The step that is too big
Linearity is the assumption underneath every figure in the first four fields of this collection. It is what makes a frequency response a complete description; it is what allows a signal to be decomposed into sinusoids and reassembled; it is what makes a transfer function exist at all. And it is testable in one line: double the input and the output must double, exactly.
An amplifier fails that test at a smaller amplitude than almost anyone expects.
The mechanism, which is not subtle
An operational amplifier’s frequency response is deliberately shaped by a capacitor inside it, charged by a stage whose output current is finite. That is the whole story. The output voltage moves at a rate equal to that current divided by that capacitance, and when the circuit asks for a faster rate than the current can supply, the amplifier does not deliver a slightly smaller rate — it delivers the maximum, and holds it, until the demand falls back within range.
While that is happening the amplifier is not amplifying. Its input stage is fully unbalanced, the feedback loop has no gain to work with, and the output is a ramp whose slope is a property of the part rather than of the signal. The output arrives late, and the shape it arrives with is a straight line rather than an exponential.
The rate is the slew rate, quoted in volts per microsecond, and half a volt per microsecond is an unremarkable value for an unremarkable part.
Where the boundary sits, and why it is so low
The demand made on the output at the instant a step arrives is the error voltage multiplied by the amplifier’s gain–bandwidth product in radians. For a follower given a step of V, that is 2π·GBW·V, and the amplifier can supply it only while
V < slew rate / (2π · GBW)
For a part with a megahertz of gain–bandwidth and half a volt per microsecond of slew rate, the answer is 79.6 millivolts.
That number is the essay. Eighty millivolts is not a large signal by any ordinary use of the words. It is a tenth of a diode drop, a hundredth of a logic swing, and far below anything a reader would hesitate over. And above it, the circuit is not linear — not slightly nonlinear, not nonlinear in a way that adds a little distortion, but operating in a regime where the transfer function does not apply and the response is not a scaled copy of anything.
The slider makes the dependence explicit:
| Slew rate | Largest step that stays linear |
|---|---|
| 0.1 V/µs | 15.9 mV |
| 0.5 V/µs | 79.6 mV |
| 3 V/µs | 477 mV |
| 40 V/µs | 6.37 V |
The last row is the one that explains why fast parts are specified the way they are. A part with forty volts per microsecond is linear for steps up to six volts, which covers most of what it will ever be asked to do — and the same part in a circuit demanding ten megahertz of bandwidth is back in the same trouble, because the boundary falls in proportion to the gain–bandwidth product.
What “no transfer function contains it” means
It is worth being precise about the claim in this essay’s summary, because “the model does not predict this” is a weaker statement than the one intended.
A linear model does not predict the slew limit approximately. It predicts a specific, definite, completely wrong answer: that the response to an eight-volt step is eight times the response to a one-volt step, arriving at the same instants, with the same shape. The real eight-volt response takes 16 µs to travel a distance the linear model says takes 0.16 µs — a factor of a hundred — and has a straight segment in it that no sum of exponentials contains.
More importantly, the linear model gives no warning. There is nothing in a Bode plot, a pole diagram or a step response that flags the amplitude at which it ceases to apply, because amplitude is exactly the variable a linear model has been constructed not to depend on. The boundary has to come from outside the model.
This is the sharpest example on the site of a general point. A model’s range is not derivable from the model. It comes from the thing the model was an abstraction of, and a collection that presents only the models has quietly discarded the information needed to know when to trust them.
Where the rate limit comes from
It is worth knowing why the limit exists, because it explains why it cannot simply be designed away.
An operational amplifier’s frequency response is deliberately shaped by a single capacitor, placed so that the open-loop gain falls at twenty decibels per decade from a low frequency — the dominant-pole compensation the feedback field describes. That capacitor has to be charged and discharged by the stage driving it, and that stage is a differential pair whose output current cannot exceed its own bias current, however large the input difference is.
So the maximum rate is the bias current divided by the compensation capacitance, and both numbers are fixed by choices made for other reasons: the capacitance sets the gain–bandwidth product, and the bias current sets the input stage’s noise and offset. The slew rate is the ratio of two quantities chosen for stability and for noise, which is why it is a number the designer inherits rather than specifies.
It also explains the one arrangement that improves it dramatically. Adding resistance in series with the input pair’s emitters — degeneration — makes the stage deliver more current for a given input difference before it runs out, at the cost of transconductance and therefore of gain–bandwidth for a given capacitance. Trading gain–bandwidth for slew rate is exactly what a “fast” amplifier has done, and it is why fast parts tend to be noisier: the same degeneration that buys the slew rate costs input-referred noise.
That trade is worth stating in the collection’s usual terms. Two boundaries on one device are set by one internal choice, and moving one moves the other. A designer choosing a part is choosing a point on that exchange, not choosing two independent specifications.
How this figure is checked
Since the slew-limited response cannot be obtained from a transfer function, it is integrated directly: the output is treated as an integrator driven by the error, with the integrator’s rate clamped, and the equation is advanced with a fourth-order Runge–Kutta step.
That raises the obvious question of whether the integration is right, and the answer is the collection’s usual one. Running the same routine with the clamp removed must reproduce the linear answer, which is known exactly — a single exponential with a time constant of 1/(2π·GBW) — and it does, to within two parts in a thousand over the whole trace. So the integrator is verified against a case with a closed form, and then used on the case that has none.
