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

Geometric mean — where it appears

The square root of a product, which is the natural centre of a quantity bounded by one thing from below and another from above. It is where a sampling radius or a bisection bracket is placed when the two ends are orders apart, since the interesting point is not near either.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A resonant circuit of Q = 8, and its measured bandwidth. The half-power points are 1.50 kHz and 1.69 kHz, a bandwidth of 198.9 Hz. The components predict f₀/Q = 198.9 Hz. They differ by 0.000%.

Resonance, and the bandwidth it sets exactly

The half-power bandwidth of a resonant circuit is f₀/Q — not approximately, but to every digit the arithmetic has, which is rare enough to be worth checking. What is not exact, and is drawn as though it were, is the idea that the band sits centred on the resonance. At a quality factor of one its middle is twelve per cent above.

frequency · Resonance The best shunt drops 7.75 mV, whatever the current is. computed by solving, not by drawing. Two errors on one axis, both from solved networks: the shunt's own drop, which lowers the current that was to be measured, and the amplifier's 5.0 µV of offset divided by the voltage the shunt develops. The first rises with the burden voltage and the second falls, so the worst case has an interior minimum at 7.7460 mV — the geometric mean of the offset and the 12 V supply — where the error is 0.1291%, being twice the root of the offset over the supply. Neither the shunt's resistance nor the current appears in either number: at 1 A the answer is 7.75 mΩ, and at a hundred times the current it is the same burden voltage across a hundredth of the resistance. What does depend on the current is the 7.7 mW the shunt then dissipates, and 40 K of self-heating at 50 ppm/K is 0.2000% on its own. The third curve is the shunt's own Johnson noise in a kilohertz of measurement bandwidth, as a fraction of the current: it is 4.5e-6% at the best burden and is the only line here that moves with the current at all, falling as one over its square root — so above about an ampere it leaves the bottom of these axes entirely and is drawn nowhere rather than flattened onto the floor.

The ammeter that is a resistor

Every direct measurement of a current is a measurement of a voltage across something the current was made to flow through, so the instrument has two errors pointing opposite ways: a larger shunt changes the current, a smaller one leaves less for the amplifier's offset to be compared with. Written in the burden voltage they are the burden over the supply and the amplifier's offset over the burden, and the best of them is at the geometric mean — 7.75 millivolts on a twelve-volt rail, with a worst-case error of 0.129 per cent. Neither number contains a resistance, and neither contains the current: ten milliamps and a hundred amps want the same 7.75 millivolts.

instruments · Current sensing What a high-side shunt's optimum is made of, at 100 dB of rejection. computed by solving, not by drawing. The same two errors as a low-side shunt, with the amplifier now standing at the rail rather than at the return. 100 dB of rejection turns 12 V of common mode into 120.0 µV of equivalent input error, which is 24 times the amplifier's own 5.0 µV of offset. The optimum keeps its form — the geometric mean of an input error and the supply, golden-sectioned on the solved worst case rather than substituted — and changes its value: 38.73 mV of burden and 0.6440% of error, against 7.75 mV and 0.129% low-side. A shunt sized by the low-side answer reads 1.678% wrong. With the common-mode term removed the optimum returns to the low-side value exactly, which is what says the term is the whole of the difference.

The rail that is an input error

The best burden voltage across a shunt is the geometric mean of the amplifier's offset and the supply, and moving the shunt to the high side does not change that form — it changes what the offset is. A hundred-decibel amplifier on a twelve-volt rail turns the rail into 120 µV of equivalent input error, twenty-four times its own five, so the optimum moves from 7.75 mV and 0.129 per cent to 38.73 mV and 0.644. The two are equal only at 128 dB, and with the common-mode term removed the optimum returns to the low-side value exactly.

instruments · Current sensing With a 2 ps reference and a 0.1 ps-a-period oscillator the best loop has a time constant of 20.0 periods, where the clock's error is 0.453 ps — 90.9 dB on a 10 MHz tone. computed by solving, not by drawing, with seeded locked clocks as dots. A free-running oscillator whose period has a spread of 0.1 ps, locked by a first-order loop to a reference whose timing error, as the phase detector sees it each period, is 2 ps rms and white. The loop passes the reference's error below its corner and the oscillator's walk above it, so the stationary error is σₒ²/(1 − a²) + σᵣ²(1 − a)/(1 + a) with a = exp(−1/τ). The oscillator's part rises as √(τ/2)·σₒ and the reference's falls as σᵣ/√(2τ); they are equal, and their sum least, at τ = 20.00 periods, the ratio σᵣ/σₒ, where the clock's error is 0.4528 ps — √(σᵣ·σₒ). Seeded clocks of 65,536 periods: τ = 2, 1.000 ps; τ = 5, 0.656 ps; τ = 20, 0.452 ps; τ = 80, 0.649 ps; τ = 300, 1.187 ps; τ = 1000, 2.105 ps.

The loop a reference allows

A sampling clock is a free-running oscillator held by a loop to a reference, and the reference has an error of its own. With 0.1 ps of period spread in the oscillator and 2 ps of white error in the reference, a first-order loop of 20 periods leaves the clock 0.453 ps from ideal — 90.9 dB on a 10 MHz tone — and that is the best any first-order loop can do: the oscillator's share rises as the square root of the loop's time constant, the reference's falls as one over it, and they are equal at σᵣ/σₒ. The clock's error there is the geometric mean of the two, so a reference a hundred times worse costs ten. On a spectrum the rule reads as one instruction: put the loop's corner where the free oscillator's skirt meets the reference's floor. And it holds only for long captures. Below about seven loop time constants the free oscillator beats every loop.

digital · Aperture jitter

Named alongside it

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

Design tradeoffInput offsetModel rangeAperture jitterBurden resistanceCommon-mode rejectionCorner frequencyCurrent sensingFour-terminal sensingHalf-power bandwidthLoadingLow-resistance measurement

All concepts