The ideal amplifier, and where it stops being one

The two golden rules of the ideal operational amplifier are that its input terminals draw no current and sit at the same potential. From those two statements every standard circuit follows in a line or two of algebra, and the answers are exact: an inverting amplifier’s gain is exactly the ratio of two resistors, a follower’s gain is exactly one, an integrator integrates exactly.

They are also two statements that no device satisfies, and the frequency at which the failure starts to cost something is computable from one number on the datasheet.

The ideal amplifier as an element rather than a rule #

The horizontal line in that figure is not drawn at the ratio of two resistors. It is the solved response of a network containing a nullor — an element with two terminals that draw no current and are constrained to the same potential, and an output branch carrying whatever current that requires.

That distinction matters. A line drawn at the resistor ratio would be a restatement of the rule being tested; a solved network containing a nullor is the rule applied to this particular arrangement of components, and it would produce a different answer if the arrangement were different. The nullor stamps into the nodal matrix as one extra unknown and one extra equation, and it is the exact expression of the two golden rules rather than an approximation to them.

The alternative that many treatments use — a controlled source with a very large gain, say a million — is a numerical trick rather than a model. It puts a 10⁶ next to a 10⁻³ in the same matrix, and then blames whatever comes out on floating-point arithmetic. The nullor has no large number in it anywhere.

What the real one is #

The real amplifier in the figure is three elements: a controlled source of gain 10⁵ driving an internal node, a resistor and capacitor there setting a pole at ten hertz, and a unity-gain source with a small series resistance presenting the result to the outside world. Solving that gives an open-loop gain of 10⁵ at direct current falling at twenty decibels per decade above ten hertz, which is what a general-purpose amplifier’s datasheet describes.

The product of those — a hundred thousand times ten hertz — is a megahertz, and it is constant everywhere above the pole. That is the gain–bandwidth product, and it is the single number that decides everything in this essay.

Expanding the model into elements rather than writing down a gain function is not a stylistic choice. It means the feedback network’s loading of the output, the amplifier’s output impedance and their interaction are all solved rather than assumed away — and those are exactly the effects a formula would drop.

Three boundaries, not one #

The figure reports three numbers and keeping them apart is most of the content.

The direct-current error. The closed-loop gain is not the resistor ratio even at zero frequency. The real one is low by a factor of 1/(1 + A₀β), and for a gain of 100 from an amplifier with 10⁵ of open-loop gain that is 0.0999% — small, but not zero, and it is the first thing a reader should know because it is present before any frequency argument is made at all.

The one-per-cent frequency. The real response departs from the ideal answer by one per cent at 1.35 kHz. This is the number that matters in practice and it is the one nobody quotes.

The bandwidth. The response is 3 dB down at 10.0 kHz, which is the gain–bandwidth product divided by the closed-loop gain, and it is the number the datasheet does quote.

The gap between the second and third is a factor of seven, and it is the same shape of gap as elsewhere in this collection: the quoted boundary is where the model has completely failed, not where it started to. A designer working to the datasheet number is working to a frequency at which the answer is already thirty per cent wrong.

Across the slider:

Every row of the right-hand column is the megahertz divided by the gain. That is the whole content of “gain–bandwidth product”, and it is a genuine constraint rather than a convention: gain and bandwidth are one resource, and asking for more of one is spending the other.

The bottom row is worth reading twice. A gain of two thousand from a one-megahertz part has a bandwidth of 510 Hz, and is already one per cent low at one hertz — because at that gain the direct-current error alone is two per cent, so the response never gets within one per cent of the ideal answer at any frequency at all.

Where the accuracy comes from #

The reason the closed-loop gain is so much more accurate than the amplifier is worth stating, because it is what feedback is for.

The excess of open-loop gain over closed-loop gain is the loop gain: a hundred thousand divided by a hundred, which is a thousand. The error in the closed-loop gain is roughly the reciprocal of one plus that, so a loop gain of a thousand gives a tenth of a per cent. Everything the amplifier gets wrong — its gain, its nonlinearity, its drift — is divided down by the same factor.

And the loop gain falls at twenty decibels a decade, so the accuracy falls with it. At 1.35 kHz the loop gain is down to about a hundred and the error is one per cent; at ten kilohertz the loop gain is unity and there is nothing left to divide anything by. The closed-loop response follows the ideal answer only while there is loop gain to enforce it, and the whole shape of the figure is a picture of loop gain running out.

