Real op-amp imperfections

The Ideal op-amp model gets you within a fraction of a percent on almost every circuit, which is why it is worth using first and correcting second. A real device departs from it in four ways, and each one has a regime where it stops being a rounding error and starts being the answer:

1. DC imperfections — the input offset voltage and the input bias currents.
2. Finite, frequency-dependent open-loop gain — $A$ is large but not infinite, and it rolls off above a few hertz.
3. Output voltage saturation — the output cannot leave the supply rails.
4. Slew-rate limiting — the output cannot get to where it is going arbitrarily fast.

The first two are small-signal problems; the last two are large-signal problems. Knowing which of the four is biting is most of the debugging.

DC imperfections

Input offset voltage $V_{OS}$: tie both inputs together and a real op-amp still does not sit at zero. A few millivolts of mismatch between the input stage’s two halves acts as a tiny DC source in series with one input, and negative feedback amplifies it by exactly the same Closed-loop gain as the signal. An inverting stage with gain $-100$ and $V_{OS}=2\text{mV}$ carries $200\text{mV}$ of DC error at the output with no input applied. That is fine for audio behind a coupling capacitor and fatal for a precision DC measurement.

Input bias current $I_B$: golden rule 1 says no current flows into the inputs, and a real input stage draws some anyway: tens of nanoamperes for a 741, picoamperes for a FET-input part. That current flows through whatever DC resistance it sees and turns into a voltage error.

The standard fix is worth knowing because it is free: make the DC resistance looking out of the inverting input equal the DC resistance looking out of the non-inverting input. Then the two roughly equal bias currents produce roughly equal voltage drops, the two inputs shift together, and only their difference survives. That difference is the input offset current $I_{OS}=|I_{B+}-I_{B-}|$, typically an order of magnitude smaller than either bias current on its own.

Finite and frequency-dependent open-loop gain

Open-loop gain $A$ is around $10^5$ at DC and falls at $-20$ dB/decade above a corner frequency $f_b$ that is often only a few hertz. Above that corner the product of gain and frequency is roughly constant, and that constant is the gain-bandwidth product $f_t$: about $1\text{MHz}$ for a 741, hundreds of megahertz for modern parts.

The consequence for a closed-loop circuit is a hard trade:

$\text{closed-loop bandwidth}\approx \frac{f_t}{|G|}$

where $G$ is the closed-loop gain. Ask for gain and you give up bandwidth one-for-one. An inverting amplifier with $G=-100$ built on a 741 is flat only to about $10\text{kHz}$; run a $100\text{kHz}$ signal through it and the output comes back attenuated and phase-shifted, even though the ideal analysis promised a clean $\times 100$.

Output saturation

The output cannot go outside the supply rails, and in practice stops a volt or two short of them: on $\pm 15\text{V}$ supplies, expect clipping near $\pm 13\text{V}$. Op-amp output saturation is not a subtle effect. If the ideal analysis predicts a $20\text{V}$ output, what you actually get is a flat-topped $13\text{V}$ waveform and every linear conclusion drawn about the circuit is void.

Sanity-check the predicted swing against the rails before trusting any gain calculation. It costs one line of arithmetic.

Slew-rate limiting

Even with the steady-state output comfortably inside the rails, the output can only move so fast. The Slew rate SR is that ceiling on $|d{v}_O/dt|$, about $0.5\text{V/}\mu \text{s}$ for a 741. A sine of peak $V_O$ and frequency $\omega$ has maximum slope $\omega V_O$, so the highest frequency at which a full-amplitude sine survives intact is the Full-power bandwidth

$f_M=\frac{\text{SR}}{2\pi V_{O,\max }}.$

A 741 asked for a $\pm 10\text{V}$ swing manages $f_M\approx 8\text{kHz}$. Above that the sine turns into a triangle: the output ramps at the slew rate between turning points because it cannot trace the steep parts.

Two bandwidths, not one

The most common mistake is to treat $f_t$ and $f_M$ as the same limit. They are independent, and which one binds depends on the signal amplitude.

• $f_t/|G|$ is the small-signal limit. It applies no matter how small the signal, because it is about gain running out.
• $f_M$ is the large-signal limit. A signal well below the rails has a small $\omega V_O$ and never approaches the slew rate, so it sails past $f_M$ untouched and is limited only by $f_t/|G|$.

A circuit can be stopped by either. Check both, and check the DC error separately, because none of the three has anything to say about the others.