L0/01 — Quasi-static reduction

Background

The full laws of electromagnetism are Maxwell's equations: four coupled field equations in which changing electric fields beget magnetic fields and vice versa, so that disturbances propagate — as waves, at the speed of light. Circuit theory knows nothing of this. When a circuit diagram says a node has "a voltage," it asserts that the whole node agrees on one value now; when Kirchhoff's current law says currents into a node sum to zero, it ignores the travel time of influence entirely. The gap between these pictures closes only because chips are small and their signals slow: if the time light needs to cross the structure is utterly negligible against the time anything meaningful changes, then every part of the field has already "caught up" at every instant, and the field configuration is just the static solution for the present boundary voltages, gliding through time. This is the quasi-static approximation, and it is the step at which "the circuit" — nodes, voltages, capacitances — comes into existence as a concept. The comparison is of feature size against wavelength (the distance a wave travels per oscillation period); at this chip's size and speed they are five orders of magnitude apart, which is as comfortable as approximations get.

Quasi-statics splits into two sub-regimes by which half of Maxwell survives. Electroquasistatics (EQS) keeps charge and capacitance and drops magnetic induction — the regime of everything L1 computes. Magnetoquasistatics (MQS) is the complementary half: current loops, inductance — the effect by which a changing current induces opposing voltage around its own loop. Inductance is the awkward one for this project's methods, for a reason worth understanding: capacitance is between nearby things (and screened — L0/10), but inductance belongs to current loops, and a loop is only closed through the signal's return path, which may run anywhere in the power grid — so there is no locality argument for it, and its extraction is a genuinely global problem. The honest treatment at this design's tens-of-MHz clock is to bound the inductive contribution and show it negligible — a checked side condition, rather than a modelling burden.

The chapter's remaining content is the shape of the error statement. Physics folklore says the quasi-static error is "second order in size-over-wavelength"; a verification needs that folklore as an inequality with a computable constant on this geometry, so it can be added to the margin budget like every other error. The mathematical form is a perturbation expansion — solve the equations as a power series in the small parameter, bound the discarded tail — and theorems of exactly this shape exist in the applied-analysis literature; what is missing is only the explicit-constant version. That gap (M6) is a fair sample of what this whole layer is: no new physics, no controversy, just the difference between "everyone knows it's fine" and a bound.

Statement

Maxwell's equations reduce to a sequence of elliptic problems, with an error bound. This is the first abstraction step and it is where "the circuit" becomes a meaningful concept at all — before it, there are only fields.

Why it is needed

Every downstream statement (RC enclosures, delay, Kirchhoff's laws) presupposes that propagation can be ignored and that charge and current are instantaneously related to potential. That is an approximation, and nobody in the flow states its error.

The content

The criterion is scale separation: feature size L ≪ λ = c/(f√ε_r).

At 130 nm in SiO₂ (ε_r ≈ 3.9), λ is roughly 1.5 m at 100 MHz and 15 mm at 10 GHz. The die is a few mm. So at this design's tens-of-MHz operating frequency the separation is five orders of magnitude and the approximation is excellent; at multi-GHz on global nets it starts to bind, which is exactly when on-chip inductance became an industry concern.

Two sub-regimes, and only one matters here:

  • Electroquasistatic (EQS) — ∇×E ≈ 0, capacitance dominates. This is the interconnect regime and the one L1 solves.
  • Magnetoquasistatic (MQS) — inductance, current loops. Relevant for long global nets and clock distribution at high frequency. Not local: there is no screening argument for inductance because return paths can be distant, which is why inductance extraction is much harder than capacitance extraction — the engineering response is the partial inductance formalism (Ruehli's PEEC; FastHenry is the reference extractor), which assigns loop-free per-segment values and recovers physical loop inductance only in closed sums. At this design's frequency it is negligible; that should be a stated and checked condition, not an assumption.

What the error bound should look like. Expand in the small parameter L/λ (equivalently ωL/c). The leading correction to the EQS solution is O((L/λ)²), so a rigorous statement has the form

‖φ_Maxwell − φ_EQS‖ ≤ C · (L/λ)² · ‖source‖

with C depending on the geometry. Theorems of exactly this shape exist, on the MQS side: Ammari–Buffa–Nédélec justified the eddy-current approximation of Maxwell with error bounds in powers of the small parameter (SIAM J. Appl. Math. 2000), and Raviart–Sonnendrücker did the same programme for the Darwin model (the combined EQS+MQS, radiation-free intermediate). The EQS analogue is easier and essentially folklore. What does not exist is the version this project needs: an explicit, computable C over realistic multi-material Lipschitz geometry, stated as a verification hypothesis rather than an asymptotic order. So the open problem is sharpening constants in a known theorem, not inventing the theorem.

Kirchhoff's laws are consequences of this reduction, not axioms. KCL is ∇·J = 0 integrated over a surface enclosing a node; KVL is ∇×E = 0 integrated around a loop. Both are exact in the quasi-static limit and carry the O((L/λ)²) error otherwise. This is worth stating explicitly because circuit theory is normally introduced axiomatically, and here it needs to be derived — see 08.

Status

Settled physics, standard asymptotic analysis, entirely unformalised in this context. The mathematics is a regular perturbation expansion; the work is stating it with explicit constants over realistic domains.

Open problems

  1. An explicit, geometry-dependent constant in the O((L/λ)²) bound — i.e. the Ammari–Buffa–Nédélec-style result with the constants made computable. Without it the reduction is qualitative and the error cannot be folded into the margin budget.
  2. A checkable side condition for when MQS may be dropped — i.e. a bound on the inductive contribution to delay, given the design's frequency and net lengths.
  3. Whether the expansion is uniform near the corner singularities of 00, where ‖source‖ in the bound above is itself unbounded.

First experiments

  • Compute L/λ for the actual design: die size, longest net, clock frequency, ε_r. Confirm the separation is as comfortable as it appears, and record it as a checked side condition rather than folklore.
  • Estimate the inductive contribution on the longest global net and the clock spine, to justify dropping MQS.

Effort

Weeks for the statement; M6's explicit constant is the open-ended part.

Reading

Haus & Melcher, Electromagnetic Fields and Energy — the standard careful treatment of the EQS/MQS split. Ammari, Buffa & Nédélec, "A justification of eddy currents model for the Maxwell equations" (SIAM J. Appl. Math. 2000). Raviart & Sonnendrücker on the Darwin model. Alonso Rodríguez & Valli, Eddy Current Approximation of Maxwell Equations — book-length treatment of the MQS side. Ruehli on PEEC / partial inductance; Kamon–Tsuk–White (FastHenry) for what inductance extraction actually computes. Any asymptotic-analysis text for the regular perturbation structure.