Linear Circuit Analysis

1. Introduction
2. Basic Concepts
3. Simple Circuits
4. Nodal and Mesh Analysis
5. Additional Analysis Techniques
6. AC Analysis
7. Magnetically Coupled Circuits
8. Operational Amplifiers
9. Laplace Transforms
10. Time-Dependent Circuits
11. Two-port networks
Appendix

The Laplace Transform

The Laplace transform of a function $f(t)$ is defined as $$\begin{equation}ℒ\left[f(t)\right]=F(\textcolor{blue}{s})=\int_{0}^{\infty}f(t)e^{-\textcolor{blue}{s} t} dt\end{equation}$$ where $\textcolor{blue}{s}$ is the complex frequency and function $f(t)$ is assumed to be defined for $t\geq 0$. The Laplace transform is linear $$\begin{equation}ℒ\left[f(t)\right]=F(\textcolor{blue}{s})\int_{0}^{\infty}f(t)e^{-\textcolor{blue}{s} t} dt\end{equation}$$ and has the uniqueness property that for any $f(t)$ there is a unique $F(\textcolor{blue}{s})$.

The Laplace transform can be used to solve integro-differential equations such as the ones that appear in electric circuit analysis. The Laplace transform reduces a linear differential, integral, or integro-differential equation to an algebraic equation, which can then be solved using standard algebraic methods. Finally, the solution of the original equation can found by applying the inverse Laplace transform to the algebraic equation.

Common Laplace transforms
Table 1. Common Laplace transforms.
$f(t)$ $F(\textcolor{blue}{s})=ℒ\left[f(t)\right]$
$1$ $\dfrac{1}{\textcolor{blue}{s}}$
$e^{-at}$ $\dfrac{1}{\textcolor{blue}{s}+a}$
$t^n $ $\dfrac{n!}{\textcolor{blue}{s}^{n+1}}$
$t^n e^{-at}$ $\dfrac{n!}{(\textcolor{blue}{s}+a)^{n+1}}$
$\cos(\omega t)$ $\dfrac{s}{\textcolor{blue}{s}^2+\omega^2}$
$\sin(\omega t)$ $\dfrac{\omega}{\textcolor{blue}{s}^2+\omega^2}$
$\cos(\omega t+\varphi)$ $\dfrac{\textcolor{blue}{s} \cos\varphi-\omega \sin\varphi}{\textcolor{blue}{s}^2+\omega^2}$
$\sin(\omega t+\varphi)$ $\dfrac{\textcolor{blue}{s} \sin\varphi+\omega \cos\varphi}{\textcolor{blue}{s}^2+\omega^2}$
$e^{-at}\cos(\omega t)$ $\dfrac{\textcolor{blue}{s}+a}{(\textcolor{blue}{s}+a)^2+\omega^2}$
$e^{-at}\sin(\omega t)$ $\dfrac{\omega}{(\textcolor{blue}{s}+a)^2+\omega^2}$
$e^{-at}\cos(\omega t+\varphi)$ $\dfrac{(\textcolor{blue}{s}+a) \cos\varphi-\omega \sin\varphi}{(\textcolor{blue}{s}+a)^2+\omega^2}$
$e^{-at}\sin(\omega t+\varphi)$ $\dfrac{(\textcolor{blue}{s}+a) \sin\varphi+\omega \cos\varphi}{(\textcolor{blue}{s}+a)^2+\omega^2}$
Laplace transforms of integrals and derivatives
Table 2. Laplace transforms of integrals and derivatives.
$f(t)$ $F(\textcolor{blue}{s})=ℒ\left[f(t)\right]$
Differentiation$\dfrac{df(t)}{dt}$ $s F(\textcolor{blue}{s}) - f(0)$
Differentiation (n-times)$\dfrac{d^n f(t)}{dt^n}$ $\textcolor{blue}{s}^n F(s) - \textcolor{blue}{s}^{n-1}f(0) - \textcolor{blue}{s}^{n-2}\dfrac{df}{dt}(0) -...-s^0 \dfrac{d^{n-1}f}{dt^{n-1}}(0)$
Integration$\int_{0}^{t} f(\tau) \,d\tau$ $\dfrac{F(\textcolor{blue}{s})}{\textcolor{blue}{s}}$
Convolution$\int_{0}^{t} f_1(\tau)f_2(t-\tau) \,d\tau$ $F_1(\textcolor{blue}{s})F_2(\textcolor{blue}{s})$
Properties of Laplace transform
Table 3. Properties of Laplace transforms.
$f(t)$ $F(\textcolor{blue}{s})=ℒ\left[f(t)\right]$
Addition/subtraction$f_1(t) \pm f_2(t)$ $F_1(\textcolor{blue}{s}) \pm F_2(s)$
Linearity$C_1 f_1(t) \pm C_2 f_2(t)$ $C_1 F_1(\textcolor{blue}{s}) \pm C_2 F_2(\textcolor{blue}{s})$
Time scaling$f(c t)$ $\dfrac{1}{c} F\left(\dfrac{\textcolor{blue}{s}}{c}\right)$
Time shifting$f(t) u (t-t_0)$ $e^{-t_0 \textcolor{blue}{s}} ℒ(f(t+t_0))$
Frequency shifting$e^{-a t} f(t)$ $F(\textcolor{blue}{s}+a)$
Multiplication by $t$$t f(t)$ $-\dfrac{dF(\textcolor{blue}{s})}{ds}$
Multiplication by $t^n$$t^n f(t)$ $(-1)^n \dfrac{d^nF}{d@s^n}$
Division by $t$$\dfrac{f(t)}{t}$ $\int_{0}^{\infty} F(x) \,dx$
Sample Solved Problems
See also
Read more

Pierre-Simon Laplace
Laplace transform