Environmental Engineering Reference
In-Depth Information
input excitation to cause the output to increase indefinitely or sustain
oscillations. Note, however, that in practice the output of an unstable circuit
cannot diverge
indefinitely,
since a limit is set by the power supply rails.
It should also be well known that stability is ensured if all the poles of a
given circuit/system lie in the left-half of the
-plane. Thus, we could check
the stability of a feedback amplifier by evaluating the closed-loop transfer
function and determining the locations of its poles. This procedure, however,
does not provide design insides and does not specify the margins by which
stability is achieved. In fact, circuit components are affected by
manufacturing tolerances, temperature and ageing phenomena, etc., which
cause a parameter to deviate from its nominal value. Under this scenario, we
need to introduce safety stability margins, which are the
phase margin
and
gain margin.
Moreover, even stable amplifiers, hence that have a bounded
response, can take too much time to reach a steady state. For this purpose,
the classical feedback circuit analysis technique derived from the well-
known Bode disclosures can be utilised [B45].
s
In the following paragraphs we will examine the frequency response of
transfer functions characterised by different combinations of poles (and
zeros) that are found usually in real practice. Starting from this, useful
definitions will be given which help designers to derive fundamental
relations to ensure closed-loop stability with adequate margins. The closed-
loop step response in the time domain, for each typology of transfer
function, is also derived.
4.1 ONE-POLE FEEDBACK AMPLIFIERS
Among the feedback properties, the closed-loop bandwidth extension to
the original open-loop amplifier is often included [G85], [SS91]. We will
show that this property applies only to one-pole amplifiers, but is not
effective in multi-pole amplifiers.
Let us consider an open-loop amplifier having the following transfer
function including a single (negative) pole, whose angular frequency is
Now connect the amplifier in feedback with a pure resistive network,
whose feedback factor is
f,
as shown in Fig. 4.1.
