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Sallen-Key Low Pass Filter Design Equations

The Sallen- Key filter is a very popular active filter which can be used to create 2nd order filter stages that can be cascaded together to form larger order filters.  The op-amp provides buffering between filter stages, so that each stage can be designed independently of the others.  These circuits are suitable for filters which have complex conjugate poles. 

When implementing a particular transfer function, a designer will typically find all of the poles, and group them into real poles and complex conjugate pairs. Each of the complex conjugate pole pares are then implemented with a Sallen-Key filter, and the circuits are cascaded together to form the complete filter.

We've created a low pass Butterworth Sallen-Key filter calculator, which automatically computes the resistor and capacitor values for a filter with a given number of poles.

We want to derive a transfer function for the Sallen-Key op-amp circuit in the following form:

The sum of the currents at node V₁.

Substituting for V₁

We have more free parameters than we really need so we can set R₁=R₂=R.