INCONSISTENT INITIAL CONDITIONS - Computer Methods for Analysis of Mixed-Mode Switching Circuits - page 105

7. Summary
We have shown in this chapter that due to ideal switching, and
may differ, leading to inconsistent initial conditions and the cre-
ation of impulses at switching instants. Because the topology of circuits
changes before and after switching, instead of must be
used to continue integration after switching. Several computer methods
that compute from have been examined in detail in this
chapter. We have shown that for linear circuits, can be computed
efficiently using the numerical Laplace inversion based two-step algo-
rithm to achieve a high degree of accuracy. The accuracy of this method
depends upon the step size used in the integration. Both too small and
too large steps should be avoided.
To handle the inconsistent initial conditions of nonlinear circuits, nu-
merical Laplace inversion based approach is ineffective because it is dif-
ficult to obtain the response of these circuits. Backward Euler
formula, on the other hand, yields the correct consistent initial condi-
tions for both linear and nonlinear circuits immediately after switching.
The initial conditions can be obtained in two consecutive forward steps
from switching instants, one forward step and one backward step of equal
step size. The latter provides better accuracy. We have also shown that
two forward step, followed by two backward steps of identical step size,
through yield the correct consistent initial conditions, does not lead to
an improvement in accuracy. The Taylor series based approach also
fails to offer better accuracy. The consistent initial conditions can also
be obtained using Volterra functional series based approach where the
consistent initial conditions are obtained from those of corresponding
Volterra circuits that are linear. An advantage of this approach is that
numerical Laplace inversion based two-step algorithm can be used for
better accuracy.
In analysis of circuits with internally controlled switches, because the
state of these switches is determined by the network variables associated
with the switches and because the strength of the impulses generated
at switching instants can be quite large, the switching of one switch
may activate the switching of other switches in the circuits. This pro-
cess continues until no switching occurs. The detection of impulses at