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7/30/2019 Chap 16 Applications of the Laplace Transform
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Chap 16 Applications of theLaplace Transform
7/30/2019 Chap 16 Applications of the Laplace Transform
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Outline
Introduction Circuit Element Models
Circuit Analysis
Transfer Functions
State Variables
Chap 16 Applications of the Laplace Transform 2
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Chap 16 Applications of the Laplace Transform 3
Introduction
A system is a mathematical model of a physical
process relating the input to the output.
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Chap 16 Applications of the Laplace Transform 4
Circuit Element Models
Steps in Applying the Laplace Transform1. Transformthe circuit from the time domain to
the s-domain.
2. Solvethe transformed circuit using nodalanalysis, mesh analysis, source transformation,
superposition, or any circuit analysis technique
with which we are familiar.
3. Take the inverse transform of the solution andthus obtain the solution in the time domain.
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Resistor Model
Chap 16 Applications of the Laplace Transform 5
( ) ( ) ( ) ( )v t i t R V s RI s
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Inductor Model
Chap 16 Applications of the Laplace Transform 6
( )
( ) ( ) ( ( ) (0 ))( ) (0 )
( ) (0 )) ( )
div t L
dt
V s V s L sV s i
sLI I s i I ss LisL s
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(
( )
( ) ( ) ( ( ) (0 )) () (0 )
( )) (0 )
dvi t C
dt
V s II s v
sCV s Cvs
s C sV s v I sC s
Capacitor Model
Chap 16 Applications of the Laplace Transform 7
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Chap 16 Applications of the Laplace Transform 8
Passive Elements with Zero Initials
Resistor
( ) ( )V s RI s
Capacitor
1( ) ( )V s I s
sC
Inductor
( ) ( )V s sLI s
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Chap 16 Applications of the Laplace Transform 9
Impedance of Elements in s-domain
1 ( )( )
( ) ( )
I sY s
Z s V s
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Chap 16 Applications of the Laplace Transform 10
Example 16.1
Q: Find vo(t) in the circuit, assuming zero initialconditions.
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Chap 16 Applications of the Laplace Transform 11
Example 16.1 (cont.)
1 3
1
F
1
1
)
H
3
(
s
s
sL
C
us
s
t
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Chap 16 Applications of the Laplace Transform 12
Example 16.1 (cont.)
1 2
1 2
2
1 2
For mesh 1:
For mesh 2
1 3 31
3 30 5
1( 5 3)
3
:
I I
s s s
I s Is s
I s s I
2
2 2
3 2
2
2 3 2
1 3 1 31 ( 5 3)
3
3 ( 8 18 )
3
8 18I
s s s
s s I I s s s
s s s I
2 2 2 2
4
3 3 2( )
8 18 2 ( 4) ( 2)
He3
( ) sin n 2 V,ce, 02
t
o
ot
V s sI s s s
v e t t
2 2
2 2
sin( )
cos( )
at
at
e ts a
s ae t
s a
L
L
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Chap 16 Applications of the Laplace Transform 13
Example 16.2
Q: Find vo(t) in the circuit. Assume vo(0)=5 V.
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Chap 16 Applications of the Laplace Transform 14
Example 16.2 (cont.)
10/( 1)2 0.5
10 10 101 1
2.5 ( 2
/2
10 10)
1 10
o o o
o o
o
s V V V
sV sV
V ss
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Chap 16 Applications of the Laplace Transform 15
Example 16.2 (cont.)
151
15
)1(
3525)()2(
101
10)2(
3525)()1(
where
21)2)(1(
3525
)2(251
10
22
11
s
so
s
so
o
o
s
ssVsB
sssVsA
s
B
s
A
ss
sV
sV
s
2(
10 15
Th
) (10 15 ) (
( )1
us
)
2
Vt
o
o
tv t
V ss s
e e u t
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Chap 16 Applications of the Laplace Transform 16
Example 16.3
Q: In the circuit, the switch moves from position a toposition b at t= 0. Find i(t) for t> 0.
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Chap 16 Applications of the Laplace Transform 17
Example 16.3 (cont.)
