Calculator Input
Example Data Table
| F(s) | Numerator | Denominator | Final Value | Comment |
|---|---|---|---|---|
| 5 / s(s + 2) | 5 | 1, 2, 0 | 2.5 | One origin pole cancels after multiplication by s. |
| 10 / (s + 4) | 10 | 1, 4 | 0 | The s factor drives the limit to zero. |
| (3s + 9) / s(s + 3) | 3, 9 | 1, 3, 0 | 3 | The lowest powers match after multiplying by s. |
| 1 / s² | 1 | 1, 0, 0 | Undefined | A remaining origin pole breaks the finite final value. |
Formula Used
The calculator uses the final value theorem:
f(∞) = lim s→0 [sF(s)]
Here, F(s) is the Laplace transform of f(t). The calculator multiplies the transfer expression by s and studies the lowest nonzero powers around s = 0.
If numerator and denominator powers match, their coefficients are divided. If the numerator has a higher power, the result is zero. If the denominator has a higher power, the result is undefined or infinite.
The theorem is valid only when all poles of sF(s) lie in the left half plane. This page uses a coefficient based Routh check for that condition.
How to Use This Calculator
- Write F(s) as a numerator polynomial over a denominator polynomial.
- Enter coefficients in descending powers of s.
- Include zero coefficients for missing powers.
- Choose the number of decimal places.
- Add a unit or note if needed.
- Press Calculate to view the result below the header.
- Use CSV or PDF download for records.
Detailed Guide
Understanding the Final Value Theorem
The final value theorem estimates the long term value of a time function. It works from a Laplace domain expression. The method is useful in control systems, signals, and applied mathematics. It avoids full inverse transformation when only the steady result is needed.
Why the check matters
The theorem is not only a limit shortcut. It also needs a stability condition. After multiplying the transform by s, all remaining poles must stay in the left half plane. A pole at the origin, or in the right half plane, can make the answer false. Oscillating systems may also fail this test. This calculator therefore reports both the numeric limit and a condition note.
What the calculator accepts
Enter numerator and denominator coefficients in descending powers of s. For example, 3, 6, 2 means 3s squared plus 6s plus 2. The tool builds F of s, then studies s times F of s near zero. It handles zero factors at the origin. It can show a finite value, zero, or an undefined result. Choose decimal places to format the answer for reports.
Interpreting the result
A finite number means the algebraic limit exists. A valid theorem note means the steady value is also trusted under the standard condition. If the condition is uncertain, use the result as an algebraic limit only. Always compare it with pole locations when designing a critical system. Repeated poles at the origin usually mean a ramping response. Right half plane poles usually mean divergence.
Practical uses
Engineers use this theorem for steady state error. Students use it to check inverse Laplace answers. Analysts use it to compare transfer functions quickly. It is also helpful for step response studies. The exported CSV keeps data ready for spreadsheets. The generated PDF gives a compact record for coursework or documentation.
Good input habits
Keep coefficients numeric. Include zero coefficients for missing powers. Write the denominator order correctly. Avoid symbolic letters inside the fields. If a result looks unexpected, inspect the formula steps. Then test the original function with another method. For best accuracy, verify units before saving exports. The theorem gives a value for the original time response, not every intermediate signal in real practice.
FAQs
What does the final value theorem calculate?
It calculates the expected long term value of a time function by using its Laplace transform. It avoids doing the full inverse transform when only the steady value is needed.
When is the theorem valid?
It is valid when all poles of sF(s) are in the left half plane. Poles on the imaginary axis or right half plane can make the result incorrect.
How should I enter coefficients?
Enter coefficients in descending powers of s. For 2s² + 3s + 4, enter 2, 3, 4. Use zero for missing powers.
Why does the calculator show undefined?
Undefined usually means the denominator has a stronger zero factor near s = 0 after multiplying by s. That suggests no finite final value.
Can I use this for transfer functions?
Yes. It is often used with transfer functions, especially for steady state response and control system studies. Make sure the pole condition is satisfied.
Does zero final value mean the system is stable?
No. A zero algebraic limit does not prove stability alone. You must also check the poles of sF(s), which the calculator summarizes.
What does the Routh first column mean?
It comes from the Routh stability table. A strictly positive first column supports left half plane pole placement for the checked denominator.
Can I export my calculation?
Yes. Use the CSV button for spreadsheet records. Use the PDF button for a compact report containing the formula, result, and condition check.