6. Convergence and Damping

The system of equations which has to be solved for mixed-mode device
simulation is non-linear and extremely sensitive to small changes in the
solution variables. While the semiconductor equations are difficult to solve
themselves the situation becomes even worse when using dynamic mixed-mode
boundary conditions. To solve these equations the Newton method is used which
is known to have quadratic convergence properties for an initial-guess
sufficiently close to the final solution. However, such an initial-guess is
hard to construct for both the distributed quantities inside the device and
the circuit equations. Hence methods have to be found to enlarge the region
of convergence to succeed even with a poor initial-guess. This is achieved by
suitable damping schemes which will be discussed in the following sections.
Damping schemes for the device equations will be discussed in
Section 6.1 while Section 6.2
focuses on damping schemes for the circuit equations.

Especially important is a reliable method to obtain a DC operating point which is needed as a starting point for a transient analysis or a static transfer characteristic. Transient simulations are far better conditioned as the time derivatives provide main-diagonal entries and act as a natural damping. As the solution of the last timestep provides a good initial-guess it is normally possible to obtain convergence for a sufficiently small timestep. Although the conditioning of the equation system does not change for DC transfer analysis the last solution again provides a good initial-guess. In case the system fails to converge for a given step the step can normally be reduced in such a way to obtain convergence. Hence the following discussion will focus solely on DC operating point calculation.

To the best knowledge of the author no useful damping scheme for mixed-mode
has been published so far. Only in [49] it was stated that the
change of the node voltages was limited to a user-specified value which is in
the range of
2^{ . }*V*_{T}. This is, as pointed out in the same paper, far
from being optimal as it guarantees a large number of iterations for larger
supply voltages as is the case for some of the circuits simulated in this
thesis. E.g., for the output stage of an OpAmp as shown in
Fig. 6.8 the supply voltages are 15 *V*,
hence it takes at least
15/0.05 = 300 iterations to build up the supply
voltages without even considering the effect of non-linearities. Furthermore
it is stated in [49] that a solution can only be obtained for an
initial-guess as close to the solution as 0.2*V* for
forward-biased junctions.

These restrictions of mixed-mode simulations seem to be generally accepted nowadays making
it a challenging task to seek for alternatives. A new method is proposed in
Section 6.2.5 which works admirably well for small circuits.
Solutions could be found for several typical analog and digital circuits
starting from the zero initial-guess for the node voltages and charge neutrality
assumptions for the semiconductor devices within 20-50 iterations
which is a comparable effort to *SPICE* which uses compact models.

1999-05-31