/
( )( ) 0
/( )
( ) / ( / )
/ /( )
/ ( / )
( ) , 0 where /
o
o
o o o o
o o o
to o
o
VI s R sL LI
sLI V I V L
I sR sL s R sL s R L s s R L
I V R V RI s
s R L s s R L
V Vi t I e t R L
R R
(( )0 )o L
I i
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Chap 16 Applications of the Laplace Transform 18
Example 16.3 (cont.)
0 0
/ /
The final value ( ) / ,/
lim ( ) lim/ /
( ) (1 ), 0
o
o o o
s s
t to
o
i V RsI V L V
sI ss R L s R L R
Vi t I e e t
R
/
/ /
( ) / ( / )
( ) , 0 where /
o o o
to o
o
I V R V R
I s s R L s s R L
V Vi t I e t R L
R R
/
If the initial condition, 0,
( ) (1 ), 0
o
to
I
Vi t e t
R
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Chap 16 Applications of the Laplace Transform 19
Circuit Analysis
Remember, equivalent circuits, withcapacitors and inductors, only exist in the s-
domain; they cannot be transformed back into
the time domain.
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Chap 16 Applications of the Laplace Transform 20
Example 16.4
Q: Consider the circuit shown in (a). Find the value of thevoltage across the capacitor assuming that the value of
vs(t) = 10u(t) V and assume that at t= 0, -1 A flows
through the inductor and +5 V is across the capacitor.
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Chap 16 Applications of the Laplace Transform 21
Example 16.4 (cont.)
1 1 1
1
1
2
1
2
1
10 / 0 [ (0) / ](0)0
10 / 3 5 1/ (0.1 )2 3 1
0.1 3 0.5
where (0) 5 V and (0) 1
( ) (35 30 ) ( )
A
( 3 2) 40 5
40 5 35 30
( 1)( 2) 1 2
So, Vt t
V s V V v si
s s s
s Vs s s
v i
s s V s
sV
v t e e u t
s s s s
KCL at node 1:
1
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Chap 16 Applications of the Laplace Transform 22
Example 16.5: Superposition
Q: For the circuit in Example 16.4 , and the initial
conditions used in Example 16.4, use superposition tofind the value of the capacitor voltage. (i(0)=-1, v(0)=5)
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Chap 16 Applications of the Laplace Transform 23
Example 16.5 (cont.)
2
1
1 1 1
1
2
1
1
10 / 0 00 0
10 / 3 5 1/ (0.1 )
2 30.1 3
( 3 2) 30
30 30 30
( 1)( 2) 1 2
So, ( ) V(30 30 ) ( )t t
V s V V
s s
s V
s s
s s V
v t e e u
Vs s s
t
s
1)
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Chap 16 Applications of the Laplace Transform 24
Example 16.5 (cont.)
2)2 2 2
2
2
2
2
2
2
0 0 0(0)0
10 / 3 5 1/ (0.1 )
2 1
0.1 3
( 3 2) 10
10 10 10
( 1)(
( ) (10 10 ) (
2) 1 2
So, V)t t
V V Vi
s s s
s Vs s
s s V
Vs s s
t e e u
s
v t
i(0)=-1
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Chap 16 Applications of the Laplace Transform 25
Example 16.5 (cont.)
3)
3 3 3
3
2
3
2
3
3( ) (
0 0 00 0
10 / 3 5 1/ (0.1 )
20.1 3 0.5
( 3 2) 5
5 5 10( 1)( 2) 1 2
So, V5 10 ) ( )t t
V V V
s s
s Vs
s s V s
sVs s s s
v t e e u t
v(0)=5
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Chap 16 Applications of the Laplace Transform 26
Example 16.5 (cont.)
1
2
2 3
2
( ) ( ) ( )
{(30 10 5) ( 30 10 10) } ( )
( )
(35 30 V)V
( )
t t
t t
v t v t v t
e
v t
e e uu t
te
Example 16 6:
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Chap 16 Applications of the Laplace Transform 27
Example 16.6:Thevenins Theorem
Q: Assume that there is no initial energy stored in the
circuit shown below at t= 0 and that is = 10 u(t). (a)
Find Vo(s) using Thevenins theorem. (b) Apply the
initial- and finial-value theorems to find vo(0+) and
vo(
). (c) Determine vo(t).
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Chap 16 Applications of the Laplace Transform 28
Example 16.6 (cont.)
