# Low-pass filter

"A low-pass filter is a filter that allows low-frequency signals to pass through while attenuating the signals with frequencies higher than the cutoff frequency. [...] The actual amount of attenuation for each frequency varies depending on specific filter design. A low-pass filter is the opposite of a high-pass filter. A band-pass filter is a combination of a low-pass and a high-pass."[1]

#### Passive low-pass filter

A passive electronic low-pass filter is a common filter type and can be created using a resistor and capacitor voltage divider as shown in Figure 1.*Figure 1: A passive electronic low-pass filter built from a resistor-capacitor voltage divider.*

The impedance of a capacitor is $\frac{1}{\omega C}$, where $\omega$ is the frequency and $C$ is the capacitance. The higher the frequency or capacitance the lower the "resistance" to oscillating current. A capacitor acts like a short for high frequencies and acts like an open circuit for low frequencies. Because of this, in Figure 1, high frequencies get pulled to ground while low frequencies are blocked from ground and so carry through to the output. This has the effect of suppressing high frequencies from the output signal.

The frequency at which the amplitude begins to attenuate is called the cut-off frequency. It is the frequency at which the signal has attenuated by 3dB as shown in Figure 2. All higher signal frequencies are attenuated at a rate of 20 dB/decade which means a drop in signal strength by 20 dB for each order of magnitude greater frequency than the cut off frequency. For example, if the cut off frequency is 1kHz, frequency of 10kHz would attenuate by 20dB and a frequency of 100kHz is attenuated by 40dB.

*Figure 2: The amplitude suppressing effect of a low-pass filter. Used with permission CC BY-SA 3.0*

The cut-off frequency for a passive RC filter can be determined using the following equation:

$$f_c = {1\over{2\pi RC}} $$

where $f_c$ is the cut-off frequency, $R$ is the resistor's resistance and $C$ is the capacitor's capacitance. A low resistance resistor may be desired in order to not overload the source signal.

#### Active low-pass filter

Another common low-pass filter type is an active filter which can be built using an op-amp with a few resistors and capacitors as shown in Figure 3. The cut-off frequency for this type of filter is:

$$f_c = \frac{1}{2\pi R_2C}$$

The gain for this type of filter is $-\frac{R_2}{R_1}$ and the attenuation rate after the cut-off frequency is 20dB per decade similarly to the passive low-pass filter.

The op-amp is in a non-inverting amplifier configuration. The output signal is an inverted and amplified version of the input signal. It provide negative feedback to the the input signal. The capacitor acts as a short for high frequencies so high frequencies are allowed to provide higher negative feedback than lower frequencies to the the input signal. Low frequencies are allowed to pass and maintain their amplitudes.

*Figure 3: An active low-pass filter built using an op-amp.*

References

[1] http://en.wikipedia.org/wiki/Low-pass_filter