Steady-State Error Calculator

• •Use the Steady-State Error 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 Steady-State Error Calculator is a precision engineering calculation tool designed for students, engineers, and technical professionals. Steady-state error of a unity-feedback loop for step, ramp and parabolic inputs, with the system type and Kp, Kv and Ka, after checking that the closed loop is stable 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.

In a unity negative-feedback loop the controller acts on the error e(t) = r(t) − y(t), and with the open-loop transfer function G(s) the error's transform is E(s) = R(s)/(1 + G(s)). The steady-state error is what remains as t grows. When the closed loop is stable, the final value theorem gives it directly: e_ss = lim(s→0) s·E(s). For the three standard test inputs this becomes a property of G(s) near s = 0. A unit step (R = 1/s) leaves e_ss = 1/(1 + Kp), with the position constant Kp = lim G(s); a unit ramp (R = 1/s²) leaves 1/Kv, with Kv = lim s·G(s); and a unit parabola t²/2 (R = 1/s³) leaves 1/Ka, with Ka = lim s²·G(s). The system type, the number of free integrators (poles at s = 0) in G(s), decides which of these are finite. A type 0 loop follows a step with a constant error but cannot follow a ramp; a type 1 loop follows a step with no error and a ramp with the error 1/Kv; a type 2 loop follows a ramp with no error and a parabola with the error 1/Ka. Each integrator removes the error for one more input, which is why the integral action of a PI controller removes a step error. All of this assumes a stable closed loop. If a closed-loop pole, a root of D(s) + N(s), is in the right half-plane, the error grows without bound, and the formulas then produce a meaningless number: 0.5/(s − 1) has Kp = −0.5 and 1/(1 + Kp) = 2, yet its closed-loop pole at +0.5 makes the error diverge. A closed-loop pole on the jω axis leaves the error oscillating or drifting, so there is no final value either. This calculator therefore finds the closed-loop poles first, with the same root finder as the site's Root Locus and Bode tools, and gives a number only for a stable loop. It finds the type and the error constants from the coefficients: factors of s are counted exactly from the trailing zeros you type, and no other factor is cancelled, because a factor that N(s) and D(s) share stays in the closed loop. (s − 1)/((s − 1)(s + 2)) looks like 1/(s + 2), but its closed loop has a pole at +1, and the tool reports it as unstable.

• •Choosing the loop type: a servo that must hold a fixed setpoint needs a finite or zero step error, one that must track a target moving at constant speed needs a type 1 loop, and constant acceleration a type 2 loop.
• •Sizing a gain for a ramp specification: in a type 1 loop the ramp error is 1/Kv, so e_ss ≤ 0.05 for a unit ramp needs Kv ≥ 20 s⁻¹, the target of the lead-compensator example on the Bode page.
• •Checking homework: the table gives Kp, Kv, Ka and all three errors at once, and says when the closed loop is unstable instead of applying the formula anyway.
• •Seeing what integral action does: add an integrator (a trailing 0 in the denominator) and watch the step error go to zero and the ramp error become finite, as long as the loop stays stable.
• •Catching hidden modes: a factor shared by N(s) and D(s) cancels in G(s) but not in the closed loop. The tool keeps it, and reports the loop as unstable or marginally stable when it is.

• !Applying e_ss = 1/(1 + Kp) without checking the closed loop: an unstable loop has no steady-state error, whatever the formula gives.
• !Counting the type from the closed-loop transfer function. The type is the number of integrators in the open-loop G(s), the forward path of the unity-feedback loop.
• !Using these formulas with a feedback path H(s) ≠ 1 without first rewriting the loop as unity feedback. This calculator assumes unity negative feedback.
• !Cancelling a factor that N(s) and D(s) share before checking stability: the cancelled mode is still in the closed loop.
• !Mixing up the parabolic input t²/2 (R = 1/s³) with t² (R = 2/s³): the second doubles the error.
• !Expecting a type 1 loop to follow a ramp with no error. Its ramp error is 1/Kv: finite, but not zero.