3.7. Transformadas de Laplace#
\(\mathbf{f (t), t > 0}\) |
\(\mathbf{\bar{f}(s)}\) |
|---|---|
Impulso unidad, \(\delta(t_0)\) |
1 |
Pulso unidad, \(\delta_A(t)\) |
\(\frac{1}{A} \frac{1 - e^{- sA}}{s}\) |
Escalón unidad |
\(\frac{1}{s}\) |
Rampa, \(f(t) = t\) |
\(\frac{1}{s^2}\) |
\(t^2\) |
\(\frac{2!}{s^3}\) |
\(t^n\) |
\(\frac{n!}{s^{n + 1}}\) |
\(e^{- a t}\) |
\(\frac{1}{s + a}\) |
\(t^n e^{- a t}\) |
\(\frac{n!}{(s + a)^{n + 1}}\) |
\(\sin (\omega t)\) |
\(\frac{\omega}{s^2 + \omega^2}\) |
\(\cos (\omega t)\) |
\(\frac{s}{s^2 + \omega^2}\) |
\(\sinh (\omega t)\) |
\(\frac{\omega}{s^2 - \omega^2}\) |
\(\cosh (\omega t)\) |
\(\frac{s}{s^2 - \omega^2}\) |
\(e^{- a t} \sin (\omega t)\) |
\(\frac{\omega}{(s + a)^2 + \omega^2}\) |
\(e^{- a t} \cos (\omega t)\) |
\(\frac{s + a}{(s + a)^2 + \omega^2}\) |
\(\mathbf{\bar{f} (s)}\) |
\(\mathbf{f (t)}\) |
|---|---|
\(\frac{1}{(s + a) (s + b)}\) |
\(\frac{e^{- a t} - e^{- b t}}{b - a}\) |
\(\frac{1}{(s + a) (s + b) (s + c)}\) |
\(\frac{e^{- a t}}{(b - a) (c - a)} + \frac{e^{- b t}}{(c - b) (a - b)} + \frac{e^{- a t}}{(a - c) (b - c)}\) |
\(\frac{s + a}{(s + b) (s + c)}\) |
\(\frac{1}{c - b} [(a - b) e^{- b t} -(a - c) e^{- c t}]\) |
\(\frac{a}{(s + b)^2}\) |
\(a t e^{- b t}\) |
\(\frac{a}{(s + b)^3}\) |
\(\frac{a}{2} t^2 e^{- b t}\) |
\(\frac{a}{(s + b)^{n + 1}}\) |
\(\frac{a}{n!} t^n b^{- b t}\) |
\(\frac{1}{s (a s + 1)}\) |
\(1 - e^{- t / a}\) |
\(\frac{1}{s (a s + 1)^2}\) |
\(1 - \frac{a + t}{a} e^{- t / a}\) |
\(\frac{\omega^2}{s (s^2 + 2 \zeta \omega s + \omega^2)}\) |
\(1 + \frac{e^{-\zeta \omega t}}{ \sqrt[]{1 - \zeta^2}} \sin \left( \omega \sqrt[]{1- \zeta^2} t - \varphi \right)\) donde \(\cos \varphi = - \zeta\) |
\(\frac{s}{(1 + a s) (s^2 + \omega^2)}\) |
\(- \frac{1}{1 + a^2 \omega^2}e^{- t / a} + \frac{1}{\sqrt[]{1 + a^2 \omega^2}} \cos (\omega t +\varphi)\) donde \(\varphi = \arctan (a \omega)\) |
\(\frac{s}{(s^2 + \omega^2)^2}\) |
\(\frac{1}{2 \omega} \sin (\omega t)\) |
\(\frac{1}{(s + a) [(s + b)^2 + \omega^2]}\) |
\(\frac{e^{- a t}}{(a - b)^2+ \omega^2} + \frac{e^{- b t} \sin (\omega t + \varphi)}{\omega\sqrt[]{(a - b)^2 + \omega^2}}\) donde \(\varphi = \arctan \left(\frac{\omega}{a - b} \right)\) |