Bode Plot to Transfer Function Calculator

Calculator

Bode data rows

Use one row per sample. Phase is optional but recommended.

Frequency unit

Pole and zero mode

Slope change sensitivity

dB per decade. Lower values detect more corners.

Maximum order per corner

Manual zero frequencies

Manual pole frequencies

Gain mode

Manual gain dB

Manual linear gain

Estimate pure delay

Unwrap phase

Example Data Table

Frequency Hz	Magnitude dB	Phase Degrees	Meaning
10	20.0	-5	Low frequency gain region
100	14.0	-45	Near first pole corner
500	0.0	-105	Falling response zone
5000	-20.1	-170	High frequency rolloff region

Formula Used

Magnitude: |G(jω)| = K × Π√(1 + (ω/ωz)²) / Π√(1 + (ω/ωp)²)

Decibels: Magnitude dB = 20 log10(|G(jω)|)

Phase: ∠G(jω) = Σ atan(ω/ωz) − Σ atan(ω/ωp) − ωT

Slope rule: each pole adds about -20 dB/decade. Each zero adds about +20 dB/decade.

Gain fit: the calculator divides measured linear magnitude by the pole-zero ratio, then uses the median value.

How to Use This Calculator

Enter frequency, magnitude, and phase rows copied from your Bode plot. Keep the rows sorted if possible, but the calculator can sort them. Select hertz or radians per second. Choose automatic detection when you do not know pole and zero frequencies.

Use manual mode when you already have corner frequencies from a graph. Enter zeros and poles as comma separated values. Select fitted gain for unknown systems. Select manual gain when a test report gives a known value.

Submit the form. Review the transfer function, fit errors, and residual table. Download the CSV for spreadsheet checks. Download the PDF for a compact report summary.

Bode Plot Transfer Function Guide

Why the Method Works

A Bode plot shows how a system responds across frequency. It gives magnitude in decibels and phase in degrees. This calculator converts those readings into a usable transfer function estimate. It is useful when only plotted test data is available.

The tool looks for slope changes in the magnitude curve. A negative change usually means a pole. A positive change usually means a zero. Each twenty decibels per decade suggests one order. Corner frequency is taken from the detected breakpoint or your manual entry.

Gain, Phase, and Delay

The fitted gain is found from the measured magnitude values. Each sample is corrected for the pole and zero factors. The median corrected value becomes the gain. This reduces the effect of noisy points. You can also enter a known gain when lab notes already provide it.

Phase data adds another check. The calculator compares measured phase with model phase. It can estimate pure time delay from the remaining phase lag. Delay is helpful for sensors, filters, drives, and control loops.

Accuracy Notes

A transfer function from a Bode plot is still an approximation. Real plots may include noise, saturation, non minimum phase behavior, hidden resonances, and measurement delay. Use more frequency points around corners for better results. Avoid reading values from a crowded graph when accuracy matters.

Start by entering frequency, magnitude, and phase rows. Use hertz or radians per second. Then choose automatic breakpoint detection or manual pole and zero lists. After submitting, review the engineering form, polynomial form, gain, corners, delay, and fit errors.

Using the Output

The exported CSV is best for spreadsheets. It includes measured values, predicted values, and residuals. The PDF summary is useful for reports. Keep the original plot with the result, because the estimate depends on sampling quality.

This calculator helps engineers move from visual frequency response to a working model. It does not replace system identification software. It gives a clear first pass for design, teaching, tuning, and documentation.

For control work, treat the result as a starting model. Verify it with a step response, simulation, or fresh sweep. Small phase errors near crossover can change stability margins. Always compare the fitted curve against the measured table before using controller settings again.

FAQs

What does this calculator estimate?

It estimates a transfer function from Bode magnitude and phase samples. It identifies possible poles, zeros, gain, and pure delay. The output is an engineering approximation.

Can it read an image of a Bode plot?

No. Enter sampled points from the plot. You can use cursor readings from plotting software or manually read values from a graph.

Why are poles and zeros only approximate?

Bode plots are often noisy and sparse. Corner frequencies may be hidden by nearby dynamics. The slope rule gives a practical estimate, not an exact system identification result.

Should I use automatic or manual mode?

Use automatic mode for a quick estimate. Use manual mode when you already know corner frequencies or when the slope detector finds too many false corners.

What does gain fitting do?

It converts magnitude dB to linear magnitude. Then it removes the effect of estimated poles and zeros. The median corrected value becomes the gain.

What is pure delay in the result?

Pure delay is a time lag term. It appears as extra phase lag that increases with frequency. The calculator estimates it from phase residuals.

Why does phase unwrapping matter?

Phase plots often jump at ±180 degrees. Unwrapping removes those artificial jumps. This helps delay fitting and phase error calculations behave more smoothly.

Can I use the result for controller design?

Yes, but verify it first. Compare predicted and measured response. For critical systems, confirm the model with simulation, step tests, and safety margins.