RC-Circuits.md

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RC-Circuits

RC-circuits are circuits involving capacitors, resistors, batteries, and switches.

Transient Equations

Capacitor Time Constant ($\tau_C$) - The time required for the current to change by 60% of it's maximum value starting at $t=0$.

$$\Large \tau_C = RC$$

Charging Capacitor

Only works for a circuit with one resistor, one voltage source, and one capacitor.

Current

$$\Large i(t) = \frac{\mathcal{E}}{R} \left(e^{-t/\tau_C} \right)$$

LEGEND: $i(t)$ - Current through the capacitor. $\mathcal{E}$ - Voltage of the power source. $R$ - Resistance of the resistor. $\tau_C$ - Capacitor time constant. $t$ - Time since charging started.

At $t=0$, the capacitor acts as a normal wire with $0$ resistance.

At $t\to\infty$, the capacitor blocks any current from going through it.

Voltage

$$\Large V(t) = \mathcal{E} \left(1 - e^{-t/\tau_C} \right)$$

LEGEND: $V(t)$ - Voltage across the capacitor. $\mathcal{E}$ - Voltage of the power source. $\tau_C$ - Capacitor time constant. $t$ - Time since charging started.

Charge

$$\Large Q(t) = C\mathcal{E} \left(1 - e^{-t/\tau_C} \right)$$

LEGEND: $V(t)$ - Voltage across the capacitor. $\mathcal{E}$ - Voltage of the power source. $C$ - Capacitance of the capacitor. $\tau_C$ - Capacitor time constant. $t$ - Time since charging started.

Discharging Capacitor

Only works for a circuit with one resistor and one capacitor.

Current

$$\Large i(t) = i_0 \left(e^{-t/\tau_C} \right)$$

LEGEND: $i(t)$ - Current through the inductor. $i_0$ - Initial current through the inductor. $\tau_L$ - Inductor time constant. $t$ - Time since discharging started.

Voltage

$$\Large V(t) = V_0 \left(e^{-t/\tau_C} \right)$$

LEGEND: $V(t)$ - Voltage across the capacitor. $V_0$ - Initial voltage across the capacitor. $\tau_C$ - Capacitor time constant. $t$ - Time since discharging started.

Charge

$$\Large Q(t) = Q_0 \left(e^{-t/\tau_C} \right)$$

LEGEND: $V(t)$ - Voltage across the capacitor. $\mathcal{E}$ - Voltage of the power source. $C$ - Capacitance of the capacitor. $\tau_C$ - Capacitor time constant. $t$ - Time since discharging started.