ten years.Furthermore we will discuss the problem of step-size control and switching betweendifferent integration algorithms (order control) from a control theoretical point ofview; this part includes also some results worked out in our corresponding project.We illustrate this material by means of some examples from circuit simulation.The final section contains something about the problem of rounding errors. Thisis an esoential subiect because the development of algorithms (operations) and thecharacterization of their properties will be discussed in the real numbers IR andin other sets, e g. CY, , which are constructed in a 'vertical manner'. Inclassical numerics we choose a suitable finite set (e.g. floating point numbers F)of M and associated operations and construct the 'higher' sets and operations ina 'vertical manner'm, too Therefore it is not clear that the nuaerical algorithm(implemented in F) works in the manner as the algorithm in lR. To circumventthis problem it seems to be useful to apply a well-defined arithmetic (Kulsch-arithmetic) and well-adapted algorithms. This approach is close related to thewave-digital filter method of Fettweis.

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-(1 At oeo f-1.

Restrictio toooto tho tIl *Pim nsonlts

Too~e~ Wstootlso~dt

*. Teyto Methods

+o 4t - (,At1-)) (4) *Ieotooo elole o2.teSoe

o . 4, slMehod + L 't(/ . P-0t

C kttoh.- If (s..t Att. s. . P-o

-.L o.-2 a (2)o +. ST.t

+z "'.) p ot)+O) (s)2

(Mop1ooot Sue hod Atd

APP-1,11 11911 M3.2 Basic Properties of DE from ODE0

cM--ni Operator. of LInes, Meitotec Methods

h NtobSi oi Scip.

- Oipieceonent .. Chronisiof of C. by Poiyotiei

FQ)* (oft) P~f Pfyroit Ficin -)So-i

* - Isiejison ef ro ney *..:y *n,4ei ii if f nfco4, ee 5 ~e

Onder -014 .£ . C- 0oyoi~

LI.. feIt~t c .1 fd -~~ E 1.e\ZS

I.. E .. ~ ~ ~ o~..i- StcIo NI -- o E 1 epW ... S

ogisng isopofo De OEnoistionedt D

- heoot etncofscce diocid ~ifiinci) -A E. piocci, LOS nch

..- 1c , Apply~ .-tfincn ftch -d B.S W.oo htce

.i 0~," L -1sc~e~ T, - . -ce Eiin Iii (3 4 ) -A

(of.))-5ocfe,)-.54,(fc f - tLrs a DE by , hi ecf 0, E

*,po tlfiei tots)- ,pss-sco EIe (LIE)/4

f,, Ecc d o..c.cI.- ~ ~ ~ ~ ~ ~ ~ ~ ~ h2 I.CtfEis(eos i e'es D

(A. o~tot) Sebi~tyN~py o De~q.,t4. Stability Propertar of DE1's from Slpecti ODE.s

P-.~j s] .- uss,1 4t A ttine Te- ODE

Csrsitof EMS by boasofy , W Ver iiotresij for pranic) o'plioC--q6.,* . 11eiotSl.ioesho desolh)

oM (-i b-. ablbllii

5 LO I') s 1 ooLorsoS sod 50.1 t-b (h

hs "'.lo~ .0- . .

S. (5 (-l L f ocIto iS 0)de s.] So

0 Sthlly rep-o ofD o the tes t ODE A,

soleint ~ ~ J(, S(S 1)'eoers-O

ldosl Cordloe 4.2 Step Sin Cotrol .od iStlfroreos..- c

E-eple C-1s N-Ionl Sl"Ibod A-irsy deoseiloe the Step S-o

U r WIE-r, Ai I

* q Dhqoiio A Sisbiffip r OrA.,Iy-il diesog siJoss(ADE) d.-.Aceogsioioss(E) (I bAle.

osinls loesot Ecci hitbod Itr - r '. OAA. I

Z Fi- Cosin. Arytooh

* tdoionsIsly A Scslotsy .....i~ A--esy - L'TbeohnIsbyii - sDt-.Ohs

E1.0-~ LS -nitrd: '.Aso to, A -iS W

-iydctenxmxx hr W 'S~t? ODVs 4 3 'Stiff' ODEteerexpi.~A.OD xt .bx .4x.tb. Asw cwxwr x/

ODE ADE-~h ODEt. Exempt.,

rip-SxeG <fix.l >2 t.'? +9f(A xt

h-xx. . - aak

2)2 Stexx~e~e~e~ex~eh- -92.'U16-1/21ti 12-Sa..G16, t

E.g.ee S I .~ Oxxex

-Ite itg theli, btm t' *4. .(t - leirpfejetx,. xp(_t) ..j exP(_log) ate xxt ci.].x(I + Aix) , +LTC,.

-2 < 0

*The. fsasetx OD',. reta d 11-

23 sx~is.ee, i.,24

* th- Iitiat- aceelee a 6e tAit?, beet.. Differext Stxbility Coxcepta for SttffxessIA Al is ... ut

ODC

N"'.~ rrz..f

N- .,,-S AeL-N,

Further Characteursat.oot of Stiffness' 5 Remarks to the zoP1tthio of ODE-Solvers

... onP obi.m. foe 'M,,,.tiog.ILMS f.,I~y

{Ssdr,6L$tos Effi-

5t Rep S-e irul*- SI'---~ 1.1~ N.eoo typ. Metlhods

Erroee Ordt Costel-- IP t-11t~ho05ptth.d dd

Wtry Usef. C.."'tsto-eC..,.1 71-yt* P~etto. etiod dettotoe stsrtog ~..A. Eosnepl., tR, hosi

I" C-.df Il"I1-l

C Step Sir. Cooeol step Sto. Coettol sith tI C

29 .~vv,30

oviie, em~rs i Jvwe~vvi~ivvi6 R evised 'Des,ptivsv Equaetivon fver

*Suvss,,~,.. ExPbc, ODE . .. .. .. svisiy fo., s ,,_,* E30~t &~, o~,, ~ htd-

E- l '5 Oute * -pbvi ODE d-tl vbt.vdEMy'es of...UNll.hi/o) TyZ -f Eqv-tvv (DAE

- vp t ODE(* DAv E

- 101-1up,, DAE

O': (xp

*Eluvv Niv',& dl-ubed by .s.vt-r vf

(R .. Dhvifl b ....vs aas ... bvl.uu,i,,,q

:be s.. .d .v..es b ss'v .... P 1 d sds.tv,,bhs.s.......

e 'sie o 0 1. -(No.,)

0026 .Og, ss20.r 1

ti-O.f v.9 0

fo 0

Fl.

