Question

Design a 3ʳᵈ order Butterworth LPF with f = ω/2π = 2000Hz and a gain of 100.

What is the transfer function?

Answer 1

The transfer function for a 3rd order Butterworth LPF with a 2000Hz cutoff frequency and a gain of 100 is H(s) = 100 / (s^3 + 2(2π(2000))s^2 + 2(2π(2000))^2s + (2π(2000))^3).

Step-by-step explanation:

To design a 3rd order Butterworth low-pass filter (LPF) with a cutoff frequency (ω) of 2000Hz and a gain (A) of 100, we need to determine the filter's transfer function H(s). A 3rd order Butterworth filter has a squared magnitude response given by:

|H(jω)|2 = A2 / (1 + (ω/ω0)6), where ω0 = 2πf0 is the cutoff angular frequency (rad/s).

The transfer function of a Butterworth filter can be found using the polynomial that defines its magnitude response. For a 3rd order Butterworth filter, the transfer function H(s) in terms of the Laplace variable s is:

H(s) = A / ( s3 + 2ω0s2 + 2ω02s + ω03 ).

For the given specifications:

f = 2000Hz, so ω0 = 2π(2000) rad/s, and A = 100.

Then, the transfer function H(s) for the 3rd order Butterworth LPF with a gain of 100 is:

H(s) = 100 / ( s3 + 2(2π(2000))s2 + 2(2π(2000))2s + (2π(2000))3 ).