Laplace Transforms.pptxhhhhhhhhhhhhhhhhh

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1. Laplace Transforms:

2. Laplace transform table:

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:

10. Example 2: