Model closed loop systems from polynomial coefficients. Check equations, orders, gains, and simplified expressions instantly. Export clean results for review, teaching, validation, and documentation.
Let G(s) = Ng(s) / Dg(s) and H(s) = Nh(s) / Dh(s).
Negative feedback: T(s) = G(s) / [1 + G(s)H(s)]
Expanded form: T(s) = Ng(s)Dh(s) / [Dg(s)Dh(s) + Ng(s)Nh(s)]
Positive feedback: T(s) = G(s) / [1 - G(s)H(s)]
Expanded form: T(s) = Ng(s)Dh(s) / [Dg(s)Dh(s) - Ng(s)Nh(s)]
The calculator multiplies and combines polynomial arrays directly. It then builds the closed loop numerator, denominator, normalized form, order, and DC gain.
Example entry format: for s² + 4s + 6, type 1, 4, 6.
| Case | G(s) | H(s) | Feedback | Closed Loop T(s) |
|---|---|---|---|---|
| Example 1 | (2s + 5) / (s² + 4s + 6) | 1 | Negative | (2s + 5) / (s² + 6s + 11) |
| Example 2 | (s + 3) / (s² + 2s + 2) | 0.5 | Positive | (s + 3) / (s² + 1.5s + 0.5) |
| Example 3 | (4) / (s + 4) | (s + 1) / (s + 2) | Negative | 4(s + 2) / [(s + 4)(s + 2) + 4(s + 1)] |
A closed loop transfer function shows how a system behaves after feedback is applied. It links the output to the reference input. This form is central in control analysis. It helps students and engineers simplify large block diagrams. It also helps them compare alternative feedback paths with speed and accuracy.
This calculator accepts polynomial coefficients for the forward path and the feedback path. It multiplies the relevant polynomials. It then combines them using the selected feedback sign. The result is a complete closed loop transfer function. The page also shows the open loop product, the characteristic equation, normalized coefficients, order, and DC gain.
Coefficient entry is practical. It avoids manual algebra. It also reduces sign mistakes. You can enter unity feedback, constant feedback, or higher order feedback models. This makes the tool flexible for textbook work, classroom checks, and project notes. It is especially useful when transfer functions have mixed integer and decimal terms.
The numerator tells you how the input enters the final model. The denominator shapes system dynamics. Its degree gives the closed loop order. The characteristic equation comes from setting that denominator equal to zero. The normalized form is helpful when comparing systems or preparing equations for reports and further analysis.
Always verify the feedback sign first. A wrong sign changes the entire denominator. Enter coefficients in descending powers of s. Keep zero coefficients when a power is missing. For example, enter 1, 0, 3 for s² + 3. After calculation, inspect the denominator carefully before using the result in design work.
This calculator supports revision, homework, lab preparation, and model validation. It is useful for unity feedback systems, sensor feedback models, and general transfer function reduction. The export options also help when saving results for records, sharing examples, or attaching outputs to technical documents.
It is the ratio of output to input after feedback is included. It shows the final dynamic behavior of the system, not just the forward path.
Enter coefficients from the highest power of s to the constant term. For s² + 4s + 6, type 1, 4, 6.
Yes. Enter 1 for the feedback numerator and 1 for the feedback denominator. That represents H(s) = 1.
Negative feedback uses 1 + G(s)H(s) in the denominator step. Positive feedback uses 1 - G(s)H(s). The sign changes the final polynomial.
It comes from the closed loop denominator. Its roots determine poles, which strongly influence stability, damping, and transient response.
It means the denominator leading coefficient is scaled to one. This is useful for comparison, reporting, and many standard analysis methods.
If the denominator becomes zero at s = 0, the ratio cannot be evaluated there. In that case, the displayed DC gain is undefined.
Yes. After calculation, use the CSV or PDF button. The exported file includes the main computed expressions and summary values.
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.