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INTRODUCTION TO LAPLACE INTRODUCTION TO LAPLACE TRANSFORMTRANSFORM
Advanced Circuit Analysis Advanced Circuit Analysis TechniqueTechnique
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TOPIC COVERAGETOPIC COVERAGE
WEEK TOPICS LABS
Week 6 Introduction to the Laplace Transform: Waveform Analysis
Lab 3: Intro to Laplace Transform
Week 7 Laplace Transform in Circuit Analysis: Transfer Function.
Lab 4: Laplace Transform Analysis
Week 8 Frequency Response: Bode Plot
Week 9 Frequency Selective Circuits: Passive Filters
Lab 5: RL and RC Filter
Lab 6: Bandpass and Bandreject Filter
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TOPIC 1: TOPIC 1: LAPLACE TRANSFORMLAPLACE TRANSFORM
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AREA OF COVERAGEAREA OF COVERAGE
WEEKWEEK TOPICSTOPICS DURATIONDURATION
Week 6Week 6 Intro LT: Definition, Excitation Intro LT: Definition, Excitation function, Functional transform, function, Functional transform, Operational transform.Operational transform.
2 hrs2 hrs
Intro LT: Operational Intro LT: Operational transform, Partial fraction.transform, Partial fraction.
1 hr1 hr
Week 7Week 7 Circuit Analysis: S-domain Circuit Analysis: S-domain circuit component, Transfer circuit component, Transfer function. function.
3 hrs3 hrs
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LEARNING OUTCOMESLEARNING OUTCOMES
Be able to calculate the Laplace transform of a function using the definition and/or Laplace transform table.
Be able to calculate the inverse Laplace transform using partial fraction expansion and the Laplace table.
Be able to perform circuit analysis in the s-domain.
Know how to use a circuit’s transfer function to calculate the circuit’s impulse, unit, and step response.
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OVERVIEWOVERVIEW
Analyze a “linear circuit” problem, in the frequency domain instead of in the time domain.
Convert a set of differential equations into a corresponding set of algebraic equations, which are much easier to solve.
Analyze the bandwidth, phase, and transfer characteristics important for circuit analysis and design.
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OVERVIEWOVERVIEW
Analyze both the steady-state and “transient” responses of a linear circuit.
Analyze the response under any types of excitation (e.g. switching on and off at any given time(s), sinusoidal, impulse, square wave excitations, etc.)
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DEFINITION OF LAPLACE DEFINITION OF LAPLACE TRANSFORMTRANSFORM
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.
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)(tf
0, te at
0,0 t
0t
)(tf
ate
0t
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DISCONTUNITY FUNCTIONDISCONTUNITY FUNCTION
When f(t) has a finite discontinuity at the origin, the Laplace transform formula is rewritten as:
0
)()()( dtetfsFtfL st
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STEP FUNCTIONSTEP FUNCTION
The symbol for the step function is Ku(t).
Mathematical definition of the step function:
0,)(
0,0)(
tKtKu
ttKu
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f(t) = K u(t)f(t) = K u(t)
)(tf
K
0t
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STEP FUNCTIONSTEP 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:
atKatKu
atatKu
,)(
,0)(
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f(t) = K u(t-a)f(t) = K u(t-a)
)(tf
K
ta0
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Ex:Ex:)(tf
2
10 2 3 4t
2
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Three linear functions at Three linear functions at t=0, t=1, t=3, and t=4t=0, t=1, t=3, and t=4
)(tf
4
10 2 3 4t
4
2
2
t2
42 t
82 t
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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=4
Step function can be used to turn on and turn off these functions
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Step functionsStep functions
)]4()3()[82(
)]3()1()[42(
)]1()([2)(
tutut
tutut
tututtf
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IMPULSE FUNCTIONIMPULSE FUNCTION
The symbol for the impulse function is (t).
