Proofs of Butterworth lowpass filter transfer function

[asdf26asdf26]

Please help me find proofs of the Butterworth lowpass filter and the bandpass filter.

for the butterworth. i need to prove (Vo/Vin) = 1 / [1 + (f/fc)^4]^(1/2)

for the bandpass: (Vo/Vin) = 1/[1 + {[(w^2) / (wo^2)] - 1} / wRC]

my professor said these proofs were on the internet, but i can't find anything useful.

any help would be greatly appreciated. thanks.

 

[berkeman]

Homework problems should be posted in the Homework Forums. This thread will hopefully get moved there.

You need to show us what you have done to try to solve the problem, and then we can try to help you. Do you know how to draw a first order Butterworth lowpass filter? How about a second order? Draw the circuit, and write the differential equation for Vo/Vi. Then solve it to see if you get the equations above.

 

[asdf26asdf26]

Sorry I didn't know there was a Homework section. This isn't really homework anyway. its just lab knowledge, but whatever, sorry about that.

I figured out the bandpass filter and am still working on the Butterworth.

For the bandpass i looked at a RLC crkt and set
Vo = (R / [jwL + (1/jwL) + R]) Vin

... work ...

then i just used algebraic manipulation and substitution and finally got to
Vo/Vin = 1 / sqrt{[(wRC)^2 / (wRC)^2] + ((w^2 LC - 1)^2) / (wRC)^2}
where LC = 1/wo^2

it looks so confusing typed out like shown above. it looks much better in Equation Editor.