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State Varaible Filter

State variable filters are so named because each of the parameters on the filter can be set independently of the others. It is also capable of very high Q, unlike other topologies, such as the Sallen Key. It also features the high-pass, band-pass, and low-pass outputs simultaneously. The drawback of this circuit over others is that it requires three op-amp stages vs one stage.

The diagram below shows the circuit diagram for the state variable filter.

The transfer functions of this circuit are:


All three transfer functions share the same solution for :


By setting Fc using R1, R2 and C1, C2, Q can be set independently using R and Ra. In fact if we let C1=C2 and R1=R2 the solution simplifies even further to: