Capacitor Charging Equations to Know for Electrical Circuits and Systems I

Capacitor charging equations are essential for understanding how capacitors behave in electrical circuits. These equations describe voltage, current, charge, and energy storage, helping us analyze transient responses and design circuits effectively in Electrical Circuits and Systems I.

1. Capacitor voltage equation: v(t) = V(1 - e^(-t/RC))

   • Describes how the voltage across a capacitor increases over time as it charges.
   • The term "V" represents the maximum voltage the capacitor will reach.
   • The exponential term indicates that the voltage approaches V asymptotically, never quite reaching it in finite time.

2. Capacitor current equation: i(t) = (V/R)e^(-t/RC)

   • Shows how the current flowing into the capacitor decreases over time as it charges.
   • The initial current is at its maximum when t=0, equal to V/R.
   • The exponential decay reflects the diminishing flow of charge as the capacitor approaches full charge.

3. Time constant: τ = RC

   • Represents the time it takes for the voltage (or current) to reach approximately 63.2% of its final value.
   • A larger time constant indicates a slower charging process.
   • It is a crucial parameter for analyzing the transient response of RC circuits.

4. Charge stored in capacitor: Q = CV

   • Relates the charge (Q) stored in the capacitor to its capacitance (C) and the voltage (V) across it.
   • Indicates that the charge increases linearly with both capacitance and voltage.
   • Essential for understanding how much energy can be stored in the capacitor.

5. Energy stored in capacitor: E = (1/2)CV^2

   • Calculates the energy (E) stored in a charged capacitor.
   • Shows that energy increases with the square of the voltage, emphasizing the importance of voltage in energy storage.
   • This equation is fundamental for applications involving energy storage and transfer.

6. Capacitor power equation: P(t) = Vi = (V^2/R)e^(-t/RC)

   • Describes the instantaneous power delivered to the capacitor as it charges.
   • Power decreases over time, reflecting the reduction in current as the capacitor approaches full charge.
   • Important for understanding energy transfer in circuits.

7. Voltage across resistor: v_R(t) = Ve^(-t/RC)

   • Indicates how the voltage drop across the resistor decreases as the capacitor charges.
   • The voltage across the resistor is initially equal to the source voltage and approaches zero as the capacitor is fully charged.
   • This relationship is key for analyzing the behavior of RC circuits.

8. Final voltage across capacitor: v_C(∞) = V

   • States that the final voltage across the capacitor will equal the supply voltage after a long time.
   • This is a critical concept for understanding steady-state conditions in circuits.
   • Reinforces the idea that capacitors can store energy up to a certain limit defined by the voltage source.

9. Initial current: i(0) = V/R

   • Indicates the maximum current flowing into the capacitor at the moment charging begins.
   • This initial current is determined by the voltage source and the resistance in the circuit.
   • Important for understanding the immediate response of the circuit when power is applied.

10. Time to reach 63.2% of final value: t = RC

    • Highlights the significance of the time constant in determining how quickly the capacitor charges.
    • This time frame is critical for analyzing transient responses in circuits.
    • Understanding this concept helps in designing circuits with desired charging characteristics.