Introduction to Laplace Transforms (Webinar)

In Introduction to Laplace Transforms (Webinar), you'll learn ...

• The fundamental principles of Laplace Transforms, including time-domain and s-domain representations, transformation equations, and mathematical properties.
• How to apply Laplace Transform techniques to solve differential and integrodifferential equations using initial conditions, inverse transformations, and partial fraction decomposition.
• The application of Laplace Transforms to Linear Time-Invariant electrical circuits, including natural, forced, transient, and steady-state responses.
• How to analyze feedback control systems using transfer functions, pole-zero relationships, stability criteria, and Root Locus techniques.

Overview

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This presentation provides the principles and characteristics of transformation of signals from the time domain to the complex frequency domain, i.e, the s-domain.

The Laplace Transform equations, as well as theorems and tables of Laplace Transform, are provided prior to illustration of a number of specific applications. A sample calculation is provided to illustrate the steps required by an actual transformation from the time to the s-domain which confirms the transformed equation provided by the appropriate Laplace Transform Table.

An example is provided that illustrates the use of Laplace Transform to solve a second-order differential equation including the use of partial fraction decomposition of the s-domain equation to arrive at the appropriate time domain solution provided by the inverse Laplace Transform operation.

The relative ease offered by Laplace Transform to solve the defining integrodifferential equation controlling the behavior of a simple Linear-Time-Invariant circuits is illustrated with specific steps required by the transformation to the s-domain. The provided example points to the significant calculation simplicity offered by Laplace Transforms vs. the use of an actual integrodifferential solution to obtain the elements of the time-invariant linear circuit. The concept of the defining “Transfer Function” and the use of the Laplace Transform Convolution Theorem to obtain generalized solutions of linear time-invariant circuits is introduced.

The key concern regarding the stability of Feedback Control Systems is provided using the application of Root Locus plots. Specific steps to draw generalized Root Locus plots and examples using these steps are provided.

This is a good introduction to the fundamental transformation of time-domain functions to the complex s domain environment which, if offered in conjunction with Introduction to Fourier Analysis for steady-state stage applications of LTI circuits, points to the advantage provided by Laplace Transform to include all transient effects prior to the steady-state condition.

Learning Objectives

Upon completion of this webinar, participants will be able to:

• Define the Laplace Transform and explain how time-domain functions are represented in the complex s-domain, including the significance of the real and imaginary components of the complex frequency variable.
• Identify and apply fundamental Laplace Transform equations, properties, and transformation tables to convert common time-domain functions into their corresponding s-domain expressions.
• Calculate Laplace Transforms of mathematical functions, including exponential, sinusoidal, impulse, step, and polynomial functions, using appropriate transformation formulas.
• Apply Laplace Transform differentiation and integration properties to convert differential and integrodifferential equations into algebraic equations while accounting for specified initial conditions.
• Solve second-order differential equations using Laplace Transformation, partial fraction decomposition, and inverse Laplace Transform techniques to obtain time-domain solutions.
• Transform resistor, inductor, and capacitor relationships into the s-domain and apply algebraic circuit analysis techniques to determine voltage and current responses in Linear Time-Invariant circuits.
• Distinguish between natural, forced, transient, and steady-state circuit responses and calculate the total response of an R-C circuit subjected to DC or sinusoidal excitation.
• Determine the transfer function and impulse response of an LTI circuit and apply the Laplace Transform Convolution Theorem to analyze circuit responses to time-varying inputs.
• Derive closed-loop transfer functions for feedback control systems and evaluate system stability using characteristic equations and the locations of poles in the complex s-plane.
• Interpret and construct Root Locus plots using applicable graphing rules to evaluate how changes in system gain affect pole locations, stability, and feedback control system performance.