How to find amplitude of given frequency ?

Hi all, I have a question like in the picture

I have tried to solve it with code below, but couldn't get correct answer.

Please need help

clc; clear;
syms x(t) A T0 n
x(t) = 2;
a_0 = (1 / T0) * int(t + 1, t, -1, 1);
a_n = (4 * sin(n * (2 * pi * T0))) / (3 * n * (2 * pi * T0));
b_n = -(4 * cos(n * (2 * pi * T0))) / (3 * n * (2 * pi * T0)) + (4 * sin(n * (2 * pi * T0))) / (3 * n * (2 * pi * T0))^2;

T0 = 0.6;
A = 85.6;
frequency = 5;
n = 1;
eval(sqrt(a_n^2 + b_n^2) * A)

Answer 1

[Dev]

Hi @zeybek,

In the code provided in the question, the Fourier coefficients (‘a_n’ and ‘b_n’) are calculated for a waveform, and then the amplitude for the nth harmonic is computed. However, I have listed some issues with the code below which we can fix to compute the right amplitude-

• Incorrect waveform definition: x(t) = 2 defines a constant function, but we need to define our actual waveform here.
• Incorrect Fourier coefficient formulas: ‘a_n’ and ‘b_n’ formulas appear to be symbolic and not directly related to standard Fourier coefficient formulas.
• Frequency mismatch: The frequency is set to 5 and ‘n’ as 1, but to match the 5/3 Hz component, we should ensure ‘n’ is chosen so that n/T_0 = 5/3.

Also, we need to find the correct harmonic number. Since the wave repeats every 2 seconds, T_0 = 2. Now, with this time period, we can calculate the harmonic, ‘n’, using the formula-

n = given_freq (5/3 Hz) / T_0

Using the above values, we can then calculate the required amplitude.

I hope the above explanation helps to solve the question.