Environmental Engineering Reference
In-Depth Information
relationship (4.29) can be written
By inspection of (4.33) we note that the second term in square brackets
represents the change caused by the doublet in the tangent of the phase
margin. Of course, if pole perfectly matches zero and (4.33)
simplifies to (4.16). Moreover if is greater (lower) than is negative
(positive), and the effect of the doublet is to decrease (increase) the phase
margin compared to the same two-pole system without the doublet.
From the above it derives that we can model the two-pole amplifier with
a pole-zero doublet by using an equivalent pure two-pole amplifier having
the same gain-bandwidth product (i.e., gain and dominant pole) and a second
pole, which guarantees the same phase margin given by (4.33). Hence,
(4.25) is approximated by
where
The second term within square brackets in (4.35) gives the deviation of
the equivalent second pole with respect to the actual second pole, which
depends on both and It can be easily shown that the deviation is at a
maximum when for a fixed value of (and
K
), that is when the
average doublet frequency is equal to the gain-bandwidth product. In
contrast, with
and
K
given, the influence of the doublet is highest in
