888 Chapter 25
25.11.21 Q and Filter Gain
Pretty much every resonant-type active filter has the
characteristic of its gain at resonance being at least
related and often directly proportional numerically to
the Q of the filter. This means a filter with a Q of 10
usually has a voltage gain of 10 (20 dB) at resonance.
Naturally, this does not make the building of practical
equalizers any easier. Nothing much does. Even speci-
fying a maximum Q of 5 (14 dB gain) only helps by
losing 6 dB of boost with respect to a Q of 10.
That represents a very sizable chunk of system head
room stolen at the filter frequency, which also makes
the sum-and-difference matrixing necessary to provide
the usual boost-and-cut facilities difficult to configure.
The obvious solution is to attenuate the signal going
into the filter by the same amount as the gain and Q
expected of the filter. Arranging a continuously variable
Q control that also attenuates the signal source appropri-
ately is not a conspicuously simple task, at least with
most filters. Perhaps the most straightforward example
is shown in Fig. 25-65C, a state-variable-type filter with
an attenuator in the retard network altering the Q
ganged with an attenuator ahead of the input/summing
amplifier. Within reasonable limits this holds the reso-
nant peak output constant over a considerably useful Q
range. A much neater and more commonly applied solu-
tion is shown in Fig. 25-65F: a single potentiometer at
the noninverting input of the summing amp that would
serve both purposes—filter Q and input level—comple-
mentarily and simply.
Most other filters are not so obliging in terms of
continuously variable Q. Switching between a few
values of Q while substituting appropriate input attenua-
tion is quite often a practical and operationally accept-
able solution, applicable to nearly any filtering
technique. Fig. 25-64E illustrates a further develop-
ment of the Wein Bridge arrangement using this method
to provide three alternative Qs. The attenuator values
are necessarily high in impedance to prevent excessive
loading of the source, a factor that in some practical EQ
circumstances can be important.
25.11.22 High-Pass Filters
Two basic single-order high-pass filters are shown in
Fig. 25-66. The keys, for the purposes of high-pass
filtering, are the reduction of inductive reactance to
ground with reducing frequency in Fig. 25-66F and the
increasing of capacitative reactance with reducing
frequency in Fig. 25-66G.
How about combining the two and omitting the
resistors as in Fig. 25-66A? As expected, the combining
of the two opposing reactances causes an ultimate
roll-off twice as fast as for the single orders; however,
they have also resulted in a resonance peak at the point
of equal reactance. Resonance Q is the ratio of
elemental reactance to resistance; deliberately intro-
ducing loss in the circuit in the form of a termination
resistor tames the resonance to leave a nice, flat, in-band
response, Fig. 25-66B.
Substituting a basic gyrator or simulated inductance
for the real one, Fig. 25-66C naturally works just as
well and even better than expected. The filter output can
be taken straight from the gyrator amplifier output,
eliminating the need to use another amplifier as an
output buffer. Further, we can automatically introduce
the required amount of loss into the inductor by
increasing the value of the bootstrap resistor and get the
resonance damping right. (Refer to the discussion of
gyrators in Section 25.11.9.)
Further yet, we can easily change the turnover
frequency of the filter by varying what was the tuning
resistor. In doing this, of course, the elemental reac-
tance-to-loss ratio will change, causing damping factor
(and so the Q) to change with it. The frequency change
and required damping change are directly related and in
the same sense and may be simultaneously altered with
a ganged control—even, if we do our sums right, with
the two ganged tracks having the same value!
A slight redraw of Fig. 25-66C gives Fig. 25-66D, a
more conventional portrayal of the classic Sallen-Key
high-pass filter arrangement. As the Sallen-Key filter
evolves, it turns out that an equal value filter (where the
two capacitors are equal and the two resistors are equal)
results in a less than adequate response shape. An expe-
dient method of tailoring and smartening up response to
become Butterworth-like (working on the assumption
that a couple more resistors are cheaper than a special
two-value ganged potentiometer) is to alter the damping
by introducing gain into the gyrator buffer amplifier
(providing also a healthier mode of operation for the
amplifier—followers are bad news), see Fig. 25-66E. A
side effect of this technique of damping adjustment
(which, incidentally, is independent of filter frequency)
is that an input-output in-band gain is introduced. The
4 dB gain introduced necessary to render the filter
frequency response maximally flat could be included in
overall system gain, or alternatively a compensating
attenuator could be instituted ahead of it. This could as
well be arranged to be a fixed-frequency, band-end,
single-order, high-pass filter to accelerate the roll-off
slope out of band; a further alternative is to make the