First, the DUT is in the positive feedback because this way you have an inverting and a non-inverting gain in the loop, so that multiplying them you obtain an overall negative loop gain.
Second, the gain is 1000 because Vin of DUT is Vos*50/(50K+50), so if you consider that V+ of the DUT should be 0, there is only the offset applied, so the feedback forces the output to be 1k times the offset voltage.
I think that you can look at the output this way: suppose that the situation is the one described, and you have a 500uV DUT offset and so 500mV output voltage.
Now, if you try to perturb the Servo Input, the feedback forces the output of the DUT to be almost the same of Servo Input, restoring the same Output voltage.
Note: Voff is the conventional name for input offset voltage, while Vos is the output voltage with Voff applied at the input pins.
why not just measure the current by letting Vos be the reference voltage in our voltmeter?
It's not really clear what you mean by this, but Vos is entirely internal to the op-amp, and it might be a combination of different errors in different stages of the amplifier. We only model it with a voltage source in series with the input. But there's no one physical place you could put a multimeter probe to measure it.
How about other easy ways to calibrate-out the effect of the offset voltage?
Usually you can calibrate it out, for example by measuring the amplifier response to 2 or 3 different fixed input voltages.
One problem, though, is that an op-amp with high Vos is also likely to have a higher drift in Vos with temperature.
Wouldn't those ways be much cheaper than buying a high-quality op amp?
Calibration requires additional test operations in manufacturing. These might require additional operator handling. That adds significantly to manufacturing cost. If you need to calibrate over temperature, it could add dollars (not pennies) to your manufacturing cost.
Then the calibration data needs to be stored in the device somehow, and retrieved to apply the calibration to each measurement. For some (many) products that's no extra cost, but for others it might mean adding an EEPROM and uC that weren't needed before.
If there's an error in the calibration process, or the stored data is corrupted, you get field returns, which are costly.
Best Answer
Offset is non-ideal behavior. For this reason an ideal opamp does not have offset by definition.
However, this is not a requirement, even with infinite gain offset can be included because opamps are usually used in a feedback configuration that reduces the gain of the closed-loop system.