Table 12-1: Properties of the Laplace transform (f(t) = 0 for t<0). Property f(t) 1. Multiplication by constant K f (t) 2. Linearity K₁ fi(t) + K2 f2(t) -> F(s) = C[f(t)] K F(s) K₁ F1 (s) + K2 F2(s) 3. Time scaling 4. Time shift f(at), a > 0 - f(t − T) u(t − T) -> e-Ts F(s). TO 5. Frequency shift e-at f(t) df 6. Time 1st derivative f' = - S dt 7. Time 2nd derivative f": 8. Time integral Ĵ f(t) dt - 9. Frequency derivative t f (t) F(s+a) s F(s) - f(0) s²F(s) - sf (0) - f'(0-) F(s) F(s) f(t) 10. Frequency integral t [F(s') ds' Table 12-2: Examples of Laplace transform pairs for T≥0. Note that multiplication by u(t) guarantees that f(t) = 0 for t <0¯. Laplace Transform Pairs f(t) F(s) = L[f(t)] 1 la 2 2a 3 نيا За 4a 8(t) 8(t - T) 1 or u(t) u(t-T) e-at u(t) e-a(t-T) u(t-T) tu(t) (t-T)u(t-T) S +a Ts s+a 11% 5 1² u(t) السلام 9 7 8 9 sin ot u(t) te-at u(t) 1²e-at u(t) tn-le-at u(t) (s+a)² 2 (s+a)³ (n - 1)! (s+a)" ய 10 sin(wt +) u(t) s sin + @cos 11 cos wt u(t) S s² + w² s²+w² 12 scos cos(wt + 0) u(t) @sine 13 e-at sin wt u(t) s² + w² ω (s+a)²+w² 14 e-at cos wt u(t) s+a (s+a)²+w² 15 2e at cos(bt-0) u(t) ejo e-jo 2"-1 16 e-at s+a+jb ejo s+a-jb (n - 1)! cos(bt-0) u(t) e-jo (s+a+jb)" (s+a-jb)" Note: (n-1)! (n − 1)(n-2)...1.
Table 12-1: Properties of the Laplace transform (f(t) = 0 for t<0). Property f(t) 1. Multiplication by constant K f (t) 2. Linearity K₁ fi(t) + K2 f2(t) -> F(s) = C[f(t)] K F(s) K₁ F1 (s) + K2 F2(s) 3. Time scaling 4. Time shift f(at), a > 0 - f(t − T) u(t − T) -> e-Ts F(s). TO 5. Frequency shift e-at f(t) df 6. Time 1st derivative f' = - S dt 7. Time 2nd derivative f": 8. Time integral Ĵ f(t) dt - 9. Frequency derivative t f (t) F(s+a) s F(s) - f(0) s²F(s) - sf (0) - f'(0-) F(s) F(s) f(t) 10. Frequency integral t [F(s') ds' Table 12-2: Examples of Laplace transform pairs for T≥0. Note that multiplication by u(t) guarantees that f(t) = 0 for t <0¯. Laplace Transform Pairs f(t) F(s) = L[f(t)] 1 la 2 2a 3 نيا За 4a 8(t) 8(t - T) 1 or u(t) u(t-T) e-at u(t) e-a(t-T) u(t-T) tu(t) (t-T)u(t-T) S +a Ts s+a 11% 5 1² u(t) السلام 9 7 8 9 sin ot u(t) te-at u(t) 1²e-at u(t) tn-le-at u(t) (s+a)² 2 (s+a)³ (n - 1)! (s+a)" ய 10 sin(wt +) u(t) s sin + @cos 11 cos wt u(t) S s² + w² s²+w² 12 scos cos(wt + 0) u(t) @sine 13 e-at sin wt u(t) s² + w² ω (s+a)²+w² 14 e-at cos wt u(t) s+a (s+a)²+w² 15 2e at cos(bt-0) u(t) ejo e-jo 2"-1 16 e-at s+a+jb ejo s+a-jb (n - 1)! cos(bt-0) u(t) e-jo (s+a+jb)" (s+a-jb)" Note: (n-1)! (n − 1)(n-2)...1.
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
Related questions
Question
12.4 Determine the Laplace transform of each of the following
functions by applying the properties given in the Tables
(a) f1(t) = 4te−2t u(t)
(b) f2(t) = 10cos(12t +60◦) u(t)
*(c) f3(t) = 12e−3(t−4) u(t −4)
(d) f4(t) = 30(e−3t +e3t ) u(t)
(e) f5(t) = 16e−2t cos4t u(t)
(f) f6(t) = 20te−2t sin4t u(t)
![Table 12-1: Properties of the Laplace transform (f(t) = 0 for t<0).
