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Laplace Transform

Laplace Transform. The Laplace transform Solution of linear differential equations Transient response example Simulink example. The Laplace Transform. Definition Time ( t ) is replaced by a new independent variable ( s ) We call s the Laplace transform variable The Laplace domain

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Laplace Transform

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  1. Laplace Transform • The Laplace transform • Solution of linear differential equations • Transient response example • Simulink example

  2. The Laplace Transform • Definition • Time (t) is replaced by a new independent variable (s) • We call s the Laplace transform variable • The Laplace domain • Often more convenient to work in Laplace domain than time domain • Time domain  ordinary differential equations in t • Laplace domain  algebraic equations in s • General solution approach • Formulate model in time domain • Convert model to Laplace domain • Solve problem in Laplace domain • Invert solution back to time domain

  3. Laplace Transform of Selected Functions • Constant function: f(t) = a • Exponential function: f(t) = e-bt • Derivatives and integrals

  4. Properties of Laplace Transforms • Superposition • Final value theorem • Initial value theorem • Time delay

  5. Linear ODE Example • ODE • Laplace transform • Substitute y(0) and rearrange • Inverse Laplace transform

  6. Linear ODE Example cont. • Table 3.1 • Our problem • Substitute and simplify

  7. General ODE Solution Procedure • Procedure • Transform to Laplace domain • Solve resulting algebraic equations • Transform solution back to time domain • Partial fraction expansion • Necessary when inverse Laplace transform not tabularized • Break complex functions into simpler tabularized functions

  8. Partial Fraction Example • Partial fraction expansion • Determination of coefficients • Inverse Laplace transform

  9. Repeated Factor Example

  10. Quadratic Factor Example

  11. Transient Response Example • Component balance • Step input

  12. Transient Response Example cont. • Laplace transform • Substitute input • Inverse Laplace transform

  13. Simulink Solution: mixing.mdl >> plot(tout,inlet) >> hold >> plot(tout,outlet,'r') >> axis([0 15 0 6]) >> ylabel('Concentration (kmol/m^3)') >> xlabel('Time (min)') >> legend(‘Input','Output')

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