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COMPANION MODELS AND BOUNDARY CONDITIONS FOR
MULTI-RATE STEADY-STATE ANALYSIS OF INTEGRATED
ANALOG CIRCUITS
M. Iordache , Lucia Dumitriu
Electrical Engineering Department, Politehnica University of Bucharest, Spl. Independentei 313,
Bucharest, ZIP 060042, ROMANIA, e-mail [email protected]
The RF-IC applications are, in general, strongly
nonlinear and have carrier frequencies into the GHzrange, with modulated signals in the kHz-range. Due to
these peculiarities, the integration time-step must be small
enough to accurately capture the fast component and
obtaining information about the slowly component needs a
large number of time-steps. Shooting methods adjust the
guess initial condition at the end of the period using a
nonlinear solver (usually Newton-Raphson) and integrate
by transient simulation. When signals with widely
separated rates are involved, reaching steady-state needs a
large number of time steps. Consequently, shooting cannot
handle efficiently circuits driven by multi-tone signals,
being definitely a single tone algorithm [5]. The finitedifference time-domain technique (FDTD) discretizes the
differential equations over a period yielding a system of
algebraic equations, which are solved simultaneously to
find solutions in all time points of the discretization
network. Another approach for the steady state simulation
is harmonic balance method (HB) that operates in the
frequency domain.
Different methods, both in time domain and in
frequency domain, were proposed for circuit simulation
under modulated carrier excitation [2, 5, 6], and a mixed
time-frequency method for the multi-tone steady state
circuit simulation has been developed [3)]. A new timedomain approach based on multiple time scales has been
developed in the last years for steady state analysis of
nonlinear circuits with broad signal spectrum [8-12].
Using multivariate functions transforms the original
ordinary differential equations (ODEs) or differentialalgebraic equations (DAEs), describing the circuit
behavior, into partial differential equation form, in the
general case resulting in multitime partial differential
equations (MPDEs). In this new formulation the
components with different rates of variation are
decoupled, each disparate signal being represented by its
own artificial time scale. Solving the MPDE numerically,
by appropriate choice of boundary conditions [12], the
quasi-periodic and envelope-modulated solutions are
obtained for the circuits with combination of strong
nonlinearities and multirate signals.
In this paper, in order to preserve the easy formulation
of the circuit equations, we use the semi-state method
(SSM) equivalent with the modified nodal method in
dynamic behaviour. For a lumped nonlinear analogue
circuit the equations corresponding to SSM, have the
following form [1, 5-7]:
M  xt   x (t )  Gx (t )  F  xt   Bbt  . (1)


t
where: x(t )  vnt 1, imt - is the independent variable
vector, with x0 initial condition; M , G are square
n  1  m n  1  m ;
matrices


F - is a nonlinear
t t
function of x; b  j t , e - is the input vector; B is a
selector matrix, with entries (1, 0 or 1), and the
superscript “t” denotes the transpose.
In the case when the circuit exhibits multirate
behaviour, its variables can be represented efficiently
using multiple time variables. Using p time-scales and
denoting the multivariate forms of x (t) and b (t) by
xˆ t1,...,t p and bˆ t1,...,t p , respectively, the MPDE
corresponding to (1) becomes:




 xˆ
xˆ 
(2)
M  xˆ 
 ... 
  Gxˆ  F  xˆ   Bbˆ t1 ,..., t p .
t p 
 t1
Using equations (2) and replacing the dynamic
elements by discrete resistive circuit models associated
with an implicit numerical integration algorithm [13, 14]
the transient analysis of the nonlinear circuit can be
reduced to the dc analysis of a sequence of equivalent
nonlinear resistive circuits, and efficiency in numerical
computing of the associated MPDE is obtained.
Considering the two-rate case, MPDE (2) becomes:
 xˆ
xˆ 
ˆ
(3)
M  xˆ 

  Gxˆ  F  xˆ   Bbt1 ,t 2  ,
 t1 t 2 
with the periodic boundary conditions (BCs)
xˆt1  T1, t2  T2   xˆt1, t2  . We take a uniform grid
{ t i, j  } of size n1  1   p2  1 on the rectangle [0,
m1T1] x [0, T2] (Fig. 1), where t i , j   t1i ,t 2 j , with:




t 2 j  i  2T2   j  1h2 .
(4)
meaning that at each integration step h1, p2 integration
steps with step size h2 are performed, and
t1i  i  1h1 , i  1, n1  1, j  1, p2  1 ;
(5)
h1  m1T1 / n1  T1 / p1 , h2  T2 / p2 .
Consider that the slow components of bt  and xt 
depend on t1 and the fast components depend on t2.
For the first periods T1 and T2 (corresponding to the
grid size  p1  1   p2  1 ), we assume that the BCs are:
x̂1, j   0.0; j  1, p2  1 ,
(6,a)
ˆ i ,1  0.0; i  1,2 , x
ˆ i  1,1  x
ˆ i , p2  1; i  2, p1 (6,b)
x
on the row t1 = 0, and on the column t2 = 0, respectively.
We continue integration with the step h1 in respect of
the slow time t1 from the point t  p1  1, p2  1 to the
point t  p1  2,2 (see Fig. 1).
Proceeding in this way for the other grids we shall
integrate the MPDE until the point


