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# Meeting w3 chapter 2 part 1

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### Meeting w3 chapter 2 part 1

1. 1. Chapter 2 Analog Control System Eddy Irwan Shah Bin Saadon Dept. of Electrical Engineering PPD, UTHM [email_address] 019-7017679
2. 2. <ul><li>Outline: </li></ul><ul><li>Introduction </li></ul><ul><li>Laplace Transform – Table/ Theorem/ Eg. </li></ul><ul><li>Common Time Domain Input Function </li></ul><ul><li>Transfer Function – Open/ Closed Loop & Eg. </li></ul><ul><li>Electrical Elements Modelling – Table & Eg. </li></ul><ul><li>Mechanical Elements Modelling - Table & Eg. </li></ul><ul><li>Block Diagram Reduction - Table & Eg. </li></ul><ul><li>System Response – Poles/ Zeros, Second Order, Steady State Error, Stability Analysis </li></ul>
3. 3. <ul><li>Intro - Objective of this chapter </li></ul><ul><li>After completing this chapter you will be able to: </li></ul><ul><li>Describe the fundamental of Laplace transforms. </li></ul><ul><li>Apply the Laplace transform to solve linear ordinary differential equations. </li></ul><ul><li>Apply Mathematical model, called a transfer function for linear time-invariant electrical, mechanical and electromechanical systems. </li></ul>
4. 4. 2. What is Laplace Transform? <ul><li>Laplace transform is a method or techniques used to transform the time ( t ) domain to the Laplace/frequency ( s ) domain </li></ul><ul><li>What is algebra & calculus? </li></ul>Time Domain Frequency Domain Differential equations Input q(t) Output h(t) Algebraic equations Input Q(s) Output H(s) Calculus Algebra Laplace Transformation Inverse Laplace Transformation
5. 5. Laplace Transform (cont.) <ul><li>The Laplace transform solution consists of the following three steps: </li></ul><ul><li>the Laplace transformation of q1(t) and (r dhldt + h = Gq) to frequency domain </li></ul><ul><li>the algebraic solution for H(s) </li></ul><ul><li>the inverse Laplace transformation of H(s) to time domain h(t). </li></ul><ul><li>The calculus solution is shown as step 4. </li></ul>
6. 6. Definition of the Laplace Transform <ul><li>Laplace transform is defined as </li></ul><ul><li>Inverse Laplace transform is defined as </li></ul>L L -1
7. 7. Laplace Theorem
8. 8. Laplace Table
9. 9. Example 1 <ul><li>Find the Laplace transform for </li></ul>Solution:
10. 10. Example 2 <ul><li>Find the Laplace transform for </li></ul>Solution:
11. 11. Example 3 <ul><li>Find the inverse Laplace transform of </li></ul>Solution: Expanding F(s) by partial fraction: Where, Then, taking the inverse Laplace transform
12. 12. Example 4 <ul><li>Given the ,solve for y ( t ) if all initial conditions are </li></ul><ul><li>zero. Use the Laplace transform method . </li></ul>Solution: Substitute the corresponding F ( s ) for each term: Solving for the response: Where, K 1 = 1 when s=0 K 2 =-2 when s=-4 K 3 = 1 when s=-8 Hence
13. 13. 3. Common Time Domain Input Functions <ul><li>Unit Step Function </li></ul>
14. 14. <ul><li>Unit Ramp Function </li></ul>cont.
15. 15. <ul><li>Unit Impulse Function </li></ul>cont.
16. 16. 4. Transfer Function <ul><li>Definition: </li></ul><ul><li>Ratio of the output to the input; with all initial conditions are zero </li></ul><ul><li>If the transformed input signal is X(s) and the transformed output signal is Y(s) , then the transfer function M(s) is define as; </li></ul><ul><li>From this, </li></ul><ul><li>Therefore the output is </li></ul>
17. 17. TF of Linear Time Invariant Systems <ul><li>In practice, the input-output relation of lines time-invariant system with continuous-data input is often described by a differential equation </li></ul><ul><li>The linear time-invariant system is described by the following n th-order differential equation with constant real coefficients; </li></ul>c(t) is output r(t) is input
18. 18. cont. <ul><li>Taking the Laplace transform of both sides, </li></ul><ul><li>If we assume that all initial conditions are zero, hence </li></ul><ul><li>Now, form the ratio of output transform, C(s) divided by input transform. The ratio, G(s) is called transfer function. </li></ul>
19. 19. cont. <ul><li>The transfer function can be represented as a block diagram </li></ul><ul><li>General block diagram </li></ul>
20. 20. Block Diagram of Open Loop System
21. 21. Block Diagram of Closed Loop System
22. 22. Example 1 <ul><li>Problem: Find the transfer function represented by </li></ul><ul><li>Solution: </li></ul><ul><li>Taking the Laplace transform of both sides, assuming zero initial conditions, we have </li></ul><ul><li>The transfer function, G(s) is </li></ul>
23. 23. Example 2 <ul><li>Problem: Use the result of Example 1 to find the response, c(t), to an input, r(t)=u(t), a unit step and assuming zero initial conditions. </li></ul><ul><li>Solution: </li></ul><ul><li>Since r(t)=u(t), R(s)=1/s, hence </li></ul><ul><li>Expanding by partial fractions, we get </li></ul><ul><li>Finally, taking the inverse Laplace transform of each term yields </li></ul>
24. 24. Example 3 <ul><li>Problem: Find the transfer function, G(s)=C(s)/R(s), corresponding to the differential equation </li></ul><ul><li>Solution: </li></ul>
25. 25. Example 4 <ul><li>Problem: Find the differential equation corresponding to the transfer function, </li></ul><ul><li>Solution: </li></ul>
26. 26. Example 5 <ul><li>Problem: Find the ramp response for a system whose transfer function is, </li></ul><ul><li>Solution: </li></ul>