Circuit Response
every model has an edge, and the figure carries it
Series

Poles — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two poles at ζ = 0.3, recovered from the matrix. The poles are at -477.5 ± j1518 hertz. Their distance from the origin is the natural frequency to six digits; the cosine of their angle from the negative real axis is the damping ratio. The step response beside them follows.

    Where the behaviour is written down

    Two numbers in the complex plane contain everything a second-order circuit will ever do. Their distance from the origin is the natural frequency, the cosine of their angle is the damping — and the fastest-settling circuit is not the critically damped one, which is the case the textbooks name.

    part 2 · transients
  2. The sensitivity of a pole against the room it has. computed by solving, not by drawing. A series R–L–C whose damping is walked from 0.3 to 0.999999, which slides its two poles together along a straight line and changes nothing else. The exact derivative of a pole with respect to the capacitor climbs from 0.5241 to 353.6 as the gap between them falls from 19078 to 28.28 radians a second. The fitted exponent over the closest four is -1.0000, and the product of the two is the natural frequency itself — 9999.6894 against 9999.6894, at every damping drawn and not merely in the limit, which a closed form gives and this computation never sees. The resistor's curve runs at 2ζ times the capacitor's — below it at 0.3 and at twice it by the time the poles have met — and the inductor's lies exactly under the capacitor's throughout.

    The gap a derivative needs

    The derivative of a pole is exact and has no step size in it, and beside the formula sits a sentence nobody had measured: it divides by a quantity that vanishes when two poles meet. Driven together, the sensitivity climbs as the reciprocal of the gap — fitted exponent −1.0000, the product a constant 1.00000 times the natural frequency — while the largest change it still describes falls as the gap *squared*. A one per cent capacitor is outside first order once the poles are 3194 radians a second apart, which is an ordinary critically damped design.

    part 3 · transients
  3. 3 equal lags in a loop closed by an inverting gain k, from k = 0 to 12.8. computed by solving, not by drawing: the closed-loop poles, rooted from the netlist's determinant at 61 gains. At k = 0 the 3 poles coincide at −1/τ; as k rises they leave along 3 straight lines at angles π(2j + 1)/3, each ∛k/τ from the start, to the imaginary axis at k = sec(π/3) to the power 3 = 8.000. A gain of a hundredth moves them 21.5% of their own frequency; a single lag would move 1%.

    The root a ring of equal stages takes

    Two coincident poles separate as the square root of whatever splits them; three as the cube root; m as the m-th. In a loop of m equal lags closed by a gain k, the m-fold pole leaves along an m-pointed star, each point the m-th root of k from where it started — fitted exponents 1.0000, 0.5000, 0.3333 and 0.2500 over six decades — so a loop gain of a hundredth moves a triple pole 21.5% of its frequency where a single lag would move 1%. The star's nearest points reach the imaginary axis at k = secᵐ(π/m): 8 for three lags, 4 for four, 1.884 for eight, falling towards one. Staggering three lags by a factor of four raises the gain they tolerate from 8 to 26.6, and at the same gain cuts a step's overshoot from 53.9% to 20.6% — or, at the same gain margin, buys nothing.

    part 4 · transients

All series