Control SystemsPreliminary engineering
Pole-Zero Analyzer & s-Plane Plot
Poles and zeros on the s-plane with left/right-half-plane classification, the stability verdict, and ζ / ωn of the dominant pole pair.
About this calculator
The s-plane is the control engineer's chart of the world: distance from the imaginary axis is decay rate, distance from the origin is natural frequency, and the angle from the negative real axis encodes damping. This tool computes the poles and zeros of H(s) and places them on that map, classifies each pole by half-plane, and renders the stability verdict that follows.
For the dominant complex-conjugate pair — the stable pair closest to the imaginary axis, which shapes most of what you see in a step response — it reports the damping ratio ζ = cos θ (θ measured from the negative real axis) and the natural frequency ωn = |p|. Those two numbers connect the geometry directly to time-domain behavior: ζ sets the overshoot, ζωn sets the settling envelope, and the damped frequency ωd = Im(p) is what actually appears as ringing.
The worked solution writes the characteristic equation, lists the roots exactly as computed, and shows the ζ/ωn extraction from the dominant pole's coordinates. The chart shades the left half-plane (decay) and right half-plane (growth) so a glance answers the first question — and the color is always backed by the text verdict, never a substitute for it.
Second-order intuition transfers to higher orders exactly when the dominant pair is well separated from the remaining poles (a factor of 5 or more further left is the usual rule of thumb); the interpretation panel says when that holds and when it doesn't.
Design notes & common mistakes
- Dominance is relative: the '5× further left' rule of thumb assumes the far poles carry comparably small residues. A far pole paired with a nearby zero can still matter.
- ζ of the dominant pair predicts overshoot only when no finite zero sits near the pair — a zero inside ~10ωn adds overshoot beyond the classic %OS(ζ) value.
- Imaginary-axis zeros produce notches, not instability; imaginary-axis poles produce sustained oscillation — students swap these constantly.
- Moving a pole deeper into the LHP speeds decay but costs control effort in closed loop; pole locations are a budget, not a free choice.
Assumptions
- Continuous-time proper SISO LTI transfer function in the Laplace domain.
- The dominant-pair ζ/ωn interpretation assumes that pair actually dominates — the interpretation panel checks the 5× separation heuristic and warns when it fails.
- Roots of degree ≥ 3 polynomials are computed numerically (documented Durand–Kerner iteration; residual-verified in the engine test suite).
When to use this calculator
Appropriate for
- Placing poles and zeros on the s-plane and reading dominant-pair damping ratio and natural frequency
- Judging whether second-order intuition applies to a higher-order system via the dominance check
- Teaching the s-plane geometry of decay, frequency, and damping
Not suitable for
- Systems where no pole pair dominates, so the ζ/ωn of the nearest pair does not summarize the response
- Discrete-time systems, where the unit circle — not the imaginary axis — is the stability boundary
- Design decisions on a real plant without validating the pole/zero model against measurement
What this calculator does not cover
- Denominator order capped at 4 in the entry form (engine supports more; UI and worked solution are tuned for hand-scale systems).
- The ζ/ωn readout describes the dominant pair only — it is not a property of the full system when dominance fails.
- No root-locus: this shows the poles of the H(s) you typed, not how they move with a gain (planned as a separate tool).
- Continuous-time only; no z-plane or delay terms.
- As with every calculator on this site: results are preliminary and educational, are not verified for any specific installation, and must be reviewed against the applicable code edition and stamped by a licensed Professional Engineer before real-world use.
Frequently asked questions
How do I read damping ratio off the s-plane?
ζ = cos θ, where θ is the angle of the pole measured from the negative real axis. Poles on the real axis have ζ = 1 (no oscillation); poles on the imaginary axis have ζ = 0 (pure oscillation); the 45° ray is ζ ≈ 0.707.
Which pole pair is 'dominant'?
The stable pair closest to the imaginary axis — it decays slowest, so its transient outlives the others and shapes what you see. The usual heuristic: second-order formulas apply when other poles are at least 5× further left and no nearby zero distorts the residues.
What does a zero in the right half-plane mean?
A non-minimum-phase system. It is stable if the poles are stable, but its step response initially moves opposite to its final value, and the zero imposes a fundamental bandwidth limit — no controller can hide it.
Why does my pole-zero map show poles exactly on the imaginary axis?
The denominator has a root with zero real part — an undamped oscillator (±jω) or an integrator (s = 0). These are marginal: the mode neither decays nor grows, and any additional pole at the same location tips the system into instability.
References
- Nise, N. S., Control Systems Engineering, 8th ed., Ch. 4 (poles, zeros, and system response; second-order pole geometry)
- Franklin, G., Powell, J. & Emami-Naeini, A., Feedback Control of Dynamic Systems, 8th ed., Ch. 3 (pole locations and time response, dominance)
- Ogata, K., Modern Control Engineering, 5th ed., Ch. 5 (transient response and the s-plane)
Related calculators
Transfer Function Analyzer
Enter H(s) as polynomial coefficients and get poles, zeros, gain, characteristic equation, system order and type, and a stability verdict — with the pole-zero map and a live step-response preview.
Second-Order System Calculator
Sweep ζ and ωn with live sliders and watch the poles move and the step response reshape — with every time- and frequency-domain characteristic computed in closed form.
Step / Impulse / Ramp Response
Simulate the step, impulse, or ramp response of H(s) and read the performance metrics — rise time, peak time, overshoot, settling time — with markers on the curve.
Stability & Margins Checker
Close the unity-feedback loop around L(s) and check it two independent ways: the Routh–Hurwitz table on the characteristic polynomial, and the gain/phase margins with their crossover frequencies.
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