From Signals and Systems with MATLAB Applications, Second Edition

11.4 Design of Butterworth Analog Low-Pass Filters

We will consider the Butterworth low-pass filter whose amplitude-squared function is


where k is a positive integer, and ? C is the cutoff ( 3 dB) frequency. Figure 11.10 shows relation (11.20) for k = 1, 2, 4, and 8. The plot was created with the following MATLAB code.

<span class="serif">w_w0=0:0.02:3; Aw2k1=sqrt(1./(w_w0.^2+1)); Aw2k2=sqrt(1./(w_w0.^4+1));...Aw2k4=sqrt(1./(w_w0.^8+1)); Aw2k8=sqrt(1./(w_w0.^16+1));...plot(w_w0,Aw2k1,w_w0,Aw2k2,w_w0,Aw2k4,w_w0,Aw2k8); grid</span>

Figure 11.10: Butterworth low-pass filter amplitude characteristics

All Butterworth filters have the property that all poles of the transfer functions that describes them, lie on a circumference of a circle of radius ? C, and they are 2 ?/ 2k radians apart. Thus, if k = odd, the poles start at zero radians, and if k = even, they start at 2 ?/ 2k. But regardless whether k is odd or even, the poles are distributed in symmetry with respect to the j ? axis. For stability, we choose the left half-plane poles to form G( s).

We can find the nth roots of a the complex number s by DeMoivre's theorem. It states that


Example 11.6

Derive the transfer function G( s) for the third order ( k = 3) Butterworth low-pass filter with normalized cutoff frequency ? C = 1 rad/ s.

Solution:

With k = 3 and ? c = 1 rad

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Topics of Interest

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