applications of the laplace transform
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APPLICATIONS OF THE LAPLACE
TRANSFORM
Chapter 16
Applications of LT
•ch16 Applications of the Laplace
Transform
•2
Introduction
Circuit Element Models
Circuit Analysis
Transfer Functions
State Variables
16.1 Introduction
•ch16 Applications of the Laplace
Transform
•3
A system is a mathematical model of a physical
process relating the input to the output.
16.2 Circuit Element Models
•ch16 Applications of the Laplace
Transform
•4
Steps in Applying the Laplace Transform:
1. Transform the circuit from the time domain to the s-
domain.
2. Solve the circuit using nodal analysis, mesh analysis,
source transformation, superposition, or any circuit
analysis technique.
3. Take the inverse transform of the solution and thus
obtain the solution in the time domain.
Current – voltage relationship
•ch16 Applications of the Laplace
Transform
•5
s
vsI
sCsV
s
isV
sLsI
sRIsV
)0()(
1)(
)0()(
1)(
)()(
Resistor
•ch16 Applications of the Laplace
Transform
•6
)()( )()(
:Resistor
sRIsVRtitv
Induktor
•ch16 Applications of the Laplace
Transform
•7
s
i
SL
sVsI
LissLIsV
issILsVdt
diLtv
)0()()(
)0()()(
))0()(()( )(
:Inductors
Inductor
•ch16 Applications of the Laplace
Transform
•8
Capacitor
•ch16 Applications of the Laplace
Transform
•9
s
v
SC
sIsV
CvssCVsI
vssVCsIdt
dvCti
)0()()(
)0()()(
))0()(()( )(
:Capacitors
Capacitor
•ch16 Applications of the Laplace
Transform
•10
Circuit with Zero Initials
•ch16 Applications of the Laplace
Transform
•11
sCsZ
sLsZ
RsZ
sI
sVsZ
1)( :Capacitor
)( :Inductor
)( :Resistor
)(
)()(
•ch16 Applications of the Laplace
Transform
•12
)()]([
)()]([
)(
)(
)(
1)(
saItai
saVtav
sV
sI
sZsY
L
L
Example 16.1
•ch16 Applications of the Laplace
Transform
•13
Find vo(t) in the circuit of Fig. 16.4, assuming zero
initial conditions.
Example 16.1
•ch16 Applications of the Laplace
Transform
•14
ssC
ssLs
tu
31F
3
1H 1
1)(
Example 16.1
•ch16 Applications of the Laplace
Transform
•15
sssI
Isss
Is
Issss
IssIIs
sIs
Is
Iss
188
3
)188(3
3)35(
3
131
1
)35(3
135
30 2,mesh For
331
1 1,mesh For
232
223
222
22
121
21
Example 16.1
•ch16 Applications of the Laplace
Transform
•16
0 V,2sin2
3)(
)2()4(
2
2
3
188
3)(
4
2222
ttetv
ssssIsV
to
o
Example 16.2
•ch16 Applications of the Laplace
Transform
•17
Find vo(t) in the circuit of Fig. 16.7. Assume vo(0)=5 V.
Example 16.2
•ch16 Applications of the Laplace
Transform
•18
)2(10
1
1010
25.2
1
1
/10105.02
10
1)10/(
sVsVV
s
s
VVVs
ooo
ooo
Example 16.2
•ch16 Applications of the Laplace
Transform
•19
151
15
)1(
3525)()2(
101
10
)2(
3525)()1(
where
21)2)(1(
3525
)2(251
10
22
11
sso
sso
o
o
s
ssVsB
s
ssVsA
s
B
s
A
ss
sV
sVs
Example 16.2
•ch16 Applications of the Laplace
Transform
•20
V )()1510()(
2
15
1
10)(
Thus
2 tueetv
sssV
tto
o
Example 16.3
•ch16 Applications of the Laplace
Transform
•21
In the circuit of Fig. 16.10(a),
the switch moves from position
a to position b at t = 0. Find i(t)
for t > 0.
Example 16.3
•ch16 Applications of the Laplace
Transform
•22
LRtR
Ve
R
VIti
LRs
RV
s
RV
LRs
IsI
LRss
LV
LRs
I
sLRs
V
sLR
LIsI
s
VLIsLRsI
otoo
ooo
oooo
oo
/ where0 ,)(
)/(
//
/)(
)/(
/
/)()(
0))((
/
Example 16.3
•ch16 Applications of the Laplace
Transform
•23
0 ),1()(
,0 condition, initial In the
0 ),1()(
/
/
/lim)(lim
,/)( valueinitial The
/
//
00
teR
Vti
I
teR
VeIti
R
V
LRs
LV
LRs
sIssI
RVi
to
o
toto
ooo
ss
o
16.3 Circuit Analysis
•ch16 Applications of the Laplace
Transform
•24
Remember, equivalent circuits, with capacitors and
inductors, only exist in the s-domain; they cannot be
transformed back into the time domain.
