Kondensator Entladevorgang
U c : = U 0 ⋅ ( e − t R ⋅ C ) {\displaystyle {U}_{c}\mathrm {\colon } ={U}_{0}\cdot \left({e}^{-{\frac {t}{R\cdot C}}}\right)} , τ : = R ⋅ C {\displaystyle \tau \mathrm {\colon } =R\cdot C}
Uo = Ladespannung
Uc = Spannung am Kondensator
τ(tau) = Zeitkonstante
C = Kapazität
R = Widerstand im Stromkreis
t = Zeit, Dauer
e = Eulersche Zahl (e = 2,718....)
Kondensator gilt nach 5τ als entladen.
Beispiel:
k Ω : = 1000 Ω {\displaystyle k\Omega \mathrm {\colon } =1000\Omega } , μ F : = 0.000001 F {\displaystyle {\mathit {\mathrm {\mu } F}}\mathrm {\colon } =0.000001F} , n F : = 0.000000001 F {\displaystyle {\mathit {nF}}\mathrm {\colon } =0.000000001F}
U 0 : = 10 V {\displaystyle {U}_{0}\mathrm {\colon } =10V} , R : = 5 k Ω {\displaystyle R\mathrm {\colon } =5k\Omega } , C : = 100 μ F {\displaystyle C\mathrm {\colon } =100{\mathit {\mathrm {\mu } F}}}
t : = 0.1 s ( 0 … 30 ) {\displaystyle t\mathrm {\colon } =0.1s\left(0\dots 30\right)}
τ : = R ⋅ C = 0.5000 s {\displaystyle \tau \mathrm {\colon } =R\cdot C={\text{0.5000}}s} , 5 ⋅ τ = 2.500 s {\displaystyle 5\cdot \tau ={\text{2.500}}s}
U c : = U 0 ⋅ ( e − t R ⋅ C ) {\displaystyle {U}_{c}\mathrm {\colon } ={U}_{0}\cdot \left({e}^{-{\frac {t}{R\cdot C}}}\right)}
p l o t x y ( t , U c ) {\displaystyle {\mathit {plotxy}}\left(t,{U}_{c}\right)}
i c : = − U 0 R ⋅ e − t R ⋅ C {\displaystyle {i}_{c}\mathrm {\colon } ={\frac {-{U}_{0}}{R}}\cdot {e}^{-{\frac {t}{R\cdot C}}}}
p l o t x y ( t , i c ) {\displaystyle {\mathit {plotxy}}\left(t,{i}_{c}\right)}
Z.B. Wann erreicht U c : = U 0 ⋅ ( e − t R ⋅ C ) {\displaystyle {U}_{c}\mathrm {\colon } ={U}_{0}\cdot \left({e}^{-{\frac {t}{R\cdot C}}}\right)} eine bestimmte Spannung?
Umwandeln der Formel
Beispiel: U c : = 4 V {\displaystyle {U}_{c}\mathrm {\colon } =4V}
t : = − R ⋅ C ⋅ ln ( U c U 0 ) {\displaystyle t\mathrm {\colon } =-R\cdot C\cdot \ln \left({\frac {{U}_{c}}{{U}_{0}}}\right)} t = 0.4581 s {\displaystyle t={\text{0.4581}}s}
Z.B. Wann erreicht i c : = − U 0 R ⋅ e − t R ⋅ C {\displaystyle {i}_{c}\mathrm {\colon } ={\frac {-{U}_{0}}{R}}\cdot {e}^{-{\frac {t}{R\cdot C}}}} einen bestimmten Strom?
Beispiel: i c : = 1 m A {\displaystyle {i}_{c}\mathrm {\colon } =1{\mathit {mA}}}
t : = − R ⋅ C ⋅ ln ( − i c ( U 0 R ) + 1 ) {\displaystyle t\mathrm {\colon } =-R\cdot C\cdot \ln \left({\frac {-{i}_{c}}{\left({\frac {{U}_{0}}{R}}\right)}}+1\right)} t = 0.3466 s {\displaystyle t={\text{0.3466}}s}