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