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, enter image description here

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,

enter image description here

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). enter image description here

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.

  • 4
    \$\begingroup\$ 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? \$\endgroup\$ Commented Aug 26, 2019 at 12:34
  • \$\begingroup\$ Fourier series works. It's easier to use angle rather than time as the independent variable. \$\endgroup\$
    – Chu
    Commented Aug 26, 2019 at 13:29

1 Answer 1


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, $$\int_{t_o + T/2}^{t_0+T}i(t)dt = 0$$

I believe you can solve the rest

  • \$\begingroup\$ Thanks a lot. One question: during first half T, there are two cycles of the actual sine wave. Will the integral will be multiplied by 2 or it will just remain the same as you mentioned? \$\endgroup\$ Commented Aug 27, 2019 at 4:59
  • \$\begingroup\$ 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. \$\endgroup\$ Commented Aug 27, 2019 at 5:57
  • \$\begingroup\$ Thank You. I will try this. \$\endgroup\$ Commented Aug 28, 2019 at 4:01

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