Bode Plot Generator

Step-by-Step Magnitude & Phase Symbolic CAS Free · No Signup

Generate Bode magnitude and phase plots for any transfer function H(s) with step-by-step analysis. Features zeros/poles identification, common transfer functions reference table, interactive dual-subplot graphs, and a built-in Python compiler. Essential for control systems, filter design, and stability analysis.

Bode Plot Generator
Enter a rational function of s, or use AI above
Type a transfer function above…
SystemH(s)Type
s^2    (s+1)^2    s^3
1/(s+1)    s/(s^2+1)
10*(s+1)/(s*(s+10))
Multiplication: Use * explicitly: 2*s not 2s
Powers: s^2 or (s+1)^2
Grouping: Use parentheses for clarity

Result

H(s)

Enter a transfer function and click Generate

Generate Bode magnitude and phase plots with step-by-step analysis.

Bode Plot — Magnitude & Phase

Generate a Bode plot to see the magnitude and phase diagrams.

Python Compiler

What is a Bode Plot?

A Bode plot is a standard way to visualize the frequency response of a linear time-invariant (LTI) system. It consists of two graphs: a magnitude plot showing |H(jω)| in decibels (dB) and a phase plot showing ∠H(jω) in degrees, both plotted against frequency ω on a logarithmic scale.

Named after Hendrik Bode, these plots are fundamental in control engineering for analyzing system stability, designing compensators, and understanding how systems respond to different input frequencies.

Key Concepts

Gain Margin

The amount of gain (in dB) that can be added before the system becomes unstable. Measured at the phase crossover frequency where phase = −180°.

Phase Margin

The additional phase lag before instability, measured at the gain crossover frequency where |H| = 0 dB. Positive margins indicate stability.

Corner Frequency

The frequency at which the asymptotic approximation changes slope. For a pole at s = −a, the corner frequency is ω = a rad/s.

Asymptotic Approximation

Each pole adds −20 dB/decade and −90° phase. Each zero adds +20 dB/decade and +90°. Straight-line approximations simplify hand sketching.

Applications

⚙

Control Systems

Design PID controllers, analyze loop gain, and determine stability margins for feedback systems.

🔧

Filter Design

Visualize low-pass, high-pass, band-pass, and notch filter frequency responses.

📐

Stability Analysis

Determine gain and phase margins to assess closed-loop stability of feedback systems.

🎵

Audio Engineering

Analyze equalizer response curves, amplifier frequency characteristics, and speaker crossover networks.

Frequently Asked Questions

A Bode plot is a graphical representation of a system's frequency response. It consists of two plots: the magnitude plot showing |H(jw)| in decibels versus log frequency, and the phase plot showing the phase angle of H(jw) in degrees versus log frequency. Bode plots are essential for analyzing stability and performance of control systems.
On a Bode plot, the x-axis is frequency on a logarithmic scale. The magnitude plot (top) shows gain in dB - positive values mean amplification, negative values mean attenuation. The phase plot (bottom) shows phase shift in degrees. Key features include the -3dB bandwidth, gain crossover frequency, and phase margin.
Gain margin is the amount of gain increase (in dB) needed to make the system unstable, measured at the phase crossover frequency (where phase = -180 degrees). Phase margin is the additional phase lag needed for instability, measured at the gain crossover frequency (where magnitude = 0 dB). Both should be positive for a stable system.
Each pole contributes -20 dB/decade slope to the magnitude plot and -90 degrees to the phase. A pole at s = -a creates a corner frequency at w = a, where the magnitude starts rolling off. Complex conjugate poles can create a resonance peak near the natural frequency, with the peak height depending on the damping ratio.
A Bode plot shows magnitude and phase separately as functions of frequency on two subplots. A Nyquist plot shows the frequency response as a single curve in the complex plane (real vs imaginary parts). Bode plots are easier to sketch by hand and read gain/phase margins, while Nyquist plots are better for analyzing encirclement-based stability criteria.
Yes, this Bode plot generator is completely free with no signup required. You get symbolic computation via SymPy, step-by-step analysis, interactive Plotly graphs, LaTeX export, and a built-in Python compiler.

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