Transformadas de laplace
2
Laplace Transforms
2.1 2.2 2.3 2.4 Definitions and Laplace Transform Formulae Properties Inverse Laplace Transforms Relationship Between Fourier Integrals of Causal Functions and One-Sided Laplace Transforms 2.5 Table of LaplaceTransforms 2.2 Table of Laplace Operations 2.3 Table of Laplace Transforms References Appendix 1
Examples • Inversion in the Complex Plane • Complex Integration and the Bilateral Laplace Transform
2.1 Definitions and Laplace Transform Formulae
2.1.1 One-Sided Laplace Transform
F( s) =
∫ f (t ) e
0
∞
− st
dt
s = σ + jω
f (t) = piecewise continuous and of exponential order
2.1.2One-Sided Inverse Laplace Transform
f (t ) = 1 2πj
σ+ j ∞
σ− j ∞
∫ F( s) e
st
ds
where the integration is within the regions of convergence. The region of convergence is half-plane σ < Re{s}.
©1999 CRC Press LLC
2.1.3 Two-Sided Laplace Transform
F( s) =
−∞
∫
∞
f (t ) e − st dt
s = σ + jω
f (t) = piecewise continuous and of exponential order
2.1.4Two-Sided Inverse Laplace Transform
f (t ) = 1 2πj
σ+ j ∞
σ− j ∞
∫ F( s) e
st
ds
where the integration is within the regions of convergence which is a vertical strip σ1 < Re{s} < σ2.
©1999 CRC Press LLC
2.2 Properties
2.2.1 Properties of the Laplace Transform (one sided)
TABLE 2.1 Laplace Transform Properties
1. 2. Linearity L {K1 f1 (t ) ± K 2 f 2 (t )} = L {K1 f1 (t )} ± L{K 2 f2 (t )} = K1 F1 (s) ± K 2 F2 (s) Time derivative d L f (t ) = sF(s) − f (0 + ) dt Higher time derivative dn L n f (t ) = s n F(s) − s n −1 f (0 + ) − s n − 2 f (1) (0 + ) − L − f ( n −1) (0 + ) dt where f (i)(0+), i = 1,2,…,n – 1 is the ith derivative of f (·) at t = 0+. 4. Integral with zero initial condition L Integral with initial conditions L
3.
∫t
0
F(s) f (ξ) dξ = s
5. 6. 7. 8. 9.
∫
F(s) f ( −1) (0 + ) + f (ξ) dξ = where f ( −1) (0 + ) = lim t→0 + s s −∞
t
∫
t
−∞
f (ξ) dξ
Multiplication by exponential L {e ± at f (t )} = F(s m a) Multiplication by t L {t f (t )} = − d F(s) ; ds L {t n f (t )} = ( −1) n dn F(s) ds s
Time shifting L { f (t ± λ ) u(t ± λ )} = e ± sλ F(s) t Scaling L f = aF( as) ; a L { f (ta)} = 1 s F a a a>0
10. 11.
t ∆ Time convolution L f1 (t − τ) f2 ( τ) dτ L { f1 (t ) ∗ f2 (t )} = F1 (s) F2 (s) 0 Frequency convolution
∫
L { f1 (t ) f 2 (t )} = 1 2 πj
∫
x + j∞
x − j∞
F1 ( z ) F2 (s − z ) dz = 1 {F1 (s) ∗ F2 (s)} 2 πj
where z = x + jy, and where x must be greater than the abscissa of absolute convergencefor f1(t) over the path of integration. 12. 13. 14. Initial value lim f (t ) = lim sF(s) provided that this limit exists.
t→0 + s→∞
Final value lim f (t ) = lim sF(s) provided that sF(s) is analytic on the jω axis and in the right half of the s plane
t →∞ s→ 0 +
f (t ) Division by t L = t
∫ F(s ′) ds ′
s T − st
∞
15.
f (t) periodic
∫e L { f (t )} =
0
f (t ) dtf (t ) = f (t + T )
1 − e − sT
2.2.2 Methods of Finding the Laplace Transform
1. 2. 3. 4. Direct method by solving (2.1.1). Expand f (t) in power series if such an expansion exists. Differentiation with respect to a parameter. Use of tables.
©1999 CRC Press LLC
2.3 Inverse Laplace Transforms
2.3.1 Properties
1. Linearity L−1 {c1 F1 (s) ± c2 F2 (s)} = c1 f1 (t ) ± c2 f2 (t ) 2.Shifting L−1 {F(s − a)} = e at f (t ) 3. Time shifting L−1 {e − as F(s)} = f (t − a) 4. Scaling property L−1 {F(as)} = 1 f t a a t>a a>0 F ( n ) ( s) = d n F( s) ds n
()
5. Derivatives L−1 {F ( n ) (s)} = ( −1) n t n f (t )
6. Multiplication by s L−1 {sF(s) − f (0 + )} = L {sF(s)} − f (0 + ) L {1} = f (1) (t ) + f (0)δ(t ) F( s) = 7. Division by s L−1 s
∫
t
f (t ′) dt...
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