Mi libro de análisis de red me dijo que el voltaje a través del condensador C2 viene dado por:
$$ \ text {U} _ {\ text {C} _2} \ left (t \ right) = \ text {U} _ {\ text {in}} \ left (t \ right) + \ text {C} _1 \ cdot \ text {L} \ cdot \ text {U} _ {\ text {in}} '' \ left (t \ right) - \ text {L} \ cdot \ text {I} _ { \ text {in}} '\ left (t \ right) $$
Donde se ve la red:
¡¿Y quiero saber si es correcto y cómo lo obtuvieron?
MI TRABAJO:
$$ \ text {I} _ {\ text {C} _2} \ left (t \ right) = \ text {U} _ {\ text {C} _2} '\ left (t \ right) \ cdot \ text {C} _2 = \ frac {\ text {d}} {\ text {d} t} \ left (\ text {U} _ {\ text {in}} \ left (t \ right) - \ text {U} _ {\ text {L}} \ left (t \ right) \ right) \ cdot \ text {C} _2 = $$ $$ \ frac {\ text {d}} {\ text {d} t} \ left (\ text {U} _ {\ text {in}} \ left (t \ right) - \ text {I} _ { \ text {L}} '\ left (t \ right) \ cdot \ text {L} \ right) \ cdot \ text {C} _2 = $$ $$ \ frac {\ text {d}} {\ text {d} t} \ left (\ text {U} _ {\ text {in}} \ left (t \ right) - \ text {I} _ { \ text {C} _2} '\ left (t \ right) \ cdot \ text {L} \ right) \ cdot \ text {C} _2 $$
Entonces, obtenemos:
$$ \ text {I} _ {\ text {C} _2} \ left (t \ right) = \ text {U} _ {\ text {C} _2} '\ left (t \ right) \ cdot \ text {C} _2 = \ left (\ text {U} _ {\ text {in}} '\ left (t \ right) - \ text {I} _ {\ text {C} _2}' '\ left (t \ right) \ cdot \ text {L} \ right) \ cdot \ text {C} _2 \ Longleftrightarrow $$ $$ \ text {U} _ {\ text {C} _2} '\ left (t \ right) = \ text {U} _ {\ text {in}}' \ left (t \ right) - \ text { I} _ {\ text {C} _2} '' \ left (t \ right) \ cdot \ text {L} \ Longleftrightarrow $$ $$ \ int \ text {U} _ {\ text {C} _2} '\ left (t \ right) \ space \ text {d} t = \ int \ text {U} _ {\ text {in}} '\ left (t \ right) \ space \ text {d} t- \ int \ text {I} _ {\ text {C} _2}' '\ left (t \ right) \ cdot \ text {L} \ espacio \ texto {d} t \ Longleftrightarrow $$ $$ \ color {rojo} {\ text {U} _ {\ text {C} _2} \ left (t \ right) = \ text {U} _ {\ text {in}} \ left (t \ right) - \ text {I} _ {\ text {C} _2} '\ left (t \ right) \ cdot \ text {L}} $$
Y sabiendo que:
$$ \ text {I} _ {\ text {C} _2} \ left (t \ right) = \ text {I} _ {\ text {in}} \ left (t \ right) - \ text {I} _ {\ text {C} _1} \ left (t \ right) = \ text {I} _ {\ text {in}} \ left (t \ right) - \ text {U} _ {\ text {in}} '\ left (t \ right) \ cdot \ text {C} _1 $$
Podemos escribir la parte roja:
$$ \ color {red} {\ text {U} _ {\ text {C} _2} \ left (t \ right) = \ text {U} _ {\ text {in}} \ left (t \ a la derecha) - \ frac {\ text {d}} {\ text {d} t} \ left (\ text {I} _ {\ text {in}} \ left (t \ right) - \ text {U} _ {\ text {in}} '\ left (t \ right) \ cdot \ text {C} _1 \ right) \ cdot \ text {L}} = $$ $$ \ color {rojo} {\ text {U} _ {\ text {C} _2} \ left (t \ right) = \ text {U} _ {\ text {in}} \ left (t \ right) - \ left (\ text {I} _ {\ text {in}} '\ left (t \ right) - \ text {U} _ {\ text {in}}' '\ left (t \ right) \ cdot \ text {C} _1 \ right) \ cdot \ text {L}} $$