and no. waveform relaxation analvsis rlcgdpaul/comp5704/papers/wr-single... · 2007-10-03 · 1394...

1394 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOI .37, NO. 11, NOVEMBER 1990

Waveform Relaxation Analvsis of RLCG J

Transmission Lines

Abstract -Branin’s method of characteristics has been extended for the transient analysis of transmission lines with constant E C G parameters. It has been further generalized for iterative waveform relaxation analysis. The sequence of waveforms generated by the iteration process has been identified as the series expansion of the transmission-line response functions. By incorporating the fast Fourier transform in the waveform relaxation analysis, a phenomenal two-order reduction of CPU time and one-order savings in computer memory have been achieved. Examples of E C G lines driven by bipolar logic gates are given to illustrate the advantage of waveform relaxation over the discrete-time simulation.

I. INTRODUCTION HE method of characteristics was introduced in 1967 T by Branin [ l l for the transient analysis of an ideal

transmission line. Subsequently, it was successfully generalized [2] and implemented in circuit simulators for the analysis of coupled transmission lines. However, the gen- eralization of the method for lossy lines is nontrivial; modified formulations [3] , [4] have been proven to be incorrect [5] for the RLCG line and no explicit solution has emerged in the past two decades. Nevertheless, by treating a lossy line as segments of ideal lines connecting with discrete resistors, the classical method of characteristics remains applicable for simulation of lossy lines [6]. Such a brute-force approach requires excessive computer memory for simulating the distributed nature of the conductor loss ( R ) and the dielectric leakage (G). Further- more, unless the integration time-step is kept small for the entire simulation process, incorrect simulation results characterized with such false ringing waveforms as shown in Fig. 1 are produced. Thus using the classical method of characteristics for simulating a lossy line as a distributed- lumped network is unattractive in terms of CPU time and memory requirement.

In this paper, a rigorous network theoretic basis for the method of characteristics is established for the discrete- time simulation of transmission lines characterized with constant E C G parameters. The general case of nonuniform transmission lines with frequency-dependent parameters will be described in separate articles [71, [81. To

Manuscript received September 25, 1989; revised June 20, 1990. This paper was recommended by Associate Editor T. K. Ishii.

F.-Y. Chang is with the General Technology Division of IBM, Hopewell Junction, NY 12533.

IEEE Log Number 9038547.

o .6 i

TIM

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Fig. 1. False ringing (000) caused by the improper use of the method of characteristics for transient simulation of a lossy transmission line. (a) Near-end terminal voltage waveform. (b) Far-end terminal voltage waveform.

reduce simulation cost, the classical method of characteristics has been generalized for waveform relaxation analysis [9]. The generalized method of characteristics has been implemented on the interactive circuit design (ICD) program [IO] on an experimental basis. Explicit formulas have been derived and coded in specially written subrou- tines for the synthesis of the characteristic impedance function and the exponential propagation function of the uniform RLCG transmission lines. The implementation requires neither modification of integration routines nor the pole-zero placement for the approximation of characteristic impedance function and the exponential propagation function as described in [ I l l and [12]. Therefore it can be implemented on any existing circuit simulator. We choose the ICD program because it differs from other simulators in that it is interactive; therefore, data sets can

0098-4094/90/1100-1394$01.00 01990 IEEE

rc- 1 T

CHANG: WAVEFORM RELAXATION ANALYSIS 1395

be easily manipulated for iterative waveform generation required by the waveform relaxation algorithm. In comparison to the classical discrete-time simulation, a phenomenal two-order reduction in computer simulation time and one-order savings in computer memory requirements have been achieved by applying the generalized method of characteristics in conjunction with the fast Fourier transform (FFT).

The organization of this paper is outlined as follows. In Section 11, the classical method of characteristics is reviewed and extended to include conductor loss and dielectric leakage. Equivalent lumped-circuit models for the RLCG lines are derived by applying the Pad6 approximation, the Mittag-Leffler's expansion theorem, and the Lambert's partial fraction expansion. For illustration, the equivalent circuit models are used for the discrete-time simulation of lossy lines driven by bipolar logic gates. In Section 111, the waveform relaxation method is briefly reviewed and the classical method of characteristics is generalized for waveform relaxation analysis. It is shown that the sequence of waveforms generated by the generalized method of characteristics can be identified as the series expansion of the network response functions ex- pressed in terms of the chain-matrix parameters of the lossy lines. It is shown that by simulating the exponential propagation function of the lossy line using the FFT, the simulation cost can be drastically reduced. The advan- tages of waveform relaxation over the discrete-time simulation are illustrated with examples of distributed-lumped parameter networks. Concluding remarks are contained in Section IV, along with a description of the ongoing research work on the generalized method of characteristics.

11. DISCRETE-TIME SIMULATION OF RLCG TRANSMISSION LINES

Transmission lines that are used for circuit interconnection' can be uniquely characterized by the chain- matrix formulation' [131:

cosh 8 Z,sinh8 V, [:]=[(l/Zo)sinh8 cosh8 ][I,] (2'1)

where (V,, Zl) and (V,, I,) are the terminal voltages and currents at the near end and the far end of the transmission line, as shown in Fig. 2. For a transmission line of length 1 and per-unit-length series resistance R , induc- tance L , shunt capacitance C, and conductance G , the

"+ R --

Fig. 2. An RLCG transmission line and its chain-matrix characterization.

Equation (2.1) can be converted into the following equivalent form:

Vl-Z,Z,=[exp( -€I)](V,-Z,Z,) (2.3a)

V,+Z,Z,= [exp(-O)](Vl+Z,Zl) (2.3b)

which can be rewritten into the characteristic formulation

(2.4a)

(2.4b)

[I], [21: V' = Z,I, + E ,

V, = - Z,Z, + E ,

where E, = - 8>1(2V, - E2) (2.5a)

E , = [exp( - 011 w 1 - E,) (2 Sb)

are the waveform generators for simulating the delay and attenuation of signals propagating on the E C G line. Thus the discrete-time simulation of an RLCG line can be carried out by using the two-port network in Fig. 3 in conjunction with (2.5) and the equivalent circuits for Z, and exp( - e), to be described next.

A. Pad6 Synthesis of the Lossy Characteristic Impedance z, = \/( R + S L ) / ( G + SC)

The characteristic impedance Z, of an RLCG transmission line can be synthesized by the driving-point impedance of a ladder network constructed by using an infinite number of symmetrical T-network sections with series and parallel impedances (ZJ, Z p } connected in tan- dem as shown in Fig. 4. To estimate the truncation error of using finite k sections for the lumped approximation of

characteristic impedance Z , and the propagation constant 8 are given by the expressions

z,(s) = \/( R + S L ) / ( G + SC) (2.2a)

e ( s ) = ,/(R + S L ) ( G + SC) 1. (2.2b)

'In this aper we are not concerned with the detailed spatial distribution of voiage and current as required by the power distribution line analysis.

*Frequency- and time-domain functions are assigned by uppercase and lowercase letters, respectively, such as (V, I } and (U, i).

Fig. 3. The characteristic model of a uniform E C G transmission line.

1396

I I 1 % I ! I I I J

$&) ;;&:,j-----' f f ' ' jl.,

Fig. 4. A periodic ladder network constructed with symmetrical T-net works.

IEEE TRANSACTIONS O N CIRCUITS A N D SYSTEMS, VOL. 37, NO. 11, N O V E M B E R 1990

Z,, the driving-point impedance function of the ladder network terminating with impedance Z, is derived in Appendix A with the result:

Z ' k ' = Z O ( Q i k + l - l ) / ( Q ~ " + ' + l ) , Z ( o ) = Z s (2.6)

where

z , ( s ) = 4- = & R + S L ) / ( G + S C )

Qo(

(2.7a)

= 1 + ( Z , / Y,) + 4 [ 2 + (2, / Z , I] ( Z, / Z , ) . (2.7b)

Thus the series and shunt impedances Z, and Z, of the T-network can be derived from (2.7a). Equation (2.6) will be used to demonstrate that the approximation of Z , by Z ( k ) is optimum in the sense of Pad6 approximation [14]. First of all, (2.7a) is solved for ( Z , , Z J , leading to the two-element-kind network structures shown in Fig. 5. The symmetrical T-network is an RC or RL network depend- ing on whether the reactive component of Z , is capacitive or inductive. The elements are tabulated below in Table I.