7. DAE's are not ODE'. N A 0

- ..l y 3 E A ape,,, ) .

- SOA3* ,3O 00.,. 3 030~~.6

- C)l...i0 11.o. by 'Zndex' Co.ept=

I l) o fd.i. U,... DA 6 il¢~t~ C~ef t

0 0 94

. . .. . . . ............. ... ..n .,) Y O U0) -.1 1, (I,. ( . ..., ..

()) a.' o (3) "h -

3..,3) (3 Al. tz 33S003,S3I=Ollae~ (3,330,a)

- 2 =..,o ,,1,r3) fie 43,0.¢|,0. 3'dtah atull

kb3 bl.o 0(,.

... .. ) - ). )3.) 0)t._,)

3),-. 0= ,3,/()). ),0) )P)- 0t.) -0 0.3

3h er ~ A,ebk (,,

7'0 0r S~

o f2 ox- 0

0 *0.~o ~ 2' ~'0 C

.0 0 t 0 t '0~ 0 ~ ~0.. .~ o0~.C.0

'~ C 2 *'

E E ~ '2 e .0:00

E ~ ~' 00 ~

2.00'~ ~ Q 0'

-

22.

0 .0 00

'2~- 0 .. 0 ~ oQ~ o2 .0 50 '00 2-

- 50 S

0

0~ 2 -

.. o~E

0 2 ~ TT O~'~ 20. 002

~ 222 E.2S ~ 00.*0~ '2-' '~E ~ ~

0 2: I I I I j

Definition of CA

Djscret, in spac: CA consist 01 a dvcret lattice of saces

Cellular Automata: Dsrt ntm'C vlei iceetm tp-Applications andl implementaltion Dsrt io Aeov odsrt ieset

Discrete states' Each state takes on a finite set of possible

values

Homogeneous All cells are identical ani ore arranged in

a regular may

hi. ir Synchironous: All cell vaues are updated to synchtony

Ur Sej .. WO.iee Deterisnistic' Each cell is uiolated according inoa fixeddetrnintistic role

Spatially local, The odle depends only oii the values ofta

local ineiglhothood

Teimporally tocal. The role depends only on values fot a

fined number of preceditg steps

Formal Definitton Related Models

Piartiat differential equations: space, lone and site values

.E (I, t) Qo({(I-d, 1 1) d E r)))) ate continuous

Finte difference equsations: site values ate contunuous

I "TeindexParticle models: particles have conttinuous positions asidI idex domain of CA

aobitary function S!" -S

neighbothood. e g 11 =3 3J E Z" A 11711 !5 r) Neural etoort models, connection patteens ate aibitajiy.

set of states. toC a(1, ) E S site values are continuous, updates are asynchrtonous

set of colifigurattons. Ite E= S11 ellrnua e5

rsst ausaecniuu1' global mtapping, soe 4, E - Cellrsaa ntok o aue t oonu

0 s et ot configurations geneciated afir f iterated Iteratise arrays: differetit purpose rtan CA

applcatons f 0 t Q' EW-' OQ Sl Array processors: sites can store ostensive infortiation

Applications of CA Physical Modelingipttation Theory Motivation b eioec pril

*self reprodidion (3v Neumann 1949. Conway 1970 phsenomena are often describedby(irlna)pra)*fonl"3 languageC (S WVolfram 1984) differential equations, two different approaches to s.,.'t'

I Viifieu cqiaation ons a msacrosoopse 30o10lisoogicl l'iodelagdiscrtneation in lime and space

*self reproduction 0I v Neumninn 1949) 2 CSnCdsrtnd d

*evolution (S Ulam 1918) microtop. diceie ode'

*mcdels of msenmory (hi bujns, 1969) large nounber of smilar components with local astcraction; '

Phlysical Moodeling

*calculating spaces (K Zuse 1940)fucinlhmgniyrletspeadtmerva-ance, locolity reflects finite speed of informtiron

*hydrodynamics (0 Hands 1973. U rrisclt 1986)

*groistli snclsainstos (I D Gunton 1983) cellular automata are simple to program aod amenable to

atiem recognition (K Prestoit 1984) parallel processing

$ pot mtodels (lii Cicute 1980) studies of collective phenomena possible (trbulence,

* ove odiels (HI Clien 1988) chaos. fractality,

I1B Salem, S Wolfraiss 'I'liettnodynraics and 11) drodyon-

nnes isiltCellular Antosnata In Theiory and ApiplicationisFudDnmc

,,I Cvlul Autoitti W01ld S-iiif, 1987 Fluid P M 92,Fr~ DyaicshPm 96

fluid inoompressible. abseoce of essemal forces Navies-

________________________________Stoikes Equation )NSE)

W+ V All -Vp + v V21'

------- -' velocity

~ P (constant) densityVviscosity

-. ~0Z\ Z<Z~/I'\ ~*fictiiiouts microscopic model,

*as simple as possible dynamics (snst necessarily follo-

___________ ting Hcamdltoniani equations for intetacting pat iv~les)

*reproduces NSE on r roscopic level

Ploss paut an obstacle (from Salons and Wellsain) .particles and their scattering ane modeled by reversible

