Analog & Mixed-Signal Design — Advanced
Overview's op-amp gain formulas assumed an IDEAL op-amp — infinite gain, infinite bandwidth, instantaneous response. Interview Q&A already named the "virtual short" idealization directly. This page quantifies the two most practically important ways real op-amps fall short of that ideal: limited bandwidth (which shrinks as gain increases, a direct, quantifiable tradeoff) and slew rate (a hard speed limit on how fast the output voltage can physically change).
Analogy — Gain-bandwidth tradeoff is a megaphone that can either amplify a whisper very loudly across a short range, or amplify it modestly across a much longer range — the megaphone has a fixed total "reach" budget, and turning up amplification (gain) directly shrinks how far (how high in frequency) it still works well. Slew rate is a completely different physical limit — the fastest an amplifier's output voltage can possibly move, full stop, regardless of gain — like a car's absolute top acceleration, which no amount of steering finesse can exceed.
Gain-Bandwidth Product (GBW) — Gain and Bandwidth Trade Off Directly
A real op-amp has a GAIN-BANDWIDTH PRODUCT (GBW) — a roughly constant
number (for a given op-amp) representing the product of gain and the
frequency at which that gain is achievable:
GBW = closed-loop gain x maximum usable frequency
Rearranged, this gives the actual practical constraint a circuit
designer works with:
f_max = GBW / closed-loop gain
Since GBW is roughly FIXED for a given op-amp, requesting more gain
(a larger Rf/Rin ratio, from Overview's gain formulas) directly and
proportionally SHRINKS the maximum frequency that gain remains valid
for — there is no way to get high gain AND wide bandwidth
simultaneously from the same op-amp; it's a real, physical tradeoff.
Annotated Example — Bandwidth at Two Different Gain Settings
An op-amp has a GBW of 1 MHz (a common, realistic value for a general-purpose op-amp).
At closed-loop gain = 10 (Overview's own Try It gain magnitude):
f_max = 1,000,000 Hz / 10 = 100,000 Hz = 100 kHz
At closed-loop gain = 100 (a 10x higher gain setting):
f_max = 1,000,000 Hz / 100 = 10,000 Hz = 10 kHz
Verified computationally: increasing gain by 10x (from 10 to 100)
reduced the usable bandwidth by exactly 10x (from 100 kHz to 10 kHz)
-- confirming GBW is genuinely constant for this op-amp, and that
gain and bandwidth trade off in exact inverse proportion, not just
approximately.
Slew Rate — A Hard Speed Limit Independent of Gain
SLEW RATE (SR) is the maximum rate the op-amp's output voltage can
physically change, in volts per second (commonly specified in V/µs
on datasheets). Unlike GBW, slew rate is a hard physical limit that
applies regardless of the closed-loop gain configuration.
For a sine wave output of peak amplitude Vpeak at frequency f, the
maximum rate of change the sine wave itself demands is:
max required slew rate = 2 x pi x f x Vpeak
If this exceeds the op-amp's actual SR spec, the output can no
longer track the ideal sine wave shape and becomes visibly distorted
(a triangle-wave-like shape instead of a smooth sine) -- this failure
mode is called SLEW-RATE LIMITING. Solving for the maximum frequency
an op-amp can output a given amplitude WITHOUT slew-rate limiting:
f_max (slew-limited) = SR / (2 x pi x Vpeak)
Annotated Example — Maximum Undistorted Frequency at a Given Amplitude
An op-amp has a slew rate of 0.5 V/µs (= 500,000 V/s), driving a 5V-peak sine wave output.
Given: SR = 0.5 V/µs = 500,000 V/s, Vpeak = 5 V
f_max (slew-limited) = SR / (2 x pi x Vpeak)
= 500,000 / (2 x pi x 5)
= 500,000 / 31.42
= 15,915 Hz ≈ 15.9 kHz
Verified computationally: above roughly 15.9 kHz, THIS op-amp
cannot output a clean, undistorted 5V-peak sine wave -- not because
of gain-bandwidth limits (a separate constraint), but purely because
the output voltage physically cannot change fast enough to trace out
the sine wave's shape at that combination of amplitude and frequency.
Try It (2 Minutes)
Using the same op-amp (SR = 0.5 V/µs), but driving a smaller-amplitude output — 2V-peak instead of 5V-peak.
1.What is the new maximum slew-rate-limited frequency?
2.Is it higher or lower than the 5V-peak case's 15.9 kHz?
3.What does this reveal about why a slew-rate-limited op-amp might work fine for a small-signal application but distort badly for a larger-amplitude one at the same frequency?
You should land on: f_max = 500,000 / (2π × 2) ≈ 39,789 Hz ≈ 39.8 kHz — meaningfully HIGHER than the 5V-peak case's 15.9 kHz, since a smaller amplitude sine wave requires a lower peak rate of change for the same frequency (the 2×π×f×Vpeak formula scales directly with Vpeak). This reveals a genuinely important practical design implication: the same op-amp, at the same frequency, can perform perfectly well driving a small-amplitude signal while badly distorting a larger-amplitude signal — slew-rate limiting isn't a fixed frequency ceiling in isolation, it's a joint limit on amplitude AND frequency together, which is exactly why op-amp datasheets specify slew rate as a single number but its real-world impact depends entirely on the specific amplitude a given application actually needs.
Study Resources
•Sedra & Smith, Microelectronic Circuits — covers gain-bandwidth product and slew rate limitations in exactly this style
•Analog Devices — "MT-053: GBW and Slew Rate" (Analog Dialogue tutorial series) — free, practically oriented reference on both non-idealities covered above
•Texas Instruments — Op-Amp Slew Rate application notes — free reference material on slew-rate-limited distortion and practical op-amp selection
