# Passive Filters

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 : ## Transfer Functions

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:

## Cutoff Frequency

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.

## RC High Pass ## RC Low Pass ## RL High Pass ## RL Low Pass ## 2nd Order Circuits

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.

### Example Using , where , and :