CA rules

J 1B Satcm,- S Wolfram TItlermodynamics and Ifydrodyna- 'mUIcs With Cellular Automata In Theory and ApplicationsLatcG sOf Cellular Automata World Scientific. 1987 Latc Ga

Particles move On a lattice anid satisfy certain symmetry- requirements Moving and Scattering by reversible rules

Derivation Of moacroscopic behaviorSolecular level motion is revertible

kineoc tevel nuneatuilibriam statistical mechaics

~'~-- -- \~. * macroscopic level continuum approximationI Wave eqaions-

~~~~~~~~10 s~~~~'* Z /'.- n mall Perturbations from eqailibriom. linear elasticProperties Of lattice gas~NNN piopdg4iioii Of a disratbation Is governed by wave

-- - equation

FlowPas an bstcle(fro Saem ndWlfrm) tudy Of wave Initerference, reflection diffraction, re-oustan bstcleCfrm Saem rd ~ulwv)fraction

.4-

'R i . 1 -v.

-4* . - - x s e"

1, V -lie171

Modeling of Wave Equations Htyugens principle any spatial pointcasb iugtI

to Cire 19551as a revw wave source with intensity

Liea wvepopgaio=C2 72,,(X, t) decay rate Of Source

' 1 DU)2Cl(,u)2 i2(x,t + 1) = t2(x,t) - 9(Z' t + 1)

energy H dt*continuous linear wave equation is recovered after ma.

wointi P=2 -V (LU)Idt ling an ensemble averaging

*result can be converted to a finite difference ecojationConcept

*two kinds of photons propagating ov a lattice a

,VO(x,t)- nuntber of phiatons vnub quantum oat Va

particular sitir x aitd tinto t mavuig Wit elocity a

lChen, S Chet, 0 Doolen, Y C Lee Sitmple Li

tucGas Mol 0l for WVaves Cuniplcs Systents 2 (1988l)

259-267Implementation of Cellular Systems

ii -. ~(S Y K.,jt, P' tDaitle. E Dopreini a ttnr,,

- Specification

*Cellular automata

(!I I ao v(l,t).d({a( -d,t1- 1) 1 c P',)

- ij~u~iCell!ar Systems

3) {u'(I d) d Vi)

it()=6t ({ut(I - d) d E 7tt11,

~ -~(at,(] - d) d E Dir

xai truei iir '' a.a uerio

Example-agtAcietr

I ndence graph dedicated hardware (CAM-6. CAM.8 (Toffoh))

*Coarse grant parallel systemsa (MtMlD, Transputer)

Sfinr grain parallel systems (SIMD. Connection Mlachair)

Mapping criteria

X commuonicationa- comtputation

*consideration of pipelared aeslltictic unils

Progr am onisderation of finte resourcea

suited for automatic comilation1,1 0 5 2 2A 12!

a)(', 0 = 01(-2(t 1,1 l),ci(s 1 1)) 11Applications

a2(,,t) =0(sal (z, t),ai(s +- 1, 1 -1)) *neural networks*Itrfive arrays

Jsiced depcndence grathcllltsuomtsolution af PDE

systolic array$

I-D

41 Un 0 Itat<I

<aae

zI ; U :-L

Implementations Mapping Problems _

irfiguration: Partitioning (limited resources)

Synchronization (control path)

(0 O( nilobanki I

Scheduling (pipelined arithmetic units)

eovoral-~

specification or embedding (softnare)

Partitioning Partitioning

Projection. Actise Clustering.

a reduce dtoieooon of DG by I matrch given number of cells

slinprjecin oDGmatch gisen dimension of array

* rcncedooosin o DOrobiruilycombine ninloiprojcoon and passive clustering

* loop control or flow controlSoiuh I/O rate and computation rate I ocal Seqouniat / Gtobal Paratlel (I.SGP):

DG Srquencmtt Sigral fl., Sriph Processor array Clustcred anay

Passise Ctustering-

MA ninoefficient array efficient Loeat Parattet I Gtobal Sequenilt (LPtGS),m ike use of pipolviod out P'rocesor array Clustercd array

AA 00,00 :-l~t 7-7

I P~Os(folding)

Hierachicl T-a-n-sormaionsHierarchical -R epresentation of Algorithms

o11 fieain spie sngtilng mnn P Iowerhirarchia levels ~ dn

S2 S4

* nirodUaion of ieniiioals o Tlere are two basic Program schernas

70Day 2 Straigl'rline code (set of oisigrr'en statements, nonreguli

bRegular algonihen (regular dependence graph)

pormasformaio

(11,A,', JEJ AK E KA E r[ K]l - PxJ- d - i1 +~ P)))

If J- d - r EJ3

Hierarchical Compilation Strategy Mathematical Tools

Icqince of provable corroct progranm traniformations *opcions on index sets

*spcfication of paranicrirable basic iranstomianoris tiling of iteration spacrs

*pumizatin iteger linear algebra

linear and integer linteir prngraniig

*geonietry of numbersprogram *coibinaorial algoribimi on periodic graphs

bas.n programn tranfornat,ons

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7tt

ANALYSIS OF NONLINEAR MICROWAVE CIRCUITS VIA THETIME-DOMAIN VOLTAGE-UPDATE METHOD

H.D. Fol, *JJI. Davis, and T. ItolhtAlthough direct transient time-domain solutions of circuit equations (for example, SPICE-type

programs) are sometimes employed to analy , microwave systems containing nonledt ar devices,teady-state methods (for example, harmonic balance techniques). preferable in cases involvingigh-Q resonant ciruits and other narrow-band structures, for which steady-state methods have a

significant advantage in computation time.

eail 'atdmtet-oin .A haou sead-tt ehius

based hceet

formulated: (1) the waveform-balanc technique, which is related to the piecewiseharmonic-balance method, and (2) the sme-doiain voltage-update technque, which is relaxation-based. In this paper, the laster technique will be examined in more detail and compared to moreconventonal approaches.

In the tme-domain version of the voltage-update, tme samples representing the steady-statevoltage waveform are applied o the nonlinear devce(s). The resuing current samples are appiedto the linear porons of the system, leading toa new set of voltage samples. Relaxaion ()awmetersare then applied to determine the staring samples for the next iteraeion, and the process ts repeateduntil convergence is obtained.

Voltage-update techniques have a marked advantage over most other approaches i simplcity andspeed per iteraton, when appled to problems in whih the frequency is known, such asamplifiers, frequency multilihers, and mixers They can also be applied, with some modifications,to variable-frequency problems such as oscillators.

