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11. S. Zhou, W. Wu, S. Campbell, P. Yeh, X. Yand, and H. Ling,‘‘Optical Implementation of Fuzzy-Set Reasoning,’’ Appl. Opt.,Vol. 33, 1994, pp. 5335]5347.

12. H. Itoh, S. Mukari, and H. Yajima, ‘‘Optoelectronic Fuzzy Infer-ence System Based on Beam-Scanning Architecture,’’ Appl. Opt.,Vol. 33, 1994, pp. 1485]1490.

13. T. Konishi, J. Tanida, and Y. Ichioka, ‘‘Visual-Area CodingŽ .Technique VACT : Optical Parallel Implementation of Fuzzy

Logic and Its Visualization with the Digital-Halftoning Process,’’Appl. Opt., Vol. 34, 1995, pp. 3097]3102.

14. K. Hirota and K. Ozawa, ‘‘The Concept of Fuzzy Flip-Flop,’’IEEE Trans. Syst., Man, Cybern., Vol. 19, 1989, pp. 980]997.

15. A. A. S. Awwal and M. A. Karim, ‘‘Polarization-Encoded OpticalShadow-Casting: Design of a J-K Flip-Flop,’’ Appl. Opt., Vol. 27,1988, pp. 3719]3722.

16. R. A. Rizvi and M. S. Zubairy, ‘‘Optical Flip-Flops in a Polariza-tion-Encoded Optical Shadow-Casting Scheme,’’ Appl. Opt., Vol.33, 1994, pp. 3775]3781.

17. C.-H. Lee, T.-H. Yong, and S.-Y. Shin, ‘‘Optical Flip-Flop UsingLight-Emitting Diodes and Photodetectors,’’ Appl. Opt., Vol. 25,1986, pp. 2244]2245.

18. Y. H. Yu, J. M. Park, and T. L. Lim, ‘‘Pitchfork Bifurcation andFlip-Flop Operation with a Coupled Nonlinear System,’’ Appl.Opt., Vol. 33, 1994, pp. 2128]2133.

19. K. M. Iftekharuddin and M. A. Karim, ‘‘Butterfly Interconnec-tion: Design of Multiplier, Flip-Flop, and Shift-Register,’’ Appl.Opt., Vol. 33, 1994, pp. 1457]1462.

20. E. L. Johnson and M. A. Karim, Digital Design: A PragmaticApproach, PWS Pub., Boston, MA, 1987.

21. W. Pedryez, Fuzzy Control and Fuzzy Systems, John Wiley & Sons,New York, 1989.

22. A. Bergeron, H. H. Arsenault, E. Eustache, and D. Gingras,‘‘Optoelectronic Thresholding Module for Winner-Take-All Op-erations in Optical Neural Networks,’’ Appl. Opt., Vol. 33, 1994,pp. 1463]1468.

Q 1997 John Wiley & Sons, Inc.CCC 0895-2477r97

AN EFFICIENT NUMERICAL SCHEMEFOR FAST PARAMETERCALCULATION OF VLSIINTERCONNECTIONYang Zhao1 and Yun-Yi Wang11State Key Laboratory of Millimeter WavesDepartment of Radio EngineeringSoutheast UniversityNanjing 210018, P.R. China

Recei ed 1 April 1997

( )ABSTRACT: This letter proposes the substructure-front S-F techniquein conjunction with the geometry-independent measured equation of

( )in¨ariance GIMEI for fast parameter extraction of a multilayer and( )multiconductor interconnect based on the finite-element method FEM

for VLSI circuit analysis. Results show that they are in good agreementwith the published results, and great computer resource can be sa¨ed¨ia the efficient numerical scheme. Q 1997 John Wiley & Sons, Inc.Microwave Opt Technol Lett 16: 48]50, 1997

Key words: finite-element method; substructure-front technique; VLSIinterconnect; multilayer and multiconductor interconnect; measuredequation of in¨ariance

1. INTRODUCTION

Recently, the fast parameter extraction of an interconnect isgaining increasing importance in high-speed VLSI chips dueto the rapid increase of operating frequencies and decreasein chip sizes. However, there are some difficulties with inter-connect analysis such as the open-boundary field computa-tion and the large matrix equation to be solved. To circum-vent these difficulties, we propose an S-F technique in anumerical scheme for FEM solution incorporating GIMEIfor mesh truncation. In this paper, the following three aspectshave been emphasized which are different from those ofw x1]3 :

1. The S-F technique is presented to solve large FE equa-tions encountered in interconnect analysis for the firsttime.

2. The meshes are truncated with GIMEI rather thanw xconventional MEI 1]3 .

3. The parameter extraction of an interconnect is empha-w x w xsized instead of scattering 1, 3 or electrostatic 2

problems via FEM.

