Laplace Transform

$F(s) = L(f) = \int_0^{\infty}e^{-st}f(t)dt$

f(t) L(f)
1 $\frac{1}{s}$
t $\frac{1}{s^2}$
$t^2$ $\frac{2!}{s^3}$
$t^n$ $\frac{n!}{s^{n+1}}$
$t^a$ $\frac{\Gamma (a+1)}{s^{a+1}}$
$e^{at}$ $\frac{1}{s-a}$
$coswt$ $\frac{s}{s^2+w^2}$
$sinwt$ $\frac{w}{s^2+w^2}$
$cosh at$ $\frac{s}{s^2-a^2}$
$sinh at$ $\frac{a}{s^2-a^2}$
$e^{at}coswt$ $\frac{s-a}{{(s-a)}^2+w^2}$
$e^{at}sinwt$ $\frac{w}{{(s-a)}^2+w^2}$
  • s-Domain Circuit Element Model
Resistor.jpg

$v(t)=Ri(t) \Rightarrow V(s)=R I(s)$
$Z_R(s)=\frac{V_R(s)}{I_R(s)}=R$

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