Analyze transfer functions with precise pole and zero mapping. Inspect stability, damping, and gain instantly. Export clean results for reports, labs, and design reviews.
| Case | System Type | Numerator Coefficients | Denominator Coefficients | Expected Finite Zeros | Expected Finite Poles | Stability |
|---|---|---|---|---|---|---|
| Lead network | Continuous | 1, 2 | 1, 3, 2 | -2 | -2, -1 | Stable |
| Second order digital model | Discrete | 1, -0.2 | 1, -1.1, 0.3 | 0.2 | 0.5, 0.6 | Stable |
The transfer function is written as H(v) = K × N(v) / D(v).
Finite zeros are the roots of N(v) = 0.
Finite poles are the roots of D(v) = 0.
For continuous systems, stability requires Re(pole) < 0.
For discrete systems, stability requires |pole| < 1.
Damping ratio for a continuous pole is ζ = -σ / √(σ² + ω²).
Natural frequency is ωn = √(σ² + ω²).
Time constant is τ = -1 / σ when σ is nonzero.
1. Choose continuous-time or discrete-time analysis.
2. Enter numerator coefficients in descending powers.
3. Enter denominator coefficients in descending powers.
4. Add overall gain if your model includes a scalar multiplier.
5. Enter sample time for discrete systems when needed.
6. Leave plot range blank for automatic scaling.
7. Press the calculate button.
8. Review the summary, plot, root table, and exports.
A pole zero map shows where system roots sit. It summarizes dynamic behavior quickly. Control engineers use it for stability checks. Filter designers use it for frequency response insight. Students use it to connect equations with motion. This calculator turns coefficient lists into useful engineering outputs.
Poles shape natural response and settling behavior. Zeros shape transmission and cancellation behavior. A left half plane pole usually supports stable continuous motion. A pole outside the unit circle signals discrete instability. Closely placed poles and zeros may reduce visible effects. They can also hide modeling errors.
The tool accepts numerator and denominator coefficients. It then solves the polynomial roots numerically. Finite zeros come from the numerator. Poles come from the denominator. The calculator reports real parts, imaginary parts, magnitude, angle, and relative degree. It also checks properness and near cancellations. Continuous systems include damping ratio and natural frequency metrics. Discrete systems include radius and equivalent mapped values when sample time is provided.
A pole zero map helps with controller tuning. It also supports compensator design and model review. You can compare plant and controller dynamics quickly. You can inspect oscillation risk before hardware testing. You can see whether extra zeros may cause overshoot. This saves time during early design iterations.
Poles near the imaginary axis often mean slow decay. Repeated poles can increase sensitivity. Complex conjugate poles often create oscillation. Zeros near dominant poles may flatten response. Large relative degree changes high frequency roll off. Always verify units and coefficient order before trusting results.
The export tools help with design records. CSV supports spreadsheets and batch comparison. PDF supports quick sharing and print review. The example table below shows common transfer functions. Use it as a validation reference before analyzing custom data.
In practice, pole zero maps support servo systems, power electronics, communication filters, and vibration models. They reveal whether a design is robust or fragile. They also help explain why two transfer functions with similar gains can behave very differently in time and frequency domains.
It shows the finite zeros and finite poles of a transfer function on the complex plane. Their positions help explain stability, oscillation, damping, and frequency response behavior.
Poles belong to the natural response. If they sit in unstable regions, the system can grow instead of decay. That is why pole locations dominate basic stability checks.
Yes. That creates an improper transfer function. The calculator reports the relative degree and properness so you can detect models that may need restructuring.
A nearby zero may partially cancel the pole's effect. This can simplify observed behavior, but it can also hide sensitivity or model mismatch in real systems.
Repeated poles can increase sensitivity and change time response shape. They often make the system more delicate during tuning, approximation, and implementation.
For continuous poles, the calculator uses ζ = -σ / √(σ² + ω²). Here, σ is the real part and ω is the imaginary part.
Enter sample time when you analyze a discrete model and want equivalent mapped values. It is not required for a continuous transfer function.
The table reports finite zeros from the numerator polynomial. If the numerator is constant, the transfer function has no finite zeros to display.
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.