Laplace transforms are an adapted integral form of a differential equation (created and introduced by the French mathematician Pierre-Simon Laplace (1749-1827)) used to describe electrical circuits and physical processes. Adapted from previous notions given by other notable mathematicians and engineers like Joseph-Louis Lagrange (1736-1812) and Leonhard Euler (1707-1783), Laplace transforms are used to be a more efficient and easy-to-recognize form of a mathematical equation.
Standard Form
This is the standard form of a Laplace transform that a function will undergo.
Sample Functions
The following is a list of commonly seen functions of which the Laplace transform is taken. The start function is noted within the Laplace symbol .
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Transfer Function
The Laplace transform of the impulse response of a circuit with no initial conditions is called the transfer function. If a single-input, single-output circuit has no internal stored energy and all the independent internal sources are zero, the transfer function is
Impedances and admittances are special cases of transfer functions.
Example
Solve the differential equation:
We start by taking the Laplace transform of each term.
The next step is to perform the respective Laplace transforms, using the information given above.
Using association, the equation is rearranged:
Continuing on using the method of partial fractions, the equation is progressed:
Plugging the above values back into the equation further up, we get:
Applying anti-Laplace transforms, we get the equation:
Applying the Laplace transforms in reverse (as the above equation utilizes inverse Laplace transforms) for the above equation, we get the solution:
References
DeCarlo, Raymond A.; Lin, Pen-Min (2001), Linear Circuit Analysis, Oxford University Press, ISBN 0-19-513666-7 .
External links
Authors
Colby Fullerton
Brian Roath
Reviewed By
David Robbins
Thomas Wooley
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