Operating point and frequency response

Solves the circuit at rest: capacitors open, inductors shorted, AC sources at their DC offset. You get a table of every node voltage and every device current — the bias check you'd do first on paper.

It is also what seeds a transient run, so playback starts from the settled state rather than jolting at t = 0. Invisible on a resistive divider; the difference between a clean start and a phantom turn-on transient on anything with a capacitor in it. Restart (⟲) re-solves it, which a plain stop-then-play does not.

Currents are read from the final solution rather than from whatever the last iteration happened to leave behind. A device that carries a branch current — a source, an inductor, a transformer, an opamp — latches its current at the end of a timestep, and an operating point never commits one, so reading it the lazy way gives a table full of 0 A that looks exactly like a solver bug.

Pick a source to drive, the outputs to plot, and a start frequency, stop frequency and point count. The circuit is linearised at its operating point and solved per frequency on a log grid, giving gain in dB and phase in degrees.

An RC low-pass comes out at −3 dB and −45° at its analytic corner, rolling off at −20 dB/decade — which is what the engine's own tests assert against, since a regression suite that only compares to a previous run will happily lock in a wrong simulator.

Two decisions show up in what you see:

A linear circuit at a fixed timestep is factored once, and every step after that is a substitution. That's what makes an interactive simulation affordable at all — and it's why changing a value costs a re-factor while dragging a variable supply doesn't wipe your traces. Any circuit with a nonlinear device in it re-factors per iteration, which is unavoidable.

The transient integration is backward Euler, chosen deliberately over trapezoidal. Trapezoidal is more accurate but rings on switching edges, and ringing that isn't in the circuit reads as a bug. Backward Euler is unconditionally stable, which matters far more when you're driving the thing by hand. The cost is a slight first-order lag — an RC curve that trails the analytic exponential a little while having exactly the right shape and time constant.