1258 on 11, no. 10, october 1992 transient analysis of ... · the analysis of a coupled...

1258 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 11, NO. 10, OCTOBER 1992

Transient Analysis of Dispersive VLSI Interconnects Terminated in Nonlinear Loads

Rui Wang, Member, IEEE, and Omar Wing, Fellow, IEEE

Abstract-A new algorithm to compute the transient response of a coupled, dispersive multiconductor system terminated in nonlinear loads is presented. The characterization of the multiconductor system is based on full-wave analysis using the spectral-domain approach, rather than the usual TEM approximation. To compute the transient response of such a system, a bilevel waveform relaxation method is used. Waveform relaxation is applied to compute a time-domain solution at the input and output interfaces. The waveforms are then transformed into the frequency domain to compute the updates in- duced by the multiconductor system and are transformed back to the time domain for the next set of relaxation at the interfaces, Techniques to improve convergence are presented and conditions for convergence are discussed. Examples of ECL, CMOS, and GaAs circuits connected by coupled lines are given for illustration.

I. INTRODUCTION HE IMPORTANCE of VLSI interconnects becomes T more and more evident as VLSI circuits are required

to operate at ever-increasing speeds to meet the demands of digital communication, high-speed computation, signal processing, and related applications. At very high switching speeds and layout densities, circuit performance is limited not so much by logic design, but by the interconnections. Accurate circuit simulation programs that are capable of predicting the behavior of interconnection networks terminated in external circuitries are of vital importance in high speed VLSI circuit design. Considering the complexity of modern VLSI circuits, it is essential that the simulation program combine accuracy and robustness with efficiency.

In 1967, the method of characteristics [ l ] for the solution of wave equations was first used by Branin in the transient analysis of a single ideal transmission line [2]. Later, Chang extended the method to lossless coupled transmission line systems [3]. A considerable amount of

Manuscript received June 5, 1990; revised October 4, 1991. This work was supported in part by the National Science Foundation under Grant NSF ECD 88-1 11 11 and by the New York State Center for Advanced Technol- ogy-Computer and Information Systems. This paper was recommended by Associate Editor A. Ruehli.

R. Wang was with the Department of Electrical Engineering and Center for Telecommunications Research, Columbia University, New York, NY 10027. She is now with Cadence Design Systems, Santa Clara, CA 95054.

0. Wing is with the Department of Electrical Engineering and Center for Telecommunications Research, Columbia University, New York, NY 10027.

IEEE Log Number 9200817.

studies have been reported since then [4]-[13]. Djorjevic, Sarkar, and Harrington introduced a general algorithm for the analysis of a coupled transmission line system [5]. Transient analysis of the system terminated in nonlinear loads is carried out by convolving the impulse response with the voltages at the line terminals. In order to shorten the duration in which convolution is evaluated, Djorjevic et al. used a quasi-matched passive network to terminate the lines. Schutt-Aine and Mittra, on the other hand, at- tack the same problem by inserting a segment of ideal transmission lines between the terminations and the multiconductor system [8].

Chang was the first to apply the waveform relaxation technique [14]-[18] combined with the method of characteristics to the transient analysis of multiconductor transmission line systems [ 191. Pad6 synthesis is then used to construct a lumped-element ladder network to approx- imate the characteristic impedance of each decoupled line. Waveform relaxation is applied to the approximated system terminated in a nonlinear circuit.

All of the above methods start from the Telegraphist’s equations. The per-unit-length parameters are used to characterize the multiconductor systems. Since these parameters are usually calculated by static (dc) analysis, such a characterization is geared to dispersionless TEM modes. However, the TEM approximation is no longer valid at high frequencies when the longitudinal field components are not negligible. Since the starting point of a transient analysis program for VLSI interconnects should be an accurate characterization of the interconnection network, we use a model that is derived from full-wave analysis of the multiconductor structure. It takes into account all possible field components and satisfies all the required boundary conditions in the structure. A bilevel waveform relaxation algorithm is used to compute the transient response of such a structure terminated in nonlinear loads. This algorithm enables the separation of the interconnection network from the termination circuits, and the most efficient method can be used to analyze each subnetwork.

We first present the computation model of the multiconductor structure, followed by a description of the bilevel waveform relaxation algorithm. It is then shown that convergence can be improved by inserting a resistive network at the input and output interfaces. The conditions for convergence are discussed. The paper is concluded with illustrations of the algorithm applied to ECL, CMOS, and GaAs circuits connected by coupled lines.

0278-0070/92$03.00 0 1992 IEEE

WANG AND WING: TRANSIENT ANALYSIS OF DISPERSIVE VLSI INTERCONNECTS 1259

11. COMPUTATIONAL MODEL BASED ON FULL-WAVE ANALYSIS

The spectral-domain approach [20] is used to characterize a multiconductor system embedded in a multilay- ered dielectric medium. Fig. 1 shows the cross section of an example of such a structure. This structure does not support pure TEM waves due to the contributions of fringing field components E, and H, at the dielectric interfaces. The computational model that takes into account the frequency dependent hybrid nature of the structure is derived in [2 11.

There are n dominant propagating modes in a n-conductor system. Each mode has its own propagation constant y;, i = 1, 2, - - - , n. These propagation constants can be calculated by the spectral-domain method and, for a system even when lossless, they are frequency dependent. Fig. 2 shows a three-conductor structure and the effective dielectric constants versus frequency for each dominant mode. The effective dielectric constant is defined as E,, = where X = 2 n / y is the wavelength and Xo represents the wavelength in free space. Un- der the TEM assumption, E,, is a constant. The discrepancies between the TEM approximation and full- wave analysis can be easily seen. In addition to the propagation constants, the current eigenamplitude vector, Mi = [mli m 2 i - - mnilT is needed for a complete characterization of a mode i wave. In the above mki is defined to be the relative amplitude of the mode i longitudinal current on line k as illustrated in Fig. 3. Once y; and Mi are known, the scattering matrix of the multiconductor system can be found as [21]

where e -rL = diag (e-Y"), i = 1 , 2 , * * * , N is the modal propagation matrix, and M = [MI M2 * - - M,] is the eigenvector matrix. One should keep in mind that for a dispersive multiconductor system, both the propagation matrix e -" and the eigenvector matrix M are frequency dependent. In general, off-diagonal terms in submatrix Me -rL M are nonzero, accounting for the couplings among the lines.

