Step-by-step Calculations for Designing a Low-pass Butterworth Filter

Designing a low-pass Butterworth filter involves calculating specific parameters to achieve desired cutoff frequency and filter order. This process ensures the filter provides a smooth response with minimal ripple in the passband.

Determine the Filter Specifications

Identify the cutoff frequency (f_c) and the acceptable ripple or attenuation in the passband. These specifications influence the filter order and component values.

Calculate the Filter Order (n)

The order of the Butterworth filter is calculated using the formula:

n = log₁₀[(10^0.1A_max – 1)] / log₁₀(f_stop / f_pass)

Where A_max is the maximum passband attenuation in decibels, and f_stop and f_pass are the stopband and passband frequencies, respectively.

Calculate the Cutoff Frequency

The cutoff frequency (f_c) is typically specified by the design requirements. It is the frequency at which the filter’s response drops to 3 dB below the passband level.

Determine Component Values

Using the calculated filter order and cutoff frequency, component values for resistors and capacitors are determined. The normalized low-pass prototype has standard values, which are then scaled to the desired cutoff frequency.

• Calculate the normalized component values for the prototype filter.
• Scale these values according to the cutoff frequency.
• Choose standard component values close to the calculated ones.