Two further assertions hold at every position of the slider. The smallest step never reaches the rate limit, which is what makes it the reference the other four are compared against. And the largest and smallest, normalised, differ by more than twenty per cent of the final value somewhere in the trace — which is the claim the figure exists to support, stated as a number rather than left to the eye.
The frequency version of the same limit
Slew rate is usually met as a frequency specification rather than an amplitude one, and the two are the same statement.
A sinusoid of amplitude A at frequency f has a maximum rate of change of 2πfA, at the zero crossing. Requiring that to stay below the slew rate gives the full-power bandwidth:
f < slew rate / (2π A)
For half a volt per microsecond and a ten-volt amplitude, that is 7.96 kHz — audio frequencies, from a part with a megahertz of gain–bandwidth. The small-signal bandwidth of the same part in a unity-gain follower is a megahertz, and the two numbers differ by a factor of a hundred and twenty because they are answers to different questions.
The failure this produces is characteristic and easy to recognise once it has been seen: a sine wave that comes out as a triangle. The peaks are reproduced faithfully, because the demanded rate there is low, and the fast parts through the zero crossings become straight ramps at the slew rate. It is one of the few distortion mechanisms with an unmistakable signature.
What breaks when linearity does
The consequences reach further than a distorted waveform, and they are worth listing because each one silently invalidates something used earlier in this collection.
Superposition fails. The response to a sum of two signals is no longer the sum of the responses. That is not an inconvenience; it is the assumption that makes a frequency response a complete description of a circuit. Once it fails, knowing what a circuit does to every sinusoid separately tells nobody what it does to a piece of music.
New frequencies appear. A linear circuit can only ever attenuate, delay or phase-shift the frequencies present at its input; it cannot create one that was not there. A slew-limited amplifier creates a whole series of them. Feed it two tones and the output contains their sum, their difference, and combinations of their harmonics — none of which any transfer function predicted.
A measurement becomes amplitude-dependent. The frequency response of an amplifier measured at one volt and at ten volts are different curves, and the difference is not calibration error. This is why measurement standards specify the amplitude as well as the frequency range, and why a response measured at a convenient amplitude can be quietly wrong about the one that matters.
A feedback loop’s margin changes. The loop gain during slewing is essentially zero, since the amplifier is not responding to its input at all. A loop that is comfortably stable in small signal can therefore behave quite differently on a large transient, and the recovery as it comes out of the limit is a real effect with a name — a settling tail that is far longer than the linear analysis predicts.
Recognising it on a bench
Three signatures identify slew limiting, and knowing them is worth more than the arithmetic in practice because the arithmetic requires a datasheet and the signatures do not.
A sine wave that comes out as a triangle. The peaks are reproduced, because the demanded rate there is low; the fast parts through the zero crossings become straight ramps at the maximum rate. Unmistakable once seen.
A rise time that does not scale. Halve the input amplitude and a linear circuit’s output takes exactly as long to get there; a slew-limited one takes half as long, because it is travelling at a fixed speed over half the distance. That test needs nothing but a generator with an amplitude control and is the quickest available diagnosis.
A distortion that improves as the signal gets smaller and worsens as the frequency rises. Both directions follow from the boundary being a rate: rate is amplitude times frequency, so reducing either reduces the demand.
The second is the one worth building a habit around, because it distinguishes slew limiting from every other cause of a slow edge. A bandwidth limit, a load capacitance and an insufficiently compensated loop all produce an edge whose duration is independent of amplitude. Only a rate limit produces one whose duration is proportional to it.
What it does to a specification
Two numbers on a datasheet describe the same device’s speed and they answer different questions, so a specification that quotes one is incomplete.
Gain–bandwidth product answers: how much gain can this part provide at this frequency? It is a small-signal number and it is the right one for a circuit handling millivolts.
Slew rate answers: how fast can this part’s output travel? It is a large-signal number and it is the right one for a circuit handling volts.
The two are related through the compensation capacitor but they are not derivable from one another, and the ratio between them varies by more than an order of magnitude between parts that have the same gain–bandwidth. A designer who checks only the first has verified the circuit for a signal amplitude that may be a hundredth of the one it will see.
The tidiest way to hold both is the full-power bandwidth, which combines them into a single frequency for a stated output amplitude: 7.96 kHz for ten volts from a part with half a volt per microsecond. Comparing that against the small-signal bandwidth of the same part in the same configuration — a megahertz for a follower — gives the factor of a hundred and twenty that this essay opened with, and it is the single number most likely to be surprising.
Two limits, one device, no common model
The pairing at the end of the previous section is the cleanest statement of the essay’s point, so it is worth ending on it.
The ideal-amplifier model fails above a frequency, and the frequency comes from a linear parameter — the gain–bandwidth product — so a better linear model fixes it. Replace the nullor with a finite-gain, single-pole model and the prediction becomes right again, over a range with a computable edge of its own.
The slew limit fails above an amplitude, and no linear model of any order fixes it, because the failure is a violation of the property all of them share. The only repair is a different kind of model, and the only way to get an answer is to integrate the equations forward rather than solve them at a frequency.
Two boundaries on one part, in two different units, with two different remedies. A collection that draws only frequency responses would find the first and never mention the second, and the second is the one that turns up in practice more often — because eighty millivolts is a signal, and a megahertz is not a frequency most circuits are asked to reach.