The refusal #

Above the frequency where the loop gain reaches unity, the ideal-amplifier model does not give an approximate answer. It gives a specific number — the resistor ratio — for a circuit that produces something entirely different, and being told the answer is “approximately” the resistor ratio is worse than being told nothing.

So the model declines. Asked for its prediction above the closed-loop bandwidth, it raises an error naming the frequency, the gain asked for and the bandwidth available, and says that the real amplifier has less than unity loop gain there so the ideal answer is not an approximation to anything. That refusal is exercised in the figure — it is run, and the message is checked — rather than described.

A refusal that is described but never triggered is a comment. Half the value of an assertion is what it rejects, and an assertion that has stopped rejecting has stopped testing anything at all.

What the nullor does and does not assume #

The ideal element used here deserves one more paragraph, because it is exact about two things and silent about several others, and knowing which is which prevents an over-reading of the flat line.

The nullor states exactly that the two input terminals draw no current and sit at the same potential, and that the output supplies whatever current is required to make that true. Everything the ideal- amplifier rules give follows from it — the virtual earth of an inverting stage, the unity gain of a follower, the exact resistor ratio.

What it does not assume, because nothing in it mentions them, is anything about supply rails, output current limits, input voltage range or offsets. So the flat line in the figure would still be flat at a gain of a million and an output of ten thousand volts, which is a reminder that the ideal model’s silence is not the same as its permission.

The real model in the same figure is silent about most of those too. It has finite gain and one pole and nothing else — no slew limit, no saturation, no input offset — so it is a better model in exactly one respect and no better in the others. That is the usual situation with a hierarchy of models, and it is why the site names the boundary each figure is about rather than implying that the better model is better everywhere.

Where the gain–bandwidth product comes from #

The constancy of the product is worth deriving in one line, because it makes the constraint feel structural rather than empirical.

Above its dominant pole the amplifier’s open-loop gain falls as 1/f, so the product of gain and frequency is a constant — that is what a slope of twenty decibels per decade means. Closing a loop for a gain G means the loop gain reaches unity where the open-loop gain has fallen to G, and by the constancy that is at a frequency of GBW/G. So the closed-loop bandwidth is GBW/G, and gain times bandwidth is GBW again.

The whole argument is one line because the single-pole shape is what makes it one line, and that shape was put there deliberately. An amplifier with two poles close together would not have a constant gain–bandwidth product, and it would also be unstable at low gains — which is the reason it is not built that way.

Measured on the model in this figure the product is constant to about four per cent across three decades of gain, and the small deviation is itself informative: it is largest at low closed-loop gain, where the loop gain is largest and the feedback network loads the amplifier’s output resistance hardest. The idealisation is that the output resistance is zero; the four per cent is what that idealisation costs.

The other limits on the same part #

Two boundaries belonging to the same device are elsewhere in this collection and it is worth placing them against this one, because they are in different units and neither predicts the other.

The slew rate limits what the output can do at an amplitude, and it is a nonlinearity: no model of any order that satisfies superposition can express it. For the same class of part the boundary is about eighty millivolts of step, which is far below anything called large.

The input impedance and offset limit what the amplifier does at direct current and with high source resistances, which is a boundary in ohms.

Three limits, three units, one device. A datasheet quotes them separately because they are separate, and a model that captures one says nothing about the others. That is the site’s rule in its most concrete form: the range of a model is not derivable from the model.

What else the loop gain buys #

Bandwidth is the most visible thing the loop gain is spent on, and it is not the only one. Every imperfection of the amplifier is divided by one plus the loop gain, which means the same falling curve governs several specifications at once.

Distortion. The amplifier’s own nonlinearity is reduced by the loop gain, so a stage with a thousand of loop gain has a thousandth of the open-loop distortion. Since the loop gain falls at twenty decibels a decade, distortion rises at twenty decibels a decade — which is why an amplifier’s distortion specification is quoted at a frequency, and why the number at twenty kilohertz is typically a hundred times the number at twenty hertz.

Output impedance. The closed-loop output impedance is the open-loop one divided by one plus the loop gain, so a stage with fifty ohms of open-loop output resistance and a thousand of loop gain presents about fifty milliohms — until the loop gain runs out, above which it climbs back toward fifty. An amplifier’s output impedance therefore rises with frequency, which looks like an inductance and is one of the two reasons a capacitive load causes trouble.

Gain accuracy. The subject of this essay, and the same division.

All four are the same curve read four ways, which is the point worth carrying: there is one resource, it falls at twenty decibels a decade, and every specification that feedback improves gets worse at the same rate. A designer who has located the frequency at which the gain accuracy becomes unacceptable has, without further work, located the frequency at which the distortion and the output impedance do too.

The second reason a capacitive load causes trouble #

The first is above: the output impedance rises with frequency, so it looks inductive, and an inductance with a capacitance across it resonates.

The second is inside the loop. The output resistance and the load capacitance form a pole, that pole is between the amplifier’s output and the feedback network’s input, and it therefore sits inside the loop where it costs margin. Fifty ohms and ten nanofarads give a pole at 318 kHz; on a part whose loop crosses unity at a megahertz, that removes about seventy degrees.

The standard remedies both work by getting the pole out of the loop. A small resistor in series with the output puts the capacitance beyond it, at the cost of some output impedance at high frequency. A capacitor from the output back to the inverting input provides a high-frequency path that bypasses the load. Both are in every applications note, and both are consequences of the single observation that a pole inside a loop is a pole in the loop gain.

The integrator, where all of this is visible at once #

A last circuit, because it makes every point in this essay simultaneously.

An ideal integrator — a resistor in and a capacitor in the feedback path — has infinite gain at direct current and falls at twenty decibels a decade for ever. A real one cannot: its gain at direct current is the amplifier’s open-loop gain, a hundred thousand rather than infinity, so it is really a very slow low-pass filter with a corner at a fraction of a hertz.

The consequence is that a real integrator does not integrate at low frequencies, and its output drifts to a rail on any input offset, because there is no feedback path at direct current to correct it. The standard remedy is to put a large resistor across the capacitor, which deliberately makes it a low-pass filter with a known corner instead of an accidental one — trading an ideal property nobody has for a real one that can be specified.

That exchange is the whole field in miniature. The ideal model gives a clean answer over a range; outside the range the real device does something else; and good design consists of choosing where the boundary sits rather than pretending there is not one.

The same finite gain–bandwidth, in four other places #

The boundary measured here is one number about one part, and four essays elsewhere in this collection are consequences of it that would not be guessed from a bandwidth.

The gain the loop closes against is the immediate one: an inverting amplifier with two equal resistors has a gain of one and a loop that closes against two, so it has half the bandwidth of a follower built from the same part — 4.99 megahertz against ten. The number on this page is therefore a property of the arrangement rather than of the amplifier, and halves before any component has been called imperfect.

How much of the amplifier gets through reads the same falling loop gain as an accuracy rather than as a bandwidth, and finds the fraction of a device change reaching the output rising above one near crossover — so feedback there makes the answer more device-dependent than no feedback at all.

The Q the amplifier decides is the one that leaves the field. Build a filter section with an amplifier a hundred times its corner — the rule of thumb, and a comfortable margin by every number on this page — and its quality factor comes out 1.97 per cent high while its pole comes out 1.97 per cent low, the same number in both directions, and a fifth-order half-decibel Chebyshev built that way has 2.1 decibels of ripple. Neither the Q expression nor the pole expression contains the amplifier, which is why the error arrives without warning.

And the frequency that is not the formula finds the same lag deciding an oscillator’s frequency: the shift is 4.45 divided by the ratio of gain–bandwidth to oscillation frequency, so a part used a thousand times below its gain–bandwidth still runs 0.44 per cent low. That essay’s crossing — the ratio at which a faster amplifier stops being worth buying — is 578, which is to say that the “hundred times the corner” rule is wrong by a factor of six for that circuit and by rather less for the filter above.

Four different symptoms — a bandwidth, an accuracy, a quality factor and a frequency — from the one pole measured here.

Where the gain-bandwidth turns up again #

One finite product decides five later results in this field, each in a different quantity. What is left at crossover is the loop gain it leaves and the margin that follows. The node that is at ground for a while is the summing node’s impedance, at one thousandth of the part’s gain-bandwidth. The load that gets inside the loop and The path that buys the error back are two arrangements that spend it on stability instead of bandwidth. The current above which there is no impedance is where the same amplifier stops having a small-signal description at all. And The step that is too big is the amplitude boundary that is independent of every one of them.