Slew rate calculator: the slew rate a signal needs and the bandwidth an op amp allows

A sine is steepest at its zero crossings, 2π·f·V_p, and an op amp whose output cannot move that fast turns it into something else. Enter a frequency and an amplitude to get the slew rate the sine needs, the full-power bandwidth a given slew rate allows at that amplitude, and, past the limit, the waveform the output actually makes: SBOA092's OPA277, at 0.8 V/µs and a 13 V peak, runs out at 9.79 kHz, and a 20 kHz sine at that amplitude would need 1.63 V/µs. Or enter a step and the closed-loop bandwidth to see whether it slews at all, and the rise time that results.

Fig 1 — A 13.0 V peak sine at 20.0 kHz needs 1.63 V/µs at its zero crossings; at 0.8 V/µs the output can only run up and down at SR and never catches the sine: it becomes a triangle of 10.0 V peak, SR/(4f), lagging the input.

Signal

Op amp slew rate SR (V/µs)

Sine frequency f (kHz)

Output amplitude V_p (V)

Amplitude given as

Slew rate the sine needs, 2πf·V_p: 1.63 V/µs · 1.63 MV/s
Op amp slew rate SR · SR over what is needed: 0.8 V/µs · 0.49 ×
Full-power bandwidth at 13.0 V peak, SR/(2π·V_p): 9.79 kHz
Largest undistorted peak at 20.0 kHz, SR/(2πf): 6.37 V peak · 12.7 V p-p · 4.50 V RMS
The sine: peak · peak-to-peak · RMS: 13.0 V · 26.0 V · 9.19 V
Output, slope clamped to ±SR: peak · fundamental: 10.0 V · 8.11 V
Output shape: a triangle, SR/(4f) = 10.0 V peak
Harmonic distortion of that output, THD (model): 12.1 %
Fundamental's lag behind the input: 39.7° · 5.52 µs

Slew-limited: the output never catches the sine and runs as a triangle, up and down at 0.8 V/µs. Its peak, 10.0 V, is set by the slew rate and the frequency alone: raising the input no longer raises the output. A triangle's distortion is 12.1 % whatever its size.

How this is calculated

Standard: TI SLOA011 §5.13, §5.16, §5.17; TI SLAA013 §4; TI SBOA092 p. 72; TI SBOA268; Tektronix ABCs of Probes (60W-6053-15)

$SR = \frac{dV}{dt}, \qquad SR = \frac{2I_E}{C_c}$: SLOA011 §5.13 and Figure 5-8: "Slew rate, SR, is the rate of change in the output voltage caused by a step input." The second form is its account of the mechanism, the input stage's tail current charging the compensation capacitor, "Essentially, SR = 2IE/Cc. However, there are op amps that work on different principles where this is not true."
$V = V_O \sin 2\pi f t, \qquad \frac{dV}{dt} = 2\pi f V_O \cos 2\pi f t, \qquad \left.\frac{dV}{dt}\right|_{max} = 2\pi f V_O$: SLAA013 §4, Figure 8 and the text below it: "The maximum slew rate occurs at the zero crossing point." SLAA013 derives it for an ADC's aperture error; it is the same slope an op amp output has to follow. V_O is the peak.
$f_{FP} = \frac{SR}{2\pi V_p}, \qquad V_{p,max} = \frac{SR}{2\pi f}$: The same relation solved for the frequency, the full-power bandwidth, and for the amplitude. Derived: no source in the library prints either. SLOA011 §5.16 names the datasheet version of the first, the maximum output-swing bandwidth B_OM, and says "The limiting factor for BOM is slew rate."
$V_{p} = \frac{V_{pp}}{2} = \sqrt{2}\,V_{RMS}$: For a sine. The formula wants the peak, measured from zero, at the op amp output.
$V_{tri} = \frac{SR}{4f}, \qquad \mathrm{THD}_{tri} = \sqrt{\frac{\pi^4}{96} - 1}$: The triangle a heavily slew-limited output becomes: SR for half a period is its peak-to-peak. Its harmonics are the odd ones at 1/n² of the fundamental (8/π² of the peak), which gives 12.1 %. It is a pure triangle once SR/(2πf·V_p) ≤ 1/√(1 + π²/4) = 0.537; above that it rejoins the sine for part of each half-cycle. Derived; between the two the calculator time-steps the output with its slope clamped to ±SR.
$\tau = \frac{1}{2\pi f_{-3dB}}, \qquad \Delta V_{crit} = SR \cdot \tau, \qquad t_{r} = \tau \ln 9 \approx \frac{0.35}{f_{-3dB}}$: A single-pole closed loop starts a step ΔV at ΔV/τ, so it slews when ΔV > SR·τ (derived). Its small-signal 10–90 % rise is τ·ln 9 = 0.3497/f; Tektronix's ABCs of Probes gives the rule as "Tr = 0.35/BW". For a gain stage f_−3dB ≈ GBW / noise gain (SLOA011 eq 37).
$v(t) = SR\cdot t \;\; (t \le t_1), \qquad v(t) = \Delta V - SR\,\tau\, e^{-(t - t_1)/\tau} \;\; (t > t_1), \qquad t_1 = \frac{\Delta V - SR\,\tau}{SR}$: The step with its slope clamped: a ramp at SR until the remaining error is SR·τ, where the exponential's own slope has fallen to SR, then the exponential. Continuous in value and slope at t_1. The rise time and settling times come from this. Derived.

