8.4. Effect of the Bias Resistor on Voltage Gain
The results above ignore the effect of resistance, Rbias. When it is included, it is no longer valid to assume that Vgs1 = Vsg2, even with ID1 = ID2. The bias-resistor effect is examined here based on a signal equivalent circuit of Fig. 8.3. The resistance at the source of M1 consists of the bias resistor and the linear-model resistance 1/gm2 (which neglects 1/gds2), looking into the source of M2.
Figure 8.3. Source circuit as viewed from M1. Source resistance includes actual resistor Rbias in parallel with input resistance at source of M2, which is 1/gm2 for the conditions of this unit.
For the noninverting mode, the circuit transconductance, Gm2 = Id2/Vi, is
Equation 8.18
Parameter gm2 is the transconductance Id2/Vs2 (i.e., from the source of M2 to the output Vo2). Note that the fraction is just a voltage divider, Vs2/Vi, since the output resistance at the source of M1 is 1/gm1, as noted in connection with (7.6). With gm1 = gm2, the multiplying fraction assumes the value of ½ for very large Rbias.
The noninverting gain, which includes the effect of the bias resistor, is
Equation 8.19
where the right-hand side uses gm1 = gm2. The gain result is technically that of the cascade of a source-follower stage (M1) and a common-gate stage, gm2RD2 (M2). The multiplying fraction is < ½.
The gain for the inverting side can be obtained by using (5.15), which is for a common-source amplifier with source resistor. This gives for the inverting-mode gain
Equation 8.20
where the right-hand side again uses gm1 = gm2. The multiplying fraction is > ½. Note that for both inverting and noninverting gains, the effect of Rbias can be neglected for most purposes, as the multiplying fractions in (8.19) and (8.20) are both close to ½.
The choice of the input at gate 1 is arbitrary and all of the results obtained here apply equally to the input taken at gate 2. The inverting output is always at the side of the input, and the noninverting output is always at the opposite side.