t n1  1, p 2  1  t1n11 ,t 2 p 21 ,
and t2 p
2 1
with
t1n
1 1
 m1T1
 n1T2 . At each time moment t i, j  we have to
solve a nonlinear algebraic equation system. For this end
we use the Newton-Raphson algorithm.
The discrete resistive circuit equations, associated to the
BDF (backward differential formula) of the first order,
when the characteristics of the nonlinear elements are
approximated by piecewise-linear continuous curves,
at t i, j  and at the (k+1)th iteration of the NewtonRaphson algorithm, corresponding to the SSM, have the
following form:
 
 
 
 
k  
Gdn1,n1 s k 
Bdn1,m sik, j   v nk11i, j    i sc
,i , j 
 k 1 ki , j 




 . (9)

Rdm,m sik, j    i mki ,1j  e mk,i , j  
 Adm,n1 si , j 

where: Gdn 1, n 1 sik, j  is the incremental node-conductance
matrix corresponding to n–1 independent nodes;
Bdn 1, m sik, j  is an (n-1)m matrix that contains the
elements –1, 0, +1 and the current gains of the CCCSs;
Adm, n 1 sik, j  represents a m(n-1) matrix containing the
elements –1, 0, +1 and voltage gains of the VCVSs;
Rdm, m sik, j  is a mm matrix having the elements made up
of: the transfer resistances of the CCVSs, the incremental
resistances of the discrete models of the current-controlled
dynamic circuit elements and the incremental resistances
of the current-controlled nonlinear resistors; v nk11i, j  is the
node-voltage vector corresponding to n–1 independent
nodes at the (k+1)th iteration and at the time t i, j  ; i mki,1j
represents the current vector corresponding to the nonNA-compatible circuit branches, and i sck,n 1i , j  , emki , j 
represent the contributions of the excitation sources
(independent current and voltage sources), of the sources
corresponding to the approximations of the nonlinear
resistors, and the initial values of the inductor currents and
of the capacitor voltages, which are determined from
previous time steps t i  1, j  of the slow time t1, and
t i, j  1 of the fast time t2.
The contribution of some dynamic circuit elements to
the Eq. (9) is presented in APPENDIX.
 
 
 
Figure 1. . A uniform grid { t i, j  } of size

n1  1   p2  1 .

We start the integration process on the row 2 from the
point t 2,2  t12 ,t 22 , with t12  h1 , t 22  h2 , in
respect of the fast time t2 from the column 2 to the column
p2+1, and so on until we arrive in the point
t 2, p 2  1  ( t12 ,t 2 p 1 ) with t12  h1 , t 2 p 1  p 2 h2 .
2
2
After that we integrate one time step h1 in respect of the
slow time t1 –assigning to x̂3,1 the value of
x̂2, p2  1 – and then we start again the integration
process on the row3 in respect of the fast time t2 from the
column 2 to the column p2+1, and so on until we arrive in
t  p1  1, p 2  1  t1 p 1 ,t 2 p 1 ,
the
point
with

t1 p
1 1
 p1h1  T1 and t2 p
2 1
1
2

 p1T2 .
Remark 1. Before passing to the integration on the next
grid (each grid having the size  p1  1   p2  1 ),
starting from the point t  p1  2,2   t1 p  2 ,t 22 , with
t1 p
1 2
  p1  1h1  T1  h1 , and

1

t 22  p1T2  h2 , we
must consider the following boundary conditions:
xˆ  p1  2,1  xˆ  p1  1, p 2  1;
xˆ  p1  i ,1  xˆ  p1  i  1, p 2  1
(7)
for i  3, p1  1, on thecolumnt 2  0 , and
x̂ p1  1, j , j  2, p2  1 i  2, p2  1
on the row t1 = T1.
(8)
 
[1] R. Achar, and M.S. Nakhla,
Simulation of High-Speed
Interconnects, Proceedings of the IEEE, 89, 5, 2000, pp. 693-728.
[2] H. G. Brachtendorf, G. Welsch, and R. Laur, A novel timefrequency method for the simulation of the steady state of circuits
driven by multi-tone signals, Proceedings of the ISCAS,1997, pp.
1508-1511.
[3] Angela Hodge, and R. W. Newcomb, Semistate Theory and Analog
VLSI Design, IEEE Circuit and Systems Magazine, 2, 2, 2002, pp.
30-49.
[4] M. Iordache, Lucia Dumitriu, and L. Mandache, Time-Domain
Modified Nodal Analysis for Large-Scale Analog Circuits, Revue
Roum. Sci. Techn. - Électrotechn. et Énerg., 48, 2-3, 2003, pp. 257268.