Concept explainers
A “half-sinusoidal” waveform is shown in Fig. 17.31, which is the output of a half-wave rectifier used to help convert a sinusoidal input to dc. Find the Fourier series representation and plot the signal and Fourier series representation for n = 10 terms.
■ FIGURE 17.31
Find the Fourier series coefficients
Answer to Problem 11E
The values of the Fourier series coefficients
Explanation of Solution
Given data:
Refer to Figure 17.31 in the textbook.
Formula used:
Write the general expression for Fourier series expansion.
Write the general expression for Fourier series coefficient
Write the general expression for Fourier series coefficient
Write the general expression for Fourier series coefficient
Write the expression to calculate the fundamental angular frequency.
Here,
Calculation:
In the given Figure 17.31, the time period is
The function
Substitute 0.4 for T in equation (5) to find
Applying equation (6) in equation (2) to find
Simplify the above equation as follows,
For half wave symmetry and even symmetry,
For all values of ‘n’,
Applying equation (6) in equation (3) to find the value of coefficient
Consider the function,
Consider,
On differentiating the above expression,
Equation (8) will be follows,
In the above equation, consider,
By applying linearity,
In equation (10),
consider,
Let,
Equation (11) will be as follows,
Similarly, in equation (10),
consider,
Let,
Equation (12) will be as follows,
Substitute the values of m and l in equation (10) as follows,
Substitute the value of y in equation (9) as follows,
Therefore,
On applying the limits,
Simplify the equation as follows,
Simplify the equation as follows,
Substitute the value of
Substitute the value of
Representing through
By considering
Matlab code for the signal
t=linspace(-0.6,0.6,1000); % vector for time over 1000 points.
Vm=1;
v0=Vm/(pi); % constant.
for i=1:1000;
sum=0;
sum=sum+ 0.5*cos(5*pi*t(i)) + (2/(3*pi))*cos(10*pi*t(i)) - (2/(15*pi))*cos(20*pi*t(i)) + (2/(35*pi))*cos(30*pi*t(i)) - (2/(63*pi))*cos(40*pi*t(i)) + (2/(99*pi))*cos(50*pi*t(i));
vt(i)=v0+sum;
end
plot(t,vt)
xlabel('Time t in seconds')
ylabel('Voltage value')
title('Fourier representation of voltage signal v(t)')
Matlab output:
The Fourier representation of voltage signal
Conclusion:
Thus, the values of the Fourier series coefficients
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