Notice that the ladder network degenerates to a single resistor: Z, = dm for the trivial cases of lossless line ( R = 0, G = 0) and distortionless line ( R C / G L = 1). The fact that the reactive component of 2, is capacitive for the case C > G L / R and inductive for L > R C / G can be verified by expressing Z,(jw) in the radical form:

Z,( jw) = \/( R / G ) ( a k j b )

= Jm [ 4 ( + a ) /2

*;,J(J,+hz- u ) / 2 ]

where

a ( # ) = (1 + w 2 T , d ~ c ) / ( 1 + w ' ~ : ) ,

( w = angular frequency, j = 4 7 ) b ( w ) = ~ ( 7 , ~ - ~ ~ ) / ( 1 + w27:) > 0, for T~ > T~

- b ( w ) = w ( T ~ - T ~ ) / ( ~ + w ~ T ~ ) > ~ , for T ~ > T ~ .

T~ = L / R and T~ = C / G are the time-constants defined by the transmission-line parameters. In the following we

(b) The symmetrical T-network corn onents for synthesis of a lossy

characteristic impedance Z , = Jh. (a) Case 1: capacitive Zo. (b) Case 2: inductive Z , .

Fig. 5.

shall prove that the synthesis of Z , by the ladder network is optimum in the sense of Pad6 approximation [14].

Consider the approximation of the capacitive Z,(S) by the driving-point impedance:

Z'k '= Z , [ ( 1 + X ) 2 k + 1 - ( 1 - X ) 2 k + 1 ] /

[ ( l + ~ ) ~ ~ + + ' + ( 1 - x ) ' ~ + + ' ] (2.8a)

of the ladder network with a resistor terminator 2, = Jw. Equation (2.8a) is derived by substituting into (2.6142.7) the circuit elements for the capacitive Z , given in Table I and

z,, = d r n ( 1 / x ) , x = \/( z + 1) /( PZ + 1) 7

z = l / s r c , p = ~~-7~. (2 Ab) Applying the binomial series expansion and long division, we derive from (2.8a) the rational function and the power-series representations of Z ( k ) ( ~ ) for k = 1,2:

Z ' " ( z ) 1 + ( 1 / 4 ) ( 3 p + l ) z \/L/c - 1 + ( 1 / 4 ) ( p + 3 ) z --

= 1+(1/2) ( p - 1 ) ~ -(1/8)( p -1)( p + 3 ) z 2

+ ( 1 / 3 2 ) ( p - l ) ( p + 3 ) ' z 3 - Z ' 2 ) ( z ) l+(1 /4) (5p +3)z+(1/16)(5p2+10p+l)zz

JL/c 1 + ( 1/4) ( 3 p + 5) Z+ (1/16) ( p2+ 10p+S)z2 -=

= 1+(1/2) ( p - l ) ~ -(1/8)( p -1)( p + 3 ) z 2 + ( 1 / 1 6 ) ( p - 1 ) ( p 2 + 2 p + 5 ) z 3

- (1 /128)(p- l ) (Sp3+9p2 +1Sp+3S)z4

+ (1/512)( p - 1)( 13p4 + 28p3 +30p2

+60p + 12.5)~' - . . . . Using the binomial expansions:

Jl+x = 1 + x /2 - ( 1/8) X' + ( 1 /16) x 3 - (5/128) x 4

+ ( 7 / 2 5 6 ) ~ ~ - ( 2 1 / 1 0 2 4 ) ~ ~ + . . . 1 /J1-tx = 1 - x /2 + ( 3 / 8 ) x ' - (5/16) x 3 + (35/128) x 4

- (63/256) X' + (693/3072) x - . . *

I1 I


TABLE I CIRCUIT ELEMENT^ OF THE SYMMETRIC T-NETWORKS (FIG. 5)

zO C a p a c i t i v e Z Inductive

(~=RC!/GL<I)

the characteristic impedance function (2.8b) is also ex- panded into power series:

Z O ( Z ) / C E

= 4 ( 1 + PZ) / ( l+ 2 )

= 1 +(1/2)( p - 1)z - (1/8)( p - 1)( p +3)z2

+ (1/16)( p - 1)( p 2 + 2 p +5)z3

- (1/128)( p - 1)(5p3 +9p2 + 15p +35)z4

+(1/256)( p-1)(7p4 +Up3 +18p2

coefficients in the power series of Fm,,(z) agree to those of F ( z ) for every term up to and including the zm+" term. Thus we have proven that Z ( k ) ( z ) is the Pad6 (k,k)-th approximant to Z , ( z ) for the capacitive case: C > G L / R . The proof for the inductive case can be carried out in the same manner by rewriting Z,(s) as

which is identical to (2.8) except for the new variable y arid parameter q. The difference between Z k ( z ) and Z ( k ) ( y ) is that they are derived by expanding Z,(s) about s =w and 0, respectively.

Since the ladder-network synthesis of Z,(s ) is optimum in accordance to the Pad6 theory, only a few (two or three) sections of the symmetrical T-networks are needed for the equivalent circuit model of 2,. The truncation error can be calculated from the following expressions:

a) Capacitive Z, , p = 7 c / 7 L > 1

+28p+63)z5- . Z(k)( j w ) / Z , ( j w ) = tanh{( k + 1/2) Thus comparing the infinite series representations of Z ( ' ) ( z ) and Z ( 2 ) ( z ) to Z,(z) , we observe that the coefficients of Z(k) (z ) and Zo(z) are identical to the constant term up to and including the z Z k term for k = 1,2. To prove that it is also true for k > 2, (2.8a) is rewritten as

' c O s h - ' [ ( P + l ) / ( ~ - 1 ) f ~ 2 0 7 c / ( P - 1 ) ] } (2.9a)

b) Inductive z,, P = 7 ~ / 7 ~ < 1

follows: z ( ~ ) ( j w ) / Z,( io)

Z'k'(2) = Z 0 - 2 i r n ( 1 - X y + ' / = tanh{( k + 1/2) .cosh-' [ (1+ p ) / ( 1 - p )

{ x [ ( l + X y k +'+ ( 1 - X ) 2 k ''1). Thus by virtue of the factor (1 - xIzk+' whose first 2 k derivatives vanish at x = 1 when z = 0, we obtain

Therefore, the power series

are identical in the first (2k + 1) terms. A rational function:

a, + a , z + a222 + a 3 z 3 + * . * + a m z m =

1 + b , z + b 2 z 2 + b,z3 + + bnzn

is said to be the Pad6 (m, n)th approximant to F ( z ) E141 if the polynomial Pm(z) and Q,<z> are so chosen that the

which are derived from (2.8a) and its modified form for the inductive Z , ( y ) . An example is given below for illustration.

Example 2.1: Consider a 10-cm long RLCG line with parameters R = 2.5 0 /cm, L = 10 nH/cm, C = 4 pF/cm, and G = 1/2000 (Cl/cm)-'. The lossy line has a capacitive Z , since p = R C / G L = 2 > 1. From Table I, we obtain the circuit elements of the symmetric T-network (Fig. 6(a)) for the ladder-network synthesis of Z,. The time constants of the transmission line are rc = 8, 7L = 4, and 7 = ml = 2 ns. Using (2.9a), the truncation errors of approximating Z, by Z ( k ) in magnitude and phase angle for k = 1,2,3,4 are plotted in Fig. 6(b) and (c). Notice that with four sections of T-networks, Z , is approximated to within 0.01% magnitude error and 0.01" phase shift. The choice of the Pad6 approximation of 2, is justified in view of its accuracy and the explicit result. Other methods require data fitting and time consuming procedure of network synthesis that may lead to circuit structures with nonphysical elements such as negative resistors. Nonphysical elements in circuit models cause severe instability problem in the transient simulation.

1398

50n 50n

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IEEE TRANSACTIONS

-0.1 L FFECEWf (lnz IN Lot10 -I+

(C) Fig. 6. (a) The symmetric T-network for synthesis of Ztk’ and the

truncation errors of approximating 2, by Z ( k ) ( k = 1,2,3,4) (Example 2.1). (b) Percentage error in magnitude. (c) Error in phase angle.

