PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 609

Example 6.3

Given the second-order fi lter shown in Figure 6.37.

a. Determine by hand the transfer function H(s) = Vo(s)/ Vi(s).

b. Verify using MATLAB that the circuit of Figure 6.37 corresponds to a normalized

LPF by obtaining the magnitude and phase plots of H(w).

c. Make the following substitutions corresponding to the frequency shifting prop-

erty: R = RN, L = LN/2, and C = CN/2 and obtain its transfer function H(s), its magni-

tude and phase plots, and its numerator and denominator polynomial coeffi cients.

d. Using the transformations of part (c), obtain magnitude and phase plots of the

nonnormalized fi lter H(w) corresponding to an LPF and verify that the fi lter’s

cutoff frequency is wc = 2 rad/s.

e. Use the MATLAB function lp2lp on the fi lter transfer function of part (b) to shift the

cutoff frequency to wc = 2 rad/s. Obtain the system transfer function H(s), its magni-

tude and phase plots, and its numerator and denominator polynomial coeffi cients.

f. Implement the fi lter specs of part (b) using a Butterworth, second-order normal-

ized LPF. Obtain its transfer function H(s), its magnitude and phase plots, and its

numerator and denominator polynomial coeffi cients.

g. Using the MATLAB function lp2lp transform the normalized LPF of part (f) into a

denormalized LPF with cutoff frequency of wc = 2 rad/s. Obtain its transfer func-

tion H(s), its magnitude and phase plots, and its numerator and denominator poly-

nomial coeffi cients.

h. Implement the normalized Butterworth LPF of part (f) using electrical elements.

i. Implement the denormilized Butterworth LPF of part (g) using electrical elements

(RLC).

FIGURE 6.36

Plots of H(f), ∠H(f), H(f)dB, and ∠H(f ) using standard circuit techniques of Example 6.2.

0

magnitude versus freq.

magnitude versus freq.

phase versus freq.

phase versus freq.

frequency in Hrz frequency in Hrz

0

0.2

magnitude

magnitude (dB)

0.4

0.6

0.8

1

 5 5  5 5

 15

 20

 25

 10

 5

0

phase (degrees)

phase in degrees

80

60

40

20

0

100

 100

 50

0

50

100

0

102 104

104 

104

106 102 104 106