Systèmes et représentations/Fiche/Table des transformées de Laplace

Fiche mémoire sur les transformées de Laplace

${\begin{array}{c|c}{\textrm {Domaine~de~Laplace}}&{\textrm {Domaine~temporel}}\\\hline F(s)&f(t)\\\\\hline F(s+a)&e^{-at}f(t)\\\\\hline 1&\delta (t)\\\\\hline \displaystyle {\frac {1}{s}}&\Gamma (t)\\\\\hline \displaystyle {\frac {1}{s^{2}}}&t\\\\\hline \displaystyle {\frac {1}{1+\tau s}}&\displaystyle {{\frac {1}{\tau }}e^{-{\frac {t}{\tau }}}}\\\\\hline \displaystyle {\frac {1}{s(1+\tau s)}}&\displaystyle {1-e^{-{\frac {t}{\tau }}}}\\\\\hline \displaystyle {\frac {1}{s^{2}(1+\tau s)}}&\displaystyle {t-\tau +\tau e^{-{\frac {t}{\tau }}}}\\\\\hline \end{array}}$

${\begin{array}{c|c}{\textrm {Domaine~de~Laplace}}&{\textrm {Domaine~temporel}}\\\hline \displaystyle {\frac {1}{(1+\tau _{1}s)(1+\tau _{2}s)}}&\displaystyle {{\frac {1}{\tau _{1}-\tau _{2}}}\left(e^{-{\frac {t}{\tau _{1}}}}-e^{-{\frac {t}{\tau _{2}}}}\right)}\\\\\hline \displaystyle {\frac {1}{s^{2}+2\zeta \omega _{n}s+\omega _{n}^{2}}}&\displaystyle {{\frac {1}{\omega _{p}}}\sin(\omega _{p}t)e^{-\zeta \omega _{n}t}}\\\\\hline \displaystyle {\frac {1}{s(s^{2}+2\zeta \omega _{n}s+\omega _{n}^{2})}}&\displaystyle {{\frac {1}{\omega _{n}^{2}}}-{\frac {1}{\omega _{p}\omega _{n}}}\sin(\omega _{p}t+\varphi )e^{-\zeta \omega _{n}t}}\\\\\hline \displaystyle {\frac {s}{s^{2}+\omega ^{2}}}&\cos(\omega t)\\\\\hline \displaystyle {\frac {\omega }{s^{2}+\omega ^{2}}}&\sin(\omega t)\\\\\hline \end{array}}$ $\omega _{p}=\omega _{n}{\sqrt {1-\zeta ^{2}}}$ $\varphi =\arccos(\zeta )$