14: Laplace Transforms
- Page ID
- 5507
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 14.1: Introduction to Laplace Transforms
- This page introduces the Laplace transform, defined as \(\bar{y}(s) = \int_0^{\infty} e^{-st} y(t) dt\), crucial for solving differential equations in electrical circuits. It emphasizes the need to comprehend forward and inverse transforms while contrasting them with complex number methods for AC circuits. The text highlights the Laplace transform's versatility in analyzing transient and steady-state conditions beyond just sinusoidal variations.
- 14.2: Table of Laplace Transforms
- This page provides a table of Laplace transforms for various functions, showcasing their practical applications while refraining from extensive mathematical details. It introduces theorems related to transforms, some stated with examples and others derived. The page encourages readers to utilize these theorems to deepen their understanding of Laplace transforms, especially in relation to circuit theory.
- 14.3: The First Integration Theorem
- The First Integration Theorem is most useful for finding an inverse Laplace transform.
- 14.4: The Second Integration Theorem (Dividing a Function by t)
- This page covers the second integration theorem for direct Laplace transforms, differentiating it from the first integration theorem used for inverse transforms. An example demonstrates its application by finding the transform of \(\frac{\sin at}{t}\), showcasing the theorem's ability to simplify the integration process. The page concludes with a specific result, \(\tan^{-1}\left(\frac{a}{s}\right)\), and motivates readers to apply both theorems for improving their Laplace integral tables.
- 14.5: Shifting Theorem
- This page covers the shifting theorem in Laplace transforms, explaining that the transform of \( e^{-at} y(t) \) results in \( \bar{y}(s+a) \). It includes examples demonstrating the application of the theorem, such as transforming \( t \) to \( te^{-at} \) and improving the Laplace transform table. The page highlights the redundancy of the entry for \( \textbf{L}(1) \) when considering \( \textbf{L}(e^{at}) \). Overall, the theorem is useful for direct and inverse transformations.
- 14.6: A Function Times tⁿ
- This page covers the Laplace transform of \( t^n y \) for positive integers \( n \), expressed as \((-1)^n \frac{d^2\bar{y}}{ds^n}\). It encourages readers to use specific theorems to enrich their Laplace transform tables with functions like \((\sin at)/t\), \( te^{-at}\), and \(t^2e^{-t}\), enhancing their understanding and application of Laplace transforms in various contexts.
- 14.7: Differentiation Theorem
- This page explains the Laplace transform of derivatives, focusing on the first and second derivatives. It presents the formulas \(\mathbf{L} \dot{y} = s \bar{y} - y_0\) and \(\mathbf{L} \ddot{y} = s^2 \bar{y} - sy_0 - \dot{y}_0\). The derivation utilizes integration by parts and underscores the significance of these transformations for simplifying complex differential equations, making them more manageable for practical application.
- 14.8: A First Order Differential Equation
- This page explains how to solve the first-order linear differential equation \(\dot y + 2y = 3te^t\) with the initial condition \(y_0=0\) using two methods: the integrating factor and the Laplace transform. Both approaches lead to the same solution \(y = te^t - \frac{1}{3}e^t + \frac{1}{3}e^{-2t}\), highlighting the importance of practice in mastering these techniques.
- 14.9: A Second Order Differential Equation
- This page provides a detailed solution to the differential equation \(\ddot y - 4 \dot y + 3y = e^{-t}\) by employing traditional methods and Laplace transforms. It outlines step-by-step derivation, applying the Laplace transform and simplifying to obtain \(\bar{y}\). After partial fraction decomposition, the final solution is \(y=\frac{1}{8}e^{-t} + \frac{7}{4}e^t - \frac{7}{8}e^{3t}\).
- 14.10: Generalized Impedance
- This page covers the analysis of circuits with sinusoidal voltages applied to inductance, resistance, and capacitance in series, detailing the equations for voltage, current, and impedance. It explains that Laplace transforms can be used for non-sinusoidal voltages to solve differential equations. The generalized Ohm's law is introduced, which incorporates resistance, inductance, and capacitance, enabling analysis irrespective of voltage time dependence. Examples are provided in later sections.
- 14.11: RLC Series Transient
- This page covers the analysis of RLC circuits with a constant EMF battery, utilizing conventional methods and Laplace transforms to derive equations for charge and current based on different damping conditions. It examines three cases defined by the discriminant and emphasizes the efficiency of Laplace transforms in simplifying circuit analysis.
- 14.12: Another Example
- This page analyzes a circuit with equal resistances and capacitances connected to a battery, applying Kirchhoff’s second rule to derive equations for the capacitor charges over time. It transforms these equations for inverse transforms, resulting in solutions for the charges \(Q_1\) and \(Q_2\). The findings highlight the exponential behavior of capacitor charging, reflecting the effects of resistive and capacitive properties.