The Laplace transform is an integral transformation of a function \(f(t)\) from the time domain into the complex frequency domain, giving \(F(s)\).
\(\boxed{\mathcal{L}\left[f(t)\right] = F(s) = \int_{0^-}^{\infty}f(t)\,e^{\displaystyle -st}\,dt}\)
where \(s\) (in frequency \(\textrm{s}^{-1}\)) is a complex variable given by \(\boxed{s = \sigma + j\omega}\).
A function \(f(t)\) may not have a Laplace transform. In order for \(f(t)\) to have a Laplace transform, the Laplace integral must converge to a finite value.
\(\displaystyle \int_{0^-}^{\infty}\left|f(t)\,e^{\displaystyle -st}\right|\,dt < \infty\)
But \(\displaystyle \left|e^{\,j\omega t}\right| = 1\) for any value of \(t\),
\(\boxed{\int_{0^-}^{\infty}e^{\displaystyle -\sigma t}\left|f(t)\right|\,dt < \infty}\)
When the Laplace transform is applied to circuit analysis, the differential equations represent the circuit in the time domain. The terms in the differential equations take the place of \(f(t)\). Their Laplace transform, which corresponds to \(F(s)\), constitutes algebraic equations representing the circuit in the frequency domain.
Properties of Laplace Transform
| Property | \(\displaystyle f(t) = \mathcal{L}^{-1}\left[F(s)\right]\) | \(\displaystyle F(s) = \mathcal{L}\left[f(t)\right]\) |
|---|---|---|
| Linearity | \(\displaystyle a_1 f_1(t) + a_2 f_2(t)\) | \(\displaystyle a_1 F_1(s) + a_2 F_2(s)\) |
| Scaling | \(\displaystyle f(at)\) | \(\displaystyle \frac{1}{a}F\left(\frac{s}{a}\right)\) |
| Time shift | \(\displaystyle f(t - t_0)\,u(t - t_0)\) | \(\displaystyle e^{\displaystyle -t_{0}s}F(s)\) |
| Frequency shift | \(\displaystyle e^{\displaystyle -at}f(t)\) | \(\displaystyle F(s + a)\) |
| Time differentiation | \(\displaystyle f^{(n)}(t)\) | \(\displaystyle s^n F(s) - \sum_{k = 1}^{n}s^{n - k}f^{(k - 1)}(0^-)\) |
| Time integration | \(\displaystyle \int_{0^-}^t f(\tau)\,d\tau\) | \(\displaystyle \frac{1}{s}F(s)\) |
| Frequency differentiation | \(\displaystyle t^n f(t)\) | \(\displaystyle (-1)^{n}F^{(n)}(s)\) |
| Frequency integration | \(\displaystyle \frac{f(t)}{t}\) | \(\displaystyle \int_{s}^{\infty}F(s)\,ds\) |
| Time periodicity | \(\displaystyle f(t) = f(t + nT)\) | \(\displaystyle \frac{1}{1 - e^{-Ts}}\int_{0}^{T}f(t)\,e^{\displaystyle -st}\,dt\) |
| Initial value | \(\displaystyle f(0)\) | \(\displaystyle \lim\limits_{s \to \infty}sF(s)\) |
| Final value | \(\displaystyle f(\infty)\) | \(\displaystyle \lim\limits_{s \to 0}sF(s)\) |
| Convolution | \(\displaystyle f_1(t) \ast f_2(t)\) | \(\displaystyle F_1(s)\,F_2(s)\) |
See: Properties of Laplace Transforms
Laplace Transform Pairs
For \(t \ge 0\); \(f(t) = 0\), for \(t \lt 0\).
| Function | \(\displaystyle f(t) = \mathcal{L}^{-1}\left[F(s)\right]\) | \(\displaystyle F(s) = \mathcal{L}\left[f(t)\right]\) |
|---|---|---|
| unit impulse | \(\displaystyle \delta(t)\) | \(\displaystyle 1\) |
| delayed unit impulse | \(\displaystyle \delta(t - t_0)\) | \(\displaystyle e^{\displaystyle -t_{0}s}\) |
| unit step | \(\displaystyle u(t)\) | \(\displaystyle \frac{1}{s}\) |
| delayed unit step | \(\displaystyle u(t - t_0)\) | \(\displaystyle \frac{1}{s}e^{\displaystyle -t_{0}s}\) |
| unit ramp | \(\displaystyle t\,u(t)\) | \(\displaystyle \frac{1}{s^2}\) |
| delayed unit ramp | \(\displaystyle (t - t_0)\,u(t - t_0)\) | \(\displaystyle \frac{1}{s^2}e^{\displaystyle -t_{0}s}\) |
| \(\displaystyle e^{-at}\) | \(\displaystyle \frac{1}{s + a}\) | |
| \(\displaystyle t^n\) | \(\displaystyle \frac{n!}{s^{n + 1}}\) | |
| \(\displaystyle t\,e^{-at}\) | \(\displaystyle \frac{1}{(s + a)^2}\) | |
| \(\displaystyle t^n e^{-at}\) | \(\displaystyle \frac{n!}{(s + a)^{n + 1}}\) | |
| \(\displaystyle \sin(\omega t)\) | \(\displaystyle \frac{\omega}{s^2 + \omega^2}\) | |
| \(\displaystyle \cos(\omega t)\) | \(\displaystyle \frac{s}{s^2 + \omega^2}\) | |
| \(\displaystyle \sinh(\omega t)\) | \(\displaystyle \frac{\omega}{s^2 - \omega^2}\) | |
| \(\displaystyle \cosh(\omega t)\) | \(\displaystyle \frac{s}{s^2 - \omega^2}\) | |
| \(\displaystyle \sin(\omega t + \theta)\) | \(\displaystyle \frac{s\sin(\theta) + \omega\cos(\theta)}{s^2 + \omega^2}\) | |
| \(\displaystyle \cos(\omega t + \theta)\) | \(\displaystyle \frac{s\cos(\theta) - \omega\sin(\theta)}{s^2 + \omega^2}\) | |
| \(\displaystyle e^{-at}\sin(\omega t)\) | \(\displaystyle \frac{\omega}{(s + a)^2 + \omega^2}\) | |
| \(\displaystyle e^{-at}\cos(\omega t)\) | \(\displaystyle \frac{s + a}{(s + a)^2 + \omega^2}\) |