ch4_continuous-time system analysis using the laplace transform

7/24/2019 Ch4_Continuous-Time System Analysis Using the Laplace Transform

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Continuous-Time System Analysis

Using The Laplace Transform

Dr. Mohamed Bingabr

Uniersity of Central !"lahomaSlides #or Lathi$s Te%tboo" &roided by Dr. &eter Cheung


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Outline

'ntroduction &roperties of Laplace Transform Solution of Differential ()uations Analysis of (lectrical *et+or"s

Bloc" Diagrams and System ,ealiation #re)uency ,esponse of an LT'C System #ilter Design by &lacement of &oles and eros of /0s1

The materials in these slides are covered in the

Lathi Te%tboo" all Ch 2 e%cept sections 2.23 2.43 2.53 2.66


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x(t) = 3e-5t

22)5(

3)(

5

3)(

++

=

+=

sX

ssX ts

i

iiesXtx

=

)()(

780s17

LT

Sigma

!mega


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u= e-st dv= dx


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HW5_Ch4: 4.1-1 (a, b, c, d), 4.1-3 (a, b, c, d, ), 4.!-1 (a, b, e, "), 4.!-3

(a,c), 4.!-#


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Where is H(s)$


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(%ample

'n the circuit3 the s+itch is in the closed position for a long time

before t9:3 +hen it is opened instantaneously. #ind the inductorcurrent y0t1 for t :.

10 V

2

t=0

5

1 H

0.2 F

y(t)

x(t)

=++t

tudyC

tRydt

dyL )(10)(

1)(

ss

dy

s

sYsYyssY

10)(5

)(5)(2)0()(

0

=+++

Ay 2510)0( == 2)0()(

0

===

CVqdy c

sss

sYsYyssY 1010)(5)(2)0()( =+++

)()6.262cos(5)( tutety ot +=


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(%ample

#ind the response y0t1 of an LT'C system described by the

e)uation

if the input %0t1 9 ;e-


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' t l St bilit


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'nternal Stability

'nternal Stability 0Asymptotic1

'f and only if all the poles are in the L/& Unstable if3 and only if3 one or both of the

follo+ing conditions e%ist>

At least one pole is in the ,/& There are repeated poles on the imaginary a%is

Marginally stable if3 and only if3 there are no

poles in the ,/&3 and there are some

unrepeated poles on the imaginary a%is.

( t l St bilit B'B!


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(%ternal Stability B'B!

The transfer function /0s1 can only indicate the e%ternal

stability of the system B'B!.

N

NN

M

MM

asas

bsbsbsH

++++++

=

...

...)(

1

1

1

10

'amle

's the system belo+ B'B! and asymptotically 0internally1 stable?

1

1

S 11

+

S

S%0t1 y0t1

B'B! stable if M*and all poles are in the L/&

Bl " Di


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Bloc" Diagrams

S t , li ti


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System ,ealiation

N

NN

M

MM

asas

bsbsbsH

++++++

=

...

...)(

1

1

1

10

,ealiation is a synthesis problem3 so there is no uni)ue +ay

of realiing a system.

A common realiation of /0s1 is using

'ntegrator Scalar multiplier Adders

Di t # ' , li ti


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Direct #orm ' ,ealiation

32

2

1

3

32

2

1

3

0

)( asasas

bsbsbsbsH +++

+++=

Divide every term y s!it"

t"e "i#"est order s3

33

221

3

3

2

210

1

)(

sa

sa

sa

s

b

s

b

s

bb

sH

+++

+++=

+++ +++=

3

3

2

2133

2210

1

1)(

s

a

s

a

s

as

b

s

b

s

bbsH

/60s1 /@0s180s1 0s1 0s1

Di t # ' , li ti


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Direct #orm ' ,ealiation

/60s1 /@0s180s1 0s1 0s1

+++

+++=

3

3

2

213

3

2

210

1

1)(

s

a

s

a

s

as

b

s

b

s

bbsH

Di t # '' , li ti


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Direct #orm '' ,ealiation

+++

+++=

3

3

2

210

3

3

2

211

1)(

s

b

s

b

s

bb

s

a

s

a

s

asH

/@0s1 /60s180s1 0s1 0s1

( l


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(%ample

#ind the canonic direct form realiation of the follo+ing transfer

functions>

56

2$%d)

&

5c)

&)

&

5')

2++

+

+

+

+

+

ss

s

s

s

s

s

s

Cascade and &arallel ,ealiations


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Cascade and &arallel ,ealiations

56

2$%

)( 2 ++

+

= ss

s

sH

Cascade *eali+ation

+

++

=+++

= 51

1

2$%

)5)(1(

2$%

)( ss

s

ss

s

sH

arallel *eali+ation

5

2

1

6

)5)(1(

2$%)(+

+

=++

+=ssss

ssH

The comple% poles in /0s1 should be realied as a second-

order system.

Using !perational Amplifier for


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Using !perational Amplifier for

System ,ealiation

(%ample


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(%ample

Use !p-Amp circuits to realie the canonic direct form of the

transfer function

10%

52)(

2 +++

=ss

ssH

HW#_Ch4: 4.3-1 (b,c), 4.3-! (b,c), 4.3-4, 4.3-, 4.3-1, 4.4-1, 4.5-!, 4.#-

ch4_continuous-time system analysis using the laplace transform

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