Question

Design a 3rd order Butterworth LPF with f​=ω/2π​​=2000 Hz and a gain of 100. What is the transfer function?

Answer 1

The transfer function of a 3rd order Butterworth Low Pass Filter (LPF) with a cutoff frequency
`\(f = (\omega)/(2\pi) = 2000\)` Hz and a gain of 100 is given by:

`\(H(s) = \frac{100}{{\left(1 + (s)/(\omega_c)\right) \left(1 + (s)/(\omega_c)\right) \left(1 + (s)/(\omega_c)\right)}}\)`

Step-by-step explanation:

To design a 3rd order Butterworth Low Pass Filter (LPF), we start with the Butterworth filter prototype and modify it according to the given specifications. The transfer function for a Butterworth filter is:

`\(H(s) = \frac{1}{{\left(1 + \left((s)/(\omega_c)\right)^2\right)^n}}\)`

Where n is the order of the filter and
`\(\omega_c\)` is the cutoff frequency.

Given
`\(f = (\omega)/(2\pi) = 2000\)` Hz and a gain of 100, the transfer function can be expressed as:

`\(H(s) = \frac{K}{{\left(1 + (s)/(\omega_c)\right) \left(1 + (s)/(\omega_c)\right) \left(1 + (s)/(\omega_c)\right)}}\)`

To achieve a gain of 100, the constant K in the transfer function is set to 100. Hence, the transfer function for the 3rd order Butterworth LPF with a cutoff frequency of 2000 Hz and a gain of 100 is as provided in the final answer.

This transfer function describes the behavior of the filter in the frequency domain, showing how the input signal passes through the filter with a specific gain and cutoff frequency while attenuating higher frequencies beyond the cutoff point.