3 Second-Order Low-Pass Filter Standard Form

The transfer function ${\mathrm{H}}_{\mathrm{LP}}$ of a second-order low-pass filter can be expressed as a function of frequency ( $f$ ) as shown in Equation 17, the Second-Order Low-Pass Filter Standard Form.

In this equation, $f$ is the frequency variable, $f_{\mathrm{c}}$ is the cutoff frequency, $FSF$ is the frequency scaling factor, and $Q$ is the quality factor. Equation 17 has three regions of operation: below cutoff, in the area of cutoff, and above cutoff. For each area, Equation 17 reduces to:

• $f\ll f_{\mathrm{c}}\Rightarrow {\mathrm{H}}_{\mathrm{LP}}\left(f\right)\approx K$
  • The circuit passes signals multiplied by the gain factor $K$ .
• $\frac{f}{f_{\mathrm{c}}}=FSF\Rightarrow {\mathrm{H}}_{\mathrm{LP}}\left(f\right)=jKQ$
  • Signals are phase-shifted 90° and modified by the $Q$ factor.
• $f\gg f_{\mathrm{c}}\Rightarrow {\mathrm{H}}_{\mathrm{LP}}\left(f\right)\approx -K{\left(\frac{FSF\times f_{\mathrm{c}}}{f}\right)}^2$
  • Signals are phase-shifted 180° and attenuated by the square of the frequency ratio.

With attenuation at frequencies above $f_{\mathrm{c}}$ increasing by a power of two, the last formula describes a second-order low-pass filter.

The frequency scaling factor $FSF$ is used to scale the cutoff frequency of the filter so that it follows the definitions given before.