Environmental Engineering Reference
In-Depth Information
7.2 HARMONIC DISTORTION IN THE FREQUENCY DOMAIN
In the previous paragraphs, both the amplifier and the feedback network
were assumed to be frequency independent. This hypothesis is clearly a
rough approximation. Transistors have parasitic capacitances which cause
the gain and even the nonlinear amplifier coefficients to vary with
frequency. Yet, high-gain feedback circuits must be frequency compensated
to ensure closed-loop stability, while the feedback network can include
reactive (usually capacitive) components. Therefore, the previous
expressions can be used with reasonable accuracy only under the hypothesis
of low-frequency input signals.
In general, evaluation of harmonic distortion of a
dynamic
system
requires complex calculation involving Volterra series or even Wiener series
[BR71], [MSE72], [NP73], [WG99]. Nevertheless, under the assumption of
low-distortion conditions -which means in practice, that the amplifier output
is not saturated and transistors do not leave their active region of operation-
we can use the usual small-signal analysis to produce accurate results. Let us
start our discussion by considering amplifiers in open-loop configuration.
7.2.1 Open-loop Amplifiers
To render the analysis sufficiently general, we will refer to two-stage
amplifiers, that adequately model real amplifiers (the obtained results could
then be extended also to multi-stage topologies, as well). Besides, we
simplify analysis by separating the effect of nonlinearities of the first and
second stage. These two cases are illustrated in Fig. 7.3a and 7.3b. Of
course, in real amplifiers both the two phenomena coexist as nonlinearity
can contemporarily come from the input and the output sections.
Nevertheless, this simplification is instructive and even representative of
actual cases. Indeed, the first scheme (Fig. 7.3a) exemplifies a conventional
op-amp or a CMOS OTA with a nonlinear output stage. In this event the
output section operates in large-signal conditions and its nonlinear behaviour
is hence exacerbated. The second scheme (Fig. 7.3b) seems uncommon.
Later, we will demonstrate that this case models the high-frequency
distortion in single-stage amplifiers. Besides, it can exemplify amplifiers
operated under large common-mode input signals, responsible for the
generation of nonlinearities in the input stage.
