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

Frequency response — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · frequency
  2. Four readings of one distance: |H| is |p| over the length of this arrow. The complex plane of a single-pole low-pass with its corner at 995 Hz. The pole sits at −995 Hz on the real axis and the faint circles are contours of constant magnitude — circles centred on the pole, because |H| = |p|/|s − p| and a constant magnitude is a constant distance. The imaginary axis crosses them, and where it crosses is the frequency response: 1.00 corners up the distance is 1.4142 times the pole's own, so the magnitude is 0.70711 and the phase is minus the 45.00° the separation subtends. Checked against the nodal solve at sixty frequencies over five decades, the two agree to 6.7e-16. So the four readings are one geometric fact: the corner is where the distance is √2 times its smallest, the roll-off is the distance becoming ω, the flat passband is the distance not yet changing, and the phase is the angle.

    Four readings of one distance

    Reactance, phase, the corner frequency and the roll-off are four readings of one complex number, and the number has a geometry. A single pole's magnitude is the pole's distance from the origin divided by its distance from the point on the imaginary axis, and the phase is minus the angle that separation subtends — agreeing with the nodal solve to a part in 10¹² over five decades. So the corner is where the distance is root two times its smallest, the roll-off is the distance becoming omega, the flat passband is the distance not yet changing, and the phase is the angle. One distance, four readings.

    part 2 · frequency

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