Why Does a Square Wave Contain Only Odd Harmonics?

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SUMMARY

A square wave contains only odd harmonics due to its odd symmetry, which results in the cancellation of even harmonics during Fourier expansion. The Fourier series representation of a square wave is given by the formula 4/π(sin(x) + sin(3x)/3 + sin(5x)/5 + ...), confirming that only odd harmonics contribute to its shape. Even harmonics are effectively destructive in this context, leading to their absence in the square wave's harmonic content. The duty cycle of the square wave does influence the presence of harmonics, but primarily, the odd nature of the waveform dictates the harmonic structure.

PREREQUISITES
  • Understanding of Fourier series and harmonic analysis
  • Knowledge of odd and even functions in mathematics
  • Familiarity with waveforms and their properties
  • Basic principles of signal processing
NEXT STEPS
  • Study Fourier series and their applications in signal processing
  • Explore the concept of harmonic distortion in waveforms
  • Investigate the effects of duty cycle on waveform characteristics
  • Learn about the mathematical properties of odd and even functions
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Students and professionals in electrical engineering, signal processing, and applied mathematics who are interested in waveform analysis and harmonic content.

likephysics
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Why does a square wave contain only odd harmonics of the fundamental?
If the even harmonics are low, how low are they.
What happens to the even harmonics.
 
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Because the resulting function is odd. Even fundamental harmonics would be destructive.
 
likephysics said:
Why does a square wave contain only odd harmonics of the fundamental?
If the even harmonics are low, how low are they.
What happens to the even harmonics.

That's the Fourier expansion for a square wave:
\frac{4}{\pi}(sin(x) + sin(3x)/3 + sin (5x)/5 + ...)

The function f(x) = |sin(x)| has only even harmonics.
 
Does the duty cycle of the square wave have anything to do with the odd harmonics?
 
I do not have a good working knowledge of physics yet. I tried to piece this together but after researching this, I couldn’t figure out the correct laws of physics to combine to develop a formula to answer this question. Ex. 1 - A moving object impacts a static object at a constant velocity. Ex. 2 - A moving object impacts a static object at the same velocity but is accelerating at the moment of impact. Assuming the mass of the objects is the same and the velocity at the moment of impact...

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