B. Lattice-Network Synthesis of the Exponential Propagation Function

can be synthesized as the voltage transfer function [151: The exponential propagation function of an RLCG-line

T I , = V,/ VI = exp( - 0) = ( R , - Z , ) / ( R, + Z , )

0 = \/( R + sL) (G + s C ) I of a symmetrical lattice network with circuit elements

Z , = R, tanh ( 0 / 2 )

Z b = Rg / Z ,

and a terminating resistor R, = dm as shown in Fig. 7(a). Z , apd z b are synthesized in t_erms of the lossy inductor ( L ) and the leaky capacitor ( C ) :

i= L1\ / (1+1/ST, ) ( l+ l /ST, ) = \ / L / c ( B / S )

d = C 1 \ / ( l + l / S 7 , ) ( 1 + 1 / S T , ) = ) / m ( 6 / S ) .

Thus applying Mittag-Leffler’s expansion formula [161:

z Z + ( a / 2 ) ’ ] + 1 / [ ~ ’ + ( 3 ~ / 2 ) ~ ]

+ + l / [ z Z + ( k + 1 / 2 ) 2 a 2 ] + * * a )

we obtain the Foster network realization of Z , and z b as

ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 11, NOVEMBER 1990

(C) Fig. 7. Lattice network synthesis of the exponential propagation func-

tion exp[ - d( R + sL)(G + sC) I ] by the Mittag-Leffler’s expansion. (a) The lattice network structure. (b) The synthesis of 2,-component. (c) The synthesis of 2,-component.

ikustfated in Fig. 7. Notice that Z , consists of parallel ( L , C ) secfiops connected in series. Since the impedance of such ( L , C ) section decreases rapidly as the result of the decreasing magnitude of t$e lossy inductor (at the rate of 1 / ( 2 k + 1)’)’ very few ( L , C ) sections are needed for the approximation of 2,. As a dual element of 2,’ z b

can also be approximated with very few serial <i ,6) sections connected in parallel.

Applying the Lambert’s continued fraction expansion formula [ 171:

1 1 1 c o t h ~ = - +

2

1 - + 2 5

1 - + 2 7 - + 4

. . . + 1

( 2 k + 1 ) Z

we obtain the Cauer network realization of Z , and as shown in Fig. 8. Notice that for the trivial case of lossless transmission line, the Cauer network structure degenerates into the Bessel filter [18] for the realization of the ideal delay function exp( - s m 0 .

The synthesis of the lossy inductor and the leaky capacitor in terms of lumped elements is described in the next section.

C. Pad6 Synthesis of Lossy Inductor and Leaky Capacitor The lossy inductor (i) and the leaky capacitor (6) are

frequency-dependent elements that differ from the ideal elements L and C by the modulation function: @(s) = \/( 1 + l / s ~ , ) ( 1 + l / s ~ , ) . i and d are synthesized as the input-impedance of the ladder networks constructed with T-networks and pi-networks as shown in Fig. 9. By

I1 ,


(C)

Fig. 8. Lattice network synthesis of the exponential propagation function exp[ - ( R + sL)(G + sC) I ] by the Lambert's partial fraction expansion. & The lattice network structure. (b) The synthesis of 2,-component. (c) The synthesis of 2,-component.

n

SRP I

(a)

I I I 1 I I I I L I I

CP gj2GP G

(b) Fig. 9. Synthesis of (a) lossy inductor i, (b) leaky capacitor 6.

identifying the characteristic impedance (admittance) of a symmetrical T(pi)-network as the impedance (admittance) of a lossy inductor (leaky capacitor):

4- = SLld( 1 + 1/S7,)( 1 + l/STL)

we obtain the network elements as tabulated in Tables I1 and 111.

I+

hS Fig. 10. An ECL gate driving a lossy transmission line.

TABLE I1 ELEMENTS OF THE T-NETWORK FOR THE SYNTHESIS OF L

Inductive Z, Capacitive Z,

TABLE I11 ELEMENTS OF THE PI-NETWORK FOR SYNTHESIS OF C

Y =G s s

The approximation of t and d by the ladder networks described above is optimum in accordance to Padt's theory. Thus a lossy inductor (leaky capacitor) can be accu- rately modeled by using only one half-section of T(pi)- network consisting of one inductor (capacitor) and two resistors. This is illustrated in the following example.

Example 2.2: The lossy transmission line of Example 2.1 is terminated with a 100-kR load resistor, and its near-end terminal is connected to the in-phase output of an emitter-coupled-logic (ECL) gate as shown in Fig. 10. The ECL gate is driven by a step-voltage generator with a 0.5-ns rise time and a kO.4 V voltage swing. Waveforms in Fig. 11 are obtained by discrete-time simulation using the disjoint two-port network of Fig. 3 for modeling the lossy line. As described in Example 2.1, four T-network sections are adequate for a near perfect simulation of the lossy characteristic impedance Z,. The waveform generators (El, E,) of the lossy line model are simulated by using thejattice networks of Figs. 7 and 8. Each lossy inductor L (and leaky capacitor C) for constructing the lattice-network elements ( Z a , Z,) is synthesized by using one half-section of T(pi)-network. As clearly illustrated in Fig. 11, the accuracy of the simulation results improves as more and more terms (and more and more L and C

1400 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 11, NOVEMBER 1990

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exponential propagation [unction. (a) Near-end waveforms. (b) Far-end waveforms. Fig. 11. Discrete-time simulation of a loss transmission line using Mittag-Leffler and Lambert’s expansions of the

elements) are used in the expansion of the exponential propagation function. Compared in Tables IV and V are the accuracy, the CPU time, and the computer memory required by using the Mittag-Leffler’s and the Lambert’s expansions of the exponential propagation function.

The root-mean-square (rms) errors of the waveforms obtained by the generalized method of characteristics tabulated above are calculated with reference to the waveforms obtained by using the classical method of characteristics. It is assumed that the waveform u ( t ) obtained from the classical method of characteristics with very small time-steps is exact and the rms difference

between the waveforms obtained by the generalized and the classical method of characteristics:

is defined as the rms error of the 6( t ) obtained from the generalized method of characteristics. Waveforms in Fig. 1 are obtained by applying the classical method of characteristics, which is formulated only for the analysis of lossless lines. Thus in applying the classical method of characteristics, the lossy line is modeled by 40 segments of

1401

I of RE18 error(mv) cpu time

terms near-end far-end (second)

1 24.63 58.05 28.2

3 8.13 13.70 120.4

5

7

7.81 199.3

5.43 247.8

4.08

2.46

CHANG: WAVEFORM RELAXATION ANALYSIS

memory (kbyte)

301

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335

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TABLE IV RMS ERROR, CPU TIME, AND COMPUTER MEMORY

REQUIREMENT VERSUS NUMBER OF TERMS BY MITTAG-LEFFLER'S EXPANSION

RE18 error(rnv) CPU time memory

terns near-end far-end (second) (kbyte)

20.27

333

347

5.14 14.54 124.7

11.54 149.4 4.24

(b)

Fig. 11. Conrinued.

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 31, NO. 11, NOVEMBER 1990

lossless lines, and the conductor loss and the dielectric leakage are simulated by resistors connecting between adjacent lossless-line segments and resistors connecting between ground and the lossless-line terminals. Such a brute-force approach requires excessive computer memory (1 Mbyte) for simulating the distributed nature of the conductor loss and dielectric leakage. Furthermore, the computer simulation time is extremely long (485 s CPU time on an IBM 3090) since the integration time-step is kept smaller than the delay time (0.05 ns) of the individ- ual lossless-line segments for the entire simulation process. Otherwise, incorrect results characterized with such false ringing as shown in Fig. 1 are produced even though 40 sections of lossless lines cascaded with as many lumped resistors are used in the simulations. From the simulation results shown in Fig. 11 and Tables IV and V, we observe that the generalized method of characteristics for the discrete-time simulation offers at least a factor of two reduction of CPU time and memory required by the classical method of characteristics. Lambert's expansion of the exponential propagation function used in the generalized method of characteristics provides more accurate a result than that of the Mittag-Leffler's expansion but also takes a longer computer simulation time.

The voltage transfer function for simulating the time- delay and wave-shape distortion can also be synthesized by applying the Bode's asymptotic expansion method of locating the poles and zeros of the exponential propagation function described in [ l l ] and [12]. The lattice-network synthesis of the exponential propagation function described above is explicit with network structures shown in Figs. 7-9 and circuit elements tabulated in Tables I1 and 111. The difficulties of traveling wave modeling in the discrete-time simulation can be avoided by using the FFT in conjunction with the waveform relaxation technique described in the next section.

