Every model in electronics is an approximation with a range. This site draws the edge of the range.
A capacitor is a capacitor below a frequency its own leads decide and an inductor above it. An ideal operational amplifier configured for a gain of a hundred is already one per cent wrong at 1.4 kHz. A small-signal model is one per cent optimistic above seven millivolts, which is a quarter of the thermal voltage rather than a fraction of a supply rail. And Kirchhoff's laws have a frequency of their own, set by nothing but the size of the board. None of those is a caveat: each is a number, computed from the model's own parameters, and no figure here is drawn without it.
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19 essays
What a network answers, and how the answer is checked
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.
Frequency, which is the same solveOne solve, read four ways
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.
Before the steady stateOne step, computed twice
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.
Filters, measured not tabulatedThree families, one corner
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.
Feedback, and the marginWhat is left at crossover
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.
Where the models stopEvery model has an edge
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.
Networks, and how a solve is checkedThe divider, and the thing it does not know about
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.
Frequency, which is the same solveThree voltages that close on one, and the steady state they assume
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.
Before the steady stateWhere 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.
Filters, measured not tabulatedWhat a steep skirt costs
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.
Feedback, and the marginTwo measurements of one margin
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.
Where the models stopKirchhoff's own frequency
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.
Networks, and how a solve is checkedThe source that is not a source
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.
Frequency, which is the same solveResonance, and the bandwidth it sets exactly
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.
Before the steady stateThe step that is too big
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.
Filters, measured not tabulatedFlat magnitude, unflat delay
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.
Feedback, and the marginThe ideal amplifier, and where it stops being one
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.
Where the models stopHow small is small signal
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.
Frequency, which is the same solveThe capacitor that is an inductor
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.
Threads running through
themes, not chapters
Every model has an edge
The ideal amplifier, the small-signal transistor, the lumped element, the capacitor: each is excellent inside a range and wrong outside it, and the range is a number rather than a warning. No figure here is drawn without the frequency, amplitude or size at which the model in it stops being true.
Two routes to a number
A phase margin from the loop gain and an overshoot from the step response. A transfer function from the matrix and the same one from the poles it was factored into. A ladder solved by nodal analysis and by a chain-matrix product. Neither route in any of those pairs can confirm itself, and they share no arithmetic.
The schematic is a label
Two drawings of the same circuit with different placement are the same circuit, so the layout carries no information — which is exactly not true of a mechanism, where the geometry is the content. Schematics here are small, drawn in one hand, and put in a corner. The canvas belongs to the response.
Refused, not extrapolated
A network with no path to ground has no answer, and a solver that returns one is lying. A model asked to work outside its range declines and says why. Half the value of an assertion is what it rejects, so every refusal on this site was produced by running it rather than by describing it.
The parasitic is the component
A capacitor is a capacitance, a resistance and an inductance, and above a computable frequency the third one is the whole part. A source is an electromotive force and a resistance. The thing nobody draws is usually the thing that decides the answer, and it is always a number.
Measured, not tabulated
Butterworth is flat, Chebyshev is steep, Bessel has good delay — three adjectives and no quantities, in the most reproduced table in the subject. Every comparison on this site is taken off a solved network instead, which is how the third column, the one that decides whether a filter can pass an edge, turns out to have been missing.
The constants decide
A divider's ratio does not predict what happens when something is connected to it — its magnitude does. A filter's family does not say how much delay distortion it costs — a measurement does. The quantity that gets discarded to make a rule memorable is repeatedly the one that decides the outcome.