First-Order Lowpass Filters

198  Designing Lowpass Filters

Second-Order Filters

Second-order filters are much more versatile. They are defined by two parameters; the cutoff frequency and the Q, and as described in Chapter 7, higher-order filters are usually made by cascading 2nd-order stages with carefully chosen cutoff frequencies and Q’s. Odd-order filters require a 1st-order stage as well.

There are many ways to make a 2nd-order filter. The simplest and most popular are the well-known Sallen & Key configuration and the multiple-feedback (MFB) filters. The latter are better-known in their bandpass form but can be configured for either lowpass or highpass operation. MFB filters give a phase inversion so can only be used in pairs (e.g. in a Linkwitz-Riley 4th-order configuration) or in conjunction with another stage that re-inverts the signal to get it back in phase.

These two filter types are not suitable for producing high Q characteristics as for Qs above about 3, such as might be needed in a high-order Chebyshev filter, as they begin to show excessive component sensitivities and also may start to be affected by the finite gain and bandwidth of the opamps involved. For higher Qs more complex multi-opamp circuit configurations are used that do not suffer from these problems. Having said that, very high Q filters are not normally required in crossover design.

Sallen & Key 2nd-Order Lowpass Filters

The Sallen & Key filter configuration was introduced by R.P. Sallen and E. L. Key of the MIT Lincoln Laboratory in 1955. [1] It became popular as it is relatively easy to design and the only active element required is a unity-gain buffer, so in pre-opamp days it could be effectively implemented with a simple emitter-follower, and before that with a cathode-follower.

An example of a 2nd-order lowpass Sallen & Key filter is shown in Figure 8.2. Its operation is very simple; at low frequencies the capacitors are effectively open circuit, so it acts as a simple voltagefollower. At higher frequencies C2 begins to shunt the signal to ground, and this reduces the output of the follower, causing the “bootstrapping” of C1 to be less effective, and so causing it to also shunt signal away. Hence there are two roll-off mechanisms working at once, and we get the familiar 12 dB/ octave filter slope.

There are two basic ways to control the Q of a 2nd-order lowpass Sallen & Key stage. In the first, the two capacitors are made unequal, and the greater the ratio between them the higher the Q; a unity-gain buffer is always used. In the second method, the two capacitors are made equal (which will usually be cheaper as well as more convenient), and Q is fixed by setting the gain of the amplifier to a value greater than one. The first method usually gives superior performance, as filter gain is often not wanted in a crossover. The Q of a Sallen & Key filter can also be controlled by using non-equal resistors in the lowpass case or non-equal capacitors in the highpass case, but this is unusual.

There are likewise two methods of Q-control for a 2nd-order highpass Sallen & Key stage, but in this case it is the two resistors that are being set in a ratio, or held equal while the amplifier gain is altered.

While the basic Sallen & Key configuration is a simple and easy to design, it is actually extremely versatile, as you will appreciate from the many different versions in this chapter. It is possible to design bandpass Sallen & Key filters, (see Chapter 12) but this chapter is confined to lowpass versions and the next chapter to highpass versions.

200  Designing Lowpass Filters

The main difference in the circuit that you will notice from textbook filters is that the resistor values are rather low and the capacitor values correspondingly high. This is an example of low-impedance design, where low resistor values minimise Johnson noise and reduce the effect of the opamp current noise and common-mode distortion. The measured noise output is−117.4 dBu if a 5532 section is used. This is after correction by subtracting the testgear noise floor.

In this case the component choice for the capacitors is simple. C2 is a preferred value and C1 can be made up of two 100 nF in parallel. The resistance required is not a preferred value; R1 = R2 are shown to four significant figures in Figure 8.2, but a more exact value is 1125.41 Ω. By pure chance this is very close to the E24 value of 1100 Ω, and using such resistors will give a nominal cutoff frequency of 1023.1 Hz which is only 2.3% in error. If this is not acceptable, there are three useful options for more accuracy. These options are described in more detail in Chapter 15:

1.Use a single E96 value. The nearest to 1125.39 Ω is 1130 Ω, which gives a nominal cutoff frequency of 995.93 Hz, 0.41 % low.