Property
f(t)
1. Multiplication by constant K f (t)
2. Linearity K₁ fi(t) + K2 f2(t) ->
F(s) = C[f(t)]
K F(s)
K₁ F1 (s) + K2 F2(s)
3. Time scaling
4. Time shift
f(at),
a > 0
-
f(t − T) u(t − T)
->
e-Ts F(s). TO
5. Frequency shift
e-at f(t)
df
6. Time 1st derivative
f'
=
-
S
dt
7. Time 2nd derivative
f":
8. Time integral
Ĵ f(t) dt
-
9. Frequency derivative
t f (t)
F(s+a)
s F(s) - f(0)
s²F(s) - sf (0)
- f'(0-)
F(s)
F(s)
f(t)
10. Frequency integral
t
[F(s') ds'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7534f13a-064e-4169-8c87-38e5472c444f%2F52422cd2-0925-408f-addc-ccaaae432f95%2Fv0xwhgb_processed.png&w=3840&q=75)
Transcribed Image Text:Table 12-1: Properties of the Laplace transform (f(t) = 0 for t<0).
Property
f(t)
1. Multiplication by constant K f (t)
2. Linearity K₁ fi(t) + K2 f2(t) ->
F(s) = C[f(t)]
K F(s)
K₁ F1 (s) + K2 F2(s)
3. Time scaling
4. Time shift
f(at),
a > 0
-
f(t − T) u(t − T)
->
e-Ts F(s). TO
5. Frequency shift
e-at f(t)
df
6. Time 1st derivative
f'
=
-
S
dt
7. Time 2nd derivative
f":
8. Time integral
Ĵ f(t) dt
-
9. Frequency derivative
t f (t)
F(s+a)
s F(s) - f(0)
s²F(s) - sf (0)
- f'(0-)
F(s)
F(s)
f(t)
10. Frequency integral
t
[F(s') ds'
![Table 12-2: Examples of Laplace transform pairs for T≥0. Note that multiplication by u(t) guarantees that f(t) = 0 for t <0¯.
Laplace Transform Pairs
f(t)
F(s) = L[f(t)]
1
la
2
2a
3
نيا
За
4a
8(t)
8(t - T)
1 or u(t)
u(t-T)
e-at u(t)
e-a(t-T) u(t-T)
tu(t)
(t-T)u(t-T)
S
+a
Ts
s+a
11%
5
1² u(t)
السلام
9
7
8
9
sin ot u(t)
te-at u(t)
1²e-at u(t)
tn-le-at u(t)
(s+a)²
2
(s+a)³
(n - 1)!
(s+a)"
ய
10
sin(wt +) u(t)
s sin + @cos
11
cos wt u(t)
S
s² + w²
s²+w²
12
scos
cos(wt + 0) u(t)
@sine
13
e-at sin wt u(t)
s² + w²
ω
(s+a)²+w²
14
e-at cos wt u(t)
s+a
(s+a)²+w²
15
2e at cos(bt-0) u(t)
ejo
e-jo
2"-1
16
e-at
s+a+jb
ejo
s+a-jb
(n - 1)!
cos(bt-0) u(t)
e-jo
(s+a+jb)"
(s+a-jb)"
Note: (n-1)! (n − 1)(n-2)...1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7534f13a-064e-4169-8c87-38e5472c444f%2F52422cd2-0925-408f-addc-ccaaae432f95%2Fhgzb2k_processed.png&w=3840&q=75)
Transcribed Image Text:Table 12-2: Examples of Laplace transform pairs for T≥0. Note that multiplication by u(t) guarantees that f(t) = 0 for t <0¯.
Laplace Transform Pairs
f(t)
F(s) = L[f(t)]
1
la
2
2a
3
نيا
За
4a
8(t)
8(t - T)
1 or u(t)
u(t-T)
e-at u(t)
e-a(t-T) u(t-T)
tu(t)
(t-T)u(t-T)
S
+a
Ts
s+a
11%
5
1² u(t)
السلام
9
7
8
9
sin ot u(t)
te-at u(t)
1²e-at u(t)
tn-le-at u(t)
(s+a)²
2
(s+a)³
(n - 1)!
(s+a)"
ய
10
sin(wt +) u(t)
s sin + @cos
11
cos wt u(t)
S
s² + w²
s²+w²
12
scos
cos(wt + 0) u(t)
@sine
13
e-at sin wt u(t)
s² + w²
ω
(s+a)²+w²
14
e-at cos wt u(t)
s+a
(s+a)²+w²
15
2e at cos(bt-0) u(t)
ejo
e-jo
2"-1
16
e-at
s+a+jb
ejo
s+a-jb
(n - 1)!
cos(bt-0) u(t)
e-jo
(s+a+jb)"
(s+a-jb)"
Note: (n-1)! (n − 1)(n-2)...1.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 13 images
Recommended textbooks for you
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:
9780078028229
Author:
Charles K Alexander, Matthew Sadiku
Publisher:
McGraw-Hill Education
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:
9780134746968
Author:
James W. Nilsson, Susan Riedel
Publisher:
PEARSON
Engineering Electromagnetics
Electrical Engineering
ISBN:
9780078028151
Author:
Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:
Mcgraw-hill Education,