Digital Signal Processing Reference
In-Depth Information
which indicates that for every 1.0%
increase
in
R
1
there is a 0.5%
decrease
in ω
p
.
Likewise, we find that
R
C
Q
R
2
2
S
p
=
−
0
.
+
Q
p
R
C
1
1
1
(4.72)
which for equal resistor and capacitor values becomes
Q
R
S
p
=
−
0
.
+
Q
p
(4.73)
1
Equation (4.73) indicates that the sensitivity of
Q
p
with respect to
R
1
can be
quite high since
Q
values can reach 50 to 100. Therefore, the selection of
R
and
C
values for the active filters plays an important part in the overall sensitivity of the
circuit. We do not have space for a full treatment of sensitivity analysis in this
text. However, references in Appendix A provide further details on sensitivity and
on methods of selecting component values that yield low sensitivities. The text by
Daryanani provides a particularly good presentation. In fact, that text presents a
different choice for the component values for a Sallen-Key active filter as listed
below. By making those selections, it can be shown by substitution into (4.70) that
Q
p
is no longer a function of
R
at all and therefore would have a sensitivity of zero
with respect to changes in resistance value:
K
=
1
R
=
R
=
R
1
2
2
⋅
Q
1
p
C
=
,
C
=
1
2
R
⋅
ω
2
⋅
R
⋅
Q
⋅
ω
p
p
p
A sensitivity analysis provides us with valuable information about the
component values used in the circuit. With that information, we can determine
which components are critical in controlling variation in key filter parameters.
However, it is also very important to perform a worst-case analysis on the circuit.
This analysis of the circuit would use the sensitivities that have already been
determined to set each component to the extreme limit of its tolerance in order to
see the effect on the overall circuit performance. For example, all components
with negative sensitivities would be set to their lower tolerance limit while all
components with positive sensitivities would be set to their upper limits. The
circuit would then be analyzed to see if it still met the specifications. Next, all
component values would be set to the opposite limit and the test would be
repeated. If a circuit passed this test, it should perform up to specifications with
