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

### Introduction.

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.

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].

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 ).

 Simulation Unity gain Sallen-Key filter Unity 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 variation 47 : 1 1.2 : 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.

### References.

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