ENGS 22 — Systems Laplace Transform Table Largely modeled on a table in D’Azzo and Houpis, Linear Control Systems Analysis and Design, 1988 F (s) 1. 1 2. 1 s 3. 1 s2 1 4. sn 1 −as 5. e s 1 −as ( 1 − e ) 6. s 7. 1 s+a 1 ( s + a) n 1 s ( s + a) f (t) 0 ≤ t δ (t ) unit impulse at t = 0 1 or u(t ) unit step starting at t = 0 t ⋅ u(t) or t ramp function 1 t n −1 ( n − 1)! n = positive integer u (t − a ) unit step starting at t = a u(t) − u(t − a) rectangular pulse e −at exponential decay s( s + a)(s + b) 1 12. (s + a)(s + b) 1 t n−1e −at n = positive integer (n − 1)! 1 (1 − e −at ) a 1 b −at a −bt (1 − e + e ) ab b−a b−a 1 b(α − a) −at a(α − b) −bt [α − e + e ] ab b−a b−a 1 (e − at − e −bt ) b−a s 13. ( s + a )( s + b) 1 ( ae − at − be −bt ) a−b 8. 9. 1 10. s(s + a)(s + b) s +α 11. Laplace Table Page 1 ENGS 22 — Systems F(s) s +α 14. ( s + a )( s + b ) 1 15. ( s + a)(s + b)(s + c) s +α 16. (s + a)(s + b)(s + c) ω 17. 2 s + ω2 s 18. 2 s + ω2 s+α 19. 2 s +ω2 0≤t f(t) 1 [(α − a)e −at − (α − b)e −bt ] b−a e−at e−bt e−ct + + (b − a)(c − a) (c − b)(a − b) (a − c)(b − c) (α − a)e−at (α − b)e−bt (α − c)e−ct + + (b − a)(c − a) (c − b)(a − b) (a − c)(b − c) sin ω t cos ω t α 2 + ω2 sin(ωt + φ ) ω s sin θ + ω cos θ 20. s2 + ω2 sin(ωt + θ ) 1 21. s ( s 2 + ω 2 ) s+α 22. s ( s 2 + ω 2 ) 1 1 23. (s + a)(s 2 + ω 2 ) ω2 φ = atan2(ω, α ) (1 − cosωt ) α α2 +ω2 − cos( ω t + φ ) φ = atan2(ω,α ) ω2 ω2 e − at 1 + sin(ωt − φ ) 2 2 2 2 a +ω ω a +ω φ = atan2(ω, α ) 1 24. (s + a) 2 + b 2 1 24a. 2 2 s + 2ζω n s + ω n s+a 25. ( s + a ) 2 + b 2 Laplace Table 1 − at e sin(bt ) b 1 ωn 1 − ζ 2 e −ζωnt sin(ωn 1 − ζ 2 t ) e − at cos( bt ) Page 2 ENGS 22 — Systems F(s) s +α 26. ( s + a ) 2 + b 2 26a. (α − a ) 2 + b 2 − at e sin( bt + φ ) b ( α ωn s +α s2 + 2ζωns +ωn 0≤t f(t) − ζωn ) 1−ζ 2 2 2 +1 ⋅ e −ζωnt sin(ω n 1 − ζ 2 t + φ ) φ = atan2(ωn 1 − ζ 2 ,α − ζωn ) 1 1 −at + e sin(bt −φ) 2 2 2 2 a +b b a +b 27. 1 s[(s + a)2 + b2 ] 27a. 1 1 s(s 2 + 2ζωn s + ωn 2 ωn 2 − φ = atan2(b,α − a) 1 ωn 2 1 − ζ 2 φ = atan2( b,− a ) e−ζωnt sin(ωn 1 − ζ 2 t + φ ) φ = cos − 1 ζ 28. 