Laplace Transform

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.

$F(s)={ℒ}\{f(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}f(t)dt$

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 ${ℒ}\left\{\right\}$.

$F(s)={ℒ}\left\{1\right\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}dt=$ $\frac{1}{s}$

$F(s)={ℒ}\left\{t^n\right\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{t}^{n}dt=$ $\frac{n!}{s^{n+1}}$

$F(s)={ℒ}\left\{e^{at}\right\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{e}^{at}dt=$ $\frac{1}{s-a}$

$F(s)={ℒ}\{sin(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}sin(\omega t)dt=$ $\frac{\omega }{s^2+{\omega }^2}$

$F(s)={ℒ}\{cos(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}cos(\omega t)dt=$ $\frac{s}{s^2+{\omega }^2}$

$F(s)={ℒ}\{t^{n}g(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{t}^{n}g(t)dt=$ $$\begin{gathered} \frac{(-1{)}^n{d}^{n}G(s)}{d{s}^n}\text{ for}n \\ \text{= 1,2,...} \end{gathered}$$

$F(s)={ℒ}\{tsin(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}tsin(\omega t)dt=$ $\frac{2\omega s}{(s^2+{\omega }^2{)}^2}$

$F(s)={ℒ}\{tcos(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}tcos(\omega t)dt=$ $\frac{s^2-{\omega }^2}{(s^2+{\omega }^2{)}^2}$

$F(s)={ℒ}\{g(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}g(t)dt=$ $\frac{1}{a}G\left(\frac{s}{a}\right)$

$F(s)={ℒ}\{e^{at}g(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{e}^{at}g(t)dt=G(s-a)$

$F(s)={ℒ}\left\{e^{at}{t}^n\right\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{e}^{at}{t}^{n}dt=$ $$\begin{gathered} \frac{n!}{(s-a{)}^{n+1}}\text{ for}n \\ \text{= 1,2,...} \end{gathered}$$

$F(s)={ℒ}\left\{t{e}^{-t}\right\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}t{e}^{-t}dt=$ $\frac{1}{(s+1{)}^2}$

$F(s)={ℒ}\{1-e^{-t/T}\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}(1-e^{-t/T})dt=$ $\frac{1}{s(1+Ts)}$

$F(s)={ℒ}\{e^{at}sin(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{e}^{at}sin(\omega t)dt=$ $\frac{\omega }{(s-a{)}^2+{\omega }^2}$

$F(s)={ℒ}\{e^{at}cos(\omega t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{e}^{at}cos(\omega t)dt=$ $\frac{s-a}{(s-a{)}^2+{\omega }^2}$

$F(s)={ℒ}\{u(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}u(t)dt=$ $\frac{1}{s}$

$F(s)={ℒ}\{u(t-a)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}u(t-a)dt=$ $\frac{e^{-as}}{s}$

$F(s)={ℒ}\{u(t-a)g(t-a)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}u(t-a)g(t-a)dt=e^{-as}G(s)$

$F(s)={ℒ}\{g^{\prime }(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{g}^{\prime }(t)dt=sG(s)-g(0)$

$F(s)={ℒ}\{g^{″}(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{g}^{″}(t)dt=s^2\cdot G(s)-s\cdot g(0)-g^{\prime }(0)$

$F(s)={ℒ}\{g^{(n)}(t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-st}{g}^{(n)}(t)dt=s^n\cdot G(s)-s^{n-1}\cdot g(0)-s^{n-2}\cdot g^{\prime }(0)-...-g^{(n-1)}(0)$

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

$H(s)=\frac{{ℒ}(responsesignal)}{{ℒ}(inputsignal)}.$

Impedances and admittances are special cases of transfer functions.

References

DeCarlo, Raymond A.; Lin, Pen-Min (2001), Linear Circuit Analysis, Oxford University Press, ISBN 0-19-513666-7 .

External links

• The Laplace Transforms.

Authors

Colby Fullerton

Brian Roath