Laplace Transforms.pptxhhhhhhhhhhhhhhhhh
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Mba Ebooks ! Edhole
The document discusses key concepts in Laplace transforms including:
1) The Laplace transform is defined as an integral transform that transforms a function of time into a function of a complex variable, simplifying analysis of differential equations.
2) Important properties include the Laplace transforms of derivatives and integrals, which allow transforming differential equations into algebraic equations.
3) The existence theorem guarantees a unique solution to initial value problems under certain conditions on the function.
Laplace transform
The document discusses the Laplace transform, which takes a function of time and transforms it into a function of complex frequency. This transformation converts differential equations into algebraic equations, simplifying solving problems involving systems. The Laplace transform has many applications in fields like engineering, physics, and astronomy by allowing analysis of linear time-invariant systems through properties like derivatives becoming multiplications in the frequency domain.
Application of Laplace Transformation (cuts topic)
This document discusses applying the Laplace transform to modeling the vocal tract. It begins with an abstract discussing the Laplace transform and its properties. It then discusses a generalized acoustic tube model of the vocal tract that includes the oral cavity, nasal cavity, and main vocal tract. The model considers a branch section where the three branches meet. The document discusses obtaining the transfer function from this generalized model and using it to formulate a pole-zero type linear prediction algorithm. It will evaluate the prediction coefficients and reflection coefficients by analyzing voiced and nasal sounds.
Adv math[unit 2]
The document discusses the Laplace transform and its applications. The Laplace transform maps functions defined in the time domain to functions defined in the complex frequency domain. It makes solving differential equations easier by converting calculus operations into algebra. Some key properties include: the Laplace transform of derivatives can be obtained algebraically instead of using calculus rules, and the transform allows shifting between time and complex frequency domains. Examples are provided to illustrate definitions, properties, and how to use Laplace transforms to solve initial value problems for ordinary differential equations.
laplace transform 1 .pdf
The document discusses Laplace transforms, which convert functions of time into functions of frequency to make differential equations easier to solve. Laplace transforms are useful for solving complex differential equations by converting them into simpler polynomial equations. The document also outlines several applications of Laplace transforms in fields like control systems, signal processing, physics, and circuit analysis.
Meeting w3 chapter 2 part 1
This document provides an overview of analog control systems and Laplace transforms. It introduces key concepts like Laplace transforms, common time domain inputs, transfer functions, and modeling electrical, mechanical and electromechanical systems using block diagrams and mathematical models. Examples are provided to illustrate Laplace transforms, transfer functions, and analyzing system response using poles, zeros and stability analysis.
Meeting w3 chapter 2 part 1
This document provides an overview of analog control systems and Laplace transforms. It introduces key concepts like Laplace transforms, common time domain inputs, transfer functions, and modeling electrical, mechanical and electromechanical systems using block diagrams and mathematical models. Examples are provided to illustrate Laplace transforms, transfer functions, and analyzing system response using poles, zeros and stability analysis.
Laplace transform
The document presents information about Laplace transforms including:
- The definition and properties of the Laplace transform. It transforms differential equations into algebraic equations.
- Examples of applications in solving ordinary and partial differential equations, electrical circuits, and other fields.
- Limitations in that it can only be used to solve differential equations with known constants.
- Conclusion that the Laplace transform is a powerful mathematical tool used across many areas of science and engineering.
Recommended
Mba Ebooks ! Edhole
The document discusses key concepts in Laplace transforms including:
1) The Laplace transform is defined as an integral transform that transforms a function of time into a function of a complex variable, simplifying analysis of differential equations.
2) Important properties include the Laplace transforms of derivatives and integrals, which allow transforming differential equations into algebraic equations.
3) The existence theorem guarantees a unique solution to initial value problems under certain conditions on the function.
Laplace transform
The document discusses the Laplace transform, which takes a function of time and transforms it into a function of complex frequency. This transformation converts differential equations into algebraic equations, simplifying solving problems involving systems. The Laplace transform has many applications in fields like engineering, physics, and astronomy by allowing analysis of linear time-invariant systems through properties like derivatives becoming multiplications in the frequency domain.
Application of Laplace Transformation (cuts topic)
This document discusses applying the Laplace transform to modeling the vocal tract. It begins with an abstract discussing the Laplace transform and its properties. It then discusses a generalized acoustic tube model of the vocal tract that includes the oral cavity, nasal cavity, and main vocal tract. The model considers a branch section where the three branches meet. The document discusses obtaining the transfer function from this generalized model and using it to formulate a pole-zero type linear prediction algorithm. It will evaluate the prediction coefficients and reflection coefficients by analyzing voiced and nasal sounds.
