Lowpass and Highpass Filter Performance  303

Distortion in Sallen & Key Filters: Looking for DAF

It is evident that if DAF theory is correct, we are going to see a large peak in distortion at some frequency.

This will not occur at the cutoff frequency but at sub-multiples of it, which have harmonics falling at the cutoff frequency. If cutoff is at 1 kHz, and we have second-harmonic distortion in the amplifier, we will see a distortion peak at an input frequency of 500 Hz, as the second harmonic of 500 Hz is 1 kHz. Likewise, with third-harmonic distortion we will see a peak when the input frequency is 333 Hz.

Perhaps rather belatedly (after working out all the DAFs) Idecided to see this effect for real. In Peter Billam’s second paper on active filter distortion [4] he uses as an example an S&K lowpass filter with a Q of 5, as shown in Figure11.3a.The response has a +14 dB peak, shown in Figure11.4.AQ of 5 is higher than any we are likely use in crossover design; even an 8th-order Butterworth filter uses a maximum Qof only 2.56. Note the high capacitor ratio of 100 needed to get Q= 5 with this type of filter. For the standard circuit, the DAF peaks at a frightening 50.9, labelled DAF 1 in Figure11.4. Itried the Billam optimisation procedure, as described in [4], which gives the modified filter shown at Figure11.3b, which has the same frequency response but a much reduced DAF peak of 11.0; it is labelled DAF 2 in Figure 11.4. The optimised filter has a much lower ratio of capacitors; 9.1 times instead of 100 times, which if nothing else will save a good deal of money on C1. But. . . to get the correct response the optimised filter has to be configured with a good deal more than unity gain; 1.91 times or+5.6 dB in fact.This is likely to be very inconvenient, causing significant problems with noise and headroom.

Trying this optimisation process with the rather more popular Butterworth 2nd-order lowpass, the peak

DAF is reduced from 2 to 1.709, which is a less than impressive improvement.Again of 1.336 times is required to get the correct response, and this is likely to be unhelpful.

Figure 11.3: Second-order lowpass Sallen & Key filters with Q = 5. Standard filter at a, after Billam optimisation at b. Cutoff frequency is 1 kHz for both.

304  Lowpass and Highpass Filter Performance

Figure 11.4: Second-order lowpass Sallen & Key filters with Q = 5; DAF 1 is for standard circuit, DAF 2 after Billam optimisation. Cutoff frequency is 1 kHz for both. DAF and dB share the same scale on the y-axis.

Going back to the Q = 5 filter, the frequency response of the optimised version is the same as for the original filter, but the gain is higher. The question is, have we actually managed to reduce the filter distortion?

The filter in Figure 11.3a was built using a 5532 section and measured with input levels of 1 Vrms and 2 Vrms. Considering that S&K filters are regularly criticised as being unsuitable for anything other than low Q’s, I was pleasantly surprised at how well this Q = 5 version behaved. The cutoff frequency and response peaking were as calculated.

And now for the distortion behaviour. Figure 11.5 shows no trace of a big distortion peak. The 1 Vrms input shows no measurable distortion, but for a 2 Vrms input there is a small distortion peak at 500 Hz, indicating that some second harmonic is being produced. This only reaches 0.0007% at its peak.

The distortion residual at this frequency could be seen to be fairly pure second harmonic as expected, while at all other frequencies only noise could be seen. Since the Q of 5 gives a +14 dB (5x) gain peak at cutoff, and the clipping level is about 10 Vrms, a 2 Vrms input is about the most demanding test possible.

Having read the Billam and Bonello papers, it was rather a surprise to find in my measurements no evidence of DAF in action at all. The conclusion at this point must be that DAF theory need not be taken into account when designing crossover filters for low distortion, and the “optimisation” process is dubious if it requires extra filter gain. Choosing a low-distortion amplifier such as the 5532 seems to be all that is necessary.