Introductory Circuit Analysis (13th Edition)
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ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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Transcribed Image Text:4. Using the Laplace transform pairs of Table 2.1 and the Laplace transform theorems of Table 2.2, derive the Laplace
transforms for the following time functions: For table 2.1 & 2.2, see appendix at the end.
e-at sin ot u(t)
a
b
с
e-at cos wt u(t)
Bu(t)
![Appendix
TABLE 2.1 Laplace transform table
Item no.
f(1)
1.
2.
3.
4.
5.
6.
7.
1.
2.
3.
4.
5.
6.
TABLE 2.2 Laplace transform theorems
Item no.
Theorem
7.
8.
9.
10.
11.
12.
L[f(t)]=F(s) = f(t)e-sdt
L[kf (1)]
L
8(1)
u(t)
=kF(s)
Lf11) +f2(1)] = F₁(s) + F2(s)
Le-atf(1)]
= F(s+a)
L[f(1-T)]
= e-sTF(s)
L[f(at)] ---F(²)
HESE
tu(t)
t"u(t)
e-at u(t)
sin cotu(t)
cos atu(t)
dt
di
[d"f
den
L[fo_f(t)dt]
f(xo)
ƒ(0+)
F(s)
S
=lim sF(s)
S-0
=lim sF(s)
F(s)
-
1
1
S
1
$2
n!
sh +1
1
s+a
=SF (s)-f(0-)
=s²F(s)- sf (0-) - f'(0-)
=s" F (s)-s-kpk-1 (0-)
k=1
(
s² + w²
S
s²+w²
Name
Definition
Linearity theorem
Linearity theorem
Frequency shift theorem
Time shift theorem
Scaling theorem
Differentiation theorem
Differentiation theorem
Differentiation theorem
Integration theorem
Final value theorem¹
Initial value theorem²
¹For this the orem to yield correct finite results, all roots of the denominator of F(s) must have negative real
parts, and no more than one can be at the origin.
2For this theorem to be valid, f(t) must be continuous or have a step discontinuity at t= 0 (that is, no
impulses or their derivatives at t= 0).](https://content.bartleby.com/qna-images/question/5c85250b-6a52-47e3-86e8-89d741652c47/09c5c760-f9e7-467f-a6a9-75246d3f2ac3/0y5oykn_processed.jpeg)
Transcribed Image Text:Appendix
TABLE 2.1 Laplace transform table
Item no.
f(1)
1.
2.
3.
4.
5.
6.
7.
1.
2.
3.
4.
5.
6.
TABLE 2.2 Laplace transform theorems
Item no.
Theorem
7.
8.
9.
10.
11.
12.
L[f(t)]=F(s) = f(t)e-sdt
L[kf (1)]
L
8(1)
u(t)
=kF(s)
Lf11) +f2(1)] = F₁(s) + F2(s)
Le-atf(1)]
= F(s+a)
L[f(1-T)]
= e-sTF(s)
L[f(at)] ---F(²)
HESE
tu(t)
t"u(t)
e-at u(t)
sin cotu(t)
cos atu(t)
dt
di
[d"f
den
L[fo_f(t)dt]
f(xo)
ƒ(0+)
F(s)
S
=lim sF(s)
S-0
=lim sF(s)
F(s)
-
1
1
S
1
$2
n!
sh +1
1
s+a
=SF (s)-f(0-)
=s²F(s)- sf (0-) - f'(0-)
=s" F (s)-s-kpk-1 (0-)
k=1
(
s² + w²
S
s²+w²
Name
Definition
Linearity theorem
Linearity theorem
Frequency shift theorem
Time shift theorem
Scaling theorem
Differentiation theorem
Differentiation theorem
Differentiation theorem
Integration theorem
Final value theorem¹
Initial value theorem²
¹For this the orem to yield correct finite results, all roots of the denominator of F(s) must have negative real
parts, and no more than one can be at the origin.
2For this theorem to be valid, f(t) must be continuous or have a step discontinuity at t= 0 (that is, no
impulses or their derivatives at t= 0).
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