Eight amplifiers, and what they add

The rung below this one built a floating inductor out of two half-gyrators, checked that it was floating, realised a fifth-order Chebyshev ladder with them, and measured what the realisation costs. Its answer was that the response converges on the passive ladder’s as the resistance scale rises — twenty-two decibels out at ten kilohms, under a twentieth of a decibel at ten megohms — and that the sensitivity advantage a ladder is built for does not survive being built that way at all.

It closed by naming two quantities it had not measured: the noise, which is eight amplifiers’ worth, and the dynamic range, since every internal node of a gyrator swings and some of them swing more than the signal does.

They are the same quantity, and both of them are made worse by exactly the thing that made the response better.

Every resistor’s own noise, one at a time #

A filter’s floor is not a property a designer can read off a schematic, because how much of each resistor’s noise arrives is a different transfer function for every resistor in the circuit. So each one is split, given a source of its own 4kTR in series with it, and solved for the transfer to the output — which is what a noise source is, written literally: a noiseless resistance with a voltage source in series, exactly as the field’s founding essay puts one in a network and as the capacitive-load essay does for three feedback arrangements.

The passive ladder realising the same response has a floor of 1.211 µV rms over 100 Hz to 100 kHz, and 97 per cent of the power in it is the load resistor. That is the least a filter can have: the terminations are the only resistances in the circuit, and one of them is the load the signal is delivered to.

The active realisation at a megohm has 78.8 µV — sixty-five times more — and the six loudest contributors are the gyrators’ own fifth resistors. Those are the megohms the rung below needed for accuracy.

Where the sixty-five comes from #

The factor between the two realisations is worth taking apart, because it is not one thing.

Two of it is the count. A passive fifth-order ladder has two resistors in it; this one has twenty-six, because each floating inductor is two half-gyrators and each half-gyrator is five resistors. Noise powers add, so twenty-six sources instead of two is a factor of about 3.6 in voltage before anything about their values is considered.

Most of it is the values. A passive ladder’s resistors are its terminations — a kilohm each at this impedance level — and the active one’s are megohms. √(4kTR) is thirty-two times larger for a megohm than for a kilohm, and that is the bulk of the sixty-five.

And a small part of it is the transfers. Not every source arrives with the same weight: the six loudest are the fifth resistors of the three inductors’ six half-gyrators, and they are loudest because each sits at the point in its gyrator where the whole synthesised current flows. The other twenty contribute measurably less, which is why the total is sixty-five times rather than the hundred the count and the values alone would give.

Taking it apart that way says which lever is available. The count is fixed by the topology. The transfers are fixed by the arrangement. The values are the only free variable, and they are the ones the rung below found the accuracy depends on — which is the trade, restated as an accounting.

The floor rises as the square root of the scale #

Sweep the resistance scale and the law is exact.

At a hundred kilohms, three hundred, one megohm, three and ten, the floor is 24.9, 43.2, 78.8, 136.4 and 249.1 µV: a factor of √10 per decade, to the digits drawn. That is not a surprise — Johnson noise is √(4kTR), the transfers are dominated by the same resistors, and the signal path is unchanged — but it is worth having as a measurement, because it is the exact counterpart of the rung below’s own law and points the other way.

The rung below found the response error falling as roughly one over the scale. This finds the floor rising as the square root of it. Every decade of resistance buys about a factor of ten in accuracy and costs a factor of 3.16 in noise, and there is no arrangement of the same topology that avoids the trade, because the resistors that set the accuracy are the resistors that make the noise.

The amplifier carries the inductor’s current #

The ceiling is the more striking half and it is not a noise argument at all.

Solve the active ladder at the frequency where its response peaks and look at every node in the netlist, not only the output. At a megohm, an amplifier output inside the first gyrator carries 780 times the input voltage.

The reason is mechanical rather than subtle. A gyrator makes an inductance by forcing the inductor’s own current through its own resistors. At 10 kHz the first inductor is 28.8 mH, its reactance is 1.8 kΩ, and a volt across it is half a milliamp — which through a megohm is five hundred and fifty volts. The amplifier that must produce that current has to produce that voltage, and the internal node swings by exactly the ratio of the resistance scale to the inductor’s reactance.