Strategies for extending the rsnge and speed of convergence for the relaxaton procedure will bediscussed, along with the relationship between frequency-domain and trie-domain relaxationparameters. The results of several repreentatve applicat ons ievolving negatve reststance devicesand SIS junctions will be presented.

The Umversity of Texas. Elecical Enginerng Research Laboratoty, Ausi.n, TX 78712mUniversif of Calfomia. Los Angeles, Department of Elec tcal Engineenng, Los Angeles, CA

90024

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Efficient Anslytical.Num I Modeling Of Uliza.WiletbeM Pulsed

Plane Wave Scatteing Front a Lapg Strip Grat

Lawrence Carin and Leopold B. PelsenWeber Research Institute/ Electrical Engineeridg Department

Polytechnic University

Farmingdale, NY 11735-

Summary. Ultra-wideband (UWE) ptulsed plane wave scattering from a largea but finite stripgrating in free space is analyzed in the frequency domain via decomposition into plane waveSpectra, implemented numerically by the method of momrenta, and then 1sivrted Into thetime domain (TD). To make this procedure practical under UWE conditions, closed formnexpressions are derived for interaction integrals Involving widely separated excpansion andtesting functions. Th'lese closed forms are based on a judicious choice of the bala unictiona,and on asymptotic methods for reducing the integrals. Although large separation distane$are assumed, the expressions have been found to be accrate for separations as small as 0.1wavelength. The TD self terms can also be evaluated effciently. To test the frequencydomain algorithms, comparisons are made with available data In the literature for surfacecurrents and far field scattering from multiple strips. New short pulse MI results are shownas well.

Transient, currents and fields of wire antennas with diodes

N. Scheffer, Telefunken SysteMtechnik VR3 E51, Sedanstr.l0, 7900 Ulm+ 1. Numerical method

The electric field integral equation for a wire antenna can be written in matrixnotation for the.time step t.) as

The unknown quantitiy in (1) is the current distribution I,.of the antenna. The antenna is devided into N segments. .The impedance matrix Z is time independent.

l(( .. ) represents a diagonal matrix containing

the nonlinear loads.The components of E' are the tangential components of the incident electricfield and Z, are the tangential components of the scattered electric field.

Introducing the admittance matrix y Z- and11" 11Z.'. =Y ('(+ K) (2)

equation (1) can be written as

Z repesentr the unity matrix.For the special case, that the antenna is loaded at exactly two segments m and nwith p_ and R, , the currents in the loaded segments are

"=(- + 1. )- . (I. ( , Y. &~r - (4),

"" + - + - (5)

The currents in the unloaded segments 'Ym,' are

(6)

2. The time detendent current distribution

The above method of calculation is used to examine a dipole. The dipole containstwo diodes and is centerfed by a source voltage with Gausssian pulse shape asshown in Fig. I.The two diodes are described by resistors on ao

A- ,lo1.Z,W11 ,<0

Fig. 2 shows the two current pulses arriving at the antenna ends after passingthe diodes almost undistortedly. The arrows mark the positions of the diodes. InF1g. the current pulses have arrived at the reversed biased diodes after theyhave been reflected at the antenna ends. Fig. 4 shows the reflection at the re-versed biased diodes.

3 The electric nearfoeld

It is possible to reduce the fieldline equation drxt.o to a potential.function

" e eHsr y: because of the c~linder symmetry.

An electric fieldliue is then described by U=4ov * , is derived in the ap-pendix. Fig. 5 shows the electric f-eldlines located on concentric spheresaround the generator. The pulse has Just arrived at the antenna end. In Fig. 6the reflection at the antenna end is considered: the fieldlines are now locatedon concentric spheres around the source and the antenna end. Fig. 7 shows the 1reflection at the diode- a new sphere around the diode is visible. In Fig. 8multiple reflections for a late time step can be observed.

N

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* Calculating Yrequency Domain Data by

byhI ~eer h.Dobe.Time Domain Methods

Techniache Ifothschule Darmstardt Tiecorie Elelromagnetisclier relder (FS 18)

Abu~ract

We show the slolati.o Qf far field pattersns old aatering Parameters by mreans of timsedotooln methods. Is order to o',talo a moods exci I Won, the elgensolutlons of the discrete reoveguide

4 tsleaval.. problem In circl -o-tton with an -deqoaute pruelaiog at the boandarieo I# Used.

Par Field TreasfornasIn general the electric field 0. ra1diating structureo con lbe written for lag* dlstssnces as

'here en -e. a1ti the waoe number nd r,e,4 denote spherical coordinates.FOe the 9CA11 e3bOtio of the far held UtrSONfoPA 0(,). one needs to dettermin e the coraple v toe

tangentil electric and aoote amplitues on a closed surface surrounding the radiating structure.

=F(,0) cl" (; ( x A))- 4, x (it x JdA ,(2)

The compurtton oef the comiexe Aeld %mplitudea by a Fast Moorier 'Thonrsforer requires smplingand sarorage of the tmisgoortial electric and morletlc Avid valus It the integration surface ant isonly feasilele for saell meeeoh sizeo Therefore %beoc aniltiudeA ate obtained by usng time harmeonicexation sources land a direct sampling of the barmonle kilds. Anotlitr way is to perform amrooorromatrr. asigle freemoency footer tromoforat of the time domain fields.

Filter D.clgra

'tbe trAnroereat olcitic held in a woAieitudt ia dtjtcrihed for the tome harmonic c-e. by the mode,xpaOsi.1O

9(r.y.0o);=Reei (rye)t(Jse Jo + (.ectl} ()

which casn be formulated for general time dependency sorb double coneolutionac:

Z4(,y.0 - (0,Y) P,f) +Ls40) e.Mt. -Z)) (4)