2.Use two E24 values combined (usually in parallel) with the values as near equal as possible to maximise the improvement in effective tolerance (see Chapter 15). The best answer depends on how you prioritise an accurate nominal value versus improvement in effective tolerance, but here a convincing result is 1800 Ω paralleled with 3000 Ω, giving a combined value of 1125.0 Ω, as in Figure 8.2. This gives a nominal cutoff frequency of 1000.35 Hz, only 0.035 % high. To get the benefit of this it will be necessary to use 0.1% tolerance resistors; the effective tolerance is improved by the use of two resistors to 0.073%.

3.Use three E24 values combined (usually in parallel) with the values as near equal as possible to maximise the improvement in effective tolerance. For maximum accuracy of the nominal value,

1500 Ω,8200 Ω, and 10000 Ω in parallel gives 1125.343 Ω and a cutoff frequency of 1000.05 Hz, a tiny 0.005 % high. This is far more accurate than is likely to ever be necessary but shows little reduction in effective tolerance (1% resistors will give 0.77% overall), as the resistors are far from equal. Somewhat better in this respect is 2000 Ω, 2700 Ω, and 56000 Ω in parallel, giving 1125.84 Ω and a nominal cutoff frequency of 999.61 Hz, which is only 0.039 % low. Using 1% resistors will give an effective tolerance of 0.70%.

This gives you something of the flavour of choosing between various ways of implementing awkward component values. In general C1 will not be an integer multiple of C2, and so a similar process will have to be applied, though the practice is rather different because capacitors come in much sparser series than resistors, usually E3, E6 or E12. This matter is dealt with in Chapter 15.

Here are the design and analysis equations for the general unity-gain Sallen & Key lowpass filter; in other words they cover all characteristics, Butterworth, Bessel, etc. The design equations give the component values required for given cutoff frequency and Q; the analysis equations give the cutoff frequency and Q when the existing component values are fed in.

Design equations:

Begin by choosing a suitable value for C2

Designing Lowpass Filters  201

Analysis equations:

Table 8.2 gives the Qs and capacitor ratios required for the well-known 2nd-order filter characteristics.

Note that these FSFs apply only to 2nd-order filters. Higher-order filters in general have different design frequencies for each stage. These are given in the tables in Chapter 7.

It is important to remember that a Q of 1 does not give the maximally flat Butterworth response; the value required is1/√2, i.e. 0.7071. The lowest Q’s you are likely to encounter are Sallen & Key filters, with a Q of 0.5 sometimes used in 2nd-order Linkwitz-Riley crossovers, but these are not favoured because the 12 dB/octave roll-off of the highpass filter is not steep enough to reduce the excursion of a driver when a flat frequency response is obtained, nor to attenuate drive unit response irregularities. [2]

The input impedance of this circuit is of importance, because if it loads the previous stage excessively, then the distortion of that stage will suffer. The input impedance is high at low frequencies, where the series impedance of the shunt capacitors is high. It then falls with increasing frequency, reaching 2.26 kΩ at the 1 kHz cutoff frequency.Above this frequency it falls further, finally levelling out around 4 kHz at 1.125 kΩ, the value of R1. This is because at high frequencies there is no significant signal at the opamp non-inverting input, so C1 is effectively connected to ground at one end; its impedance is now very low, so the input impedance of the filter looks as if R1 was connected directly to ground at its inner terminal.

Another consideration when contemplating opamp loading and distortion is the amount of current that the filter opamp must deliver into the capacitor C1. This is significant. If the circuit of Figure 8.2 is driven with 10 Vrms, the current through C1 is very low at LF, and then peaks gently at 1.9 mA rms at 650 Hz. The C1 current then reverses direction as frequency increases, and C1 is required to absorb

Table 8.2: Second-order Sallen & Key unity-gain lowpass Qs and capacitor ratios for various filter types.