Circuit Response
every model has an edge, and the figure carries it
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 4 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 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.

frequency · Frequency response A pole pair at Q = 2: the response peaks where the circle centred on the pair's real part and through both poles meets the axis, 0.9354 of ω₀, at 2.0656. computed by solving, not by drawing. The poles of a series RLC low-pass (10 mH, 100 nF, and the resistance for Q = 2), solved from the netlist, at −0.2500 ± j0.9682 in units of ω₀ = 1/√(LC) = 5.03 kHz: on the circle of radius ω₀, and 1/(2Q) of it from the imaginary axis. The magnitude at a point jω is ω₀² over the product of that point's two distances to the poles, to 9e-16 of the solve at 241 frequencies. The circle centred at −σ with radius ωd passes through both poles; it meets the axis at 0.935414 of ω₀, which is where the product of the two distances is least and the response peaks — 2.065591, which is ω₀²/(2σωd), found by a search on the solve at 0.935414.

The circle a peak sits on

A pole pair's response at a frequency is ω₀² over the product of two distances — from the point on the imaginary axis to each pole — to a part in 10¹⁵ of the nodal solve. That turns a resonance into geometry. The peak is where a circle centred on the pair's real part and through both poles meets the axis: at Q = 2, 0.9354 of ω₀ and a height of 2.0656, found the same by a search. The quality factor is an angle, Q = 1/(2 cos θ), and a peak exists exactly when the pair is closer to the imaginary axis than to the real one — past forty-five degrees. The half-power width is where the short leg has grown by √2, twice the pair's distance from the axis. And the step, marched, rings at the pole's height and decays at its distance: a projection, not a length.

frequency · Frequency response

Named alongside it

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

Asymptotic approximationBode plotCorner frequencyReactanceHalf-power bandwidthKirchhoffs current lawModified nodal analysisPole pairPolesQuality factorResonanceSingular matrix

All concepts