(a)
1
oc TH
sc
1 1
1
Since 0,10 50
5
/ 2
( 2 ) 0 0100
5 2
100
2 3
x
x
x
I
s s
I V
V V
I s
V I V
s s
Vs
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Chap 16 Applications of the Laplace Transform 29
Example 16.6 (cont.)
1sc
oc
sc
T
TH
H
Hence,
100 /(2 3) 50
2 2 (2 3)
50/50 /[ (2 3)]
5 5 50 250
5 5 2
2 3
125
(( ) 4)3 2 8THo
V sI
s s s s
V s
I s s
V
Z s s
Z s
V
s ss s
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Chap 16 Applications of the Laplace Transform 30
Example 16.6 (cont.)
(b) Using the initial-value theorem we find
Using the finial-value theorem we find
01
0
/41
/125lim
4
125lim)(lim)0(0
s
s
sssVv
sso
s
V25.314
125
4
125lim)(lim)(
00
sssVv
so
so
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Chap 16 Applications of the Laplace Transform 31
Example 16.6 (cont.)
(c) By partial fraction,
00
47
4
By (a)
125
( ) 31.254
125( 4) ( ) 31.25
31.25 3
125
( 4) 4
( ) 31.
1.25
4
25(1 ( V) )
o s
s
o s
s
o
t
o
o
A
A sV s s
B
BV
s s s
s V ss
V
s
v t e
s
t
s
u
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Chap 16 Applications of the Laplace Transform 32
Transfer Functions
)(
)()(
sX
sYsH
The transfer functionH(s) is the ratio of the output
response Y(s) to the input excitationX(s), assuming all
initial conditions are zero.
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Chap 16 Applications of the Laplace Transform 33
Four Possible Transfer Functions
( )( ) Voltage gain( )
( )( ) Current gain
( )
( )( ) Impedance
( )
( )( ) Admittance
( )
o
i
o
i
V sH sV s
I sH s
I s
V sH s
I s
I sH s
V s
Unit Impulse Response
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Chap 16 Applications of the Laplace Transform 34
Unit Impulse Responsev.s. Transfer Function
( ) ( ) ( )Y s H s X s
1
Let ( ) ( ) ( ) 1
Hence,
( ) ( ) or ( ) ( )
whe (r (e ) [ )]
x t t X s
Y s H s y t h
h t s
t
H
L
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Chap 16 Applications of the Laplace Transform 35
Example 16.7
Q: The output of a linear system isy(t) = 10e-tcos4tu(t)
when input isx(t)=e-tu(t). Find the transfer function of
the system and its impulse response.
Ifx(t)=e-tu(t) andy(t) = 10e-tcos4tu(t), then
172
)12(10
16)1(
)1(10
)(
)()(
Hence,
4)1(
)1(10)(and
1
1)(
2
2
2
2
22
ss
ss
s
s
sX
sYsH
s
ssY
ssX
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Chap 16 Applications of the Laplace Transform 36
Example 16.7 (cont.)
To find h(t),
2
2 2 2
10( 2 1) 4( ) 10 40
2 17 (
( ) 10 ( ) 40 sin
4
(
1)
4 )th t t e t
s sH s
s s s
u t
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Chap 16 Applications of the Laplace Transform 37
Example 16.8
Q: Determine the transfer functionH(s) = Vo(s) /Io(s) ofthe circuit.
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Chap 16 Applications of the Laplace Transform 38
Example 16.8 (cont.)
2
2
2
( 4)
4 2 1/ 2
But
2( 4)2
5 1/ 2
Hence,
( ) 4 ( 4)( )2
12 1( )
o
o
o
o
o
s II
s s
s IV I
s sH ss s
s s
V sI s
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Chap 16 Applications of the Laplace Transform 39
Example 16.9
Q: For the s-domain circuit, find: (a) the transferfunctionH(s) = Vo/Vi, (b) the impulse response, (c)
the response when vi(t) = u(t) V, (d) the response
when vi(t) = 8cos2tV.
E l 6 9 ( )
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Chap 16 Applications of the Laplace Transform 40
Example 16.9 (cont.)