Adv math[unit 2]
The document discusses the Laplace transform and its applications. The Laplace transform maps functions defined in the time domain to functions defined in the complex frequency domain. It makes solving differential equations easier by converting calculus operations into algebra. Some key properties include: the Laplace transform of derivatives can be obtained algebraically instead of using calculus rules, and the transform allows shifting between time and complex frequency domains. Examples are provided to illustrate definitions, properties, and how to use Laplace transforms to solve initial value problems for ordinary differential equations.
laplace transform 1 .pdf
The document discusses Laplace transforms, which convert functions of time into functions of frequency to make differential equations easier to solve. Laplace transforms are useful for solving complex differential equations by converting them into simpler polynomial equations. The document also outlines several applications of Laplace transforms in fields like control systems, signal processing, physics, and circuit analysis.
Meeting w3 chapter 2 part 1
This document provides an overview of analog control systems and Laplace transforms. It introduces key concepts like Laplace transforms, common time domain inputs, transfer functions, and modeling electrical, mechanical and electromechanical systems using block diagrams and mathematical models. Examples are provided to illustrate Laplace transforms, transfer functions, and analyzing system response using poles, zeros and stability analysis.
Meeting w3 chapter 2 part 1
This document provides an overview of analog control systems and Laplace transforms. It introduces key concepts like Laplace transforms, common time domain inputs, transfer functions, and modeling electrical, mechanical and electromechanical systems using block diagrams and mathematical models. Examples are provided to illustrate Laplace transforms, transfer functions, and analyzing system response using poles, zeros and stability analysis.
Laplace transform
The document presents information about Laplace transforms including:
- The definition and properties of the Laplace transform. It transforms differential equations into algebraic equations.
- Examples of applications in solving ordinary and partial differential equations, electrical circuits, and other fields.
- Limitations in that it can only be used to solve differential equations with known constants.
- Conclusion that the Laplace transform is a powerful mathematical tool used across many areas of science and engineering.
Nandha ppt
This document discusses the Laplace transformation, which converts a function into another function through mathematical manipulation. Specifically, it transforms a differential equation into an algebraic expression that can be easily solved. The document outlines the key topics that will be covered, including the definition of the Laplace transform, using tables of transforms to solve problems, inverse transforms, step functions, and using Laplace transforms to solve initial value problems, especially those involving step functions. It aims to illustrate how Laplace transforms simplify solving differential equations.
Using Laplace Transforms to Solve Differential Equations
This document discusses using Laplace transforms to solve differential equations. It begins with an introduction to Laplace transforms and their inventor Pierre-Simon Laplace. The author then explains that Laplace transforms can be used to transform differential equations into simpler algebraic equations that are easier to solve. As an example, the document walks through using Laplace transforms to solve a first order differential equation. It concludes that Laplace transforms are an important mathematical tool for solving differential equations.
Simplex Algorithm
The document discusses the simplex algorithm for solving linear programming problems. It begins with an introduction and overview of the simplex algorithm. It then describes the key steps of the algorithm, which are: 1) converting the problem into slack format, 2) constructing the initial simplex tableau, 3) selecting the pivot column and calculating the theta ratio to determine the departing variable, 4) pivoting to create the next tableau. The document provides examples to illustrate these steps. It also briefly discusses cycling issues, software implementations, efficiency considerations and variants of the simplex algorithm.
Laplace Transform and its applications
The Laplace transform is an integral transform that converts a function of time (often a function that represents a signal) into a function of complex frequency. It has various applications in engineering for solving differential equations and analyzing linear systems. The key aspect is that it converts differential operators into algebraic operations, allowing differential equations to be solved as algebraic equations. This makes the equations much easier to manipulate and solve compared to the original differential form.
Multiple scales
This document describes the method of multiple scales for extracting the slow time dependence of patterns near a bifurcation. The key aspects are:
1) Introducing scaled space and time coordinates (multiple scales) to capture slow modulations to the pattern, treating these as separate variables.
2) Expanding the solution as a power series in a small parameter near threshold.
3) Equating terms at each order in the expansion to derive amplitude equations for the slowly varying amplitudes, using solvability conditions to constrain the equations.
4) The solvability conditions arise because the linear operator has a null space, requiring the removal of components in this null space for finite solutions.
