Operation with a Common-Mode Input voltage

To see how the differential pair works, consider first the case of the two gate terminals 

interjoined together and connected to a voltage vCM called the common-mode voltage. That is as shown in Fig (2.2)                

 

Since Ql and Q2 are matched, it follows from symmetry that the current I will divide equally between the two transistors. Thus

and the voltage at the sources, vs will be

 

where V_GS is the gate-to-source voltage corresponding to a drain current of I/2. Neglecting channel-length modulation, VGs and I/2 are related by

or in terms of the overdrive voltage   

  

                                                                                                                             

The voltage at each drain will be

Thus, the difference in voltage between the two drains will be zero.

Fig. 2.2      The MOS differential pair with a common-mode input voltage

Now, let us vary the value of the common-mode voltage v_CM Obviously, as long as Q1 and Q2 remain in the saturation region, the current I will divide equally between Q1 and Q2 and the voltages at the drains will not change. Thus the differential pair does not respond to (i.e., it rejects) common-mode input signals.

       An important specification of a differential amplifier is its input common-mode range.This is the range of v_CM over which the differential pair operates properly. The highest value of v_CM is limited by the requirement that Q1 and Q2 remain in saturation, thus

The lowest value of v_CM is determined by the need to allow for a sufficient  voltage across current source I for it to operate properly. If a voltage Vcs is needed across

the current source, then

References