with P,(1, :) beking the inverse 'Fourier truoufoersa of eap( ~(er.The bowledge of 44,eje) z:I MM + LI,-1) aesoeial for the calculation of the tanxennis boundarY field 1(r, 0,iw) at the

enrerfaco , = 0, The otinoIsuteow.m.se (j) ts kuoon, but b() bag to ho calculated fromr

&(Jw.) can be obtaied by nerdstexpaonoson of tbefield to the plant zm = 4. Therimodes &,(s.pqeje)and the Popgtonatno ar Aluad by a 21) eigenealue sotlver. Tbe case of frequencydlepetedenit L, ib problematic because the exapansion of sampled fieidi,s.'~~ is only poosoibleilo the atoady atate. In the other cant D.(t,) is yielded directly and

rae calculated by single convoluetions. Thrs method to applicable for horogeeoslY filled Wae -oileo nd for miodes, sith mesk freqeqncy dependency. A oimplificatiorn of the algorithui to possible,

0'if the convolution east be approximated in the destred freqenecy range by a lom order digitA filter.

fITIME DOMAIN ANALYSIS OF INHOMOGENEOUSLY LOADED

STRUCTURES USING EIGENFUNCTION EXPANSION

MICHAL MROZOWSKITECHNICAL UNIVESITY Or GDANSK, TEsccoMMUNICATION INSTITUTE

80-952 GDA9SK, POLAND, TEL,: +(048 58) 472 549, FAX: +(048 58) 47 10 71

SUMMARY

At present there are two algorithms, namely the TLM and FD-TD, which are used to solvefMaxwell's equations in time domain. In this contribution we shall present new methods which

may broaden the range of options available in time domain analysis of 2-D and 3-D structures.A wave Is treated as a superpositlon of elgenmodes (elgenfunctions) of the homogeneous Laplaceequation. An inhomogenesty in the structure perturbs the field and causes the coupling of eigen-modes. Eigenmodes are chosen so that they fulfil the Helmholtz equation either on the entirehomogeneous domain or on homogeneous subdomalns. An advantage of this approach is that itallows to obtain time domain algorithms which, in contrast to TLM and FD.TD methods, do notexhibit the numerical dispersion.

i* Outline of time domain eigenfunction expansion algorithms.

Based on the concept briefly described above, a number of algorithms can be proposed. Weshall start with an algorithm called a complete eigenfunction expansion (CEE). Let us consider aset of coupled differential equations reflecting the form of Maxwell equations

d

where £t, L, are linear operators.In the FD-TD algorithm the above equations are discretized both in time and space. In the CEEalgorithm the discretization is only in time. As a result we get

= jns + Atsg"1-1l e'+1/2 =

9-

1 /2 + Atic2f' (2)

The unknown functions f,g are now expanded into series of complete set of orthonormalfunctions. f = °,/, 9 = Zg., (3)Expansion function are defined on the entire domain. A sensible choice for the electromagneticfields is are the eigenfunctions of Laplace equation with suitable boundary conditions. Substituting(3) into (2) we get

ZF4, = o,',, + ALC1 - (4)

'- b' 1g, = Ev"1 + Atic, E , ofTaking the inner product with the expansion functions results in

A" + t

b. = b- l

+ At < £4f",, >

The above equatiqns can be cast into the following matrix form

A .- I + AtA_"" 2 (6)

An1/ in-1 + 4t~an

where p and J are the vectors containing expansion coefficients and A and B8 are dense matriceswith elements

Aqj = < C 9j, fj > B,,j = < L2 j, 9 > (7)

Another version of the eigenfunction expansion algorithm is obtained if the discretization is intime and one spatial coordinate and the expansion Is done with respect to two remaning spatialcoordinate. This algorithm we shall call partial eigenfunction expansion (PEE). In this techniquethe space is sliced into subdomains and the fields are ecpanded on each subdoman (slice) into

series of local expansion functions. In the PEE method one obtains a set of equations similar to(6) except that matrices A and . are sparse.

Compared with the FD-TD method the CEE and PEE algorithms show the time evolution ofthe expansion coefficients rather then field components at nodes. Such an approach allows one toinvestigate propagation of particular modes and their mutual interactions. Moreover, in contrastto FD.TD and TLM techniques, both algozithms proposed in this contribution do not exhibitnumerical dispersion.

Efficient numerical implementation of elgenfunction expansion algorithms: CEE-FFTand PEE-FFT.

One drawbact of the CEE asd PEE algorithms that they may lead to higher numerical costthen FD-TD and TLM. The CEE involves matrix multiplication hence. assuming that expansion isdone using L elgenfuncions, the cost of one lime step is of order O(V). For the PEE this cost is loweras the matrices involved are sparse. In the FD.TD and TLM method with N nodes, the numericalcost is of order O(N). Consequently, eigenfunction expansion techniques may be regarded ax analternative to classical time domain algorithms only when V - N. This condition will be fulfilledin slightly and moderately perturbed homogeneous structures. Nevertheless, much more efficientversion of CEE and PEE may be obtained if the expansion functions are sine and cosines. Equations(6) imply that at each step one evaluates the inner products < £sgn1/1,2 f > and < £x ,gi >.For sine and cosine functions the inner product can be computed in a very efficient way usingthe technique described in (1). In this technique the Inner products are computed In a sequenceof inverse and forward FFTs. The numerical cost of such computations is low and therefore theoverall performance of the CEE-FFa and PEE-FFT algorithms is better than original CEE andPEE methods.

