1 ekt919 electric circuit ii chapter 2 laplace transform
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EKT919EKT919ELECTRIC CIRCUIT ELECTRIC CIRCUIT IIII
Chapter 2Laplace Transform
Definition of Laplace Transform Definition of Laplace Transform
The Laplace Transform is an integral transformation of a function f(t) from the time domain into the complex frequency domain, giving F(s)
s: complex frequencyCalled “The One-sided or unilateral
Laplace Transform”.In the two-sided or bilateral LT, the
lower limit is -. We do not use this.
3
Definition of Laplace Transform Definition of Laplace Transform
Example 1
Determine the Laplace transform of each of the following functions shown below:
4
Definition of Laplace Transform Definition of Laplace Transform
Solution:
a) The Laplace Transform of unit step, u(t) is given by
0
11)()(
sdtesFtuL st
5
Definition of Laplace Transform Definition of Laplace Transform
Solution:
b) The Laplace Transform of exponential function, e-tu(t),>0 is given by
0
1)()(
sdteesFtuL stt
6
Definition of Laplace Transform Definition of Laplace Transform
Solution:
c) The Laplace Transform of impulse function,
δ(t) is given by
01)()()( dtetsFtuL st
Functional Functional TransformTransform
TYPE f(t) F(s)
Impulse
Step
Ramp
Exponential
Sine
Cosine
δ(t)
u(t)
t
ate
s1
1
2s1
as1
tsin
tcos
2
2s
22ss
TYPE f(t) F(s)
Damped ramp
Damped sine
Damped cosine
atte
te at sin
te at cos
21
as
22
as
22
asas
Properties of Laplace Transform Properties of Laplace Transform
0,)(
0,0)(
tKtKu
ttKu
Step Function
The symbol for the step function is K u(t).Mathematical definition of the step
function:
f(t) = K u(t)f(t) = K u(t)
)(tf
K
0t
Properties of Laplace Transform Properties of Laplace Transform
atKatKu
atatKu
,)(
,0)(
Step Function
A discontinuity of the step function may occur at some time other than t=0.
A step that occurs at t=a is expressed as:
f(t) = K u(t-a)f(t) = K u(t-a)
)(tf
K
ta0
Ex:Ex:)(tf
2
10 2 3 4t
2
Three linear functions at t=0, Three linear functions at t=0, t=1, t=3, and t=4t=1, t=3, and t=4
)(tf
4
10 2 3 4t
4
2
2
t2
42 t
82 t
Expression of step functions Expression of step functions
Linear function +2t: on at t=0, off at t=1
Linear function -2t+4: on at t=1, off at t=3
Linear function +2t-8: on at t=3, off at t=4Step function can be used to turn on and
turn off these functions
Step FunctionsStep Functions
)]4()3()[82(
)]3()1()[42(
)]1()([2)(
tutut
tutut
tututtf
Properties of Laplace Transform Properties of Laplace Transform
0,0)(
1)()(
tt
tdt
Impulse Function
The symbol for the impulse function is (t).Mathematical definition of the impulse
function:
Properties of Laplace Transform Properties of Laplace Transform
Impulse Function
The area under the impulse function is constant and represents the strength of the impulse.
The impulse is zero everywhere except at t=0.
An impulse that occurs at t = a is denoted K (t-a)
f(t) = K f(t) = K (t)(t)
)(tf
K
0t
K)(tK )( atK
a
21
Properties of Laplace Transform Properties of Laplace Transform
)()()()( 22112211 sFasFatfatfaL
Linearity
If F1(s) and F2(s) are, respectively, the Laplace Transforms of f1(t) and f2(t)
Example:
22
)(2
1)()cos(
s
stueeLtutL tjtj
22
Properties of Laplace Transform Properties of Laplace Transform
)(1
)(a
sFa
atfL
Scaling
If F (s) is the Laplace Transforms of f (t), then
Example:
22 4
2)()2sin(
s
tutL
23
Properties of Laplace Transform Properties of Laplace Transform
)()()( sFeatuatfL as
Time Shift
If F (s) is the Laplace Transforms of f (t), then
Example:
22
)())(cos(
s
seatuatL as
24
Properties of Laplace Transform Properties of Laplace Transform
)()()( asFtutfeL at
Frequency Shift
If F (s) is the Laplace Transforms of f (t), then
Example:
22)(
)()cos(
as
astuteL at
25
Properties of Laplace Transform Properties of Laplace Transform
)0()()(
fssFtudt
dfL
Time Differentiation
If F (s) is the Laplace Transforms of f (t), then the Laplace Transform of its derivative is
Example:
22
)sin(
s
u(t)ωtL
26
Properties of Laplace Transform Properties of Laplace Transform
)(1
)(0
sFs
dttfLt
Time Integration
If F (s) is the Laplace Transforms of f (t), then the Laplace Transform of its integral is
Example:
1
!n
n
s
ntL
27
Properties of Laplace Transform Properties of Laplace Transform
ds
sdFttfL
)()(
Frequency Differentiation
If F(s) is the Laplace Transforms of f (t), then the derivative with respect to s, is
Example:
2)(
1)(
astuteL at
28
Properties of Laplace Transform Properties of Laplace Transform
)(lim)0( ssFfs
Initial and Final Values
The initial-value and final-value properties allow us to find the initial value f(0) and f(∞) of f(t) directly from its Laplace transform F(s).
