Laplace transform of a sinusoidal pulse (semiperiod)

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 11:56

*Enlace a Spanish version

Summary:

This video presents two techniques for calculating the Laplace transform of a sinusoidal pulse (a half-period of the sine function).

Apart from calculating it with the laplace command of Matlab’s Symbolic Toolbox, it proposes calculating it (a) as the superposition of a sinusoid and the same sinusoid delayed one semi-period using the lag operator $e^{-ds}$, or (b) directly by making the integral that defines the Laplace transform, integrating by parts twice.

Obviously, all results are identical.

NOTE: although these types of questions are typical in some exams, it turns out that by applying ”superposition”, the Laplace transform of the input calculated here is often NOT necessary to calculate the time response, since, for example, we may calculate the response to a sinusoid (complete, from $t=0$ to infinity, transformed $\omega ∕(s^2+{\omega }^2)$ ) and apply the same displacement+superposition on the output. In fact, this is done in quite a few of the examples with “piecewise” inputs in other videos.

For instance, time response to this sinusoidal pulse of an electric circuit is computed in videos [sinpulRCREN] and [sinpulRCR2EN].

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.