Characterization of Analog Circuits Using Transfer Function Trajectories

Abstract

A methodology is presented to characterize and model strongly nonlinear behavior of analog circuits with a compact set of nonlinear differential equations. While simulating a circuit in the time domain, the nodal matrix is extracted at each time step, similar to trajectory piecewise sampling (TPW). The circuit snapshots projected on a frequency-state space domain to facilitate the regression problem, based upon the vector fitting algorithm. Strongly nonlinear function approximation of the pole-residue trajectories render analytical models with very high accuracy, even with a limited training sequence. The proposed method proves to scale very well for large circuits with speedups that exceed the TPW implementation. The method is compatible with SPICE3f5 netlisting and commercial CMOS technologies and the models are easily exported as a state space description to behavioral languages such as VHDL-AMS or Verilog-AMS. The method is validated for analog circuits of medium and large size.

Extracting Analytical Nonlinear Models from Analog Circuits by Recursive Vector Fitting of Transfer Function Trajectories

“…The focus of this paper is to automatically extract a set of analytical equations from a TPW approach by means of Transfer Function Trajectories [3], [4] in a more general fashion. The TFT approach enhances the transition from Modified Nodal Analysis (MNA) matrix samples toward analytical equations by transforming the samples to a mixed state-space/frequency domain [4].…”

“…D ∈ R N ×Mo is the output matrix and y = y(t) ∈ R Mo the output variables. A nonlinear system approximation for (1) is derived by approximating the Jacobians at each location k in state space by a rational function in the frequency domain [3], [4]. If one is able to fix the polesâ p of the model over the entire state space, then the nonlinear functionality of the system approximation is fully embedded in the residues, which in turn is approximated by a rational function.…”

“…The extraction of MNA samples in the time domain and their transform to the frequency domain is described in detail in [3], [4]. The latter results in a state-dependent transfer function H (k) lm (s) between input l and output m around state k (t = t k ):…”

“…The static nonlinear function v = i −1 (u) can be reconstructed up to a constant from this set of small-signal conductance samples by indefinite integration over the input trajectory [4]. The remaining dynamical nonlinear part equals H…”

“…A TFT approximation of the state-dependent transfer function H (k) lm (s) with P ≪ N poles is given by its pole-residue form [4]:…”