Euler's Formula and FFT

Hi guys, from the .csv file, I performed an FFT analysis in Simulink and obtained the following equations (pdf file). Take this equation (screen 1) for example, can I expand it to this form (screen 2)? I know, I can use e^ix = cos(x) + isin(x) but i don't know how to do that. Thanks in advance!

Best regards,

Seweryn

Answer 1

[Sulaymon Eshkabilov]

If understood correctly, you are trying to find a non-linear fit of the large flacturations using sine and cosine functions. If so, here is how it can be attained:

% Data import
D = readmatrix('tek0005.csv');
D(1:15, :) = [];  % Data cleaning
f1  = 2;   % found using fft
Model = @(a, t)(a(1)*sin(2*pi*f1*t)+a(2)*cos(2*pi*f1*t));
t  = D(:,1);
y1 = D(:,2); % Channel 1
a0 = [1 1];  % Initially estimated guess values for coeff a
Coeffs1 = nlinfit(t, y1, Model, a0);
a = Coeffs1; % Found fit model coeffs
y_fit = (a(1)*sin(2*pi*f1*t)+a(2)*cos(2*pi*f1*t));
figure(1)
plot(t, y1, 'r')
hold on
plot(t, y_fit, 'k--', 'LineWidth',2)
legend('Data: CH1', 'Fit Model')
ylabel('Channel 1')
xlabel('Time')
hold off

% Similarly, you can do for the other channel data
f1  = 2;   % found from fft
t  = D(:,1);
y2 = D(:,3);
% NB: mean of y2 is NOT "0". Thus, it should be removed from y2 or added to the fit model! 
a0 = [1 1]; % Initially estimated guess values for coeff a
Coeffs = nlinfit(t, y2, Model, a0)

Coeffs = 1×2

-0.0335 -0.0016

a = Coeffs;
y_fit = (a(1)*sin(2*pi*f1*t)+a(2)*cos(2*pi*f1*t))+mean(y2);
figure
plot(t, y2, 'b')
hold on
plot(t, y_fit, 'k--', 'LineWidth',2)
legend('Data: CH2', 'Fit Model')
ylabel('Channel 2')
xlabel('Time')
hold off

[Seweryn]

Hi @Sulaymon Eshkabilov, I have the following code, can I get the Fourier equations like in the "screen2" for maybe 50 harmonics with correct coefficients and get correct fitting in plots? In this code, I get Fourier equations for only one harmonic component and only sin without cos part. Thanks in advance!

data = readmatrix('tek0005.csv', 'Delimiter', ';', 'HeaderLines', 21);

time = data(:, 1);
voltage = data(:, 2);
current = data(:, 3);

fs = 1 / (time(2) - time(1));

N = length(time);
frequencies = (0:N-1) * fs / N;
voltage_fft = fft(voltage);
current_fft = fft(current);

a1_voltage = abs(voltage_fft(2)) / N;
a1_current = abs(current_fft(2)) / N;

total_harmonic_distortion_voltage = sqrt(sum(abs(voltage_fft(2:end)).^2)) / abs(voltage_fft(2));
total_harmonic_distortion_current = sqrt(sum(abs(current_fft(2:end)).^2)) / abs(current_fft(2));

f1 = frequencies(2);
equation_voltage = @(t) a1_voltage * sin(2 * pi * f1 * t) + mean(voltage);
equation_current = @(t) a1_current * sin(2 * pi * f1 * t) + mean(current);

fprintf('Fourier equation for voltage: %.4f * sin(2 * pi * %.4f * t) + %.4f\n', a1_voltage, f1, mean(voltage));
fprintf('Fourier equation for current: %.4f * sin(2 * pi * %.4f * t) + %.4f\n', a1_current, f1, mean(current));

fprintf('For voltage: a1 = %.4f, THD = %.4f\n', a1_voltage, total_harmonic_distortion_voltage);
fprintf('For current: a1 = %.4f, THD = %.4f\n', a1_current, total_harmonic_distortion_current);

t = linspace(min(time), max(time), 1000); % Create 1000 time points
voltage_fit = equation_voltage(t);
current_fit = equation_current(t);

figure;
subplot(2, 1, 1);
plot(time, voltage, 'b', t, voltage_fit, 'r');
title('Voltage over time');
xlabel('Time [s]');
ylabel('Voltage [V]');
legend('Data', 'Fitting');
subplot(2, 1, 2);
plot(time, current, 'b', t, current_fit, 'r');
title('Current over time');
xlabel('Time [s]');
ylabel('Current [A]');
legend('Data', 'Fitting');

phase_voltage = angle(voltage_fft);
phase_current = angle(current_fft);

phase1_voltage = phase_voltage(2);
phase1_current = phase_current(2);

fprintf('Phase angle for voltage: %.4f radians\n', phase1_voltage);
fprintf('Phase angle for current: %.4f radians\n', phase1_current);

[Seweryn]

I need the form of equations like you used above: (a(1)*sin(2*pi*f1*t)+a(2)*cos(2*pi*f1*t))+mean(y2);

*I make a new topic here: https://www.mathworks.com/matlabcentral/answers/2042907-fft-thd-and-fourier-equations *

[Sulaymon Eshkabilov]

I posted my answer code for a shorter fit model with sine fcn only in your new post.