CUESTIONES FUNDAMENTALES.pdf

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)