The inductor that is an amplifier

Two essays in this collection have named a gyrator and neither has built one.

The delay equaliser’s prose says that an all-pass section at audio frequencies wants inductors of several henries, that such filters were built with gyrators for exactly that reason, and that a gyrator is an amplifier with its own three boundaries. The magnetics field’s two-port essay names it as the element that breaks reciprocity — the one thing a network of resistors, capacitors and inductors cannot do, and the reason the off-diagonal impedance parameters of everything else in this collection come out equal.

It is time it was in the netlist. What follows is one, solved rather than described, and the three boundaries measured rather than named.

Five components and one line of algebra #

The arrangement is Antoniou’s, and it is five impedances with two amplifiers holding four nodes equal in pairs. The input impedance is

$$Z_\mathrm{in} = \frac{Z_1 Z_3 Z_5}{Z_2 Z_4}$$

so making $Z_4$ a capacitor and the other four resistors gives $Z_\mathrm{in} = sC\,R_1R_3R_5/R_2$ — an inductance, standing between the input node and ground, made of parts that are none of them inductive.

With four ten-kilohm resistors and ten nanofarads that is $10^{-8} \times 10^4 =$ one henry. As a wound component, one henry is a few thousand turns on a core, several hundred ohms of wire and a self-resonance somewhere in the tens of kilohertz; here it is five parts that cost nothing and occupy a square centimetre.

It is worth saying plainly what it is not. Nothing in this circuit stores energy in a magnetic field. The current lags the voltage because an amplifier is holding a capacitor’s voltage somewhere else in the loop, and every property a real inductor has that this one does not — and every property this one has that a real inductor does not — follows from that one substitution.

Measuring an impedance rather than asserting one #

The measurement is the definition. A current source of one ampere is placed at the input node, the network is solved, and the node voltage that comes back is the impedance. Its imaginary part divided by $\omega$ is the inductance the arrangement behaves as; its real part is the series resistance; the ratio is the quality factor.

At a hundred hertz the answer is 1000.2 millihenries against a design value of 1000.0, two parts in ten thousand — and the two tenths of a per cent that is missing is not a numerical error, it is the amplifiers’ finite open-loop gain of $10^5$. Replacing them with ideal ones in the same netlist gives the design value to six figures at every frequency drawn, which is the check that separates a property of the topology from a property of the parts.

The band over which it is an inductor to within one per cent runs from below a hertz to 3.65 kilohertz for one-megahertz amplifiers: three and a half decades. It is bounded above and not below, which is the first difference from a wound coil — a real inductor’s impedance stops being $\omega L$ at both ends, resistive below and capacitive above, and this one has no wire to be resistive.

The first boundary: the amplifiers run out #

The upper edge is where the amplifiers stop holding the nodes they are holding.

Both amplifiers in this arrangement are used as unity-gain-ish followers of the input node, so what matters is their loop gain at the frequency in question, and a one-megahertz part has a loop gain of a hundred at ten kilohertz and ten at a hundred. When the loop gain falls low enough that the node voltages are no longer equal to the precision the impedance transformation needs, the transformation stops being $Z_1Z_3Z_5/Z_2Z_4$ and becomes something with the amplifier’s own dynamics in it.

Measured, the departure reaches one per cent at 3.65 kilohertz for a one-megahertz part — which is 0.37 per cent of the gain–bandwidth, not one per cent or ten. It is a factor of two hundred and seventy below the frequency a data sheet suggests. That factor is worth carrying: the loop here is not a simple follower, it is two nested ones with the transformation in between, and the accuracy demanded of each is the accuracy demanded of the impedance.

So the useful design statement is a ratio rather than a frequency. A synthetic inductor is good to one per cent about two and a half decades below the amplifiers’ gain–bandwidth, and it scales with the parts: ten-megahertz amplifiers give 12.5 kilohertz, and a hundred give 40.

The second boundary: the resistance changes sign #

The second edge is the one no passive component has, and it arrives long before the first.

The series resistance of this inductor — the real part of that solved impedance — is 0.098 ohms at ten hertz, which is already a hundred times better than any wound coil of a henry. It crosses zero at 63 hertz. Above that it is negative: −25 ohms at a kilohertz, where the inductance is still within 0.23 per cent of its design value.

