Example
A simple example of a Butterworth filter is the third-order low-pass design shown in the figure on the right, with C2 = 4/3 F, R4 = 1 Ω, L1 = 3/2 H, and L3 = 1/2 H. Taking the impedance of the capacitors C to be 1/Cs and the impedance of the inductors L to be Ls, where s = σ + jω is the complex frequency, the circuit equations yield the transfer function for this device:
The magnitude of the frequency response (gain) G(ω) is given by
and the phase is given by
The group delay is defined as the derivative of the phase with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different frequencies. The gain and the delay for this filter are plotted in the graph on the left. It can be seen that there are no ripples in the gain curve in either the passband or the stop band.
The log of the absolute value of the transfer function H(s) is plotted in complex frequency space in the second graph on the right. The function is defined by the three poles in the left half of the complex frequency plane.
These are arranged on a circle of radius unity, symmetrical about the real s axis. The gain function will have three more poles on the right half plane to complete the circle.
By replacing each inductor with a capacitor and each capacitor with an inductor, a high-pass Butterworth filter is obtained.
A band-pass Butterworth filter is obtained by placing a capacitor in series with each inductor and an inductor in parallel with each capacitor to form resonant circuits. The value of each new component must be selected to resonate with the old component at the frequency of interest.
A band-stop Butterworth filter is obtained by placing a capacitor in parallel with each inductor and an inductor in series with each capacitor to form resonant circuits. The value of each new component must be selected to resonate with the old component at the frequency to be rejected.
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