Example 16.4
•ch16 Applications of the Laplace
Transform
•25
Consider the circuit in Fig.
16.12(a). Find the value
of the voltage 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.
Example 16.4
•ch16 Applications of the Laplace
Transform
•26
V )()3035()( So,
2
30
1
35
)2)(1(
540
540)23(
A 1)0( and V 5)0( e wher
5.0132
31.0
0)1.0/(1
]/)0([)0(
5
0
3/10
/10
21
1
12
1
111
tueetv
ssss
sV
sVss
iv
ssV
ss
s
svV
s
i
s
VsV
tt
Example 16.5
•ch16 Applications of the Laplace
Transform
•27
For the circuit in Fig. 16.12,
with the initial conditions used
in Example 16.4, use
superposition to find the
value of the capacitor
voltage.
Example 16.5
•ch16 Applications of the Laplace
Transform
•28
For Fig. 16.13(a), we get
V )()3003()( So,
2
30
1
30
)2)(1(
30
30)23(
3231.0
0)1.0/(1
00
5
0
3/10
/10
21
1
12
1
111
tueetv
ssssV
Vss
sV
ss
s
V
s
VsV
tt
Example 16.5
•ch16 Applications of the Laplace
Transform
•29
For Fig. 16.13(b), we get
V )()0110()( So,
2
10
1
10
)2)(1(
10
10)23(
1231.0
0)1.0/(1
01
5
0
3/10
0
22
2
22
2
222
tueetv
ssssV
Vss
sV
ss
s
V
ss
VV
tt
Example 16.5
•ch16 Applications of the Laplace
Transform
•30
For Fig. 16.13(c), we get
V )()015()( So,
2
10
1
5
)2)(1(
5
5)23(
0.52
31.0
0)1.0/(1
00
5
0
3/10
0
23
3
32
3
333
tueetv
ssss
sV
sVss
Vs
s
s
V
s
VV
tt
Example 16.5
•ch16 Applications of the Laplace
Transform
•31
V )()3035(
V )(})101030()51030{(
)()()()(
2
2
321
tuee
tuee
tvtvtvtv
tt
tt
Example 16.6
•ch16 Applications of the Laplace
Transform
•32
Assume that there is no initial energy stored in the
circuit of Fig. 16.14 at t = 0 and that is = 10 u(t). (a)
Find vo(s) using Thevenin’s theorem. (b) Apply the
initial- and final-value theorems to find vo(0+) and
vo(). (c) Determine vo(t).
Example 16.6
•ch16 Applications of the Laplace
Transform
•33
(a)
32
100
02
0
5
0)2(10
2/
50105
,0 Since
1
11
1sc
THoc
sV
s
VIV
s
sVII
ssVV
I
x
x
x
Example 16.6
•ch16 Applications of the Laplace
Transform
•34
)4(
125
)82(
25050
325
5
5
5
32)]32(/[50
/50
)32(
50
2
)32/(100
2
Hence,
TH
sc
ocTH
1sc
ssssssV
ZV
sss
s
I
VZ
sss
s
s
VI
THo
•ch16 Applications of the Laplace
Transform
•35
(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
V 25.314
125
4
125lim)(lim)(
00
sssVv
so
so
Example 16.6
•ch16 Applications of the Laplace
Transform
•36
Example 16.6
•ch16 Applications of the Laplace
Transform
•37
(c) By partial fraction,
V )()1(25.31)(
4
25.3125.31
25.31125
)()4(
25.314
125)(
4)4(
125
4
74
00
tuetv
ssV
ssVsB
sssVA
s
B
s
A
ssV
to
o
sso
sso
o
16.4 Transfer Functions
•ch16 Applications of the Laplace
Transform
•38
The transfer function H(s) is the ratio of the output
response Y(s) to the input excitation X(s), assuming
all initial conditions are zero.
)(
)()(
sX
sYsH
•ch16 Applications of the Laplace
Transform
•39
)(
)(Admittance)(
)(
)(Impedance)(
)(
)(gainCurrent )(
)(
)(gain Voltage)(
sV
sIsH
sI
sVsH
sI
sIsH
sV
sVsH
i
o
i
o
•ch16 Applications of the Laplace
Transform
•40
)]([)( where
)()(or )()(
1)( that so ),()(
)()()(
1 sHth
thtysHsY
sXttx
sXsHsY
L
Example 16.7 •41
The output of a linear system is y(t) = 10e-t cos4t u(t)
when input is x(t)=e-tu(t). Find the transfer function of
the system and its impulse response.