Mathematical definition of the impulse function:
0,0)(
)()(
tt
KtdtK
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f(t) = K f(t) = K (t)(t)
)(tf
K
0t
K)(tK )( atK
a
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IMPULSE FUNCTIONIMPULSE 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)
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FUNCTIONAL FUNCTIONAL TRANSFORMTRANSFORM
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TYPE f(t) (t>0-) F(s)
Impulse
Step
Ramp
Exponential
Sine
Cosine
δ(t)
u(t)
t
ate
s1
1
2s1
as1
tsin
tcos
2
2s
22ss
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TYPE f(t) (t>0-) F(s)
Damped ramp
Damped sine
Damped cosine
atte
te at sin
te at cos
21
as
22
as
22
asas
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OPERATIONAL OPERATIONAL TRANSFORMTRANSFORM
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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 constant
2. Addition/subtraction
3. Differentiation
4. Integration
5. Translation in the time domain
6. Translation in the frequency domain
7. Scale changing
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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
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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
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OPERATION f(t) F(s)
Scale changing
First derivative (s)
n th derivative
s integral
0),( aatf asFa1
dssdF )()(ttf
s
duuF )(ttf )(
)(tft nn
nn
dssFd )()1(
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Translation in time domainTranslation in time domain
If we start with any function:
we can represent the same function translated in time by the constant a, as:
In frequency domain:
)()( tutf
)()( atuatf
)()()( sFeatuatf as
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Ex:Ex:
21)(s
ttuL
2)()(s
eatuatLas
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Translation in frequency Translation in frequency domaindomain
Translation in the frequency domain is defined as:
)()( asFtfeL at
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Ex:Ex:
22)(
cos
as
asteL at
22
cos
s
stL
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Scale changingScale changing
The relationship between f(t) and F(s) when the time variable is multiplied by a constant:
oaasFaatfL ,1)]([
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Ex:Ex:
222 1)(
1cos
s
s
s
stL
1
cos2
s
stL
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APPLICATIONAPPLICATION
dcI
0t
R CL
)(tv
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ProblemProblem
Assumed 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.
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Integrodifferential EquationIntegrodifferential Equation
A single node voltage equation:
)()(
)(1)(
lglg
0
tuIdt
tdvCdxxv
LR
tv
KCLIaIa
dc
t
outin
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s-domain transformations-domain transformation
sIvssVCs
sV
LR
sVdc1)0()(
)(1)(
)()(
)(1)(
0
tuIdt
tdvCdxxv
LR
tvdc
t
s
IsC
sLRsV dc
11)(
=0
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)()( 1 sVLtv
)1()1()(
2 LCsRCsC
I
sVdc
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INVERSE LAPLACE INVERSE LAPLACE TRANSFORM (LTRANSFORM (L-1-1))
PARTIAL FRACTION EXPANSION PARTIAL FRACTION EXPANSION (PFE)(PFE)
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Rational function of s: expressed in a form of a ratio of two polynomials.
Called proper rational function if m≥nInverse transform rational functions of F(s),
can solve for v(t) or i(t).
011
1
011
1
)(
)()(
bsbsbsb
asasasa
sD
sNsF
mm
mm
nn
nn
RATIONAL FUNCTIONSRATIONAL FUNCTIONS
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PARTIAL FRACTION PARTIAL FRACTION EXPANSIONEXPANSION
1) Distinct Real Roots of D(s)
)6)(8(
)12)(5(96)(
sss
sssF
s1= 0, s2= -8s3= -6
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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
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Find KFind K11
0
3
0
21
068)6)(8(
)12)(5(96
ssss
sK
s
sKK
ss
ss
120)6)(8(
)12)(5(961 K
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Find KFind K22
8
32
8
1
8)6(
)8(
)6(
)8(
)6(
)12)(5(96
sssss
sKK
ss
sK
ss
ss
72)2)(8(
)4)(3(962
K
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Find KFind K33
3
6
2
6
1
6)8(
)6(
)8(
)6(
)8(
)12)(5(96K
ss
sK
ss
sK
ss
ss
sss
48)2)(6(
)6)(1(963
K
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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
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2) Distinct Complex Roots 2) Distinct Complex Roots
)256)(6(
)3(100)(
2
sss
ssF
S1 = -6 S2 = -3+j4 S3 = -3-j4
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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.
