The second-order multipurpose filter described here can perform as a low-pass, bandpass, high-pass, or notch filter at audio frequencies. What makes this filter unique is that all of its characteristics can be varied independently with potentiometers.
To calculate the transfer function of the second-order filter, the following equation is used:
where s = jω
In basic filters, only one k value is non-zero. Figure 1's circuit includes the three basic types of filters. The characteristic frequency, Q value, and transfer gains are:
When resistors R3, R4, and R8 have the same value, the parenthetical expressions are valid. R1 and R2 cause the characteristic frequency to change independently. To independently vary the Q value, R6 is adjusted. All of the gains change when R5 is modified.
There are three useful outputs in the circuits. They are low-pass, high-pass, and bandpass outputs. In the circuit diagram, these are marked with abbreviations LP, HP, and BP. Replacing resistor R5 with the circuit in Figure 2 makes the gain adjustable. This method can also be used with the Q value. It's possible to achieve adjustability by replacing resistor R6 with Figure 2's circuit.
The characteristic frequency adjustment is more complicated. To keep the adjustments independent, resistors R1 and R2 should be adjusted at the same time. This can be accomplished using the dual potentiometer as demonstrated in Figure 3. Although the pots are linear, the adjustment is logarithmic due to the virtual ground at the op amp's negative input. The component values are selected so that the characteristics dependent on the potentiometer rotation follow Table 1. It must be emphasized that all the adjustments are independent. Gain adjustment has no effect on Q value or frequency. This is true for the Q value and characteristic frequency adjustments as well.
There are several other types of filters that can be derived using this basic configuration. All can be made using linear combinations of the three basic filters, as summarized in Table 2. Figure 4 illustrates a circuit that can be used to implement the linear combination filter. It allows both polarities, and each of its inputs has a variable voltage gain from 0 to 1.
The second-order filter also is a building block for higher-order filters. By cascading these filters, even-order filters can be built.