PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 617

The implementation of a Butterworth normalized (wc = 1 rad/s) LPF prototype using

electrical components is illustrated below, obtained by equating the coeffi cients of the

obtained transfer function with the second-order Butterworth polynomial as indicated

in the following (the solution in term of electrical components is shown in Figure 6.43).

R = 1 Ω, RLC = 1 and L = 1.412 H

then

C = 1/(R * C)= 1/1.41 F

The implementation of a Butterworth denormalized (wc = 2 rad/s) LPF prototype in

terms of electrical components is illustrated in Figure 6.44, where R = 1 Ω, L = 2.8284 H,

then C = 1/(R * C) = 1/(4 * 2.8284) F.

FIGURE 6.43

Synthesis of Butterworth normalized (wc = 1 rad/s) LPF prototype of Example 6.3.

1/1.41 F

Vi(s) Vo(s)

C

1.41 H

L

1 Ω

R

FIGURE 6.44

Synthesis of Butterworth denormalized (wc = 2 rad/s) LPF prototype of Example 6.3.

C

L = 2.8284 H

R = 1Ω

Vi(s) =^1 Vo(s)

11.3136

F

Example 6.4

Test the performance of the LPF and HPF of Examples 6.1 and 6.2 by applying the fol-

lowing input: x(t) = (^5) cos(2 pi (^1000) t) + 12.5 sin(2 pi (^35000) t) to each fi lter and
by obtaining and observing the resulting plots of their respective outputs. Discuss the
results.
The solution of Example 6.4 is given by the script fi le testfi lter as follows:
MATLAB Solution
% Script File: test filter
% testing LPF & HPF
R = 1e3/(2pi); % electrical elements
C =.1e-6;
a = 1/(RC);