Conclusions. New algorithms of the time domain analysis of inhomogeneously loaded microwavestructures have been described. The methods proposed are based on the expansion of fields intocomplete series of orthogonal elgenfunctions. The resulting equations show the time evolution of theexpansion coefficients and consequently allow one to investigate propagation of separate modes andtheir mutual interactions The algorithms proposed in this contribution do not exhibit numericaldispersion and allow coarser time discretization than the equivalent FD-TD or TLM program.[I) M. Mrozowski, "IEEM FFT- A fast and efficient tool for rigorous computations of propa-

gation constants and field distributions sn dielectric guides with arbitrary crosssection andpermittivity profiles", IEEE Trants. Microwave Theory Tech., vol. MT'T-39, Feb. 1991.

2 4_7 -I

The Hilbert Space Formulation of the TLMMethod

Peter Russer, Michael Krumpholz II Abstract

The Hilbert space representation of the TLM method for time do-main computation of electromagnetic fields and the algebraic compu.tation of the discrete Green's function are Investigated. The completefield state is represented by a Hilbert space vector. The space and time

evolution of the field state vector is governed by operator equations inHilbert space. The discrete Green's functions may be represented by aNeumann series in space- and time-shift operators. The Hibert apacerepresentation allows the description of the geometric structures by pro-jection operators, too. The system of difference equations governing thetime evolution of the electromagnetic field in configuration space is de-rived from the operator equation for the field state vector in the Hilbertspace.

I Introduction

The TLM (transmission line matrix) meth,.d developed and first published in1971 by Johns and Beurle is a discrete time domain method for electron'ag-netic field computation (1,2,31. In this paper, the Hilbert space representation

of the TLM method is presented and applied to the algebraic computation ofdiscrete Green's functions. The Hilbert space representation is a very general

and powerful concept in field theory [41. Whereas in the electromagnetic the-ory Hilbert s ),ce methods are mainly used for solving the field equations asfor example in the moment method (5], in quantum theory, the fundamentaltheoretical concepts have been formulated in Hilbert space (6,71.

The state of a discretized field can be represented by a vector in the Hilbertspace. The !pecification of the mesh node connections and the boundary con-

'Lehrstuhl far Hochfrequenztechnik, Technische Univer;'tit Minchen, Ar-cisstrasse 21, D-8000 Munich 2, Fed. Rep. Germany

!L

Ii- 2ditions is done by operators in the Hilbert space. The Hilbert space representa.

t|on also allows the description of geometric structures by projection operators.

The space and time evolution of the field state vector is governed by operator

equations.

In field theory, the field propagation in spatial domains may be treated usingGreen's functions [8). The concept of Green's functions may also be applied

to discrete time domain field computation (9]. Discrete time domain Green'sfunctions allow to model the relation between the field values on the boundaries

if the knowledge of the field in the spatial domains beyond the boundaries isnot required.

In this paper, the algebr6€ computation of the dis-rete Green's function is

investigated. Our approach is based on a Hilbert space represent&.tion of thespace- and time discretized electromagnetic field. The discrete Green's func-

tions may be represented by a Neumann series in space- and time-shift oper-ators. The system of difference equations governing the time evolution of the

electromagnetic field in configuration space ii derived from the operator equa-tion for the field state vector in the Hilbert vpace. First results are presented

for the two-dimevsional case.

2 The Two-dimensional TLM Method

The electromagnetic field is discretized within space and time. The space

is modelled by a mesh of transmission lines connecting the sample points inspace. The field computation algorithm consists of two steps:

• The propagation of wave pulses from the mesh nodes to the neighbouringnodes.

* The scattering of the wave pulses in the mesh nodes.

In the following, we restrict our considerations to the two-dimensional casewith the transverse electric field. In the shunt TLM model, voltage wave

amplitudes are used instead of total voltage and current. The voltage wave

amplitudes of the incident and the reflected waves are given by ka,, and

kb, The left index k denotes the discrete time coordinate and the right

__ __ _

indices m and n denote the two discrete space coordinates. We consider the

TLM mesh to be composed by elementary TLM shunt node four-ports as

shown in Fig. 1, where each of the four arms is of length At/2. The scattering

in this elementary four-port is connected with the time delay At.

The scattering of the wave pulses is described by

Kb., t;4 5b2 ra2(1

k+ b4 a4

with the scattering matrix S given by

s:= 2-2 (2)

I I 1 -1J

With the scattering, a time delay of At is associated and therefore, the time

index k is incremented by one. The scattered pulses are the incident pulses ofthe neighbouring elementary cell. This is described by

a=,. = b,+i,. (3)ka3,,, = N...I

3 The Discrete Field State Space

In the TLM model, the field state at a given discrete time is described com.pletely by specifying the amplitudes of the tour wave pulses incident to eachmesh node. The space of the voltage wave amplitudes of the incident and thereflected waves ka,,,,, and kbi,,, is the four-dimensional real vector space24. In order to develop our formalism in a more general way we introduce the

I-!

,m7m4

14

4

n

3 mn

Figure 1: A two-dimensional TLM shunt node four-port.

four-dimensional complex vector space C for representing the wave amplitudes

ka,,, and b,,.