Laplace transform
The document discusses the Laplace transform and its applications. Specifically:
- The Laplace transform was developed by mathematicians including Euler, Lagrange, and Laplace to solve differential equations.
- It transforms a function of time to a function of complex frequencies, allowing differential equations to be written as algebraic equations.
- For a function to have a Laplace transform, it must be at least piecewise continuous and bounded above by an exponential function.
- The Laplace transform can reduce the dimension of partial differential equations and is used in applications including semiconductor mobility, wireless networks, vehicle vibrations, and electromagnetic fields.
Laplace transform
The document discusses the Laplace transform and its applications. Specifically:
- The Laplace transform was developed by mathematicians including Euler, Lagrange, and Laplace to solve differential equations.
- It transforms a function of time to a function of complex frequencies, allowing differential equations to be written as algebraic equations.
- For a function to have a Laplace transform, it must be at least piecewise continuous and bounded above by an exponential function.
- The Laplace transform can reduce the dimension of partial differential equations and is used in applications including semiconductor mobility, wireless networks, vehicle vibrations, and electromagnetic fields.
Receptor kinetics Laplace transform method PDF
1) The document describes using the Laplace transform method to derive equations for receptor-ligand binding kinetics models.
2) The Laplace transform allows time-dependent differential equations to be solved using simple algebra by transforming them into another mathematical domain.
3) Four binding models are presented: ligand association, dissociation, unlabeled ligand pre-incubation and washout, and competition kinetics. The method is applied step-by-step to derive the analytic equations for each model.
Receptor kinetics Laplace transform method Word
1) The document describes using the Laplace transform method to derive equations for receptor-ligand binding kinetics models.
2) The Laplace transform allows time-dependent differential equations to be solved using simple algebra by transforming them into another mathematical domain.
3) Four binding models are used as examples: ligand association, dissociation, unlabeled ligand pre-incubation and washout, and competition kinetics. Through taking the Laplace transform, substituting terms, and taking the inverse transform, analytical solutions are obtained for each model.
Maths ppt partial diffrentian eqn
This document discusses applications of Taylor series and partial differential equations. Taylor series can be used to represent complicated functions as infinite polynomials, making their properties easier to study. They also allow differential equations to be solved more easily. Some applications discussed include using Taylor series to evaluate definite integrals of functions without anti-derivatives, study the asymptotic behavior of electric fields, and solve partial differential equations like the heat equation. Taylor series provide a way to approximate functions with polynomials.
Derivative rules.docx
The document discusses techniques for calculating derivatives of functions, including:
- Using formulas and theorems to calculate derivatives more efficiently than using the definition of a derivative.
- Applying rules like the power rule, product rule, and quotient rule to take derivatives.
- Using derivatives to find equations of tangent lines and instantaneous rates of change.
Laplace transform
The Laplace transform is a mathematical tool that is useful for solving differential equations. It was developed by Pierre-Simon Laplace in the late 18th century. The Laplace transform takes a function of time and transforms it into a function of complex quantities. This transformation allows differential equations to be converted into algebraic equations that are easier to solve. Some common applications of the Laplace transform include modeling problems in semiconductors, wireless networks, vibrations, and electromagnetic fields.
Laplace transformation
The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
Laplace Transformation & Its Application
This document presents an overview of the Laplace transform and its applications. It begins with an introduction to Laplace transforms as a mathematical tool to convert differential equations into algebraic expressions. It then provides definitions and properties of both the Laplace transform and its inverse. Examples are given of how Laplace transforms can be used to solve ordinary and partial differential equations, as well as applications in electrical circuits and other fields. The document concludes by noting some limitations of the Laplace transform method and references additional resources.
hsu-Chapter 6 Laplace transform.pdf
The document provides an introduction to the Laplace transform and its applications in engineering analysis. It discusses key topics including:
- The history and development of the Laplace transform by mathematician Pierre-Simon Laplace.
- How the Laplace transform is used to transform a variable like time or position into a parameter to solve differential equations.
- Examples of using Laplace transform properties like linearity, shifting, and change of scale to calculate transforms of functions.
- How Laplace transforms can be used to solve ordinary and partial differential equations that model physical systems.
- Tables of common Laplace transforms to help with transforming functions.
