Transient Response of RLC Circuits

Introduction

• Focus on transient response for RLC circuits (series and parallel).
• Importance in applications:
  • Tuning circuits in radio communications.
  • Voltage multipliers.
  • Passive filters.

Series RLC Circuit

Circuit Description

• Resistor (R), Inductor (L), and Capacitor (C) connected in series.
• DC voltage applied at time t = 0.
• Initial conditions are zero (no stored energy).

Applying Kirchhoff's Voltage Law (KVL)

• At time t = 0+:
  • KVL: V = V_R + V_L + V_C
  • Current (I) flowing through the circuit.
• Voltage across elements:
  • V_R = I * R
  • V_L = L * (di/dt)
  • I = I_C = C * (dV_C/dt)

Differential Equation

• Rearranging leads to:
  • d²V_C/dt² + (R/L)(dV_C/dt) + (1/LC)V_C = V/(LC)
• Second-order linear differential equation.

General Solution

• Total solution = Complementary Function (Transient Response) + Particular Integral (Steady State Response).
• Steady state: V_C(t → ∞) = V_V.

Finding Transient Response

• Complementary function derived by setting excitation to zero:
  • d²V_C/dt² + (R/L)(dV_C/dt) + (1/LC)V_C = 0.
  • Characteristic equation: D² + (R/L)D + (1/LC) = 0.

Roots and Damping Response

1. Overdamped:
   • If (R/2L)² > (1/LC):
   • Roots are real and negative.
2. Critically Damped:
   • If (R/2L)² = (1/LC):
   • Roots are real and equal.
3. Underdamped:
   • If (R/2L)² < (1/LC):
   • Roots are complex conjugates.
4. Pure Oscillation:
   • If R = 0:
   • Roots are purely imaginary.

Graphical Representation

• Overdamped: Sluggish response, slow to reach final value.
• Critically damped: Faster than overdamped, no oscillations.
• Underdamped: Fastest response, overshoots and oscillations.

Parallel RLC Circuit

Circuit Description

• Current source connected to RLC components in parallel.
• Initial conditions are zero.

Applying Kirchhoff's Current Law (KCL)

• KCL equation: I = (V/R) + C(dV/dt) + (1/L)∫V dt.
• Rearranging leads to:
  • d²V/dt² + (1/RC)(dV/dt) + (1/LC)V = 0.

Roots and Damping Response (Parallel)

• Similar analysis with cases as in series circuit:

1. Overdamped:
   • If (1/2RC)² > (1/LC).
2. Critically Damped:
   • If (1/2RC)² = (1/LC).
3. Underdamped:
   • If (1/2RC)² < (1/LC).
4. Pure Oscillation:
   • If R = 0.

Examples

• Example 1: Series RLC Circuit
  • Find current I(t) and evaluate at t = 0.1s.
• Example 2: Parallel RLC Circuit
  • Find voltage V(t) and analyze behavior.

Conclusion

• Understanding transient responses for both series and parallel RLC circuits is crucial for electronics applications.
• Encouragement to experiment with component values to observe changes in circuit behavior.