408 Equalisation
Figure 14.22: Biquad equaliser circuit designed for f0 = 100 Hz, Q0 = 1.5, and fp = 45 Hz, Qp = 1.3.
generate a peak and an independently controlled dip in the response. It is referred to as a biquad equaliser because its mathematical description is a fraction with one quadratic equation divided by another.
While it is possible to alter all of the ten passive components independently, in practice it is found that the easiest way to get manageable design equations is to set:
R1a = R1b = R1
R2a = R2b = R2
14.1, 14.2, 14.3, 14.4
C2a = C2b = C2
R3a = R3b = R3
This approach is illustrated in Figure 14.22; it reduces the number of degrees of freedom to four, and the amplitude response is conveniently defined by f0, Q0 (the frequency and Q of the dip in the response) and fp, Qp (the frequency and Q of the peak in the response). Note that Q values greater than 0.707 (1/√2) are required to give peaking or dipping. The circuit in Figure 14.22, derived from a crossover design I did for a client, has f0 = 100 Hz, Q0 = 1.5, and fp = 45 Hz, Qp = 1.3, and so helpfully shows both a peak and a dip with different Q’s; its response is shown in Figure 14.23.You will see that the gain flattens out +13.9 dB at the LF end, considerably greater than the HF gain of 0 dB. The further apart the f0 and fp frequencies are, the greater in difference the gain will be, and care is needed to make sure it does not become excessive. The actual peak frequency in Figure 14.23 is 36 Hz and the dip frequency 115 Hz, differing from the design values because f0 and fp are sufficiently close together for their responses to interact significantly.
The value of R2a and R2b (i.e. R2) has been kept reasonably low to reduce noise, but as a consequence of the low frequencies at which the equaliser acts, the capacitors C1 and C3 are already sizable, and it is not very feasible to reduce R2 further. C1 in particular will be expensive and bulky if it is a nonelectrolytic component.
The gain of this circuit always tends to unity at the HF end of the response, so long as R2a = R2b, because at high frequencies the impedance of C2a, C2b becomes negligible and R2a, R2b set the gain.
At the LF end all capacitors may be regarded as open circuit, so gain is set by R3/R1 and may be either
Equalisation 409
Figure 14.23: Frequency response of the biquad equaliser in Figure 14.22.
much higher or much lower than unity, depending on the values set for f0 and fp. Note that the stage is phase-inverting, and this must be allowed for in the system design of a crossover.
We saw in the previous section that the bridged-T configuration has the disadvantage of high noise gain because at high frequencies C1 has negligible impedance and R1b is effectively connected from the virtual earth point to ground. In this case the value of R2a cannot be neglected, as it is comparable with
R1b, so the parallel combination of the two is used to give a more accurate equation for the noise gain:
Noisegain = |
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Thus the noise gain for Figure 14.22 comes to 5.04 times or +14.0 dB, which is certainly uncomfortably high but nothing like as bad as the +21.8 dB given by the bridged-T equaliser in the previous section.
The design procedure given here is based on the design equations introduced by Siegfried Linkwitz.
[8] The procedure is thus:
1.First calculate the design parameter k.
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The parameter k must come out as positive, or the resistor values obtained will be negative.
410 Equalisation
2.Choose a value for C2 (a preferred value is wise, because it’s probably the only one you’re going to get, and it occurs twice in the circuit), then calculate R1:
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3. Calculate R2, C1, C3 and R3 |
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The HF gain is always unity, because R2a = R2b, but the LF gain in dB is variable and is given by:
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These equations can be very simply automated on a spreadsheet, which is just as well, as several iterations may be required to get satisfactory answers.