Simplex algorithm
The document provides an overview of the simplex algorithm for solving linear programming problems. It begins with an introduction and defines the standard format for representing linear programs. It then describes the key steps of the simplex algorithm, including setting up the initial simplex tableau, choosing the pivot column and pivot row, and pivoting to move to the next basic feasible solution. It notes that the algorithm terminates when an optimal solution is reached where all entries in the objective row are non-negative. The document also briefly discusses variants like the ellipsoid method and cycling issues addressed by Bland's rule.
M1 unit viii-jntuworld
This document outlines the contents and structure of a course on Mathematics-I. It covers topics such as ordinary differential equations, linear differential equations, functions of several variables, vector calculus, and Laplace transforms. It lists textbooks and references for the course. It provides an overview of the 12 lectures in the Laplace transforms unit, covering definitions, properties, and applications of the Laplace transform to solve differential equations.
Inverse Laplace Transform
This document discusses the inverse Laplace transform, which finds the original function given its Laplace transform. It defines the inverse Laplace transform and proves it is unique. The key points are:
1. The inverse Laplace transform of a function F(s) is the function f(t) whose Laplace transform is F(s).
2. The uniqueness theorem proves there is only one function f(t) that corresponds to a given F(s).
3. The inverse is only defined for t ≥ 0, as the Laplace transform only uses information from the positive t-axis.
Partial fraction decomposition for inverse laplace transform
This document discusses partial fraction decomposition for inverse Laplace transforms. It begins with an introduction to partial fraction decomposition and why it is useful for integration. It then covers various cases for partial fraction decomposition of inverse Laplace transforms, including when the denominator is a quadratic with two real roots, a double root, or complex conjugate roots. It also covers the case when the denominator is a cubic with one real and two complex conjugate roots. The goal is to decompose the function into simpler forms that can be easily inverted using the Laplace transform table.
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Nandha ppt
This document discusses the Laplace transformation, which converts a function into another function through mathematical manipulation. Specifically, it transforms a differential equation into an algebraic expression that can be easily solved. The document outlines the key topics that will be covered, including the definition of the Laplace transform, using tables of transforms to solve problems, inverse transforms, step functions, and using Laplace transforms to solve initial value problems, especially those involving step functions. It aims to illustrate how Laplace transforms simplify solving differential equations.
Using Laplace Transforms to Solve Differential Equations
This document discusses using Laplace transforms to solve differential equations. It begins with an introduction to Laplace transforms and their inventor Pierre-Simon Laplace. The author then explains that Laplace transforms can be used to transform differential equations into simpler algebraic equations that are easier to solve. As an example, the document walks through using Laplace transforms to solve a first order differential equation. It concludes that Laplace transforms are an important mathematical tool for solving differential equations.
Simplex Algorithm
The document discusses the simplex algorithm for solving linear programming problems. It begins with an introduction and overview of the simplex algorithm. It then describes the key steps of the algorithm, which are: 1) converting the problem into slack format, 2) constructing the initial simplex tableau, 3) selecting the pivot column and calculating the theta ratio to determine the departing variable, 4) pivoting to create the next tableau. The document provides examples to illustrate these steps. It also briefly discusses cycling issues, software implementations, efficiency considerations and variants of the simplex algorithm.
Laplace Transform and its applications
The Laplace transform is an integral transform that converts a function of time (often a function that represents a signal) into a function of complex frequency. It has various applications in engineering for solving differential equations and analyzing linear systems. The key aspect is that it converts differential operators into algebraic operations, allowing differential equations to be solved as algebraic equations. This makes the equations much easier to manipulate and solve compared to the original differential form.
Multiple scales
This document describes the method of multiple scales for extracting the slow time dependence of patterns near a bifurcation. The key aspects are:
1) Introducing scaled space and time coordinates (multiple scales) to capture slow modulations to the pattern, treating these as separate variables.
2) Expanding the solution as a power series in a small parameter near threshold.
3) Equating terms at each order in the expansion to derive amplitude equations for the slowly varying amplitudes, using solvability conditions to constrain the equations.
4) The solvability conditions arise because the linear operator has a null space, requiring the removal of components in this null space for finite solutions.
Laplace transform
The document discusses the Laplace transform and its applications. Specifically:
- The Laplace transform was developed by mathematicians including Euler, Lagrange, and Laplace to solve differential equations.
- It transforms a function of time to a function of complex frequencies, allowing differential equations to be written as algebraic equations.
- For a function to have a Laplace transform, it must be at least piecewise continuous and bounded above by an exponential function.
- The Laplace transform can reduce the dimension of partial differential equations and is used in applications including semiconductor mobility, wireless networks, vehicle vibrations, and electromagnetic fields.
