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

Capacitor ratio — where it appears

The quotient of two capacitances on one chip, which is what a switched-capacitor circuit's gain and corner frequency are made of. Two capacitors drawn from the same layers track each other to a fraction of a per cent where either alone varies by twenty, so a ratio is the precise quantity and a single value is not.

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

A switched capacitor is a resistor below a ratio, not below a frequency. computed by solving, not by drawing. The exact response of a capacitor shuttled between the input and a holding capacitor at 1.00 MHz — a difference equation with one pole, evaluated on the unit circle — against the continuous R–C its equivalent resistance is supposed to make. The corner is 3.15 kHz against the model's 3.18 kHz, 0.99% out, and the discrepancy is set by the capacitor ratio alone: one per cent needs a ratio under 0.0201, which is a clock 315 times the corner. The second difference has no counterpart at all — the sampled response repeats at the clock, so the image rising on the right of this plot is signal at 997 kHz arriving as though it were at the corner. The third is settling: 1 pF charged through 1 kΩ gets 500.0 time constants a half period at this clock, and being short of full charge raises the equivalent resistance by 0.000%, which puts a ceiling on the clock at 108 MHz.

A resistor made of a clock

A capacitor shuttled between two nodes at a megahertz behaves as a megohm, and a tenth of a picofarad shuttled at ten kilohertz behaves as a gigohm — which is how a filter with a one-hertz corner fits on a chip. What the equivalence costs is three conditions, and they bind on three different quantities: a capacitor ratio under 0.0201, a signal below half the clock, and a clock below the frequency at which the charge stops arriving.

filters · Switched-capacitor A stray of a tenth of the sampling capacitor moves a one-plate integrator's coefficient by 9.85%; switched at both plates, by 0.14%, nearly all of it the amplifier's own gain error. computed by solving, not by drawing. A switched-capacitor integrator with a 1 pF sampling capacitor, 4 pF of feedback and 0.5 pF at the amplifier's input, and a stray capacitance to ground at the sampling capacitor's switched plate; the error of the per-cycle coefficient against Cs/Cf, against the stray as a fraction of Cs. With one plate switched (solid) the stray is charged to the input and dumped with the capacitor, so it is part of the capacitor: the error is the stray's fraction less the amplifier's small gain error, 9.85% at a tenth. With both plates switched (dashed, A = 1000; dotted, A = 10⁵) the plate that meets the summing node moves between ground and −Vo/A, so the stray's charge carries a factor 1/A: its sensitivity falls from 1/Cf to Cs/(A·Cf²), 4000 times smaller, and what is left, 0.14%, is the amplifier's gain error (1 + (Cx + Cs)/Cf)/A, which the one-plate arrangement has as well.

The capacitance at the switched plate

A switched-capacitor filter's corner is a ratio of two capacitors, and every plate on a chip has a capacitance to the substrate that is on no schematic. Switch one plate of the sampling capacitor and that stray is charged to the input and dumped with it: a stray a tenth of the capacitor is a tenth of error, and a bottom plate's fifteen per cent is fifteen per cent at any size. Switch both plates and integrate into a virtual ground and the stray at the plate that meets the summing node moves only between ground and −Vₒᵤₜ/A — its sensitivity falls from 1/Cf to Cs/(A·Cf²), four thousand times less at a gain of a thousand, and what is left is the amplifier's own gain error, 0.14%. The stray has not vanished: it has moved out of the coefficient that sets the corner and into the integrator's leak, where the direct-current gain becomes A·Cs/(Cs + Cp).

filters · Switched-capacitor

Named alongside it

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

Equivalent resistanceModel rangeSampled dataSwitched-capacitorAliasingGain errorImpedance scalingLoop gainParasiticsSettling time

All concepts