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Predictor/Corrector Newton-Raphson (PCNR): A Simple, Flexible, Scalable, Modular, and Consistent Replacement for Limiting in Circuit Simulation

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Scientific Computing in Electrical Engineering (SCEE 2018)

Part of the book series: Mathematics in Industry ((TECMI,volume 32))

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Abstract

Modern circuit simulators predominantly use Newton-Raphson (NR) iteration to solve circuit equations. To improve NR convergence, circuit simulators use a practice called “limiting”. This ensures that sensitive circuit quantities (such as diode voltages) do not change too much between successive NR iterations. However, in most simulators, the implementation of limiting tends to be inflexible, non-modular, inconsistent, and confusing. We therefore propose Predictor/Corrector Newton-Raphson (PCNR), a replacement for limiting that overcomes these disadvantages while incurring modest computational overhead. The key ideas behind PCNR are, (1) to add each limited circuit quantity as an extra unknown to the circuit’s Modified Nodal Analysis (MNA) system of equations, (2) to split each NR iteration into a “prediction” phase followed by a “correction” phase, and (3) to mitigate the computational cost of the extra unknowns by eliminating them from all Ax = b solves using a Schur complement based technique.

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Notes

  1. 1.

    PCNR works for differential-algebraic equations as well, but for simplicity, we only consider algebraic equations in this write-up.

  2. 2.

    Equivalently, one can also compute the Jacobian dgdx, and an “RHS” function given by \(\mathrm {RHS}(\mathbf {x}) = \left ( \frac {d\mathbf {g}}{d\mathbf {x}} \right ) .\, \mathbf {x} - \mathbf {g}(\mathbf {x})\), at each iteration x i, which is the approach traditionally followed by SPICE simulators.

  3. 3.

    In practice, each diode also returns a Boolean flag to the simulator telling it whether limiting was or was not applied, which the simulator uses to determine whether NR has truly converged or not.

  4. 4.

    https://en.wikipedia.org/wiki/Schur_complement.

References

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Acknowledgements

This work was sponsored by the Laboratory Directed Research and Development (LDRD) Program at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

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Correspondence to Karthik V. Aadithya , Eric R. Keiter or Ting Mei .

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Aadithya, K.V., Keiter, E.R., Mei, T. (2020). Predictor/Corrector Newton-Raphson (PCNR): A Simple, Flexible, Scalable, Modular, and Consistent Replacement for Limiting in Circuit Simulation. In: Nicosia, G., Romano, V. (eds) Scientific Computing in Electrical Engineering. SCEE 2018. Mathematics in Industry(), vol 32. Springer, Cham. https://doi.org/10.1007/978-3-030-44101-2_19

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