Introduction:

Laplace transforms are a powerful mathematical tool used in electrical engineering to analyze linear time-invariant (LTI) systems, particularly in circuit analysis. They allow engineers to convert time-domain differential equations into algebraic equations in the complex frequency domain, simplifying the process of solving and understanding the behavior of electrical circuits.

Laplace Transform Definition:

The Laplace transform of a time-domain function $f(t)$ is defined as:

$$ ⁍ $$

where s is the complex frequency variable, and F(s) is the Laplace transform of $f(t)$. The Laplace transform is valid for t ≥ 0, and it is assumed that f(t) = 0 for t < 0.

Inverse Laplace Transform:

The inverse Laplace transform converts a function from the complex frequency domain back to the time domain. It is denoted as:

$$ ⁍ $$

Properties of Laplace Transforms:

1. Linearity: $\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)$
2. Time shifting: $\mathcal{L}\{f(t-a)u(t-a)\} = e^{-as}F(s)$
3. Frequency shifting: $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$
4. Differentiation in time domain: $\mathcal{L}\{f'(t)\} = sF(s) - f(0^-)$
5. Integration in time domain: $\mathcal{L}\left\{\int_0^t f(\tau)d\tau\right\} = \frac{F(s)}{s}$

Laplace Transforms of Common Functions:

1. Unit step function: $\mathcal{L}\{u(t)\} = \frac{1}{s}$
2. Exponential function: $\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$
3. Sine function: $\mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2}$
4. Cosine function: $\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$