Since the frequency range of interest for high-speed VLSIis often below 20 GHz, herein the quasi-TEM assumption isadopted, and therefore, the governing equation is

2 Ž .= U s 0. 1

2. MESH TRUNCATION SCHEME

The geometry-independent measured equation of invariancew x4 is a new extended concept of MEI to overcome thetedious deduction of the closed-form Green’s function andthe time-consuming calculation of MEI coefficients because,for complex structures, the Green’s functions are generallyderived in the spectral domain and then transformed to thespace domain by inverse Fourier transformation which areinfinite integrals; further, many Sommerfeld integrals will beencountered in MEI coefficients calculation. By using GIMEI,an artificial measuring loop is introduced to isolate the MEIelements from the region containing conductors, and thenthe MEI coefficients are determined by the metrons on themeasuring loop rather than on the conductors. Additionally,the outer space outside the loop is assumed to be free space;thus, the free-space Green’s function can still be utilized.

The implementation of GIMEI in FEM has been com-pleted. Furthermore, we suggest that the high-order isopara-metric finite elements be adopted in GIMEI because numeri-cal experiences show that they are more accurate, and canenable us to truncate meshes more closely to the body than

w xbilinear elements. Since it has been well demonstrated 5 ,the description of the method is not provided here.

3. APPLICATION OF S-F TECHNIQUE

In recent years, the most widely employed method for solvinglarge matrix equations has been based on the iterativemethod, but sometimes it has a convergence problem, and itsiteration is time consuming. For this reason, the S-F tech-nique is employed incorporating FEM. Although the S-F

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 16, No. 1, September 199748

Figure 1 Numerical example of interconnect analysis

Figure 2 Illustration of substructure-front technique in intercon-nect analysis

technique was originally presented for mechanics calculation,it can still be utilized for a VLSI interconnect calculationvery efficiently, as shown in this letter.

As seen in Figure 1, the corresponding illustration of theS-F technique is given in Figure 2. In this case, the nodevariables enclosing two conductors are sufficient for capaci-tance extraction; thus, we call S , S substructures, i.e., theI IIinteresting region to be solved. The S-F process has foursteps. First, the whole region is divided into an MEI regionŽ . Ž .U and an FE region including two substructures U , U ,G 1 2

Ž . Ž .one terminal interface U , and left variables U . Second,C V

the elements in the FE region are dealt with using the frontmethod, where only those variables of U , U , and U are1 2 C

Ž .considered to be kept in the ‘‘front’’ a triangular matrixduring the time when all element matrices are loaded intothe ‘‘front.’’ Third, the present ‘‘front’’ is transformed to afull matrix to prepare for combining with MEI conditions.Finally, the MEI elements are dealt with, namely, the MEIcoefficients corresponding to the boundary node and the FEequations corresponding to the adjacent interior node areadded on their own locations, respectively.

In Figure 2, the FEM equations in the interior region aresymmetric; thus, we have

w xUK K K K w x0VV , V V , 1 V , 2 V , C

w x w xU FK K K 1 11, 1 1, 2 1, C Ž .s . 2w x w xU FK K 2 22, 2 2, C � 0� 0 � 0w x w xK U 0C , C C

Since the substructure S , S and the terminal interface GI II Care isolated, therefore

w x w x w x w x Ž .K s K s K s 0 . 31, 2 1, C 2, C

By using the front method, the volume vector of the uninter-w xested region variable U can be eliminated during theV

Ž .process. Then Eq. 2 becomes

X X X w x XUK K K F11, 1 1, 2 1, C 1X X Xw xU Ž .K K Fs 422, 2 2, C 2

X X� 0 � 0� 0w xK FUC , C CC

w xwhere, if necessary, U can be determined byV

y1 y1w x Žw x w x w x w x w x w xU s y K K U q K K UV V , V V , 1 1 V , V V , 2 2

y1w x w x w x. Ž .q K K U . 5V , V V , C C

Ž .Now, we transform 4 to a full matrix equation and loadMEI element coefficients into it. Thus, we obtain

