Cosh sinh eat sin damped sine wave cos damped cosine wave table continued function est sinh cosh from table sin cos sin cos example use table determine the laplace transforms the following waveforms step voltage which starts time step voltage which starts time from and table sexample find the laplace transform sin cos solution

208	Chapter 8

1.

δ (unit impulse)

1
s

eat (exponential function)

1

s	�	a

5.

Sin ωt (sine wave)

s2	�	ω 2

s2	� ω 2

7.

t (unit ramp function)

1
2s

s	3

tn ( n� 1, 2, 3 … )

n	!
sn�1

10.

s2	� ω 2

ω

s2	�	ω 2

12.

	n !
( s	�	an�1

( s	� a ) 2 �	ω 2

14.

e� at cos ωt (damped cosine wave)

	s		�	a
( s	�	a )2		�	ω2

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Transients and Laplace Transforms

15.

(	s	� a ) 2 �	ω 2

		s	�	a
(	s	�	a )2	�	ω2

(b) � { 5 e2t� 3 e� t } � 5 � {e2t } � 3 � {e� t }

� { 6 sin 3 t� 4 cos 5 t } � 6 � { sin 3 t } � 4 � { cos 5 t }

�	6	⎛⎜⎜⎜⎝						⎞⎟⎟⎟⎟⎠		�	4	⎛⎜			s		⎞⎟⎟
		⎛⎜⎜⎜⎝	s	2	�		3 2	⎞⎟⎟⎟⎟⎠		�	4	⎜⎜⎝	s	2	�	5 2	⎟⎟⎠
�		18				�		4			s			2	�	5 2	⎟⎟⎠
	s	2	�		9	�		s	2	�		25