- 12th July 2010, 13:25 #1

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

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

anyone knows?

Added after 23 minutes:

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

- 12th July 2010, 13:25

- 12th July 2010, 13:58 #2

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

Originally Posted by**edafisher**

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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- 12th July 2010, 15:16 #3

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## 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

- 12th July 2010, 15:16

- 12th July 2010, 16:31 #4

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

Originally Posted by**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) ?

- 12th July 2010, 16:31

- 12th July 2010, 16:45 #5

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## 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) )

- 12th July 2010, 17:03 #6

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

Congratulations! This formula is correct!

- 12th July 2010, 17:08 #7

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

Originally Posted by**LvW**

- 12th July 2010, 17:14 #8

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## 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.

- 13th July 2010, 19:27 #9

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## cutoff frequency of seond order low pass filter

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