CHAPTER 11 (1): PHASORS AND COMPLEX NUMBERS

Chapter 11: Phasors and Complex Numbers (Part 1)

Introduction to Phasors

• Definition: A phasor is a vector with a constant magnitude that rotates at a constant angular velocity.

• Usefulness:

  • Graphic representation of sine waves in terms of magnitude and phase angle.

  • Analysis of AC circuits with sinusoidal sources.

Objectives of Part 1

• Understand how a phasor represents a sine wave.

• Draw phasor diagrams for single and multiple sine waves.

• Write angular and time domain expressions for sine waves.

Phasor Diagrams

• Characteristics:

  • Abstract representations of quantities with both magnitude and direction.

  • Magnitude represents amplitude.

  • Direction corresponds to phase angle.

  • 

• Sinusoidal Representations:

  • AC waveforms can be illustrated using phasor diagrams.

Example of Phasor Representation

• Visualize with angles:

  • Magnitude and phase angles illustrated.

• Example phasors:

  • Phasor with magnitude = 2, phase angle = 45°.

  • Phasor with magnitude = 3, phase angle = 180°.

  • Phasor with magnitude = 1, phase angle = -45°.

Sine Wave Representation with Phasors

• Full cycle of sine wave: Represented by a phasor rotating 360°.

• Instantaneous Value: Given by the vertical distance from the phasor tip to the horizontal axis.

• Phasor Position and Instantaneous Value Relationship:

  • Vertical distance indicates the sine wave’s instantaneous value at any point.

Positive and Negative Angles

• Phasor angles can be expressed:

  • Positive angles: Counter-clockwise from 0°

  • Negative angles: Clockwise from 0°

Examples of Phasor Analysis

• Example calculation of instantaneous value for different phase angles.

• Formula used: Instantaneous value = VP * sin(θ).

  

Phasors for Multiple Sine Waves (can be more than 2)

• Phasor diagrams can illustrate the relationships between multiple sine waves of the same frequency:

  • A fixed position phasor represents a complete sine wave.

    

  • Constant phase angle between sine waves throughout cycles.

    

Understanding Phase Relationships

• Example: Wave B leads Wave A by 30° with a smaller amplitude indicated by the phasor lengths.

Angular Velocity of Phasors

• Cycle Representation: One cycle represented by rotation through 2π radians (360°).

• Time Relationship: Time to complete one cycle is the period (T) of the sine wave.

• Angular Velocity (ω): Related to the frequency of the phasor.

• Formula:

  

General Expressions for Sine Waves

• Instantaneous Voltage: (

  • Formula:

    • v(θ) = Vp sin(θ) — angular domain expression

    • v(t) = Vp sin(2πft) — time domain expression

• Examples of Time Domain Expressions:

  • Specific values calculated from phasor diagrams and frequencies given.

    

RMS vs Peak Values

• Phasors often represented in RMS values for AC voltage or current in circuit analysis.

• Notation to clarify that RMS values are used unless specified otherwise.

Summary of Phasors

• Phasor diagrams effectively represent sine waves:

  • Angular position indicates the sine wave angle relative to a reference;

  • Length of phasor indicates amplitude.

• Demonstrates phase relationships across multiple sinusoidal quantities.

• Time domain expression: v(t) = Vp sin(2πft + Φ).