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

Dynamic range — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The usable range of one stage, in a 10 kHz measurement. computed by solving, not by drawing. The floor is the Johnson noise of a 1 kΩ source in the measurement's own noise bandwidth — 501.6 nV, using 15.7 kHz rather than the 10 kHz corner. The ceiling is the drive at which an exponential's distortion reaches one per cent, 1.03 mV. Between them is 66.3 dB, and nothing a designer does moves either number without changing the circuit.

    A floor and a ceiling

    Every other boundary these essays measure is a ceiling. This one puts a floor underneath and measures the distance between them — 4.00 nanovolts per root hertz at the bottom, one per cent of harmonic distortion at 1.03 millivolts at the top, and 66.3 decibels of range in a ten-kilohertz measurement. Both ends are computed, neither is on a datasheet, and the arrangement of the stages decides which one moves.

    part 3 · noise
  2. Every node of a Chebyshev 5, and the one that clips first. computed by solving, not by drawing. The largest signal each node of the realised network ever carries, over the whole frequency sweep, relative to the input. The output reaches 1.000 times the input and F0b reaches 4.537, so on a ±15 V supply the input can be driven to 3.31 V rather than 15.00 before something clips — and the thing that clips is not the output. With the floor at 249.7 nV that is 142.44 dB of range against the 155.57 dB an instrument on the output would report, a difference of 13.14 dB that no measurement at the output can see.

    The ceiling is not at the output

    A fifth-order Chebyshev's noisiest node is its output and its largest signal is not. F0b carries 4.537 times the input, so on ±15 V the range is 142.44 decibels and not the 155.57 an instrument on the output reports. Across fifteen realised filters the output reading spans 3.97 dB and the range they actually have spans 22.34 — and one family of the three has no internal peaking at all, at any order.

    part 4 · noise
  3. The Chebyshev 5's sections, each node's peak before and after equal-maxima scaling. computed by solving, not by drawing. Unscaled (lighter), section by section the largest signal anywhere inside reaches 4.537×, 3.109×, 1.000× the input, so the ceiling is 4.537× at F0b. With a gain of 0.220 at the input, 1.000 and 1.602 between sections and 2.833 at the output, every section peaks at 1× (darker) and the response is unchanged. The floor rises from 249.7 to 883.8 nV, so the range gains 2.16 dB of the 13.14 dB the internal peak was costing.

    The ceiling a gain gives back

    A fifth-order Chebyshev loses 13.14 dB of its range because its first section's capacitor node carries 4.537 times the input. The textbook repair is to scale: divide the signal in front of the loud section, restore it later, so that every node peaks at the same level. Done exactly, with silent gains of 0.220, 1.000, 1.602 and 2.833 and the response unchanged to the last digit, it lowers the ceiling to the input and raises the floor from 249.7 to 883.8 nV — and gives back 2.16 dB of the 13.14. Every section between the division and the restoration has its noise multiplied by what the signal was divided by. On the even-order Chebyshevs scaling loses half a decibel; on the seventh-order it recovers 3.77 of 22.45. Reordering the sections and scaling them together gives 4.75 dB. The rest of the ceiling belongs to the high-Q section and no gain moves it.

    part 5 · noise
  4. Scaling a Chebyshev 5 with real 4 nV/√Hz amplifiers everywhere: 118.18 dB at equal maxima and 118.32 dB with no gain stages at all, so the gains cost 0.14 dB, where silent ones promised 2.16 dB. computed by solving, not by drawing: the dynamic range on ±15 V of a fifth-order Chebyshev with gains d^α in front of each section and after the last, against α. Every amplifier has a voltage noise of 4 nV/√Hz, the sections' own and the stages', and each stage carries a kilohm in its divider or lower feedback arm (solid). With every amplifier and gain silent (dashed), equal maxima recovers 2.16 dB over no scaling, 142.44 dB to 144.60 dB. With real ones: 118.32 dB with no stages, and with stages 115.75 dB at α = 0.2, 116.27 dB at α = 0.4, 116.86 dB at α = 0.6, 117.50 dB at α = 0.8, 118.18 dB at α = 1. Merely having the stages is a step down at the smallest α, since each is a real amplifier in the path, and the curve climbs from there to equal maxima. The response is unchanged to a part in a billion at every α.

    The floor the gains bring

    Equal-maxima scaling recovers 2.16 decibels of a fifth-order Chebyshev's range when its gains make no noise and its amplifiers make none either. Give every amplifier in the circuit a voltage noise of 4 nV/√Hz, the sections' own and the stages', and the unscaled filter reads 118.32 dB over a 1 MHz band and the scaled one 118.18: the last gain has nothing after it to filter its noise, so it multiplies 4 µV of wide-band noise by 2.83, and the lower ceiling is spent paying for it. At 1 nV/√Hz scaling costs 6.61 dB, and at 16 nV/√Hz it gains 0.99, so a noisier amplifier makes it look better. A narrower measuring band does not rescue it. On a Chebyshev 7, with a node 13.26 times the input, the same arrangement gains 6.15 dB.

    part 6 · noise

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