Pertemuan Pertemuan 1313Transformasi - ZTransformasi - Z

Z-TransformIntroduction

G(s)

Linear system

Y(s) y(t) U(s) u(t)

Tools to analyse continuous systems : Laplace transformIt could be used for sampled or discrete systems

t

)t(f

t

)t('f

X

t

)t(f )t('f

T

Factors like Exp(-sT) are involvedUnlike the majority of transfer functions of continuous systemsIt will not lead to rational functions

Z-Transform

0k

Tsk

0

st

0k

e)Tk(f2

)0(fdte)Tkt()t(f)]t('f[L

Apply Laplace transform of f’(t)

0k0k

)Tk(f)Tkt()t(f)t('f

t)t('f

Tsez

Definition

0k

k

0k

k z)k(fz)kT(f)z(F

)k('F)0(f2

1)s(F'

Z-Transform

)Tsin(e)zIm(

)Tcos(e)zRe(

j)zln(T

1s

T

T

)]zln(T

1s)[s('F)z(F

Summary

The operation of taking the z-transform of a continuous-datafunction, 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)

0k

Tkse)Tk(f)s('F

3- Replace by z in F’(s) to get Tse

0k

kz)Tk(f)z(F

Mapping between the s-plane and the z-plane

Tsez S-plane

z-plane

T

2s

2s

2s

Primary strip

j

Rez

Imz

1

23

4 5

The left half of the primary strip is mapped inside the unit circle

12

5

3

4 1

Mapping between the s-plane and the z-plane

Tsez S-plane

Z-plane

2s

2s

Primary strip

j

1

2 3

45

1 Rez

Imz

2

5

3

4 1

The right half of the primary strip is mapped outside the unit circle

Mapping between the s-plane and the z-plane

S-plane

Z-plane

)k2

1(s

)k2

1(s

Complementary strip

j

Rez

Imz

1

The right half of the complementary strip is also mapped inside the unit circle

Tskj2TsTjkTsT)jks( eeeeee ss

s-plane properties of F’(s)

Primary strip

j

2/s

2/s

2/3 s

2/5 s

2/3 s

2/5 s)s('F)jms('F s

0s

s0 js

s0 2js

s0 2js

s0 js

Complementary strip

Complementary strip

Complementary strip

Complementary strip

s-plane properties of F’(s)

Primary strip

j

2/s

2/s

2/3 s

2/5 s

2/3 s

2/5 s

0s

s0 js

s0 2js

s0 2js

s0 js

Complementary strip

Complementary strip

Complementary strip

Complementary strip

X

X

X Poles of F’(s) in primary strip

X

X

X

s-plane properties of F’(s)

Primary strip

j

2/s

2/s

2/3 s

2/5 s

2/3 s

2/5 s

0s

s0 js

s0 2js

s0 2js

s0 js

Complementary strip

Complementary strip

Complementary strip

Complementary strip

X

X

X Poles of F’(s) in complementary strips

X

X

X

Folded back poles

The constant damping loci

s-plane z-plane

1

2

j

TjTeez 1

TjTeez 2

The constant frequency loci

s-plane z-plane

1j

1j

jTj 1ez

Tj 1ez

2jT1

T2

The constant damping ratio loci

s-plane z-plane

j

2j s

2s

1

2

4 5

125

3

34

Rez

Imz

The constant damping ratio loci

s-plane z-plane

j

2j s

2s

Rez

Imz

jtans

4s

2s

4

3 s s

Mapping between the s-plane and the z-plane

Conclusion:

All points in the left half of the s-plane are mapped into theRegion inside the unit circle in the z-plane.

The points in the right half of the s-plane are mapped into theRegion outside the unit circle in the z-plane

Example: discrete exponential function

0k

k** z)k(f)z(F

0k

k1

0k

kk* )ze(ze)z(F

ez

z

ze1

1)k(F

1*

k* e)k(f

0k,0)k(f *

0

k

1

Apply z-transform

Series

n32

0n

nn y........yyy1ys

n32n y........yyy1s

)1n(n32n yy........yyys.y

)1n(nnn y1)y1(ss.ys

y1

1

y1

y1s

)1n(

n

Reminder

2

ee)kcos()k(f

kjkj*

)ez

z

ez

z(

2

1)z(F

jj*

)1)ee(zz

)ee(z2(

2

z)z(F

jj2

jj*

1cosz2z

)cosz(z)z(F

2*

Example: discrete Cosine function

ez

z]e[Z k

])ez)(ez(

)ez()ez([

2

z)z(F

jj

jj*

Another approach

jke)ksin(j)kcos()k(y

sinjcosz

z

ez

z)z(Y

j

)sinjcosz)(sinjcosz(

)sinjcosz(z)z(Y

1cosz2z

)sinjz)cosz(z)z(Y

2

1cosz2z

)cosz(z[cos]Z

2

1cosz2z

)sin(z[sin]Z

2

Dirac function

1)]t([F

)t(

1)]t([Z

1)0(z)t()]t([F0k

Ts

Sampled step function

t

u(t)

T T2 T3 T4 T5

1

0

1z

z

z1

1)z(U

e1

1e)s(U

1

Ts0k

Tsk

1z

z

z1

1z)z(U

e)s(U

10k

k

0k

Tsk

NB: Equivalent to Exp(-k) as 0

t

k

T )Tkt(

t

T

T

0k

Tske

Ts'e e)T)k[(xe),s(X

k

e'e ]T)k(t[]T)k[(xx

Delayed pulse train

Complete z-transform

0k

k

0k

k z),k(fz]T)k[(f),z(F

Example:exponential function

0,e),k(f )k(

e

ez

zzeeze),z(F

0k

kk

0k

k)k(

eez

z),z(F

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