It is necessary to operate all the MOSFETs of an op-amp in the saturation region in order to prevent unwanted output distortion etc. Therefore, the following equation must be satisfied:
|VDS| > |VGS| - |Vth|
Qp1, which acts as a current source, enters the linear region when the input voltage is between VDD and (VSD of the circuit’s current source) + (VSG of the differential input pair). An op-amp enters the linear region when an input voltage within (VSD of the circuit’s current source) + (VSG of the differential input pair) is below the supply voltage. This voltage range is out of the common-mode input range. An input voltage in this range results in an output distortion or a reduction in gain. It is therefore necessary to operate op-amps in the common-mode input voltage range.
The differential input pair of a rail-to-rail op-amp is composed of a pair of differential P-channel MOSFETs and a pair of differential N-channel MOSFETs connected in parallel. Therefore, a rail-to-rail op-amp can be used with an input voltage between GND and VDD.
The following documents also contain related information:
Basics of Operational Amplifiers and Comparators
What is the purpose of using a differential amplifier? (Common-mode rejection ratio: CMRR)
Are there any considerations for using an op-amp at low voltage?
What is the common-mode input voltage of an op-amp?
What does rail-to-rail mean?
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Operational Amplifiers and Comparators