The topology in Figure 14.22 appears to have been first put forward by Siegfried Linkwitzin 1978; [3] it is not clear if he invented it, but it seems he was certainly the first to develop useful design equations. Its use for equalising the LF end of loudspeaker systems was studied by Greiner and Schoessow in 1983. [9] Rather surprisingly, Greiner and Schoessow laid very little emphasis on the flexibility and convenience of this topology. The shunt-feedback configuration means that the two halves of the circuit are completely independent of each other and there is no interaction of the peak and dip parameters. However, it is important to understand that if f0 and fp are close together, the summation of their responses at the output may make the centre frequencies appear to be wrong at a first glance.
We will now see how this equaliser can be used for LF response extension. Figure 14.24 shows the low-end response of the loudspeaker we used in the previous section, with the relatively high Q of 1.20 and a −3 dB cutoff point of 74 Hz, which corresponds to a −3 dB point that would be at 100 Hz if the Q was reduced to 0.7071 for maximal flatness. The loudspeaker response peaks by 2.4 dB at 124 Hz. We now design a biquad equaliser with f0 = 100 Hz—note that is the frequency for the Q = 0.7071 version of the loudspeaker—and a Q of 1.20. This will cancel out the peaking in the loudspeaker response. The choice of fp and Qp depends on how ambitious we are in our plan to extend the LF response, but in this case fp has been set to 40 Hz and Qp to 0.5.
Equalisation 411
Figure 14.24: Low-frequency extension using a biquad equaliser for a loudspeaker with an unequalised Q of 1.20. The dotted line is at −3 dB.
Figure 14.24 shows that the response peak from the original loudspeaker has been neatly cancelled, and the overall LF response extended. The −3 dB point has only moved from 74 Hz to 62 Hz, which does not sound like a stunning improvement, but the gentler roll-off of the new response has to be taken into account. The −6 dB point has moved from 63 Hz to 40 Hz, a more convincing change. The LF roll-off takes on the Q of Qp, which at 0.5 would be considered overdamped by some critics, but I adopted it deliberately, as it shows how equalisation can turn an underdamped response into an overdamped one.
It is worth noting that it is not possible to set fp to 50 Hz instead of 40 Hz while leaving the other setting unchanged. This gives a negative value for the parameter k and a negative value of −3537 Ω for R2. While it is possible to construct circuitry that emulates floating negative resistors, it is not a simple business. If you run into trouble with this you should use another kind of equaliser.
The improvements may not be earth-shattering, but bear in mind that the equaliser has a gain of 9.0 dB at 40 Hz, so if the maximum SPL is going to be maintained down to that frequency, the output of the associated power output amplifier will have to increase by almost eight times. Driver cone excursion increases rapidly—quadrupling with each octave of decreasing frequency for a constant
sound pressure—so great care is required to prevent it becoming excessive, leading to much-increased non-linear distortion and possibly physical damage. Thermal damage to the voice coil must also be considered; a sobering thought.
The equaliser circuit for Figure 14.24 is shown in Figure 14.25. The noise gain is now a much more acceptable 2.1 times or +6.4 dB, which emphasises that the biquad equaliser is much superior to the bridged-T. C3 could be a 100 nF capacitor in parallel with 1.5 nF without introducing significant error.
412 Equalisation
Figure 14.25: Schematic of a biquad equaliser designed for LF extension with exact component values.
Figure 14.26: An ambitious attempt at low-frequency extension with the equaliser fp = 20 Hz and Qp = 0.5. The combined response is now −3 dB at 31 Hz, but the equaliser gain to achieve this is excessive. The dotted line is at −3 dB.
C1 is an awkward value, but if you are restricted to the E6 capacitor series it could be made up of 470 nF +150 nF + 15 nF in parallel. This is a relatively expensive solution, especially if you are using polypropylene capacitors to avoid distortion, and will also use up a lot of PCB area. A solution to this problem is provided in the next section.
It is instructive to try some more options for LF extension. Once the design equations have been set up on a spreadsheet and the circuit built on a simulator (including a block to simulate the response of the loudspeaker alone, which can in many circumstances be a simple 2nd-order filter of appropriate cutoff frequency and Q), variations on a theme can be tried out very quickly.
