Answer1

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	WHAT'S YOUR ENGINEERING QUOTIENT?
Problem 1—Anyone who has worked with spectral analysis knows that a squarewave contains all of the odd harmonics, where the amplitude is one over the frequency ratio. In other words, if the fundamental frequency has an amplitude of one, the third harmonic has an amplitude of one third, and so on. What is special about the following waveform, which can be constructed by taking two squarewaves and adding them together after shifting one by one-sixth of a period relative to the other? Draw a phasor diagram that explains the special characteristics of the waveform. Answer: The spectrum of the waveform contains no third harmonic at all, nor any multiples of the third harmonic (9th, 15th, etc.). The waveform contains only the fundamental, and the 5th, 7th, 11th, etc. harmonics. If we set the 0° point of the waveform at the center of the rising zero crossing, as shown below, the symmetry of the phasor diagram becomes more obvious. Relative to the 0° point, the A waveform's fundamental crosses zero 30° earlier (–30°) and the B waveform's fundamental does so 30° later (+30°). The sum of these two components aligns with the 0° axis, and has a magnitude equal to 1.732 × the amplitude of A or B alone. The third harmonics have phase shifts that are 3× that of the fundamentals, putting them at –90° and +90° on the phasor diagram. Clearly, they directly cancel each other, leaving none of that component to appear in the output. The fifth harmonics have phase shifts of 5× the fundamentals, so they add in the same proportion as the fundamental, resulting in no net change in amplitude relative to the original squarewave alone. Contributor: Dave Tweed 12-01 — NEXT Q&A For questions or comments about Test Your EQ, e-mail eq@circuitcellar.com.