Linear Circuit Analysis
1. Introduction
2. Basic Concepts
- Currents and voltages
- Linear circuits
- Linear components
- Loops and nodes
- Series and parallel
- R, L & C combinations
- V & I combinations
- Power and energy
3. Simple Circuits
- Ohm's law
- Kirchhoff's current law
- Kirchhoff's voltage law
- Single loop circuits
- Single node-pair circuits
- Voltage division
- Current division
4. Nodal and Mesh Analysis
5. Additional Analysis Techniques
6. AC Analysis
7. Operational Amplifiers
8. Laplace Transforms
9. Time-Dependent Circuits
- Introduction
- First-order transients
- Nodal analysis
- Mesh analysis
- Laplace transforms
- Additional techniques
10. Two-port networks
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
| $f(t)$ | |
|---|---|
| $1$ | $\dfrac{1}{\textcolor{blue}{s}}$ |
| $e^{-at}$ | $\dfrac{1}{\textcolor{blue}{s}+a}$ |
| $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
| $f(t)$ | ||
|---|---|---|
| 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
| $f(t)$ | ||
|---|---|---|
| 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 | $f(t) u (t-t_0)$ | $e^{-t_0@s} ℒ(f(t+t_0))$ |
| 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
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Laplace transforms
Power function
Exponential function
Cos/Sin functions
Cos/Sin functions with non-zero angle