8.8. Body Effect and Voltage Gain
In Project 9 we are able to connect the sources of the transistors to the chip body. Certainly, in general this cannot be done such that there is a body effect associated with the differential-amplifier-stage transistors. The necessary alterations to the gain equations are determined in the following. At the end of this unit, we will have the complete, precision-gain calculation equations of Level 1 SPICE. It will be informative to consider numerical results that are based on various degrees of approximations, and this is done below.
8.8.1. Common-Source Stage and Body Effect
With the body effect present, a component of current, gmb1Vs (Fig. 8.6), is subtracted from gm1Vgs1 such that (8.28) for this case is modified to become
Equation 8.41
Figure 8.6. Circuit for obtaining the inverting gain of the differential stage with body effect included. Body effect is accounted for by the current source added to the transistor signal (linear) equivalent circuit.
Additionally using Vs = Id1Rs and Vgs1 = Vi – Id1Rs (8.30)] in (8.41), a relation between Id1 and Vi is obtained, which is
Equation 8.42
Solving for Id1, the circuit transconductance is, for the body-effect case,
Equation 8.43
It follows that the gain for the inverting mode, with the addition of body effect, is
Equation 8.44
The effect of ηn in the denominator tends to make the gain smaller. However, the body effect, as shown below, will decrease Rs such that the two effects tend to cancel one another.
8.8.2. Common-Gate Stage and Body Effect
The voltage applied to the common-gate stage is Vs2 = Vsg2 (Fig. 8.7). Recall that in the g model for the transistor, as discussed for the common-source mode, current sources gmbVsb and gmVgs are in parallel but in opposite directions. Thus, the current sources gmVsg and gmbVsb are, for the common-gate mode, in the same direction since Vsb = Vsg = –Vgs; the common-gate has an effect transconductance of (1 + η)gm. It follows that the input resistance of the common-gate stage is as obtained before the body effect was included [(8.38)], except for the addition of ηn, as in the following:
Equation 8.45
Figure 8.7. Circuit that includes the voltage-dependent current source due to body effect associated with M2. Body effect affects the input resistance into the source of M2 and the gain of the common-gate stage of M2.
Similarly, the common-gate gain, (8.38), is readily modified with the addition of the multiplying factor (1 + ηn). This is
Equation 8.46
Note that in the absence of gds2, the gain for the common-source stage reduces to avcd = gm2 (1 + ηn)RD2, where, with body effect, the effective transconductance is, again, (1 + ηn) gm2.
8.8.3. Source-Follower Stage with Body Effect
The transconductance relation obtained for the common-source stage given by (8.43) also applies to the source-follower stage. Combining this with Vs1 = Id1Rs leads to the source-follower gain associated with M1, which is
Equation 8.47
The overall gain is again
av2 = avsf avcg
Note that the body effect for the source-follower stage increases the denominator of (8.49) while it increases the numerator in the common-gate result, (8.48). These tend to cancel, as in the case of the inverting gain.