Lecture Notes: Understanding Impedance

Introduction to Impedance

• Impedance: Opposition to the flow of AC (Alternating Current) in a circuit.
• Relation to Resistance:
  • Resistance opposes both DC (Direct Current) and AC.
  • Impedance is specific to AC circuits.
• Symbol and Unit:
  • Symbol: Z
  • Unit: Ohms (Ω)

Elements Affecting Impedance

• Resistors, Inductors, and Capacitors impede AC current.
• Capacitive Reactance (Xc): Opposition provided by a capacitor.
• Inductive Reactance (Xl): Opposition provided by an inductor.
• Units: Both reactances are measured in Ohms.

Formula for Impedance

• Equation:
  • [ Z = \sqrt{R^2 + (Xl - Xc)^2} ]
  • R: Resistance
  • Xl: Inductive reactance
  • Xc: Capacitive reactance

Reactance and Frequency Relationships

• Inductive Reactance (Xl):
  • Increases with frequency.
  • Formula: [ Xl = 2\pi fL ]
  • L: Inductance, f: Frequency
• Capacitive Reactance (Xc):
  • Decreases with frequency.
  • Formula: [ Xc = \frac{1}{2\pi fC} ]
  • C: Capacitance
• Resonant Frequency:
  • Occurs when Xl = Xc.
  • Formula: [ f_r = \frac{1}{2\pi \sqrt{LC}} ]

Example Problems

Example 1: Circuit with Resistor and Capacitor

• Given:
  • Capacitance: 5µF
  • Resistance: 400Ω
  • AC Signal: 60Hz, 120V
• Calculate Current (I):
  • Use impedance to find current: [ I = \frac{V_{rms}}{Z} ]
  • Capacitive Reactance (Xc): 530.5Ω
  • Impedance (Z): 539.8Ω
  • Current: 222mA_

Example 2: RLC Circuit

• Given:
  • Resistance: 100Ω
  • Capacitance: 20µF
  • Inductance: 200mH
  • AC Signal: 60Hz, 120V
• Calculate:
  • Capacitive Reactance (Xc): 132.6Ω
  • Inductive Reactance (Xl): 75.4Ω
  • Impedance (Z): 115.2Ω
  • Current: 1.04A

Calculate Resonant Frequency

• Formula: [ f_r = \frac{1}{2\pi \sqrt{LC}} ]
• Given in Example 2:
  • Inductance (L): 0.2H
  • Capacitance (C): 20µF
• Resonant Frequency: 79.6Hz

Conclusion

• Understanding impedance helps determine current flow in AC circuits.
• Calculating the impedance, reactance, and resonant frequency are essential for analyzing AC circuits.