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Comparison of Sallen-Key and State variable topology for a Linkwitz-Riley cross-over network.



The most common general purpose cross-over filter today is the 4.order Linkwitz-Riley filter.
This can be implemented in different ways.
The 2 most common are compared.

Linkwitz-Riley filter.

Sallen-Key cross-over filter schematic.
Fig.1: Sallen-Key cross-over filter.

This filter uses 4 OP-AMPs and has a total of 16 components to determine the cross-over frequency.
To obtain identical component values for all components, 0.5*RF can be 2 resistors in parallel and 0.5*CF can be 2 capacitors in series. 20 components in total.
The cross-over frequency is: Fx = 1 / ( 2 * π * √0.5 * RF * CF ).

It is possible to design this filter with all components of identical value by adding a gain of 1.59 to each of the amplifiers.
The cross-over frequency is then: Fx = 1 / ( 2 * π * RF * CF ). [1].

State variable cross-over filter schematic.
Fig.2: State variable cross-over filter.

This filter is described in detail in ref. [2].
This filter uses 5 OP-AMPs and has a total of 8 components to determine the cross-over frequency. All frequency determining components are of the same value.
A continuously variable cross-over (for experimental purposes) can be built be replacing the 4 RF with a 4-gang potmeter.
The cross-over frequency is: Fx = 1 / ( 2 * π * RF * CF ).

Table 1: Comparison between Sallen-Key and state variable filter. All simulations are from 20 Hz to 20 kHz.
SimulationUnity gain Sallen-Key filterUnity gain state variable filter
Summation error, 5% capacitors, 1% resistors-1.25/+1 dB±0.4 dB
Summation error, 1% capacitors, 1% resistors±0.35 dB±0.4 dB
Input impedance variation47 : 11.2 : 1
Summed output noise, NE5532 OP-AMPs-116 dBu-103 dBu
Summed output noise, OPA2134 OP-AMPs-110 dBu-97 dBu
Summed output noise, TL072 OP-AMPs-103 dBu-91 dBu

Table 1 shows that the worst case summation error for the Sallen-key filter is 1.25 dB. This will be much less when the components are from the same batch/tape (the reason to use 4 extra components to obtain identical values).
The state variable filter is less sensitive to its source impedance than the Sallen-Key filter.
The Sallen-Key filter must be driven from a low impedance source.
The state variable filter is 13 dB more noisy than the Sallen-Key filter. The reason is that the summation amplifier in the state variable filter has a gain of 16 dB.


[1] Texas Instruments: Analysis of the Sallen-Key Architecture.
[2] Dennis A. Bohn: A Fourth-order state variable filter for Linkwitz-Riley active crossover designs.

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