Q Factor & Bandwidth: Derive BW=fr/Q Equation

[merlyn]

Could some please explain to me how the equation BW=fr/Q is derived?
Where BW=bandwidth, fr=resonance freq and Q is Quality factor.
I understand this relationship suppose to hold, I just don't understand why.
I basically am looking for a proof.

thank you all a priori.
drforbin

 

[Bystander]

http://en.wikipedia.org/wiki/Q_factor
Origin rather than proof enough?

 

[merlyn]

http://en.wikipedia.org/wiki/Q_factor
Origin rather than proof enough?

Thank you so much for your reply.
Unfortunately I already read that Wiki page.
I'm really looking for a proof and I can't seem to find one ANYWHERE.
I understand intuitively how the relationship could work (or does work) but I really would like to see the math.

 

[NascentOxygen]

Start with the general 2nd order transfer function for a bandpass filter,
remember Q = 1/(2ζ )
replace s by jω
calculate ω where the magnitude of the real part equals that of the imaginary part,
you should be close to getting your answer
http://thumbnails112.imagebam.com/37333/0363e9373324851.jpg

 

[meBigGuy]

Googled q factor proof:
This may answer your question.
http://ocw.mit.edu/courses/electric...pring-2006/lecture-notes/resonance_qfactr.pdf

Your problem will be dealing with this:
"By combining Equations (1.9), (1.10), 1.11) and (1.18) we obtain the relationship between the bandwidth and the Q factor"

which, of course, he does not do in the paper.
This paper has some proofs, but I didn't go through it rigorously to see if it actually answered your question. It takes a different approach.
http://www.lightandmatter.com/html_books/lm/ch18/ch18.html