Laplace Transform

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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 .

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 .

Zill, Dennis G. (2005), A First Course in Differential Equations with Modeling Applications Ninth Edition, Brooks/Cole Cengage Learning, ISBN 0-495-10824-3 .

External links

Authors

Colby Fullerton

Brian Roath

Reviewed By

David Robbins

Thomas Wooley

Read By

Jaymin Joseph

John Hawkins

Comments

John Hawkins:

  • Nice list of transforms! Where did you find it? Inside the back cover of the textbook has a good list, but none including transforms of . I see your reference to the textbook. What page?
    • Oops! It seems I forgot to add a reference to the list, thanks for mentioning it. I got the others from a list in the back of the ODE textbook. I'll list the reference up above.


  • I believe that

should be

    • Thank you for pointing that out. It should be fixed now.
  • Also, the solution does not match the initial condition of .
    • Yeah, you're right. A minor sign error was what messed it up. It should be fixed now.