What transfer function forms does this tool support?

First-order with an optional zero: H(s) = K(s/z + 1)/(s/p + 1). Second-order standard form: H(s) = K·ωn²/(s² + 2ζωn·s + ωn²). Both cover the most common textbook and real-world compensator shapes.

How is magnitude in dB calculated?

Magnitude in dB = 20·log10(|H(jω)|), where H(jω) is the complex-valued transfer function evaluated on the imaginary axis at angular frequency ω = 2πf.

What is the gain margin and how do I find it here?

Gain margin is the additional gain (in dB) the system can tolerate before going unstable. Find it by locating the frequency where phase = −180° and reading the negative of the magnitude at that frequency.

What is the phase margin and how do I find it here?

Phase margin is how many degrees of additional phase lag would cause instability. Find it by locating the gain crossover frequency (magnitude = 0 dB) and reading 180° + the phase angle at that point.

What does the damping ratio ζ control in a second-order system?

ζ 1 is overdamped (two real poles). Typical control designs aim for ζ ≈ 0.5–0.7.

Are the results exact or approximate?

Exact, within floating-point precision. The tool evaluates the transfer function directly in the complex domain at s = jω — no asymptotic Bode approximation is used.

Bode Plot Helper

Enter a transfer function (first or second order) and a frequency to get the exact magnitude in dB and phase in degrees.

Transfer Function Type

First-order

H(s) = K · (s/z + 1) / (s/p + 1)

Second-order

H(s) = K · ωn² / (s² + 2ζωn·s + ωn²)

First-Order Parameters

DC Gain K

Pole frequency p (rad/s) — 0 = pure integrator

Zero frequency z (rad/s) — 0 = no zero

Frequency to Evaluate

Frequency (rad/s)

Input in Hz (converted to rad/s)

Result at Entered Frequency

Press Calculate to see the result.

Frequency Sweep

Start (rad/s)

End (rad/s)

Points

Run sweep to populate this table.

Stability Indicators

Summary

Bode Plot Helper computes the frequency response of a linear time-invariant (LTI) system described by a first-order or second-order transfer function. Enter the DC gain, pole, zero, and natural frequency parameters, then sweep across a frequency range to see magnitude (dB) and phase (degrees) at each point — the building blocks of gain margin and phase margin analysis.

How it works

Select the transfer function type: first-order (pole/zero) or second-order (underdamped/overdamped).
Enter the transfer function parameters such as DC gain K, pole frequency, damping ratio, and natural frequency.
Enter a single frequency or use the sweep to evaluate the response across a range.
The tool computes |H(jω)| in dB and ∠H(jω) in degrees using exact complex-number arithmetic.
Read the magnitude and phase values to assess stability margins.

Use cases

Verify hand-calculated Bode plot magnitude and phase at a spot frequency.
Find the gain crossover frequency where magnitude = 0 dB.
Find the phase crossover frequency where phase = −180°.
Estimate gain margin and phase margin for a compensated loop.
Study how damping ratio affects the resonant peak in second-order systems.
Check whether a PID-tuned plant meets gain/phase margin specifications.
Quickly explore how changing pole/zero locations shifts the Bode plot.
Teach or learn frequency-domain control concepts without MATLAB.

Frequently Asked Questions

Last updated: 2026-06-18 · Reviewed by Nham Vu