One inductor, and ten components

The rung below this one built an inductor out of four resistors, a capacitor and two amplifiers, and measured it: one henry, accurate to a per cent from below a hertz to 3.65 kilohertz, with a quality factor that peaks at 47,000 and a series resistance that goes negative at 63 Hz. It ended with a third boundary that is not a frequency at all — one end of it is soldered to ground and cannot be lifted.

That is not a footnote. A low-pass ladder’s inductors are in series, between two signal nodes, and neither end is ground. The one arrangement this field would most like to build out of gyrators is exactly the one a grounded gyrator cannot make.

This essay makes the floating version, checks that it is floating, builds a filter out of it, and asks the question the rung below named: does the sensitivity advantage that makes a ladder worth building survive being built this way?

The answer is no, and the shape of the no is worth the essay.

Two amplifiers do not make a floating inductor #

The obvious construction is the rung below’s gyrator with its fifth resistor returned to the far terminal instead of to ground. It is one component moved, and it does not work.

The test is the one a two-terminal element has to pass and nothing else does: drive one terminal with a current and see how much of it arrives at the other. A resistor passes it, a capacitor passes it, and an arrangement containing amplifiers has four other places to send it, because every amplifier’s output is a source referenced to ground.

With one half-gyrator, the current that arrives is 6.3 per cent of the current that left at a hundred hertz, and at three kilohertz 189 per cent of it arrives — more comes out than went in. It is not an element at all; it is a two-port that happens to have an inductive input impedance.

Two halves back to back — each taking its reference from the other terminal, four amplifiers rather than two — deliver 1.00000 at every frequency drawn, to five decimal places.

That the amplifiers are ideal in that measurement is the point of it. Nothing about the failure of the one-half version is a part’s fault: it is the topology, and no amplifier makes it a two-terminal element.

What the arrangement costs before any part does #

The four-amplifier version is floating and it is an inductor — 1000.00 mH at a hertz against a design value of 1000 — and it has a loss that the grounded version did not.

Its series resistance rises as the square of frequency, measured at the 2.00 power with perfect amplifiers, so its quality factor falls as one over frequency: 1,592 at a hertz and 15.9 at a hundred. The grounded gyrator a rung below had a quality factor that peaked at 47,000 and a series resistance of 0.098 Ω at ten hertz.

So the pair is two or three orders worse as an inductor than the single, before any amplifier’s gain–bandwidth is involved, and the reason is structural: each half sees the other half’s terminal as its reference, and each half’s own input network is then a load on the other.

There is a design variable that moves it, and it is the one everything here turns on. The synthesised inductance is $C r^2$, so a given inductance can be built at any resistance scale with the capacitance that follows — and the arrangement’s own corner, $r/2\pi L$, moves with it. At ten kilohms that corner is 5.4 kHz; at a megohm it is 537 kHz; at ten megohms, 5.4 MHz.

The filter, and the scale that makes it the same filter #

Replace both series inductors of a third-order Chebyshev ladder with floating gyrators, leave the shunt capacitor and the two terminations alone, and compare the response.

At the bottom it is a different filter. At the top it is the same filter to three thousandths of a decibel — and the amplifiers are ideal, so what is converging is the arrangement’s own loss falling out of the way rather than a part getting better.

Each decade of resistance scale is worth a decade of departure, which is the loss falling as $1/r$.

So the answer to “can a ladder’s inductors be amplifiers?” is yes, at a price in impedance level, and the price is payable: a megohm of resistor and thirty picofarads of capacitor is an ordinary integrated circuit.

That is the easy half of the question, and it is the half that gets asked.

The sensitivity, which does not converge at all #

This field has said since its ladder essay that a doubly terminated ladder’s response is stationary in every reactive element at its passband maxima, and the networks field has now measured that directly with the adjoint network: at the ripple peak, every reactive element of the passive ladder has a magnitude sensitivity of order $10^{-8}$.

The gyrator ladder has the same response to three thousandths of a decibel. Point the same measurement at it:

The middle row is seven orders above the top one and it does not move with the resistance scale: 0.675 at ten kilohms, 0.522 at a hundred, 0.502 at a megohm, 0.500 at ten. The response converges and the sensitivity does not.

