252 Chapter 9 C1 100 nF R1 Input 12.7 K A = passband gain C = C 1 = C 2 R1 = Q A . 2 f 0 C R2 = 1 2Q - A 2 f 0 C Q R3 = 2Q 2 f 0 C 100 nF R2 1.82 K C2 - + A1 R3 25.5 K Out Figure 9.1: An MFB or Rauch bandpass multiple-feedback filter with f 0 = 250 Hz, Q = 2, and a gain of 1. for f 0 = 250 Hz, Q = 2, and A = 1 using the equations given, and the usual awkward resistor values emerged. The resistors shown are the nearest E96 values, and the simulated results come out as f 0 = 251 Hz, Q = 1.99, and A = 1.0024, which, as they say, is good enough for rock'n'roll. The Q of the filter can be quickly checked from the response curve as Q is equal to the centre frequency divided by the -3 dB bandwidth, that is, the frequency difference between the two -3 dB points on either side of the peak. This configuration in either Rauch or Deliyannis form, is not suitable for Q's greater than about 10, as the filter characteristics become unduly sensitive to component tolerances. If independent control of f 0 and Q are required the state-variable filter should be used instead. Similar configurations can be used for lowpass and highpass filters; see Chapter 8. The lowpass version does not depend on a low opamp output impedance to maintain stop-band attenuation at high frequencies, and so avoids the oh-no-it's-coming-back-up-again behaviour of Sallen & Key lowpass filters. 9.2 High-Q Bandpass Filters As we have just seen, the simple MFB/Rauch filter is not suitable for high Q's. When these are required (which is not likely to be very often in crossover design) there are many, many kinds of active filter that can be used. We will just take a quick look at one of the most useful, the Double-Amplifier BandPass or DABP filter; this is a good example of the way that active filter performance can sometimes be transformed by adding one more inexpensive opamp section. Figure 9.2 shows the circuit with values for a centre frequency of 1 kHz and an impressively high Q of 70, with A2 is being used to provide one of the feedback paths. Note the high value of R1 with respect to the rest of the circuit; this derives from the high Q.