Operating point and sweep

// Two of the first circuits anybody builds, side by side, and the two
// readings this editor can take of them.
//
// LEFT, columns 8 to 18: a divider. 5 V through 10 k into 10 k.
//
//   I   = 5 V / 20 k          = 250 uA
//   tap = 5 V x 10k / 20k     = 2.500 V
//
// RIGHT, columns 22 to 29: an RC. 5 V through 10 k into 10 uF.
//
//   tau  = 10 k x 10 uF       = 100 ms
//   v(t) = 5 V (1 - e^(-t/tau))
//
// -------------------------------------------------------------- 1. Shift+O
//
// Press Shift+O, or the meter button on the canvas toolbar. Every net gets its
// voltage. With nothing running you are reading the circuit at the instant it
// is plugged in, which is why the two halves read differently:
//
//   the divider tap      2.500 V     a resistor has no memory
//   the RC node          7.500 uV    a capacitor starts discharged
//
// Seven microvolts, not a round zero, and that is not a rounding error: every
// node in this engine carries a 1 pS leak to ground so that a node with no DC
// path to the reference still has an answer instead of a singular matrix. It
// is a 1 Tohm resistor, three orders of magnitude leakier than nothing and
// three orders of magnitude tighter than any real part. The readout prints
// what the solver holds rather than tidying it away, which is the whole point
// of four significant figures in engineering notation.
//
// Click a resistor while the labels are up and the properties panel shows its
// current. The divider's two carry 250 uA each. The RC's resistor carries
// 500 uA, because at t = 0 the whole supply is across it and none of it is
// across the capacitor yet.
//
// Now press Run, wait a second, and press Shift+O again. The RC node is at
// 5.000 V, its resistor is carrying nothing, and the divider has not moved.
// Five time constants is half a second and gets you to 99.3%; a whole second
// is ten of them.
//
// -------------------------------------------------------- 2. Sweep the supply
//
// More menu, then Sweep. Step v1.voltage from 0 to 5 in steps of 0.25 and
// watch the divider tap. You get a straight line of slope exactly one half,
// because that is what a divider is: the ratio does not depend on the supply.
//
//   at 1 V   the tap is 0.500 V
//   at 3 V   the tap is 1.500 V
//   at 5 V   the tap is 2.500 V
//
// Download CSV and the whole table is there, every net and every current, at
// full precision.
//
// ------------------------------------------------------- 3. Sweep a resistor
//
// Same sheet. Step r2.value from 1000 to 20000 in steps of 1000, same probe.
// This one is not a straight line, and the arithmetic says why:
//
//   tap = 5 V x R2 / (10k + R2)
//
//   at 1 k    0.455 V
//   at 10 k   2.500 V
//   at 20 k   3.333 V
//
// It is steep where R2 is small and flattens off as R2 grows past 10 k, which
// is the practical lesson about dividers: the half of the ratio you can feel
// is the smaller resistor.
//
// ------------------------------------------------ what the sweep does not do
//
// Every point of a property sweep is a fresh power-on, so the capacitor is
// discharged at every one of them. Sweep the supply and watch the RC node
// instead of the tap and you get a flat line at zero, which is true and is
// the note printed under the plot. Put 500000000 in "settle each point for"
// (half a second, in nanoseconds) and the RC node follows the supply, at the
// cost of running half a second of simulated time twenty-one times over.
//
// ----------------------------------------------------------- 4. The time step
//
// The bottom of the same sheet sets the transient step. Leave it at 0 and the
// solver picks each step from the error it is making. Set 1000 (one
// millisecond) and tick "exactly this step", and the RC still charges on the
// same curve: a hundred steps a time constant is far more than the trapezoidal
// rule needs. Set 50000 (fifty milliseconds, half a time constant) and it is
// still stable, and now visibly coarse.