Where the stationarity went #

The bottom row says what happened, and it is not that the stationarity was destroyed.

Every component in one half of a gyrator has a relative sensitivity of about $+\tfrac12$ and every component in the other half about $-\tfrac12$. They are equal and opposite, so a change that moves both halves together very nearly cancels: the combination is stationary to $2.5\times10^{-4}$ at ten megohms, and it falls as one over the resistance scale exactly.

That combination is the one that changes the synthesised inductance. A gyrator’s inductance is $C r^2$ per half, so changing both halves’ capacitors by one per cent changes the inductance by one per cent and nothing else — and the ladder is stationary against that, as the theorem says it must be.

Changing one half’s capacitor by one per cent changes the inductance by half a per cent and makes the two halves different, which is a perturbation of a kind the passive network has no analogue for. The arrangement is no longer symmetric, the two-port is no longer the lossless one the theorem is about, and what comes out is first order.

So the honest statement is this: the stationarity has become a statement about a combination of components, and a manufacturer does not buy combinations. One inductor with a one per cent tolerance has become ten components with ten independent one per cent tolerances, and nine of the ten ways they can be wrong are ways the passive ladder had no way to be wrong at all.

Why the two-terminal test is the right test #

It is worth a paragraph on why “does the current come out the other end” is the measurement, because it looks like a formality and it is the whole difference between the two constructions.

An impedance between two nodes is only a quantity if the element carrying it is two-terminal. Ask what “the impedance between a and b” is for a network with a third connection and the answer depends on what is attached to the third one — so the number a designer wants is not defined, and any figure of it is a figure of one particular arrangement.

The one-half construction has that third connection, and it is invisible in a schematic: the amplifiers’ outputs are drawn as arrows and their supply returns are not drawn at all. A small-signal netlist has no such convention — an amplifier is a controlled source between its output node and ground, which is the truth about what a real one does — so the third connection is there in the arithmetic whether or not it is in the drawing.

That is why the measurement is a current driven in and read out, and why the answer at three kilohertz is 189 per cent. There is nothing wrong with the number; a two-port can perfectly well deliver more current than it is given. What is wrong is calling it an inductor.

The resistance scale, and what it is spending #

Every improvement in this essay comes from raising the resistance scale, and it is worth saying what that costs, because the sweep runs to ten megohms and stops there for a reason.

At a fixed inductance, $C = L/r^2$: the capacitance falls as the square of the scale. A one-henry inductor at ten kilohms needs ten nanofarads; at ten megohms it needs ten femtofarads, which is smaller than the stray capacitance of the node it would be connected to. The stray becomes the component.

So the practical ceiling on $r$ is where the intended capacitance stops being distinguishable from the parasitic one, and on an integrated circuit that is around a picofarad — which for a one-henry inductor is a scale of about a megohm, the middle of the sweep here. The three thousandths of a decibel at ten megohms is available in arithmetic and not on a die.

That is the same shape of boundary this field measured in its impedance-scaling argument, where a realisation’s band of possible impedance levels is closed at the top by stray capacitance and at the bottom by amplifier output resistance. This arrangement sits inside that band and the same two things close it.

What the trade actually is #

Putting the three realisations of one response side by side, at the passband maxima:

• a buffered cascade — one amplifier and a handful of parts per pole, first-order sensitive, no inductors;
• a passive ladder — two inductors and a capacitor, second-order sensitive, and the inductors are wound components with tolerances, losses and a size;
• a gyrator ladder — eight amplifiers and twenty parts, first-order sensitive, no inductors.

The gyrator ladder has the cascade’s sensitivity and the ladder’s topology, and it is much the most expensive of the three. On the one quantity the ladder was chosen for, it is the cascade.

That is a clean answer to the question the rung below asked, and it is worth saying what it does not mean. Gyrators are used, widely, and not because anybody believed this: they are used because a wound inductor at audio frequencies is large, expensive, picks up mains hum and saturates. Replacing it with an integrated circuit is a good trade for all of those reasons. It is not a good trade for the reason a ladder is chosen over a cascade in the first place, and choosing the ladder topology because of its sensitivity and then realising it this way is buying nothing with eight amplifiers.

