Tuesday, March 24, 2020

Dynamic Formulas for the Capacitor and for the Inductor


Charging the capacitor from the CV(=Vs) src. ::
The voltage drop on the series resistor \[V_R=V_S-V_C\]
\[t_{0_{↑}^{↑}→V_C}=-R·C·ln\left({1-\frac{V_C}{V_S}}\right)=R·C·ln\frac{V_S}{V_R}\]
\[t_{V_{1_C}{}_{↑}^{↑}→V_{2_C}}=R·C·ln\frac{V_S-V_{1_C}}{V_S-V_{2_C}}=R·C·ln\frac{V_{1_R}}{V_{2_R}}\]
Dis-charging the capacitor through a fixed value resistor ::
\[t_{V_{2_C}{}_{↓}^{↓}→V_{1_C}}=-R·C·ln\frac{V_{1_C}}{V_{2_C}}=R·C·ln\frac{V_{2_C}}{V_{1_C}}\]

"Charging" the inductor from the CV(=Vs) src. ::
The peak current \[I_{MAX}=\frac{V_S}R\]
\[t_{0_{↑}^{↑}→I_L}=-\frac LR·ln\left({1-\frac{I_L}{I_{MAX}}}\right)\]
\[t_{I_{1_L}{}_{↑}^{↑}→I_{2_L}}=\frac LR·ln\frac{I_{MAX}-I_{1_L}}{I_{MAX}-I_{2_L}}\]
"Dis-charging" the inductor through a fixed value resistor ::
\[t_{I_{2_L}{}_{↓}^{↓}→I_{1_L}}=\frac LR·ln\frac{I_{2_L}}{I_{1_L}}\]


[Eop]

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