Transient Process in RC Circuits with a DC Source: Complete Analytical Derivation of Switching Behavior

Abstract

The transient response of resistor-capacitor (RC) circuits under direct current (DC) excitation represents a fundamental topic in electrical engineering and applied physics. This article provides a rigorous, and fully derived of the charging and discharging processes, emphasizing formulas, physical interpretation, and engineering relevance.

Introduction

RC circuits are first-order linear dynamic systems whose transient behavior arises when a switch connects or disconnects a DC source. These processes describe how energy is stored and released in a capacitor and are mathematically governed by differential equation of the first order with constant coefficients.

Two key regimes are analyzed:

• Charging (switch ON)
• Discharging (switch OFF)

Mathematical Model of the RC Circuit

Applying Kirchhoff’s Voltage Law (KVL):

V0 = vR(t) + vC(t)

Using:

vR(t) = R * i(t)

i(t) = dq/dt = C * (dvC/dt)

Substitute:

V0 = R * C * (dvC/dt) + vC

Charging Process (Switch Closing at t = 0)

Differential Equation

Initial condition:

vC(0) = 0

Governing equation:

R * C * (dvC/dt) + vC = V0

Step-by-Step Derivation

Rearrange:

dvC/dt = (1 / (R * C)) * (V0 – vC)

Separate variables:

dvC / (V0 – vC) = dt / (R * C)

Integrate:

– ln(V0 – vC) = t / (R * C) + K

Solve:

V0 – vC = A * e^(-t/(R*C))

Apply initial condition:

vC(0) = 0 → A = V0

Final Voltage Expression

vC(t) = V0 * (1 – e^(-t/(R*C)))

Current Expression

i(t) = C * (dvC/dt)

Result:

i(t) = (V0 / R) * e^(-t/(R*C))

Physical Interpretation

• At t = 0:

vC = 0, i = V0 / R

• At t → ∞:

vC → V0, i → 0

• Capacitor behaves:
  • Initially as a short circuit
  • Eventually as an open circuit

Time Constant (Key Parameter)

τ = R * C

Meaning:

• At t = τ:

vC ≈ 0.632 * V0

Hence comes the definition of the time constant, which is the time for which the voltage (in general, the signal) reaches 63.2% of its maximum value.

• At t = 5 * τ:

vC ≈ 0.993 * V0

Discharging Process (Switch Opening)

Initial condition:

vC(0) = V0

Differential Equation

R * C * (dvC/dt) + vC = 0

Solution

Rearrange:

dvC/dt = -vC / (R*C)

Separate variables:

dvC / vC = -dt / (R*C)

Integrate:

ln(vC) = -t/(R*C) + K

Solve:

vC = A * e^(-t/(R*C))

Apply initial condition:

A = V0

Final Voltage Expression

vC(t) = V0 * e^(-t/(R*C))

Current Expression

i(t) = -(V0 / R) * e^(-t/(R*C))

Interpretation

• Voltage decreases exponentially
• Current reverses direction
• Energy is dissipated as heat in resistor

Unified General Solution

All first-order systems can be expressed as:

x(t) = x_inf + (x0 – x_inf) * e^(-t/τ)

Where:

τ = R * C

Energy Analysis

Energy stored in capacitor:

W = (1/2) * C * V0^2

Important result:

• 50% energy stored
• 50% dissipated in resistor during charging

Engineering Applications

• Low-pass filters
• Timing circuits
• Pulse shaping networks
• Power supply smoothing
• Communication systems

Advanced Insight: Exponential Nature

The exponential response:

e^(-t/(R*C))

appears due to:

• Linear differential equation (differential equation of the first order with constant coefficients)
• Energy storage element (capacitor)
• Dissipative element (resistor)

This same mathematical structure appears across:

• Thermal systems
• Mechanical damping
• Population decay models

Conclusion

The transient response of RC circuits to DC switching is one of the most elegant examples of first-order system dynamics. Through rigorous derivation and interpretation, we see how exponential laws emerge naturally from physical principles.

Mastery of these equations enables:

• Accurate circuit design
• Efficient signal analysis
• Deep understanding of dynamic systems