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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.#-