Lecture on Analyzing AC Circuits Using Phasor Diagrams

Introduction

• Topic: Analyzing AC circuits using phasor diagrams.
• Goal: Understand the relationship between applied voltage and current in AC circuits.
  • Focus on series LRC circuits.
• Method: Use vector diagrams (phasors) to represent amplitude and phase of voltages.

Basic Concepts

AC Generator

• Symbol: Circle with a sine wave inside, represents electromotive force (EMF).
• Voltage:
  • Denoted as ( V_{max} \cos(\omega t) ).
  • Amplitude: ( V_{max} ), Period defines one oscillation.
• Frequency:
  • Angular frequency (( \omega )) is related to frequency (f) by ( \omega = 2 \pi f ).

Phasor Representation

• Phasor: Vector representing voltage as ( V_{max} ) and angle ( \omega t ).
• Projection:
  • ( V_{applied} ) is the cosine projection along the x-axis.
  • Sine functions would be along the y-axis.

Simple AC Circuits

Resistor in AC Circuit

• Ohm's Law: Voltage drop across resistor = EMF of generator.
• Current: ( I(t) = \frac{V_{max}}{R} \cos(\omega t) ).
• Phasor Diagram: Current and voltage vectors are in phase._

Capacitor in AC Circuit

• Capacitance: Related to charge on plates and voltage.
• Current: Derived from rate of change of charge.
  • Phase Shift: Current leads voltage by ( \frac{\pi}{2} ).
• Capacitive Reactance: ( X_C = \frac{1}{\omega C} ).

Inductor in AC Circuit

• Inductive Reactance: ( X_L = \omega L ).
• Current: Derived from integration, current lags voltage by ( \frac{\pi}{2} ).
• Phase Relationship: Voltage leads the current.

Analysis of Series LRC Circuits

Kirchhoff's Voltage Law

• Equation: Sum of potential drops = applied voltage.
• Phase Considerations: Potential drops are not in phase.

Phasor Diagram Analysis

• Current Phasor: Defined as ( I(t) = I_{max} \cos(\omega t - \Delta) ).
• Components:
  • Resistor: In phase with current.
  • Capacitor: Voltage lags current by ( \frac{\pi}{2} ).
  • Inductor: Voltage leads current by ( \frac{\pi}{2} ).
• Phasor Addition: Use vector addition to solve for overall impedance and phase angle._

Impedance and Phase Angle

• Impedance (Z): ( Z = \sqrt{R^2 + (X_L - X_C)^2} ).
• Phase Angle ( \Delta ): Calculated from reactances.

Summary

• Phasor diagrams simplify complex AC circuit analysis.
• Key Tool: Phasors represent phase relationships effectively.

Conclusion

• Understanding phasors provides a better appreciation of AC circuit analysis.
• Encouragement to apply phasor diagrams in practical scenarios.

This lecture emphasized the role of phasors in simplifying the analysis of AC circuits, particularly in understanding the phase relationships between current and voltage across different circuit elements.