Sallen & Key 4th-Order Lowpass: Single-Stage Linkwitz-Riley

222  Designing Lowpass Filters

In the practical filter of Figure 8.15, C3 is made up of 220 nF + 68 nF + 18 nF = 306 nF, which assumes your capacitors are available in the E12 series. By pure luck this comes out very well, being only 0.093% low. C4 is simply made to be 27 nF; this is 1.03% low but, this makes a negligible difference to the response when it is compared with a conventional two-stage 4th-order Linkwitz-

Riley filter as described in Table 8.13, despite the fact that C4 shows the highest component sensitivity.

The component sensitivities for Figure 8.14 and Figure 8.15 are shown in Table 8.16; you will note that they are rather more favourable than for those for the single-stage Butterworth filter in Table 8.14 because of the lower Q’s in a Linkwitz-Riley filter. Things look pretty good, with only C2, C4, and R4 needing a mild eye to be kept on them.

Alternatively, it may be more cost-effective to make C1 = C3. Figure 8.16a shows the precise values that emerge when R1–R4 is 1 kΩ and the gain is adjusted to give the desired filter characteristic, while this time making C1 and C3 very nearly equal. Figure 8.16b shows the result of scaling the circuit values of Figure 8.16a to make C1 = C3 = 220 nF, while keeping the same 1 kHz cutoff frequency. Resistors R1–R4 now have the awkward value of 1.016 kΩ. This can be obtained using the 2xE24 combination of 1300 Ω and 4700 Ω in parallel, which has a nominal value only 0.23% low.Again of 1.33x (+2.5 dB) is now required, which is likely to be less convenient. The economics of capacitor choice in the 4th-order single-stage circuit are explored in more detail in the previous section on the

Butterworth version of this filter.

The overall response of the circuit is exactly the same as two cascaded 2nd-order Butterworth filters with the same cutoff frequency, which is the usual way of making a 4th-order Linkwitz-Riley filter.

We have saved the cost of an opamp and probably reduced noise and distortion somewhat due to its absence, but the extra series resistances (four instead of two) may cause trouble with increased common-mode distortion. See Chapter 11 on filter performance.

Table 8.16: Fourth-order single-stage Sallen & Key Linkwitz-Riley lowpass cutoff frequency sensitivities. Gain = 1.13x.

Designing Lowpass Filters  223

Figure 8.16: Fourth-order Sallen & Key Linkwitz-Riley 1 kHz lowpass filters implemented as a single stage: (a) equal-C with gain = 1.33 and exact capacitor values; (b) equal-C with

gain = 1.33 and preferred values for C1 and C3.

The component sensitivities for the filter in Figure 8.16b are shown in Table 8.17; they are slightly better than for the 4th-order Butterworth because of the lower Q’s involved in a Linkwitz-Riley characteristic but worse than the low-gain Linkwitz-Riley of Figure 8.14 and Figure 8.15. The worst, for C4 again, is 2.15, which is now only about four times worse than a 2nd-order Butterworth stage, and with appropriate capacitor sourcing may make this approach a viable alternative. However, overall it may not be the optimal strategy; we have saved an opamp but need more precise capacitors to achieve the same accuracy, and the net cost may be greater.

The loading effects of these filters on the preceding stage is much as for the 2ndand 3rd-order Sallen & Key filters.At low frequencies the input impedance is high because the capacitors are effectively open circuits. As the frequency rises to the cutoff point, the input impedance falls until it is

224  Designing Lowpass Filters

Table 8.17: Fourth-order single-stage Sallen & Key equal-C Linkwitz-Riley lowpass cutoff frequency sensitivities.

only slightly greater than the series input resistor R1. Above that the input impedance rises again by about 10% and then stays flat with increasing frequency.

Sallen & Key 4th-Order Lowpass: Single Stage With

Non-Equal Resistors

Earlier in this chapter we saw that by using non-equal resistors, it was possible to make a filter with three equal capacitors. This principle can be extended to 4th-order filters, as shown in Figure 8.17. The required gain is still 2.2 times, but note that the resistors now range over a ratio of 13.5 instead of 4.81.

These circuits also have potentially worse problems with the response coming back up again in the audio band. If you are living in the past with TL072s, the use of the 5534/2 instead prevents any problems.

Sallen & Key 4th-Order Lowpass: Single Stage With

Other Filter Characteristics

The component values for other filter characteristics such as Bessel and Chebyshev are laid out in

Table 8.18, together with a summary of the Butterworth and Linkwitz-Riley options given earlier.

Using a gain of 1.090 times (+0.75 dB) gives reasonable capacitor values with only tiny compromises on noise and headroom. For the filters that incorporate higher Q’s, such as the Chebyshev versions, the capacitor ratios become unwieldy, and this is probably not a good route to take; a 2-stage implementation may require another amplifier, but this will almost certainly be outweighed by lower capacitor costs.