In order to evaluate the transient repsonse of the above system with both input and output ports terminated in nonlinear loads, we must find a characterization of the system in the current-voltage domain. We have for the input port

II = (ar - bI) (2.2a)

VI = &(a, + bI) (2.2b) where 2, = Z,M-' is the system characteristic impedance matrix, and 2, is the modal characteristic impedance matrix. The definition of Z,, again, is based on matching the total power. Similarly, for the output port

(2.3a)

(2.3b)

10 = (a0 - bo)

Vo = Zc(a, + bo).

Fig. 1. Cross section of a typical multiconductor system.

t h, I

6:L_/ ' 0 10 20 30 40 50 60 70 80 90 100

F(GHz)

(b) Fig. 2. (a) Cross section of a three-conductor system. (b) Normalized

propagation constants versus frequency.

Combining (2.1)-(2.3), we can represent the input voltages and currents by the output voltages and currents and vice versa:

VI - ZCI, = @(V, + ZJ,) (2.4a) (2.4b)

9 = M-'e-'LM'. (2 .4~)

Vo - zczo = @(V, + 2,ZI)

1260 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 11, NO. IO, OCTOBER 1992

J

Fig. 3. Current eigenamplitude vector.

Fig. 4. Computational model of a multiconductor system.

By a similar formulation given in [3], we define two voltage-controlled voltage sources

U, = q 2 v 0 - U,) (2.5a)

U, = +(2V, - U,). (2.5b) Equations (2.4a) and (2.4b) can then be written as

V d j 4 = Z C ( j 4 M j 4 + U A j u ) (2.6a) V,(jo) = Z,(ju)Z,(ju) + U,(jw). (2.6b)

Based on the above, we can represent the multiconductor system by the computational model shown in Fig. 4. The model consists of two identical networks, Z,, and two sets of voltage-controlled voltage sources, U, and U,. It should be emphasized that the equivalent circuit is given in the frequency domain.

111. TRANSIENT ANALYSIS BY BILEVEL WAVEFORM RELAXATION

Computing transient response of a multiconductor transmission-line system terminated in nonlinear loads is very expensive in computer time. Essentially, one has a system consisting of two different subsystems, namely, the termination subnetwork, which is nonlinear and thus can be characterized only in the time domain, and the linear transmission line subsystem modeled in the frequency domain. Most available analysis methods convert the frequency -domain transmission-line model into the time domain and then apply the conventional discrete-time solution method to solve the whole system. Several draw- backs are inherent from algorithms falling into this cate-

gory. First of all, from the preceding section it has been shown that computationally, only multiplications and summations are required in the frequency domain to characterize a multiconductor system. In the time domain, however, the simplicity of the original multiconductor transmission line model is destroyed. First, convolution, which has a computational complexity of O(m2) by itself (m is the number of time points to be computed), must be employed. Second, the attempt to analyze two different subsystems as a whole creates a difficulty in step size control. Generally speaking, a stepsize suitable for the nonlinear terminations may not be an optimum choice for the interconnection network. Third, because of the difficulties involved in treating a frequency-domain model in the time domain, a common procedure is first to decouple the original multiconductor system, deal with each decoupled line separately, and then use time-domain linear combination to obtain the final solution of the coupled system [3], [20]. For a dispersive multiconductor system, however, this is not possible since the transformation between the coupled and decoupled systems is also frequency dependent as pointed out in the previous section. (The transformation is realized through the eigenvector matrix M(w).) This leaves most of the available methods valid only for non- dispersive TEM models.

In this section, we introduce a new analysis method for the abovementioned system. Rather than trying to find a single algorithm for the entire circuit, we separate the multiconductor system from its termination network, analyze the multiconductor system in the frequency domain and the termination subsystem in the time domain. By doing so, the most efficient analysis method can be adopted for each subsystem. The responses thus computed, however, must be transformed from one domain to the other. We choose to use FFT because of its computational efficiency. The immediate problem arising from this choice is that FFT can transform only sampled data of a whole waveform between the two domains. To over- come this problem as well as to retain the efficiency we intend to pursue, we propose a hierarchical bilevel waveform relaxation (BWR) algorithm for the transient analysis. The method differs from the one introduced in [19] in that no decoupling, recombining, and network syn- thesizing are needed. The algorithm is completely general, valid for both TEM and full-wave models of both lossless and lossy multiconductor systems. Simple mul- tiplication and summation suffice for the linear subsystem. Different analysis methods as well as the multirate properties can be taken for the linear and nonlinear subnetworks to improve efficiency.

Consider a nonlinear circuit with its interconnections modeled by multiconductor transmission lines. The interconnects are represented by the voltage-based equivalent circuit introduced in Section 11. Refer to Fig. 5(a) for the complete system. The “F” inside the dotted box represents a Fourier transform. The input and output nonlinear circuits are shown in the time domain and the multiconductor system in the frequency domain. In order to effi-


Thdonuini t-- Fquencydomain - T i m domaiii

1,tio)

U. b 1--‘-7 V,cj U) (b)

Fig. 5 . (a) Multiconductor system with terminations. (b) Frequency-domain interconnect model.

ciently analyze such a system, we utilize the natural par- itions between the input circuit to the multiconductor system interface and the output circuit to the multiconductor system interface. The partitions are defined as the input interface and output interface, respectively. The transient response of the system is computed by a bilevel waveform relaxation method to be outlined in this section. At the first level, waveform relaxation is used to set up the equivalent voltage-controlled voltage sources for the multiconductor system in the frequency domain. At the second level, waveform relaxation is applied at the input and output interfaces of the system to compute the transient response.