That is not a degraded inductor. It is an active element. A real inductor’s series resistance is wire and is positive at every frequency; this one is the residue of two amplifier loops that do not quite cancel, and the residue changes sign. A resonator built around it at a kilohertz does not have a very high quality factor — it has a negative loss, and it will start oscillating on its own noise.

The quality factor as usually computed says nothing useful about this at all. It peaks at 47 000 just below the crossing, passes through infinity at it, and comes back as a large negative number above — which in a specification would be read as “excellent” and is the number changing sign as it goes through the point where the component stops being passive.

So the useful boundary here is not a $Q$ and not a bandwidth: it is the frequency at which the sign of the real part changes, and it is a decade and a half below the frequency at which the inductance stops being right. A designer who checks only the inductance will build the oscillator by accident.

Why the resistance goes negative, and not positive #

An error that comes out negative when the obvious guess is positive deserves a mechanism rather than an observation, and the mechanism here is a phase shift rather than a loss.

The transformation depends on two amplifiers holding two node voltages equal to the input’s. Each holds its node with a finite loop gain that falls with frequency and, crucially, lags: a one-pole amplifier’s loop gain has ninety degrees of phase at every frequency well above its own dominant pole. So the node the amplifier is holding is not merely a little short of where it should be — it is a little short and a little late.

An impedance built out of quantities that are a little late has its own phase shifted, and shifting the phase of a mostly-inductive impedance towards ninety-plus degrees is precisely what moves its real part below zero. The lag is a gain of phase in the wrong direction, so the residue it leaves in the real part has the sign of a source rather than of a load.

Which also says why it is not a small effect. The imaginary part is $\omega L$ and is large; the real part is the difference of two nearly equal things and is small; so a fraction of a degree of phase error appearing in a large imaginary part is a substantial number in the small real part. At a kilohertz the inductance is 0.23 per cent out — nothing — and the real part is −25 ohms, which is the whole of the component’s loss budget several times over.

That is the general shape of the trap and it is worth carrying beyond this component. A quantity computed as the small residue of two large ones inherits the phase errors of both, and a specification written about the large quantity says nothing at all about the small one.

The third boundary: one end of it is soldered down #

The third is not a frequency at all, and it is the one that decides where the component may be used.

The arrangement measures the current out of one node and returns a voltage there. There is no second terminal: the fifth resistor goes to ground and the whole transformation is defined with respect to ground. So this is a grounded inductor, and any filter that needs a floating one — which is every series arm of every low-pass ladder in this field — cannot use it.

That is why synthetic inductors appear in high-pass and band-pass ladders, where the inductors are shunt elements, and not in the low-pass ladders this field’s sensitivity essays are about. The transformation that gives a floating inductance needs two of these arrangements, back to back, with everything that implies for the number of amplifiers, the matching between them and the number of ways it can be made to oscillate.

It is also the sharpest possible statement of the difference between this and a coil. A wound inductor is a two-terminal component and knows nothing about ground; this one is a three-terminal network with a reference, and the reference is not optional.

What it is exactly right about #

Everything above is a list of ways this component is not an inductor, so it is worth ending on what it is exactly.

It is scalable without limit in impedance, which is the thing this field’s realisability essays spend their time on. One henry needs a thousand turns; a thousand henries needs a hundred thousand and is not a component. Here it is $C R_1R_3R_5/R_2$, so a thousand henries is a hundred nanofarads and the same four resistors, at exactly the same accuracy, in the same square centimetre.

It has no core, so nothing about it saturates, walks, or depends on temperature through a permeability — the three boundaries the magnetics field spends four essays on do not exist for it. Its temperature dependence is a resistor ratio’s, which is parts per million per kelvin and cancels between the numerator and denominator.

And it has no external field, which is a property nothing else in this collection measures but which decides layouts: a wound inductor of a henry couples to everything near it, and the crosstalk essay in the lines field is about a far smaller mutual inductance than that.

So the trade is legible. A gyrator buys size, scalability and the absence of a core, and pays with a finite band, a series resistance that changes sign inside it, and a terminal that cannot be lifted from ground.

What the alternative costs, in this collection’s own numbers #

The comparison worth making is with the component this replaces, and this collection has already measured most of it.

