Filter Design – Butterworth Low Pass

Find the order of an active low pass Butterworth filter whose specifications are given as: A_max = 0.5dB at a pass band frequency (ωp) of 200 radian/sec (31.8Hz), and A_min = -20dB at a stop band frequency (ωs) of 800 radian/sec. Also design a suitable Butterworth filter circuit to match these requirements.

Firstly, the maximum pass band gain A_max = 0.5dB which is equal to a gain of 1.0593, remember that: 0.5dB = 20*log(A) at a frequency (ωp) of 200 rads/s, so the value of epsilon ε is found by:

Secondly, the minimum stop band gain A_min = -20dB which is equal to a gain of 10 (-20dB = 20*log(A)) at a stop band frequency (ωs) of 800 rads/s or 127.3Hz.

Substituting the values into the general equation for a Butterworth filters frequency response gives us the following:

Since n must always be an integer ( whole number ) then the next highest value to 2.42 is n = 3, therefore a “a third-order filter is required” and to produce a third-order Butterworth filter, a second-order filter stage cascaded together with a first-order filter stage is required.

From the normalised low pass Butterworth Polynomials table above, the coefficient for a third-order filter is given as (1+s)(1+s+s²) and this gives us a gain of 3-A = 1, or A = 2. As A = 1 + (Rf/R1), choosing a value for both the feedback resistor Rf and resistor R1 gives us values of 1kΩ and 1kΩ respectively as: ( 1kΩ/1kΩ ) + 1 = 2.

We know that the cut-off corner frequency, the -3dB point (ω_o) can be found using the formula 1/CR, but we need to find ω_o from the pass band frequency ω_p then,

So, the cut-off corner frequency is given as 284 rads/s or 45.2Hz, (284/2π) and using the familiar formula 1/CR we can find the values of the resistors and capacitors for our third-order circuit.

Note that the nearest preferred value to 0.352uF would be 0.36uF, or 360nF.

Third-order Butterworth Low Pass Filter

and finally our circuit of the third-order low pass Butterworth Filter with a cut-off corner frequency of 284 rads/s or 45.2Hz, a maximum pass band gain of 0.5dB and a minimum stop band gain of 20dB is constructed as follows.

So for our 3rd-order Butterworth Low Pass Filter with a corner frequency of 45.2Hz, C = 360nF and R = 10kΩ