Assumptions

The slew rate is symmetric and constant: the same rising and falling, at every output level. Enter the slower edge where a datasheet gives two.
Below the slew rate the output follows its input perfectly (sine mode) or as a single pole (step mode), and at the slew rate it switches to a straight line with no transition. A real input stage leaves its linear range gradually: SLOA011 puts full slew rate at an error voltage "on the order of 120 mV" for a bipolar input and up to 1 V to 3 V for a JFET or MOSFET one.
No output-swing limit: the amplitude or step entered is assumed to fit between the rails. Check the op amp's output swing separately; the op amp gain calculator does.
The closed loop is a single pole. A stage with little phase margin overshoots and rings, and SLOA011's settling time includes that ringing; the settling times here are a floor.
The distortion figures are those of the slope-clamp model alone: the harmonics a slew-limited waveform has. They leave out the op amp's own linear distortion and noise.

What slew rate is and what sets it

TI's Understanding Operational Amplifier Specifications(SLOA011) defines it in one line: "Slew rate, SR, is the rate of change in the output voltage caused by a step input", measured in volts per microsecond or per millisecond. Its Figure 5-8 shows the measurement, a step into a unity-gain follower and the ramp that comes out, labelled SR = dV/dt. The glossary at the back of the same note adds a word that matters: slew rate is "The average time rate of change of the closed-loop amplifier output voltage for a step-signal input." It is a straight-line rate, the one the output settles into when the input asks for more than the amplifier can give, and not the slope at any particular instant.

SLOA011 then gives the reason, for the simplified op amp it draws in its Figure 4-1: the "voltage change in the second stage is limited by the charging and discharging of capacitor Cc. The maximum rate of change occurs when either side of the differential pair is conducting 2IE. This is the major limit to slew rate. Essentially, SR = 2IE/Cc. However, there are op amps that work on different principles where this is not true." C_c is the internal compensation capacitor, the part that makes the op amp stable at unity gain, and 2I_E is all the current the input stage has to charge it with. Two consequences follow directly. A faster op amp needs more current, and SLOA011 says so: "To increase slew rate, the bias currents within the op amp are increased." And a smaller C_c is faster too, which is what a decompensated op amp is: "This increases realizable bandwidth and slew rate, but the engineer must ensure the stability of the circuit by other means."

The same section explains why slewing is a large-signal effect. The input stage only delivers its full current when it is driven hard: "An error voltage on the order of 120 mV is required for an op amp with a bipolar input to realize full slew rate. This can be as high as 1 V to 3 V for a JFET or MOSFET input." In linear operation the difference between the inputs is the output divided by the open-loop gain, which SLOA011's ideal op amp takes to zero as its "virtual short". When the output falls so far behind that the inputs are a tenth of a volt or more apart, the loop has stopped controlling the output, and the output is simply ramping at the rate its internal current allows.

The slew rate formula for a sine wave

A sine of peak V_p and frequency f is V = V_p·sin 2πft. Its slope is the derivative, 2πf·V_p·cos 2πft, which is largest where the cosine is 1: at the zero crossings. TI's Understanding Data Converters (SLAA013) works exactly this for an ADC's aperture error, and states the result: "The maximum slew rate occurs at the zero crossing point and is given by" dV/dt|_max = 2πfV_O, with V_O the peak. For an op amp it is the slope the output must reach for the sine to come out undistorted, so the slew rate calculation for an op amp driving a sine is:

SR needed = 2π·f·V_p, for a given sine;
f_FP = SR / (2π·V_p), the full-power bandwidth: the highest frequency at which a given op amp can produce that peak;
V_p,max = SR / (2π·f), the largest peak it can produce at a given frequency.