X X X w x XUK K K 0 F11, 1 1, 2 1, C 1X X X Xw xUK K K 0 F22, 1 2, 2 2, C 2 Ž .s 6X X X X Xw xUK K K K FCC , 1 C , 2 C , C C , oG C� 0 � 0� 0w xU0 0 MEI coeff 0G

where

w X x w X x w X x w X x w X x w X xK s K K s K K s K2, 1 1, 2 C , 1 1, C C , 2 2, C

Ž .7

and the number of variables to be solved finally is

Ž .N s N q N q N q N . 8S-F 1 2 C G

In fact, the number of N is small enough when comparedS-FŽ .with the total number of variables. Now, by solving Eq. 6 ,

w x w xwe obtain the value of U , U promptly, and the capaci-1 2tance matrix can then be derived easily.

It is worth mentioning that the S-F technique has some.advantages, as follows. 1 Since it is unnecessary to know the

uninterested variables, their eliminated coefficients are notw x Ž .stored in the memory; sequentially, the global U in Eq. 2V

.certainly does not appear in the computation. 2 The ‘‘front’’width may be limited to a small value by controlling theprocess of numbering the elements, which is easily realized in

. Ž .multilayer structures. 3 It can be seen from Eq. 6 that thecoupling relationship between the FEM equations and theMEI conditions is simplified because, finally, for MEI ele-ments, if the node is an interior node adjacent to the bound-ary node, the standard FEM equation is still filled in, whilefor the MEI boundary node, the MEI conditions should befilled in on corresponding locations.

4. NUMERICAL RESULTS

For verification and comparison purposes, a 2-D interconnectŽ .calculation of a multiconductor transmission line see Fig. 1

has been done. In Table 1, it is seen that our results are ingood agreement with published results, from which ourmethod is well proved. In Table 2, we find that both memoryand CPU time can be saved dramatically by using the S-Ftechnique, and the greater the number of variables, the morecomputer resources are saved.

TABLE 1 Comparison of Capacitance in Our Method and( )Other Methods Unit: pF ///// m

w x w x w xOur Method Hong 4 Cao 6 Weeks 7

C11 93.69 93.80 91.65 92.24C12 y8.27 y8.32 y8.22 y8.50C21 y8.27 y8.32 y8.22 y8.50C22 93.69 93.80 91.65 92.24

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 16, No. 1, September 1997 49

TABLE 2 Comparison of Computer Resources on PC 486

a1 a2

Memory CPU Memory CPUŽ . Ž . Ž . Ž .Example No. kbits Time s kbits Time s

Gauss method 250 98.76 s 527 456.21S-F technique 23 12.36 s 40 26.42

Note: a1 No. of truncated meshes s 390, no. of nodes s 496.a2 No. of truncated meshes s 750, no. of nodes s 806.

5. CONCLUSION

In this letter, we have presented a numerical scheme for fastparameter extraction of VLSI interconnection. Analysis andresults show that the substructure-front technique combinedwith the geometry-independent measured equation of invari-ance via FEM is suitable for interconnect calculation. It isaccurate, simple, and versatile with high efficiency.

ACKNOWLEDGMENT

This work was supported by the Chinese Postdoctoral Foun-dation.

REFERENCES

1. A. Boag, R. Mittra, and Y. Leviatan, ‘‘A Numerical BoundaryCondition for Finite-Difference and Finite-Element Analysis forOpen Structure,’’ Microwa e Opt. Technol. Lett., Vol. 7, No. 9,1994, pp. 395]398.

2. G. K. Gothard, S. M. Rao, T. K. Sarkar, and M. S. Palma, ‘‘FiniteElement Solution of Open Region Electrostatic Problems Incor-porating the Measured Equation of Invariance,’’ IEEE Microwa eGuided Wa¨e Lett., Vol. 5, No. 8, 1995, pp. 252]254.

3. D. B. Wright and A. C. Angellaris, ‘‘Finite Element Grid Trunca-tion Scheme Based on the Measured Equation of Invariance,’’Radio Sci., Vol. 29, No. 4, 1994, pp. 907]921.

4. W. Hong, W. Sun, and D. Dai, ‘‘Fast Parameter Extraction ofMultilayer and Multiconductor Interconnect Using Geometry In-dependent Measured Equation of Invariance,’’ IEEE MCMC’96.

5. Y. Zhao and Y. Y. Wang, ‘‘A New Finite Element Solution forParameter Extraction of Multilayer and Multiconductor Intercon-nects,’’ IEEE Microwa e Guided Wa¨e Lett., to be published.

