Moment Matching: AWE and PRIMA

The idea

An extracted net at sign-off is thousands of RC (Resistance-Capacitance) elements; simulating each with full accuracy per timing arc is unaffordable, and Elmore's single moment is too crude. Moment matching splits the difference. The network's response expands as a Taylor series whose coefficients, the moments, are computable by cheap recursive tree traversals (each moment is an Elmore-like pass over the results of the previous one). AWE (Asymptotic Waveform Evaluation) fits a low-order pole-zero model, a handful of exponentials, whose moments match the first 2q of the network via the Padé approximation, then reads delay and slew off the small model. Accuracy at q of three or four approaches full simulation on typical nets at a tiny fraction of the cost; Elmore is exactly the q equals one case.

From AWE to PRIMA

Raw Padé fitting is numerically treacherous: it can produce unstable or non-physical models (poles in the wrong half-plane) from perfectly passive wiring. The Krylov-subspace generation that followed cured the numerics, and PRIMA (Passive Reduced-order Interconnect Macromodeling Algorithm) added the decisive structural guarantee: its congruence- transformation projection provably preserves passivity, so reduced models remain stable in any surrounding circuit, which made reduction safe to automate at scale.

Where you meet it today

Inside every sign-off timer's interconnect engine: delay calculators reduce each net's parasitics to a compact model (moment-based methods and their descendants, with effective-capacitance techniques for the gate side), and the same mathematics underlies model-order reduction across electronics far beyond timing.