Circuit Response — every model has an edge, and the figure carries it Illustrated essays on electrical networks, with every model drawn together with the frequency, amplitude or size at which it stops being true. Modified nodal analysis over the complex plane solves each circuit and checks its own answer before anything is drawn. https://www.circuit-response.com/ 2026-08-06T15:08:35.483Z What a network answers, and how the answer is checked https://www.circuit-response.com/essays/what-a-network-answers/ 2026-08-06T15:08:35.483Z A circuit has exactly one answer and a matrix finds it. The part that matters is not that the answer exists but that it can be checked twice, by routes that share no arithmetic — and that a circuit with no answer is refused by name rather than returned as a large plausible number. One solve, read four ways https://www.circuit-response.com/essays/one-solve-read-four-ways/ 2026-08-06T15:08:35.483Z Reactance, phase, the corner frequency and the roll-off are not four ideas. They are four readings of one complex number, obtained from the same matrix that answers direct-current questions — and the straight-line sketch every engineer draws of them is itself a model, three decibels wrong exactly where it is read. One step, computed twice https://www.circuit-response.com/essays/one-step-computed-twice/ 2026-08-06T15:08:35.483Z A step response from the poles is exact. The same step walked forward in time is not, and the difference between them is the trapezoidal rule's own error rather than anything about the circuit. It falls by a factor of four every time the step is halved, which is a claim about a method and can be watched. Three families, one corner https://www.circuit-response.com/essays/three-families-one-corner/ 2026-08-06T15:08:35.483Z Butterworth is flat, Chebyshev is steep, Bessel has good delay. None of those is a number, so the table they appear in cannot answer the question anybody has. Here each family's poles are computed from its definition, built as an actual network, and then measured — starting with the step every comparison skips. What is left at crossover https://www.circuit-response.com/essays/what-is-left-at-crossover/ 2026-08-06T15:08:35.483Z A feedback loop is stable or not according to one number read at one frequency — how much phase remains before −180° at the point where the loop gain passes unity. The loop gain here is obtained the way it is obtained on a bench: cut the loop, drive one side of the cut, and measure what comes back to the other. Every model has an edge https://www.circuit-response.com/essays/every-model-has-an-edge/ 2026-08-06T15:08:35.483Z Four assumptions this collection runs on, with the frequency at which each stops being true, on one axis. The ordering is not the one most readers would guess — an ordinary amplifier circuit runs out of model at 1.42 kHz, three thousand times sooner than a ten-centimetre circuit board does. The divider, and the thing it does not know about https://www.circuit-response.com/essays/the-divider-and-its-load/ 2026-08-06T15:08:35.483Z A two-resistor divider's output is set by the ratio of its resistances — with nothing connected. Connect anything at all and what decides the answer is the quantity the ratio was built to discard: the magnitude. Two dividers of identical ratio give six volts and one volt into the same load. Three voltages that close on one, and the steady state they assume https://www.circuit-response.com/essays/three-voltages-that-close/ 2026-08-06T15:08:35.483Z Kirchhoff's voltage law drawn as a polygon in the complex plane. The three element voltages of a series circuit add head to tail to the source exactly — while their magnitudes add to five times it. And the whole picture is a statement about a settled circuit, which takes a computable number of cycles to arrive. Where the behaviour is written down https://www.circuit-response.com/essays/where-the-behaviour-is-written-down/ 2026-08-06T15:08:35.483Z 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. What a steep skirt costs https://www.circuit-response.com/essays/what-a-steep-skirt-costs/ 2026-08-06T15:08:35.483Z A filter's order buys attenuation at a known rate — twenty decibels per decade per pole, and no arrangement of components changes it. What varies between families is how quickly the slope is reached, and the currency it is paid for in is delay: the steepest of the three distorts delay eight hundred times more than the gentlest. Two measurements of one margin https://www.circuit-response.com/essays/two-measurements-of-one-margin/ 2026-08-06T15:08:35.483Z A phase margin is computed from the loop gain in the frequency domain, without ever looking at a step. An overshoot is measured from the closed-loop step response in the time domain, without ever looking at a Bode plot. Inverting the standard relation on the second returns 34.9° against the first's 34.9°, and the residue is the third pole. Kirchhoff's own frequency https://www.circuit-response.com/essays/kirchhoffs-own-frequency/ 2026-08-06T15:08:35.483Z The current law says the current entering a node equals the current leaving it at the same instant, which assumes the signal crosses the circuit in no time. It crosses at about two-thirds the speed of light, so the law has a frequency of its own — set by nothing but the physical size of the board. The source that is not a source https://www.circuit-response.com/essays/the-source-that-is-not-a-source/ 2026-08-06T15:08:35.483Z An ideal voltage source holds its voltage at any current, which makes it the flattest line in the subject and the most commonly assumed model in it. Its edge is a current, set by one resistance nobody draws — and past that current the model is not approximately right, it is describing a different object. Resonance, and the bandwidth it sets exactly https://www.circuit-response.com/essays/resonance-and-its-bandwidth/ 2026-08-06T15:08:35.483Z The half-power bandwidth of a resonant circuit is f₀/Q — not approximately, but to every digit the arithmetic has, which is rare enough to be worth checking. What is not exact, and is drawn as though it were, is the idea that the band sits centred on the resonance. At a quality factor of one its middle is twelve per cent above. The step that is too big https://www.circuit-response.com/essays/the-step-that-is-too-big/ 2026-08-06T15:08:35.483Z A linear circuit scales — double the input and the output doubles, exactly. A real amplifier does not, because its output can only move at a fixed rate, and the amplitude at which the two stop agreeing is about eighty millivolts for an ordinary part. No transfer function contains that number, because no transfer function can. Flat magnitude, unflat delay https://www.circuit-response.com/essays/flat-magnitude-unflat-delay/ 2026-08-06T15:08:35.483Z A filter that passes every frequency in its band at the right amplitude and the wrong time has not passed the signal. Group delay is the measurement that says so, it is absent from the classical comparison, and it varies by fifty per cent across the passband of the two families everybody uses. The ideal amplifier, and where it stops being one https://www.circuit-response.com/essays/the-ideal-amplifier-and-its-bandwidth/ 2026-08-06T15:08:35.483Z An ideal operational amplifier's closed-loop gain is set by two resistors and nothing else — a horizontal line at every frequency. The real one is already a tenth of a per cent low at direct current, one per cent low by 1.35 kHz, and above 10 kHz has no loop gain left, at which point the ideal answer is not an approximation to anything. How small is small signal https://www.circuit-response.com/essays/how-small-is-small-signal/ 2026-08-06T15:08:35.483Z Linearising an exponential replaces a curve by its tangent, which is exact at a point and progressively wrong away from it. The amplitude at which it is one per cent wrong is 7.3 millivolts at room temperature — 28 per cent of the thermal voltage, not a small fraction of it, and a good deal smaller than "small signal" suggests. The capacitor that is an inductor https://www.circuit-response.com/essays/the-capacitor-that-is-an-inductor/ 2026-08-06T15:08:35.483Z A hundred-nanofarad capacitor follows 1/(2πfC) for four decades and then turns round and climbs. Above 14.5 MHz it is an inductor, and a decade past that its impedance is ninety-nine times what its capacitance predicts — all of it caused by about a nanohenry of lead and via that nobody chose and nobody drew.