Slew rate and bandwidth: why real op-amps sound different

What you're hearing

A real op-amp has two speed limits the ideal one doesn't. The ideal op-amp in the op-amp's two rules has endless gain at every frequency, and its output can jump instantly. A real one has less gain at high frequencies (its gain-bandwidth), and its output can only move so many volts per microsecond (its slew rate). SynthMaker's real op-amp models both, with the data sheets' typical figures: a TL072 has 3 MHz and 13 V/µs, a 741 1 MHz and 0.5 V/µs. You write it into the op-amp's line: opamp TL072, opamp 741, or your own figures, such as opamp slew 13V/us gbw 3MHz.

Gain-bandwidth: the op-amp's own gain falls with frequency. A 741's gain is 200,000 at DC, but it falls so that gain times frequency stays at 1 MHz: 1,000 at 1 kHz, 100 at 10 kHz. A TL072's 3 MHz gives 1,000 at 3 kHz. Rule 1, "the output makes the inputs equal", only works while the op-amp has gain to spare over the stage's own gain. Where it runs out, the stage's gain falls with it.

The higher the gain, the lower the corner. This stage's gain is 1 + Rf/Rg = 1 + 99 kΩ / 1 kΩ = 100 (40 dB). Its top corner, where it's 3 dB down, is the gain-bandwidth divided by the gain: 1 MHz / 100 = 10 kHz for the 741, 3 MHz / 100 = 30 kHz for the TL072. Above the corner, the gain falls 6 dB per octave. The gain knob runs from 10 to 1,000:

At the gain of 100, here's how much each takes off the top, measured in SynthMaker:

That's up to about 1 dB more than the textbook single pole at 20 kHz, which puts the 741's −3 dB point nearer 9 kHz. So the 741 acts like a gentle treble cut, and the TL072 loses only the very top. That's the difference you hear on the saw's upper harmonics.

Slew rate: how fast the output can move. It's measured in volts per microsecond (V/µs). A 741 at 0.5 V/µs needs at least 16 µs to swing 8 V, from one ±4 V rail to the other. A TL072, at 13 V/µs, needs 0.6 µs. A sine's steepest slope is 2π × f × Vpeak. Once that's more than the slew rate, the output can't keep up: it moves in straight lines, and the sine turns into a triangle. The highest frequency a part can play at full size, its full-power bandwidth, is f = slew rate / (2π × Vpeak):

A 741 buffer asked for a 15 kHz sine of 8 V needs 2π × 15 kHz × 8 V = 0.75 V/µs, more than it has, so it plays a triangle. In a 9 V pedal, though, even a 741 follows full-size sines to about 20 kHz.

Does slew rate change a distortion's sound? Less than you might expect, in a 9 V pedal. Pedal builders often argue about swapping a slow op-amp for a fast one. At ±4 V, slew rate can only touch the steep edges of a clipped wave. In a high-gain stage, the 741's gain-bandwidth slows those edges first: at a gain of 1,000 its corner is already down at 1 kHz. Measured here, a saw's reset edge at that gain takes 52 µs on the 741 and 17 µs on the TL072, while slew alone would allow 13 µs: with only 20 mV in, the op-amp's input stage never saturates, so the edge moves at the pace its gain-bandwidth sets, and the 741's is a third of the TL072's. Slew rate matters most with big swings and little gain, such as a buffer on a ±15 V supply. The model covers speed only; real chips also differ in noise and in how close they swing to their rails.

The ideal op-amp stays the reference. Every other page in this course uses it, because its sums come out exact. The real ones are for hearing what speed does.

Try this

• Play a bright note and switch TL072, 741, Ideal. The 741 is a little duller, from its 10 kHz corner; the TL072, only 0.5 dB down at 10 kHz, may be hard to tell from ideal at this gain.
• Turn gain to 999 kΩ (a gain of 1,000) and input down to 2 mV. The output stays clean at 2 V, but the corners drop to 1 kHz and 3 kHz. Now both real parts are darker, the 741 much more so. Turn input back up and the stage clips at its rails as well (see rails and headroom).
• Open it in SynthMaker to change the parts, and find out which limit you're hearing. Write opamp TL072 slew 0.5V/us: a TL072 with a 741's slew rate. At a gain of 100, if it still sounds like the TL072, slew rate wasn't the difference; the gain-bandwidth was.

The parts list

circuit RealOpAmp  setting input: 20 mV, from 2 mV to 200 mV  setting output: 5 V  setting gain: 99k, from 9k to 999k  U1 in m out opamp TL072 rails 4V  Rg m gnd 1k  Rf out m gainend

For the 741, write opamp 741; for the ideal op-amp, opamp alone. setting output: 5 V makes 5 V at the output full volume, so even the rails at 4 V fit.

How we checked it

Checked against ngspice, a standard circuit simulator, running the same op-amp model: a saw's reset at gain 1000 takes 52.1 µs (741) and 16.9 µs (TL072) in both, to the tenth of a microsecond. At the 4× rate SynthMaker runs it at, its high-frequency response is up to about 1 dB below the continuous model's at 20 kHz.

Build it step by step

• The lesson: A screaming synth filter, built from parts (opens in a new tab). Its amplifier is an ideal op-amp at a gain of about 58. Open it in SynthMaker to change the parts: add TL072 or 741 after opamp on its line to hear a real one. At that gain, their corners are about 51 kHz and 17 kHz.
• The video: Build a screaming 70s synth filter from 11 parts: a real circuit, simulated. It loads only when tapped.

Build a screaming 70s synth filter from 11 parts: a real circuit, simulated

Next and previous

• Start of the chapter: The op-amp's two rules

All circuits

Further reading

• What slew rate is, and the full-power bandwidth: Wikipedia: Slew rate.
• Gain times bandwidth stays constant: Wikipedia: Gain–bandwidth product.
• The ideal op-amp against a real one: Wikipedia: Operational amplifier.
• The model SynthMaker's real op-amp is built on (its slew and bandwidth part): G. R. Boyle, B. M. Cohn, D. O. Pederson and J. E. Solomon, "Macromodeling of integrated circuit operational amplifiers", IEEE Journal of Solid-State Circuits, vol. 9, no. 6, 1974.
• D. T. Yeh, "Digital Implementation of Musical Distortion Circuits by Analysis and Simulation", PhD thesis, Stanford University, 2009 (the DK method, used to simulate this circuit). PDF