172  Lowpass and Highpass Filter Characteristics

unwanted peaking in the passband. The Q of the 2nd-order stages also needs to be just right to get the requisite amount of peaking. The higher the filter order, the more precise this matching needs to be, and the greater the demands on component accuracy.

Because we have a 2nd-order stage cascaded with a 1st-order stage, the ultimate roll-off rate is 18 dB/ octave.

The 4th-order filter is made up of two 2nd-order stages cascaded. One of these has a relatively low Q of 0.54, while the other has a relatively high Q of 1.31. Figure 7.30 shows that the latter gives a sharper peak in the passband than the 2nd-order stage did in the 3rd-order filter, and this sharper peak is cancelled out by the other 2nd-order stage, which rolls off faster than a 1st-order stage. Because we have two 2nd-order stages cascaded, the ultimate roll-off rate is 24 dB/octave.

The 5th order Butterworth filter of Figure 7.31 is a bit more complicated, being made up of three stages.As with the 4th-order filter, there are two 2nd-order stages, one with high Q and the other with low Q. The peaking is higher but interacts with the low Q stage and the 1st-order stage to once more give a maximally flat passband.

Linkwitz-Riley Filters up to 8th-Order

Linkwitz-Riley filters, with Butterworth filters, are the only ones that have all stages set to the same cutoff frequency, as shown in Table 7.4. Unlike Butterworth filters, only a few different values of Q are used. Using the table with the stage cutoff frequencies shown will give an amplitude response −6 dB down at the cutoff frequency; this is the “natural” cutoff attenuation for a Linkwitz-Riley filter. If you want to use the more familiar −3 dB criterion for cutoff, then you will need to change all the stage frequencies in Table 7.4 to 1.543.

Acomparison of 2nd-, 3rd-, and 4th-order Linkwitz-Riley filters is shown in Figure 7.32.

Table 7.4: Frequencies and Q’s for Linkwitz-Riley filters up to 8th-order. Stages are arranged in order of increasing Q; odd-order filters have the 1st-order section at the end with no Q shown.

Lowpass and Highpass Filter Characteristics  173

Figure 7.32: Amplitude response of 2nd-, 3rd-, and 4th-order Linkwitz-Riley lowpass filters. Note that 1 kHz cutoff is at −6 dB.

You will note that the 4th-order filter is made up of two cascaded Butterworth filters with Q = 1/√2 (= 0.7071), and this is the most common arrangement for 4th-order Linkwitz-Riley crossovers. For Linkwitz-Riley filters above 4th-order, higher Q values must be used.

Bessel Filters up to 8th-Order

Bessel filters have different cutoff frequencies as well as different Q’s for each stage in the higherorder filters. The cutoff frequencies here are based on the amplitude response being 3 dB down at the required frequency. Thus, to design a 2nd-order lowpass Bessel filter that is −3 dB at 1 kHz, you use the standard 2nd-order filter design equations in Chapter 8 to make a stage with a cutoff frequency of

1.27 kHz and a Q of 0.5773.

Acomparison of 2nd-, 3rd-, and 4th-order Bessel filters is shown in Figure 7.33.

Table 7.5 shows the cutoff frequencies and Q’s for each stage. These are no longer all the same, as they were for Butterworth and Linkwitz-Riley filters, so a bit more care is required. The frequencies are normalised on 1.000, so if you want a 3rd-order Bessel lowpass filter with a cutoff of 1.00 kHz, you design a 2nd-order stage with a cutoff frequency of 1.452 kHz and a Q of 0.691, and cascade it with a 1st-order section having a cutoff frequency of 1.327 kHz.A vital point is that if you want

174  Lowpass and Highpass Filter Characteristics

Figure 7.33: Amplitude response of 2nd-, 3rdand 4th-order Bessel lowpass filters. Cutoff frequency­ 1 kHz. Compare the Butterworth filters in Figure 7.28.

Table 7.5: Frequencies and Q’s for Bessel lowpass filters up to 8th-order. For highpass filters use the reciprocal of the frequency. Stages are arranged in order of increasing Q; odd-order filters have the 1st-order section at the end with no Q shown.

a highpass Bessel filter with a cutoff of 1.00 kHz for the other half of a crossover, you must use the reciprocal of the frequency in each case, because a highpass filter is a lowpass filter mirrored along the frequency axis, if you see what I mean. Thus a 3rd-order highpass Bessel filter a is made up of a 2nd-order stage with a cutoff frequency of 1/1.452 kHz = 688.5 Hz and a Q of 0.691, cascaded with a 1st-order section having a cutoff frequency of 1/1.327 kHz = 753.6 Hz. These two 3rd-order filters are both −3 dB at 1 kHz.

Lowpass and Highpass Filter Characteristics  175

Chebyshev Filters up to 8th-Order

Like Bessel filters, Chebyshev filters have different cutoff frequencies as well as different Q’s for each stage in the higher-order filters. Once again, if you want a highpass Chebyshev filter, you must use the reciprocal of the frequency in each case.

The stage frequencies and Q’s for Chebyshev filters with passband ripple of 0.5, 1, 2, and 3 dB are shown in Tables 7.6 to 7.9.You can see that as the passband ripple increases (and the steepness of roll-off also increases), higher Q’s are required. For the high-order filters, these high Q’s are a serious problem, as they require great component precision to implement them with the required accuracy.

The usual Sallen & Key and multiple-feedback filter configurations are not up to the job, and more sophisticated circuitry is necessary. It is fair to say that filters like these nowadays are avoided like the plague; if an 8th-order 3 dB-Chebyshev filter is the answer, then you might want to look at changing

Table 7.6: Frequencies and Q’s for 0.5 dB-Chebyshev lowpass filters up to 8th-order. For highpass filters use the reciprocal of the frequency. Stages are arranged in order of increasing Q; odd-order filters have the 1st-order section at the end with no Q shown.

Table 7.7: Frequencies and Q’s for 1 dB-Chebyshev lowpass filters up to 8th-order. For highpass filters use the reciprocal of the frequency. Stages are arranged in order of increasing Q; odd-order filters have the 1st-order section at the end with no Q shown.