Resonate Frequency of Bandpass Filter

[p75213]

Homework Statement

See attached.

Can somebody explain how the resonant frequency w0=1/RC. I worked it out by setting imaginary Z(s)=0. The answer I get is j√2 which is obviously wrong. Is it wrong to calculate the resonant frequency in this manner in this case?

Homework Equations

The Attempt at a Solution

See Attached

 

[NascentOxygen]

For a bandpass function, as with any filter, we are interested in Vout/Vin. So they derived this transfer function as H(s) near the bottom of the solution. Are you able to examine that denominator and by relating it to the general second order system describe Ѡ_n and Q here? (Or the damping ratio, ζ zeta)?

As for your approach, I'm cautious about endorsing shortcuts. But I think it might be valid, thereby allowing you to determine Ѡ_n at least. But you haven't indicated how you changed Z(s) to something with imaginary terms, so that needs to be checked.

 

[p75213]

For a bandpass function, as with any filter, we are interested in Vout/Vin. So they derived this transfer function as H(s) near the bottom of the solution. Are you able to examine that denominator and by relating it to the general second order system describe Ѡ_n and Q here? (Or the damping ratio, ζ zeta)?

As for your approach, I'm cautious about endorsing shortcuts. But I think it might be valid, thereby allowing you to determine Ѡ_n at least. But you haven't indicated how you changed Z(s) to something with imaginary terms, so that needs to be checked.

Are you able to examine that denominator and by relating it to the general second order system describe Ѡ_n and Q here? (Or the damping ratio, ζ zeta)?

No. Are you aware of a good internet resource which explains it?
Thanks for your help

 

[NascentOxygen]

Towards the end of this article: http://www.swarthmore.edu/NatSci/echeeve1/Ref/FilterBkgrnd/Filters.html

they state that the second-order filter denominator takes the form:

https://www.physicsforums.com/images/icons/icon2.gif s² + (Ѡ_n/Q)s + Ѡ²_n

where (Ѡ_n/Q) is the bandwidth, with Q being the "Q-factor" of the system.

It's well worth memorizing this expression, and what the co-efficients mean.

If you prefer the corresponding one from control theory, it's: s² + 2ζѠ_ns + Ѡ²_n
where zeta is the co-efficient of damping and you can see ζ=1/(2Q)

 

[p75213]

Thanks for that. Makes things somewhat easier than how I was doing it.