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152 2. Tutorial
2.9.3 Device Mismatch
Applying Matching Information
Many analog circuit designs, including our CMOS differential amplifier, rely upon the theoretical
assumption that some semiconductor devices are perfectly matched. For instance, the transistors M1
and M2, as well as M3 and M4, must have equal small-signal characteristics to make the differential
stage function ideally, i.e. with zero offset voltage and infinite common-mode rejection ratio (CMRR).
For applying matching information Analog Insydes provides the function
MatchSymbols (Section 3.6.15). All symbols of a matching group are replaced by a common symbol.
In our example, the matching group specification {"$M1", "$M2", "12"} causes all symbols which
end with "$M1" or "$M2" to be replaced by a symbol which ends with "12". Thus, gm$M1 and gm$M2
are replaced by gm12, as well as Gds$M1 and Gds$M2 are replaced by Gds12. For the amplifier stage
this yields an enormous reduction of the expression size for the differential gain:
In[12]:= dgmatch = Simplify @ MatchSymbols[diffgainDC,
{{"$M1", "$M2", "12"}, {"$M3", "$M4", "34"}}]
Out[12]=
gm12 Gds34 2 gm34
2 Gds12 Gds34 Gds34 gm34
Computing Common-Mode Gains with Mismatch
In practice, however, the matching condition cannot always be fulfilled completely because process
tolerances or temperature gradients can cause slight differences in transistor parameters that were
originally meant to be identical. The result is a deviation from the nominal circuit behavior which
may even be unacceptably large. Therefore, it is necessary to take possible performance degradations
due to device mismatch into account during circuit design.
With a symbolic analyzer we can derive formulas which express circuit characteristics in terms of
nominal parameter values plus mismatch contributions. In Analog Insydes, accounting for mismatch
is simply achieved by assigning the same symbolic element value to two nominally equal components
and adding a "delta term" to one of the element values. Two resistors R1 and R2 with the same
nominal value R would thus be assigned the values R and R + dR, respectively, where dR represents
the mismatch contribution.
To demonstrate this procedure let’s examine the influence of device mismatch on the common-mode
gain of the CMOS amplifier from Section 2.9.2 (see Figure 9.4). To consider the corresponding
mismatch information, we set up a list of replacement rules. Since M1 and M2 should match, both
transconductances gm$M1 and gm$M2 are assigned the value gm12. Mismatch is then accounted
by adding the delta term dgm12 to the transconductance of gm$M2. Similarly, all other transistor
parameter values are expressed in terms of a nominal value plus a mismatch term.
In[13]:= mismatchparams = {
gm$M1 −> gm12, Gds$M1 −> Gds12,
gm$M2 −> gm12 + dgm12, Gds$M2 −> Gds12 + dGds12,
gm$M3 −> gm34, Gds$M3 −> Gds34,
gm$M4 −> gm34 + dgm34, Gds$M4 −> Gds34 + dGds34};