Delivering global DC convergence for large mixed-signal circuits via homotopy/continuation methods

Abstract

Abstract-Homotopy/continuation methods are attractive for finding dc operating points of circuits because they offer theoretical guarantees of global convergence. Existing homotopy approaches for circuits are, however, often ineffective for large mixed-signal applications. In this paper, we describe a robust homotopy technique that is effective for solving large metaloxide-semiconductor (MOS)-based mixed-signal circuits. We demonstrate how certain common circuit structures involving turning-point nesting can lead to extreme inefficiency, or failure, of conventional probability-one homotopy methods. We also find that such situations can lead to numerical ill-conditioning and homotopy paths that fold back upon themselves, leading to algorithm failure. Our new homotopy model for MOS devices, dubbed Arc-tangent Schichman-Hodges (ATANSH), features decoupled continuation parameters that are instrumental in avoiding these problems. ATANSH-based homotopy methods in production use have led to the routine solution of large previously hard-to-solve industrial circuits, several examples of which are presented.

Model reduction via projection onto nonlinear manifolds, with applications to analog circuits and biochemical systems

“…Since maniMOR has trained all the possible equilibrium points, it also automatically eliminates possible multiple unrealistic/non-physical DC solutions. Continuation methods [15], which is well known to deliver global DC convergence, is also easier to apply because of the fact that we are only search in a 1-D subspace.…”

“…• Project the initial condition x 0 into the nonlinear manifold subspace, and get z 0 as the new initial condition of reduced system; • Search the k nearest points around z 0 , among sampled points stored in the LUT in maniMOR; • Compute the function values using equation (15), feed into any nonlinear solver to solve the transient integration equations, e.g., backward Euler equations; • Repeat the above two steps until time t = t end ;…”

Homotopy method with a formal stop criterion applied to circuit simulation

“…The homotopy method reported by [4] is a highly efficient multiparameter method [5] that locates the operating point for large circuits containing just MOS transistors. In [6], an excellent revision of globally convergent probability-one homotopy methods applied to circuit simulation with bipolar transistors is presented.…”

“…In [6], an excellent revision of globally convergent probability-one homotopy methods applied to circuit simulation with bipolar transistors is presented. All the homotopy methods [4,5,6] described above have been proved useful to locate one or more operating points that converge to solutions where the NR method is unable to calculate. Nonetheless, such methods lack a formal stop criterion [7]; the stop criterion allows to complete the simulation with the mathematical certainty that no more solutions are left to be found along the traced homotopy path.…”

An Effective Embed Algorithm for Newton Fixed-Point Homotopy Method in MOS Transistor Circuits

“…[13], a homotopy method based on ATAN-SH (Arc-tangent Schichman-Hodges) is proposed for MOS-based mixed-signal circuits. In this homotopy method, two homotopy parameters are used.…”