The previous chapter showed that simple piecewise linear transistor models can produce good predictions of the behavior of digital circuits. However, once a circuit uses piecewise linear models it may no longer reduce to an RC tree and its response may not be well approximated by an exponential. In fact our experience has been that as transistor models increase in complexity, so do the responses of the circuits containing them. Therefore a more flexible technique for approximating waveforms is required than Rsim’s single time constant delay estimation.
Fortunately, Rsim’s single time constant techniques have been extended to allow more accurate waveform estimates with multiple time constants. The generalized moments
matching procedure mentioned in Chapter 2 allows Mom to produce more sophisticated
estimates of the responses of circuits containing piecewise linear models. This technique possesses several promising characteristics. In contrast to numerical integration algorithms, a single computation yields a function of time,
v
(t
), representing the response for all futuretime,
t
2 01]. In contrast to single time constant algorithms, it allows the computationof waveform estimates of arbitrary accuracy.
This chapter begins with a brief review of the theory behind the moments matching procedure. It then discusses some practical aspects that must be considered in an imple- mentation. Next the utility of the procedure is demonstrated by showing a number of simulations. The chapter concludes with a description of some of the limitations of the procedure.
4.1
Waveform Approximation
One general approach to waveform approximation begins with the derivation of a low order model of the (presumably high order) system of interest. If the low order model is sufficiently simple it becomes practical to compute its response exactly. That response serves as an approximation of the response of the original system. The model order
reduction problem has been studied extensively by linear control theorists. Of the many
methods proposed, one of the simplest is based upon moments matching[BL72]. Although more advanced techniques demonstrating superior convergence and stability properties were subsequently derived[Cha91], none of these appears to be efficient enough for use in our particular application.
The moments matching waveform approximation procedure involves two steps[PR90]. First the asymptotic final voltages of each node are computed by finding the DC solution of the network (assuming all piecewise linear devices remain in their present regions). This DC solution corresponds to the particular solution. Then all DC sources in the circuit are set to zero and an estimate for the homogeneous solution is generated by matching moments. The total estimate is the sum of the two solutions.
4.2
Pad´e Approximation
It has been observed that moments matching is, in fact, just a particular application of the
Pad´e approximation[Zak73]. In general, the Laplace transform of the impulse response of
a lumped linear time-invariant circuit takes the form of a ratio of polynomials in
s
:H
(s
)= 0+ 1s
+ 2s
2 +:::
+ ms
m 0+1s
+2s
2 +:::
+ns
n:
(33)where the order of the denominator polynomial,
n
, is usually equal to the number of storage elements (capacitors and inductors) in the circuit. However, ifn
is very large it can be prohibitively expensive to computeH
(s
) exactly. Instead, a lower order Pad´eapproximation is constructed: ˆ H(
s
)=b
0+b
1s
+b
2s
2 +:::
+b
j ;1s
j;1 1+a
1s
+a
2s
2 +:::
+a
js
j (34)(
j
n
)and used to model the response of the system.The parameters of the Pad´e approximation are obtained using a two step process. First the 2
j
low order terms of the series expansion ins
ofH
(s
)are computed:H
(s
)=m
0+m
1s
+m
2s
2 +m
3s
3 +:::
+m
2j ;1s
2j;1 +O
(s
2j ) (35)Here, the
m
iare the moments of the impulse response1andO
(
s
2j
)represents all other terms
of order 2
j
or higher.Moments are of interest because they are easily computed directly from the circuit even though it is usually impractical to compute
H
(s
)in closed form. Pillage and Rohrer[PR90]point out that moment computation can be viewed as a sequence of DC solutions. First the voltages and currents at
t
= 1are found by computing the DC solution of the networkassuming that capacitors are open circuits and inductors are short circuits. The difference between the initial and final voltages across capacitors and the initial and final currents through inductors are the 0th order moments. Then the(
k
+1)st moments are recursivelycomputed from the
k
th moments by finding the DC solution of the network derived by (Figure 36):1. Setting all independent sources to zero. This has the effect of subtracting out the particular solution.
2. Replacing capacitors with current sources equal to the product of the capacitance times the
k
th moment of the capacitor voltage.3. Replacing inductors with voltage sources equal to the inductance times the
k
th mo- ment of the inductor current.Once the DC solution is found, the resulting capacitor voltages and inductor currents represent the(
k
+1)st moments. Note it is quite straight forward to find the DC solution1Strictly speaking, the Laplace coefficients,
m
i, are equal to the moments, b