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

Sampled data — where it appears

A system whose signals exist at instants rather than continuously. Its response is periodic in the clock and every frequency above half the rate arrives folded onto one below it, so what it does between the instants is decided by the arrangement rather than by any continuous approximation.

Named by 3 essays across 2 fields — 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 Same 2 ps of jitter: a floor at -121 dB, or a line at -91. computed by solving, not by drawing. A 3.337 MHz sinusoid sampled at 10 MHz with two clocks of identical root-mean-square jitter. One clock's timing error is independent gaussians and the other's is a sinusoid at 130 kHz. The total error power is the same — -87.43 and -87.55 decibels below the signal, against the closed form's -87.55 — and the pictures are not. The random clock spreads it over 2048 bins, -120.5 dB each; the modulated one puts it into two lines at 3.337 ± 0.130 MHz, -91.5 dB down. A specification in picoseconds does not choose between them.

A floor, or a line

Two clocks with identical two-picosecond jitter sample the same sinusoid, and the total error power is the same to a tenth of a decibel — the closed form the rung below computed, right for both. One of them puts that error across two thousand bins at −120 dB each; the other puts it into two lines at −91.5. The gap is the processing gain, it grows with the record length because a density falls and a line does not, and nothing in a specification quoted in picoseconds root-mean-square distinguishes the two cases.

digital · Aperture jitter The "offset" a switch leaves moves by 1.863 mV across a volt of signal. computed by solving, not by drawing. The charge a switch leaves on a 1 pF hold capacitor, as the input it was sampling changes. It is W·L·C_ox·(V_gs − V_th) with the clock's high level for the gate, so it falls as the input rises for two reasons at once: less gate drive, and a threshold that rises with the source potential through the body effect. The dashed line is the part that does not know what the signal is — the clock coupling through the gate overlap capacitance, -1.649 mV — and it is the only part of this that is honestly an offset. The curve stops where the switch does: an n-channel device passes nothing within a threshold of its own gate, and the model refuses rather than returning a zero.

The offset that knows the signal

The charge a switch leaves on the capacitor it was sampling is the gate oxide capacitance times the channel area times the overdrive, and both terms in that overdrive depend on the input — one directly and one through the body effect. So the 5.67 mV a data sheet would call an offset moves by 1.86 mV across a volt of signal, which is a gain error of 0.186 per cent and only 3.4 µV of anything else. Opening the summing-node switch first divides the gain error by the amplifier's own gain, exactly, and what is left goes from twenty-nine times the sampled-noise floor to nineteen times below it.

filters · Switched capacitor

Named alongside it

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

Switched capacitorAliasingAperture jitterCapacitor ratioCharge injectionDevice mismatchDynamic rangeEquivalent resistanceGain errorImpedance scalingKt over cLoop gain

All concepts