# cutoff frequency of seond order low pass filter

1. ## Re: cutoff frequency of seond order low pass filter

how to calculate the cutoff frequency of seond order low pass filter mathematically?
anyone knows?

to be more exact, i mean how to calculate fc (cutoff) from the transfer function, similar to the Q, Wp calculation.

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2. ## Re: cutoff frequency of seond order low pass filter

Originally Posted by edafisher
how to calculate the cutoff frequency of seond order low pass filter mathematically?
anyone knows?

to be more exact, i mean how to calculate fc (cutoff) from the transfer function, similar to the Q, Wp calculation.
The answer is simple - however the calculation itself can be involved:

1.) Calcualte the magnitude of the transfer function
2.) Set the magnitude equal to 0.7071 (for 3-dB cutoff) and solve for w.
3.) For Chebyshev and elliptic (Cauer) response: Set the magnitude equal to the value at dc and solve for w.

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3. ## cutoff frequency of seond order low pass filter

thank LvW. I also got the answer at
http://www.engin.umich.edu/group/ctm/freq/wbw.html

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4. ## Re: cutoff frequency of seond order low pass filter

Originally Posted by edafisher
thank LvW. I also got the answer at
http://www.engin.umich.edu/group/ctm/freq/wbw.html
Hi edafisher,

are you really sure that the above link contains the information you want
(relationship between cut-off and pole frequency for a second order lowpass) ?

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5. ## Re: cutoff frequency of seond order low pass filter

hi Lvw,

thanks for your notice. i found the link gives the wrong formula (lacks a coefficient).

I re-do the derivation, and gives the right formula in below

W-3dB = Wn * sqrt[1-2ζ^2+sqrt(4ζ^4-4ζ^2+2)]

where Wn is the pole frequency and ζ is the damping factor (equals 1/(2Q) )

6. ## Re: cutoff frequency of seond order low pass filter

Congratulations! This formula is correct!

7. ## Re: cutoff frequency of seond order low pass filter

Originally Posted by LvW
Congratulations! This formula is correct!
Yeah!

8. ## Re: cutoff frequency of seond order low pass filter

Just one additional comment (in case you don't know yet):

The formula gives you the 3-dB cut-off.
However, for Chebyshev and elliptic (Cauer) responses it is common (also) to use another definition for cut-off (end of pass band):
The frequency where the peak (ripple) in the pass band crosses the transfer function value at w=0.
In most textbooks, this definitin is used to table the pole data.

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9. ## cutoff frequency of seond order low pass filter

How do we choose teh natural frequency of a closed loop??

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