Transfer Function Poles and Zeros Calculator
Calculate zeros, poles, and basic stability for a second-order transfer function from numerator and denominator coefficients.
Calculate
H(s) = (n₂s² + n₁s + n₀) / (d₂s² + d₁s + d₀)
What this calculator teaches
A transfer function describes how an input signal becomes an output signal in the s-domain. Zeros come from the numerator and poles come from the denominator.
Poles are especially important because their real parts help determine stability and transient response.
Formula & Symbols
| Concept | Formula | Meaning |
|---|---|---|
| Transfer function | H(s)=N(s)/D(s) | Ratio of output to input in the s-domain. |
| Zeros | N(s)=0 | Values of s that make the numerator zero. |
| Poles | D(s)=0 | Values of s that make the denominator zero. |
| Stability idea | Re(poles)<0 | Continuous-time system is stable when all pole real parts are negative. |
Step-by-Step Examples
- N(s)=s²+3s+2.
- Factor: (s+1)(s+2).
- Zeros are -1 and -2.
Where students use this
Common Mistakes to Avoid
Zeros come from the numerator; poles come from the denominator.
Second-order systems often have complex-conjugate poles.
Stability depends primarily on pole locations, not zero locations.
Frequently Asked Questions
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Formula Explorer connections
Interpretation: This formula describes an algebraic relationship among variables, functions, equations, roots or sequences. Assumption: Respect the expression’s domain and excluded values. Check roots in the original equation because transformations can introduce extraneous solutions.