Butterworth Poles Steven Butterworth, a British engineer, 1885–1958, discovered a method of designing electric filters. He was quoted saying ‘‘An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies.’’ His algorithms are widely used in filter design, as we will see in Chapter 14.

He based his design on locating the poles of his filters in a unique pattern around a circle of radius vC. The number of poles on the circle constitutes the order of the filter. The more poles, the better the filter. Odd -order filters include one real pole at vC and pairs of complex-conjugate poles placed on a circle of radius vC at equal angular spacing.

Even-order filters locate complex-conjugate poles placed on a circle of radius vC at equal angular spacing. Figure P9–69 shows the location of the poles in a third- and in a fourthorder Butterworth filter. Assuming an vC of 1 rad/s, what is the denominator of the Laplace transform, F(s), associated with each of the two Butterworth filters shown in the figure?

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3rd.Order Butterworth jo 60 WC 60