The last two are the first rearranged, and the page labels them derived: none of the library's sources prints them. SLOA011 does name the datasheet quantity. Among the frequency specifications in its section 5.16 is the "Maximum output-swing bandwidth (BOM)", which "specifies the bandwidth over which the output is above a specified value", and "The limiting factor for BOM is slew rate. As the frequency gets higher and higher the output becomes slew rate limited and can not respond quickly enough to maintain the specified output voltage swing." Full-power bandwidth is that idea with the specified value set to the full output swing.

Note what the formula does not contain: the op amp's gain-bandwidth product, and the stage's gain. The frequency and amplitude are those at the output, and the slope follows from them alone. A sine that needs 6.28 V/µs at 1 V peak and 1 MHz needs it whether the op amp is a follower or a gain of 100.

Full-power bandwidth chart

The highest sine frequency each slew rate can carry at four output peaks, from f = SR/(2π·V_p). The first four slew rates are those of parts in TI's own documents in the library; the last two are round numbers. Double the peak and the bandwidth halves; the chart is two straight proportions, which is why it is worth running the numbers rather than trusting a datasheet's small-signal bandwidth.

Slew rate	1.00 V peak	2.50 V peak	5.00 V peak	10.0 V peak
0.4 V/µs OPA170 (SBOA268)	63.7 kHz	25.5 kHz	12.7 kHz	6.37 kHz
0.8 V/µs OPA277 (SBOA092)	127 kHz	50.9 kHz	25.5 kHz	12.7 kHz
2.5 V/µs OPA512 (SBOA092)	398 kHz	159 kHz	79.6 kHz	39.8 kHz
5 V/µs OPA1671 (SBOA268)	796 kHz	318 kHz	159 kHz	79.6 kHz
20 V/µs	3.18 MHz	1.27 MHz	637 kHz	318 kHz
100 V/µs	15.9 MHz	6.37 MHz	3.18 MHz	1.59 MHz

At the top of the chart, the OPA170 slews at 0.4 V/µs and has a unity-gain bandwidth of 1.2 MHz in SBOA268's table, but it can only swing a 10 V peak up to 6.37 kHz. Between those two frequencies it is fast enough for small signals and too slow for large ones.

Worked example: the OPA277 at 20 kHz, and SBOA092's compound amplifier

TI's Handbook of Operational Amplifier Applications (SBOA092) lists, on page 72, the slew rate and output swing of two op amps it combines: the precision OPA277 at 0.8 V/µs and ±13 V, and the OPA512 power op amp at 2.5 V/µs and ±35 V. The calculator's defaults are the OPA277 asked for its full 13 V peak at 20 kHz.

needed      2π × 20 kHz × 13 V                       = 1.63 V/µs 
have        OPA277, SBOA092 Table 1                  = 0.8 V/µs  
ratio       0.8 / needed                             = 0.49 ×    
FPBW        0.8 V/µs / (2π × 13 V)                   = 9.79 kHz  
max peak    0.8 V/µs / (2π × 20 kHz)                 = 6.37 V    
triangle    0.8 V/µs / (4 × 20 kHz)                  = 10.0 V    
fund.       8/π² × triangle peak                     = 8.11 V

The OPA277's full-power bandwidth at 13 V peak is 9.79 kHz. At 20 kHz the sine needs 1.63 V/µs, 2.04 times what the part has, and the output does not come out as a smaller sine. It becomes a triangle: the output ramps up at 0.8 V/µs until it meets the falling input, then ramps down at 0.8 V/µs, and in half a period it can only cover 0.8 V/µs × 25.0 µs = 20.0 V, a 10.0 V peak. Its fundamental is 8.11 V and its harmonic distortion, from the waveform alone, is 12.1 %. To stay undistorted at 20 kHz the OPA277 can deliver 6.37 V peak.

Nearer the limit the damage is smaller but not zero. At 15 kHz the sine needs 1.23 V/µs: the output follows the sine near its peaks and runs straight at 0.8 V/µs through each zero crossing, slewing for 84 % of the period, with 11.0 %distortion. At 10 kHz, just past the 9.79 kHz full-power bandwidth, it is 0.19 %. The figure in the calculator draws each case from the same model.

