Pertemuan 13 Transformasi - Z.

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Presentation transcript:

Pertemuan 13 Transformasi - Z

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

Z-Transform t 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

Z-Transform Definition

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) 2- Take the Laplace transform of f’(t) 3- Replace by z in F’(s) to get

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

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

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

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

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

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

The constant damping loci s-plane z-plane

The constant frequency loci s-plane z-plane

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

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

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

Example: discrete exponential function k 1 Apply z-transform

Series Reminder

Example: discrete Cosine function

Another approach

Dirac function

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

Delayed pulse train T t t

Complete z-transform Example:exponential function

Terima kasih Terima kasih

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