DESIGN OF BUTTERWORTH ACTIVE (ANALOG) FILTERS

EDITOR: B. SOMANATHAN NAIR

1. BUTTERWORTH FILTER STRUCTURES

Figures 1 and 2 show the first- and second-order filter structures called Sallen and Key structures, named after the originators. These structures are also known as voltage-controlled voltage-source (VCVS) and KRC (constant-gain RC) filter circuits. By cascading these basic structures, filters of any order can be obtained. We now numerically design a Butterworth active LPF of the KRC type.

Example 1:  Design a Butterworth first-order LPF of the KRC type, shown in Fig. 1, for cut-off frequency of 1000 Hz.

Step 1: Determination of C and R

We have the cut-off frequency of an LPF given by

                                                                 

                                     f_c = 1/2πRC  (1)                                                   

Substituting for f_c = 1000 Hz and C = 0.1 mF (a standard value of capacitors widely available in the open market) in (1), we find

                                                                      

                                                R = 1/2πx1000x0.1x10^‒6 = 1.59 kW

R can be realized by using a standard resistor of 1 kW in series with a potentiometer of 1 kW for precise adjustment of resistor values.

Step 2:  Determination of amplifier components R₁ and R₂

For the non-inverting amplifier used, the gain is given by

                                                   A = 1 + (R₂/R₁) (2)                                                         

Assuming a typical gain of 100, we find

                                                           R₂/R₁ = 99                                                   

Choosing arbitrarily

                                                             R₁ = 1 kW                                                    

we obtain

                                                            R₂ = 99 kW                                                   

Use a standard resistor of 82 kW in series with a potentiometer of 22 kW. The designed circuit is shown in Fig. 2.

Example 2:  Design a Butterworth second-order LPF shown in Fig. 3 for a cut-off frequency f_c of 1000 Hz.

  

Step 1: Determination of C and R

From Example 1, we obtain the values of R and C as 1.59 kΩ and 0.1 mF, respectively, where R is realized by using a standard resistor of 1 kW in series with a potentiometer of 1 kW.

Step 2:  Determination of Amplifier components R₁ and R₂

As in Example 1, for the non-inverting amplifier used, the gain A is given by

                                                          A = 1 + (R₂/R₁) (1)                                                     

Using the coefficient of s of a second-order filter (see Table 1 in the previous blog), we have

                                                            3 ‒ A = 1.414         (2)

From (1) and (2), we get

                                                                  R₂/R₁ = 0.586                                                   

Choosing

                                                               R₁ = 1 kW (arbitrarily)          

we obtain

                                                                 R₂ = 0.586 kW           

In practical cases, we choose a 560 W in series with a pot of 100 W as R₂. The fully designed second-order low-pass filter is shown in Fig. 4.