Calculator Inputs
Enter coefficients in descending powers of s. Example: denominator 1, 10, 100 means s² + 10s + 100.
Formula Used
Transfer function: H(s) = N(s) / D(s) × e-sT
Frequency substitution: s = jω
Magnitude: |H(jω)|dB = 20 log10(|H(jω)|)
Phase: ∠H(jω) = atan2(Im(H), Re(H)) × 180 / π
Phase margin: measured at the frequency where magnitude crosses 0 dB.
Gain margin: the negative magnitude at the frequency where phase crosses -180°.
Bandwidth: the first sampled frequency near DC gain − 3 dB, when such a point exists.
The calculator evaluates the exact polynomial ratio numerically at each logarithmically spaced frequency sample, rather than drawing a hand-sketch approximation.
How to Use This Calculator
- Enter numerator coefficients in descending powers of
s. - Enter denominator coefficients in descending powers of
s. - Set any transport delay in seconds, or keep it at zero.
- Choose whether your frequency range is in
Hzorrad/s. - Provide start frequency, end frequency, and the total sample count.
- Click Calculate Bode Plot to place results above the form.
- Review the summary cards for gain crossover, phase margin, gain margin, and bandwidth estimates.
- Download the sampled response table as CSV or PDF when you need to save, compare, or document results.
Example Input Cases
| Case | Numerator | Denominator | Delay (s) | Frequency span | Unit | Notes |
|---|---|---|---|---|---|---|
| Second-order low-pass | 100 | 1, 10, 100 | 0 | 0.1 to 1000 | rad/s | Smooth roll-off and clear crossover behavior. |
| Lead compensator | 1, 5 | 1, 50 | 0 | 0.01 to 1000 | rad/s | Useful for studying phase lift over a band. |
| Delayed plant | 10 | 1, 3, 10 | 0.08 | 0.1 to 200 | Hz | Shows how delay shifts phase downward. |
Frequently Asked Questions
1) How do I enter transfer function coefficients?
Use descending powers of s, separated by commas. Example: numerator 100 and denominator 1, 10, 100 means 100 / (s² + 10s + 100).
2) Should I choose Hz or rad/s?
Choose the unit that matches your design notes. The calculator converts Hz to angular frequency internally before evaluating the transfer function.
3) Why does the phase keep dropping below -180°?
Higher-order poles and time delay add negative phase. The plot is unwrapped so the curve stays continuous instead of jumping at ±180°.
4) What does the delay field change?
Delay multiplies the response by e-jωT. Magnitude stays the same, but phase becomes more negative as frequency rises.
5) How are gain and phase margins estimated?
Gain crossover is where magnitude crosses 0 dB. Phase margin is measured there. Phase crossover is where phase reaches -180°, and gain margin is the negative magnitude at that point.
6) Can I model improper transfer functions?
Yes. The calculator evaluates any entered polynomial ratio. Improper systems may show rising high-frequency magnitude and can represent nonphysical or approximate models.
7) Why might crossover fields show N/A?
N/A appears when the sampled response never reaches the required threshold inside your chosen range. Widen the range or increase points for better detection.
8) Are these exact or straight-line Bode plots?
These are exact sampled frequency-response values from the entered transfer function, not hand-drawn asymptotic straight-line approximations.