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Pertemuan 13

Pertemuan 13. Transformasi - Z. Y(s)  y(t). U(s)  u(t). G(s). Linear system.  T. t. t. t. X. Z-Transform. Introduction. Tools to analyse continuous systems : Laplace transform It could be used for sampled or discrete systems. t. Z-Transform. Apply Laplace transform of f’(t).

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Pertemuan 13

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  1. Pertemuan 13 Transformasi - Z

  2. Y(s)  y(t) U(s)  u(t) G(s) Linear system T t t t X Z-Transform Introduction Tools to analyse continuous systems : Laplace transform It could be used for sampled or discrete systems

  3. t Z-Transform Apply Laplace transform of f’(t) Factors like Exp(-sT) are involved Unlike the majority of transfer functions of continuous systems It will not lead to rational functions

  4. Z-Transform Definition

  5. 2- Take the Laplace transform of f’(t) 3- Replace by z in F’(s) to get Summary The operation of taking the z-transform of a continuous-data function, f(t), involves the following three steps: 1- f(t) is sampled by an ideal sampler to get f’(t)

  6. 1 Mapping between the s-plane and the z-plane S-plane Primary strip Imz z-plane Rez The left half of the primary strip is mapped inside the unit circle

  7. Mapping between the s-plane and the z-plane S-plane Primary strip Imz Z-plane Rez 1 The right half of the primary strip is mapped outside the unit circle

  8. Mapping between the s-plane and the z-plane S-plane Complementary strip Imz Z-plane Rez 1 The right half of the complementary strip is also mapped inside the unit circle

  9. s-plane properties of F’(s) Complementary strip Complementary strip Primary strip Complementary strip Complementary strip

  10. s-plane properties of F’(s) Complementary strip X Complementary strip X Primary strip X Complementary strip X Complementary strip X X Poles of F’(s) in primary strip

  11. s-plane properties of F’(s) Complementary strip X Folded back poles Complementary strip X Primary strip X Complementary strip X Complementary strip X X Poles of F’(s) in complementary strips

  12. The constant damping loci s-plane z-plane

  13. The constant frequency loci s-plane z-plane

  14. The constant damping ratio loci Imz Rez s-plane z-plane

  15. The constant damping ratio loci Imz Rez s-plane z-plane

  16. Mapping between the s-plane and the z-plane Conclusion: All points in the left half of the s-plane are mapped into the Region inside the unit circle in the z-plane. The points in the right half of the s-plane are mapped into the Region outside the unit circle in the z-plane

  17. 1 k Example: discrete exponential function Apply z-transform

  18. Series Reminder

  19. Example: discrete Cosine function

  20. Another approach

  21. Dirac function

  22. u(t) 1 t Sampled step function NB: Equivalent to Exp(-k) as  0

  23. Delayed pulse train T t t

  24. Complete z-transform Example:exponential function

  25. Terima kasih

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