Concept

Complex frequency — where it appears

The variable s, whose imaginary part is a frequency and whose real part is a growth or decay, at which a network is solved once for every question about it. Direct current is its value at zero, a frequency sweep is a walk along the imaginary axis, and a pole is where the matrix loses rank.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

One 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.

frequency · Frequency response

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.

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.

networks · Nodal analysis

Named alongside it

The objects these essays reach for when they reach for this one.

Asymptotic approximation Bode plot Corner frequency Kirchhoffs current law Modified nodal analysis Reactance Singular matrix Verification

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