Lowpass and Highpass Filter Characteristics 153
The Butterworth response was introduced by Stephen Butterworth (1885–1958) in 1930. [8] Compared with other filters, it is remarkable that not only was Butterworth a physicist /engineer rather than a mathematician, but in this case the filter is actually named after the person that put it into use.
Figure 7.4 shows the amplitude response, with a sharper “knee” than other filters such as the Bessel. The ultimate slope is 12 dB/octave, determined by the fact that this is a 2nd-order filter. The phaseshift is shown in Figure 7.5; it is 90° at the cutoff frequency.
As Figure 7.6 demonstrates, the Butterworth response gives a modest degree of overshoot when it is faced with a step input. Maximal flatness in the frequency response does not mean no overshoot and does not mean a flat group delay (Figure 7.7). For that you need a Bessel filter.
Linkwitz-Riley Filters
The lowest Q normally encountered in a 2nd-order filter is 0.50, which is used for 2nd-order LinkwitzRiley (LR-2) crossovers. This is less than the Q of 0.58 used for 2nd-order Bessel filters. Note that 0.5 is the square of the Butterworth Q which is0.7071 (1/√2). They are sometimes called Butterworthsquared filters because the 4th-order version is often implemented by cascading two identical 2ndorder Butterworth stages. They are also called Butterworth-6 dB filters because there is a −6 dB level at the cutoff frequency of the two cascaded filters.
Designing for 1.00 kHz using the equations given later, you will actually get −6.0 dB at 1.00 kHz; the −3 dB point is at 633.8 Hz. The phase-shift is 90° at 1.00 kHz. Use a frequency scaling factor (FSF) of
Figure 7.4: Amplitude response of 2nd-order Butterworth lowpass filter (Q = 1√2 = 0.7071).
Cutoff frequency is 1.00 kHz. Attenuation at 10 kHz is −40 dB.
Figure 7.5: Phase response of 2nd-order Butterworth lowpass filter. Cutoff frequency is 1.00 kHz. The phase-shift is 90° at 1.00 kHz. Upper trace is amplitude response.
Figure 7.6: Step response of 2nd-order Butterworth lowpass filter. Cutoff frequency is 1.00 kHz. There is significant overshoot of about 4%.
Lowpass and Highpass Filter Characteristics 155
Figure 7.7: Group delay of 2nd-order Butterworth lowpass filter. Cutoff frequency is 1.00 kHz. The delay peaks by 15% near the cutoff frequency.
1.578, i.e. design for 1.578 kHz, and you will then get −3 dB at 1.00 kHz. The phase-shift is then 90° at 1.578 kHz.
The cutoff frequency is defined as the −3-dB point.
Figure 7.8 demonstrates the slower roll-off of the Linkwitz-Riley compared with the Butterworth filter. The attenuation at 10 kHz is −32.5 dB instead of −40 dB for the Butterworth. The ultimate slope is still 12 dB/octave, as this is determined by the fact that this is a 2nd-order filter and is not affected by the Q chosen. Figure 7.9 shows the phase response. There is no step-response overshoot (Figure 7.10) and no group delay peak (Figure 7.11).
Bessel Filters
As is common in the world of filters, Bessel filters as such were not invented by Bessel at all. Friedrich Wilhelm Bessel (1784–1846) was a German mathematician and astronomer; he was long dead by the time that anyone thought of applying his mathematics to electrical filtering. He systematised the Bessel functions, which, to keep the level of confusion up, were actually discovered by Daniel Bernoulli
(1700–1782).
The first man to put Bessel functions to work in filters was W. E. Thomson, [9] and that is why Bessel filters are sometimes called Thomson filters or Bessel-Thomson filters.According to Ray Miller, [10] Thomson was actually anticipated by Kiyasu [11] working in Japan in 1943, but given the date
Figure 7.8: Amplitude response of 2nd-order Linkwitz-Riley lowpass filter (Q = 0.50); −3 dB at 1 kHz. Attenuation at 10 kHz is −32.5 dB.
Figure 7.9: Phase response of 2nd-order Linkwitz-Riley lowpass filter;−3 dB at 1.00 kHz. The phase-shift is 90° at 1.578 kHz. Upper trace is amplitude response.
Figure 7.10: Step response of 2nd-order Linkwitz-Riley lowpass filter;−3 dB at 1.00 kHz. There is no overshoot.
Figure 7.11: Group delay of 2nd-order Linkwitz-Riley lowpass filter;−3 dB at1.00 kHz. There is no peaking in the delay.
158 Lowpass and Highpass Filter Characteristics
it is not surprising that communications with the West were somewhat compromised. It is important to remember that the term Bessel-Thomson does not refer to a hybrid between Bessel and Thomson filters, because they are the same thing. This is in contrast to Butterworth-Thomson transitional filters, which are hybrids between Butterworth and Thomson (i.e. Bessel) filters.
A Q of 1/√3 = 0.5773 is used for 2nd-order Bessel filters. The Bessel filter gives the closest approach to constant group delay (Figure 7.15); in other words the group delay curve is maximally flat.As a result the time response is very good (Figure 7.14), and the overshoot of a step function is only 0.43% of the input amplitude. This is often not visible on plots and has led some people to think that there is no overshoot at all; it is certainly very small and unlikely to cause trouble in most applications, but it does exist.
The downside of the Bessel filter is that the amplitude response roll-off is slow—actually very slow compared with a Butterworth filter; see Figure 7.12. The cutoff frequency is defined as the −3-dB point. Figure 7.13 gives the phase response.
If you design for 1.00 kHz using the equations, you will actually get −4.9 dB at 1.00 kHz; the −3 dB point is at 777 Hz. There is however a 90° phase-shift at 1.00 kHz. Use a frequency scaling factor (FSF) of 1.2736, i.e. design for 1.2736 kHz, and you will get −3 dB at 1.00 kHz.
The attenuation at 10 kHz is −36 dB, which is worse than the Butterworth (−40 dB) but better than the Linkwitz-Riley (−32.5 dB).
Note that the flat part of the group delay curve extends further up in frequency than for the LinkwitzRiley filter, and yet the group delay does not peak like the Butterworth filter. If you want to combine
Figure 7.12: Amplitude response of 2nd-order Bessel lowpass filter (Q = 1/√3 = 0.5773). Cutoff frequency is 1 kHz. Attenuation at 10 kHz is −36 dB.
Figure 7.13: Phase response of 2nd-order Bessel lowpass filter; −3 dB at 1.00 kHz. The phase-shift is 90° at 1.3 kHz. Upper trace is amplitude response.
Figure 7.14: Step response of 2nd-order Bessel lowpass filter; −3 dB at 1.00 kHz. The 0.43% overshoot is just visible.