Why do we use transfer function in filters? and How to obtain cut-off frequency?

[Michael George]

I need to know what the equation of cut-off frequency of third order filter is.


I found the transfer function everywhere on the internet:


I don't understand Why we don't simply use the cut-off frequency equation? Why do we have to use the transfer function?

One more question, How can I get the cut-off frequency form the transfer function?

I found the above circuit and its transfer function on Wikipedia: https://en.wikipedia.org/wiki/Butterworth_filter

Thank you very much,

[awallin]

transfer functions are useful because they completely describe the response of a linear system.
roughly stated a sine-wave at the input will be that same sine-wave at the output with possibly a change to the amplitude and phase.

for deriving the cut-off frequency you need to study the definition of cut-off frequency!

[T3sl4co1l]

For a general 3rd order system, there need not be a single, well defined cutoff frequency.

For an intentionally well-designed one, there is a tolerance of passband attenuation (where the tolerance is smaller than the cutoff threshold), a cutoff threshold (usually -3dB), and a stopband (of the required attenuation slope).

Going from transfer function to cutoff frequency is nontrivial, because you must solve a cubic equation for where it intersects the -3dB point (or whatever the threshold is defined as).  There may be one, two or three frequencies where this occurs!

Tim

[bson]

One more question, How can I get the cut-off frequency form the transfer function?

Solve for Vo(s)/Vi(s) = -6dB = 0.5

[Michael George]

Thank you all very much, 

[TimFox]

A more conventional definition of cut-off is the -3 dB point, where Vout/Vin = 0.707 = SQRT(1/2) and Pout/Pin = 1/2 (for equal impedances).