Environmental Engineering Reference
In-Depth Information
Chapter 4
STABILITY
FREQUENCY AND STEP RESPONSE
In the previous chapter we enunciated the basic feedback concepts and
described efficient techniques for analysing feedback amplifiers.
Particularly, we defined the open-loop gain, the loop-gain (or return ratio),
and other quantities, all as
DC values.
However, these quantities are in
general a function of frequency and they should be better referred to as
transfer functions
instead of gains. Moreover, the feedback factor could also
be frequency dependent (to this end, the best example is perhaps the well-
known RC-active integrator made up of an op-amp and a feedback network
constituted by a resistor and a capacitor). Thus, all these effects should be
taken into account in the Rosenstark and Choma relationships, (3.8) and
(3.16), which allow us to accurately obtain the closed-loop transfer function.
In addition, for a first-order model, they should also be considered in (3.1).
Similarly, in the Blackman equations, (3.23) and (3.24), we need to use the
appropriate return ratio transfer function to obtain input and output
impedances instead of resistances.
For the sake of simplicity, in this chapter we will assume that the
feedback factor is constant, at least in the frequency range of interest. In
addition, we will assume that the feedback network is designed so as to not
introduce further poles in the loop gain. Such a condition is fortunately often
verified in feedback amplifiers with a purely resistive feedback network.
It should be well known to the reader that an electronic circuit and system
are said to be
stable
if all bounded excitations yield bounded responses.
Otherwise, if bounded excitations produce an unbounded response the
system is said to be
unstable.
Passive RLC circuits are by nature stable.
Active networks contain internal energy sources that can combine with the
