# Laplace Transform Circuit Element Model **A model of circuits that allows for the use of the [[Laplace transform]] for circuit analysis.** To use the [[Laplace transform]] for circuit analysis, circuit elements must be converted to a model in the *complex frequency domain* before applying steady-state analysis methods. The [[Laplace Transform#Inverse Laplace transform|inverse transform]] of the solution can then be taken to retrieve the solution in the *time domain*. The conversion is done by taking the Laplace transform of equations established in the time domain, such as [[Ohm's law]]. ## Resistor > [!Resistor model] > The properties of a resistor do not depend on frequency. Thus, > $V(s)=RI(s)$ ## Inductor > [!Inductor model] > For an [[inductor]], > $\begin{align*} > V(s)&=sLI(s)-Li(0^+) \\ > I(s)&=\frac{1}{sL}V(s)+\frac{i(0^+)}{s} > \end{align*}$ > > ![[InductorLaplace.svg]] ## Capacitor > [!Capacitor model] > For a [[capacitor]], > $\begin{align*} > I(s)&=sCV(s)-Cv(0^{+})\\ > V(s)&=\frac{1}{sC}I(s)+\frac{v(0^+)}{s} > \end{align*}$ > > ![[CapacitorLaplace.svg]]