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What is the best way to tell the maximum differential voltage swing that an op amp will handle? This does not seem to be a common specification listed in the datasheets.

The new TI OPAX192 series of op-amps explicitly address this in their datasheet:

The OPAx192 uses a unique input architecture to eliminate the need for input protection diodes but still provides robust input protection under transient conditions. Conventional input diode protection schemes shown in Figure 53 can be activated by fast transient step responses and can introduce signal distortion and settling time delays because of alternate current paths, as shown in Figure 54. For low-gain circuits, these fast-ramping input signals forward-bias back-to-back diodes, causing an increase in input current, and resulting in extended settling time, as shown in Figure 55.

opa192 input protection

Some situations where I would like to have an op amp with these characteristics known are in integrator circuits. In a thermal system for example, the integrator will be out of feedback range until the oven warms up. During this warm-up period the inputs will be held apart, possibly damaging the op amp. Also, the op amp no longer presents high input impedance to the rest of the board (such as a reference on the non-inverting input) and may pull excessive current.

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First, the number you're looking for is, in my experience, almost always found in the section labeled "Absolute Maximum Ratings", or something similar.

Second, the problem of integrator saturation is usually dealt with by using a clamp (such as a zener diode or diodes) on the feedback path. This keeps the op amp in the linear region, and differential limits simply aren't a consideration. In PID loops, in particular, paying attention to integrator saturation is often important to keep windup from playing hob with the behavior of the system when it comes out of overload.

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