Fourier Series Expansion of PWM Signal with Time-Varying Duty Cycle for LCL Filters

Hi, 
I'm working on a project for my university which involves LCL filters. The input proposed to eliminate harmonics is a PWM Signal, using the function generator to simulate the carrier (square wave)  and the modulating (sine wave) and one of the steps is to expand this signal in Fourier Series. So, I'm confused in how I do expand this signal in Fourier, since the duty cycle is variant (a function in time). Would I have a double Fourier Series changing in n (to compose my square wave) and other variating on d?
http://www.fourwalledcubicle.com/files/freaks/timerstutorial/Waves.png

Thank you.Rafael

Is your carrier frequency (the PWM repetition rate) exactly a multiple of the frequency of your modulating signal? If so, you can do a Fourier Series in terms of the modulating sinewave and its harmonics, over a time interval equal to one cycle of the modulating sinewave. This works because the PWM pulses occur at exactly the same phases of every cycle of the modulating sinewave. Calculate Fourier components to well beyond the carrier frequency. You will see the Fourier component at the modulation frequency. This is the desired filtered output. You will see Fourier components at the carrier frequency and its harmonics. You will see Fourier components at the carrier frequency plus and minus the modulating frequency (these are called sidebands).

If your carrier frequency is between two exact multiples of your modulating frequency, I suggest doing the Fourier analysis twice, once for each of the nearest exact multiples. Alternatively, you could use a Windowed Fourier Transform.

If you have not looked up how to build analog radio receivers, I suggest you do so, as the filter issues are similar to those in your problem.

Hi  Peter,
Thank you for your response.My carrier is a multiple of the frequency of my modulating signal, indeed. So, the carrier is a square wave of 60kHz and my modulating signal is 6kHz. I guess I lost the track - Are you telling me that if I calculate an and bn using my carrier and put the Fourier Serie in terms of my modulating it'll result on my modulated PWM signal? Sorry, I think I  didn't followed , I'm still studying all this. Anyway, I appreciate your help and support.Rafael.

I've been doing Fourier series for so long that I don't know today's notation very well.  For consistency, I am referring to https://en.wikipedia.org/wiki/Fourier_series for terminology and background. As an aside, one of the illustrations in this article graphically shows that the Greek astronomers' epicycles were terms in a complex Fourier series.
Set the Fourier Series period P to 1/6000 of a second, starting and ending on rising zero-crossings of the 6 kHz modulation sinewave. If you calculate the Fourier coeficients of the modulation sinewave alone, you should get one nonzero Fourier component, b1 the sinewave component at 6 kHz, and all the other Fourier components zero.Keep the same period and the same starting and ending points, and calculate the Fourier Series for the PWM signal. The modulating signal you want will show up in b1. The DC component a0 will be half of VCC. The 60-kHz PWM carrier fundamental-frequency component will show up in a10 and b10. The carrier third harmonic will show up in a30 and b30, and so on. The carrier m-th harmonic will show up in a10m and b10m. The even harmonics will be zero for the symmetrical PWM in your referenced Waves.png image.
If you then insert the Fourier components into the definition of the Fourier series sN(x)  and calculate, you should get back your PWM waveform with some wrinkles (the Gibbs phenomenon).If you turn down the modulation to zero amplitude, the PWM becomes a squarewave, whose Fourier series and Fourier coefficients are discussed in the Wikipedia article and in textbooks. This is a useful way to check your understanding and your calculating process.

Hi Peter,
I see, now it's making more sense for me. I've plotted the Fourier serie I calculated and it has worked, I've obtaneid the expected result.Thank you very much for your help.

Sincerely,Rafael.