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2 Pages

### LT Tables_rev3

Course: EECE 301, Fall 2011
School: Binghamton
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Word Count: 458

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Transform Laplace Table Time Signal u(t ) u(t ) - u(t - c), c &gt; 0 t N u(t ), N = 1, 2, 3,K Laplace Transform 1/ s (1 - e - cs ) / s, c &gt; 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(...

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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
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