Low-pass filter

A low-pass filter is a system with a frequency response that filters out high frequencies and passes through low frequencies (i.e., it has a low frequency passband and a high frequency stopband). First-order low-pass transfer functions are generally given by:

$$T(s)=\frac{K}{s+\alpha }=\frac{K/\alpha }{1+s/\alpha }$$

where $K$ is the gain and $\alpha$ is the corner frequency. Accordingly, its gain plot looks like this. And its phase plot looks like:

Circuits

A simple low-pass filter can be constructed with an RC circuit. Its transfer function is given by:

$$T(s)=\frac{1/sC}{R+1/sC}=\frac{\frac{1}{RC}}{s+\frac{1}{RC}}$$

The achieved gain plot is due to the property of the capacitor. As $\omega \to 0$, the impedance approaches infinity. As $\omega \to \infty$, the impedance approaches 0. All of this means that lower frequencies are functionally filtered out.

Operational amplifiers also by default act as low-pass filters when not in their feedback configuration (i.e., open-loop). We can write their open-loop gain as the transfer function:

$$A(j\omega )=\frac{A_0}{1+j\omega /{\omega }_C}$$ $$V_O(s)=A(s)[V_p(s)-V_n(s)]$$

where $A_0=A(j\omega ){|}_{\omega =0}$ is the DC gain.

Active low-pass filters have the transfer function:

$$T(s)=\frac{K}{(s/{\omega }_0{)}^2+2\zeta (s/{\omega }_0)+1}$$

See also

• High-pass filter
• Bandpass filter
• Bandstop filter