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A STEPPING ALGORITHM FOR TRANSIENT ANALYSIS OF DISTRIBUTED NETWORKS Linchao Lu, Michel S . Nakhla and Qi-jun Zhang Department of Electronics, Carleton University, Ottawa, Canada K1S 5B6 Abstract- A reinitialization technique based on the numerical inversion of Laplace transform (NILT) is described to improve the simulation accuracy of distributed networks and systems with long transients. frequency dependent or distributed elements that are best described in the frequency domain. Without loss of generality the modified nodal admittance matrix [9] equation of the network x can be written as dv, N. (f) + C , - + dl G,v,(f) I -Introduction Evaluation of the transient responses of distributed networks is critical in the design of high-speed VLSI interconnections [l-81. With subnanosecond rise times, the electrical length of interconnects can become a significant fraction of a wavelength. Consequently the conventional lumpedimpedance interconnect model is not adequate in this case. Instead a distributed transmission line model should be used. An efficient method based on numerical inversion of Laplace transform (NILT) has been developed for time-domain analysis of lossy coupled transmission line networks [2]. A recent expansion of the method is the high degree NKT technique which provides improved accuracy using high-order derivative information [3]. Such NILT based simulation is more reliable and efficient than previously published techniques based on fast Fourier transform [6] and is particularly useful in the analysis of mismatched coupled systems where reflections result in very long response times [2,6]. v,(t) E iXNs is the vector of node voltage waveforms appended by independent voltage source current and inductor current waveforms, Dk j E d i , j E (0,1 ) {l,z,...,Nn} 9 { 1,2, ...,2 N k } with a maximum of one nonzero in = [d,,,]3 9 each row or column is a selector matrix that maps i k ( t ) E R"', the vector of currents entering the linear subnetwork k, into the node space YINs of the network x, N , is the number of linear subnetworks and e, ( 1 ) E sN" is the vector of source waveforms. Assuming zero initial conditions, the frequency domain representation can be obtained by taking the Laplace transform of (1): N. (sc, + G,) v, (s) + D k I k (s) = E , ($1 Assume the frequency domain equations of the linear subnetwork k to be in the form Akvk (3) = where Vk (s) and I,, (s) represent the frequency domain terminal voltages and currents of the subnetwork k and A , represents the modified nodal admittance matrix of the subnetwork. Combining (2) and (3) produces Y,V, = E, (a) Formulation of the Nework Eauations Consider a linear network II which contains linear lumped components and arbitrary subnetworks. The linear lumped components can be described by equations in either the time or frequency domain. The arbitrary linear subnetworks may contain (2) k=l - will present the new method and provide accuracy and computation cost comparisons. (1) c,E i ~ ' Gn E~ Y I ~ ~ x' " ~are ~constant ~ matrices ~ with entries determined by the lumped linear components, 11 Outline of the Step!, ine Algorithm In this section we will first review the numerical inversion of Laplace transform method reported in [2]. Next we =0 1 where In this paper an altemative approach for enhancing the accuracy of numerical inversion of Laplace transform is presented. The new method uses a stepping algorithm to reinitialize the reference time point for NILT within a distributed network environment. The approach bridges the gap between NILT and the integration approach for distributed networks. Compared to the inversion techniques reported in [2,3], the new method is more accurate for calculating long transients. D k i k ( f )- e , ( t ) k= where 1957 Nr Y, = sC,+G,+ C DkAkD1k k=l (4) where v, ( t o ) contains voltages across the capacitors and currents through the inductors at time to. In the special case where the subnetwork k consists of a multiconductor transmission line system, A , can be described in terms of the line parameters as follows. The transmission line is assumed to be uniform along its length with an arbitrary cross section. The cross section of transmission line k with N, signal conductors, can be represented by the following N, x N, matrices of line parameters: the inductance per unit length L, the resistance per unit length R, the capacitance per unit length C, and the conductance per unit length G. The approach as proposed in [10,11] can not handle distributed elements. In the next section, an alternative resetting technique is presented which is applicable for lumped networks and multiconductor transmission lines. (c) li.wdm Let $ be an eigenvalue of the matrix ZLYL with an associated eigenvector Xi,where Z, = R + s L and Y , = G + s C , i.e., (r;2v-zLYL)xi = 0 The New Method In the presence of distributed transmission lines, the NILT can be reinitialized using the equation (6) The admittance matrix stamp for the distributed transmission line is where D is the length of the transmission line. v , ( x , t , ) and i, ( x , to) contain the voltages and currents at location x of the transmission line and at time io, (7) where E, and E, are diagonal matrices calculated from the lcngth of the transmission lime D and the eigenvalues of (6). Sy is a matrix