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Complex Plane

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About This MicroSim

This interactive simulation demonstrates the geometric representation of complex numbers in the complex plane (also known as the Argand diagram). Students can manipulate the real and imaginary parts of a complex number and observe how the rectangular and polar forms change in real-time.

Embedding This MicroSim

You can include this MicroSim on your website using the following iframe:

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<iframe src="https://dmccreary.github.io/signal-processing/sims/complex-plane/main.html" height="532px" scrolling="no"></iframe>

Key Features

Visual Components:

  • Coordinate Axes: Blue horizontal axis represents the Real axis, red vertical axis represents the Imaginary axis
  • Complex Number Vector: Green arrow from the origin to the plotted point z
  • Right Triangle: Shows the geometric relationship between real part, imaginary part, and magnitude
  • Angle Arc: Green arc showing the phase angle θ from the positive real axis
  • Dashed Projections: Lines showing how the complex number projects onto both axes

Information Displays:

  • Left Panel: Shows Re(z), Im(z), magnitude |z|, and phase angle θ
  • Right Panel: Displays both rectangular (a + bi) and polar (r∠θ) forms
  • On-Graph Labels: Magnitude label on the vector

Mathematical Concepts

Rectangular Form

A complex number z can be written as:

\[z = a + bi\]

where: - \(a\) = Real part (Re(z)) - \(b\) = Imaginary part (Im(z)) - \(i\) = Imaginary unit (\(i^2 = -1\))

Polar Form

The same complex number can be expressed as:

\[z = r \angle \theta = r(\cos\theta + i\sin\theta) = re^{i\theta}\]

where: - \(r = |z| = \sqrt{a^2 + b^2}\) (magnitude/modulus) - \(\theta = \tan^{-1}(b/a)\) (phase angle/argument)

Euler's Formula Connection

The polar form connects to Euler's famous formula:

\[e^{i\theta} = \cos\theta + i\sin\theta\]

This relationship is fundamental to signal processing, allowing sinusoidal signals to be represented as rotating phasors in the complex plane.

Lesson Plan

Learning Objectives

By the end of this lesson, students will be able to:

  1. Understand the geometric interpretation of complex numbers as points in a 2D plane
  2. Convert between rectangular (a + bi) and polar (r∠θ) forms
  3. Calculate magnitude and phase from real and imaginary components
  4. Explain how changing real and imaginary parts affects the polar representation
  5. Connect complex plane concepts to phasor representation of signals

Grade Level

College-level introductory signal processing or pre-calculus/calculus courses

Duration

15-20 minutes for exploration, 30-45 minutes with instruction

Prerequisites

  • Basic understanding of the Cartesian coordinate system
  • Familiarity with trigonometry (sine, cosine, tangent)
  • Knowledge of the Pythagorean theorem

Suggested Activities

Activity 1: Quadrant Exploration (5 min)

Have students move the sliders to place the complex number in each quadrant and observe:

  • How does the phase angle change in each quadrant?
  • When is the phase positive? Negative?

Activity 2: Special Cases (5 min)

Set the sliders to these values and discuss:

  • Real = 1, Imag = 0 (pure real number)
  • Real = 0, Imag = 1 (pure imaginary number)
  • Real = 1, Imag = 1 (45° angle case)
  • Real = 0, Imag = 0 (origin)

Activity 3: Pythagorean Relationship (5 min)

Find combinations where the magnitude equals exactly 5 (like 3+4i, 4+3i, -3+4i, etc.)

Activity 4: Unit Circle (5 min)

Find real/imaginary combinations that produce a magnitude of exactly 1 (points on the unit circle)

Assessment Questions

  1. If z = 3 + 4i, what is the magnitude |z|?
  2. What is the phase angle of z = -1 + i in degrees?
  3. Convert z = 2∠60° to rectangular form
  4. Why is the complex plane useful for representing signals in signal processing?

Applications in Signal Processing

The complex plane representation is essential in signal processing for:

  • Phasor Analysis: Representing sinusoidal signals as rotating vectors
  • Fourier Transform: Complex exponentials form the basis of frequency analysis
  • Filter Design: Pole-zero plots in the complex plane characterize filter behavior
  • Modulation: AM, FM, and PM signals can be visualized as trajectories in the complex plane

References

  • Oppenheim, A. V., & Schafer, R. W. (2010). Discrete-Time Signal Processing
  • Proakis, J. G., & Manolakis, D. G. (2006). Digital Signal Processing