Frequent switchings between the frequency-domain and the time-domain representations make it necessary to clar- ify the notations. All frequency-domain voltage and current vectors are denoted throughout the paper by capital letters (i.e, V ( j w ) and Z(jw)) and the time-domain quan- tities are represented by lower case letters (v(t) and i ( t ) ) . The subscript “I” is used to indicate the input interface and “0” is used for the output interface. Superscripts are for iteration counts.

The computational model developed in the last section shows that the frequency domain voltages and currents at one end of the multiconductor system affect the voltages and currents at the other end of the system by changing the equivalent voltage-controlled voltage sources (U, ( j w ) or U,(jw)) , (see (2.5) and (2.6)). On the other hand, the terminal voltages and currents also depend on the external circuits connected to the multiconductor system. The computation of transient response of the system, therefore, consists of two interrelated processes. One is local- ized to the input or output interface responsible for the computation of terminal voltages from the termination circuit for a given voltage-controlled voltage source U , ( j w ) or U,(jo). The other process is a global one in the sense that it handles the influence of terminal voltages

Gauss-Seidel LWR

(b) Fig. 6 . (a) Input/output part equivalent circuit. (b) Local waveform relax-

ation.

at one interface to the equivalent voltage-controlled voltage source at the other interface (U, ( j w ) or U, ( jw ) ) , and consequently, the terminal voltages.

Let us focus, first on the frequency-domain multiconductor system computational model shown in Fig. 5(b). For a given Z [ ( j w ) , the terminal voltage V,( jw) can be computed by (2.6a), assuming U,(jw) is known. The frequency-domain voltage-controlled voltage source at the output port, U,(jw), can then be updated from (2.5b). The output port voltage V,( jw) is determined by the equivalent voltage source U, ( j w ) , the characteristic impedance matrix Z, ( j w ) , together with the boundary condition mandated by the external circuit connected to the output port, which is, in general, nonlinear (see Fig. 6(a)). A local waveform relaxation (LWR) process is uti- lized to evaluate the voltage v, (t) . The process starts with an initial guess waveform of (v:(t), t E [0, TI ) supplied to the output port termination circuit No. The LWR process is illustrated in Fig. 6(b). The subscript “o” is omit- ted in the figure, since the same algorithm applies to both input and output ports. The current flowing out of the nonlinear circuit, i A (t) is computed by solving the nonlinear circuit in the time domain. It is then transformed into the frequency domain to give I , ( jw) . Equation (2.6b) is used to compute the terminal voltage V,(jw), which is transformed back into the time domain to give a new voltage waveform z, : (t). z, : (t) is then used in No to compute the next iteration of i,(t) and the process continues until the difference between the waveforms of two iterations, namely, lut( t ) - vk,-’(t)l; t E [o, T I is satisfactorily small.

The frequency-domain output terminal voltage V, ( j w ) is available when the SLWR is considered to have reached convergence. This voltage is now applied to update the voltage-controlled voltage source at the input port, U,(jw), by means of (2.5a). The updated V , ( j w ) pro- duces a new frequency-domain input terminal voltage

I262

io

+

Ni VO

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 11, NO. IO, OCTOBER 1992

No

,p< to wmputc

U (a) (b)

Fig. 7 . (a) Global relaxation flowchart. (b) Local waveform relaxation flowchart.

V, ( jw) , which is computed by the same LWR performed at the input interface as at the output interface. We name the iterative process of updating the input and output port equivalent voltage-controlled voltage sources U,( j w ) and U,( j w ) for the whole frequency range global waveform relaxation (GWR). The computation continues until both GWR and LWR reach convergence. The process that combines the GWR and LWR is referred to as bilevel waveform relaxation. A flow chart of the GWR algorithm is shown in Fig. 7(a) and LWR in Fig. 7(b). The formal description of GWR will be given next, followed by the description of local waveform relaxation (LWR). Con- vergence problems of the algorithms are discussed in the subsequent sections.

Let the superscript r indicate the rth GWR iteration. The algorithm is given as follows.

Algorithm GWR 0. Initialization:

Connect the input circuit Ni to the output circuit No (see Fig. 8), compute vo( t ) and io@) to initialize U y ( j w ) by U ? ( j w ) = @,[F(vo(t) + ZcF(io(t))] . (F indicates a Fourier transformation). Set r = 1.

1. Compute v:( t ) by LWR, and compute V ; ( j w ) = F [ ; @ ) I .

2. Update U ; ( j w ) = 'Pu[2V;(jw) - U i ( j w ) ] . 3. Compute uL(t) by LWR, and compute VL(jw) =

4. Update U ; ( j w ) = @,[2V; ( jw) - UL( jw) ] , set r =

Remarks : 1) The relaxation process starts from the original non-

linear circuit without the interconnections. It may also start from zero initial condition (iy(t) = 0,

2) Convergence is checked in the second level waveform relaxation process. GWR convergence is de-

Flu m. r + 1 and repeat steps 1-5 until convergence.

v;(t) = 0).


fined as:

(3.1)

wherep = I , 0 and E is the tolerable error,

Algorithm L WR 0. Set k = 0 and v:"(t) = F-'[V;-'(jw)], t E [0, TI

by analyzing the termination circuit in the time domain.

1) Connect a voltage source of value v f ' ( t ) to N ~ . Compute iF'(t) f o r t E [0, TI. (See Fig. 6(b)).

2) Compute U : + ' , '( t) = ~ - ' { ~ , ( j w ) ~ [ i F ~ ( t ) ] + U;- ( j w ) } .

3) Check for convergence. 3a) If

exit ; 3b) Else if max,.[O,Tl I lv:+'* ' ( t ) - vFr(t)II < E , and

k > 0, update Vi( j w ) = F [ v : + '(t)] and exit to GR;

3c) Else set k = k + 1 and repeat steps 1-3. Remarks: 1) Condition 3a) is equivalent to convergence for both

GWR and LWR since it indicates that the new waveform does not change from the previous GWR iteration and the correction of the nonlinear termination circuit on the waveform is negligible. In other words, conditions (3.1) and (3.2) are satisfied simultaneously.

2) Windowing strategy [17] can be applied to LWR to improve computational speed. We start with one window of size [0, TI; as the waveform within the window converges, the window is reduced to smaller and smaller sizes. The full waveform, however, is always used in the frequency domain for easy utili- zation of FFT.

3) The convergence condition for LWR can be set relatively loose for the first few GR iterations. More accurate conditions should be used as the number of GWR iterations increases.

Examples Fig. 9(a) shows a pair of CMOS inverters connected by

a lossless stripline with 0.4-ns delay and 5 0 4 characteristic impedance. Transient response of the circuit is shown in Fig. 9(b). In the plot, q ( t ) and u2(t) are the voltage waveforms at the near and far ends of the conductor, and U&) is the final output voltage waveform. If 50% of the total signal swing is regarded as the logic one threshold, (2.5 V in this case), the final output barely reaches the logic one level. Further increase in line delay causes the output pulse to disappear completely, as plotted in Fig. 9(c).

Fig. 10(a) shows an example with coupled lines where the first and the third inverter pairs are driven by pulse

inputs and the second inverter pair is connected to ground. The computed results for the near and far ends of the multiconductor transmission lines are plotted in Fig. 10(b) and the transient output voltages of the final stage is also given. Both the coupling effect and the logic error are clearly shown in these plots.

This example is used merely to show the applicability of the introduced algorithm. We have intentionally chosen long interconnects, and the circuit under consideration operates at relatively low frequency. TEM approximation of the interconnects can be used, and therefore it is possible to apply the algorithm introduced in [3] to analyze the circuit. The correctness of the bilevel waveform relaxation algorithm is demonstrated by comparing results. It has also been found that only 10- 15 % of the computation time is used on FFT.

IV. IMPROVED CONVERGENCE BY OVERLAPPED PARTITION

The bilevel waveform relaxation algorithm described in the last section has one drawback, namely, the necessary and sufficient condition for the second level waveform relaxation to converge can be satisfied only by a very limited class of circuits [22]. Moreover, one has no control over the rate of convergence. Experiments show that when the algorithm is used to analyze a multiconductor system, the convergence slows down if the system is well matched by the termination network. For a system terminated in ECL and GaAs transistors, the algorithm does not always converge. A theoretical study of the convergence condition can be found in [22] and [25].

Obviously, a more robust algorithm is needed for the analysis method to be generally useful, and it is desirable to have some control over the rate of convergence. To improve the convergence of the GWR algorithm introduced in the last section, an investigation of the cause for nonconvergence and slow convergence is necessary. Careful study of the conventional waveform relaxation method [14]-[18] shows that there are usually overlaps between the circuit partitions corresponding to the assign- ment-partition process of waveform relaxation. When using waveform relaxation to analyze a multiconductor system, however, there is no overlap between the time domain nonlinear circuit and the frequency -domain multiconductor system. It is intuitively obvious that if overlaps can be created, faster convergence will be achieved. In light of the above discussion, the following algorithm is proposed.

First, we insert three networks consisting only of resis- tors on the diagonal into the input interface. The resistive network is denoted by -R, 2R, and -R, R = diag (Ti), i

, n, where ri can have any positive values. In = 1, e - . computing the transient response of a multiconductor system, we choose ri to be equal to the value of the irh diagonal element of z,(O). The system after insertion is shown in Fig. l l(a). The new system thus created is electrically equivalent to the original system, yet an overlap partition is now possible due to the insertion. The overlap

1264 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. I I , NO. 10, OCTOBER 1992

T W (C)

Fig. 9. (a) A pair of CMOS inverters with interconnect. (b) Transient response for CMOS inverter with interconnect. (c) Logic error created by the interconnect.

WANG AND WING: TRANSIENT ANALYSIS OF DISPERSIVE VLSI INTERCONNECTS

..... . . . ............................................

s >

I265

-0 1 2 3 4 5 6 7 8

T(m) 0)

Fig. 10. (a) Three-conductor system terminated in CMOS inverters. (b) Three-conductor system output waveforms.

partition is illustrated in Fig. l l (b) and (c). For conven- ience, we refer to the subsystem of Fig. l l (b) as the left partition and the subsystem of Fig. l l ( ~ ) as the right partition. The two partitions overlap on the network 2R. As a direct consequence of the overlap, we are able to use only voltage sources in the iterative process instead of switching between voltage and current sources as required by LWR. Notice that the resistive network can be pulled into the frequency domain directly, as shown in the right partition. Also, since the total insertion to both the left and right partitions is positive, instability is not introduced.

Based on this overlap partition, we start the waveform relaxation process by giving an initial guess waveform of v?(t) and apply it to the left partition. Voltage v;'(t) can

as the input voltage source to the right partition. v;'(t) is transformed into the frequency domain, and by analyzing

the linear system we get the frequency-domain voltage V: ( j w ) . The voltage is transformed back to the time domain and a new input voltage waveform for the left partition is thus obtained. The iterative process continues until convergence.

The just outlined algorithm is named the improved local waveform relaxation (ILWR). The flow chart of ILWR is given in Fig. 12. Using notations similar to those adopted in Section 111, let r be the GWR iteration count. Refer to Fig. l l (a) for the system under consideration. Let r, be the i th diagonal element of zc (0). A formal description of the algorithm is given as follows.

Algorithm ZL WR 0. Initialization: set k = 0, R = diag (Ti = rCJ, and

1. Connect a voltage source of value @(t) to NL through the positive resistive network 2R, compute

then be computed from the nonlinear circuit and is used v:.'(t) = F- l [V; - l ( jw) ] .

-

1266 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 11, NO. IO, OCTOBER 1992

n

Fig. 1 1

n

-1 v,.k+'(j 0) v p ( j 0)

NI'

(C)

partition. (c) Right partition. (a) Input port of multiconductor system after insertion. (b) Left

v;k+ ( t ) for t E [0, TI by analyzing the termination circuit in the time domain (see Fig. l l(b)).

2. Compute vik+'(ju) = ~ [ v ; ~ + l ( t ) l . 3. Connect a frequency-domain voltage source of value

Vjk + to N,! through the positive resistive network, 2R, and compute V f ' l x r ( j w ) (see Fig. l l(c)) .

