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).