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

Fields

Eleven areas, in the order the argument builds — what a network is and how a solved one is checked, then the same solve read along the frequency axis, and outward from there until every model in the collection has met the thing that ends it.
A network solved, and checked: a bridge, which no series-parallel reduction reaches. Node potentials from modified nodal analysis. The branch currents are then recomputed from each element's own law and summed at every node; the residual is 2.7e-16 of the largest current in the circuit, which is floating-point rounding and nothing else.

Networks, and how a solve is checked

A circuit has one answer and a matrix finds it. What matters is not that the answer exists but that it can be checked: the branch currents are rebuilt from the element laws and summed at every node, and the energy is counted twice. Networks with no answer are refused by name rather than returned as a large plausible number.

22 essays A single-pole low-pass with its corner at 995 Hz. Solved at 209 frequencies. The straight-line sketch, drawn faintly, is 3.01 dB wrong at the corner and within a tenth of a decibel only below 152 Hz. The phase is already −5.7° a decade before the corner and −84° a decade after it.

Frequency, which is the same solve

Reactance, phase, resonance and the corner frequency are four readings of one number. The models that fail here are the straight-line sketch every engineer draws, which is three decibels wrong exactly where it is read, and the capacitor, which is a capacitor only below a frequency its own leads decide.

24 essays One step response, computed twice: from the poles, and by walking the network forward. A damping ratio of 0.22, so the overshoot is 49.2%. The two curves are drawn on top of each other; the panel below is the difference between them, which is the trapezoidal rule's error at 500 steps and reaches 1.70e-3 V.

Before the steady state

Everything on the frequency axis assumes a settled circuit. These are the figures about getting there — and about the one limit no transfer function contains, where a step becomes large enough that the response stops being a scaled copy of a smaller one.

24 essays Three filter families at order 5, all with the same half-power point. At three times the corner the Chebyshev is -64.0 dB down, the Butterworth -47.7 dB and the Bessel -28.3 dB. The inset is the passband at forty times the vertical magnification, which is the only place the Chebyshev's half-decibel of ripple is visible at all.

Filters, measured not tabulated

Butterworth is flat, Chebyshev is steep, Bessel has good delay. None of those is a number. Each family's poles are computed from its definition, built as a network, and then the ripple, the skirt, the delay variation and the ringing are measured on the network that results.

24 essays Loop gain of a three-pole amplifier closed for a gain of 100. Unity loop gain at 5.73 kHz, where 34.9° of phase remains before −180°. The phase reaches −180° at 89.6 kHz, where the loop gain is 46.1 dB below unity.

Feedback, and the margin

The site's strongest construction, and the subject supplies it free: a phase margin computed from the loop gain in the frequency domain, an overshoot measured from the step response in the time domain, one circuit, and a required agreement between two numbers that share no arithmetic.

22 essays Where four of this site's models stop being true. In order: the ideal operational amplifier at 1.42 kHz, a 10 V output at full amplitude at 7.96 kHz, Kirchhoff's laws on 10.0 cm at 3.97 MHz, the ideal 100 nF capacitor at 4.69 MHz. The fifth boundary is an amplitude rather than a frequency and cannot share this axis: a small-signal model is 1% wrong above 7.3 mV, at every frequency there is.

Where the models stop

The boundaries as the subject rather than as a mark on something else. Four of them are frequencies and one is an amplitude; the last is the frequency at which Kirchhoff's laws themselves become an approximation, and it is set by nothing but the size of the board.

24 essays A 20 Ω, 50 mH load on 230 V at 50 Hz. computed by solving, not by drawing. The load draws 1636 W and 1285 var, an apparent power of 2080 VA at a power factor of 0.786. The reactive side is confirmed by a route that touches no impedance: 2ω times the energy stored in the inductor gives 1285 var. The cable carries 9.04 A and only 7.11 A of it does anything.

Power, and the part that does no work

A nodal solve computes real power as a check on its own answer and then throws it away. This field reads it out — and then the imaginary half beside it, which sizes the cable, heats the transformer and is billed for. A correction capacitor is exact at the load it was computed for and at no other; a power factor is cos φ only while the current is a sinusoid, and a rectifier's is not.

25 essays A diode fed from 5 V through 1.0 kΩ. computed by solving, not by drawing. The operating point is where the exponential meets the load line: 0.692544 V and 4.3075 mA, reached in 13 damped Newton steps from a cold start. The one-line Newton on Vs = v + R·i(v), which touches no matrix, gives 0.692544 V. The "drop" is not a constant: it moves 59.53 mV per decade of current, measured between two solved operating points.

Devices, and the amplitude they stop being linear at

An operating point is where a transcendental equation and a linear network agree, and finding it is Newton's method on the whole netlist. Past that, the thing a linear model cannot express at all: distortion. It arrives seven times sooner than gain error does, its harmonics are Bessel functions of the drive, and a differential pair removes every even one of them exactly.

25 essays A 1 V step onto 1.00 m of 50 Ω line into an open circuit. computed by solving, not by drawing as a sum of 81 arrivals. The source drives 0.8333 V into the line immediately — set by 10 Ω against the line's 50 Ω, and not by the load, which it cannot yet know about. One delay of 4.83 ns later the far end reaches 1.6666 V. The staircase settles at 0.999990 V, which is what the resistive divider gives.

Lines, where a wire has a length

On the far side of the frequency at which Kirchhoff's laws give out. What a source drives into is decided by geometry before the load has any say; what comes back one delay later decides the rest. The wave picture is checked against a lumped ladder that has never heard of a wave — and the useful result is how badly the ladder does.