Initial-value theorem
)(lim)(0
ssFfs
Final-value theorem
29
The Inverse Laplace Transform The Inverse Laplace Transform
Suppose F(s) has the general form of
The finding the inverse Laplace transform of F(s) involves two steps:
1. Decompose F(s) into simple terms using partial fraction expansion.
2. Find the inverse of each term by matching entries in Laplace Transform Table.
polynomialr denominato)...(
polynomialrator ......nume)()(
sD
sNsF
30
Example 1
Find the inverse Laplace transform of
Solution:
The Inverse Laplace TransformThe Inverse Laplace Transform
4
6
1
53)(
2
ssssF
0 t),()2sin(353(
4
6
1
53)(
2111
tute
sL
sL
sLtf
t
Partial Fraction ExpansionPartial Fraction Expansion
)6)(8(
)12)(5(96)(
sss
sssF
1) Distinct Real Roots of D(s)
s1= 0, s2= -8s3= -6
1) Distinct Real Roots1) Distinct Real Roots
To find K1: multiply both sides by s and evaluates both sides at s=0
To find K2: multiply both sides by s+8 and evaluates both sides at s=-8
To find K3: multiply both sides by s+6 and evaluates both sides at s=-6
68)6)(8(
)12)(5(96)( 321
s
K
s
K
s
K
sss
sssF
Find KFind K11
120)6)(8(
)12)(5(961 K
0
3
0
21
068)6)(8(
)12)(5(96
ssss
sK
s
sKK
ss
ss
Find KFind K22
72)2)(8(
)4)(3(962
K
8
32
8
1
8)6(
)8(
)6(
)8(
)6(
)12)(5(96
sssss
sKK
ss
sK
ss
ss
Find KFind K33
48)2)(6(
)6)(1(963
K
3
6
2
6
1
6)8(
)6(
)8(
)6(
)8(
)12)(5(96K
ss
sK
ss
sK
ss
ss
sss
Inverse Laplace of F(s)Inverse Laplace of F(s)
)(4872120)(
6
48
8
72120
68
1
tueetf
sssL
tt
6
48
8
72120)(
ssssF
2) Distinct Complex Roots 2) Distinct Complex Roots
)256)(6(
)3(100)(
2
sss
ssF
S1 = -6 S2 = -3+j4 S3 = -3-j4
Partial Fraction ExpansionPartial Fraction Expansion
43436
43436
)256)(6(
)3(100)(
221
321
2
js
K
js
K
s
K
js
K
js
K
s
K
sss
ssF
Complex roots appears in conjugate pairs.Complex roots appears in conjugate pairs.
Find KFind K11
1225
)3(100
256
)3(100
621
sss
sK
Find KFind K2 2 and Kand K22**
13.53
43
2
1086
)8)(43(
)4(100
)43)(6(
)3(100
j
js
ej
jj
j
jss
sK
13.5310862
jejK
Coefficients Coefficients associated associated
with with conjugate conjugate pairs are pairs are
themselves themselves conjugates.conjugates.
