Butterworth Low-Pass Filters.

Figure 1.  Circuit Diagram for a First-Order Butterworth Low-Pass Filter

The circuit shown in Figure 1 is a first-order Butterworth low-pass filter. A low-pass filter is a circuit that blocks signals with frequencies greater than a cut-off frequency fc.  The circuit in Figure 1 uses an op-amp configured as a non-inverting amplifier, with an RC circuit at the non-inverting input to do the filtering of the high-frequency signals. The cut-off frequency fc of this circuit is determined by R and C, i.e., fc = 1/{2p(RC)}.

The pass-band gain Gp of this filter is given by: Gp = 1 + (RF/R1).  Thus, if the frequency f of the input signal is lower than fc, Vo ˜ Gp x Vin.  If f = fc,  Vo ˜ 0.707 Gp x Vin.  If f > fc, Vo < Gp x Vin.

Figure 2.  Circuit Diagram for a Second-Order Butterworth Low-Pass Filter

As the frequency of the input signal goes higher than fc, the gain of the first-order Butterworth low-pass filter in Figure 1 decreases at a rate of -20 dB/decade.  If one desires a better low-pass frequency response than this, the second-order Butterworth low-pass filter in Figure 2 can be used.  This circuit exhibits a -40 dB/decade roll-off at f>fc, wherein fc = 1/{2p x sqrt(R2R3C2C3)}. Also, for this circuit, magnitude of Vo/Vin = (1+RF/R1)/(sqrt(1+(f/fc)4)).