4. Computer v:+l . ' ( t ) = F - I [ V ~ + ' ~ ~ ( ~ U ) I . 5. Check for convergence.

5a) If waveform has converged and k = 0, exit; 5b) else if waveform has converged and k > 0, return to GWR; 5c) else set k = k + 1 and repeat steps 1-4.

The ILWR algorithm is tested in a multiconductor system terminated in TTL, ECL, and GaAs transistors. Sev- eral examples are presented here.

Sketched in Fig. 13(a) is an ECL inverter circuit. Two such inverters are used to terminate a single transmission line system. The LWR algorithm introduced in Section I11 fails to converge for this circuit. When ILWR is used to analyze the system, convergence has always been achieved in no more than three iterations. The computed waveforms are plotted in Fig. 13(b). To illustrate the pos-

r-- k = k+l +

Fig . 12. Improved local waveform relaxation flowchart

sibility of causing logic error by badly matched terminations, we intentionally change the characteristic impedance of the line from 50 to 175 Q. The voltage waveforms on both ends of the line as well as the output of the second inverter are given in Fig. 13(c). As expected, a logic error is created.

In this example, a five-conductor transmission line system is connected to GaAs inverters at both input and output ports. The multiconductor system is shown in Fig. 14(a) and the circuit in Fig. 14(b). For the GaAs transistors, the SPICE default model is used. Without overlap, the SLWR algorithm cannot reach convergence. Using the ISLWR algorithm, the computed voltage waveforms are recorded in Fig. 15.

V . CONVERGENCE CONDITION FOR ILWR In order to understand the reason for the better conver-

gence of the ILWR algorithm in comparison to LWR and also to explore the possibility of extending the algorithm to the analysis of general nonlinear systems, the condition for convergence is studied in this section. For clarity, we start with linear systems. A multiconductor system with linear termination can be treated as a special case of the above. Numerical examples are also included to illustrate the extension of the algorithm to nonlinear systems.


f v out I

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T(sec.) xl0-8 (C)

Fig. 13. (a) ECL inverters connected by transmission line. (b) Voltage response of circuit in Fig. 13(a). (c) Logic error caused by mismatch.

1268 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 11, NO. 10, OCTOBER 1992

€1 = 1 I lor

VDD

VDD 3-7 , "01 ~ -

(b) Fig. 14. (a) Five-conductor system. (b) Five-conductor system terminated

in GaAs inverters.

Given a linear system consisting of two general dy- namic systems, L1 and &. Partition the system along the L1 and & interface and insert -R, 2R, and - R between the partition as shown in Fig. 16(a). The new system is electrically equivalent to the original system. Next, partition the system by overlapping on the positive insertion 2R as shown in Fig. 16(b). The partitioned subnetworks are called the left and right partition. When the ILWR algorithm is applied to the analysis of this system, a slight modification of the algorithm is necessary to make it more general, namely, the resistor values chosen for the overlap will not be restricted to the diagonal elements of the

multiconductor characteristic impedance matrix as given in the original ILWR algorithm. Let the state variables of L1 be xI and & be x2. The terminal voltage vector of L1 and & is denoted by U, and the currents flowing into L1 and & are called i l and iz, respectively. The system given in Fig. 16(b) can be characterized by

XI = alx l + b lu l + el ~2 = ~ 1 x 1 + diu1 + f i x2 = a2x2 + b2u2 + e2 V I = ~ 2 x 2 + d 2 ~ 2 + f2. (5.1)

WANG AND WING: TRANSIENT ANALYSIS OF DISPERSIVE VLSI INTERCONNECTS I269

i

T((seC.) x10-'0

(b)

Fig. 15. (a) Voltage waveforms on line 3 of Fig. 14(b). (b) Voltage waveforms on line 2 of Fig. 14(b). ( c ) Voltage waveforms on line 1 of Fig. 14(b).

A convergence theorem of ILWR can be stated as follows.

where

Theorem I: The necessary and sufficient condition under which ILWR applied to the overlapping partitioned

e(h) = h2C1(z - hal)-'b,cz(i -

+ hd'C2(I - ha2)-'b2 linear system shown in Fig. 16(b) converges to the correct solution is that the spectral radius: and h is the stepsize.

The methodology for the proof of Theorem 1 is first given by Wing in [23]. The actual proof is given in the P ( W ) + 4 d 2 ) c 1 (5.2)

1270 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 1 1 , NO. IO, OCTOBER 1992

T(sec.) (C)

Fig. 15. (Conrinued.)

x10-'0

(b)

Fig. 16. (a) Creating overlap. (b) Overlap partition.

Appendix and also in [25]. We concentrate next on find- ing practical guidance in choosing values for the overlapping insertion. We note that to this end, the spectral radius is a norm, so that if in a given system, p(d, d2) < 1, a value of h can always be found such that condition (5.2) is satisfied, and waveform relaxation will converge on that system. In the rest of this section, we derive the properties

that the subsystems LI , L2, and R must have in order that p(d, d2) < 1 .

Let Ri, i = 1, 2, be the input impedance matrix of Li with xi and all independent sources set to zero, and Gi be the corresponding conductance matrix. Notice that Ri or Gi may not exist for a given system. We define three types of linear systems.


1) A type A system is one in which none of the components of vector i(t) or v(t) at the boundary of L1 and b is a state variable of the system.

2) A type B system is one in which some or all of the components of vector i(t) are state variable(s) of the system.

3) A type C system is one in which some or all of the components of vector v(t) are state variable(s) of the system.

We make the following observations. Observation I: If Li is a type A system, both Ri and Gi

(5.3a)

exist and

di = (Ri - R)(R, + R)-1 = (RjR-' - I)(RiR-' + I ) - ' (5.3b) = ( I - GiR)(I + GiR)-'. (5.3c)

di = I. (5 -4)

Observation 2: If Li is a type B system (xi = ii), then

Since ii = Givi I x i = o = 0 for a type B system, we have Gi = 0. Substituting this into ( 5 . 3 ~ ) gives us (5.4).