So the swing rises as the scale itself — fitted at 1.012 across two decades — while the floor rises as its square root. Dynamic range on a fixed supply therefore falls as the three-halves power:

The passive ladder realising the same response has 141 decibels, and its worst internal node carries 1.10 times the input.

The rung below’s cure is this rung’s disease #

Put the two laws together and the anchor’s design problem is fully stated for the first time.

The rung below’s response error falls with the scale. This rung’s dynamic range falls with the scale faster. There is no scale at which both are good: at ten kilohms the response is twenty-two decibels wrong, and at ten megohms it is right to a twentieth of a decibel and the filter has eighteen decibels of dynamic range, which is three bits.

The usable window is in between and it is narrow. A megohm gives a response error the rung below measured in tenths of a decibel and forty-eight decibels of range — a filter that is accurate and can handle a signal swing of about a fiftieth of the supply before an amplifier nobody is looking at clips. That is the honest specification of this realisation, and neither half of it appears on either of the two rungs below.

It also explains a fact about the technique that is otherwise puzzling: gyrator filters are used at audio frequencies and almost nowhere else. At audio, the inductances wanted are henries, their reactances are kilohms at the frequencies of interest, and the ratio of the resistance scale to the reactance — which is the internal swing — is manageable. At radio frequencies the inductances are microhenries with reactances of tens of ohms, and the same arrangement asks its amplifiers for four orders of magnitude.

The same reading explains the other observed habit, which is that gyrator filters are built at the lowest impedance level that will do. Impedance scaling moves every reactance and every resistance together and leaves the response alone, so it looks like a free choice — and for a passive ladder it very nearly is, bounded only at the two ends the impedance-scaling anchor measures. Here it is not free in either direction: scaling the filter down lowers the inductances and raises their currents, and scaling the gyrators’ own resistance up is the axis this essay has just priced. The two are separate knobs on the same design and both of them run into this ceiling.

What a dynamic range is, and why it needs two solves #

It is worth being explicit about what has been computed, because the phrase covers two different measurements in ordinary use.

A dynamic range is a ceiling divided by a floor, and neither of them is a property of the output. The floor here is an integrated noise voltage referred to the output; the ceiling is the largest input that keeps every internal node inside the supply, referred to the output through the passband gain. They come from two independent solves of the same netlist — one with a source at each resistor and no signal, one with a signal and no sources — and neither can be inferred from the other.

That is the shape the noise field’s own dynamic-range essay establishes and it is worth repeating here because the active ladder is a case where the two ends belong to different components. The floor is set by the gyrators’ fifth resistors and the ceiling by the gyrators’ amplifier outputs, and a designer who improved either one alone would find the other unmoved.

The passive ladder’s 141 decibels is the same arithmetic applied to a circuit where both ends are almost trivial, and it is a useful yardstick precisely because it is so nearly the theoretical best: a filter whose only resistances are the ones its design requires, and whose largest internal signal is its own passband ripple.

The clipping is invisible #

There is a practical hazard in the ceiling that the number understates.

An overload at an internal node does not look like an overload. The output does not clip — it is 78 dB below the internal node — so what a user sees is a filter that is linear for small signals and develops distortion as the level rises, with no flat top anywhere and no obvious threshold. It is the same failure mode the dynamic-range essay describes for a cascade, where the first stage decides the floor and some later stage decides the ceiling, made worse by the fact that here the deciding stage is inside a two-terminal element that the schematic draws as an inductor.

That is the strongest argument in this essay against the technique, and it is an argument about debuggability rather than about performance. A passive ladder’s largest internal signal is 1.1 times its input and its failure is visible at the output. An active one’s is a thousand times its input, at a node with no name on the schematic, and its failure is a distortion figure that gets slowly worse.

What is not modelled #

The amplifiers are ideal. Every number here uses nullors, so the amplifiers contribute no noise of their own, have infinite gain and never run out of anything. That is deliberate and it makes the answer a lower bound: a real amplifier adds its own voltage noise at each of eight inputs, and the rung below has already measured what a real one costs a grounded gyrator in bandwidth. The floor here is what the arrangement cannot get below, not what it achieves.

