circuit applications of laplace transform
DESCRIPTION
Circuit Applications of Laplace Transform. Chairul Hudaya, ST, M.Sc. Electric Power & Energy Studies (EPES) Department of Electrical Engineering University of Indonesia http://www.ee.ui.ac.id/epes. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/1.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Circuit Applications of Laplace Transform
Electric Power & Energy Studies (EPES)Department of Electrical Engineering
University of Indonesiahttp://www.ee.ui.ac.id/epes
Chairul Hudaya, ST, M.Sc
Depok, October, 2009 Electric Circuit
![Page 2: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/2.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Circuit applications
1. Transfer functions2. Convolution integrals3. RLC circuit with initial conditions
sCC
sLLRR
1
![Page 3: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/3.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Transfer function
h(t) y(t)x(t)
)()()( txthty
)(Input)(Output
)()()(,functionTransfer
ss
sXsYsH
In s-domain, )()()( sXsHsY In time domain,
Network
System
![Page 4: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/4.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Example 1
For the following circuit, find H(s)=Vo(s)/Vi(s). Assume zero initial conditions.
![Page 5: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/5.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Solution
Transform the circuit into s-domain with zero i.c.:
)(sVs )(sVo
s
s10
![Page 6: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/6.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
ss
sso
Vss
Vss
Vs
s
sVs
s
sV
309220
)52)(2(2020
252
2052
20
210//4
10//4
2
Using voltage divider
309220
)()()( 2
sssVsVsH
s
o
![Page 7: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/7.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Example 2
Obtain the transfer function H(s)=Vo(s)/Vi(s), for the following circuit.
![Page 8: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/8.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Solution
Transform the circuit into s-domain (We can assume zero i.c. unless stated in the question)
)(sVs )(sVo
)(sI s2
)(2 sI
s
![Page 9: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/9.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Iss
IsIs
V
IIIV
s
o
9323)3(2
9)2(3
We found that
2939
9329
)()()( 2
ss
s
ss
sVsVsH
s
o
![Page 10: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/10.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Example 3
Use convolution to find vo(t) in the circuit ofFig.(a) when the excitation (input) is thesignal shown in Fig.(b).
![Page 11: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/11.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Solution
Step 1: Transform the circuit into s-domain (assume zero i.c.)
)(sVs )(sVos2
Step 2: Find the TF
)(2)(2
21)/2(
/2)()()( 21
tuethss
ssVsVsH t
s
o
L
![Page 12: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/12.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Step 3: Find vo(t)
dvthtvthtv
sVsHsV
s
t
so
so
)()()()()(
)()()(
0
)(20)1(20
2020
102)(
22
02
0
2
0
)(2
tttt
tttt
t to
eeee
eedee
deetv
For t < 0 0)( tvo
For t > 0
![Page 13: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/13.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Circuit element models
Apart from the transformations
we must model the s-domain equivalents of the circuit elements when there is involving initial condition (i.c.)
Unlike resistor, both inductor and capacitor are able to store energy
sCCsLLRR 1,,
![Page 14: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/14.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Therefore, it is important to consider the initial current of an inductor and the initial voltage of a capacitor
For an inductor :– Taking the Laplace transform on both sides of eqn gives
or
dttdiLtv L
L)()(
)a1.....()0()()()]0()([)( LLLLL LisIsLissILsV
)b1.....()0()()(s
isL
sVsI LLL
![Page 15: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/15.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
)0()()()( LLL LisIsLsV s
isL
sVsI LLL
)0()()(
![Page 16: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/16.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
For a capacitor Taking the Laplace transform on both sides of eqn gives
or
dttdvCti C
C)()(
)a2.....()0(/1
)()]0()([)( CC
CCC CvsCsVvssVCsI
)b2.....()0()(1)(s
vsIsC
sV CCC
![Page 17: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/17.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
)0(/1
)()( CC
C CvsCsVsI
svsI
sCsV C
CC)0()(1)(
![Page 18: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/18.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Example 4
Consider the parallel RLC circuit of the following. Find v(t) and i(t) given that v(0) = 5 V and i(0) = −2 A.
![Page 19: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/19.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Solution
Transform the circuit into s-domain (use the given i.c. to get the equivalents of L and C)
)(sI
)(sVs4
161
s80s4
810
)(sV
![Page 20: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/20.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Then, using nodal analysis
208)96(516
961616
80)208(
1614
80)8(
48
01614
80//10
2
2
sssV
ss
ssVss
sVs
sV
ss
VI
![Page 21: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/21.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Since the denominator cannot be factorized, we may write it as a completion of square:
22222 2)4()2(230
2)4()4(5
4)4()96(5)(
sss
sssV
V)()2sin2302cos5()( 4 tuetttv t
Finding i(t),
sssss
sVI 2
)208()96(25.1
48
2
![Page 22: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/22.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
A)(])2sin375.112cos6(4[)( 4 tuettti t
Using partial fractions,
sssCBs
sA
ssssssI 2
2082
)208()96(25.1)( 22
It can be shown that 75.46,6,6 CBA
Hence,
22222 2)4()2(375.11
2)4()4(64
20875.4664)(
sss
ssss
ssI
![Page 23: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/23.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Example 5
The switch in the following circuit moves from position a to position b at t = 0 second. Compute io(t) for t > 0.
0t
V 42
5
1F 1.0H 625.0
)(tio
a b
![Page 24: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/24.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Solution
The i.c. are not given directly. Hence, at firstwe need to find the i.c. by analyzing the circuitwhen t ≤ 0:
V24 )0(Li
)0(Cv
5
V0)0(,A8.4524)0( CL vi
![Page 25: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/25.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Then, we can analyze the circuit for t > 0 by considering the i.c.
1025.6625.0)10(3
625.03
1//625.03
210
1010
ss
sss
Iss
3)0( LLi
s625.0
s10 1
)(sIo
Let
I
![Page 26: Circuit Applications of Laplace Transform](https://reader034.vdocuments.site/reader034/viewer/2022050809/56815dde550346895dcc058d/html5/thumbnails/26.jpg)
Depok, October, 2009 Laplace Transform Electric Circuit
Using current divider rule, we find that
)8)(2(48
161048
1025.6625.030
1010
1
2
210
10
ssss
ssI
sII
s
so
Using partial fraction we have
28
88)(
sssIo
A)()(8)( 28 tueeti tto