containing all eigenvectors Xi, i=1,2, ..., N , . r is a diagonalmatrix with r. . = yi, and Si = ZZSJ. I, 1 ( b ) N w i c a l Inversion of [email protected] TraaEform Using Pa& Auuroximation Details of the method is given in [10,11] and its application to the time-domain analysis of high-speed interconnects is described in [2]. For correct understanding of this paper a brief summary is presented here. The transient voltages va ( 1 ) are obtained from V , (s) through numerical inversion of Laplace transform, i.e., v, ( 1 ) = 5 2t i , l (8) R [KiV, (+] Notice that the initial condition of the distributed transmission line is dependent upon the voltages and currents at all locations along the line. lhI!uua The s-domain voltages and currents of the transmission line V , ( x , s) and I , ( x , s) is computed at all locations x of the line through where Zi and K,, i=1,2 ,..., M ' , are predetermined poles and residues of a Pade rational function approximating e' [10,11]. The response computed from (8) is more accurate for t close to its initial value, i.e., t=O, and less accurate for t far away from t=O. This approach is called NILTO, i.e, NILT with degree 0. In [3], high degree NILT techniques were developed. For example, the NILT2 technique provides enhanced accuracy by using up to second-order derivative information of the circuit response. The Resetting Techniaue for LumDed Networks For networks consisting of lumped elements, a resetting technique described in [ 10,111can be used to reinitialize the NILT at t = io. The result at to is considered as the initial point for the new time step. The response at t > to can be calculated from the equation (G, + sC,) V , = J, ($1 + C,V, ('0) (9) where Q = 1 We call the new technique NILTR. ProrJerties of the New Method The new mcthod is equivalent to high-order integration of distributed networks. It can be proven that the method is absolutely stable. 1958 l3xmuld [41 networks,” Proceedings of IEEE Int. Symp. Circuits Syst.,pp. 888-891, June 1991. R. A. Sainati and T. J. Moravec, “Estimating high speed circuit interconnect performance,” IEEE Trans. Circuits Syst., vol. 36, pp. 533-541, April 1989. The circuit in Fig.1 has 7 lossless transmission lines. The results from NILTR is compared with that of HSPICE and NILTO, as shown in Fig. 2. Both NILTO and NILTR use M ’ = 2 . In Fig. 2, two step reinitializations are taken at tol = 0.84ns and fo2 = 1.3ns, respectively. A good agreement between the new method and HSPICE is obtained. [51 H. Hasegawa and S. Seki, “Analysis of interconnection delay on very high-speed LSI/VLSI chips using a microstrip line model,” IEEE Trans. on Electron Devices, vol. ED-31, pp. 1954-1960. Dec. 1984. I2xmQ-u [61 A. R. Djordjevic, T. K. Sarkar and R. F. Harrington, “Tmedomain response of multiconductor transmission lines,*’Proceedings of IEEE, Vol. 75, pp. 743-764, June 1987. [7] E Y. Chang, “The generalized method of characteristics for waveform relaxation analysis of lossy coupled transmission lies,” IEEE Trans. on Microwme Theory and Tech., pp. 2028-2038, Dec. 1989. [81 S. Gao, A. Yang and S. Kang, “Modelling and simulation of interconnection delay and crosstalk in high-speed integrated circuits,”IEEE Trans. Circuits Syst., vol. 37, pp. 1-9, Jan. 1990. PI C. W. Ho, A. E. Ruehli and P. A. Brennan, “The modified nodal approach to network analysis,” IEEE Trans. Circuits Syst., vol. CAS-22, pp. 504-509, June 1975. Consider the circuit shown in Fig. 3 which contains lossy coupled transmission lines. Since HSPICE can not simulate such a distributed circuit, we compare results of NILTR with that of NET0 and NILT2. Fig. 4 and Fig. 5 show the solutions obtained from the three methods. All three methods use M ’ = 3 . In Fig. 4, two step reinitializations are taken at tol = 1.1ns and to2 = 3.3ns, respectively. In Fig. 5, three step reinitializations are taken at tol = l.lns, to2 = 3.3ns and tO3 = 511s.From Fig. 5 we can see that using the new proposed approach we obtain comparable accuracy to NKT2. However, the new approach does not require high-order derivative information as is the case with NILT2. References [l] M. S.Nakhla, “Analysis of pulse propagation on high speed VLSI chips,” IEEE Journal of Solid-state Circuits, vol. 25, [2] [31 [lo] K.Singhal and J. Vlach, “Computation of timedomain response by numerical inversion of Laplace transform,” Journal of the Franklin Institute, vol. 299, pp. 109-126, Feb. 1975. pp. 490-494, April 1990. R. Griffith and M. Nakhla, “Time-domain analysis of lossy coupled transmission lines.” IEEE Trans. on M ~ C “ + w v [11] K. Singhal, J.Vlach and M. Nakhla, “Absolutely stable, Theory and Tech., vol. M’IT-38, pp. 1480-1487, October high order method for time domain solution of networks,” Archiv fur Electronik und Uebertragungstechnik, vol. 30, 1990. pp. 157- 166, 1976. M. Nakhla and L. Lu, “Time-domain analysis of distributed & Fig.1 Circuit schematic for example 1 1959 0.7 06 05 $ 4 , 04 -Pmpcwdm h o d 03 P 0.2 01 0 -0I -02 0 2 4 6 8 10 12 14 16 18 20 TMW TiUE(P.5.) Fig.4 Comparison of HSPICE, NILTO and two step reinitializations for the circuit of Fig. 3 Fig. 2 Comparison of HSPICE, NILTO and two step reinitializations for the circuit of Fig. 1 TransmissionLine #3 .0.21 0 " 2 4 ' 6 " 8 IO " 12 14 " 16 18 Fig.5 Comparison of HSPICE, NILTO and three step reinitializations for the circuit of Fig. 3 1960 I 20