1 (α − a) 2 + b2 −at + e sin(bt + φ) 2 2 2 2 a +b b a +b φ = atan2( b , α − a ) − atan2( b , − a ) 28a. α 1 α −ζω t 2 2 2 + ( − ζ ) + ( 1 − ζ ) ⋅ e sin( ω 1 − ζ t +φ) n 2 2 ωn ωn 1−ζ ωn α s +α s[(s + a)2 + b2 ] s +α n s(s2 + 2ζωn s + ωn ) 2 29. 1 (s + c)[(s + a)2 + b2 ] Laplace Table φ = atan2(ωn 1 − ζ 2 ,α − ω nζ ) − atan2( 1 − ζ 2 ,−ζ ) e −ct e −at sin(bt − φ ) + (c − a) 2 + b 2 b (c − a) 2 + b 2 φ = atan2(b, c − a) Page 3 ENGS 22 — Systems F(s) 30. 1 s(s + c)[(s + a)2 + b2 ] 31. s +α s(s + c)[(s + a)2 + b2 ] 0 ≤1 f(t) 1 e−ct e−at sin(bt − φ) − + c(a2 + b2 ) c[(c − a)2 + b2 ] b a2 + b2 (c − a)2 + b2 φ = atan2(b,−a) + atan2(b, c − a) (c − α )e −ct α + c(a 2 + b 2 ) c[(c − a) 2 + b 2 ] (α − a) 2 + b 2 + e −at sin(bt + φ ) b a 2 + b 2 (c − a) 2 + b 2 φ = atan2(b, α − a) − atan2(b,−a) − atan2(b, c − a) 1 32. s 2 ( s + a ) 1 33. s(s + a)2 s +α 34. s(s + a)2 s 2 + α1s + α 0 35. s(s + a)(s + b) s 2 + α1s + α 0 36. s[(s + a) 2 + b 2 ] 1 (at−1+ e−at ) 2 a 1 (1− e−at − ate−at ) 2 a 1 −at −at [ α − α e + a ( a − α ) te ] 2 a α0 a2 −α1a + α0 + e −at b2 −α1b + α0 −bt e − b(a − b) ab a(a − b) α0 1 2 2 2 + [( a − b − α a + α ) 1 0 c 2 bc 1 2 2 + b (α1 − 2a) ] e−at sin(bt + φ) φ = atan2[ b(α 1 − 2a ), a 2 − b 2 − α 1 a + α 0 ] − atan2( b,− a ) 2 c2 = a2 +b2 Laplace Table Page 4 ENGS 22 — Systems 0 ≤1 F(s) f(t) 37. (1 / ω ) sin(ωt + φ1 ) + (1 / b)e − at sin(bt + φ 2 ) 1 (s2 +ω2 )[(s + a)2 +b2 ] 38. s +α (s2 +ω2 )[(s + a)2 +b2 ] s +α 39. s2[(s + a)2 + b2 ] s 2 + α1s + α0 40. s 2 (s + a)(s + b) 2 2 2 φ1 = atan2(−2aω, a2 + b2 − ω2 ) φ2 = atan2(2ab, a 2 − b 2 + ω 2 ) 1 α 2 + ω2 2 ( ) sin(ωt + φ1 ) ω c 1 1 (α − a)2 + b2 2 −at + [ ] e sin(bt +φ2 ) b c c = (2aω)2 + (a2 + b2 −ω2 )2 1 φ1 = atan2(ω , α ) − atan2( 2aω , a 2 + b 2 + ω 2 ) φ2 = atan2(b,α − a) + atan2(2ab, a 2 − b 2 − ω 2 ) 1 2 1 2α a [b + (α − a ) ] − at (α t + 1 − )+ e sin( bt + φ ) c bc c 2 2 c = a 2 + b2 φ = 2atan2(b, a) + atan2(b,α − a) α1 +α0t α0 (a + b) 1 α1 α0 ab − Laplace Table 1 2 2 2 [ 4a ω + ( a + b − ω ) ] 2 − (ab) 2 − a −b (1− a + 2 a )e−at α α 1 (1− 1 + 20 )e−bt b−a b b Page 5