Since vi = RI ii Ix, = = 0 for a type C system, (xi = vi), we have R1 = 0, or

Observation 3: If Li is a type C system, then,

di = -I. (5.5) In partitioning the original system L into L1 and & we

assume that no state variable of L is assigned to both L1 and &. In other words, we do not split capacitors or in- ductors in the partition process.

Observation 4: L1 and & cannot be both type B (type

observation 5: If Li is of type A, then both Gi and Ri

The last observation is true since v fRiv > 0 and i f Gii

Lemma 1: If Li is a type A system, then

C).

are positive definite.

> 0 are always true for nonzero v and i.

d d i ) < P(I - GiR). W (5.6) In order to prove the lemma, we must recall from ma-

1) If A and B are two diagonalizable real matrices, then

2) A = BD is positive definite if B is a real positive

Returning to the proof of Lemma 1, we have from

(5.7) GiR is positive definite so both (I - GiR) and (I + G,R)-' are diagonalizable. Using Fact 1 above, we have

Since GiR is positive definite, its eigenvalues are all positive. Let A, be the smallest eigenvalue of (I + GiR), we have

trix theory the following facts.

P(A@ = P ( 4 definite matrix and D is a real positive diagonal matrix.

(5.3c) :

di = ( I - GiR)(I + GiR)-'

p(di) I p(I - GiR)p(I + GiR)-'. (5.8)

A, > 1.

Thus the maximum eigenvalue of ( I + GiR)-', A,, is 1

A , = - C l A s

and Lemma 1 follows. Lemma 1 will be used to derive sufficient conditions for convergence.

Theorem 1 suggests that in order that the algorithm converges to the correct solution, one should choose such resistor values that p(dld2) < 1 if this is at all possible.

Suppose L 1 is of type A. If L 2 is also of type A, let I,,, be the maximum of the diagonal elements of R1 and R2, and let &A be the maximum of the diagonal elements of GI and G2. If L2 is a type B system, R2 does not exist and r,, is the maximum diagonal element of RI . If L2 is a type C system, G2 does not exist and g,, is the maximum diagonal element of G1. Finally, let r,, (rl) be the maximum (minimum) element of R = diag (T i ) .

Theorem 2: (Sufficient conditions) If L 1 is of type A, WR applied to the overlapped partition is guaranteed to converge to the correct solution if every resistor ri chosen to be inserted to the system satisfies

ri 2 r, > - (5.9) ruA 2 or if every resistor ri satisfies

W (5.10)

Proof: If L2 is not a type A system, by Observations (2) and (3) we have p(dld2) = p ( d l ) . If L2 is a type A system, we have p(dl d2) I p(dl ) p(d2). In any case, the following is true

ddl d2) I P@l) P(d2) *

2 guA

ri I r,, < -.

Since L 1 is of type A, by Lemma 1 we have to prove only that (5.9) and (5.10) both imply that

p ( I - G1R) I 1 or equivalently,

p(R ,R- ' - I ) I 1.

Let A represent R1 R- ' . Let Xu be the upper bound on the eigenvalues of A and let a k k be the maximum diagonal element of A. By Gersgorin's disc theorem, all eigenvalues of the system are inside the circle with orign akk and radius less than a&. However, akk is less than or equal to rUA/rl. We have A,, c a k k I ruA/rl. Therefore,

p(R'R-1 - I ) I - - 1 < 1 lr: I requires that

ruA 2

rl > -.

Applying the same procedure to p(Z - G 1R) results in 2

r,, C - &A

which concludes the proof of Theorem 2.

1272 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 11. NO. 10, OCTOBER 1992

rl cl

0.8

0 1 2 3 4 5 6 7 8 9 10

T(Kc.) (b)

Fig. 17. (a) A simple circuit without grouding capacitor. (b) Convergence of Fig. 17(a) by ILWR.

It should be emphasized that both conditions given by (5.9) and (5.10) are sufficient conditions and need not be satisfied simultaneously. Moreover, they are not necessary conditions for convergence. In the case of a one-di- mensional system, for example, if R = rand Gi = gi , the necessary and sufficient condition for convergence re- duces to

which is always satisfied for 0 < gi < 00 by any r > 0. In practice, if the input resistance matrix of the partitioned system, Rp = [rjj] is known, we should choose the overlapping network such that R = diag (ri = rii). This could effectively reduce the spectral radius of d, d2 so that faster convergence can be achieved. In other words, if the overlapping network is chosen so that a closely matched condition is created, faster convergence can be expected.

When neither L 1 nor L2 is of type A, by Observations (2)-(4), we have

For the WR process to converge, in this case, it is required that

This condition can usually be satisfied by a proper choice of the stepsize.

Notice that requirements such as grounded capacitors at every node and weak couplings across the partitions are not necessary for waveform relaxation to converge if the - R -t 2R - R network is inserted. Without the inserted network, these requirements are essential. For example, applying conventional WR (without overlap) to a simple circuit shown in Fig. 17 does not converge if no grounded capacitor is added to the system, whereas waveform relaxation with overlap converges to the correct solution in three iterations.

Next, let us look at Fig. 18, a high-pass circuit. When we partition the circuit by cutting a strong feedback loop as shown, WR with overlap partition again converges in three iterations.

We have demonstrated, both theoretically and experi- mentally, that for linear systems the ILWR has better


Partition Here 2,- I

-0.41 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

T( sec.) (b)

Fig. 18. (a) A high-pass circuit. (b) Convergence of Fig. 18(a) by ILWR.

convergence property than the SLWR algorithm introduced in the last section. One great advantage of the algorithm is that one can actually choose the overlapping network to accelerate convergence. In the remainder of this section, we apply the algorithm to a nonlinear circuit to demonstrate the possible extension of the algorithm to general circuit analysis.

