Tengo el siguiente problema:
Tenemos un circuito RLC en serie y me gustaría encontrar la función: $$ \ text {V} _ {C} (t) $$ el voltaje a través del capacitor.
Dadas las siguientes cosas:
- $$ \ text {V} _ {\ text {in}} (t) = 20 \ espacio \ text {V} $$
- $$ \ text {C} = 4 \ espacio \ text {F} $$
- $$ \ text {L} = 1 \ espacio \ text {H} $$
- $$ \ text {R} = 5 \ space \ Omega $$
- $$ \ text {I} _ {C} (0) = - 2 \ space \ text {A} $$
- $$ \ text {V} _ {C} (0) = 10 \ space \ text {V} $$
- $$ \ text {I} _ {T} (t) = \ text {I} _ {R} (t) \ espacio \ text {es el total actual} $$
Mi trabajo:
$$ \ text {V} _ {\ text {in}} (t) = \ text {V} _ {R} (t) + \ text {V} _ {C} (t) + \ text {V} _ {L} (t) \ Longleftrightarrow $$
Sabiendo que:
- $$ \ text {V} _ {L} (t) = \ text {LI} '_ {L} (t) = \ text {LI}' _ {T} (t) $$
- $$ \ text {I} _ {C} (t) = \ text {I} _ {T} (t) = \ text {CV} '_ {C} (t) \ to \ text {I } '_ {C} (t) = \ text {I}' _ {T} (t) = \ text {CV} '' _ {C} (t) $$
$$ \ text {V} _ {\ text {in}} (t) = \ text {RI} _ {T} (t) + \ text {V} _ {C} (t) + \ text {LI} '_ {T} (t) \ Longleftrightarrow $$ $$ \ text {V} _ {\ text {in}} (t) = \ text {R} \ cdot \ text {CV} '_ {C} (t) + \ text {V} _ {C} ( t) + \ text {L} \ cdot \ text {CV} '' _ {C} (t) \ Longleftrightarrow $$ $$ \ text {V} _ {\ text {in}} (t) = \ text {CRV} '_ {C} (t) + \ text {V} _ {C} (t) + \ text {CLV } '' _ {C} (t) $$
Entonces el problema se convierte en:
$$ \ begin {cases} 20 \ text {V} '_ {C} (t) + \ text {V} _ {C} (t) +4 \ text {V}' '_ {C} (t) = 20 \\ \ text {V} _ {C} (0) = 10 \\ 4 \ text {V} '_ {C} (0) = - 2 \ end {cases} $$