Op amps are active filters because they require power. Passive filters use passive components (Resistors, Inductors, Capacitors) to achieve a similar effect. They are constructed using a potential divider with reactive components. The diagram below shows a potential divider with two impedances, and :
The transfer function is the ratio of input to output (see ES197 - Transfer Functions for more details.). For a passive filter, this is the ratio of output voltage to input voltage, as shown above. For a filter, this will be a function of the input waveform, . When and are both identical resistors :
However, if was a capacitor , :
The gain and phase of the output are then the magnitude and argument of the transfer function, respectively:
Similar to active filters, passive filters also have a cutoff frequency . This is the point at which the power output of the circuit falls by , or the output gain falls by -3dB, a factor of . Using the above example again (a low pass RC filter):
This is also the point at which
The filter bandwith is the range of frequencies that get through the filter. This bandwith is 0 to for low pass filters, or and upwards for high pass.
For circuits more complex than those above, to find the transfer function, either:
- Find a thevenin equivalent circuit, as seen from the element
- Combine multiple elements into single impedances
Note that any of the above techniques only work for simple first order circuits.
Using , where , and :