Series RLC Circuit Analysis

Introduction

• Focus: Series RLC Circuit
• Components: Capacitor, Inductor, Resistor, AC Signal

Circuit Details

• Resistor: 30 Ω
• Inductor: 100 mH
• Capacitor: 40 µF
• AC Signal: 15 V, 60 Hz

Part A: Capacitive Reactance

• Formula: [ X_C = \frac{1}{2 \pi f C} ]
  • Frequency ( f = 60 ) Hz
  • Capacitance ( C = 40 \times 10^{-6} ) F
• Capacitive Reactance: ( X_C = 66.3 ) Ω

Part B: Inductive Reactance

• Formula: [ X_L = 2 \pi f L ]
  • Inductance ( L = 100 \times 10^{-3} ) H
• Inductive Reactance: ( X_L = 37.7 ) Ω

Impedance Calculation

• Formula: [ Z = \sqrt{R^2 + (X_L - X_C)^2} ]
• Given:
  • Resistance ( R = 30 ) Ω
  • Inductive Reactance ( X_L = 37.7 ) Ω
  • Capacitive Reactance ( X_C = 66.3 ) Ω
• Impedance (Z): 41.45 Ω

Part C: RMS Current

• Formula: [ I_{RMS} = \frac{V}{Z} ]
• RMS Current: 0.3619 A_

Part D: Voltage Across Components

• Resistor (V_R):

  • Formula: ( V_R = I_{RMS} \times R )
  • Voltage: 10.857 V

• Inductor (V_L):

  • Formula: ( V_L = I_{RMS} \times X_L )
  • Voltage: 13.644 V

• Capacitor (V_C):

  • Formula: ( V_C = I_{RMS} \times X_C )
  • Voltage: 23.994 V

• Verification:

  • Formula: ( V_S = \sqrt{V_R^2 + (V_L - V_C)^2} )
  • Calculated Source Voltage: ~15 V_

Part E: Power Consumption

• Power Formula:

  • [ P = I_{RMS}^2 \times R ]
  • Power: 3.929 W

• Alternative Formula:

  • [ P = V_{RMS} \times I_{RMS} \times \text{Power Factor} ]
  • Power Factor: ( \frac{R}{Z} = 0.7238 )
  • Result: 3.929 W (confirms previous calculation)_

Resonant Frequency

• Formula: [ f_0 = \frac{1}{2 \pi \sqrt{LC}} ]
  • ( L = 0.1 ) H
  • ( C = 40 \times 10^{-6} ) F
• Resonant Frequency: 79.6 Hz
• Key Point: At resonant frequency, ( X_L = X_C ) and ( Z = R )