Equalisation  415

Any ingenious new circuitry that adds more active devices needs to be carefully scrutinised to make sure that neither the noise or distortion performance is unduly compromised. The capacitancemultiplier method shown here does not significantly degrade either noise or distortion. The lack of extra noise is explained by the fact that if a signal is injected into the inverting input ofA3 (which is a virtual earth point) through a resistor of equal value to R5, the gain to the output of the stage is −11 dB, and so the extra noise contribution fromA2 andA3 is of little significance.

The technique does require careful consideration of signal levels. Given the condition R1a = R1b referred to earlier, the signal voltage at the top of C1 is half that at the input of the equaliser. Therefore, if multiplication factors greater than two are used, clipping may occur in the capacitance multiplier at the output ofA3 before it does in any other part of the circuit.

Equalisers With Non-Standard Slopes

Since lowpass and highpass filter slopes come only in multiples of 6 dB/octave, creating any other slope requires some approach other than a straightforward filter. Lesser slopes can be made over small frequency ranges by putting single highpass and lowpass time-constants close together, but this will not work over larger ranges. There are more sophisticated ways of combining time-constants to get

a non-standard response slope, which I shall illustrate with a classic non-loudspeaker example. Pink noise is much more useful than white noise for audio measurement because it gives a flat line on a spectrum analyser, but the various methods of noise generation all give white noise. White noise is therefore converted into pink noise by what is called a “pinkening filter”, which has a −3 dB/octave slope over the whole audio band from 20 Hz to 20 kHz.

Equalisers With −3 dB/Octave Slopes

A−3 dB/octave slope is proverbially difficult, or at least non-obvious, to obtain, as the ultimate slopes of filters come in multiples of −6 dB/octave only. The standard solution when you need a pinkening filter is a series of overlapping lowpass and highpass time-constants that approximate to the required slope. This gives a response that wiggles up and down around a −3 dB/octave line, and the more pairs of poles and zeros used, the less the wiggle and the more accurately the response approximates to the line.

Two possible versions are shown in Figure 14.29; in both cases capacitor values have been restricted to the E6 series. The simpler version uses three RC networks with a final unmatched pole introduced by C4. Its response is shown in Figure 14.30, with an exact −3 dB/octave line (dotted) added for comparison; the error trace at the top has been multiplied by ten times to make the deviations more visible. The worst errors in the 100 Hz–10 kHz range are +0.22 dB at 259 Hz, −0.09 dB at 2.5 kHz, and +0.36 dB at 8.2 kHz, which I hope you will agree is not bad for a simple circuit.

This 3xRC circuit was originally published in Small Signal Audio Design, [11] but with R2 = 12k, R3 = 3k9, and R4 = 1k2. This had errors of up to ± 0.65 dB because of a historical requirement to use

416  Equalisation

Figure 14.29: Two −3 dB/octave equalisers built from repeated RC networks. The version at

(b) is more accurate over a wider frequency range.

only E12 resistor values. The 3x RC circuit is not the best option unless you are really closely counting the cost of every component, for reasons that will now appear.

Figure 14.31 shows the response of the version with four RC networks shown in Figure 14. 29b. It has four pole-zero pairs, with a final unmatched pole. It gives rather better performance for two reasons: one obvious, the other less so. Clearly, the more RC networks are used, the closer is the pole-zero spacing, and so the less the wobble on the frequency response. Less obvious is the fact that four RC networks allow the pole-zero frequencies required to match E6 capacitor values much better—you will note the tidy pattern of component values in Figure 14.29b, where 100/33 = 3.03 and 33/10 = 3.3, giving near-equal spacings on a log-frequency scale.

The error plot now shows a pleasing sine wave undulation around zero, with one cycle per decade of frequency, indicating that the errors (± 0.28 dB) are as low as they can be with this number of RC networks. The frequency range over which the errors are well controlled is also much greater; a very useful 20 Hz–20 kHz span. This is a good return from adding just one resistor and one capacitor. Finally, you will note that our “premium” four RC network has an overall lower impedance to reduce Johnson noise and the effect of device current noise in the resistors. This means that C1 is relatively

418  Equalisation

Figure 14.32: A −3 dB/octave equaliser with better accuracy using seven RC networks that fit in with the E6 capacitor series. Accuracy is within +0.1 dB over a 10 Hz–20 kHz frequency range.

Figure 14.33: The response of the 7x RC −3 dB/octave equaliser in Figure 14.32. The error is now within ±0.1 dB over a 10 Hz–20 kHz frequency range.

large at 1 uF, but I see no reason why the capacitance multiplication concept described for biquad equalisers should not work nicely in this application; I should however say that I have not yet tried it.

As we have just seen, it makes sense to go with the flow of the E6 capacitor series. The 4xRC network we have just looked at uses just two values from the series; 10 and 33. This principle can be extended by using three values instead of two; 10,22, and 47. This gives ratios of 100/47 = 2.10, 47/22 = 2.14, and 22/10 = 2.20. The resulting 7xRC circuit is seen in Figure 14.32, where exact resistor values derived from the capacitor ratios are shown rather than preferred values.

This gives considerably improved accuracy over the 4x RC network. The error is now within ±0.1 dB over a frequency range extended to 10 Hz–20 kHz, though the error curve does not have the same symmetry; see Figure 14.33. In each stage both the resistor and capacitor values halve. It will of course