Inverse Laplace of F(s)Inverse Laplace of F(s)
43
13.5310
43
13.5310
6
12
)256)(6(
)3(100)(
2
jsjss
sss
ssF
Inverse Laplace of F(s)Inverse Laplace of F(s)
)(10
1012
43
10
43
10
6
12
)43(13.53
)43(13.536
13.5313.531
tuee
eee
js
e
js
e
sL
tjj
tjjt
jj
Useful Transform PairsUseful Transform Pairs
)()1 tuKeas
K at
)()(
)22
tuKteas
K at
)()cos(2)3 tuteKjs
K
js
K t
)()cos(2)()(
)422
tuteKtjs
K
js
K t
44
• Given two functions, f1(t) and f2(t) with Laplace Transforms F1(s) and F2(s), respectively
• It is defined as
The Convolution Integral The Convolution Integral
tt ethety 25)( and 4)(
0 t),(201
4
2
5)()()(*)( 211
tt eess
LsXsHLtxth
)(*)()()( 2121 tftfLsFsF
• Example:
)(*)()(or )()()( thtxtydthxty
Operational Operational
TransformTransform
Operational TransformsOperational Transforms
Indicate how mathematical operations performed on either f(t) or F(s) are converted into the opposite domain.
The operations of primary interest are:1. Multiplying by a constant2. Addition/subtraction3. Differentiation4. Integration5. Translation in the time domain6. Translation in the frequency domain7. Scale changing
OPERATION f(t) F(s)
Multiplication by a constantAddition/SubtractionFirst derivative (time)Second derivative (time)
)(tKf )(sKF
)()()( 321 tftftf )()()( 321 sFsFsF
dttdf )(
2
2 )(dt
tddt
dfsfsFs )0()0()(2
)0()( fssF
OPERATION f(t) F(s)
n th derivative (time)
Time integral
Translation in timeTranslation in frequency
n
n
dttd )(
1
123
21
)0()0(
)0()0()(
n
nn
nnn
dtdf
dtdfs
dtdfsfssFs
t
dxxf0
)(s
sF )(
0
),()(
a
atuatf
)(tfe at
)(sFe as
)( asF
OPERATION f(t) F(s)
Scale changing
First derivative (s)
n th derivative
s integral
0),( aatf asFa1
dssdF )()(ttf
s
duuF )(ttf )(
)(tft n n
nn
dssFd )()1(
Translation in time domainTranslation in time domain
)()( tutfIf we start with any function:
we can represent the same function translated in time by the constant a, as:
In frequency domain:
)()( atuatf
)()()( sFeatuatf as
Ex:Ex:
21)(s
ttuL
2)()(s
eatuatLas
Translation in frequency Translation in frequency domaindomain
Translation in the frequency domain is defined as:
)()( asFtfeL at
Ex:Ex:
22
cos
s
stL
22)(
cos
as
asteL at
Ex:Ex:
1
cos2
s
stL
222 1)(
1cos
s
s
s
stL
APPLICATIONAPPLICATION
dcI
0t
R CL
)(tv
ProblemProblemAssumed no initial energy is
stored in the circuit at the instant when the switch is opened.
Find the time domain expression for v(t) when t≥0.
Integrodifferential EquationIntegrodifferential Equation
A single node voltage equation:
)()(
)(1)(
lglg
0
tuIdt
tdvCdxxv
LR
tv
KCLIaIa
dc
t
outin
s-domain transformations-domain transformation
sIvssVCs
sV
LR
sVdc
1)0()()(1)(
)()(
)(1)(
0
tuIdt
tdvCdxxv
LR
tvdc
t
s
IsC
sLRsV dc
11)(
=0
)()( 1 sVLtv
)1()1()(
2 LCsRCsC
I
sVdc
ExEx
Obtain the Laplace transform for the function below:
0 1 2 3
t
2
h(t)
4 5
Find the expression of f(t):Find the expression of f(t):
Expression for the ramp function with slope, m =2 and period, T=2:
For a periodic ramp function, we can write:
ttf 2)(1
)1()(2)(1 tututtf
Expanding:Expanding:
)1(2)(2
)1()(2)(1
ttuttu
tututtf
Different time occurred:Different time occurred:t=0t=0 and and t=1t=1
Equal time shift:Equal time shift:
)1(2)1()1(2)(2)(
)1()11(2)(2
)1(2)(2
)1()(2)(
1
1
tututttutf
tutttu
ttuttu
tututtf
Inverse Laplace using Inverse Laplace using translation in time property:translation in time property:
ss
ss
sees
s
e
s
e
ssF
tututttutf
12
222
)(
)1(2)1()1(2)(2)(
2
221
1
Time periodicity property:Time periodicity property:
sss
Ts
seees
e
sFsF
1)1(
21
)()(
22
1
Tse
sFsFnTtftf
1
)()()()( 1
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