Designing Highpass Filters 243
At lower frequencies C2 and R2 pass less signal, reducing the output of the voltage-follower A1, causing the “bootstrapping” of R1 to be less effective, and so causing it to shunt signal away. There are two roll-off mechanisms working at once, and so we get the familiar 12 dB/octave filter slope.
There are two basic ways to control the Q of a 2nd-order highpass Sallen & Key stage. In the first, the two resistors are made unequal, and the greater the ratio between them the higher the Q; a unitygain buffer is always used. In the second method, the two resistors are made equal 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 highpass filter can also be controlled by using non-equal capacitors, but this is unusual, as it makes the calculations more complicated.
While the basic Sallen & Key configuration is simple and easy to design, it is also extremely versatile, as you will appreciate from the many different versions in this chapter and Chapter 8. It is even possible to design bandpass Sallen & Key filters; see Chapter 12.
Sallen & Key Highpass Filter Components
Highpass Sallen & Key filters are basically the same as their lowpass brothers, with the R’s and C’s swapped in their circuit positions.All the considerations described for the lowpass filters, such as the component sensitivities of particular configurations, are applicable so long as this is kept in mind.
As mentioned earlier, highpass filters have the advantage that the capacitors are usually all the same value; it is the resistors that come in awkward values, and paralleling them is cheap and takes up little PCB space. Nonetheless, in an ideal world we would have as many identical resistors as possible to ease sourcing, and some of the highpass filters in this section are configured to achieve this.
Sallen & Key 2nd-Order Highpass: Unity Gain
Sallen & Key highpass filters are very much the same as the lowpass filters, with the R’s and the C’s swapped over.
Figure 9.2 shows a 2nd-order highpass Sallen & Key unity-gain filter with a cutoff frequency of
1 kHz and a Q of 0.707 to obtain a Butterworth characteristic. Now the capacitors are equal, while the resistors have a ratio of 2 to define the required Q. The measured noise output of this filter is −115.2 dBu (corrected), with its design equations:
Design equations:
Choose R2:
C = |
2Q |
R1 = |
R2 |
9.3, 9.4 |
(2π f0 )R2 |
4Q2 |
244 Designing Highpass Filters
Analysis equations:
f0 = |
1 |
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Q = |
1 |
R2 |
9.5, 9.6 |
2πC R R |
2 |
R |
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1 |
2 |
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1 |
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The input impedance of this circuit is relatively high at low frequencies, where the impedance of the series capacitors C1, C2 is high. It then falls with increasing frequency, reaching a minimum just above the −3 dB cutoff frequency.Above this it levels out to the value of R2, as the capacitor impedance is now negligible. R1 does not add to the loading because it is bootstrapped by the voltage-follower A1.
The TL072 is a very poor choice for S&K highpass filters. Its common-mode problems are sharply revealed if it is used in highpass configurations; on clipping, the output hits the top rail and then shoots down to hit the bottom rail as the opamp internals go into phase reversal, which is about the worst sort of clipping you could imagine. In the Bad Old Days when TL072s had to be used in audio paths for cost reasons, if a highpass filter was required, the signal would be attenuated by 6 dB so the commonmode limits could never be reached, and the gain was recovered elsewhere. The TL072 also has a lot of CM distortion and generates a lot of extra distortion when its output is loaded even lightly, and it must be regarded as obsolete for almost all audio applications.
Sallen & Key 2nd-Order Highpass: Equal Resistors
You can make a lowpass Sallen & Key filter with conveniently equal capacitor values if you configure the amplifier to give voltage gain instead of acting as a unity-gain buffer.Ahighpass filter can be made in exactly the same way with equal resistor values, though there is less of an advantage, because awkward resistor values are much less of a problem than awkward capacitor values. However, this approach is valuable when you need to make a variable-frequency filter using two-gang pots with equal sections; see the end of this chapter. Figure 9.3 shows a 2nd-order equal-R Butterworth highpass
Figure 9.3: Equal-R 2nd-order highpass Butterworth filter with a cutoff frequency of 1 kHz. Gain must be 1.586 times for maximally flat Butterworth response (Q = 0.7071).
Designing Highpass Filters 245
S&K filter; you will note that the component values are exactly the same as in the lowpass equal-C filter in Figure 8.3, and the same gain of 1.586 times (+4.0 dB) is required to get a Butterworth Q of 0.7071. The awkward value of 586 Ω can be obtained in three ways. The nearest E96 value is 590 Ω, with a nominal error of 0.68%. The best 2xE24 parallel pair is 750 Ω and 2700 Ω, with a nominal error of 0.16%. Using the 3xE24 format by putting 680 Ω, 4700 Ω, and 43 kΩ in parallel reduces the nominal error to 0.007%, but it is unlikely that this degree of accuracy will be useful, bearing in mind the usual resistor tolerances.
Due to the equal-value R and C components, the design and analysis equations are just the same as for the lowpass equal-C 2nd-order Butterworth described in Chapter 8.
Design equations:
Choose a value for C (= C1 = C2)
1 |
1 |
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R1 = R2 = |
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Passband gain A = 3 − |
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Q |
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(2π f0 )C |
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Then choose R3, R4 to get the required passband gain A for the chosen Q Analysis equations:
f0 |
= |
1 |
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(R’s and C’s are both equal) |
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2π RC |
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Passband gain A = |
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R3 + R4 |
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Q = |
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1 |
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R4 |
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3 |
− A |
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9.7, 9.8
9.9
9.10, 9.11
Sallen & Key 2nd-Order Butterworth Highpass: Defined Gains
We saw in the previous section that we could select convenient resistor values for a 2nd-order Sallen & Key highpass filter by selecting the gain in the filter. We can choose both the gain and the Q by altering the resistor values. Filter gain is often unwelcome, but it is sometimes required, as in the HF path of the crossover design example in Chapter 23, and building it into a filter will save an amplifier stage and reduce cost, while possibly also reducing noise and distortion, as the signal has gone through
one less stage. On the other hand, distortion may increase somewhat because the filter stage has less negative feedback; this depends on the opamp types used and the exact circuit conditions.
As for the lowpass case, I will focus on the Butterworth filter characteristic. The exact resistor values required for gains from 0 to +8 dB are shown in Figure 9.4, where the cutoff frequency in each case is 1 kHz, and the capacitors have been set to 100 nF; other frequencies can be obtained by scaling the component values. The resistor values summarised in Table 9.1 also include the standard cases of
unity gain and of equal resistors. For equal resistors the gain required is extremely close to +4 dB. It is assumed that R4 in the gain network is fixed at 1 kΩ; in the higher-gain cases it would be advisable to reduce its value to lower the impedance of the gain network and thus minimise noise.
Figure 9.4: Highpass 1 kHz Butterworth S&K filters with defined passband gains from 0 to +8 dB.