While waveform relaxation when applied successfully is a very efficient numerical technique, its major problem has been the lack of robustness when adopted for general circuit simulation. Advantages over conventional discrete time circuit simulation methods have been demonstrated for systems such as MOS circuits. In generalizing the method to systems such as bipolar or GaAs circuits, however, the convergence problem becomes a bottleneck [24]. If not partitioned properly, WR applied to these systems may not converge to the correct solutions, or, it may converge very slowly due to strong couplings and feedbacks

between partitioned stages. In this section, the ISLWR algorithm is applied to such “badly behaved” systems to demonstrate that by inserting diagonal resistive overlapping networks, waveform relaxation can be employed ef- ficiently.

The following examples illustrate that even when a system is not partitioned properly, the overlap partition technique helps to improve the convergence of WR.

In [17], a NMOS latch, sketched in Fig. 19(a), is used to illustrate the convergence problem associated with the conventional waveform relaxation method. It is shown that after 20 iterations, the waveform relaxation process still does not converge to the exact solution. In applying ISLWR to the same circuit, we get very close to the correct solution in four iterations (see Fig. 19(b)).

Fig. 20(a) shows a cross-coupled GaAs inverter circuit. The convergence process of the overlap WR is plotted in Fig. 20(b).

1274 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 1 1 , NO. 10. OCTOBER 1992

T(sec.) xl0-8

(b)

Fig. 19. (a) An NMOS latch. (b) Transient response of circuit in Fig. 19(a).

VI. CONCLUSION

We have developed and implemented a new algorithm to compute the transient response of a coupled, dispersive multiconductor system terminated in nonlinear loads such as transistors. The characterization of the multiconductor system is obtained from full-wave analysis based on the spectral domain approach, and it is suitable for circuit simulation. To compute the transient response of such a system, a bilevel waveform relaxation method is used. This algorithm allows the interconnects to be separated

from the rest of the system so that both the nonlinear termination circuit and the multiconductor system can be analyzed in the most efficient way. The method has been applied to multiconductor systems terminated in MOS, ECL, and GaAs transistors, and it is shown that reflec- tions can create logic errors in the system. Coupling ef- fects are also demonstrated. Both theory and examples are included to show the convergence property of the algorithm. We have also demonstrated by an example the possibility of applying the overlap partition method introduced in this paper to general circuit analysis.


9

1275

0.55 I

0.1 ' T(nd x10-9

(b) Fig. 20. (a) A GaAs circuit. (b) Transient response of circuit in Fig. 20(a).

APPENDIX a2x2(m - 1 ) + x2(m) - P ~ v , ( m ) = a&~(m) vl(m> - c2x2(m) - d2vdm) = f i (m)

PROOF OF THEOREM 1 Backward Euler integration (or any other implicit

method) is applied to the systems (4 .1) at each timestep t = mh, m = 1 , 2 , - - * M. Let xl (m) be the solution at have

ai = ( I - hai)-'

pi = ( I - hui)-'hbi, i = 1 , 2 . ( A I ) t = mh and similarly for x2(m), v l (m) , and v2(m). We

a lxdm - 1 ) + xl(m) - Plvdm) = alhel(m) v2(m) - c l x m - dl vdm) = f i (m)

If we write out the four equations for m = 1 , 2 , - - , M, we obtain a system of 4M linear equations in 4M unknowns which are the variables xl (m) , x2(m), v l (m) , and

1276

v2(m), i . e . , Ay = b

where

Y = M O ) - - X l ( M ) U 2 ( 0 ) * U 2 ( M ) X 2 ( 0 ) * - X 2 ( M ) U 1 ( 0 ) - * * ul(M)]‘

are the unknowns,

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN, VOL. 11, NO. 10, OCTOBER 1992

Since

with r 1 0 . 0 0 1

where Pi, i = 1 * * - 3 are functions of A l , Bl , C1, and D 1 , and

P4 = -(CiAIBI - Di)(C2AZB2 - 0 2 ) . The necessary and sufficient condition for convergence is that matrix ( L + Z)-’U is convergent

p ( ( L + z>-’u> = p(P4) < 1 . Expanding the above equation gives us condition (3). W

REFERENCES

[l] M. Lister, “The numerical solution of hyperbolic partial differential equations by the method of characteristics,” in Muthemutical Meth- ods for Digital Computer, A. Ralston and H. S. Wilf, Eds. New York: Wiley, 1960.

[2] F. H. Branin, Jr., “Transient analysis of lossless transmission lines,” Proc. IEEE, vol. 55, pp. 2012-2013, Nov. 1967.

[3] F. Y. Chang, “Transient analysis of lossless transmission lines in a nonhomogeneous dielectric medium,” IEEE Trans. Microwave The- ory Tech., vol. MTT-18, pp. 616-626, Sept. 1970.

[4] F. Y. Chang, “Computer-aided characterization of coupled TEM transmission lines,” IEEE Trans. Circuits Syst., vol. CAS-27, pp. 1194-1205, Dec. 1980.

[5] A. R. Djordjevic, T. K. Sarkar, and R. F. Hamngton, “Analysis of lossy transmission lines with arbitrary nonlinear terminal networks,” IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 660-666,

A i = ji; 1 Bi = diag (pi); Ci = diag (q).

. . -ai

Decompose A into a Strictly Upper triangular matrix U, an identity matrix I , and a strictly lower triangular matrix L:

A = (U + z + L ) where

r 0 o 01 (L + I ) = 1 -ocl -L2 oA2 : 1

L O -D2 -C2 ZJ

0 0 -D

L i - 1 0 0 0 1 . L o o o J

Now apply the Gauss-Seidel iteration to solve the linear system

(L + Z)yk = -Uyk-’ + b. (A21 (A2) can be written as

yk+l = -(L + Z)-’Uyk + (L + Z)-’b (A3a) y k = -(L + Z)-’Uyk-’ + (L + Z)-’b. (A3b)

Subtract (A3b) and (A3a) and let 6 k f I = y k + --‘,then

(L + (A4) 6 k + l =

June 1986. [6] A. R. Djordjevic, T. K. Sarkar, and R. F. Hamngton, “Time-do-

main response of multiconductor transmission lines,” Proc. IEEE, vol. 75, pp. 743-764, June 1987.

