# Finding a Fourier series of a Burst Fired sine wave having consecutive 2 cycles on & 2 cycles off?

I have been trying to find the solution of the following problem but with no luck.

I have a system that generates a waveform of sinusoidal type with the only difference being that this wave has 2 cycles on, 2 cycles off, 2 cycles on and so on...

Here is a picture of wave,

The x-axis is time. Each cycle is 20 ms or 50 Hz. After two ON cycles, there are two OFF cycles.

I want to find out the harmonic content of this waveform through Fourier analysis. But I don't have any idea where to start. I am confused with the time period. For example, I have to find a0, for that we have this,

Now I am confused with what should be T and what shout be I (t). I computed it with T = 8*Pi. and i(t) = sin(wt). But I couldn't get answers. Because I don't know what limits should be applied to the integrals.

Please help me on this one or direct me to the source where I can find the answer. My goal is to find the Fourier series of this function and I already have an idea that this wave contains Sub-harmonics and Inter-Harmonics.

Edit (In case you're wondering why do I want this):

This actually is a test waveform in one of the IEC standards to test harmonics immunity of electronics devices. According to the standard, this waveform is supposed to contain Sub-Harmonics and Inter-Harmonics.

This is the spectrum of this wave according to the standard (IEC 62053-21).

I want to find out how this spectrum is achieved through Fourier series. This test also mentions some other waveform like half-wave rectified, 90-degree phase fired sine wave, etc. I did the Fourier analysis for these and I could clearly achieve the results (Harmonics Contents spectrum) same as Standards.

Though I can't seem to reach the results for this specified case (2 cycles ON, 2 Cycles OFF).

Any help is highly appreciated.

Thank you.

• Remember that multiplication in one domain is convolution in the other. Now model your waveform as a sine wave at frequency F multiplied by a square wave going from 0 to 1 at a frequency of F/4. Does that help enough? – Cristobol Polychronopolis Aug 26 '19 at 12:34
• Fourier series works. It's easier to use angle rather than time as the independent variable. – Chu Aug 26 '19 at 13:29

Since your waveform repeats itself after 4 cycles of the sine wave, your overall time period is $$T = 8*\pi/\omega$$
Coming to solving the integrals, Since there are two parts when you look at a single period of the repeating wave, you break the integral into two parts, $$\int_{t_0}^{t_0+T}i(t)dt = \int_{t_o}^{t_0+T/2}i(t)dt + \int_{t_o + T/2}^{t_0+T}i(t)dt$$
In the first half cycle, You have a sine wave, $$\int_{t_o}^{t_0+T/2}i(t)dt = \int_{t_o}^{t_0+T/2}sin(\omega*t)dt$$ In the second half cycle, There is no wave, $$\int_{t_o + T/2}^{t_0+T}i(t)dt = 0$$
• It remains the same since the integral goes from 0 to T/2, where T is 4 full cycles of the wave. You could do something such as $$\int_{t_0}^{t_0+T/2}i(t)dt = 2\int_{t_0}^{t_0+T/4}i(t)dt$$ if you want to evaluate the integral over a single cycle. Either way the answer must remain the same. – Prateek Dhanuka Aug 27 '19 at 5:57