Control SystemsDesign workbenchPreliminary engineering
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.
About this calculator
The transfer function is the identity card of a linear time-invariant system: everything about its dynamics — how fast it responds, whether it rings, whether it is stable at all — is encoded in the roots of two polynomials. This analyzer takes H(s) = N(s)/D(s) as you would write it on paper and produces the complete record: poles (roots of the characteristic equation D(s) = 0), zeros (roots of N(s)), the leading-coefficient gain, DC gain H(0), system order, and system type (the number of pure integrators).
Stability is read directly from the pole locations: every pole strictly in the left half of the s-plane means every natural mode decays and the system is BIBO stable. A single right-half-plane pole is one growing exponential — no input choice can hide it. Simple poles exactly on the imaginary axis sit on the boundary: a sustained oscillation or a constant offset that never decays, classified here as marginally stable, and repeated poles on the axis grow without bound.
The pole-zero map shows this geometry at a glance, and the step-response preview shows what those root locations mean in time — change any coefficient and both recompute live. When the analysis points somewhere deeper, the related tools take over: the pole-zero analyzer for dominant-pair damping, the time-response tool for performance metrics, the Bode and Nyquist plots for frequency-domain design, and the stability checker for the Routh–Hurwitz table.
Everything here treats the continuous-time Laplace-domain H(s) of a proper system (numerator degree ≤ denominator degree); the entry form enforces properness by construction.
Design notes & common mistakes
- Poles decide stability; zeros never do. A right-half-plane ZERO is not instability — it is a non-minimum-phase system whose step response starts in the wrong direction and whose achievable bandwidth is fundamentally limited.
- A near pole-zero cancellation looks clean algebraically but the hidden mode is still there physically — if it is slow or unstable, the practical behavior will not match the reduced model.
- The classic sign error: writing the characteristic polynomial with a negative coefficient somewhere and not noticing. Any sign change among the denominator coefficients guarantees at least one RHP root.
- Entering a measured plant? Coefficients spanning many orders of magnitude are poorly conditioned for root finding — rescale time (s → s/ω₀) first so coefficients stay comparable.
Assumptions
- Continuous-time, linear, time-invariant SISO system in the Laplace s-domain.
- Proper transfer function (numerator degree ≤ denominator degree) — enforced by the entry form.
- Coefficients are exact as entered; no uncertainty or parameter tolerance is propagated.
- Poles and zeros of degree ≥ 3 polynomials are computed numerically (Durand–Kerner iteration) and are accurate to far better than display precision for the supported order range (≤ 4).
When to use this calculator
Appropriate for
- Analyzing a continuous-time, linear, time-invariant SISO transfer function: poles, zeros, gain, order, type, and stability
- Confirming a hand-derived transfer function parses to the poles and stability you expect
- Teaching how pole and zero locations shape response and stability
Not suitable for
- Nonlinear, time-varying, or multi-input/multi-output systems, which this s-domain model does not represent
- Discrete-time (z-domain) systems or plants with transport delay, unless the delay is first approximated by a rational term
- Certifying the stability of a real plant without validating the model against measured behavior
What this calculator does not cover
- Denominator order is capped at 4 in this tool. Higher-order systems are planned together with the root-locus tool; the underlying engine already computes them but entry and worked-solution display are tuned for n ≤ 4.
- Coefficient entry only — factored zero-pole-gain entry is not exposed in the UI yet (the ζ/ωn parametric form is available on the second-order calculator).
- Continuous-time only: no discrete-time (z-domain) analysis, no transport delay e^(−sT).
- Exact pole-zero cancellations are reported as coincident roots, not cancelled — deciding whether a cancellation is physically meaningful requires knowing where the model came from.
- 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
What makes a transfer function stable?
BIBO stability requires every pole — every root of the denominator — to lie strictly in the left half of the s-plane, so all natural modes decay. One right-half-plane pole is one growing exponential, and no input shaping can remove it; simple imaginary-axis poles are the marginal boundary case.
What is the difference between system order and system type?
Order is the denominator degree — how many states the system carries. Type counts only the poles at the origin (pure integrators), which is what sets steady-state tracking ability in a unity-feedback loop: type 1 tracks a step exactly, type 2 also tracks a ramp.
Why is my DC gain shown as unbounded?
The denominator has a root at s = 0 (a pure integrator), so H(0) = N(0)/0. Physically, a constant input is integrated forever and the steady output grows without limit — the honest answer is 'unbounded', not a large number.
Do zeros affect stability?
No — stability is decided entirely by the poles. Zeros shape the response: they redistribute mode amplitudes, can add overshoot or undershoot, and a right-half-plane zero marks a non-minimum-phase system with an initial wrong-way response, but a system with all poles in the LHP is stable regardless of its zeros.
References
- Nise, N. S., Control Systems Engineering, 8th ed., Ch. 2 (transfer functions) and Ch. 6 (stability via pole locations)
- Ogata, K., Modern Control Engineering, 5th ed., Ch. 2 (mathematical modeling, transfer functions)
- Franklin, G., Powell, J. & Emami-Naeini, A., Feedback Control of Dynamic Systems, 8th ed., Ch. 3 (dynamic response, poles and zeros)
Related calculators
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.
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.
Bode Plot Generator
Magnitude (dB) and unwrapped phase of H(jω) on a log-frequency axis, with gain and phase crossovers located and both stability margins annotated.
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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