Laplace transform
The document discusses the Laplace transform and its applications. Specifically:
- The Laplace transform was developed by mathematicians including Euler, Lagrange, and Laplace to solve differential equations.
- It transforms a function of time to a function of complex frequencies, allowing differential equations to be written as algebraic equations.
- For a function to have a Laplace transform, it must be at least piecewise continuous and bounded above by an exponential function.
- The Laplace transform can reduce the dimension of partial differential equations and is used in applications including semiconductor mobility, wireless networks, vehicle vibrations, and electromagnetic fields.
Receptor kinetics Laplace transform method PDF
1) The document describes using the Laplace transform method to derive equations for receptor-ligand binding kinetics models.
2) The Laplace transform allows time-dependent differential equations to be solved using simple algebra by transforming them into another mathematical domain.
3) Four binding models are presented: ligand association, dissociation, unlabeled ligand pre-incubation and washout, and competition kinetics. The method is applied step-by-step to derive the analytic equations for each model.
Receptor kinetics Laplace transform method Word
1) The document describes using the Laplace transform method to derive equations for receptor-ligand binding kinetics models.
2) The Laplace transform allows time-dependent differential equations to be solved using simple algebra by transforming them into another mathematical domain.
3) Four binding models are used as examples: ligand association, dissociation, unlabeled ligand pre-incubation and washout, and competition kinetics. Through taking the Laplace transform, substituting terms, and taking the inverse transform, analytical solutions are obtained for each model.
Maths ppt partial diffrentian eqn
This document discusses applications of Taylor series and partial differential equations. Taylor series can be used to represent complicated functions as infinite polynomials, making their properties easier to study. They also allow differential equations to be solved more easily. Some applications discussed include using Taylor series to evaluate definite integrals of functions without anti-derivatives, study the asymptotic behavior of electric fields, and solve partial differential equations like the heat equation. Taylor series provide a way to approximate functions with polynomials.
Derivative rules.docx
The document discusses techniques for calculating derivatives of functions, including:
- Using formulas and theorems to calculate derivatives more efficiently than using the definition of a derivative.
- Applying rules like the power rule, product rule, and quotient rule to take derivatives.
- Using derivatives to find equations of tangent lines and instantaneous rates of change.
Laplace transform
The Laplace transform is a mathematical tool that is useful for solving differential equations. It was developed by Pierre-Simon Laplace in the late 18th century. The Laplace transform takes a function of time and transforms it into a function of complex quantities. This transformation allows differential equations to be converted into algebraic equations that are easier to solve. Some common applications of the Laplace transform include modeling problems in semiconductors, wireless networks, vibrations, and electromagnetic fields.
Laplace transformation
The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
Laplace Transformation & Its Application
This document presents an overview of the Laplace transform and its applications. It begins with an introduction to Laplace transforms as a mathematical tool to convert differential equations into algebraic expressions. It then provides definitions and properties of both the Laplace transform and its inverse. Examples are given of how Laplace transforms can be used to solve ordinary and partial differential equations, as well as applications in electrical circuits and other fields. The document concludes by noting some limitations of the Laplace transform method and references additional resources.
hsu-Chapter 6 Laplace transform.pdf
The document provides an introduction to the Laplace transform and its applications in engineering analysis. It discusses key topics including:
- The history and development of the Laplace transform by mathematician Pierre-Simon Laplace.
- How the Laplace transform is used to transform a variable like time or position into a parameter to solve differential equations.
- Examples of using Laplace transform properties like linearity, shifting, and change of scale to calculate transforms of functions.
- How Laplace transforms can be used to solve ordinary and partial differential equations that model physical systems.
- Tables of common Laplace transforms to help with transforming functions.
Simplex algorithm
The document provides an overview of the simplex algorithm for solving linear programming problems. It begins with an introduction and defines the standard format for representing linear programs. It then describes the key steps of the simplex algorithm, including setting up the initial simplex tableau, choosing the pivot column and pivot row, and pivoting to move to the next basic feasible solution. It notes that the algorithm terminates when an optimal solution is reached where all entries in the objective row are non-negative. The document also briefly discusses variants like the ellipsoid method and cycling issues addressed by Bland's rule.
M1 unit viii-jntuworld
This document outlines the contents and structure of a course on Mathematics-I. It covers topics such as ordinary differential equations, linear differential equations, functions of several variables, vector calculus, and Laplace transforms. It lists textbooks and references for the course. It provides an overview of the 12 lectures in the Laplace transforms unit, covering definitions, properties, and applications of the Laplace transform to solve differential equations.