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Find KFind K11
1225
)3(100
256
)3(100
621
sss
sK
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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.
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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
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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
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Undesirable imaginary components Undesirable imaginary components in the time domainin the time domain
)13.534cos(20
10
1010
1010
1010
3
)13.534()13.534(3
)13.534(3)13.534(3
4313.534313.53
)43(13.53)43(13.53
te
eee
ee
ee
eeee
t
jtjtjt
jtjtjtjt
tjtjtjtj
tjjtjj
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Inverse Laplace of F(s)Inverse Laplace of F(s)
)()13.534cos(2012)(
43
10
43
10
6
12
36
13.5313.531
tuteetf
js
e
js
e
sL
tt
jj
)()cos(2)( tuteKtf t
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3) Repeated Real Roots3) Repeated Real Roots
3)5(
)25(100)(
ss
ssF
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Partial Fraction ExpansionPartial Fraction Expansion
5)5()5()5(
)25(100 42
33
213
s
K
s
K
s
K
s
K
ss
s
20)5(
)25(100
031
ss
sK 20
)5(
)25(100
031
ss
sK
400)25(100
52
ss
sK 400
)25(100
52
ss
sK
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Find KFind K33
First, multiply both sides by (s+5)3. Next, differentiate both sides once with respect to s and evaluate at s=-5.
5)5()5()5(
)25(100 42
33
213
s
K
s
K
s
K
s
K
ss
s
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KK33
52
4
53
52
5
31
5
)5(
)5(
)5()25(100
s
s
s
ss
sKds
d
sKds
d
Kds
d
s
sK
ds
d
s
s
ds
d
![Page 61: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/61.jpg)
KK33
10)25(
100 35
2
Ks
ss
s
![Page 62: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/62.jpg)
Find KFind K44
First, multiply both sides by (s+5)3. Next, differentiate both sides twice with respect to s and evaluate at s=-5.
5)5()5()5(
)25(100 42
33
213
s
K
s
K
s
K
s
K
ss
s
![Page 63: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/63.jpg)
KK44
Simplifying the first derivative, the second derivative becomes:
20
240
)5(20
)52()5(25100
4
4
5453
5
2
2
15
2
K
K
sKds
dK
ds
d
s
ss
ds
dK
sds
d
ss
ss
![Page 64: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/64.jpg)
F(s) and f(t)F(s) and f(t)
5
20
)5(
100
)5(
40020)(
23
sssssF
)(2010020020)( 5552 tueteettf ttt
)(tuKte at
![Page 65: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/65.jpg)
4) REPEATED COMPLEX 4) REPEATED COMPLEX ROOTSROOTS
22 )256(
768)(
sssF 22 )256(
768)(
sssF
432,1 js
![Page 66: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/66.jpg)
Partial Fraction ExpansionPartial Fraction Expansion
)43()43()43()43(
)43()43()43()43(
)43()43(
768)(
21
22
21
42
322
1
22
js
K
js
K
js
K
js
K
js
K
js
K
js
K
js
K
jsjssF
![Page 67: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/67.jpg)
Find KFind K11 and K and K22
12)8(
768
)43(
7682
4321
jjsK
js
9033
)8(
)768(2
)43(
)768(2
)43(
768
343
3
4322
j
jjs
jsds
dK
js
js
![Page 68: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/68.jpg)
F(s) and f(t)F(s) and f(t)
)43(
903
)43(
903
)43(
12
)43(
12)(
22
sjs
jsjssF
)43(
903
)43(
903
)43(
12
)43(
12)(
22
sjs
jsjssF
)()]904cos(64cos24[)( 33 tutettetf tt )()]904cos(64cos24[)( 33 tutettetf tt
)()cos(2)( tuteKttf t
![Page 69: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/69.jpg)
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
![Page 70: INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique](https://reader035.vdocuments.mx/reader035/viewer/2022081503/56649e8f5503460f94b93c96/html5/thumbnails/70.jpg)
InspirationInspiration
Do not pray for easy lives,
Pray to be stronger men!
Do not pray for tasks equal to your powers,
Pray for powers equal to your tasks!