TRANSFORMADA DE LAPLACE ∞ F ( s ) = TL [ f (t )]= ∫ f (t ) ⋅ e − st dt 0 Pares transformados: Propiedades: suponiendo siempre f j (t < 0 ) = 0 ; ∀j f (t ) F (s ) δ(t ) 1 K ⋅ u(t ) K s e − at ⋅ u (t ) sen kt ⋅ u (t ) 1 s+a k coskt ⋅ u (t ) s2 + k 2 s • Producto por exponencial: TL e s2 + k 2 b • Acción sobre 1ª derivada: TL ⎢ ⎥ = s ⋅ F ( s ) − f (0) ⎣ dt ⎦ e − at ⋅ senbt ⋅ u (t ) e − at ⋅ cosbt ⋅ u (t ) (s + a) 2 + b 2 s+a (s + a) 2 + b 2 1 s2 1 t ⋅ u (t ) t ⋅ e − at ⋅ u (t ) (s + a) 2 t n −1e − at ⋅ u (t ) (n − 1)! 1 ( s + a)n ; n = 1,2,3,... • Linealidad: TL [a1 ⋅ f1 (t ) + a2 ⋅ f 2 (t )] = a1 ⋅ TL[ f1 (t )] + a2 ⋅ TL[ f 2 (t )] −1 • Invertibilidad: f (t ) = TL [F ( s )] • Escalado de variable: TL[ f ( a ⋅ t )] = 1 ⎛s⎞ ⋅ F⎜ ⎟ a ⎝a⎠ [ m at ⋅ f (t )]= F (s ± a) ⎡ df ⎤ • Acción sobre 2ª derivada: ⎡d 2 f ⎤ 2 df = s ⋅ F ( s ) − s ⋅ f (0) − TL ⎢ ⎥ 2 dt ⎢⎣ dt ⎥⎦ t =0 ⎡t ⎤ F ( s) • Acción sobre integrales: TL ⎢ ∫ f (τ) ⋅ d τ⎥ = s ⎣0 ⎦ • Acción sobre constantes: TL[ f (t ) = K ⋅ u (t )] = K s IMPEDANCIAS TRANSFORMADAS + V (s) I (s) + + IR(s) VR(s) Z ( s) = 1/sC VC(s) R 1v s - sL VL(s) IC(s) + - C(0) LiL(0) - - VR ( s ) = R ⋅ I R ( s ) ⇒ Z R (s) = R VC ( s ) = IL(s) + 1 1 ⋅ I C ( s ) + ⋅ vC (0) sC s VL ( s ) = sL ⋅ I L ( s ) − L ⋅ i L (0) 1 sC ⇒ Z L ( s ) = sL ⇒ Z C ( s) = ADMITANCIAS TRANSFORMADAS Y (s) = I ( s) V ( s) + IC(s) IR(s) VR(s) 1/sC R + VC(s) - IL(s) CvC(0) sL + VL(s) - 1 s iL(0) - I R (s) = 1 ⋅ VR ( s ) R ⇒ YR ( s ) = 1 R I C ( s ) = sC ⋅ VC ( s ) − C ⋅ vC (0) ⇒ YC ( s ) = sC I L ( s) = 1 1 ⋅ V L ( s ) + ⋅ i L ( 0) sL s ⇒ YL ( s ) = 1 sL 1.- LEY TRANSFORMACIONES DE LEYES DE KIRCHHOFF DE MALLAS N v1 (t ) + v2 (t ) + v3 (t ) + ... + v N (t ) = ∑ v j (t ) = 0 j =1 TL N V1 ( s ) + V2 ( s ) + V3 ( s ) + ... + V N ( s ) = ∑ V j ( s ) = 0 j =1 Entendida, por supuesto, como una suma algebraica, donde cada tensión transformada V j ( s ) = TL v j (t ) ; ∀j contribuye en el sumatorio con su signo [ ] correspondiente, dependiendo de la polaridad. 2.- LEY DE KIRCHHOFF DE NUDOS N i1 (t ) + i2 (t ) + i3 (t ) + ... + i N (t ) = ∑ i j (t ) = 0 j =1 TL N I1 ( s ) + I 2 ( s ) + I 3 ( s ) + ... + I N ( s ) = ∑ I j ( s ) = 0 j =1 Entendida, por supuesto, como una suma algebraica, donde cada corriente transformada I j ( s ) = TL i j (t ) ; ∀j contribuye en el sumatorio con su signo [ ] correspondiente, dependiendo del sentido. 3.- LEY DE COMPORTAMIENTO RESISTIVO v R (t ) = R ⋅ i R (t ) 4.- LEY vC (t ) = DE TL ⎯⎯→ VR ( s ) = R ⋅ I R ( s ) COMPORTAMIENTO CAPACITIVO 1 t 1 1 TL ⋅ ∫ iC (τ)dτ + vC (0) ⎯⎯→ VC ( s ) = ⋅ I C ( s ) + ⋅ vC (0) C 0 sC s 5.- LEY DE v L (t ) = L ⋅ COMPORTAMIENTO INDUCTIVO diL dt TL ⎯⎯→ VL ( s ) = sL ⋅ I L ( s ) − L ⋅ iL (0)