The OPA512 alone has a full-power bandwidth of 11.4 kHz at its 35 V peak. SBOA092's point is the compound amplifier, where the OPA277's precision drives the OPA512's output: "The OPA512 has the highest slew rate and therefore is operated within a local closed loop. The slower OPA277 is operated within the outer loop." And for the slew rate: "The slew rate of the OPA277 is 0.8V/µs. That slew rate is gained up times three in the OPA512 so that there is an effective slew rate for the compound amplifier of 2.4V/µs." The OPA512's local loop in the schematic, 10 kΩ over 4.7 kΩ, is a gain of 3.13, and 0.8 V/µs × 3.13 would be 2.50 V/µs, the OPA512's own figure; the handbook's 2.4 V/µs rounds the gain to three. Either way the compound amplifier's full-power bandwidth at 35 V peak,10.9 kHz at 2.4 V/µs, is close to the OPA512's alone. The principle is the one to take away: an op amp's own slew rate does not scale with the gain around it, but a slew-limited signal from one stage is multiplied by the gain of the next.

Worked example: a step through SBOA268's transimpedance amplifier

TI's Analog Engineer's Circuit: Transimpedance Amplifier(SBOA268) converts 0 to 50 µA into 0 to 5 V with a bandwidth of 10 kHz, on ±15 V, using an OPA170: a 100 kΩ feedback resistor and a feedback capacitor it sizes as C1 ≤ 1/(2π·R1·f_p) and rounds to 150 pF. Its table gives the OPA170 a UGBW of 1.2 MHz and an SR of 0.4 V/µs. A full-scale step in the input current is a 5 V step at the output, and the stage's bandwidth is set by R1 and C1, not by the op amp.

C1 max      1 / (2π × 100 kΩ × 10 kHz)               = 159 pF     SBOA268: ≤ 159 pF
bandwidth   1 / (2π × 100 kΩ × 150 pF)               = 10.6 kHz  
tau         100 kΩ × 150 pF                          = 15.0 µs   
slope       5 V / τ, with no slew limit              = 0.333 V/µs
crit step   0.4 V/µs × τ                             = 6.00 V    
rise        10–90 %, ln 9 × τ                        = 33.0 µs   
settle      to 0.1 %, ln 1000 × τ                    = 104 µs

The step starts at 0.333 V/µs and the OPA170 can manage 0.4 V/µs, so it does not slew: any step up to 6.00 V is handled by the loop alone, and the full 0 to 5 V range fits. The rise time is the feedback network's, 33.0 µs from 10 % to 90 %. The calculator's step mode with 5 V, 0.4 V/µs and 10.61 kHz shows this case.

The same OPA170 used as a unity-gain follower is a different story. Its closed-loop bandwidth is then close to the 1.2 MHz UGBW, and a step larger than 53.1 mV slews. A 5 V step takes 10.0 µs from 10 % to 90 %, where the bandwidth alone would allow 291 ns: the slew rate is 34 times the limit. That is the ordinary situation for a buffer handling large steps, and it is why a datasheet's rise time for a small step says nothing about a large one.

Slew rate vs small-signal bandwidth

The two limits describe different things. The small-signal bandwidth is linear: every step, however small, rises through the closed loop's time constant τ = 1/(2π·f_−3dB). A single pole reaches 10 % at τ·ln(10/9) and 90 % at τ·ln 10, so its 10–90 % rise time is τ·ln 9 = 0.3497/f_−3dB. That is the rule Tektronix's ABCs of Probes gives for scopes and probes: "In cases where rise time isn’t specified, you can derive rise time (Tr) from the bandwidth (BW) specification with the following relationship: Tr = 0.35/BW". Its 350 MHz example comes out at 999 ps, the 1 ns the primer pairs with it.

Slewing is non-linear: it depends on the size of the step. The linear response ΔV·(1 − e^−t/τ) is steepest at the start, ΔV/τ, so a step slews when ΔV/τ exceeds SR, that is when ΔV > SR·τ. Below that critical step the bandwidth sets the rise time and slew rate is irrelevant; above it the output ramps at SR until the remaining error is SR·τ, and the exponential finishes the job. For a gain stage the closed loop bandwidth is roughly the gain-bandwidth product over the noise gain (SLOA011 equation 37, and SBOA092's constant gain-bandwidth sketch), so the critical step grows with the gain. For SBOA092's example op amp, "open loop bandwidth of 1 MHz", given the OPA277's 0.8 V/µs:

Noise gain	f_−3dB	Largest step without slewing	Small-signal 10–90 %
1	1.00 MHz	127 mV	350 ns
2	500 kHz	255 mV	699 ns
10	100 kHz	1.27 V	3.50 µs
100	10.0 kHz	12.7 V	35.0 µs

A follower slews on any step over 127 mV. At a gain of 100 the loop is so slow that only a step of more than12.7 V at the output would slew. High-gain stages are usually bandwidth-limited; followers and low-gain drivers are usually slew-limited on large signals.