In order to describe the whole mesh state, we introduce the Hilbert space 7,which allows to map each mesh node onto an ortonormal set of base vectors of7.. The time states are represented by the Hilbert space W4. With each pairof discrete spatial coordinates (mr,n), a basis vector of 71, is associated andwith each k, a basis vector of 1t is associated. We now introduce the state

space 7 given by the Cartesian product of C4, 7, and 74

?i = C4 7 0 j, (4)

The space W is a Hilbert space, too. The complete time evolution of the fieldstate within the whole three-dimensional space-time may now be representedby a single vector in 7. Using the bra-ket notation introduced by Dirac [6],

the orthonormal basis vectors of *" are given by the bra-vectors [k; m, n). Theket-vector (k; m, nI is the Hermitian conjugate of Jk; m, n). The orthogonality

relations are given by

(ks~ms,nsilk2;ma:,us) m 6ss35j63%. (5)

i 9

The incident and reflected voltage waves are represented by

tx-oo 1001t-0 jo O3

and[a 10 0 +0: k; m, n) (6)k a 4

,

and

1)0Ib) =1[k; m, n) (7)/l ° m ' ~ m ° k b 14 ,

in the Hilbert space 1h. We define the shift operators X, Y and their Hermitianconjugates X

t and Yt by

X 1k;m,n) = 1k;m + 1,n)

Xt lk;m,n) = Ik;m-l,n) (8)YIk;m,n) = fk;m,n+l)Y

tlk;m,n) = Ik;m,n-1)

The operators X and Y shift the field state by one interval Al in the positivem- and n-direction, respectively. Their Hermitian conjugates X

t and Y

t shift

the field state in the opposite direction.

We define the time shift operator T. The time shift operator increments k by

1 i.e. it shifts the field state by At in the positive time direction. If the timeshift operator is applied to a vector 1k; ,n), we obtain

T Ik;m,n) = Ik+ 1; m, n) (9)

We introduce the connection operator r given by

r Ox 010r x t 0 0 0 (10)

10 00 Yj0 0 Y

With the connection operator r, equation (3) yields the operator equation

1b) - ) (11)

Ui __

6

describing the mcnh connections. The operator r is hermitian and unitary:

V r = r-' i

Therefore we obtain from eqs. (11) and (12)

1a) = r ib) (13)

We now express eq. (1) in the Hilbert space notation by

1b) = T S 1a) (14)

This equation describes the simultaneous scattering within all the mesh node

four-ports sctording to Fig. L The scattering by a mesh node causes the unit

time delay At.

Fig. 2 shows an example of a spttial domain within a TLM mesh. This spatiai

domain is specified by a given set of mesh four-ports. A spatial domain D

in our TLM mesh may be specified by projection operators. We define thedomain projection operator PD> which projects a state vector Jo > on the

domain D: P'o (a) = ja) (l&)

This projection operator may be written in dyadic notation as the sum of the

projection operators on the nodes belonging to the domain D:

PD Ik;m, n)(k;m, nl (16).EDnED

In the same way, we define the inner domain projection operator P, and the

boundary projection operator by

P ta) = 1a), (17)

Peta) = 1o)B (18)

with

P) = PIPD (19)

Pa: POPD (20)

PB + PI = PD (21)

i... ..............

Figure 2: A spatial domain within the TLM mesh,

The inner domain projection operator projects the circuit space Nf on the innerports of the domain D. Since the projection operator P and the connectionoperator r are commuting, i.e.

1P,, ri = 0 (22)

we obtain

1b, = r a), (23)

Applying diakoptics to TLM structures requires the computation of the wavepulses scattered at the domain boundaries. The initial conditions or boundary

conditions are given by the wave pulses incident on the boundary ports. Weapply the projection operators PtPs, and PBPD in order to seperate the field

states is) and Ib) into the inner field states Ia), and 1b), and the boundary statesiG)B and lb)B. From eq. (14) we obtain

16)8 = T SBB la)s + T SB, 1a), (24)

[b), = T SIB Ia)a + T Si [a),

SBB = PB S PBSBI = PB s P, (2)

SIB = PI S PBS11 = PI S PI

8

Figure 3: The inner ports of a TLM domain.

Using eqs. (23) and (24), we eliminate the inner domain states [a), and 1b),and obtain

Ib)B = [TS nB + :Sz ( -rIs.)" OsB] 1.), (26)

This is the relation between the incident and scattered boundary state. Itdescribes the evolution of the boundary field state without knowledge of theinner field state. It has to be considered that the operator equation (26)is nonlocal with respect to both space and time. We expand the operator

(1 - rTSII)- into a Neumann series 110,11) and obtain

(1 - rTsI)- = 74 (Sn) (27)t.0

Inserting this into eq. (26) yields the boundary state evolution equation

Ib)B=GIa)B (28)

with the boundary field evolution operator G given by

G = [TSB + SBl ( +1 (rS)) rs.I (29)

The boundary field operator G gives the relation between the boundary statevector [a)B representing the wave pulses incident on the boundary and the

9

boundary state vector 1b)B representing the wave pulses reflected through theboundary. Eq. (28) is the general formulation of the boundary element problem

4 in the Hilbert space. Since the Neumann series is an infinite geometrical seriesin space- and time-shift operators, the boundary field operator is nonlocalwith respect to space and time.

4 The Discrete Two-dimensional Green's

function

As an example, we derive the discrete Green's function for the half-plane. Thediscrete Green's function for the half-plane is given by the projection of theboundary state evolution operator equation (28) onto configuration space fora point-like initial state in}.E The half-plane (Fig. 4) is defined by th, domainprojection operator Po given by

PD = E E 1k;ss,sn) (k;m, nj (30)

As in the shunt TLM-model, voltage wave amplitudes instead of total voltages

are used, a new Green's function for wave amplitudes has to be defined. Fora boundary problem, the Green's function in discrete formulation is given bythe convolution