111. WAVEFORM RELAXATION ANALYSIS OF RLCG TRANSMISSION LINES

Large scale integrated (LSI) circuits are interconnected with metal strips of finite conductivity and extremely small cross-sectional areas. Such metal interconnections are lossy transmission lines, which are often ignored in circuit simulation since most circuit simulators are pro- grammed solely for the analysis of lumped-element circuits and discrete-time simulation of LSI circuit is extremely time consuming. Ironically, including lossy transmission lines in the LSI circuit simulation can actually reduce the simulation cost in terms of both CPU time and computer memory requirement. The cost reduction is accomplished by partitioning the LSI circuits into many subcircuits, which can be simulated more efficiently by the iterative waveform relaxation technique [9]. By modeling the metal interconnections with the disjoint 2-port network (Fig. 3), the partitioning process is simplified since each metal interconnection becomes a natural boundary for the system decomposition.

- - Fig. 12. A lossy transmission line with Thevenin's terminations.

The idea of simulating a transmission line by iteration originates from the observation that the voltage (current) waveform at each transmission-line terminal is composed of an infinite sequence of incident and reflected waves unless the transmission line is lossless and terminated in its characteristic impedance. To be more specific, let us assume that the lossy transmission line is terminated into the Thevenin's sources (EA, E,) and impedances (ZA, 2,) as shown in Fig. 12. Thus, substituting the terminal conditions: (VI = EA - ZAI,, V2 = E, + z B I 2 } into (2.1) for eliminating I , and 12, we obtain

1+ZO/ZA - ( - zO / z J 3 > ( - [ - ( l - zO / z A ) exp ( - e)

.[;I = [ ( zO / z A ) exp ( - e)

l+ZO/Z,

ZO /ZA ( zO / z B ) exp ( - e) ] [ 21 ZO / Z B

which can be solved for the terminal voltages leading to the canonical expressions:

Vi = 4 / ( I - Q ) = uA + QUA 4- Q2uA 4- Q3U, + * *

+ QkUA + . . . (3.la)

V2 = UB /( 1 - Q ) = U, + QUE + Q2U, + @U, + . . .

+ QkUB + . * (3.lb)

( 3 . 1 ~ ) where the geometric progressive factor

Q = PAP, ~ X P ( -28) and the equivalent voltage sources

= ( 1/2) { ( - PA) [ + P E exp ( - 2e> 1 EA

+ ( 1 + PA ( 1 - P E ) ~ X P ( - 0) E B I (3.ld)

U, = (1/2) { (1 - PA) ( 1 + P E ) exp ( - EA + [ 1+ PA exp ( -2e)] ( l - P E ) (3.1e)

are defined in terms of the reflection coefficients PA= ( 'A- ' O ) / ( ' A + ' 0 ) (3.lf)

P E = ('E - z O ) / ( z B + ' 0 ) . (3.lg) Referring to Fig. 13(a) and (b), the iteration algorithm

for generating the sequence of incident and reflected waves of (3.la, b) using a circuit simulator is described as follows.

CHANG: WAVEFORM RELkYATlON ANALYSIS 1403

I t I

PORT 1

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- I 1

PORT 2

(b) Fig. 13. The transmission-line model for iterative waveform relaxation

analysis (Gauss-Seidel type). (a) Step 1-top: Circuit simulation of port-1 network; bottom: Generation of new waveform Z i k ) ( r ) . (b) Step 2-top: Circuit simulation of port-2 network; bottom: Generation of new waveform ~ ' , ~ ) ( r ) .

Waveform Relaxation Algorithm 3. I (Gauss - Seidel Type):

Step 0: Initialize the iteration counter (k = 1) and the waveform generator: eio)(t) = E,,u(t) where E,, is obtained from the dc analysis described in Appendix B and u ( t ) is a unit-step function.

Step I: Connect the terminating network ( E A , Z,) and the voltage generator {ejk- ' ) ( t ) ) to port 1 (Fig. 13(a)) of the disjoint two-port network and carry out the circuit simulation for the entire time interval. In terms of the new terminal voltage waveform (v\k)(t)} and the old voltage generator waveform {e jk- ' ) ( t ) } , generate the new waveform {Zik)(t 1) by simulating the exponential wave propagation function exp( - e). Store both new waveforms (v\k)(t)) , ( Z $ k ) ( t ) ) , 0 < r < T .

Step 2: Connect the terminating network (€ , ,Z , ) to port 2 (Fig. 13(b)) of the disjoint 2-port network, use the new waveform {e'ik)(t)) obtained in Step 1 for the voltage generator (e ik)( t>) , and carry out the circuit simulation for

the entire time interval. In terms of the new terminal voltage waveform { vik)(t 1) and the new voltage generator waveform {ef')( t ) } , the voltage generator waveform {e' ik)( t ) ) is updated through the simulation of the exponential propagation function. Store both new waveforms { u i k ) ( t ) ) , (Z ik ) ( t ) ) , 0 < t < T for next iteration.

Step 3: Stop the iteration if the difference between the new and the old terminal voltage waveforms { v i k ) - v ik- ' ) ) , { v i k ) ( t ) - v ik- 'Yt)) is sufficiently small. Other- wise, set k = k + 1, (e$k- l ) ( t ) ) =(Z\k- l ) ( t ) ) , 0 < t < T , go to Step 1, and repeat the iteration process.

Theorem 3.1 -Convergence Theorem of Waveform Re- laxation Algorithm 3.1: For an E C G transmission line with Thevenin's terminations, the Waveform Relaxation Algorithm 3.1 generates the sequences of waveforms (v\")(t)), {v$k)( t ) ) , which converge to the exact waveforms given in (3.la, b) if IpApB exp( - 2e)l < 1.

Proofi Referring to Fig. 13, the kth iteration step is transformed into the following system of difference equations in the s-domain':

Vjk'= (1/2) [( 1- P A ) E A + ( 1 + P A ) E I k - ' ) ]

~ $ k ) = [exp(-e)][21/,'k)- E I ~ - * ) ] v p = (1/2) [( 1 - P B ) E B + (1 + p , ) E $ k ) ]

E ! ~ ) = [exp( - e ) ] [2v jk ) - E $ ~ ) ]

which yield the following first-order difference equations:

v{k+l) = U, + QV(k) (3.2a)

V$k+ 1) = U + QV(k) 2 (3.2b)

after { E l k - ' ) , E lk ) , E lk ) } are eliminated. In (3.2), {Q,U,,U,) are exactly as defined in (3.lc, d, e). Thus, starting with the initial solution:

'l(') = ( 1/2)(1 - P A ) EA + ( 1/2)( + P A ) / s

we obtain from (3.2a) the following sequence of waveforms:

Vj2) = U, + QV{"

V{3) = U, + Q( U, + QV{") = U, + QUA + Q2Vl("

V t k ) = (1 + Q + Q' + Q3 + * * . + Q"-')U, + Qk-'V{')

= UA/( l - Q ) + Qk- ' [ V,C')- U, /( 1 - Q ) ]

which is identical to (3.la) except for the residue term: Qk-'[ Vi1) - U, /(1- Q)] related to the difference between the initial and exact waveforms. Thus, for I Q 1 = IpAp,exp(-2B)I < 1, the residue term vanishes as k approaches infinity and the sequence of waveforms converges to the exact waveform {v,(t)} defined by (3.la) in the s-domain. Similarly, ( v ik ) ( t ) } converges to {v2(t )I.

Algorithm 3.1 is categorized as the Gauss-Seidel (GS) type of iteration [9] since the terminal voltages at the opposite ends of the transmission line are solved sequentially. By simultaneously solving the terminal voltages at

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t

I 1

IEFE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 11, NOVEMBER 1990

t I

PORT 1 P O R T 2 (a)

(b) Fig. 14. The transmission-line model for iterative waveform relaxation

analysis (Gauss-Jacobi type). (a) Simultaneous simulation of port-1 and port-2 networks. (b) Simultaneous generation of new waveforms t?ik)(t) and P l k ) ( t ) .

both ends of the transmission line, we have the Gauss-Jacobi (GJ) type of iteration, which is described next with reference to the equivalent circuit shown in Fig. 14.

