Pole-Zero Map Calculator

• •Use the Pole-Zero Map Calculator to check your hand calculations on homework or exam problems — comparing its result with your own working helps identify arithmetic errors before they propagate.
• •Use it during the early design phase to rapidly iterate on parameters and narrow down feasible configurations before committing time to detailed finite element simulations or full design packages.
• •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility — a quick sanity check can prevent costly downstream errors.
• •Use it to generate reference data for a technical report or presentation without manual computation, ensuring consistent, reproducible numbers throughout the document.
• •Use it in the field when a quick estimate is needed and a full engineering software package is not available.

The Pole-Zero Map Calculator is a precision engineering calculation tool designed for students, engineers, and technical professionals. Plot the poles and zeros of a transfer function G(s) = N(s)/D(s) on the s-plane, with each root's multiplicity, natural frequency and damping ratio, and the stability of G(s) All calculations are performed using established engineering formulas from the relevant scientific literature and standards. Follow the units printed beside each field. Where a unit selector is provided, the selected unit is converted internally. Results are computed in your browser without sending data to a server, which keeps your inputs private. This calculator is intended as a supplementary tool for learning and design exploration; always verify results against authoritative references for safety-critical applications.

A transfer function G(s) = N(s)/D(s) is a ratio of two polynomials in the Laplace variable s. Its zeros are the roots of N(s), where G(s) = 0, and its poles are the roots of D(s), where |G(s)| grows without bound. Plotting both on the complex s-plane, with the real part σ across and the imaginary part ω up, gives the pole-zero map: the most compact picture of a linear system's dynamics. Each real pole p contributes a mode e^(pt) to the impulse response, and a complex pair σ ± jω contributes e^(σt)·sin(ωt + φ). So a pole's horizontal position sets how fast its mode decays (or grows), and its vertical position sets how fast it oscillates. For a pole p = σ + jω, the distance from the origin is the natural frequency ωn = |p|, and the damping ratio is ζ = −σ/|p|, the cosine of the pole's angle from the negative real axis. For a proper G(s), whose numerator degree is no higher than its denominator's, stability is read from the poles alone. G(s) is stable when every pole lies strictly in the left half-plane; marginally stable when no pole is in the right half-plane and every pole on the jω axis is simple (an integrator 1/s, an undamped oscillator 1/(s² + 1)); and unstable when any pole is in the right half-plane or a pole on the jω axis is repeated, because 1/s² has the impulse response t, which grows without bound. An improper G(s), with a numerator of higher degree, is never BIBO-stable, whatever its poles: its gain grows without bound at high frequency. Where the zeros lie does not decide stability, but it shapes the response: a zero close to a pole nearly cancels that pole's mode, and a zero in the right half-plane makes the step response start in the wrong direction. This calculator reads the coefficients strictly: a token that is not a number is reported, never dropped, because dropping it would shift every later coefficient to a different power of s. It finds the roots with the same root finder as the site's Root Locus, Bode and Transfer Function tools, Aberth–Ehrlich iteration on the original polynomial, with a group of nearly equal roots tested and refined as one repeated root, so (s + 1)² is listed once as −1 (×2). If the root finder cannot produce trustworthy roots, for example a root beyond double-precision range, the tool draws no map and calls the stability unknown, because an empty or partial map would look like a stable system. The map shows the poles of G(s) itself. To see where the poles move when G(s) is closed in a feedback loop with gain K, use the Root Locus Plotter.

• •Checking a plant model before control design: plot the poles and zeros of a motor, thermal or suspension model to see its natural frequencies, its damping and any right-half-plane pole before choosing a controller.
• •Spotting near-cancellations: a zero close to a pole means that mode barely shows in the output. The table lists both to six significant figures, so you can see how close they are.
• •Identifying non-minimum-phase behaviour: a right-half-plane zero makes the step response start in the wrong direction and limits the bandwidth any controller can reach.
• •Reading damping from a complex pole pair: ωn and ζ in the table give the speed and damping of that mode without simulating it, the quantities second-order specifications are written in.
• •Learning the s-plane: the worked examples cover distinct real poles, repeated roots, a complex pair and a double integrator, the cases that decide stability.

• !Entering roots instead of coefficients: (s + 1)(s + 3) is entered as 1 4 3, not as −1 −3.
• !Leaving out a zero coefficient: s² + 1 is 1 0 1. Entering 1 1 gives s + 1, a different system.
• !Calling a system with poles on the jω axis stable: simple ones make it marginally stable and repeated ones unstable.
• !Judging stability from where the zeros are. Only poles decide it, unless G(s) is improper (more zeros than poles), which is never BIBO-stable; a right-half-plane zero still limits the response.
• !Reading the closed loop from this map. It shows the poles of G(s); feedback moves them, so use the Root Locus Plotter or the Steady-State Error Calculator for a closed loop.
• !Cancelling an unstable pole with a zero on paper. The cancelled mode is still inside the system and still grows.