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154 2. Tutorial
In[17]:= cmgSAG = ApproximateTransferFunction[
cmgmismatch, s, dpcmos, 0.05] // Simplify
Out[17]=
dgm12 Gds12 dGds12 gm12
gm12 Gds12 Gds34 CL s
The numerator of the result indicates that the nonideal behavior is largely due to mismatch between
M1 and M2. On the other hand, tolerances between M3 and M4 have little impact on common-mode
operation.
Computing CMRR
Using the same design point we can also simplify the differential DC gain a bit further, which yields
the known approximate gain formula for this amplifier topology:
In[18]:= dgSAG = ApproximateTransferFunction[
dgmatch, s, dpcmos, 0.05] // Simplify
Out[18]=
gm12
Gds12 Gds34
Having derived expressions for both the differential and the common-mode gain we can now
calculate the common-mode rejection ratio which is defined as the quotient of both quantities. We
let s ⩵ to focus on the DC component only.
In[19]:= CMRR = (dgSAG / cmgSAG) /. s −> 0
Out[19]=
gm12 2
dgm12 Gds12 dGds12 gm12
The conclusion that can be drawn from this result is that increasing CMRR requires making the
transconductances gm12 of M1 and M2 larger.
2.9.4 Input and Output Impedances
Circuit Configuration for Impedance Calculations
Since input and output impedances are transfer functions just like voltage gains we can compute
the former using the same circuit analysis procedures as for any other transfer function. An input
impedance is the transfer function from the current flowing into a port to the voltage across the
same port (see Figure 9.6).