Designing Lowpass Filters  225

Figure 8.17: Fourth-order Sallen & Key 1 kHz lowpass filters implemented as a single stage, with four equal capacitors, gain = 2.2 and exact values: (a) Butterworth; (b) Bessel.

Sallen & Key 5th-Order Lowpass: Three Stages

A5th-order filter is usually built by cascading two 2nd-order stages and one 1st-order stage. Output bufferA3 ensures that R5-C5 are not affected by external loading; if the following stage has a suitably high impedance, thenA3 can be omitted. For the Butterworth version the three cutoff frequencies are the same, but one 2nd-order stage has a Q of 0.620 and the other a Q of 1.620. The latter is an increase on the highest Q in a 4th-order filter (which is 1.3065), and so the ratio C3/C4 is now even greater at 10.5. If C4 was set to 100 nF, then C3 becomes 1.05 uF, which is inconveniently large.

In the 4th-order two-stage lowpass filter, the resistors were set to be 1 kΩ, and the capacitor values came out as whatever they did. This makes the operation of the filter clearer, but in practice it is easier

226  Designing Lowpass Filters

Table 8.18: Component values for 4th-order unity-gain single-stage S&K lowpass filters with 1 kHz cutoff. All cutoffs at −3 dB, except Linkwitz-Riley cutoff is at −6 dB, Chebyshev cutoff is at 0 dB.

Values all checked by SPICE simulation.

Figure 8.18: Fifth-order Sallen & Key Butterworth 1 kHz lowpass filter, implemented in three stages.

and cheaper to use awkward resistor values rather than awkward capacitor values, and so I have demonstrated that approach in this example. By suitable choice of resistor values three out of the five capacitors can be chosen to be preferred values; see Figure 8.18.

C4 was chosen as 68 nF rather than 100 nF to reduce the size of C3, but it is still rather large. The input impedance of a lowpass Sallen & Key filter falls almost to the value of R3 at high frequencies, and care must be taken that it does not excessively load the previous stage. Setting C4 to 68 nF rather than 100 nF reduces C3 from 1.049 uF to 714 nF and also makes R3 and R4 722 Ω rather than 491 Ω, which should reduce the loading to the point where the distortion performance of a good opamp is not compromised. If a lighter loading onA1 is desired, then C4 could be made 47 nF, which gives C3 = 493 nF and R3 = R4 = 1045.15 Ω.Alternatively C3 can be made a preferred value, which

is usually more convenient. C4 is small enough to be made up of a number of inexpensive 1%

Designing Lowpass Filters  227

polystyrene capacitors. Table 8.19 gives some more useful component options for the second stage.

With C5 = 100 nF, R5 is large enough to present no loading difficulties forA2.

The first 2nd-order stage has a Q less than 0.7071 (1/√2) and so shows no gain peaking. Clearly the 1st-order third stage cannot show any peaking. The second stage has a substantial gain peak of +4.6 dB at 900 Hz; however, with the stage order shown the headroom loss is only +1.7 dB because of the preliminary attenuation in the first stage. Rearranging the stage order again, so the 1st-order stage comes before the low-Q stage, followed by the high-Q stage, eliminates any headroom loss. I call this the New Order. It may however impact the noise performance, because one advantage of having a 1storder final stage is that it effectively attenuates the noise from previous stages.

Component values for the common types of two-stage 5th-order lowpass filter are given in Table 8.20. C2, C4, and C5 have been selected as preferred values, giving non-preferred values for all the resistors and for C1 and C3. The value of C4 has been adjusted as required to prevent C3 from becoming

too large. The amount of gain peaking is different for each type, increasing as the filter moves from Linkwitz-Riley to a 3 dB-Chebyshev characteristic. The linear-phase filter has a 5% group delay ripple.

The component values were calculated using the tables of stage frequency and Q given in Chapter 7, and checked by simulation. The design process depends on what definition of cutoff attenuation is

Table 8.19: Options for component values in third stage of a 5th-order

Butterworth lowpass filter.

Table 8.20: Fifth-order three-stage Sallen & Key lowpass: component values for various filter types. Cutoff 1 kHz.

228  Designing Lowpass Filters

Table 8.21: Fifth-order three-stage Sallen & Key Butterworth lowpass: cutoff frequency sensitivities.

used. Throughout this book I have employed the most common definition, set out in the rightmost column of the table.

The maximum Q of 8.81 occurs in the second stage of the 3 dB-Chebyshev filter. This is perhaps a bit high for practical use with the Sallen & Key configuration, as the component sensitivity becomes high at high Qs; on the other hand, in practical measurements I found a Sallen & Key filter with a Q of 10 to be quite tractable and easy to use; see Chapter 11. When really accurate high-Q stages are required, other filter configurations such as the Tow-Thomas can be used, requiring more amplifiers but having lower component sensitivities.

The cutoff frequency sensitivities for the Butterworth version in Figure 8.18 are given in Table 8.21; as before, the worst sensitivity of 1.62 appears for the second capacitor in the 2nd-order stage with the highest Q. Comparing this with the worst-case sensitivities of 1.47 for the two-stage 4th-order and 1.33 for the two-stage 3rd-order, it can be seen that the increase is fairly slow.

Sallen & Key 5th-Order Lowpass: Two Stages

Earlier in this chapter it was shown how to make a 3rd-order filter in one stage, using only one amplifier. The same process can be used to create a two-stage 5th-order filter. The first stage gives a complex pole pair with the low Q of 0.618, plus a single pole; this is equivalent to the combination of the first stage and the last stage in Figure 8.18. The second stage gives another pole pair with the higher Q of 1.618 and does the same job as the second stage in Figure 8.18. The two stages cascaded together give a 5th-order response in the same way as the three-stage 5th-order filter, but do it with two amplifiers rather than three.

In Figure 8.19 the values of R1, R2, R3 have been twiddled to make C3 the preferred value of 68 nF, and R4, R5 have been twiddled to make C5 the preferred value of 47 nF. These values were chosen so