## 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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IC Electronic Filters
IC electronic filters are frequency-selective circuits that consist of devices such resistors, capacitors, inductors, transistors, or operational amplifiers coupled with reactive components.
Passive Filters
Passive filters are implemented using only passive components such as resistors, capacitors and inductors. These filters do not produce any amplification of the input signal.
Active High Pass Filters
Active high pass filters pass signals from high frequencies and reject signals from low frequencies.
Active Band Pass Filters
Active band pass filters are used to attenuate frequencies below and above a range of frequencies (i.e., the bandwidth or passband of the filter).
Active Band Reject Filters
Active band reject filters are tuned circuits that prevent the passage of signals within a specified band of frequencies. These devices are also known as bandstop or notch filters.

##### Topics of Interest

11.5 Design of Type I Chebyshev Analog Low-Pass Filters The Type I Chebyshev filters are based on approximations derived from the Chebyshev polynomials C k( x) which constitute a set of orthogonal...

11.7 High-Pass, Band-Pass, and Band-Elimination Filters Transformation methods have been developed where a low-pass filter can be converted to another type of filter simply by transforming the...

11.4 High-Pass, Band-Pass, and Band-Elimination Filter Design Transformation methods have been developed where a low-pass filter can be converted to another type of filter simply by transforming the...

11.6 Other Low-Pass Filter Approximations We will briefly discuss two other filter types, the Inverted Chebyshev, and the Cauer or Elliptic. The Inverted Chebyshev, also known as Type II Chebyshev,...

11.4 High-Pass, Band-Pass, and Band-Elimination Filter Design Transformation methods have been developed where a low-pass filter can be converted to another type of filter simply by transforming the...

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