A one-henry wound inductor is a few thousand turns. Its winding resistance is set by the wire and the mean turn length and is hundreds of ohms — the magnetics field’s own ohms-per-henry figure of forty is for small values and rises with the count. Its self-resonance, from the winding capacitance, is in the tens of kilohertz, so the upper edge of a wound henry and the upper edge of this synthetic one are not far apart. Its temperature coefficient is copper’s, which is four thousand parts per million per kelvin in the resistance. And it saturates.

Against that, the synthetic one has a hundredth of the series resistance where the sign is right, an edge at the same order of frequency, a temperature coefficient three orders smaller, and no saturation at all. It costs two amplifiers, their supplies, their noise, and the negative resistance above 63 hertz.

The honest summary is that this is not a substitute for a small inductor and is an excellent substitute for a large one — which is exactly the region where wound components stop being components. A microhenry is a few turns of wire and nothing here competes with it; a henry is an object, and five parts and two amplifiers is a better object.

What a synthesised component is measured against #

A gyrator is a component built out of things that are not one, and this field builds two of them. A resistor made of a clock is the other, with a quite different list of what it gives up, and The filter that samples is where that list becomes a filter’s own behaviour. What both are measured against is the passive object they stand in for: A ladder is not a cascade establishes what a doubly terminated ladder’s sensitivity is worth, The floor that outlives the arithmetic establishes the floor that ladder itself has, and The inductor that is a capacitor establishes what a wound inductor stops being above a frequency. The reading that does not care which way round it is is the property a gyrator deliberately breaks.

What is checked #

The inductance is asserted against the five components’ own product, to one per cent, at the middle of the band. The upper edge is asserted to lie well below the amplifiers’ gain–bandwidth, so that it is the parts and not the arrangement that ends it — and the same netlist with ideal amplifiers is asserted to have no such edge inside the range drawn, which is the check that separates the two.

The sign change is asserted to exist, to be the first one, and to happen at a frequency where the inductance is still within one per cent of its design value: that ordering is the essay’s central claim and the one a designer would be hurt by if it were the other way round. And the ideal-amplifier netlist is asserted to have nothing there to change sign, so the negative resistance belongs to the parts.

What is not built here: the floating version, which needs two of these; the noise of the arrangement, which has four resistors and two amplifiers in it where a coil has wire; and a filter actually realised with one, which would let the sensitivity comparison this field is built on be made against a structure whose inductors are amplifiers.

All three were built, and each answered its question with a number that changes what this component is for.

One inductor, and ten components builds the floating version and finds it exactly floating to five figures, reproducing a doubly terminated ladder’s response to a hundredth of a decibel — and then finds the thing that matters more than the response: it does not reproduce the stationarity. Each of the gyrator’s ten components moves the response by exactly a half, where the inductor it replaced moved it by $10^{-8}$. The whole reason to build a ladder rather than a cascade is that its response is stationary in every element at its passband maxima, and a ladder whose inductors are gyrators has given that away while keeping the topology.

Eight amplifiers, and what they add then measures the noise and the headroom, and finds them to be one quantity rather than two. 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, and the arrangement’s dynamic range closes from both ends as the accuracy improves. That essay also settles the accuracy question this one leaves open: the response converges on the passive one as the resistance scale rises, twenty-two decibels out at ten kilohms and a twentieth of a decibel at ten megohms.

Read together with the negative resistance measured here, the three say what a synthetic inductor actually is. It is an excellent inductance over a wide band, it is a poor inductor in every respect that made a ladder worth building, and the frequency below which its loss changes sign is inside the band where the first sentence is still true.

That essay — flat delay, bought with more delay — is also the place where the negative resistance measured here does the most damage, and it is worth saying why rather than leaving it as a caution. An all-pass section’s defining property is that its magnitude is one at every frequency, measured there on a solved network as 1 to within $9\times10^{-16}$ over six decades. That holds because the section’s pole and zero are exact mirror images, which in turn holds because the resonator’s loss appears identically in numerator and denominator. A resonator whose loss changes sign somewhere in the band does not have mirrored pole and zero there, so the magnitude departs from one — and it departs at frequencies where the delay curve, which is what the section was fitted for, looks entirely correct. The one component that would make a delay equaliser buildable at audio frequencies is the one that quietly removes the property the equaliser was chosen to have.