

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 (V_{1}), the voltage at the tenth harmonic (V_{10}), 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 (V_{1} and V_{10}), 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 (V_{1} and V_{10}).
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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, halfwave, complex waves.
