2nd Order Filters - Practical EE

Filter stages can be cascaded to make higher-order filters. Cascading two low-pass filters makes a 2nd order low-pass filter that attenuates high-frequency signals at twice the rate in terms of dB/decade. Connect a high-pass and a low-pass filter in series, and a bandpass filter is created. Connect a low-pass and a high-pass filter in parallel and a notch filter is created.

Transfer Functions of 2nd Order Filters

Second order filters have transfer functions with second order denominator polynomials. Here is the standard form for a 2nd order filter transfer function. N(s) is a polynomial of s of degree less than or equal to 2.

$H(s) = \frac{N(s)}{\frac{s^2}{\omega_n^2} + \frac{s}{Q\omega_n} + 1}$

$If N(s) = k$ , constant, the filter is lowpass with low-frequency gain of k
$If N(s) = k \cdto \frac{s}{Q\omega_n}$ , the filter is bandpass with max gain of k
$If N(s) = k \cdto \frac{s^2}{\omega_n^2}$ , the filter is highpass with high frequency gain of k
$If N(s) = k \cdot (1 - \frac{s^2}{\omega_n^2})$ , the filter is a notch filter with gain of k