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Waveforms : Fourier and Laplace Transforms

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Have the following question regarding extracting information from a waveform

a. Write down an expression, in the time domain, for the signal in the diagram above.

b.Derive the Laplace transform for this signal.

c.Use Laplace transform analysis to derive the Fourier transform in its simplest form.

All details in the attached sheet.

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https://brainmass.com/math/fourier-analysis/waveforms-fourier-laplace-transforms-165614

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Solution Summary

Waveforms and Fourier and Laplace Transforms are investigated. Laplace transforms for this signal is dervived. The simplest form for Fourier transforms are determined.

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Fourier analysis using Excel

The supply current was sampled 1024 times over a very short time interval. The data so obtained is given in column B of the accompanying Excel worksheet (see attached).
This worksheet has been set up to give a graph showing the spectral components of the data.

Question 3
i) Obtain the Fourier Transform for the data using the Fourier Analysis tool of Excel. The transformed data should commence in cell D2.
ii) Identify the principal frequencies in the current waveform.
iii) Estimate the total harmonic distortion [THD] present in the current waveform using the formula:
(See attached file for equation)
Where I1 is the r.m.s. value of the fundamental current, In the r.m.s value of the nth harmonic and n(max) is the number of the highest measurable or significant harmonic.
[Note the vertical axis of the spectrum graph is scaled in (current)^2.]
iv) Attempt to synthesise the shape of the original waveform from its principal harmonics [e.g. sketch the waveforms of the harmonics on the same time axis and add them together].
Q4. Sketch, on a set of common axes, waveforms to represent the transient response of circuits having transfer functions with the following parameters:
(see attached file for more information)

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