What a grounded gyrator is still good for #

None of this says the rung below’s arrangement is not worth having, and it is worth being explicit about where it stands, because the two are separated by one connection and by everything else.

A shunt inductor in a filter has one end at ground, and a grounded gyrator makes it: two amplifiers, a quality factor that peaks at 47,000, a per cent of accuracy over three and a half decades. That is a good component. A high-pass ladder’s inductors are shunt elements, and so is the inductor in a single-resonator notch, and so is the one in an equaliser section — all of which this field builds.

What has no grounded form is the series inductor of a low-pass ladder, and that is exactly the element the sensitivity argument is about. So the collision here is not between gyrators and ladders in general; it is between the arrangement that costs least and the topology that is chosen for its stationarity, and those two happen to want the inductor in different places.

The practical reading is short. Use a grounded gyrator wherever the inductor is grounded, and expect it to behave. Where the inductor is not grounded, the four-amplifier arrangement will reproduce the response and will not reproduce the sensitivity — so choose the topology on some other ground, because the one that usually decides it has stopped applying.

What the trade is measured against #

Every claim on this page is a distance from something measured elsewhere in the field. A ladder is not a cascade is the sensitivity result the gyrator ladder is trying to inherit, and Every derivative, and the one that is zero is where that result stops being a comparison and becomes an exact statement about a derivative — which is the statement this arrangement does not satisfy. The floor that outlives the arithmetic is the passive ladder’s own ceiling, so the honest comparison is against a floor rather than against perfection. The band that closes with the order is the cascade’s version of the same accounting, and A resistor made of a clock is the other way this field builds a component out of something that is not one — with a quite different list of things it gives up.

What is checked #

That the four-amplifier arrangement is floating is asserted as a measurement rather than as a property of the drawing: the current arriving at the far terminal must equal the current driven into the near one to four decimal places, at every frequency swept. That the two-amplifier version is not is asserted the same way, so the check would fail if the topology were ever quietly changed back.

The arrangement’s own loss is asserted with ideal amplifiers, and its exponent fitted: the series resistance rises as the square of frequency, which places the loss in the arrangement rather than in the parts. One half is asserted to present twice the design inductance and the pair to present it, which is the arithmetic of two in parallel and would catch a scaling error that a single measurement would not.

The response is asserted to converge on the passive ladder’s as the resistance scale rises, to under a twentieth of a decibel at the top, and to be decibels rather than tenths out at the bottom — three assertions, because a realisation that were always right or always wrong would pass a weaker one.

And the sensitivity carries the finding as three separate claims: the passive ladder is under $10^{-6}$; every single component of a gyrator is above 0.3 at every resistance scale, so the failure is not a matter of building it better; and the combination of both halves is three orders below a single component at the top of the sweep and falls as one over the scale.

What is not modelled: the amplifiers, in most of this — they are ideal wherever the argument is about the arrangement, and the inductor that is an amplifier has already measured what a real one costs a grounded gyrator, which is an inductance good to one per cent over three and a half decades with a series resistance that goes negative at 63 hertz, well inside the band where it is still excellent; the resistors’ own tolerances, which are ten more independent parts and would make the comparison worse rather than better; 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.

The last two of those turn out to be one quantity, and eight amplifiers, and what they add is where that is measured. The resistors that buy the accuracy are the noise, and the amplifiers inside the gyrators carry the inductor’s own current through them — so the floor rises as the square root of the impedance scale while the ceiling falls as the scale. Which lands directly on this essay’s own escape route: the cure for the twenty-two decibels of departure at ten kilohms is a higher impedance level, and a higher impedance level is exactly what closes the dynamic range from both ends at once.

That leaves the comparison this essay makes in an awkward and honest place. The band that does not close finds a doubly terminated ladder still realisable across four decades of impedance level at order nine, where a cascade of active sections has none at all at eight — and the whole of that advantage rests on the stationarity this essay has just measured a gyrator-realised ladder as not having. A gyrator ladder is therefore on the cascade’s side of that comparison rather than the ladder’s, while paying the ladder’s component count.