6. W. Cao, R. Harrington, J. Mantz, and T. Sarkar, ‘‘MulticonductorTransmission Line in Multilayered Dielectric Media,’’ IEEE Trans.Microwa e Theory Tech., Vol. MTT-32, 1984, pp. 439]450.

7. W. T. Weeks, ‘‘Calculation of Coefficients of Capacitance ofMulticonductor Transmission Line in the Presence of DielectricInterface,’’ IEEE Trans. Microwa e Theory Tech., Vol. MTT-18,1970, pp. 35]43.

Q 1997 John Wiley & Sons, Inc.CCC 0895-2477r97

MODELING OF PLASMA EFFECT ONTHE DIFFRACTION RADIATION OFRELATIVISTIC BEAM MOVING OVERA GRATING OF FINITE EXTENTGennadiy I. Zaginaylov,1 Yuri V. Gandel,1 and Petr V. Turbin21Kharkov State UniversityKharkov 310077, Ukraine2Scientific Center of Physical TechnologiesKharkov 310145, Ukraine

Recei ed 31 March 1997

ABSTRACT: The rigorous numerical treatment of the relati istic beamradiation o¨er a finite-length grating both in the presence and without a

background plasma is presented. The most attention is paid to theexcitation near the frequency of the p-type resonance used in backward

( ) ( )wa¨e oscillators BWO and multiwa e Cherenko¨ generators MWCG .Based on the obtained results, the simultaneous influence of plasmafilling and mode con¨ersion at the grating end on radiation efficiency isdiscussed. A new, more accurate treatment of the generation frequency-locking phenomenon and some other effects obser ed in BWO earlier issuggested. Q 1997 John Wiley & Sons, Inc. Microwave Opt TechnolLett 16: 50]54, 1997.

Key words: diffraction and Cherenko¨’s radiation; relati istic electronbeam; backward wa¨e oscillator; singular integral equations; numericalmethods

1. INTRODUCTION

Recent intensive experimental and theoretical studies ofw xhigh-power microwave sources such as relativistic BWO 1]3 ,

multiwave Cherenkov, and relativistic diffraction generatorsw x4, 5 have displayed a significant role in the mode conversion

Ž .effects at the ends of a slow-wave structure SWS operation.Due to the notable dependence of the output power and thegeneration frequency on the phase of reflections from bothends of the interaction chamber, one can get a wider band-

Ž .width hundreds of megahertz and a greater variation in theŽ . w xoperation efficiency by 80% 2, 3 . Great enhancement of

efficiency also can be achieved due to the presence of aw xbackground low-density plasma in the interaction chamber 6

that has no clear theoretical treatment yet. A strong depen-dence of the output power on the end reflections is observedfor MWCG as well. Note that the latter may have severalsections of SWS. However, no rigorous theory exists now toaccount for the mode conversion effects at the ends of SWS.A time-dependent nonlinear theory for operating near and

w xfar from the cutoff was developed in 1 . But it accounts forthe above-mentioned effects nonself-consistently, which ap-pears to be the origin of various disagreements betweenpredicted and measured data. One variant of a ‘‘full-wave’’approach free of such deficiencies and enabling one to cor-rectly take this effect into consideration is suggested below.As will be shown, some of the experimental facts difficult totreat by means of conventional approaches can be correctlypredicted, even within the simplest models of a beam andSWS.

2. FORMULATION OF THE PROBLEM

For simplicity, we shall consider a planar finite-length gratingŽ .with a rectangular element of periodicity see Fig. 1 im-

mersed in a homogeneous ‘‘cold’’ collisionless magnetizedplasma with the tensor of dielectric permittivity given by

1 0 00 1 0« sˆ ž /Ž .0 0 « v

Ž . 2 2 2 2where « v s 1 y V rv , V s 4p e nrm is the plasmae efrequency, and e, m and n are the charge, mass, and density

Ž .of the electrons ions of plasma are assumed to be immobile .For the beam, we will use the simplest model: if it is assumed

ªŽ .as thin in the transverse direction modulated current I x, zŽ Ž .. Ž . Ž . Ž ŽŽ .s 0, 0, I x, z where I x, z s I g x exp i krb z yz z 0..iv t , I and v are the amplitude and frequency of the0

modulation, k s vrc, b s Vrc, V is the beam velocity, c isŽ .w Ž .xthe speed of light in vacuum, and drdx ln g x ; 1rD 4

kr2p .

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 16, No. 1, September 199750