As illustrated in Figure 5.2, the relationship between current and voltage is linear for small input voltages. Furthermore, the curves are perfectly symmetrical. This is due to the symmetry of the circuit. In this section, the distortion of differential pairs is investigated. We will show that the symmetry influences positively on the distortion, when we compare the differential pair to a single MOS transistor with similar transconductance and control region.
denotes the current factor of transistors M1 and M2 in
the differential pair of Figure 5.1.
The control region is 
.
We wish to compare the differential pair to a single MOS transistor/
The single MOS transistor must have a transconductance given by Equation (5.8), and the control region , i.e.\
. The required current factor is 
:
For the single transistor, an effective gate voltage
yields a signal current
, which can be calculated as:
Apparently, the current shows linear dependence of 
, whereas
the undesired second order term will result in distortion.
In Figure 5.3, Equation (5.10) of the single
MOS transistor is depicted with the corresponding Equation (5.7)
of the MOS differential pair. In the Figure,
and 
were used.
Evidently, the curve of the differential pair is more linear.
Figure 5.3: Comparison of the output current 
dependency of the
input voltage for a) a MOS differential pair and
b) a single MOS transistor.
In Figure 5.4, the response to a sinusoidal input voltage
with angular frequency 
is shown for an amplitude
(i.e. 50% of the permitted swing):
Evidently, some distortion is seen from the differential pair. However,
the distortion of the single transistor is much stronger.
Figure 5.4: Distortion for a sinusoid input signal for a) the ideal
(desired) response, b) a MOS differential pair and c) a single
MOS transistor.
Frequently, analysis of the distortion is based on series expansion of
terms. It can be shown that the output current of the MOS differential
pair, i.e. Equation (5.7), by 5. order Taylor expansion is
approximated by the following expression:
Note that no even order terms are found. This follows from the fact,
that 
is an odd function, i.e.\
for
any . Therefore, no even order harmonic distortion can
occur. Using Equations (5.11-5.13) and the
identities
the following expressions emerge for the output current with a sinusoid
input voltage:
The coefficients of 
and 
in the above
expression express the third and fifth order harmonic distortions,
respectively, as functions of the degree of control
.
In Figure 5.5, the coefficients are compared to the second
order harmonic distortion of the single MOS transistor. Note that the
exponents of the harmonic distortions in general are calculated as the
order of the distortion minus 1. Thus, higher order distortions decreases
more rapidly, when the amplitude is reduced. In Figure 5.5,
we observe that the single transistor produces second order harmonic
distortion, i.e. distortion proportional to the input amplitude.
Figure 5.5: MOS differential pair output current as a function
of the input voltage amplitude, VA.