Waceform Relaxation Algorithm 3.2 (Gauss - Jacobi

Step 0: Initialize the iteration counter ( k = 1) and the Type):

waveform generators:

e$')( t ) = ~ , , u ( t ) , e\')( t ) = ~ , , u ( t )

where E,, and E,, are obtained form the dc analysis described in Appendix B.

Step 1: Connect terminating networks ( E A , Z,), (E,, Z,) and the waveform generators { e { k - l ) ( t > ) , {e\k-')(t)) to the disjoint two-port network (Fig. 14) and carry out the circuit simulation for the entire time interval. In terms of the new terminal voltage waveforms { L i i k ) ( t ) ) and ( c i k ) ( t ) ) and the voltage transfer function exp( - e), new waveforms are generated for the waveform generators ( E l k ) ( t ) ) , ( E i k ) ( t ) ) . Store the new terminal and generator waveforms.

Step 2: Stop the iteration if the difference between the new and the old terminal voltage waveforms ( ~ $ 1 ~ ) -

L . $ ~ - ~ ) ) , ( c i k ' ( t ) - ~ : ~ - ' ) ( t ) ) is sufficiently small. Other- wise, set k = k + 1, { e ( k - l ) ( t ) = E l k - ' ) ( t > L

{ e ( k - ' ) ( t ) = 2 i k - I ) ( t ) L O < t < T and go to Step 1, and repeat the iteration process.

Theorem 3.2 - Conr>ergence Theorem of Waveform Re- laxation Algorithm 3.2: For an E C G transmission line with Thevenin's terminations, Algorithm 3.2 generates the sequences of waveforms {L$ik)(t)), { c i k ) ( t ) ) , which converge to the exact waveforms given in (3.1(a), (b)) if

Proof: Referring to Fig. 14 and following the same procedure used for proving Theorem 3.1, we obtain the following second-order difference equations:

V / k + 2 ) = U + QV(k) (3.3a) V$k+2) = U + QV(k) (3.3b)

with parameters {Q, U,, U,) exactly as defined in ( 3 . 1 ~ ~ d, e). Thus starting with the initial solution:

lPAPBexp(-28)1 < 1.

Vi1) = (1/2)( 1 - PA 1 E A + (1/2)( 1 + P A ) El0 /s

VJ') = (1/2) ( 1 - P S I E, + ( 1/21 ( 1 + P B ) Em /S we obtain from (3.3) the following sequence of waveforms:

V ~ 2 k ~ ' ) = U A / ( 1 - Q ) + Q " - ' [ ~ ~ ' ) - U A / ( 1 - Q ) ]

V$2k)= U , / ( 1 - Q ) + Q k - ' [ Vj2) - U,/'( 1 - Q ) ] , k = 2 , 3 , 4 . . .

which is identical to (3.la) except for the residue terms Qk- I [V i1 ) - U' /(1- @I, Q'-'[V$') - UB/(l- Q)l related to the difference between the initial and exact waveforms. Thus for IQ1 = lpApB exp(-28)1 < 1, the residue terms vanish as k approaches infinity and the sequences of waveforms converge to the exact waveforms defined by (3.1(a), (b)) in the s-domain.

Corollary 3. I : The alternate sequence of waveforms (V /2k -1 ) , VjZk)) generated by the Gauss-Jacobi iteration is identical to the sequence of waveforms {Vi",, V$"} generated by the GS iteration. Thus the GJ iteration converges twice as slow as the GS iteration unless parallel proces- sors are used to implement the GJ algorithm.

Corollary 3.2: For a lossy transmission line with re- laxed initial condition and passive far-end termination, the GJ and the GS iterations generate the following sequences of identical waveforms:

V/2"'< GJ) = V : 2 k - 1) (GJ) = V{"( GS) (GJ) = Vjk'( GS) . ~ $ 2 k + I ) ( GJ) = ~ $ 2 k )

Corollaries 3.1 and 3.2 can be easily deduced from Theorems 3.1 and 3.2. The proof is omitted here for brevity.

The waveform relaxation algorithms described above can be implemented on any existing circuit simulator since the lossy characteristic impedance and the exponential propagation function of the lossy transmission lines are both synthesized by lumped equivalent circuits. With

I1 ,

CHANG: WAVEFORM KtLAXATION ANALYSIS 1405

the addition of the FFT, a conventional circuit simulator can be converted into an extremely efficient tool for the waveform relaxation simulation of LSI circuits interconnected with lossy transmission lines. Here, we take advantage of the extreme efficiency of the FFT for deriving the waveforms of the voltage generators:

GS iteration:

(3.4a)

eik)( t ) = F - ' { [exp ( -

GJ iteration:

elk)( t ) = F'{ [exp

e i k ) ( t ) = ~ - ' { [ e x p

e ) ] F [ 2 ~ $ ~ ' ( t ) - e$&'( t ) ] } (3.4b)

where { F , F - ' ) denote the Fourier and the inverse Fourier transforms. Unlike the discrete-time analysis which requires invoking the FFT routine at every time-step, the waveform relaxation analysis calls for the FFT routine after the completion of each iterative circuit simulation and therefore reduces the overall computer simulation time. Replacing the lattice-network simulation of the exponential propagation function by the FFT, the computer memory requirement is also drastically reduced.

In (3.4) and ( 3 3 , the voltage generator waveforms are computed at periodic time intervals dictated by the FFT algorithm whereas terminal voltage waveforms are often computed at nonuniform intervals since the numerical integration of circuit equations with nonuniform step sizes is more efficient. Thus data interpolation is required. Calculation of Fourier spectrum of linearly interpolated waveforms using the FFT is described in Appendix C. It is shown that the Fourier spectrum of linearly interpolated waveforms decays inversely proportional to the square of the frequency if the waveforms are sufficiently smooth. Thus the almost band-limited characteristics of interpolated waveforms can be used for reducing the spectrum aliasing effect that causes noise in the waveform transformation process. The time-domain aliasing effect [ 191 caused by converting the transient waveforms into periodic waveforms in applying the FFT for computing the convolution integrals ((3.41, (3.5)) is also minimized by padding at the end of the transient waveforms with "zero-amplitude'' waveforms and the repetition period of the augmented periodic waveforms is at least twice the time duration of the original transient waveforms. It is observed that the accuracy of the waveform relaxation analysis is also improved by such a time-window enlargement since more low-frequency spectra are included in the construction of the time-domain waveforms as a result of spectra sampling at a lower fundamental frequency (f, = 1/ T ' , T' > 2T). An example is given below.

Example 3.1: We repeat Example 2.2 by applying the waveform relaxation technique. To make the example more interesting, the per-unit-length resistance of the lossy line is reduced eight times such that the lossy-line characteristic impedance becomes inductive and is approximated by 4 T-network sections (Fig. 5(b)) with parameters R,=25 0, L,=300 nH, and Gp=1/37.5 R calculated using Table I. Compared in Fig. 15 are two sequences of terminal voltage waveforms generated by applying the GS iterations initialized with e\') = 0 and e',') = Elou( t ) , which is derived from the initial condition as described in Appendix B. The two sequences of iterative waveforms start with different initial conditions. But, at the end, they all converge to the "exact waveforms." Combined with the waveform relaxation analyses, a time- window enlargement factor of 2 and 1024 samples are used in the FFT simulation of the exponential wave propagation function of the lossy line. Seven FFT-routine calls consume only 3.6% of the total CPU time for com- pleting four iterative waveform relaxation analyses. Dis- crete-time analysis using the classical method of characteristics has also been carried out by modelling the lossy line with 40 sections of lossless lines connected with 120 lumped resistors for simulating the distributed nature of the conductor loss and the dielectric leakage. Extremely small time-steps (maximum stepsize = 0.02 ns) are used in the discrete-time simulation to eliminate the false-ringing phenomenon. In comparison to the classical method of characteristics, the waveform relaxation analysis consumes only one-tenth the CPU time and uses half the computer memory. The waveform relaxation simulation time can be further reduced by exploiting waveform latency. Referring to Fig. 15, we observe that by initializing the iteration with E,,u(t) for matching the initial condition, subsequent circuit simulations need not be started at the time origin. For this simple example, a 20% reduction of CPU time is achieved by exploiting the waveform latency.

The advantage of using the waveform relaxation analysis escalates as the circuits become more and more complex as illustrated in the following example.