Solution:
If x(t)=e-tu(t) and y(t) = 10e-t cos4t u(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
Example 16.7
•ch16 Applications of the Laplace
Transform
•42
To find h(t),
)(4sin40)(10)(
4)1(
44010)(
22
ttuetth
ssH
t
Example 16.8
•ch16 Applications of the Laplace
Transform
•43
Determine the transfer function H(s) = Vo(s) / Io(s) of
the circuit in Fig.3
Example 16.8
•ch16 Applications of the Laplace
Transform
•44
1122
)4(4
)(
)()(
Hence,
2/15
)4(22
But
2/124
)4(
2
2
2
ss
ss
sI
sVsH
ss
IsIV
ss
IsI
o
o
oo
o
Example 16.9
•ch16 Applications of the Laplace
Transform
•45
For the s-domain circuit in Fig. 16.19, find: (a) the
transfer function H(s) = Vo/Vi, (b) the impulse
response, (c) the response when vi(t) = u(t) , (d) the
response when vi(t) = 8cos2t V.
Example 16.9
•ch16 Applications of the Laplace
Transform
•46
(a)
(b)
32
1)( Thus, .
32 So,
32
1
)2/()1(1
)2/()1(
)1(11
)1(1
but ,1
1
sV
VsH
s
VV
Vs
sVV
ss
ssV
s
sV
Vs
V
i
oio
iabiiab
abo
)(2
1)(
1
2
1)(
2/3
23
tuesh
ssH
t
Example 16.9
•ch16 Applications of the Laplace
Transform
•47
(c)
3
1
2
1)(
2
3
3
1
)(2
1)(
where
)(2
2)()()(
/1)( ),()(
2/32/3
0230
23
23
ss
o
s
so
io
ii
ssVsB
sssVA
s
B
s
A
sssVsHsV
ssVtutv
Example 16.9
•ch16 Applications of the Laplace
Transform
•48
(d)
V )()1(3
1)(
11
3
1)(
),()(for Hence,
2/3
230 tuetv
sssV
tutv
to
i
25
24
4
4)(
2
3
where
4)4)((
4)()()(
and ,4
8)( then ,2cos8)(When
2/32
2/3
2
232
23
2
ss
o
io
ii
s
ssVsA
s
CBs
s
A
ss
ssVsHsV
s
ssVttv
Example 16.9
•ch16 Applications of the Laplace
Transform
•49
2
3
2
3)4(4 22 sCssBsAs
ABBAs
CBs
ACCA
0:2
34:
3
8
2
340:Constants
ts,coefficien Equating
2
Example 16.9
•ch16 Applications of the Laplace
Transform
•50
V )(2sin3
42cos
25
24)(
4
2
25
32
425
24)(
V, 2cos8)(for Hence,
64/25.24/25, ,25/24 gives theseSolving
2/3
22
23
2524
tuttetv
ss
s
ssV
ttv
CBA
to
o
i
16.5 State Variables
•ch16 Applications of the Laplace
Transform
•51
•ch16 Applications of the Laplace
Transform
•52
A state variable is a physical property that
characterizes the state of a system, regardless of
how the system got to that state.
vectors.state ingresponsent vector state
)(
)(
)(
)(
where
2
1
n
tx
tx
tx
t
n
x
BzAxx
•ch16 Applications of the Laplace
Transform
•53
input. ngrepresentior input vect
)(
)(
)(
)(
and
)(
)(
)(
2
1
2
1
m
tz
tz
tz
t
tx
tx
tx
n
n
z
x
•ch16 Applications of the Laplace
Transform
•54
DzCxy
BzAxx
matrices. and ly respective are and
matrices. andly respective are and
outputs ngrepresentitor output vec the
)(
)(
)(
)(
where
2
1
m pnp
mnn n
p
ty
ty
ty
ty
p
DC
BA
•ch16 Applications of the Laplace
Transform
•55
matrix dfeedforwar
matrixoutput
matrix couplinginput
matrix system re whe
)()(
)()(
)()()(
matrix.identity theis e wher
)()()(
)()()( )()()(
1
1
D
C
B
A
DBAICZ
YH
DZCXY
I
BZAIX
BZXAIBZAXX
ss
ss
sss
sss
sssssss
Steps to Apply the State Variable Method to
Circuit Analysis:
•ch16 Applications of the Laplace
Transform
•56
1. Select the inductor current i and capacitor voltage v as the state variables, making sure they are consistent with the passive sign convention.
2. Applying KCL and KVL to the circuit and obtain circuit variables (voltage and currents) in terms of the 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.
Example 16.10
•ch16 Applications of the Laplace
Transform
•57
Find the state-space representation of the circuit in
Fig. 16.22. Determine the transfer function of the
circuit when vs is the input and ix is the output. Take R
= 1Ω, C = 0.25 F, and L = 0.5 H.