(a)
(b)
1 , but1
1 ( 1) ( 1) / ( 2) 1
1 1 ( 1) 1 ( 1) / ( 2) 2 3
So, . Thus,2
1( )
33 2
o ab
ab i i ab i
i o
o
i
V Vs
s s s sV V V V V
s s s s
V VHV
ss
V s
3 /
32
21( ) ( )
1 1)
2
(2
th t e u
H ss
t
E l 16 9 ( )
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Chap 16 Applications of the Laplace Transform 41
Example 16.9 (cont.)
(c)
3 32 2
0 32 0
3/23/2
( ) ( ), ( ) 1/
2( ) ( ) ( )
2 ( )
where
1 1( )2( ) 3
3 1 1( )
2 2 3
i i
o i
o s
s
o
ss
v t u t V s s
A BV s H s V s
s s s s
A sV ss
B s V ss
3 /2
0 32
Hence, for ( ) ( ),
1 1 1( ) V
1( ) (1 ) ( )
33o
i
t
v t u t
V ss s
v t e u t
3 /
3
2
21( ) ( )
1 1)
2
(2
th t e u
H ss
t
E l 16 9 ( )
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Chap 16 Applications of the Laplace Transform 42
Example 16.9 (cont.)
(d)
25
24
4
4)(
2
3
where
4)4)((
4)()()(
and,4
8)(then,2cos8)(When
2/32
2/3
2
232
23
2
sso
io
ii
s
ssVsA
s
CBs
s
A
ss
ssVsHsV
s
ssVttv
2 2
2 2
sin( )
cos( )
at
at
e ts a
s ae t
s a
L
L
E l 16 9
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Chap 16 Applications of the Laplace Transform 43
Example 16.9
2 23 3
2 2
2 2
4 24 / 25( )
( )( 4) 424 3 3
4 ( 4)25 2 2
o
s Bs C V s
s s s s
s s B s s C s
2
Equating coefficients,
24 3 64
Constants: 0 4 25 2 25
3: 4
2
24 24: 0
25 25
C C
s B C
s B B
3 /
2425
2
2
232
Hence, for ( ) 8cos2 V,
24 3
24 4( ) cos2 sin 2 ( )
2
2 2( )
25 4 25 4
5
V
3
o
o
i
t
v
v t
t t
s
e t t u
V ss s s
t
St t V i bl
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Chap 16 Applications of the Laplace Transform 44
State Variables
A linear system with m inputs andp outputs.
St t V i bl ( t )
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Chap 16 Applications of the Laplace Transform 45
State Variables (cont.)
1
2
( )
( )( ) responsenting state vastate riav bleec s.
( )
tor
n
x t
x tt
x
n
t
x Ax Bz
x
A state variableis a physical property that characterizes
the state of a system, regardless of how the system got to
that state.
St t V i bl ( t )
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Chap 16 Applications of the Laplace Transform 46
State Variables (cont.)
1 1
2 2
( ) ( ) / ( ) ( ) /
( ) ( ) / n n
x t dx t dt
x t dx t dt
x t dx t dt
x
1
2inpu
( )
( )( ) representint vector inpu .tsg
( )n
z t
z t
z
mt
t
z
St t V i bl ( t )
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Chap 16 Applications of the Laplace Transform 47
State Variables (cont.)
x Ax Bz
y Cx Dz
1
2
( )
( )( ) the representing
( )
and are respectively and matrices.
and are respectively and mat
output vector
rices
output
.
s
p
y t
y tt
y t
n n n m
p n
p
p m
y
A B
C D
St t V i bl ( t )
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Chap 16 Applications of the Laplace Transform 48
State Variables (cont.)
1
1
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( )
( )
where system matrix
input coupling matrix
ou
(
t u
(
t
)
( ) )
p
s s s s s s s
s s
s s s
s
s
s
s s
I A B
X AX BZ I A X
H C I A B
BZ
X Z
Y CX DZ
Y
Z
A
B
C
D
matrix
feedforward matrixD
x Ax Bz
y Cx Dz
Steps to Apply the State Variable
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Chap 16 Applications of the Laplace Transform 49
Steps to Apply the State VariableMethod to Circuit Analysis
1. Select the inductor current i and capacitor voltage vas the state variables, making sure they are
consistent with the passive sign convention.
2. Apply KCL and KVL to the circuit and obtain
circuit variables (voltage and currents) in terms ofthe state variables. This should lead to a set of first-
order differential equations necessary and sufficient
to determine all state variables.