25 essays Two probes on a 2.0 kΩ source. computed by solving, not by drawing twice per frequency: the node alone, and the node with the probe's elements across it. The one-to-one probe's 115.0 pF makes the reading one per cent wrong at 6.79 kHz. The ten-to-one probe puts 12.8 pF in series with the cable, so its tip sees 11.5 pF and the same error arrives at 69.2 kHz — 10 times further up, bought with a factor of ten in signal — the two edges stand in the ratio of the tip capacitances, 10.00. At direct current neither probe is capacitive at all and the ten-to-one still reads 0.02% low, because 10 MΩ across 2.0 kΩ is a divider.

Measurement, which is a circuit on a circuit

An instrument is not an observer; it is an element, it goes in the netlist, and every reading is a reading of the circuit that includes it. A probe is a capacitance with a bandwidth. A divider with capacitance in it has two ratios and one equation that makes them agree. Two terminals measure the leads as well, and below ten ohms that is most of the answer.

22 essays The noise of a 1.6 kΩ resistor through a 10.0 kHz filter. computed by solving, not by drawing. A seeded white sequence of 5.06 nV/√Hz marched through the network gives 619.3 nV across six seeds, spread 1.79%. Integrating the same density against the solved |H(f)|² gives 620.6 nV — -0.21% apart, well inside the spread. The noise bandwidth is 15.03 kHz against a −3 dB point of 10.00 kHz.

The floor, which bounds from below

Every other boundary here is an upper one. This is the other end, and gain does not help because gain amplifies it too. The only figures on this site whose content is a sample — so every number is run across seeds and quoted with its spread — and the one place a bandwidth is not the −3 dB point but π/2 times it.

22 essays A 1:1 transformer at k = 0.99, and the band it is a turns ratio over. computed by solving, not by drawing. Two 10 mH windings coupled at 0.99, driven from 50 Ω into 50 Ω, with 0.5 Ω of winding resistance and 100 pF across the secondary. The response is flat at 0.4901 — which is 98.02% of the 0.5000 an ideal transformer of this ratio would give, and that shortfall is the coupling itself: the flat part is k times the turns ratio, times what the two winding resistances leave of the loop, to four figures at every k on the slider — between 400 Hz and 81.3 kHz, which is 2.31 decades. Both edges are bisected on the solved network. Below the first, the magnetising inductance is a short across the source; above the second, the leakage inductance is in series with the load. The slider moves the coupling, and it moves the upper edge only.

Two windings, and the band between them

The first two-sided model on this site. Everything else here is right below a number or above one; an ideal transformer is wrong at both ends and right in the middle, and what a designer buys is the distance between the two — measured on the solve, not taken from a T-model. Beside it, a core whose energy is almost all in its air gap, a saturation limit that is an integral rather than a frequency, and a flux that walks to it however small the imbalance. And a core that can get warm: a single-valued curve has no area, so it cannot dissipate, and giving the material a second branch turns its loss into an area, its Steinmetz exponents into local slopes, and its inductance into two numbers at one current.

34 essays The closed-loop poles at a gain of 3.05. The locus of the two poles as the amplifier's gain runs from 2.7 to 3.3. It crosses the imaginary axis at a gain of 3.000000 — bisected on the netlist, not quoted — and at 3.05 the real part is 2.500e+2 radians a second, which is an envelope multiplying by 1.17015 every cycle. The crosses are the closed form ω₀(k−3)/2 and they sit on the measured circles.

Circuits that do a job, and the range they do it over

Thirteen fields measure an element and the edge it has. This one composes several of them into a circuit with a purpose and asks the same question of the whole, where the answer is almost never the worst of the parts: an oscillator whose design condition is an exact equality no resistor can hold, a regulator that is a voltage source below three kilohertz and a capacitor above it, a threshold whose hysteresis is set by the noise underneath it, and a bridge that is linear near one point. It needed machinery the other twelve did not — a netlist with a nonlinearity in it, marched forward in time — because an oscillator's frequency comes from the linear part and its amplitude from the nonlinear part, and no analysis that drops either one returns both.

24 essays A 9.0 kHz input sampled at 10 kHz arrives as 1.0 kHz. computed by solving, not by drawing. The dots are the samples. The input at 9.00 kHz is above half the 10 kHz rate, and every dot also lies on the 1.00 kHz curve drawn beside it — the two sample sequences differ by 9.3e-15, which is the arithmetic and not a small effect. Nothing is attenuated and nothing is distorted: the samples are the samples of a different signal, at full amplitude, and there is no measurement of them that could say which one was there.

Where a signal becomes a number

A converter is an element in the netlist like any other: it has an anti-alias filter made of the same components as every filter here, a reference a resistor's Johnson noise sits on, and an aperture. What it also has is three boundaries of kinds the rest of the site does not carry — one with no gradient at all, one that stops being a floor and becomes distortion, and one whose variable is a duration.

22 essays

A field says what an essay is about, and on this site it also says which model is being taken apart: each of the eleven was opened because it contains something that visibly fails inside it, and a field with no failure in it would be a textbook chapter. The order above is the order the argument builds rather than the order of difficulty — the last four fields are all consequences of the first, and three of them were opened by noticing that the solver had been computing the answer since the first commit and discarding it.

Three other axes cut across these. A ladder says how deep one idea goes, which is a different question from which field it is filed under; the objects is the index of named things, so an essay in noise and an essay in filters that both talk about the equivalent noise bandwidth meet there; and the threads are what an essay rhymes with. The full list is the index of everything.