Linear Circuit Analysis

1. Introduction
2. Basic Concepts
3. Simple Circuits
4. Nodal and Mesh Analysis (DC)
5. Additional Analysis Techniques (DC)
6. AC Analysis
7. Magnetically Coupled Circuits
8. Polyphase Systems
9. Operational Amplifiers
10. Laplace Transforms
11. Time-Dependent Circuits
12. Two-Port Networks
Appendix

The Inverse Laplace Transform

The inverse Laplace transform is defined as $$\begin{equation}ℒ^{-1}\left[F(s)\right]=\frac{1}{2\pi j}\int_{\sigma_1-j\infty}^{\sigma_1+j\infty}F(s)e^{st} ds\end{equation}$$

Although it is possible to apply the above formula to compute the $ℒ^{-1}\left[F(s)\right]$, in electric circuits we usually compute the inverse Laplace transform using the method of partial fraction decomposition.

Properties of Inverse Laplace Transform
Table 1. Properties of inverse Laplace transform.
$F(s)$ $f(t)=ℒ^{-1}\left[F(s)\right]$
Addition/subtraction$F_1(\class{mjblue}{s}) \pm F_2(s)$ $f_1(t) \pm f_2(t)$
Linearity$C_1 F_1(\class{mjblue}{s}) \pm C_2 F_2(\class{mjblue}{s})$ $C_1 f_1(t) \pm C_2 f_2(t)$
Frequency scaling$F(c \class{mjblue}{s})$ $\dfrac{1}{c} f\left(\dfrac{t}{c}\right)$
Frequency shifting$F(\class{mjblue}{s}-a)$ $e^{a t} f(t)$
Time shifting$e^{-a@s}F(s)$ $f(t-a)u(t-a)$
Division by $s$$\dfrac{F(\class{mjblue}{s})}{\class{mjblue}{s}}$ $\int_{0}^{t} f(x) \,dx$
Partial Fractions Decomposition

A partial fraction is a fraction in which the numerator is a constant (real or complex) and the denominator is a linear polynomial raised to a positive power. Partial fractions have the form $$\frac{A}{(\class{mjblue}{s}+a)^n}$$ where $A$ and $a$ are real or complex numbers and $n$ is a positive integer.

Any proper rational function (where the degree of the numerator is less than the degree of the denominator) can be expressed as a sum of partial fractions using the method of partial fraction decomposition.

Inverse Laplace transforms of partial fractions
Table 2. Inverse Laplace transforms of partial fractions.
$F(\class{mjblue}{s})$ $f(t)=ℒ^{-1}\left[F(\class{mjblue}{s})\right]$
$\dfrac{1}{\class{mjblue}{s}}$$1$
$\dfrac{1}{\class{mjblue}{s}+a}$$e^{-at}$
$\dfrac{1}{\class{mjblue}{s}^{n}}$ $\dfrac{t^{n-1}}{(n-1)!}$
$\dfrac{1}{(\class{mjblue}{s}+a)^{n}}$ $\dfrac{t^{n-1} e^{-at}}{(n-1)!}$
Examples of Solved Problems
See also
Read more

Inverse Laplace transform
Euler's formula
Pierre-Simon Laplace
Leonhard Euler