CHAPTER 11 (2): PHASORS AND COMPLEX NUMBER

Complex Numbers and Phasors

Overview

  • Focus on phasors and complex numbers, critical in analyzing AC circuits.

  • Representation of AC voltages and applications in circuit analysis.

Objectives

  • Use phasors to represent sine waves.

  • Represent phasors using complex numbers.

  • Perform calculations with rectangular and polar forms.

  • Convert between polar and rectangular forms.

  • Determine total output voltage of AC sources connected in series.

Complex Number System

  • Phasors initially graphical; complex numbers allow mathematical manipulation.

  • Complex numbers facilitate addition, subtraction, multiplication, and division of phasors.

Rectangular and Polar Forms

Rectangular Form

  • Represents phasor as: A + jB

    • A: real value

    • jB: imaginary component

Polar Form

  • Represents phasor as: C ∠ ±q

    • C: magnitude of the phasor

    • q: angular position relative to the positive real axis

Mathematical Operations (similar operations to EM1)

Addition & Subtraction

  • Complex numbers must be in rectangular form.

  • Add or subtract real and imaginary parts separately.

Multiplication & Division

  • For polar forms: multiply magnitudes and add/subtract angles.

Example of Sine Waves

  • Different sinusoidal AC voltages represented: (rectangular form)

    • v1 = 10 sin(ωt + 120°)

    • v2 = 4 sin(ωt + 30°)

    • v3 = 8 sin(ωt - 30°)

    • v4 = 6 sin(ωt - 130°)

  • Converted to polar form using peak voltages: (polar form)

    • v1(peak) = 10∠ 120°

    • v2(peak) = 4 ∠ 30°

    • v3(peak) = 8 ∠ -30°

    • v4(peak) = 6 ∠ -130°

Total Voltage Calculation

  • When connected in series, total voltage formula:

    • Vtotal(peak) = V1(peak) + V2(peak) + V3(peak) + V4(peak)

  • Resulting calculation includes converting values to rectangular form and summing: (just use calculator to convert)

    • Total represented as: Vtotal = 2.57 ∠ 53.4° = 2.57 sin (ωt + 53.4°)

Important Note on RMS Values

  • AC voltages and currents typically described by RMS (Root Mean Square) values.

  • Magnitudes of phasors in AC circuit analysis represent RMS values unless stated otherwise.

Summary of Phasor and Complex Number Characteristics

  • Phasor Diagrams:

    • Represent sine waves with angular positions indicating phase and lengths indicating amplitude.

  • Forms of Complex Numbers:

    • Rectangular: A + jB, with real and imaginary parts.

    • Polar: C ∠ θ, with magnitude and angle.

End of Chapter 11

  • Concludes discussions on phasors and complex numbers.