Magnitude of Transfer Function Calculator

Calculator

Input Method

Frequency Input

Frequency Unit

Sweep Scale

Single Frequency

Start Frequency

End Frequency

Sweep Points

Frequency List

Numerator Coefficients

Denominator Coefficients

Gain K

Zeros

Poles

Use commas, spaces, semicolons, or new lines between values. For complex-conjugate factors, use polynomial coefficient mode.

Example Data Table

Case	Input Method	Transfer Function Setup	Frequency Pattern	Practical Use
Example 1	Polynomial	Numerator: 1, 4 \| Denominator: 1, 5, 6	0.1 to 100 rad/s, log sweep	Basic frequency response review
Example 2	Zero Pole Gain	K = 1 \| Zeros: -4 \| Poles: -2, -3	1, 5, 10, 20, 50 Hz	Compare design checkpoints
Example 3	Polynomial	Numerator: 2 \| Denominator: 1, 0.8, 4	Single frequency at 10 rad/s	One-point magnitude check
Example 4	Zero Pole Gain	K = 5 \| Zeros: none \| Poles: -1, -8	0.5 to 60 Hz, linear sweep	Filter trend comparison

Formula Used

The calculator evaluates the transfer function at s = jω. It then computes the complex response and reports the magnitude, magnitude in decibels, and phase angle.

Polynomial Form

H(s) = N(s) / D(s)

H(jω) = N(jω) / D(jω)

|H(jω)| = √(Re(H)^2 + Im(H)^2)

Magnitude dB = 20 log10(|H(jω)|)

Zero Pole Gain Form

H(s) = K × ∏(s - zi) / ∏(s - pi)

At s = jω:

|H(jω)| = |K| × ∏|jω - zi| / ∏|jω - pi|

Use coefficient mode when the function is already expanded. Use zero-pole-gain mode when the design is easier to describe through roots and a gain value.

How to Use This Calculator

Select either polynomial coefficients or zero-pole-gain input.
Enter the transfer function data using commas, spaces, or new lines.
Choose a single frequency, a custom list, or a sweep range.
Pick the frequency unit as rad/s or Hz.
Use logarithmic sweep for broad response studies.
Press the calculate button to place results above the form.
Review the table, summary metrics, and plotted magnitude curve.
Use the CSV or PDF buttons to export the current result set.

FAQs

1. What does transfer function magnitude show?

It shows the size of the output-to-input ratio at a selected frequency. It ignores sign direction and focuses on gain strength.

2. Why does the calculator use jω?

Frequency response is evaluated on the imaginary axis. Replacing s with jω converts the transfer function into a frequency-domain response.

3. When should I use coefficient mode?

Use coefficient mode when you already have the numerator and denominator expanded into polynomial form. It is also the best option for complex-conjugate systems.

4. When should I use zero-pole-gain mode?

Use zero-pole-gain mode when your design data is defined by roots and gain. It is fast for filters, control models, and stability studies.

5. What is the difference between magnitude and dB?

Magnitude is the direct absolute value. Decibels compress that value into a logarithmic scale, which is often easier to compare across wide ranges.

6. Should I choose Hz or rad/s?

Choose the unit that matches your source data. If your problem statement uses cycles per second, use Hz. Control problems often use rad/s.

7. Why might the result become extremely large?

If the selected frequency falls near a pole, the denominator approaches zero. That can produce very large magnitude values or undefined behavior.

8. Can I export the generated results?

Yes. The CSV button downloads the full table. The PDF button builds a clean report with the same rows and summary information.