**Second Order Low Pass Butterworth Filter Derivation:**

A stop band response having a 40 db/decade roll off is obtained with a Second Order Low Pass Butterworth Filter Derivation

A second order filter can be obtained by coupling two active filters having a — 20 db/decade roll off to produce a roll off of — 40 db/decade, but this would not be economical because it requires two opamps.

A second order filter can be obtained by the use of a single opamp first order low pass filter by simply using an additional RC network, as shown in Fig. 15.10.

The gain of the second order filter is set by R_{1} and R_{F}, while the high cutoff frequency f_{H} is determined by R_{2}, C_{2}, R_{3} and C_{3} as given below.

The voltage gain magnitude equation for a second order low pass Butterworth response is given by

where

A_{F} = 1+ R_{F}/R_{1}= pass band gain of the filter

f = frequency of the input signal, in Hz

f_{H} = High cutoff frequencies, in Hz

The normalised Butterworth polynomials are given in Table 15.1.

where s = jω and coefficient of s = 2 k, where k is the damping factor.

**Mathematical Proof for a Second Order Low Pass Filter**

Consider a prototype second order low pass filter shown in Fig. 15.11.

Referring to the circuit in Fig. 15.11, we see that

Substituting Eqs (15.19), (15.20) and (15.21) in Eq.(15.18)

But the open loop voltage gain A_{νo}= V_{o}/V_{in} therefore

Substituting for V_{in} Eq.(15.23)

Substituting for V’ in Eq. (15.22) we have

But this is equal to the transfer function(s).

(This is the general equation.)

Therefore Z_{1} = R_{1},Z_{2} = R_{2}, Z_{3} = 1/sC_{1}, Z_{4} = 1/sC_{2}. Now for a low pass filter, Z_{1} and Z_{2} are resistors and Z_{3} and Z_{4} are capacitors, where s is the transfer function. Therefore