Detailed Calculation Formula for Voltage and Time in Capacitor Charging Process

Technical Discussion Analog Electronics
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Detailed Explanation of Capacitor Charging Voltage and Time Calculation Formulas

Capacitor charging is one of the fundamental operating processes of RC circuits, and its voltage variation over time follows an exponential law. Below are the core formulas for voltage and time during capacitor charging, along with parameter definitions and practical calculation examples.

I. Core Parameter Definitions

  • V_0 : Initial voltage on the capacitor (capacitor voltage at the start of charging)
  • V_u : Charging voltage (charging voltage provided by the power supply; when the capacitor starts charging from 0, V_u=E , where E is the power supply voltage)
  • V_t : Voltage on the capacitor at any time \( t \)
  • R : Resistance value in the circuit
  • C : Capacitance of the capacitor
  • RC : Time constant of the RC circuit (the key parameter determining charging speed)

II. Capacitor Charging Voltage Formula

When a capacitor is charging, the voltage at any time t follows an exponential growth law, with the core formula being:

V_t = V_0 + (V_u - V_0) \times \left(1 - e^{-\frac{t}{RC}}\right)

When the capacitor starts charging from 0 (\( V_0=0 \)) and the charging voltage \( V_u=E \) (power supply voltage), the formula can be simplified to:

V_t = 0 + (E - 0) \times \left(1 - e^{-\frac{t}{RC}}\right) \tag{Formula 1}

Further simplified to the commonly used form:

V_t = E \times \left(1 - e^{-\frac{t}{RC}}\right) \tag{Formula 2}

III. Capacitor Charging Time Calculation Formula

From the voltage formula (Formula 2), we can derive the formula to calculate the corresponding time from a known capacitor voltage. The steps are as follows:

  1. Isolate the exponential term:
e^{-\frac{t}{RC}} = \frac{E - V_t}{E} \tag{Formula 3}
  1. Take the natural logarithm of both sides of the equation:
\ln\left( \frac{E - V_t}{E} \right) = -\frac{t}{RC} \tag{Formula 4}
  1. Rearrange to obtain the expression for time t :
-t = RC \times \ln\left( \frac{E - V_t}{E} \right) \tag{Formula 5}