At a megohm the ideal-amplifier assumption is a good one for the noise, and that is worth saying because it is not obvious: a megohm’s Johnson noise is 128 nV/√Hz against a good amplifier’s 4, so the resistors dominate by thirty to one and the amplifiers’ own contribution is under a tenth of a per cent of the power. At ten kilohms the ratio is three to one and the amplifiers matter. So the ideal assumption is best exactly where the arrangement is worst, which flatters it.

The resistors have no tolerance. The rung below measured what a per cent on each of ten components does to the response and found every one of them above 0.3 at every scale. Those tolerances are not in this calculation and would make the floor slightly worse and the ceiling considerably less predictable.

Nothing here is a distortion measurement. The ceiling is quoted as the level at which an internal node reaches the supply, which is where the arrangement stops working at all. Real distortion starts well below that and rises smoothly, so the usable range is smaller than the number above by whatever the specification’s distortion limit costs.

And there is no comparison against a cascade. The active ladder’s competitor in practice is not the passive ladder — which needs inductors, which is why the gyrator exists — but a cascade of biquads, whose dynamic range is set by an entirely different mechanism and whose sensitivity the tolerance essay measured at fifty-four times the ladder’s. That three-way comparison is the one a designer actually faces and this essay gives one leg of it.

What the gate checks #

The active realisation’s floor is asserted to be above the passive one’s at every scale drawn, which is the weakest form of the claim and would catch a transfer function computed the wrong way round.

The passive ladder’s floor is asserted to be the load resistor and almost nothing else — over ninety-five per cent of the power — which is what establishes the baseline as a physical minimum rather than as a number this particular design happened to reach.

The active one’s loudest contributors are asserted to be the gyrators’ fifth resistors by name, so a version whose largest noise came from somewhere else would fail rather than be plotted.

The two exponents are asserted separately and against their predicted values: the floor as the square root of the scale to five per cent, and the swing as the scale itself to five per cent. Either alone would be consistent with a mistake in the other, and the three-halves power that follows is the product of the two rather than a third fitted number.

And the internal swing is asserted to rise monotonically with the scale, which is what says the two ends of the dynamic range close together rather than trading against each other.

What this anchor now says #

Three rungs, and the arc is worth stating plainly because it is not the one the first rung suggested.

The first built an inductance out of an amplifier and measured the band over which it is one: a henry, accurate to a per cent from below a hertz to 3.65 kilohertz, with a quality factor peaking at 47,000. That is a good component.

The second made it floating, built a filter with it, and found the sensitivity advantage gone: every single component of a gyrator moves the response by more than 0.3 where the passive ladder’s every element moves it by 10⁻⁸.

This one prices the two quantities that were left. The answer is that the realisation is accurate where it is noisy and quiet where it is inaccurate, that its ceiling is set by a node the schematic does not draw, and that the window between the two is about forty-eight decibels wide at the scale where the response is right.

None of that says the technique is wrong. It says what it is for: a filter at audio frequencies, at a signal level well below the supply, where an inductor of the required value would be large, expensive and worse. What it is not is a drop-in realisation of a passive ladder, and the three rungs together are the measurement of the difference.

Which is worth stating as three numbers rather than as a conclusion, because each was measured separately and none of the three follows from the others. The inductor that is an amplifier found the component excellent as an inductance — one henry to within one per cent over three and a half decades — and found its series resistance going negative at 63 hertz, so a resonator built there starts on its own noise. One inductor, and ten components found the floating version exactly floating to five figures and reproducing the ladder’s response to a hundredth of a decibel, while each of its ten components moves that response by exactly a half against the inductor’s $10^{-8}$. And this essay finds the two quantities that pay for the accuracy to be the same quantity, moving in opposite directions with the same parameter.

Read against the band that does not close, that is the whole of what the substitution costs. The doubly terminated ladder’s advantage over an active cascade is four decades of realisable impedance level at order nine against none at all at order eight, and the entire advantage rests on the stationarity the second of those measurements found absent. A gyrator ladder keeps the topology, the component count and the design procedure of a ladder, and inherits the cascade’s sensitivity — which is the one property the topology was chosen for.