Calculator Inputs
Use angular frequency in rad/s. Enter comma-separated break frequencies for real and second-order factors.
Example Data Table
| Parameter | Example Value | Purpose |
|---|---|---|
| Gain K | 10 | Sets the starting magnitude offset. |
| Zeros at origin | 0 | Controls low-frequency rising slope. |
| Poles at origin | 0 | Controls low-frequency falling slope. |
| Real zeros | 100 | Adds +20 dB/dec after 100 rad/s. |
| Real poles | 10, 1000 | Each adds -20 dB/dec after its breakpoint. |
| Second-order poles | 5000 | Adds -40 dB/dec after 5000 rad/s. |
| Frequency range | 0.1 to 100000 rad/s | Defines the horizontal plot span. |
| Plot points | 300 | Improves plot smoothness and export detail. |
Formula Used
General transfer model
G(s) = K · s^(Nz0-Np0) · Π(1 + s/ωz) · Π(1 + 2ζz(s/ωnz) + (s/ωnz)^2) / [Π(1 + s/ωp) · Π(1 + 2ζp(s/ωnp) + (s/ωnp)^2)]
This calculator uses asymptotic slopes and standard phase transition bands. It estimates the straight-line magnitude response and the common piecewise phase approximation.
Magnitude asymptote in dB
Start with 20 log10(|K|).
Add +20 dB/dec for each zero at origin.
Add -20 dB/dec for each pole at origin.
After every real zero breakpoint, add +20 dB/dec.
After every real pole breakpoint, add -20 dB/dec.
After every second-order zero, add +40 dB/dec.
After every second-order pole, add -40 dB/dec.
Phase approximation
A real pole or zero changes phase over two decades.
The transition starts near 0.1ωb and ends near 10ωb.
A real zero contributes approximately +90°.
A real pole contributes approximately -90°.
A second-order factor contributes approximately ±180°.
How to Use This Calculator
- Enter the system gain K. Use a negative value if your transfer function has a sign inversion.
- Add zeros or poles at the origin if your transfer function contains factors like
sor1/s. - Type real breakpoint frequencies as comma-separated values. Example:
10, 100, 1000. - Type second-order natural frequencies when quadratic factors are present.
- Set the minimum and maximum angular frequency range in rad/s.
- Choose the number of plot points. More points produce smoother exported data.
- Press Generate Asymptotic Bode Plot to calculate the response.
- Review the summary, breakpoint table, plotted magnitude and phase, then export CSV or PDF as needed.
Frequently Asked Questions
1. What is an asymptotic Bode plot?
An asymptotic Bode plot is a straight-line approximation of a system’s magnitude and phase response. It highlights gain trends, slope changes, and corner frequencies without computing the exact curved response.
2. Why are break frequencies important?
Break frequencies mark where poles or zeros change the response slope. They help engineers quickly estimate bandwidth, stability trends, and filter behavior from a transfer function.
3. Does this calculator show the exact Bode curve?
No. This tool plots the asymptotic magnitude and the standard phase approximation. It is designed for quick engineering estimation, teaching, and transfer-function inspection.
4. How are poles and zeros at the origin handled?
Each zero at the origin adds +20 dB/dec and +90°. Each pole at the origin adds -20 dB/dec and -90°. Their effects begin immediately from the lowest plotted frequency.
5. Can I enter multiple poles and zeros?
Yes. Enter comma-separated breakpoint values for real poles, real zeros, second-order poles, and second-order zeros. The calculator combines all effects into one asymptotic response.
6. What frequency unit should I use?
This page uses angular frequency in rad/s. Keep all breakpoints and the plotting range in the same unit so the slope changes occur at the correct positions.
7. Why does the phase change across two decades?
The common Bode approximation spreads phase change from one decade below the breakpoint to one decade above it. This gives a practical straight-line estimate for manual analysis.
8. When is the asymptotic method less accurate?
Accuracy drops near corner frequencies, near resonant second-order behavior, and for tightly clustered poles and zeros. Exact numerical Bode analysis is better when precision matters.