BUTTERWORTH FILTERS- INTRODUCTION

EDITOR: B. SOMANATHAN NAIR

1. INTRODUCTION

Filters are a very important class of circuits in communication engineering. There are mainly four types of filters. They are low-pass, high-pass, band-pass, band- reject filters, respectively. Consider Fig. 1, which shows a passive RC circuit. We prove that this network can act as a low-pass filter.

From Fig. 1, we find

                               

                                            V_i = IR + I(1/jωC)  (1)

                                                  V_o  = I(1/jωC)  (2)

The transfer gain of the circuit is now obtained using (1) and (2) as

                                                H(ω) = V_o/ V_i =(1/jωC)/ [R + (1/jωC)]

                                                                 = 1/(1+ jωCR)

                                                                        = 1/[1+ j(ω/ω_c)]   (3)      

where ω_c = 1/CR is the cut-off frequency. Equation (3) may be expressed in the form

│H(ω) │² = 1/[1+ (ω/ω_c)²] ∟-tan^-1 (ω/ω_c)  (4)  

Equation (4) represents the transfer gain of a first-order passive low-pass filter.

            Consider (4). We observe that the magnitude H(ω) will be almost constant at 1 upto ω < ω_c. At ω = ω_c, we find that the magnitude drops to 0.707. Above this value, the gain gets reduced drastically at the rate of ‒20 dB/decade. This means that the gain gets reduced by 20 dB when the frequency gets multiplied by a factor of 10. For example, if gain at 1 kHz is 0 dB, at 10 kHz, it reduces to ‒20 dB, at 10 kHz, it reduces to ‒40 dB, and so on. Thus the circuit permits all frequencies less than ω_c to pass through it and prevents frequencies higher than ω_c from passing through it. Hence we call this circuit as a first-order low-pass filter, as there is only one RC section in it. It may be noted that the cut-off of the single-section RC filter is not sharp. To get sharper cut-off, we have to increase the number of RC sections. Thus, a two-section RC filter, called as second-order filter, has a gain drop at the rate of ‒40 dB/decade. Similarly, a third-order filter has three RC sections with a gain drop off of ‒60  dB/decade. In general, with an n-section RC filter, we have a gain drop off of ‒20n dB/decade. Equation (4) may now be modified as

                                                │H(ω) │² = 1/[1+ (ω/ω_c)²ⁿ]^1/2   (5)

Equation (5) represents the transfer gain of an n^th-order Butterworth filter.

Figure 2 shows the frequency-response characteristics of first-order, second-order, and third-order Butterworth low-pass filters. Butterworth filters have the following features:

·       The pass band can be seen to be flat upto ω = ω_c and thereafter drops off uniformly in the stop (attenuation) band.

·       The higher the order of the filter, the sharper the drop off.

·       The expression for the transfer gain of the n^th-order filter shows that the filter can be designed by knowing the values of the order of the filter n. 

2. SECOND-ORDER RLC LOW-PASS FILTER

It is customary to design and construct filters of first and second orders. It is also customary to construct filters of higher order by cascading suitable number of first- and second-order filters. For example, a third-order filter is obtained by cascading a second-order with a first-order filter. Similarly, a sixth-order filter is obtained by cascading three second-order filters.

            Figure 3 shows a second-order low-pass RLC filter. The transfer gain (function) of this filter can be obtained as follows:

            From Fig. 3, we find

                                                V_i = I(R+jωL+1/jωC)    (6)

And

                                                   V_i = I/jωC    (7)

The transfer function, therefore, is

                                    H(ω) = V_o/ V_i =(1/jωC)/ [R +jωL+(1/jωC)]   (8)

                                                                  

Now, we use Laplace transform in (8) by writing s = ω. Performing this operation yields        

                                    H(s) =1/sC)/ [R +sL+ (1/sC)]

                                       

                                        =1/(sCR+s²LC +1)

                                            = (1/lLC)/[s²+(R/L)s+ (1/LC)]  (9)

Equation (9) is expressed in a more convenient form by choosing 1/LC = ω_c² (cut-off frequency) and R/L = 2δω_c (attenuation factor). Substituting these new terms in (9) yields

                                   H(s) = ω_c²/[s²+ 2δω_cs+ ω_c²]  (10)

Equation (10) is a design equation.

Example 1: Design a second-order RLC LPF for a cut-off frequency of 1 kHz. Assume damping factor to be 0.5.

Solution: From (10), we obtain

1/LC = ω_c² (11)

R/L = 2δω_c (12)

Substituting for ω_c from (10) into (11) yields

                                                R/L = 2δ/√LC  (13)

Manipulating (13), we find

                                                 R = 2δ√(L/C)  (14)

Since f_c = 1 kHz, we find

                                                ω_c² = (2πx1000)² = 1/LC (15)

                                                 

Assume the value of capacitor

  C = 0.01 μF (16)

 we get

                                                           

                                    L = 1/(2πx1000)² x 0.01x10^‒6 = 2.53 H  (17)

The value of R can now be computed as

                                     R = 2δ√(L/C) = 2x0.5x√2.53/0.01x10^‒6 = 15.9 kΩ (18)

Equations (16), (17), and (18) give the desired values of the filter components.

            In passive filter design, the values of the components are chosen from their availability in the market. Usually, we first choose C as it is difficult to get capacitors of all values. However, we an get inductors of desired values by winding them to specifications. Resistors can be obtained by using trimmer potentiometers.  

            In the forthcoming blogs, we shall discuss the design of analog and digital Butterworth filters in detail.