Lec.5 OpenLoopSys

7/28/2019 Lec.5 OpenLoopSys

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o e a e o o a n rans er unc on o

open loop systemsTo be able to solve difference equations

CISE318_Topic 4 2(c) Khoukhi 2010


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For



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Sam lin and Reconstruction

Input)(teo samper

outputOf zero

orderhold



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sG)(te )( kTe

The Output signal of an ideal sampler is the signal whose Laplace transform is

0

( ) ( ) n s

n

E s e nT e



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The star transform is so named because it is frequently

represented by an asterisk or "star" in the notation

The star transform, is a discrete-time variation of

.

e apace rans orm a represen s anideal sampler with period of time T.

Similar to Z-Transformwith a simple change of variables,

but the star transform ex licitl identifies each sam le interms of the sampling period (T), while the Z transformonly refers to each sample by integer index value.

CISE318_Topic 4 (c) Khoukhi 2010 6


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Relationshi between X* s and X z

X s X* s

( ) ( ) stX s x t e dt

The function called the star transform is defined as:*( )X s

0k

s x e

CISE318_Topic 4 7( ) *( ) sTe zX z X s (c) Khoukhi 2010


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X s X* s

Star transform is related to Laplace Transformby taking

1)(sX (3-10) from Textbook

andA endix 3

the residues of the Laplace transform of a function

1)(Residues)(

1 TXofzX

)(1111

)( 2

2

11

2,1

TT

TT

TT

eez

zX

)()(*

2

21

TTsTeee

sX

CISE318_Topic 4 8

eeee (c) Khoukhi 2010


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polesall

kzzEofresidueskx

k

1)()(

1

polessimplefor

azaz

mtymultipliciwithpolesfor

zzazzzores ues

az

km

m

m

az

k zzEaz

dz

d

m

zzEofresidues

11

11 )()(

)!1(

1)(



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X s X* s

-The change of variable Tsz e

)2)(1()( sssX

)()(

2

2

TT

TTeez

zX

TsezsXzX

)(*)(

)(*

2TTsTeee

sX

CISE318_Topic 4 10))((

ss

eeee

(c) Khoukhi 2010


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Example



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Example



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G(s)

X(s) X*(s) Y(s)p

*****

)(*)()( sXsGsY

)()()( zXzGzY

The pulse transfer function is the transfer function between thesampled input and the output at the sampling instants

G(s)



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X(s) X*(s) Y(s)A(s)

1 2

zXzGGzY



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GGGG

Tsez

sGzGsGsGsG )(*)()()()( 11123

TsTsezez

sGzGsGzGG

)(*)()(*)( 22312



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X(s) X*(s) Y(s)A(s)

1 2

12 zzzz



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Y(s)A(s)B(s)X(s)

2 31

)()()()()( 123 zXzGzGzGzY



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Y(s)A(s)B(s)X(s)

2 31

zXzGzGGzY



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X(s) Y(s)A(s)

1 2

zXGzGzY We can not write a transfer function for



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Y(s)A(s)B(s)X(s)

2 31

zXGzGGzY



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ore xamp es



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More Examples



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Check the course webCT for the Homework

Assignment


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