Example 16.10
•ch16 Applications of the Laplace
Transform
•58
C
i
RC
vv
R
Vi
dt
dvCiii
dt
dvCi
dt
diLv
Cx
C
L
gives 1 nodeat KCL Applying
i
v
Ri
vi
v
i
v
R
vi
L
v
L
vi
vvdt
diLvvv
x
s
LL
CRC
s
sLs
01
0
0
,
11
11
Example 16.10
•ch16 Applications of the Laplace
Transform
•59
s
s
s
ss
R
LCR
LL
CRC
2
44
02
44
0
0
0101
,2
00 ,
02
44
0
obtain we, and , ,1 If
11
11
21
41
AI
C
B A
Example 16.10
•ch16 Applications of the Laplace
Transform
•60
Taking the inverse of this gives
84
8
84
82
801
84
2
0
42
401
)()(
84
42
4
)-(sI oft determinan
)-(sI ofadjoint )(
222
1
2
1
ssss
s
ss
s
s
ss
ss
s
s
s
BAICH
A
AAI
Example 16.11
•ch16 Applications of the Laplace
Transform
•61
Consider the circuit in Fig. 16.24, which may be
regarded as a two-input, two-output. Determine the
state variable model and find the transfer function of
the system.
•ch16 Applications of the Laplace
Transform
•62
KVL di loop kiri
Karena i1 harus dibuang,
dengan KVL pada loop vs,
R1Ω, R2Ω, dan C 1/3 F,
1
1
66
06
1
ivi
iiv
s
s
vviv os 1
(1)
(2)
s
ss
vivi
vviiiviv
iivv
ii
442
3
223
)(22
11
100
1
KCL di node 1
Substitusikan ke (2), selanjutnya ke (1):
(3)
(4)
(5)
Example 16.11
•ch16 Applications of the Laplace
Transform
•63
Di node 2,
Substitusikan (4) ke (3) untuk memperoleh
Kemudian substitusikan (7) dan (8) ke persamaan (6)
)(3
2
3
22
32
3
3
1
2
ss
o
oooo
vivivvi
v
ivvivv
(6) (7)
(8)
3
io
vvi
KVL di loop sebelah kanan:
is vvivv 2
Example 16.11
•ch16 Applications of the Laplace
Transform
•64
i
s
o
o
i
s
v
v
i
v
i
v
v
v
i
v
i
v
31
23
31
32
32
0
0
0
04
11
42
12
State equations yang diperoleh:
Example 16.12
•ch16 Applications of the Laplace
Transform
•65
Assume we have a system where the output is y(t) and
the input is z(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.
)(5)(2)(
3)(
2
2
tztydt
tdy
dt
tyd
Example 16.12
•ch16 Applications of the Laplace
Transform
•66
32
1
32
10
10
01
01)(
)(5
0
32
10
)(532)(5)(3)(2)(
)(let Now
)( ),(Let
2
1
2
1
2
1
212
12
1
s
sss
x
xt
tzx
x
x
x
tzxxtztytytyx
tyxx
tyxtyx
AI
y
Example 16.12
•ch16 Applications of the Laplace
Transform
•67
)2)(1(
5
2)3(
5
501
2)3(
5
0
2
13)01(
)()(
isfunction transfer The
2)3(
2
13
)( is inverse The
1
1
ssss
s
ss
s
s
ss
ss
s
s
s
BAICH
AI
Example 16.12
•ch16 Applications of the Laplace
Transform
•68
23
5
)(
)()(
)(5)(]23[
2
2
sssZ
sYsH
sZsYss
Application: Network Stability
•ch16 Applications of the Laplace
Transform
•69
Example 16.13
•ch16 Applications of the Laplace
Transform
•70
Determine k, so that the
Circuit is stable!
Applying mesh analysis:
•ch16 Applications of the Laplace
Transform
•71
In matrix form
And the determinant is
Network Synthesis
•ch16 Applications of the Laplace
Transform
•72
•ch16 Applications of the Laplace
Transform
•73
Example i
•ch16 Applications of the Laplace
Transform
•75
Jika kapasitor tidak bermuatan saat t=0,
tentukan vo(t) saat t > 0
•ch16 Applications of the Laplace
Transform
•76
KCL:
I + 4I = s.Vo
5I = sVo
Dimana
I = (5-Vo)/2 sehingga
5(5-Vo) = 2sVo Vo = 12.5/(s+5/2)
vo(t) = 12.5 e -2.5t V
Example ii
•ch16 Applications of the Laplace
Transform
•77
Convolution of 2 signals
•ch16 Applications of the Laplace
Transform
•78
•ch16 Applications of the Laplace
Transform
•79
•ch16 Applications of the Laplace
Transform
•80
•ch16 Applications of the Laplace
Transform
•81
•ch16 Applications of the Laplace
Transform
•82
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