Introduction to Time Domain and Frequency Domain

Time Domain

• Physical Meaning:
  • Commonly used in algebra for variables like time, velocity, and acceleration.
  • Example: Distance (D) = Velocity (V) × Time (T)
• Concept of Dimensions:
  • Traveling distance involves passing through time (2nd dimension).
  • Plotting dependent variable (distance) vs. independent variable (time) shows how distance changes over time.

Simple Harmonic Oscillator

• Linear Spring:
  • Properties: Restoring force doubles when stretch distance doubles.
  • Force (F) is negatively proportional to distance from neutral point (x = 0).
  • Examples:
    • x = 1 → Force = F
    • x = 2 → Force = 2F
    • x = -1 → Force = F (positive restoring force)
    • x = -2 → Force = 2F
• Mass on Spring:
  • Set mass in motion (impulse = instantaneous velocity).
  • Observed motion resembles a sinusoid (e.g., jack in the box).
  • Mathematical Description:
    • Newton's Second Law: Force = Mass × Acceleration
    • Differential equation describing motion.
    • Solution: Amplitude × sin(Natural Frequency × Time + Phase)

Frequency Domain

• Sinusoidal Signals:
  • Plotting amplitude and frequencies.
  • Single peak at frequency 2πΩ.
  • Typically, phase is discarded.
• Combining Sinusoids:
  • Creating a signal from multiple sinusoids.
  • Example: Two sinusoids with different amplitudes and periods creating a composite signal.
• Fourier Series:
  • Joseph Fourier (1807): Time domain signal with period T can be represented by an infinite sum of sinusoids.
  • First harmonic frequency, second harmonic, etc.
  • Example: Sawtooth wave described by summation of sinusoids.
• Continuous Fourier Transform:
  • Transition from discrete frequencies to continuous as period approaches infinity.
  • Fourier transform for non-repeating signals.
  • Includes all frequencies, amplitudes, and phase components.
• Fourier Transform:
  • General approach to understanding frequency response.
  • Exponential term: e raised to an imaginary exponent (Euler's formula).

Laplace Transform

• Overcoming Fourier's Limitations:
  • Real-world systems often involve differential equations with exponential terms.
  • Example: Simple harmonic oscillator with damping term (coefficient B).
  • Solution includes exponential terms and sinusoidal terms.
• Causality and Exponential Content:
  • Causal systems: Cause precedes effect (no negative time).
  • Laplace transform includes exponential growth/decay.
  • Pre-multiplying Fourier transform by e^(-σt) gives complex exponent (σ + jΩ).
  • Transforms time domain to S domain.
• Applications:
  • Quantify stability margin in control design.
  • Simplifies convolution integrals to algebraic steps in the S plane.

Future Discussions

• Further detailed lecture on the Laplace Transform and its application in designing control systems.