Lecture on RLC Circuit Analysis by Dr. P Montgomery

Introduction

• Focus: RLC circuit problem-solving session
• Circuit configuration: Parallel RLC circuit
• Initial Condition:
  • Switch is open, inductor is disconnected
  • Voltage source present, providing current and charge

Problem Set-up

• At time (T = 0):
  • Switch is closed, connecting inductor
  • Evaluating changes in voltages and currents

Initial Conditions

• Assume circuit at steady state before switch is closed
• Key parameters to evaluate:
  • Voltage across capacitor
  • Inductor current
  • Capacitor current
  • Derivative of voltage at (T = 0^+)

Evaluating Initial Conditions

Voltage Across Capacitor at (T = 0^+)

• Voltage at (T = 0^+) = Voltage at (T = 0^-)
• Circuit configuration:
  • Capacitor in parallel with 4-ohm resistor
  • 24V supply
• Capacitor acts as an open circuit at steady state
• Voltage divider method:
  • Voltage = (\frac{4}{8} \times 24 = 12V)

Inductor Current at (T = 0^+)

• No instantaneous change in inductor current
• Current at (T = 0^-) = 0 A (inductor disconnected)
• Therefore, inductor current at (T = 0^+) = 0 A

Capacitor Current at (T = 0^+)

• Instantaneous change possible in current
• Evaluate with switch closed (T = 0)
• Use source transformation:
  • Transformed circuit: Resistor in parallel with capacitor and inductor
  • Resistance = 2 ohms (parallel combination of two 4-ohm)
  • Current source: 6A (from 24V/4-ohm)
• Apply KCL at the node:
  • (-6 + \frac{12}{2} + I_{C}(T = 0^+) = 0)
  • (I_{C}(T = 0^+) = 0 A) (current cancellation)

Derivative of Voltage at (T = 0^+

• Use relationship: (I_C = C \frac{dV_C}{dt})
• Given (I_C(T = 0^+) = 0 A)
  • Therefore, (\frac{dV_C}{dt} = 0 V/s)

Final Voltage Condition

• As time approaches infinity
• Inductor in steady state acts as a short circuit
• Voltage across inductor = 0
• Voltage across capacitor = 0

Frequency and Damping Analysis

• General formulas for RLC circuit:
  • Damping frequency (\alpha = \frac{1}{2RC})
  • Natural frequency (\omega_n = \frac{1}{\sqrt{LC}})
• Calculations:
  • (R = 2 \text{ ohms}, C = \frac{1}{8} )
  • (L = 1) H
  • (\alpha^2 < \omega_n^2): underdamped response

Conclusion

• Recap of problem-solving approach
• Further analysis possible through frequency evaluations
• Anticipation for further learning in upcoming videos