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

  • Thread starter Thread starter merlyn
  • Start date Start date
  • Tags Tags
    Bandwidth
Click For Summary

Discussion Overview

The discussion centers around the derivation of the equation BW=fr/Q, where BW represents bandwidth, fr denotes resonance frequency, and Q is the Quality factor. Participants are seeking a mathematical proof of this relationship, exploring its theoretical underpinnings and implications in the context of bandpass filters.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests a proof for the equation BW=fr/Q, expressing an understanding of the relationship but seeking mathematical validation.
  • Another participant suggests referring to a Wikipedia page for the origin of the Q factor, implying it may provide sufficient context.
  • A subsequent reply indicates that the Wikipedia page has already been reviewed, reiterating the need for a mathematical proof rather than just an origin explanation.
  • A different participant proposes starting with the general second-order transfer function for a bandpass filter and suggests specific substitutions and calculations to approach the derivation.
  • Another participant shares links to external resources, including lecture notes and a paper, which may contain relevant proofs or approaches, but notes that the specific derivation may not be rigorously addressed in those sources.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the derivation of the equation. There are multiple approaches and resources suggested, but the discussion remains unresolved regarding a definitive proof.

Contextual Notes

Participants express varying levels of familiarity with existing resources, indicating that some foundational assumptions or steps in the derivation may not be fully addressed in the suggested materials.

merlyn
Messages
23
Reaction score
4
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
 
Engineering news on Phys.org
Bystander said:
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.
 
  • Like
Likes   Reactions: Demforiq
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
 
Last edited by a moderator:
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
 
Last edited by a moderator:

Similar threads

Replies
5
Views
2K
  • · Replies 10 ·
Replies
10
Views
4K
Replies
9
Views
3K
  • · Replies 28 ·
Replies
28
Views
3K
Replies
3
Views
4K
  • · Replies 1 ·
Replies
1
Views
2K
Replies
1
Views
2K
Replies
29
Views
19K
  • · Replies 2 ·
Replies
2
Views
3K
Replies
4
Views
7K