Transform Laplace Table
Time Signal u(t ) u(t ) - u(t - c), c > 0 t N u(t ), N = 1, 2, 3,K Laplace Transform 1/ s (1 - e - cs ) / s, c > 0 N! , N = 1, 2, 3,K s N +1 1 e - cs , c real 1 , b real or complex s+b N! , N = 1, 2, 3, K ( s + b) N +1 s 2 2 s + o
(t ) (t - c), c real
e - bt u (t ), b real or complex t N e - bt u(t ), cos(o t )u(t ) sin(o t )u(t ) N = 1, 2, 3, K
o 2 s + o
2 2 s 2 + 2o 2 s( s 2 + 4o )
cos2 (o t )u(t ) sin 2 (o t )u(t ) e - bt cos(o t )u(t ) e - bt sin(o t )u(t )
t cos(o t )u(t ) t sin(o t )u(t )
2o2 2 s( s 2 + 4o ) s+b 2 ( s + b) 2 + o
2 ( s + b ) 2 + o 2 s 2 - o 2 ( s 2 + o ) 2
o
Ae -nt sin n 1 - 2 t u(t ) where : A =
[(
)]
2o s 2 ( s + o ) 2
2
n 1 - 2
2 s + 2n s + n
2
Ae -nt sin n 1 - 2 t + u(t )
[(
) ]
1- 2 - n
s + 2 s + 2n s + n
2
( - n )2 + 1 = tan -1 n A= 2 2
n (1 - )
te - bt cos(o t )u(t ) te - bt sin(o t )u(t )
( s + b) 2 - o2 (( s + b) 2 + o2 ) 2 2o ( s + b) (( s + b) 2 + o2 ) 2
Laplace Transform Properties
Property Name Linearity Right Time Shift (Causal Signal) Time ) Scaling
ax(t + bv(t ) x(t - c ), c > 0 x ( at ), a > 0
t n x (t ), n = 1, 2, 3, K eat x (t ), a real or complex sin(o t ) x (t ) cos(o t ) x (t )
Property aX ( s ) + bV ( s )
e - cs X (s )
1 a
X ( s / a ), a > 0
Multiply by t
n
( -1) n
Multiply by Exponential Multiply by Sine Multiply by Cosine Time Differentiation 2nd Derivative nth Derivative
dn X ( s ), n = 1, 2, 3, K ds n X ( s - a ), a real or complex
& x(t ) &&(t ) x x ( N ) (t )
j [X ( s + jo ) - X ( s - jo )] 2 1 [X ( s + jo ) + X ( s - jo )] 2 sX ( s ) - x (0) & s 2 X ( s ) - sx(0) - x (0) N N -1 & s X ( s ) - s x ( 0) - s N - 2 x ( 0) -
L - sx ( N -2 ) (0) - x ( N -1) (0)
Time Integration Convolution in Time Initial-Value Theorem
-
x ( ) d
s
t
1 X ( s) s
x (t ) * h (t ) x(0) = lim[sX ( s )]
& x (0) = lim s 2 X ( s ) - sx (0)
s s
X ( s) H ( s)
[
] ]
lim x (t ) = lim sX ( s )
t s 0
& x ( N ) (0) = lim s N +1 X ( s ) - s N x (0) - s N -1 x (0) - L - sx ( N -1) (0)
[
Final-Value Theorem
If lim x (t ) exists, then
t