What is a Butterworth Filter?

What is a Butterworth Filter?

A Butterworth filter is a type of signal processing filter designed to have a frequency response as flat as possible in the passband. Hence the Butterworth filter is also known as “maximally flat magnitude filter”. It was invented in 1930 by the British engineer and physicist Stephen Butterworth in his paper titled “On the Theory of Filter Amplifiers”.

The frequency response of the Butterworth filter is flat in the passband (i.e. a bandpass filter) and roll-offs towards zero in the stopband. The rate of roll-off response depends on the order of the filter. The number of reactive elements used in the filter circuit will decide the order of the filter.

The inductor and capacitor are reactive elements used in filters. But in the case of Butterworth filter only capacitors are used. So, the number of capacitors will decide the order of the filter.

Here, we will discuss the Butterworth filter with a low pass filter. Similarly, the high pass filter can be designed by just changing the position of resistance and capacitance.

Butterworth Low Pass Filter Design

Designers aim for a response close to the ideal filter when crafting a Butterworth filter. Matching the exact ideal characteristics is challenging, requiring complex, higher-order filters.

If you increase the order of the filter, the number of cascade stages with the filter is also increased. But in practice, we cannot achieve Butterworth’s ideal frequency response. Because it produces excessive ripple in the passband.

A Butterworth filter provides a mathematically flat frequency response up to the -3dB cutoff frequency without any ripple. Frequencies above this cutoff will decrease at a rate of -20 dB/decade in a first-order filter.

If you increase the order of the filter, the rate of a roll-off period is also increased. And for second-order, it is -40 dB/decade. The quality factor for the Butterworth filter is 0.707.

The below figure shows the frequency response of the Butterworth filter for various orders of the filter.

The generalized form of frequency response for nth-order Butterworth low-pass filter is;

Where,
n = order of the filter,
ω = operating frequency (passband frequency) of circuit
ω_C = Cut-off frequency
ε = maximum passband gain = A_max

The below equation is used to find the value of ε.

Where,
H₁ = minimum passband gain
H₀ = maximum passband gain

First-order Lowpass Butterworth Filter