Similar to the DC analysis convergence problems occur during the
transient analysis (see section 3.3.2 on page
) as well. In order to improve the overall
convergence behaviour it is possible to improve the models on the one
hand and/or to improve the algorithms on the other hand.
The implications during Newton-Raphson iterations solving the linear equation system
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(6.165) |
are continuous device model equations (with continuous derivatives as
well), floating nodes (make the Jacobian matrix singular) and the
initial guess
. The convergence problems which in fact occur are
local minimums causing the matrix
to be singular, nearly singular
matrices and overflow problems.
The modified (damped) Newton-Raphson schemes are based on the
limitation of the solution vector in each iteration.
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(6.166) |
One possibility to choose a value for
is
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(6.167) |
This is a heuristic and does not ensure global convergence, but it can
help solving some of the discussed problems. Another possibility is
to pick a value which minimizes the
norm of the right
hand side vector. This method performs a one-dimensional (line)
search into Newton direction and guarantees global convergence.
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(6.168) |
The one remaining problem about that line search method for convergence improvement is its iteration into local minimums where the Jacobian matrix is singular. The damped Newton-Raphson method ``pushes'' the matrix into singularity as depicted in fig. 6.11.
The basic idea behind this Newton-Raphson modification is to generate a sequence of problems such that a problem is a good initial guess for the following one, because Newton basically converges given a close initial guess.
The template algorithm for this modification is to solve the equation system
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(6.169) |
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(6.170) |
with the parameter
given that
is sufficiently smooth.
starts the continuation and
ends the continuation. The
algorithm outline is as follows: First solve the problem
, e.g. set
and try to solve
. If Newton-Raphson converged then increase
by
and double
,
otherwise half
and set
. Repeat this until
.
Applied to the solution of (non-linear) electrical networks one may
think of
as a multiplier for the source vector
yielding
. Varying
form 0
to 1 and solve at each
. The actual circuit solution is done
when
. This method is called source stepping. The
solution vector
is continuous in
(hence the name continuation scheme).
Another possibility to improve convergence of almostly singular
electrical networks is the so called stepping, i.e. adding a
tiny conductance to ground at each node of the Jacobian
matrix.
The continuation starts e.g. with
and ends with
reached by the algorithm described in section
6.6.2. The equation system is slightly modified by
adding the current
to each diagonal element of the matrix
.