Pole Zero Plot Calculator

Calculator Input

System type Continuous-time Discrete-time

Gain

Rounding digits

Numerator coefficients 1, 5

Enter highest-power coefficient first. Example: 1, 5

Denominator coefficients 1, 6, 11, 6

Enter highest-power coefficient first. Example: 1, 6, 11, 6

Sample time

Used for discrete systems and equivalent damping metrics.

Plot poles and zeros Load example Reset form

How to use this calculator

1. Select a continuous-time or discrete-time model.
2. Enter the scalar gain.
3. Provide numerator coefficients from highest power to constant term.
4. Provide denominator coefficients in the same order.
5. Enter sample time when you analyze a discrete model.
6. Submit the form to compute poles, zeros, damping values, and stability.
7. Review the table, transfer function, and pole-zero graph.
8. Export the results as CSV or PDF when needed.

Formula used

Transfer function form

The calculator models a transfer function as H(x) = K × N(x) / D(x). For continuous systems, x = s. For discrete systems, x = z.

Poles and zeros

Zeros are the roots of N(x). Poles are the roots of D(x). The calculator finds those roots numerically from the coefficient sets you enter.

Continuous-time stability

A continuous system is stable when every pole has a negative real part. A pole on the imaginary axis gives marginal stability. Any pole in the right half-plane makes the system unstable.

Discrete-time stability

A discrete system is stable when every pole lies inside the unit circle. A pole on the unit circle gives marginal stability. Any pole outside the unit circle makes the system unstable.

Damping ratio and natural frequency

For a pole p = σ + jω, the calculator uses ωn = √(σ² + ω²) and ζ = -σ / ωn. For discrete models with sample time, it first maps z to an equivalent s-plane value using s = ln(z) / T.

Example data table

FAQs

1. What does a pole-zero plot show?

It shows where transfer function roots sit in the complex plane. Zeros come from the numerator. Poles come from the denominator. Their locations help explain stability, oscillation, and transient shape.

2. Why are poles important in control design?

Poles dominate natural response. They shape decay rate, oscillation, rise time, and stability. Designers usually watch pole movement first, then use zeros and gain to refine performance.

3. Do zeros affect stability?

Zeros do not set internal stability by themselves. Pole locations decide stability. Still, zeros strongly affect overshoot, phase, notch behavior, and how the output responds to commands or disturbances.

4. How is stability checked here?

Continuous models are stable when all poles have negative real parts. Discrete models are stable when all poles have magnitudes below one. Boundary cases are marked marginally stable.

5. Why do complex poles appear in pairs?

With real coefficients, nonreal roots occur as conjugate pairs. If one pole is a + jb, the other is a - jb. That keeps the polynomial coefficients real.

6. What is the meaning of damping ratio?

Damping ratio measures how quickly oscillation dies out. Larger positive values usually mean less ringing. Values near zero suggest sustained oscillation. Negative values indicate unstable growth.

7. Does changing gain move poles and zeros?

Simple scalar gain multiplies the numerator only. That usually changes the transfer function amplitude and static gain, but it does not move existing poles or zeros.

8. Why can repeated roots look numerically sensitive?

Repeated or near-repeated roots make polynomial problems ill-conditioned. Small coefficient changes can shift the estimated root locations. That is normal in numerical root solving.