Normalized Second-Order Continuous-Time Lowpass Filter

The transfer function of a normalized second-order lowpass can be written as

$$H_l(s) \;=\;\frac{1}{\tilde{s}^2+\frac{1}{Q}\tilde{s}+ 1}$$

where the normalization maps the desired -3dB frequency $\omega_c$ to 1, i.e.,

$$\tilde{s}\;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\frac{s}{\omega_c },$$

and the ``quality factor'' $Q$ is defined as

$$Q \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\frac{\omega_c }{2\alpha}$$

where $\alpha$ is a convenient definition for bandwidth in radians per second, given by minus the real part of the complex-conjugate pole locations $p$ and $\pc$ in the $s$ plane:¹

$$p,\pc\;=\;-\alpha \pm j\sqrt{\omega_c ^2-\alpha^2}$$

Here we assume $0<\alpha<\omega_c$ , so that the poles have nonzero imaginary parts.

A second-order Butterworth lowpass filter is obtained for $Q=1/\sqrt{2}$ . Larger $Q$ values give the ``corner resonance'' effect often used in music synthesizers.