Example 3.2: Consider the transient simulation of a bipolar logic system consisting of three identical ECL gates interconnected with three metal conductors of identical length ( I = 10 cm) as shown in Fig. 16(a). Using the disjoint two-port models of the metal conductors as the natural boundaries for system decomposition, we obtain four subcircuits as shown in Fig. 16(b). In the subcircuits, the characteristic impedances (Z,,, Z,,, Z,, being inductive, capacitive, and resistive) are synthesized by the lumped-element circuits of Fig. 16(c) with circuit elements calculated by substituting the transmission-line parameters:

(R,) = {5/16,5/2,5/4)R/cm { L,} = 10 nH/cm { C,) = 4pF/cm

{ G,} = 1/2000( R .cm) ~ ', m = 1,2,3

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-0 2 -* - 0 . 2 -


- 3

o . 6 [ .

0 . 6

0 . 4

0 . 2

0.0

...... :ITERATION NUMEER I

0 . 6 - -

...e*.~:ITERF)TION NUMEER 4 ***....:ITERATION NUMEER 4

0 . 4 - -

0 . 2 - -

1 ) I I l I ' I ' I " 1 , ' ' a a

c O . ' l . *.....'.."..' c'-\cr *"..*.*."......."... ...... ....................................... '

I O 15

- 0 . 2 - 0 . 2 -

0.0

-

0.4ii"\ ...... ............................................. I - ........ .....*

0 . 2

***.***:ITERATION NUMEER I @ ' 6 1 o , z t ."*.**.*"....... .............................. ......" ......." .......

******.:ITERATION NUMEER Z

O'L F ..............................................

0 . 4 - ...... I c

15 2 0 " I " " '

- 0 . 2 -

*......:ITERATION NUMEER 3

o , 6 F ****...:ITERATION NUUEER 3

o ' 6 F

(a) Fig. 15. Iterative waveform simulation of a lossy transmission line initialized with e',"'(?) = 0 (left sequence of waveforms)

and e{"(t) = El ,u( t ) (right sequence of waveforms) (Example 3.1). (a) Near-end terminal voltage waveforms.

. ,


e.e.r:ITERATION NUMBER 1

0.6

...... I ...... .. ............... " ...... " ...... I ................

0 . 6

0 . 4 1

0.2

..... I..... f-1 \. .............. T- I,......I....,.~......-.....*.-....... W I

O ' Y .....:ITERATION NUMBER 3

..... :ITERfiTION NUMBER 4

U 8 -

O I + * ~ : I T E R A T I O N NUMBER I

0 . b -

0 . 4 -

0 . 2 -

I 6 I ~ ' " ' ' ' ' 15 20 10 0.0 " " "

-..+-:ITERATION NUMBER 2 :::I 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 . 2

Fig. 15. (Continued) Iterative waveform of a lossy transmission line initialized with e(')(t) = 0 (left sequence of waveforms) and elo)(t) = E,&) (right sequence of waveforms) (Example 3.1). (b) Far-end terminal voltage waveforms.

1408 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 11, NOVEMBER 1990

1

I 1 II I I

253 50 50 50n 50

Fig. 16. (a) A bipolar logic system interconnected with lossy lines, (b) its partitioned subcircuits, and (c) the equivalent circuits of the lossy transmission line characteristic impedances.

into Table I. The initial values {Em,uo, Em,bO, m = 1,2,3) of the voltage generators (eg,;’)( t ) , eg,)b(t), m = 1,2,3) are obtained by the dc-analysis of the subcircuits (Fig. 1fXb)) following the iterative procedure described below.

Iterative Solution of = emJt = o), Em,b0 =

Step 0: Initialize the iteration counter ( k = 1 ) and set eZ,;’)(t)=O, m = 1 , 2 , 3 .

Step I : Perform dc-analysis of subcircuit # 1 and obtain the near-end terminal voltage u, , ( t ) ( t = 0) of the first transmission line. Update the voltage generator:

em, b (t = 0)):

( t = 0 ) .

Step 2: Perform dc-analysis of subcircuit #2 and obtain the terminal voltages u&),v2,(t) ( t = 0) of the first and second transmission lines. Update the voltage generators:

( t = O ) .

Step 3: Perform dc-analysis of subcircuit #3 and obtain the terminal voltages v&), u3,(t) ( t = 0) of the second and third transmission lines. Update the voltage generators:

= [ e x p ( - 4 m 1 ) ] [ 2 u 2 b ( t ) - e $ i ’ ( f ) ]

( t = O ) .

Step 4: Perform dc-analysis of subcircuit #4 and obtain the far-end terminal voltage U36 ( t = 0) of the fourth

transmission line. Update the voltage generator:

e$:’(t) = [ e x p ( - d v l ) ] [ 2 u 3 b ( t ) - e $ k , - l ’ ( t ) ] 9

t = 0. Step 5: Stop the iteration if the difference between

successive iteration solutions: (e(k) ( t ) - e ( k - 1 ) m , b (‘)I, t ) - m , u ( N , m , b

( u ( k ) ( t ) - m.0 m , u ( t ) } , (ug,nk,b(t) - v $ , i ’ ) ( t ) } ,

( t = O ) is sufficiently small. Otherwise, set k = k + 1 and go to Step 2 and repeat the iteration process.

After five iterations, the dc solution (Egluo} = (0.2873, -0.1875, -0.2129)

=(Em,uO) =(0.2858, -0.1875, -0.2129) (EE!bO} = (0.3276, -0.2670, -0.2736)

(Em,bO} = (0.3260, -0.2670, -0.2736) converges to within 0.5% of the exact solution, which is obtained from the dc analysis of the original logic system, the lossy transmission lines of which are replaced by the symmetrical, resistive T-networks as described in Ap- pendix B. The waveform relaxation analysis of the logic system can now proceed with the initial step-voltage generators:

(e!:,’( t ) 9 e$?( t ) 9 e$:)( t ) }

= { El, laU(t) 9 E,, luu(t) 9 E3,laU( The iterations follow exactly the procedure as described for the iterative dc-analysis except that in each iteration step, the FFT routine is called at the completion of each transient simulation for updating the waveforms of the voltage generators as defined in (3.4). The voltage waveforms of the logic-circuit outputs that are not connected to transmission lines converge in at most two iterations as shown in Fig. 17. The voltage waveforms at the transmission-line terminals converge to the “exact waveforms” in four iterations, as shown in Fig. 18. The “exact wave-

' I ,

0 , -

0 2 -

0 0

- 0 2 -

T I C

CHANG: WAVEFORM RELAXATION ANALYSIS

I

~ ~ ~ ~ ~ e ~ : I T E F S T I ( E I N M € R 1 I . ' . & ' &

'

1409

0 . 4

0 . 2

0.0

- 0 . 2

o.ob I 1 1 1 1

2 0 SO 4 0 10

0 . 6

0 . 4

0 .2 . . . . . . e :ITERATION NUMBER 2

I I I I 30 4 0 IO 2 0

INDUCTIVE 20

INDUCTIVE 20

... ....................... - o ' 2 t

INDUCTIVE 20

Fig. 18. The iterative terminal voltage waveforms of (a) the lossy line with an inductive Z , , (b) the lossy line with a capacitive Z, , and (c) the lossy line with a resistive Z o (Example 3.2).

1410

0 . 2

0 . 0

-0.2

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 37, NO. 1 1 , NOVEMBER 1990

.yly, --8

0 . 2 - - * * * . * ' . : I T E R A T I O N NUMBER 4 * ' . . * . . : I T E R F I T I O N NUMBER 2

1 I up- 0 . 0 I I I I IO 2 0 30 4 0 1 1 I0 2 0 3 0 4 0

-

.ll..l...P......_OD....I ....... .. ....... - 0 . 2 - -

_^___.. .....

0 . 4 O I

0 . 4

0 . 2

0 . 0

- 0 . 2

INDUCTIVE 20

0 . 4 - - ..................................................... ........ .......

e . e . . . * . I T E R A T I O N NUMBER 2 *e.......... ......l...... 0.2-

2 0 30 ,b 0.0 l i

"......"......_ -

D - . . * . * : I T E R A T I O N NUUBER I

I I I I I 10 20 30 4 0

I 1 I O

- 0 . 2 - -

ryyl k--

0 . 4

I N D U C T I V E 20


(a)


C F I P A C I T I V E 20

(b) Fig. 1 8 . Continued.

forms" are obtained by the discrete-time simulation using the classical method of characteristics with very small time-steps (0.02 ns), and each of the three lossy lines is modeled by using 40 sections of lossless lines and 120 lumped resistors. Compared in Table VI are the CPU times and the computer memory required by using the classical and generalized method of characteristics.