3. Obtain the output equation and put the final result
in state-space representation.
E ample 16 10
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Chap 16 Applications of the Laplace Transform 50
Example 16.10
Q: Find the state-space representation of the circuit.Determine the transfer function of the circuit when vs
is the input and ixis the output. TakeR = 1, C=
0.25 F, andL = 0.5 H.
Example 16 10 (cont )
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Chap 16 Applications of the Laplace Transform 51
Example 16.10 (cont.)
Applying KCL at node 1:
Applying KVL around the outer loop:
,
L
C
x C
s L s
s
div L
dtdv
i Cdt
dv V
i i i C idt R
div v v L v
v iv
RC C
vv
L L
vdt
i
1 1
11
0
01
0
RC C
s
LL
x
v vv
i iv
iiR
Example 16 10 (cont )
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Chap 16 Applications of the Laplace Transform 52
Example 16.10 (cont.)
1 14 2
1 1
11
If 1, , and
04 4 0, ,
0 2 0 2
10 1 0
0 4 4 4 40 2 0 2
RC C
LL
R C L
R
s sss s
A B
C
I A
x Ax Bz
y Cx Dz1
1
( )
( ) ( )
( ) ( )
( )
( )
s
s s
s s
s
s
I A B
H C I A
X Z
ZB
YD
Example 16 10 (cont )
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Chap 16 Applications of the Laplace Transform 53
Example 16.10 (cont.)
Taking the inverse of this gives
1
2
2
1
2 2
4
2 4adjoint of( )
determinant of 4 8
( )
4 0 8
1 0 1 02 4 2 2 8
( )
8
4 8 4 8 4 8
s
ss
s s
s
s
s
s
s s
s s s s s
AI A
A
C I A BH
x Ax Bz
y Cx Dz1
1
( )
( ) ( )
( ) ( )
( )
( )
s
s s
s s
s
s
I A B
H C I A
X Z
ZB
YD
Example 16 11
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Chap 16 Applications of the Laplace Transform 54
Example 16.11
Q: Consider the circuit, which may be regarded as atwo-input, two-output. Determine the state variable
model and find the transfer function of the system.
Example 16 11 (cont )
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Chap 16 Applications of the Laplace Transform 55
Example 16.11 (cont.)
1 1
1
K1
0VL 6 6
6
:s s
s o
v i i i v i
v i v v
01 0 1
12 4
2( )
22
3 23
KCL at node 1
4
:
s
s s
vi i v i i
i vi v ii v
vv i v i
1 3 32
KCL
3 2 3
2 32 ( )
3 2
at node 2:
o io o o o
s
o s
v v vv i v v i i
i v vv i v i v
State variables: andv i
2s i
v v i v v
Example 16 11 (cont )
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Chap 16 Applications of the Laplace Transform 56
Example 16.11 (cont.)
2 1 1 1
2 4 4 0
s
i
vv v
vi i
2 4 4
2
s
s i
i v i v
v v i v v
3
3( )
2
i
o
o s
v vi
v v i v
32 23 3 2
1 13 3
0
0 0
o s
o i
v vv
i vi
Example 16 12
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Chap 16 Applications of the Laplace Transform 57
Example 16.12
Q: Assume we have a system where the output isy(t)and the input isz(t). Let the following differential
equation describe the relationship between the input
and output
Obtain the state model and the transfer function of the
system.
2
2( ) ( )3 2 ( ) 5 ( )d y t dy t y t z t
dt dt
Example 16 12 (cont )
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Chap 16 Applications of the Laplace Transform 58
Example 16.12 (cont.)
1
2 1
2
1 1
2 2
1
2
1 2
( ), ( )Two state variables:
( )
0 1 0( )
2 3 5
(
( ) 2 ( ) 3
) 1 0
1 0
(
0 1
0 1 2
) 5 ( ) )
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Example 16 12 (cont )
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Chap 16 Applications of the Laplace Transform 59
Example 16.12 (cont.)
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Example 16 12 (cont )
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Chap 16 Applications of the Laplace Transform 60
Example 16.12 (cont.)
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HW Ch16 (Due Day: 8/24/2012)
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HW Ch16 (Due Day: 8/24/2012)