Analyze first order filters from coefficient inputs. View cutoff values, pole location, and response instantly. Exports, formulas, examples, and guidance support faster accurate analysis.
Enter coefficients for the first-order transfer function H(s) = (b1s + b0) / (a1s + a0).
| Example | Transfer Function | Computed Pole | Angular Cutoff | Cutoff Frequency |
|---|---|---|---|---|
| Low-pass example | H(s) = 1 / (0.002s + 1) | -500 | 500 rad/s | 79.577472 Hz |
| High-pass example | H(s) = 0.002s / (0.002s + 1) | -500 | 500 rad/s | 79.577472 Hz |
| General first-order example | H(s) = (0.5s + 2) / (0.01s + 5) | -500 | 500 rad/s | 79.577472 Hz |
General first-order transfer function: H(s) = (b1s + b0) / (a1s + a0)
Pole: p = -a0 / a1
Angular cutoff frequency: ωc = |p| = |a0 / a1|
Cutoff frequency in hertz: fc = ωc / (2π)
Time constant: τ = a1 / a0
For standard first-order low-pass and high-pass forms, the pole magnitude defines the cutoff point. The page also samples the frequency response using:
|H(jω)| = √(b0² + (b1ω)²) / √(a0² + (a1ω)²)
Cutoff frequency marks the boundary where a filter starts reducing signal strength. It is usually defined at the point where output magnitude falls to 0.707 of the passband value. This level equals minus 3 decibels. In first order systems, the cutoff depends directly on the pole location in the transfer function. That makes coefficient based calculation fast and useful.
A transfer function shows how output changes with frequency. It uses the complex variable s. For a first order form, the denominator often looks like a1s plus a0. The pole appears at minus a0 divided by a1. Its magnitude gives angular cutoff frequency. Dividing that value by 2π converts radians per second into hertz. This page automates that process and reduces manual errors.
The calculator accepts numerator and denominator coefficients. It computes pole location, time constant, angular cutoff, and cutoff frequency in hertz. It also samples the magnitude response across a frequency range. The graph helps you inspect the response shape. CSV export supports spreadsheet work. PDF export helps with reports, homework, and documentation. The layout stays simple, bright, and easy to scan.
Use this tool when you have a first order transfer function and need a quick estimate of cutoff behavior. It is useful for classroom practice, control examples, signal filtering, and circuit models. It also helps compare coefficient changes. If coefficients change, the pole moves. When the pole moves, the cutoff changes as well. That relation becomes clear in both the result cards and the graph.
This method is best for first order denominators. Higher order systems may have several poles and more than one break point. In those cases, one simple cutoff value may not describe the whole response. Always check the model form before using the result. For standard first order low pass or high pass systems, this calculator gives a reliable and practical estimate.
Because the outputs are shown above the form, you can review the latest values without scrolling through inputs again. That saves time during repeated testing, tuning, and teaching demonstrations with many coefficient combinations daily.
It is the frequency where a filter begins to reduce output noticeably. For many first-order filters, it matches the minus 3 dB point.
This page is designed for first-order denominator forms such as a1s + a0. It is most reliable for simple low-pass, high-pass, and similar first-order systems.
The pole controls the main break point of a first-order response. Its magnitude gives angular cutoff frequency for standard first-order filters.
The calculator shows angular cutoff in radians per second and standard cutoff in hertz. Both values are useful for analysis and design work.
You can enter them, but the result only makes physical sense when the model matches a meaningful first-order system. Stable positive-denominator forms are usually preferred.
For standard first-order cutoff estimation, the denominator pole sets the cutoff. The numerator still changes gain shape and the plotted response.
They include the main calculated values and the sampled response table. This helps with reports, checking, and reuse in other tools.
No. It is a quick analysis tool for first-order forms. Complex multi-pole systems need broader methods and deeper frequency response checks.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.