Pulled out my favorite meter of all time, the lime green TIP 194 II. Now keep in mind this meter is currently damaged and has high leakage on one of the controllers port pins that effects the resistance measurements.
In DCmV the meter over ranges at +/-500mV.
In ACmV the meter over ranges at 1.417Vp-p or again 500mV RMS (60Hz sinewave)
With 490mVRMS biased with 400mV DC, both the AC and DC millivolts readings are correct.
Raising the DC bias to 499.9mV with 490mV RMS applied, the meter provides the correct readings for both.
Next I apply 600mV of bias to our 490mV RMS signal and of course the DCmV reading is now out of range. But the ACmV reading is now effected again with no indication that there is a problem. With 1VDC bias the ACmV is all the way down to 374mV.
The TPI 194II does support the AC+DC and is a tri-display. But sadly this is not supported in mV ranges. It's too bad I could not get a controller to replace the one on the meter. I would have then attempted to improve it like I show with that low end UT61E.
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Looking at the Extech EX540
In DCmV the meter over ranges at +/-412mV.
In ACmV the meter over ranges at 1.168Vp-p or again 412mV RMS (60Hz sinewave)
With 409mVRMS biased with 400mV DC, both the AC and DC millivolts readings are correct.
Raising the DC bias to 500mV, the meter over ranges on the DCmV but there is a slight effect on the ACmV.
With 1VDC bias the ACmV is down to 399mV which is better than the TPI 194 II but it's an error and there again is no indication that there is a problem.
This meter also supports AC+DC calculations and appears to show an over range for any combination that goes above 412mV.
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And I'm sure everyone wants to know the UNI-Ts top of the line UT181A does. Again this meter was damaged like the TPI with one hit from the grill starter. I was able to repair it and made some pretty drastic mods to the PCB. This meter is not original but I do not believe I changed anything that would effect this test.
In DCmV the meter over ranges at +/-612mV.
In ACmV the meter over ranges at 1.731Vp-p or again 4612mV RMS (60Hz sinewave)
With 610mVRMS biased with 600mV DC, both the AC and DC millivolts readings are correct.
Raising the DC bias to 700mV, the meter over ranges on the DCmV but the ACmV is correct.
With 1VDC bias the ACmV is down to 587mV and again there is no indication that there is a problem.
This meter will calculate AC+DC and has a tri-display. In this mode with the same 1VDC bias, the AC+DC and mVDC are both over ranged but the mVAC shows the 587mVAC which is of course not correct. But if the OL were not enough to sway the user into thinking there may be a problem with their measurement, they go one step further and enable the yellow triangle with the lightning bolt. Still, really it's not full proof and the ACmV should show an error as well or you just know someone is going to trust that number...
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Scott's going to ask about the Fluke 97. That's a special snowflake as it auto ranges both the AC and DC values. You exceed the mV range, it will switch.
I don't disagree with the OP that there is a problem with some meters showing inaccurate results under certain conditions without any warning to the user. Hopefully running these few meters shows that this is not something unique to one brand.