2nd Order Filters

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.

Bandpass Filter AC Analysis
Notch Filter AC Analysis

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