[7] A. R. Djordjevic and T. K. Sarkar, “Analysis of time response of lossy multiconductor transmission line networks,” IEEE Truns. Mi- crowave Theory Tech., vol. MTT-35, pp. 898-908, Oct. 1987.

[8] J . E. Schutt-Aine and R. Mittra, “Nonlinear transient analysis of coupled transmission lines,” IEEE Trans. Circuits Syst., vol. 36, pp. 959-967, July 1989.

[9] F. Romeo and M. Santomauro, “Time-domain simulation of n coupled transmission lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-35, pp. 131-136, Feb. 1987.

[lo] V. K. Tripathi and R. J. Bucolo, “Analysis and modeling of multi- level parallel and crossing interconnection lines, ’’ IEEE Trans. Elec- tron Devices, vol. ED-34, pp. 650-658, Mar. 1987.

[ l l] 0. A. Palusinski and A. Lee, “Analysis of transients in nonuniform and uniform multiconductor transmission lines,” IEEE Trans. Micro- wave Theory Tech., vol. MTT-37, pp. 127-138, Jan. 1989.

1121 G. Ghione, I. Maio, and G. Vecchi, “Modeling of multiconductor buses and analysis of crosstalk, propagation delay, and pulse distor- tion in high-speed GaAs logic circuits,” IEEE Trans. Microwave Theory Tech., vol. MTT-37, pp. 445-456, Mar. 1989.

[13] H. Grabinski, “An algorithm for computing the signal propagation on lossy VLSI interconnect systems in the time domain,” Integru- tion, VLSIJ., vol. 7, pp. 35-48, 1989.

1141 E. Lelarasmee, A. E. Ruehli, and A. L. Sangiovanni-Vincentelli, “The waveform relaxation method for time-domain analysis of large- scale integrated circuits, ” IEEE Trans. Computer-Aided Design, vol.

[I51 E. Lelarasmee, A. E. Ruehli, and A. L. Sangiovanni-Vincentelli, “A new circuit simulator for large-scale MOS integrated circuits,” presented at ACM/IEEE 19th Design Automation Conf., Las Vegas, NV, May 1982.

[16] P. Debefve, J . Beetem, W. Donath, H. Y. Hsieh, F. Odeh, A. E. Ruehli, P. Wolff, Sr., and J. White, “A large-scale MOSFET circuit

CAD-1, pp. 131-145, July 1982.


analyzer based on waveform relaxation,” presented at ZEEE In?. Con5 Computer Design, Rye, NY, Oct. 1984.

[I71 J. K . White and A. L. Sangiovanni-Vincentelli, Relaxation Tech- nique for the Simulation of VLSI Circuits. Norwell, MA: Kluwer Academic, 1987.

[18] P. Saviz, “Circuit Simulation by Hierarchical Waveform Relaxa- tion,’’ Ph.D. dissertation, Columbia Univ., New York, 1990.

[19] F. Y. Chang, “The generalized method of characteristic for waveform relaxation analysis of lossy coupled transmission lines,” ZEEE Trans. Microwave Theory Tech., vol. 37, pp. 2028-2038, Dec. 1989.

[20] T. Itoh and R. Mittra, “Spectral-domain approach for calculating the dispersion characteristics of microstrip lines,” ZEEE Trans. Micro- wave Theory Tech., vol. MTT-21, pp. 496-499, 1973.

[21] R. Wang and 0. Wing, “A circuit model of a system of VLSI interconnects for time response computation,” ZEEE Trans. Microwave Theory Tech., vol. 39, pp. 688-693, 1991.

[22] R. Wang, “Computation of transient response of dispersive multiconductor systems,” Ph.D. dissertation, Columbia Univ., New York, 1990.

[23] Omar Wing, private communication, Mar. 1990. [24] P. Saviz, “Circuit simulation by hierarchical waveform relaxation,”

[25] R. Wang and 0. Wing, “Waveform relaxation on tightly coupled Ph.D. dissertation, Columbia Univ., New York, 1990.

systems,” to be presented at ISCAS91, Singapore.

Rui Wang (S’80-M’90) received the B.S. degree from South China Institute of Technology, Guangzhou, China, in 1982, and the M.S. and the Ph.D. degrees from Columbia University, New York, in 1985 and 1990, respectively.

She was a Graduate Research Assistant at Co- lumbia University from 1984 to 1990. She joined Advanced Micro Devices, Sunnyvale, CA, in 1990 as a Senior Design Engineer. She is currently a Senior Member of the Technical Staff of Cadence Design Systems, Inc., Santa Clara, CA.

Her research interests include VLSI interconnects, high speed VLSI circuit design, delay, and timing analysis, transient analysis methods for VLSI circuits and interconnection networks, and computational microwave techniques.

Omar Wing (S’50-A’53-M’58-SM’68-F’73) received the B.S. degree from the University of Tennessee in 1950, the M.S. degree from Mas- sachusetts Institute of Technology, Cambridge, MA, in 1952, and the Eng.Sc.D. degree from Co- lumbia University, New York, in 1959.

Since 1959 he has been a faculty member of the Department of Electrical Engineering, Columbia University, where he is presently Professor of Electrical Engineering. He served as chairman of the department from 1974-78 and from 1983-86.

From 1952-56, he was a Member of the Technical Staff at AT&T Bell Laboratories. He is currently Dean of the Engineering School at Chinese University of Hong Kong. His current research interests include computer- aided design of high speed electronic circuits, analysis of VLSI interconnects, modeling of heteojunction devices for circuit simulation, and optimum design of gigabits-per-second circuits for telecommunications applications.

He is the author or co-author of four texts on circuit theory and a book on GaAs digital circuits. He served as President of the IEEE Circuits and Systems Society in 1976, Editor of the IEEE TRANSACTIONS ON CIR- CUITS AND SYSTEMS (1973-76), and General Chairman of the 1968 IEEE International Symposium on Circuits and Systems.

Dr. Wing received the IEEE Centennial Medal in 1987 and the IEEE Circuits and Systems Society Award in 1989.

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