Analysis of Linear Systems by Laplace Transforms Using MATLAB

Theoretical work

The circuit diagram of an active filter is shown below.

(a) Determine an expression for the transfer function of the filter.
(b) Hence find an expression for the frequency response of the filter.
(c) Calculate the ratio when low-frequency gain of the filter is 40 dB.

(d) Calculate R and C such that the break frequency on the Bode plot occurs at , (suggest commercially available values).
(e) For the values of the components obtained in parts (c) and (d) simplify the  transfer function of the filter.
(f) Hence, using inverse Laplace transform, find h(t), the unit impulse response of the system.
(g) If the input to the filter is find an expression for the output of the  system.
(h) For the input , obtain an expression for the steady-state sinusoidal response of the filter.

Practical work

1. Invoke MATLAB.

2. The enclosed m-file is a MATLAB program that simulates the above problem. In a new m-file type in the entire commands in the m-file

3. Save the m-file, giving it a suitable file name.

4. In MATLAB command window execute the program by typing its file name.

5. If you have made no typing error(s), you should obtain a plot of the Magnitude and the Phase of the Frequency Response of the filter.

%Author: clear
close all

%Declare the numerator and the denominator coefficients
num=[0,50000]; %declare numerator coefficients
den=[1,500]; %declare denominator coefficients

%Compute Frequency Response
[H,w]=freqs(num,den); %compute frequency response

%Provide plots for the Magnitude and Phase of the frequency response
figure(1);

%plot magnitude of the frequency response in dB
subplot(211);
semilogx(w,20*log10(abs(H)));
title('Magnitude of the frequency response');
xlabel('Frequency (rad/s)');
ylabel('Gain (dB)');
grid;

%plot phase of the frequency response in degree
subplot(212);
phi=angle(H);
phi=(phi/pi)*180; %convert into degree
semilogx(w,phi);
title('Phase of the frequency response');
xlabel('Frequency (rad/s)');
ylabel('Phase (deg)');
grid;

return