Equation Of Charging And Discharging Of Capacitor

The equation of charging and discharging of acapacitor describes how voltage and current change over time when a capacitor is connected to a resistive circuit, forming the foundation for understanding transient behavior in electronics. This relationship is governed by an exponential function that depends on the resistance, capacitance, and the applied voltage, and it appears in countless applications from timing circuits to power supplies.

Introduction

Capacitors store energy in an electric field and release it when needed. When a capacitor is placed in series with a resistor and a voltage source, the voltage across the capacitor does not jump instantly to the source value; instead, it follows a predictable curve described by the equation of charging and discharging of capacitor. Grasping this equation allows engineers to design delays, filters, and pulse‑shaping networks with precision.

Basic Concepts Before diving into the mathematics, it helps to recall a few key definitions:

• Capacitance (C) – measured in farads (F), it quantifies how much charge a capacitor can hold per volt of potential difference.
• Resistance (R) – measured in ohms (Ω), it opposes the flow of electric current.
• Time constant (τ) – the product τ = R·C, representing the time required for the capacitor’s voltage to reach about 63.2 % of its final value during charging, or to fall to about 36.8 % during discharging. - Instantaneous voltage (V(t)) – the voltage across the capacitor at any moment t.
• Instantaneous current (I(t)) – the current flowing through the resistor‑capacitor branch at time t.

These quantities are interrelated through Kirchhoff’s voltage law and the capacitor’s defining equation I = C·dV/dt.

Charging Equation Derivation

Consider a series RC circuit connected to a DC source V₀ at time t = 0. Applying Kirchhoff’s voltage law:

[ V₀ = V_R(t) + V_C(t) ]

where V_R(t) = I(t)·R and V_C(t) = V(t). Substituting I(t) = C·dV/dt gives:

[ V₀ = RC \frac{dV(t)}{dt} + V(t) ]

Re‑arranging:

[ \frac{dV(t)}{dt} + \frac{1}{RC}V(t) = \frac{V₀}{RC} ]

This is a first‑order linear differential equation. Its solution, assuming the capacitor starts uncharged (V(0) = 0), is:

[ \boxed{V(t) = V₀ \left(1 - e^{-t/RC}\right)} ]

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The corresponding current follows from I(t) = C·dV/dt:

[ \boxed{I(t) = \frac{V₀}{R} e^{-t/RC}} ]

Both expressions contain the exponential term e^{-t/RC}, highlighting the role of the time constant τ = RC.

Discharging Equation Derivation

For discharging, the capacitor is initially charged to a voltage V₀ and then connected across a resistor with no external source. Kirchhoff’s law now reads:

[ 0 = V_R(t) + V_C(t) \quad \Rightarrow \quad V_R(t) = -V_C(t) ]

Using V_R = I·R and I = C·dV/dt:

[ RC \frac{dV(t)}{dt} + V(t) = 0 ]

Solving with the initial condition V(0) = V₀ yields:

[ \boxed{V(t) = V₀ e^{-t/RC}} ]

The current during discharge is:

[\boxed{I(t) = -\frac{V₀}{R} e^{-t/RC}} ]

(The negative sign indicates current flows opposite to the charging direction.)

Time Constant and Its Significance

The time constant τ = RC appears repeatedly in the charging and discharging equations. It provides an intuitive measure of how fast the capacitor responds:

• After one τ, the voltage reaches ≈63.2 % of its final value while charging, or drops to ≈36.8 % while discharging.
• After 5τ, the voltage is within 1 % of its steady‑state value, which engineers often treat as “fully charged” or “fully discharged.”

Thus, selecting appropriate R and C values lets designers set precise timing intervals for applications like debouncing switches, generating PWM signals, or creating delay lines.

Practical Examples

Example 1: LED Flasher

A simple astable multivibrator uses two RC networks to alternately charge and discharge capacitors, producing a flashing LED. If each RC pair has R = 10 kΩ and C = 100 µF, then τ = 1 s. The LED will be on for roughly 0.69τ and off for a similar interval, giving a flash period of