Notice that in comparing to the classical method of characteristics, the generalized method of characteristics consumes 14 times less CPU time and uses less than

C F I P F I C I T I V E 20

one-half of main memory. The CPU times can be further reduced by taking advantage of waveform latency as observed from the waveforms of Fig. 18. In using the interactive circuit design program [lo] for the iterative waveform relaxation simulation, each subcircuit is stored in the secondary memory called " workspace." Sequentially, each subcircuit is copied into the main memory (in which the simulation program occupies 156 kbytes) for transient analysis. The storage requirement increases as more and more waveforms are generated and stored for later hard


0 . 6

0 . 4

0 . 2

0.0

- 0 . 2

o . 2 t I *..**.*:ITERATION NUMBER 3

- 0 . 6 -

- 0 . 4 - ................................................

**-:ITERATION w n s m 3 - 0 . 2 - .... ....... :ITERATION NUMBER 4

I I I I I 1 I 10 2 0 30 4; 0.011 10 2 0 30 4 0

- - 0 . 2 -

lyyI Lyyyy)

*****..:ITERATION HUMBER 2

o ' 2 t I

0 . 4

0 . 2

0.0

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-

.r~oeeo:ITERATION NUMBER 4 -

I I I I 10 20 30 4 0

- Lyyyy)

I I 1 I I IO 2 0 so . 4 0

0 . 0

- 0 . 2 u

CAPACITIVE 20 CAPACITIVE 20

.....,_ ".......I...... 1 ... .... *.-. ............ ....... _, 0 . 4

*...**.:ITERATION NUMBER I o . 2 t I

- 0 . 2 O.Olf-----

CAPACITIVE 20

0 . 6

CAPACITIVE 20

(b) Fig. 18. Continued

CRPRCITIVE 20

copy outputs. For routine simulation, only waveforms of two consecutive iterations need to be stored for comparison and the memory requirement is reduced.

IV. CONCLUSIONS The method of characteristics [ 13 has been successfully

extended for the transient analysis of transmission lines with constant RLCG parameters. To circumvent the diffi- culty of a direct time-domain formulation that led to inconsistent solutions [3]-[5], a rigorous network theoretic basis for the method of characteristics has been estab-

lished from the frequency-domain characterization of the RLCG transmission lines. Thus the irrational transmission-line characteristic impedance function has been synthesized with two-element-kind (RC, RL) ladder networks, and the exponential propagation function has been simulated as the voltage transfer function of a lattice network. It is shown that such equivalent ladder and lattice networks can be used for constructing a disjoint 2-port characteristic model for both discrete-time and iterative waveform relaxation analyses of RLLG lines.

Transmission lines are often characterized in the frequency domain in terms of scattering parameters, inser-

1412

1 1 1


0 b

0.q

0 . 2

0.0

-0.2

0 . b

0 . 4

0.2

0 . c

- 0 . i

"."_."."".".I".".""."."".-."-."."-."."".".""." L-- *.. .*** : l T E R a T l O N NUMBER I

I I I I IO 20 30 4 0

R E S I S T I V E 20

"."."".".""."."".".""."."".".""."."".- * * . . * . . : I T E R a T I O N NUnBER 2

1 I I I 2 0 30 4 0 IO

t

C


I 4 0


(C)

Fig. 18. Continued.



Generalized method of

characteristics(with FFT)

Iter. cpu(second6) memory

# total FFT (kbytes)

1 31.12 1.62 369.6

2 32.25 1.78 373.7

3 34.61 1.59 377.9

4 34.53 1.56 381.2

total 132.51 6.55 max.=381.2

~-~ -. Classical method of

characteristics

cpu=1861.57 seconds

rnemory=993.5 kbytes

tion loss, open-circuit impedance, and short-circuit admittance parameters. Thus the Fourier transform is often used for transformation of waveforms between time and frequency domains. In particular, the FFT is convenient for the computation of convolution integrals for simulating the exponential propagation function of an RLCG line. However, discrete-time simulation of transmission lines using FFT is extremely time consuming since the FFT computation routine is repeatedly called for at every time-step during the entire period of transient simulation. Such an inefficient use of FFT is avoided in the waveform relaxation analyses since the FFT routine is called for only at the completion of each iterative circuit simulation. Thus, combining the waveform relaxation analysis with the FFT provides the most efficient method for the transient analysis of VLSI circuits interconnected with lossy transmission lines. The generalized method of characteristics combined with FFT has been used for the waveform relaxation analyses of circuits far more complex than the examples described in this paper. A two-order reduction in CPU time and an order saving in computer memory have been achieved even for circuits with moderate com- plexity, such as a simple all-bipolar %bit adder interconnected with lossy lines.

As integrated-circuit designers strive for higher circuit density and faster logic switching speed, skin-effect becomes a dominating factor that causes degradation of signal transmission through the VLSI interconnects. Thus frequency-dependent parameters in the discrete-time transmission-line analysis has been reported in literature 1111. r121. In SeDarate articles we shall describe the wave-

APPENDIX A THE DRIVING-POINT IMPEDANCE OF A PERIODIC

LADDER NETWORK Applying the Kirchhoff law at the terminals of a sym-

metrical T-network, we obtain the chain matrix:

1 A B l+Z,Y, Z,(2+ZSY,)

[ c .I=[ 5 1 + z,r, where Z , and Y, ( = l / Z , ) are the series impedance and the shunt admittance of the symmetrical T-network shown in Fig. 4. The chain matrix can be diagonalized:

I;" : ] = [ Y , i H -Y,/H

0 l + Z , Y , - H

' [ Y , j H -Y,/H 1' where H = ~ ( l + Z s Y p ) * - 1 .

Thus raising the diagonal form of the chain matrix to the k th power, we obtain the overall chain matrix:

for the terminal characterization:

- _ . - form relaxation analysis of nonuniform lossy transmission lines characterized with frequency-dependent RLCG parameters [71, [81.

of a ladder network constructed with k sections of such symmetrical T-networks.

Eliminating I, from (A1.2) and using the terminating condition I, = V, / Z 5 , we obtain

- - ( V , /2) [ ( 1 + H/Z,Y,)Q: + ( 1 - H / Z y k ; , ) Q i k ]

z k = V 1 / z l J 1 ~ = v ~ ~ z ~ ( v 2 / 2 ) ( Y , / H ) [ ( l c H/Z,Y,)Q,k - ( I - H / Z J Y P ) Q i k ]

= Z,(Qikt ' - l ) / ( Q i k + ' + 1 )

1414

where

Z , = H / Y, = Jm Q , = I + Z,Y, + H = 1 + ( z , / z , )

+ 4 [ 2 + (z, / z p ) ] ( z , and use has been made of the relation

(1+ H / Z , Y , ) / ( l - H / Z , Y p ) = - Q o .

APPENDIX B DETERMINING THE INITIAL VALUES OF THE

SATISFYING THE INITIAL BOUNDARY CONDITIONS For dc analysis, an RLCG transmission line can be

replaced by a single symmetric T-network section of Fig. 4 with resistive elements Z , = \/R/G tanh &?l/2, Z , = JR/G cosech =I derived from the z-parameters of the transmission line. Thus in using the disjoint 2-port model (Fig. 13) for the waveform relaxation analysis of an E C G transmission line, the initial condition is always satisfied at the transmission-line terminals if the initial values of the voltage waveform generators are assigned with the magnitudes:

VOLTAGE WAVEFORM GENERATORS FOR

e\"')(O) = E,, = u l ( t = 0) - il( t = 0 ) d m (A2.1a)

e$,)(O) = E 2 , = u 2 ( t = O ) + i 2 ( t = O ) ~ R / G . (A2.lb)

However, the dc analysis for obtaining the terminal voltages and currents {u,(O), im(0), m = 1,2) can be time-consuming if the transmission line is only a component of a VLSI circuit. To reduce the CPU time and the computer memory requirement, the dc analysis must be carried out on the partitioned subcircuits by iterations. Starting with E$) = 0 and performing the iterative dc analyses following the sequence as described in Algorithm 3.1, we obtain:

(A2.2a)

I 1


1 EA

coefficients and the propagation function:

PA0 = (RA - m)/( R A + m)? pBO = ( R E - d W ) / ( R, + dm), e, = m~.

