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L303.A.R1 A-1 APPENDIX SUMMARY of BUTTERWORTH FILTERS The Butterworth filter is constructed with passive elements; R, L, and C. The voltage transfer function of any filter is defined as the ratio of the output voltage to the input voltage, expressed in the terms of the complex frequency variable s as follows, H(s) = V 0 (s)/V IN (s) = N(s)/D(s) (1) The transfer function can always be expressed as the ratio of two polynomials, N(s) and D(s). The roots of the numerator are the zeros and the roots of the denominator are the poles of the transfer function. We will consider a low pass filter for our discussion. The low pass filter, as depicted in Fig. 1, will pass low frequency signals up to the corner (or break) frequency, f c . Then, the output of the filter will drop at a rate of -20n dB/dec, where n is the order of the denominator. The ideal filter has perfect response. To obtain the maximum possible slope, for a real filter, we let the order of the numerator be zero, i.e., N(s) = A. This will result in an all-pole transfer function. For an effective filter the order of the denominator n must be as large as possible. Fig. 1 Low Pass Filter Characteristics.
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L303.A.R1 A-2 Hence, the Butterworth transfer function of an n-th order low pass filter can be written as, H n (s) = A/B n (s) (2) where the denominator B n is called the Butterworth polynomial of order n and is defined in Eq. 4 below. Transfer Function Magnitude The magnitude squared of the transfer function, with s = j ω , can also be expressed as follows, |H n (j ω )| 2 = H(j ω ) H(-j ω ) = A( ω 2 )/B( ω 2 ) (3) Since A n (s) = A, then A n ( ω 2 ) = A. The denominator can be written as a polynomial in ω 2 , B n ( ω 2 ) = B 0 + B 2 ω 2 + B 4 ω 4 + . . . + B 2n ω 2n (4) where B n ( ω 2 ) is the Butterworth polynomial of order n. We see from Eq. 3 that the transfer function is an all-pole function, no zeros. If we let B 0 = A 0 then the transfer function at ω = 0 is H n (0) = 1. If B 2n = 1/ ω c 2n , then we have the special case shown below. The transfer function of the Butterworth filter of order n can now be written, H n j ω ( ) 2 = A 0 B 0 + B 2n ω 2n = 1 1 + ω ω c ( ) 2n or H n j ω ( ) = 1 1 + ω ω c 2n = 1 1 + f f c 2n (5) Eq. 5 is a simple formula for calculating the magnitude of the Butterworth transfer function in terms of the corner frequency! It is easily plotted in Maple. See Fig. 2. We see the properties of the Butterworth filter; at low frequencies the transfer function is 1, and at high frequencies the rolloff is -20n dB/dec. Also at ω = ω c the magnitude of the transfer function is 1/ 2 (or -3 dB) for any n. These are simple properties of the Butterworth filter.
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