Slew-induced distortion: what the output does past the limit

SLOA011 is plain about the result: "Two major reasons for distortion in an op amp are the limit on output voltage swing and slew rate." The calculator models the output as following the input exactly while it can, and running at ±SR in a straight line when it cannot. Once SR/(2πf·V_p) drops below 1, the output leaves the sine a little before each zero crossing, at the phase where the sine's slope first exceeds SR, and rejoins it late. As the ratio falls further it rejoins later, until at 0.537 it never does: the output is then a triangle of peak SR/(4f), set by the slew rate and the frequency alone, and turning the input up does nothing. The triangle's harmonics are the odd ones, falling as 1/n², for a total harmonic distortion of 12.1 %, and its fundamental lags the input by up to a quarter period.

Slew-rate distortion differs from clipping in one useful way: it depends on frequency. Clipping is the same at any frequency; slew limiting disappears if the frequency or the amplitude comes down. A waveform that looks like a sine at low frequency and a triangle at high frequency, at the same amplitude, is slewing.

Where the slew-rate model stops being valid

The corner is not sharp. The model has the output follow the sine perfectly up to the slew rate and then switch to a ramp. A real input stage reaches full slew rate only at the error voltage SLOA011 quotes, about 120 mV for a bipolar input and 1 V to 3 V for a JFET or MOSFET input, and it is already some way out of its linear range on the way there. Expect distortion to rise before the sine's slope reaches the full slew rate, keep a margin, and read the datasheet's THD+N curve against frequency where there is one.

Slew rate is not symmetric in every part. Analog Devices' tutorial MT-035 on op amp outputs: "While all modern op amps have push-pull output stages of some sort, many are still asymmetrical, and have a greater slew rate in one direction than the other. Asymmetry tends to introduce distortion on ac signals". Where a datasheet gives a rising and a falling figure, enter the slower one. The same note says "Input stage gm determines the slew rate and the unity-gain crossover frequency of the amplifier", and that in a rail-to-rail input stage response time degrades slightly within about 1 V of either rail, where one of its two input pairs is cut off.

The output swing is a separate limit. The calculator does not know the rails. A sine whose slope is within the slew rate still clips if its peak exceeds the output swing; the op amp gain calculator checks that.

A single pole does not ring. SLOA011's settling time is "the time required for the output voltage to settle to within a specified percentage of the final value given a step input", and its Figure 5-13 shows the slewing ramp followed by overshoot and "damped oscillation". The single-pole tail used here has no overshoot, so its settling times are a floor, reached only by a stage with a generous phase margin. A capacitive load erodes that margin; see the capacitive-load calculator.

Typical is not minimum. Where a datasheet gives a minimum slew rate as well as a typical one, design to the minimum, and check the conditions it is specified under against the application's.

Current-feedback and other architectures. SLOA011's SR = 2I_E/C_c is for the conventional voltage-feedback op amp, and it warns that "there are op amps that work on different principles where this is not true." The sine and step relations on this page assume only a fixed maximum output slope; for an amplifier whose datasheet does not describe its large-signal behaviour with a single SR figure, use the curves it gives instead.

Common slew rate mistakes

Using the peak-to-peak or RMS amplitude as V_p. The formula needs the peak. A 10 V peak-to-peak sine is a 5 V peak; entering 10 doubles the slew rate asked for. A 10 V RMS sine is a 14.1 V peak; entering 10 asks for 29 %too little.
Checking the gain-bandwidth product instead of the slew rate. A 1.2 MHz UGBW does not mean the OPA170 can deliver a 1 MHz sine of any useful size: at 0.4 V/µs its largest undistorted peak at 1 MHz is 63.7 mV.
Using the input amplitude. The slope that matters is at the output, so a 100 mV input into a gain of 50 is a 5 V sine as far as slew rate is concerned.
Assuming the slew rate scales with the stage's gain. The datasheet figure is measured in a stated configuration: SLOA011's section is titled "Slew Rate at Unity Gain", and its Figure 5-8 measures it on a follower. On SLOA011's account it is set by the input stage's current and C_c, which belong to the op amp, not to the gain around it, and a gain of 10 does not make it ten times faster. What changes with the gain is the closed-loop bandwidth, which decides whether a given step slews at all.
Treating the full-power bandwidth as a frequency response. Below it the output is not attenuated; past it the output is not a smaller sine but a distorted one, turning into a triangle.
Using a small-signal rise time for a large step. The OPA170 follower above rises in 291 ns on a small step and 10.0 µs on a 5 V one.
Forgetting square waves. A comparator made from an op amp, or an op amp buffering a PWM signal, has an output that is all edges; SBOA092 says of its comparator circuit, "The slope of the transition is determined by the slew rate of the operational amplifier."