k bnfk Gn * k on, (31)

where k an is the column vector of the incident impulse functions at the timekAt and at the boundary node number n. k ba is the column vector of the

scattered output wave impulses at the time kAt and at the n"A boundarynode. k G. is the discrete Green's function for an arbitrary boundary withn boundary nodes. It describes the relation between the incident ant the

scattered wave amplitudes in the boundary ports.

For the half-plane, the boundary is given by m i 0 and n =

-oo... ,-1,0,1,...oo. Thereforeeq. (31) yields

ab.= a-a' G..L, a a,' (32)n1=- V'-

10

-4 ---- - -- -- - - -.--

Fiur 4: Th hooeeu w-iesoa af3ae

SI - 4. -

5 ' ... . ... .. .

5 4i

t .. ........ ....

Figure 4: The homogeneous two-dimensiona half-space.

The boundary state evolution equation (28) may be expressed by the discrete

Green's function, eq. (32), via

lb)B = G Ia)B (33)

where the boundary field evolution operator is given by

G 0 0 0 0 E .,G..,Ik;O,n)(k';o,n'I (34)0= 0 0 0 ,=®k-

0000]

In order to calculate the Green's function for the boundary of the half-plane,we start from an impulsive excitation at n' = 0, P? = 0 given by

la)Bk.O = 111 0;0,0) (35)

L0j4

Mapping eqs. (28) with (29) ard (35), (33) with (34) and (35) to configurationspace by multiplying both equations from the left side with (k; 0, ni, we obtain

-

I1

a system of partial difference equations which can be solved by transforming

it to frequency- and momentum-space.

We obtain an algebraic expression for the Green's function

jG. = 5a,.+s - , + a+s. -

2 1 1 .+ r 8 k-i-jIn+2- - I k-l-l-

.o4

+ g

'.+ gk-2-j-3 (36)

with n =0, 1,2,.,,. oo; k = 2,3,4,.,..oo and

AG_,.= G, (37)

for n < 0.

The function J, is given by

a 11k-i ,+ , -2s)

kI. = 2E E -.s 3+,( 2

.-- , i 2 '() (I Isx2r) (k -2+r )(21 -4., + 2r (8

X(r ( r -l 2s+r) (38)

In Fig. 4, k G, is depicted for n =-9..., -1,0,,... 9; k = 1,2,... 10.

12

kG.

Figure 5: Values of the Green's function

References

(1) P.B. Johns, R.L. Beurle, "Numerical Solution of 2-Dimensional ScatteringProblems using a Transmission-Line Matrix", Proc. lEE, vol.118, no. 9,pp 1203.1208, Sept 1971.

[2) W.J.R. Hoefer, "The Transmission Line Matrix Method-Theory andApplications",, IEEE Trans. Microwave Theory Tech., vol. MTT-33,

no.1O,pp.882-893, Oct 1985.

(3) W.J.R. Hoefer, 'The Transmission Line Matrix (TLM) Method", Chapter8 in "Numerical Techniques for Microwave and Millimeter Wave Pa~ssiveStructures', edited by T. Itoh, New York, 1989, J. Wiley, New York 1989,pp. 496.591.

[4) K.E. Gustafson, 'Partial Differential Equations and Hilbert Space Meth-

ods", J. Wiley, New York 1987.

- -e , -

13

[5] ILF. Harrington, "Field Computation by Moment Methods", Krieger,

Malabar, Florida, 1982.

[6] P.A.M. Dirac, "Quantum Mechanics", fourth edition, Oxford UniversityPress, Oxford

[7) J.v. Neumann, "Mathematische Grundlagen der Quantenmechanik",

Springer, Belin, 1932.

(8] R.E. Collin, "Field Theory of Guided Waves", second edition, IEEE Press,New York 1991, pp. 55-172.

[9] W.J.R. Hoefer, "The Discrete Time Domain Green's Function or John'sMatrix - a new powerful Concept in Transmission Line Modelling", Inter-national Journal of Numerical Modelling : Electronic Networks, Devices

and Fields, Vol. 2, 215-225, 1989.

[10] J. Weidmann, "Lineare Operatoren in Hilbertriumen", B.G. Teubner,Stuttgart, 1976, pp. 96-105.

[11] H. Heuser, "Funktionalanalysis', B.G. Teubner, Stuttgart, 1986, pp. 106-

113.

$ [

iL_ .C .1

l- Solving eigenvalue and steady-state problemsusing time-domain models

Gunnar NitscheLehrstuhl f'ir Nachrichtentechnik

Ruhr-Universitgt lochumD-4630 Bochum 1

Germany

Time-domain modelling of partial differential equations has becomepopular in the last years for several reasons. The detailed time-domainbehaviour, however, is often not of primary concern, but one is more Inter-ested in the eigenvalues and eigenfunctions of a system or its steady-stateresponse to a sinusoidal excitation. These kinds of problems are usuallyapproached by methods based on the discrete Fourier transform.

A new alternative approach to efficiently compute the low frequencyeigeumodes of a time-domain model approximating a physical system willbe proposed. The method is based on principles known from digital signalprocessing, in particular from parametric spectrum estimation, so it is notburprising that the achievable accuracy is much higher than the accuracyof the non-parametric Fourier transform approach. The algorithm works inthe geueral lossy case even for a very large number of unknowns and caneasily be extcnded to calculate steady-state solutions for several differentfrequencies simultaneously.

I

-A fl~ ~ ~ I

-~ Restaurant GuideNear the TU there are some restaurants and coffee-houses were you can go to. Someof these are listed below. The number is related to the numbers on the map. So youcan easily find your location.

1. CANTON chzeese restaurant, Theresienstr. 49

2. BEI MARIO pizzeria, italian restaurant, Luisenstr.

3. BELLA ITALIA pizzeria, italian restaurant, Tsirkenstr.

4. HIMN Vietnamese food, Schellingstr. 91

5. CAFE ALTSCH WADING coffee house, bistro, Schellingstr.

6. WEINSCHATULLE restaurant, Theresienstr.

7. MC DONALDS fast food, Augustenstr.

pxep4Z ''4O76'C5 "~ WAe

0 A

-, v4'. Ilk

On thursday evening there is a social event consisting of a Song Recital with Pianoand dinner for all workshop participants at the restaurant "Seehaus", EnglischerGarten (see the map below). We recommend you to reach the Seehaus by car or taxicoming from the Mittlerer Ring.

Me ~~~

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.1 (318 ' * ~ *" -9,e/.

SCDietz . 5e~~' *g~.-.'

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