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Homework 1 - Harmonics

Problem 1.
Using the provided graphic in Figure 1, show in detail how to determine the amplitude of the terms of the series which represents the third harmonic.
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Problem 2.
For Figure 1, is an Ao term required in the Fourier series describing the function? Explain your answer.
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Problem 3.
Using the graphic in Figure 2, calculate the duty cycle (DS), the voltage at the first harmonic (V1), the voltage at the tenth harmonic (V10), and sketch the frequency spectrum. (Please note: duty cycle is often denoted as DC, but for some unknown reason, my former instructor used DS; therefore in the context of this homework solution I will be referring to duty cycle as DS to match my calculated solutions).
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Problem 4.
In Figure 2, if the duty cycle is equal to ten percent (DS = 10%), and time is equal to five milliseconds (T = 5ms) determine the pulse width (tau), the voltage at the first and tenth harmonics (V1 and V10), and sketch the frequency spectrum.
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Problem 5.
Repeat Problem 4, but recalculate your solutions for a duty cycle equal to three percent (DS = 3%).
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Problem 6.
Using the rectangular waveform in Figure 2, set the duty cycle equal to ten percent (DS = 10%), and the pulse width equal to one millisecond (τ = 1ms), determine the period of the rectangular waveform, and the voltage at the first and tenth harmonics (V1 and V10).
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Homework Solutions and Calculations

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Engineering key words: harmonics, duty cycle, pulse width, rectangular wave, waveform, pulse period, voltage minimum and maximum, Fourier Series, time domain, frequency spectrum, time varying, peak amplitude, fundamental frequency, radians, nth harmonic, odd, even, square wave, nonsinusoidal periodic waveforms, wave symmetry, half-wave, complex waves.

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