Thus for 9, < 1 and k becomes infinity, (A2.2) converges to the exact solutions:

In Example 3.1, we have, after two iterations:

{ E ~ ~ ) , E $ ) = { -0.144 988, -0.204 503)

{E$;), E$:))={ -0.164375, -0.231 848).

Thus from (A2.2) and the far-end terminal condition E, = 0, pe = 0.999 500 125, we obtain:

WA = E!')= -0.144 988, WE = E$;)= -0.164 375

q,, = (E!:' - E!h))/E$,) = 0.410 482 246 8

E,, = WA /( 1 - qn) = - 0.245 943

E,, = WE /( 1 - qo) = - 0.278 83

which are in good agreement with {E,,, E2J = { - 0.245 932, -0.278 816) derived from (A2.1) using the result from the full-circuit analysis, which took twice the CPU time of that by the iterative analyses. It is interesting to observe that the ECL-gate under the turn-off condition (gate-input = -0.4 v) can be represented by the Thevenin's circuit with elements:

EA = pAoWA / qO( 1 - pAO) = -0.39406 v

RA = ( 1 + pAO) ( dm)/( 1 - pAO) = 80.7828 R .

APPENDIX C FOURIER SPECTRUM OF A LINEARLY

INTERPOLATED WAVEFORM From a set of discrete data points obtained from a

circuit simulator, a piecewise-linear function can be constructed by connecting adjacent data points ( t , , U , )

(k = 0,1,2, . * . N ) by straight lines:

c( t ) = U , + mk( t - t , ) , t , G t < t k + l (A.3.la)

(A.3 .lb)

The Fourier integral of the linearly interpolated function v ( t > can be derived by substituting (A.3.1) into (A.3.2):

mk = ( U k + 1 - 4) /( t,+ 1 - t k ) *

F [ U ( t ) ] = j r N u ( t ) exp( - j 2 ~ f t ) d t . (A.3.2) 10

CHANC: WAVEFORM RELAXATION ANALYSIS

By carrying out the piecewise integration, we obtain

F [ L , ( t ) ] = V ( w ) = R( o) + j I ( w ) ,

j = J-1, w = angular frequency = 27rf

R ( w ) =(l /w)( i~,s inor , , , -ir.,,sinwt,)

+(1 /02) (mNp1coswtN - m,coswr,) N - 1

+ ( I / @ * ) (W2k-l-?72k)COSWfk (A.3.3a) k = l

[ ( U ) = (l/w)(c,coswr, - c,coswt,)

- ( 1 / w 2 ) ( mN- I sin ut, - rn, sin ut,)

N - I

- ( l / w 2 ) (rn,!-,-rnk)Sinwtk (A.3.3b) k = l

N - l

R(o)=(1/2) c (fk+l-tk)(L‘k+lt‘Ck) (A.3.3c) k = (1

I ( 0 ) = 0. (A.3.3d)

For a data set that is linearly interpolated at periodic time intervals

t , = k T / N , k = 0,1,2;. . , N

and where the Fourier spectrum is sampled at multiple units of the fundamental frequency f,, = 277 / T , we have:

w, = 27rn / T , wntk = 27rnk / N

1

1415

- ( 1 / ~ , ) ~ ( m k - , - m,) sin(27rnk/N) k = l

which can be further related to the standard form of FFT:

V( nf,) = ( T / 2 ~ n ) ~ akWLnk (A.3.4a) N - l

k = 0

a , = m , - , - r n , = ( N / T ) ( 2 i ~ , - i ~ , ~ , - L ‘ ~ + ~ )

(A.3.4b)

( A .3.4c) W, = exp( j 2 7 r / N )

if the function is periodic: = L ’ ~ , , m N p , = mcl.

REFERENCES F. H. Branin, Jr., “Transient analysis of lossless transmission line,” Proc. IEEE, vol. 55, pp. 2012-2013, Nov. 1967. F. Y. Chang, “Transient analysis of lossless coupled transmission lines in a nonhomogeneous dielectric medium,” IEEE Trans. Microwave Theory Tech., vol. MTT-18, pp. 616-626, Sept. 1970. Y. K. Liu, “Transient analysis of TEM transmission lines,” Proc. IEEE, vol. 56, pp. 1090-1092, June 1968. F. H. Branin, Jr., “Transient analysis of lossy transmission line,” in Proc. 6th Annual Allerton Conf. Circuit and System Theory, pp. 276-285, Nov. 1968. G. R. Haack, “Comment on ‘Transient analysis of lossy transmission lines,”’ Proc. IEEE, vol. 59, pp. 1022-1023, June 1971. H. W. Dommel, “Digital computer solution of electromagnetic transients in a single and multiphase networks,” IEEE Trans. Power App. Syst., vol. PAS-88, pp. 388-399, Apr. 1969. F. Y. Chang, “Image-parameter formulation of the method of characteristics and the waveform relaxation technique,” submitted for publication. -, “Waveform relaxation analysis of nonuniform lossy transmission lines characterized with frequency-dependent parameters,” submitted for publication. E. Lelarasmee, A. E. Ruehli, and A. L. Sangiovanni-Vincentelh, “The waveform relaxation method for time-domain analysis of large scale integrated circuits,” IEEE Trans. Computer-Aided De- sign, vol. CAD-1, pp. 131-145, July 1982. Interactice Circuit Design User’s Guide, IBM Corp., Data Proc. Div., White Plains, NY. A. J. Gruodis and C. S. Chang, “Coupled transmission line characterization and simulation.” IBM J. Res. Dervlop., vol. 25, pp. 25-41, Jan. 1981. J. R. Marti, “Accurate modeling of frequency-dependent transmission lines in electromagnetic transient simulations,” IEEE Trans. Power Apparatus and Systems, vol. PAS-101, pp. 147-157, Jan. 1987

[I31 New York:

1141 H. S. Wall, Analytic Theory of Continued Fraction. Princeton,

F.M. Reza and S. Seely, Modern Network Analysis. McGraw-Hill, 1959, Chap. 8.

. . NJ: Van Nostrand, 1948, chap. -20, sec. 95.

[151 N. Balabanian and T. A. Bickart. Electrical Network Theon. . - Malabar, FL.: Krieger, 1985, chap. 6. M. R. Spiegel, Theory and Problems of Complex Variables: With An Introduction to Conformal Mapping and Applications. New York:

[16]

Schaum, 1964, p. 175. [17] H. S. Wall, Analytic Theory of Continued Fraction. Princeton,

NJ: Van Nostrand, 1948, p. 349. [I81 L. Storch, “Synthesis of constant-time delay ladder network using

Bessel polynomials,” Proc. IEEE, vol. 42, pp. 1666-1675, 1954. [19] G. D. Berland, “ A guided tour of the fast Fourier transform,”

IEEE Spectrurn, pp. 41-52, July 1969.

2 r rnk/N) Fung-Yuel Chang (M’68-SM’88) received the B.S. degree in electrical engineering from Na- tional Cheng Kung University, Taiwan, China, in 1959, the M.S. degree in electronics from the National Chiao Tung University, Taiwan, in 1961, and the M.S. and Eng. Sc.D. degree, both in electrical engineering, from Columbia Uni- versity, New York, in 1964 and 1968, respectively.

H e has been with the IBM General Technol- ogy Division in East Fishkill since 1968. His first

job assignment at IBM led to the pioneering work on the method of characteristics for transient simulation of integrated circuits interconnected with multiconductor coupled transmission lines. He received the IBM corporate outstanding innovation award for his design and modeling of bipolar transistors implemented in three generations of IBM mainframe computers. In 1984, he was a visiting Associate Professor of Electrical Engineering at Columbia University, where he taught courses on bipolar device modeling and computer-aided circuit analysis. Prior to his sabbatical leave from IBM, he was a senior engineering manager of the Bipolar Device Design and Modeling Department. Since 1985, he has been an adjunct professor at Columbia University. At IBM, he is presently engaged in applying waveform relaxation technique to packag- ing analysis. His areas of interest include distributed network theory, device modeling, computer-aided circuit analysis, and VLSI interconnection theory.

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