Inverse Laplace Transform
This document discusses the inverse Laplace transform, which finds the original function given its Laplace transform. It defines the inverse Laplace transform and proves it is unique. The key points are:
1. The inverse Laplace transform of a function F(s) is the function f(t) whose Laplace transform is F(s).
2. The uniqueness theorem proves there is only one function f(t) that corresponds to a given F(s).
3. The inverse is only defined for t ≥ 0, as the Laplace transform only uses information from the positive t-axis.
Partial fraction decomposition for inverse laplace transform
This document discusses partial fraction decomposition for inverse Laplace transforms. It begins with an introduction to partial fraction decomposition and why it is useful for integration. It then covers various cases for partial fraction decomposition of inverse Laplace transforms, including when the denominator is a quadratic with two real roots, a double root, or complex conjugate roots. It also covers the case when the denominator is a cubic with one real and two complex conjugate roots. The goal is to decompose the function into simpler forms that can be easily inverted using the Laplace transform table.
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Microbial characterisation and identification, and potability of River Kuywa ...
Microbial characterisation and identification, and potability of River Kuywa ...Open Access Research Paper
Water contamination is one of the major causes of water borne diseases worldwide. In Kenya, approximately 43% of people lack access to potable water due to human contamination. River Kuywa water is currently experiencing contamination due to human activities. Its water is widely used for domestic, agricultural, industrial and recreational purposes. This study aimed at characterizing bacteria and fungi in river Kuywa water. Water samples were randomly collected from four sites of the river: site A (Matisi), site B (Ngwelo), site C (Nzoia water pump) and site D (Chalicha), during the dry season (January-March 2018) and wet season (April-July 2018) and were transported to Maseno University Microbiology and plant pathology laboratory for analysis. The characterization and identification of bacteria and fungi were carried out using standard microbiological techniques. Nine bacterial genera and three fungi were identified from Kuywa river water. Clostridium spp., Staphylococcus spp., Enterobacter spp., Streptococcus spp., E. coli, Klebsiella spp., Shigella spp., Proteus spp. and Salmonella spp. Fungi were Fusarium oxysporum, Aspergillus flavus complex and Penicillium species. Wet season recorded highest bacterial and fungal counts (6.61-7.66 and 3.83-6.75cfu/ml) respectively. The results indicated that the river Kuywa water is polluted and therefore unsafe for human consumption before treatment. It is therefore recommended that the communities to ensure that they boil water especially for drinking.
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Laplace Transforms.pptxhhhhhhhhhhhhhhhhh
- 3. Inverse LaPlace transform table: Alternatively, you can find an inverse Laplace transform by using a Laplace Transforms table, reading from the right column to the left column.
- 4. Step 1 – Laplace transform the whole differential equation: Use the transform table to convert everything in the differential equation into LaPlace form. Some examples are: initial values – i.e. Replace x0 and x1 with values (often 0) Step 2 – Insert initial values:
- 5. Step 3 -Rearrange to make the subject of the equation Factorise with outside a bracket on LHS initially, then rearrange to make the subject of the equation. This put all s on the RHS. Make RHS a single fraction. If required multiply RHS by the Step function 1/s Factorise the denominator of the resulting fraction (this will reveal the poles).
- 6. Step 4 -Partial Fractions – Split the fraction and find Residues values A, B, etc. Several types of factor in the denominator are shown below. Rewrite these as shown, then find A, B, C, etc. This can also be done in MATLAB using the residue function. Step 5 - Inverse Laplace Transform of the whole equation = ANSWER!
- 7. Why use LaPlace transforms? The Laplace transform is a method used to solve differential equations and model engineering systems. It helps derive both the complementary function (transient response) and the particular integral (steady-state response). Initial or boundary conditions can be applied to find a complete solution, though they are often set to zero for simplicity. Deriving over time?
- 8. Notation: The Laplace variable, s, is a complex variable, which replaces time, t, (time domain). s consists of a real and imaginary part. Where σ is the real part related to stability of system ω is the imaginary part related to frequency Often only the steady state response is required so we set σ = 0 and s = jω is used, where and ω is the frequency (in rad/s). X bar or X(s) - When the variable x is transformed by a Laplace transform it is renamed x bar , or X(s)
- 9. Example 1:
- 13. Example 2: