Bode Plot Online Tool Documentation

Introduction

The Bode plot is a graphical representation of the frequency response of a dynamic system. It consists of two plots:

1. Magnitude plot: Shows the gain (in decibels, dB) of the system as a function of frequency.
2. Phase plot: Shows the phase shift (in degrees) of the system as a function of frequency.

The Bode plot is widely used in control system analysis to evaluate stability, performance, and frequency-domain characteristics of a system.

Mathematical Definition

For a linear time-invariant system with a transfer function:

$$G(s) = \frac{N(s)}{D(s)}$$

where the numerator $N(s)$ and the denominator $D(s)$ are functions of a complex variable $s$, the Bode plot represents the magnitude and phase of $G(j\omega)$ as a function of frequency $\omega$.

• Magnitude:

  $$|G(j\omega)| = 20 \log_{10} \left| \frac{N(j\omega)}{D(j\omega)} \right| \, \text{(in dB)}$$

• Phase:

  $$\angle G(j\omega) = \angle N(j\omega) - \angle D(j\omega) \, \text{(in degrees)}$$

Bode Plot Analysis

The Bode plot provides insights into:

1. Gain Margin (GM): The amount of gain increase required to make the system unstable.
2. Phase Margin (PM): The amount of phase lag required to make the system unstable.
3. Bandwidth: The frequency range over which the system responds effectively.
4. System Stability: Determined by the gain and phase margins.

Interpretation of the Bode Plot

• Magnitude Plot:

  • A flat region indicates constant gain.
  • A slope of $-20 \, \text{dB/decade}$ indicates a pole, while $+20 \, \text{dB/decade}$ indicates a zero.
  • The crossover frequency (where the magnitude is 0 dB) is critical for stability analysis.

• Phase Plot:

  • A flat region indicates constant phase.
  • A phase shift of $-90^\circ$ indicates a pole, while $+90^\circ$ indicates a zero.
  • The phase margin is calculated at the crossover frequency.

Applications of the Bode Plot

• Stability Analysis: Determine gain and phase margins.
• Controller Design: Design PID controllers and compensators.
• Frequency Response Analysis: Evaluate system behavior across different frequencies.
• Filter Design: Analyze and design low-pass, high-pass, and band-pass filters.

How to Use the Bode Plot Online Tool?

1. Enter the transfer function $G(s)$. A transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input. It is typically expressed as:

   $$G(s) = \frac{N(s)}{D(s)}$$

   The numerator $N(s)$ and denominator $D(s)$ are typically represented as polynomials of a complex variable $s$. In this application, $N(s)$ and $D(s)$ are defined as lists of coefficients, as in the examples below.

Example 1

Given the numerator and denominator lists:

numerator = [1, 2, 1] denominator = [1, 3, 3, 1]

This corresponds to the transfer function:

$$G(s) = \frac{s^2 + 2s + 1}{s^3 + 3s^2 + 3s + 1}$$

where:

• $s^2 + 2s + 1$ comes from [1, 2, 1]
• $s^3 + 3s^2 + 3s + 1$ comes from [1, 3, 3, 1]

Example 2: First-Order System

For a first-order system:

numerator = [1] denominator = [1, 5]

This represents:

$$G(s) = \frac{1}{s + 5}$$

where:

• The numerator is just 1 (constant gain).
• The denominator represents a first-order lag system. The polynomial $s + 5$ comes from [1, 5].

2. Click Run. The Bode plot will be generated automatically.
3. Analyze the Bode plot:
   • Magnitude Plot: Check the gain at different frequencies.
   • Phase Plot: Check the phase shift at different frequencies.
   • Stability Margins: Determine the gain and phase margins.