Example H3.6

Matlab code for example 3.6 from the book "Regeltechniek voor het HBO"

Determine the transfer function in the ω-domain

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Table of Contents
Determine the transfer function in the -domain The given situation Determine the relations Example 1 with numbers (sinusoidal input) Example 2 with numbers (sinusoidal input from t=0)

The given situation

Determine the transfer function in the ω-domain of the electrical system as given in the figure 3.14

Determine the relations

In the time domain the following relations hold:
and
In the s-domain this becomes:
and
And in the ω-domain this becomes:
and
The transfer functions then become:
and so that:
And in the ω-domain they become:
and so that:
The equations in the ω-domain have the same form as the ones in the s-domain. It is a matter of simply replacing s with jω.

Example 1 with numbers (sinusoidal input)

Assume that in the example before the following is valid:
% clear all variables from Workspace and close all figures
clear variables;
close all;
 
% Replace s of type tf with s of type sym (symbolic)
syms s;
% create symbolic functions
syms t e1(t) e2(t);
amp=5*sqrt(2);
omega=1000;
e1(t)=amp*sin(omega*t)
e1(t) = 
Then select the values of R and C to be equal to 10kOhm and C=0.1uF. Or else to see what happens then.
R=10; % kOhm
C=0.1; % uF
L=0; % H
 
% define transferfunction of RC network
j=1i; % Define complex number
H=(j*omega*R*1000*C*1e-6)/(1+j*omega*R*1000*C*1e-6);
% Calculate the gain and the phase change of the system for the given
% frequency
gain=abs(H)
gain = 0.7071
arg=angle(H) % [Rad]
arg = 0.7854
Then calculate the output signal e2(t)
e2(t)=gain*amp*sin(omega*t+arg)
e2(t) = 
% Plot the output signal e2(t)
t_interval = [0 1e-1];
figure(101);
fplot(e2(t),t_interval);
grid on;
title('outputsignal e_2(t) with sinusoidal input');
xlabel('t [s]');
ylabel('e_2(t) [V]');
ylim([-10 10]);
xlim(t_interval);
If you had chosen R = 10 kOhm and C = 0.1 uF the magnitude or gain of H equals and the phase equals =.
And the output signal would then be equal to:

Example 2 with numbers (sinusoidal input from t=0)

In the example before the sine at the input is always there. In this example we choose an inputsignal in which the sine is switched on at t=0
Because we have a transient at t=0 we can no longer work in the ω-domain only. We need to go to the s-domain:
% create symbolic time domain functions
syms t e1(t) e2(t);
% create symbolic s domain functions
syms s H(s) E1(s) E2(s);
 
amp=5*sqrt(2);
omega=1000;
e1(t)=amp*sin(omega*t)*heaviside(t)
e1(t) = 
% Convert e1(t) to s-domain
E1(s)=laplace(e1(t),s)
E1(s) = 
Then select the values of R and C to be equal to 10kOhm and C=0.1uF. Or else to see what happens then.
R=10; % kOhm
C=0.1; % uF
 
% define transferfunction of RC network
H(s)=(s*R*1000*C*1e-6)/(1+s*R*1000*C*1e-6)
H(s) = 
 
Then calculate the output signal E2(s)
E2=E1*H
E2(s) = 
Then calculate the output signal e2(t)
e2(t)=ilaplace(E2(s))
e2(t) = 
% Plot the output signal e_2(t)
t_interval = [0 5e-2];
figure(102);
fplot(e2(t),t_interval);
grid on;
title('outputsignal e_2(t) with sinusoidal input from t=0');
xlabel('t [s]');
ylabel('e_2(t) [V]');
ylim([-10 10]);
xlim(t_interval);
It can be seen that the last equation for is equal to the one found in the ω-domain as soon as the transient of is over. In the ω-domain you could imagine that the transient already took place at .