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
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.
One possibility to choose a value for is
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.
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
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 .