Tengo un circuito de la serie R y C, con una fuente de voltaje de CC. Quiero encontrar la función de carga del condensador. Mi pregunta es: ¿es correcto el método que uso?
Mi trabajo (usando la transformada de Laplace):
$$ \ begin {cases} \ text {U} _ {\ text {in}} (t) = \ text {U} _ {\ text {C}} (t) + \ text {U} _ {\ text {R}} (t) \\ \\ \ text {U} _ {\ text {R}} (t) = \ text {I} _ {\ text {R}} (t) \ cdot \ text {R} \\ \\ \ text {I} _ {\ text {C}} (t) = \ text {U} _ {\ text {C}} '(t) \ cdot \ text {C} \\ \\ \ text {I} _ {\ text {in}} (t) = \ text {I} _ {\ text {C}} (t) = \ text {I} _ {\ text {R}} (t) \ end {cases} \ espacio \ espacio \ espacio \ espacio \ espacio \ espacio \ Longrightarrow ^ {\ mathcal {L}} \ espacio \ espacio \ espacio \ espacio \ espacio \ espacio \ begin {cases} \ text {U} _ {\ text {in}} (\ text {s}) = \ text {U} _ {\ text {C}} (\ text {s}) + \ text {U} _ {\ texto {R}} (\ text {s}) \\ \\ \ text {U} _ {\ text {R}} (\ text {s}) = \ text {I} _ {\ text {R}} (\ text {s}) \ cdot \ text {R} \\ \\ \ text {I} _ {\ text {C}} (\ text {s}) = \ text {C} \ cdot \ text {s} \ cdot \ text {U} _ {\ text {C}} (\ text {s}) - \ text {C} \ cdot \ text {U} _ {\ text {C}} (0) \\ \\ \ text {I} _ {\ text {in}} (\ text {s}) = \ text {I} _ {\ text {C}} (\ text {s}) = \ text {I} _ {\ texto {R}} (\ text {s}) \ end {cases} $$
Entonces, obtenemos:
$$ \ text {U} _ {\ text {in}} (\ text {s}) = \ frac {\ text {I} _ {\ text {in}} (\ text {s}) + \ text {C} \ cdot \ text {U} _ {\ text {C}} (0)} {\ text {C} \ cdot \ text {s}} + \ text {I} _ {\ text {en }} (\ text {s}) \ cdot \ text {R} \ Longleftrightarrow \ text {I} _ {\ text {in}} (\ text {s}) = \ frac {\ text {U} _ {\ text {in}} (\ text {s}) - \ frac {\ text {U} _ {\ text {C}} (0)} {\ text {s}}} {\ text {R} + \ frac {1} {\ text {C} \ cdot \ text {s}}} $$
Entonces, cuando quiero encontrar U_c (s):
$$ \ text {I} _ {\ text {in}} (t) = \ text {U} _ {\ text {C}} '(t) \ cdot \ text {C} \ espacio \ espacio \ space \ space \ space \ space \ Longrightarrow ^ {\ mathcal {L}} \ space \ space \ space \ space \ space \ space \ frac {\ text {U} _ {\ text {in}} (\ text { s}) - \ frac {\ text {U} _ {\ text {C}} (0)} {\ text {s}}} {\ text {R} + \ frac {1} {\ text {C} \ cdot \ text {s}}} = \ text {C} \ cdot \ text {s} \ cdot \ text {U} _ {\ text {C}} (\ text {s}) - \ text {C} \ cdot \ text {U} _ {\ text {C}} (0) $$
Resolviendo U_c (s), me da:
$$ \ text {U} _ {\ text {C}} (\ text {s}) = \ frac {\ frac {\ text {U} _ {\ text {in}} (\ text {s }) - \ frac {\ text {U} _ {\ text {C}} (0)} {\ text {s}}} {\ text {R} + \ frac {1} {\ text {C} \ cdot \ text {s}}} + \ text {C} \ cdot \ text {U} _ {\ text {C}} (0)} {\ text {C} \ cdot \ text {s}} $$
Sabiendo que la fuente de voltaje es DC:
$$ \ text {U} _ {\ text {in}} (\ text {s}) = \ frac {\ text {U} _ {\ text {in}}} {\ text {s}} $$
Entonces:
$$ \ color {red} {\ text {U} _ {\ text {C}} (\ text {s}) = \ frac {\ frac {\ frac {\ text {U} _ {\ text {en}}} {\ text {s}} - \ frac {\ text {U} _ {\ text {C}} (0)} {\ text {s}}} {\ text {R} + \ frac {1} {\ text {C} \ cdot \ text {s}}} + \ text {C} \ cdot \ text {U} _ {\ text {C}} (0)} {\ text {C} \ cdot \ text {s}} = \ frac {\ frac {\ text {U} _ {\ text {in}} - \ text {U} _ {\ text {C}} (0)} {\ text {R } \ cdot \ text {s} + \ frac {1} {\ text {C}}} + \ text {C} \ cdot \ text {U} _ {\ text {C}} (0)} {\ text {C} \ cdot \ text {s}}} $$
Usando la transformada inversa de Laplace, encontré:
$$ \ text {U} _ {\ text {C}} (t) = \ text {U} _ {\ text {in}} + e ^ {- \ frac {t} {\ text {CR }}} \ left (\ text {U} _ {\ text {C}} (0) - \ text {U} _ {\ text {in}} \ right) $$