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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). - PowerPoint PPT Presentation
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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