Ziegler–Nichols PID Tuning Calculator — Kp, Ki, Kd from Ultimate Gain Ku and Oscillation Period Tu
Calculate classic Ziegler–Nichols P, PI and PID tuning settings from ultimate gain Ku and ultimate oscillation period Tu, including Kp, Ki, Kd, Ti and Td.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Ultimate gain
Ultimate oscillation period
Results
Calculation tools Save, export and manage your results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
Classic Ziegler–Nichols closed-loop tuning formulas
This calculator uses the ultimate gain Ku and ultimate oscillation period Tu to calculate the classic Ziegler–Nichols settings for P, PI and PID controllers.
P controller
Kp,P = 0.5 · Ku
PI controller
Kp,PI = 0.45 · Ku
Ti,PI = Tu / 1.2
Ki,PI = Kp,PI / Ti,PI
Therefore: Ki,PI = 0.54 · Ku / Tu
Classic PID controller
Kp = 0.6 · Ku
Ti = 0.5 · Tu
Td = 0.125 · Tu
The calculator also converts the integral and derivative times to the equivalent parallel PID gains:
Ki = Kp / Ti = 1.2 · Ku / Tu
Kd = Kp · Td = 0.075 · Ku · Tu
Controller form
The primary PID results correspond to the parallel form:
C(s) = Kp + Ki / s + Kd · s
The same PID tuning can also be described by the standard or ideal parameters Kp, Ti and Td.
Symbols and units
- Ku — ultimate gain magnitude [output/error].
- Tu — ultimate oscillation period [s].
- Kp — proportional gain [output/error].
- Ti — integral time [s].
- Td — derivative time [s].
- Ki — integral gain [output/(error·s)].
- Kd — derivative gain [output·s/error].
The numerical units of the gain terms depend on how the controller output and error signals are scaled. Ku and the calculated proportional gains must use the same output-to-error gain convention.
When to use this calculator
Use this Ziegler–Nichols PID tuning calculator when you already know the ultimate gain Ku and ultimate oscillation period Tu of a feedback-control loop and want to convert them into classic P, PI or PID controller settings.
Typical uses
- Calculate classic PID values Kp, Ki and Kd from Ku and Tu.
- Calculate the equivalent PID integral time Ti and derivative time Td.
- Calculate proportional-only tuning Kp,P.
- Calculate PI tuning Kp,PI, Ti,PI and Ki,PI.
- Compare the classic P, PI and PID Ziegler–Nichols closed-loop tuning rules from the same ultimate-gain test data.
- Convert standard or ideal PID time parameters into the equivalent parallel-form Ki and Kd values used by many controller implementations.
When not to use this calculator
This calculator does not determine Ku or Tu from a plant model, step-response dataset or frequency-response measurement. Those values must already be known before calculation.
It also does not implement the Ziegler–Nichols open-loop reaction-curve method. If your available data consists of process gain, dead time and time constant from a step response, use an open-loop or FOPDT-based PID tuning method instead.
The calculation does not simulate the resulting closed-loop response or verify stability margins, overshoot, settling time, actuator saturation, measurement noise, derivative filtering, integral windup or discrete-controller implementation. The calculated gains should therefore be treated as initial tuning values that require verification on the actual system or a representative model.
How to interpret the result
The primary results are the classic Ziegler–Nichols PID gains Kp, Ki and Kd in parallel-controller form. The calculator also reports Ti and Td for the equivalent standard or ideal PID form, plus the corresponding P and PI tuning values.
How Ku affects the result
Increasing Ku increases all calculated proportional gains directly. For a fixed Tu, Ki, Kd and Ki,PI also increase in direct proportion to Ku.
How Tu affects the result
The proportional gains do not depend on Tu. A longer oscillation period increases Ti and Td, decreases the parallel integral gain Ki, and increases the derivative gain Kd.
For the classic PID rule:
- Kp = 0.6Ku and is independent of Tu.
- Ki = 1.2Ku/Tu, so shorter periods produce larger integral gain.
- Kd = 0.075KuTu, so longer periods produce larger derivative gain.
P and PI reference results
Kp,P is the proportional gain for the P-only rule. Kp,PI, Ti,PI and Ki,PI describe the corresponding PI controller. These are separate controller configurations and should not be combined with the PID gains.
Validity of the calculated result
The calculation requires finite values with Ku > 0 and Tu > 0. Zero, negative or non-finite inputs are returned as invalid.
A valid numerical result does not confirm that the resulting controller is stable, robust or suitable for the application. The calculator applies the classic tuning correlations only; closed-loop performance must still be verified for the actual plant, actuator, sensor, sampling system and controller implementation.
Calculation example
A control loop has already been tested and the measured ultimate-gain data are:
- Ku = 8.0 output/error
- Tu = 3.0 s
1. P controller
Kp,P = 0.5 · Ku
Kp,P = 0.5 · 8.0 = 4.0 output/error
2. PI controller
Kp,PI = 0.45 · 8.0 = 3.6 output/error
Ti,PI = 3.0 / 1.2 = 2.5 s
Ki,PI = Kp,PI / Ti,PI = 3.6 / 2.5 = 1.44 output/(error·s)
3. Classic PID controller
Kp = 0.6 · 8.0 = 4.8 output/error
Ti = 0.5 · 3.0 = 1.5 s
Td = 0.125 · 3.0 = 0.375 s
4. Convert PID time parameters to parallel gains
Ki = Kp / Ti = 4.8 / 1.5 = 3.2 output/(error·s)
Kd = Kp · Td = 4.8 · 0.375 = 1.8 output·s/error
Result
The classic parallel-form PID starting values are Kp = 4.8, Ki = 3.2 and Kd = 1.8, using the stated output/error signal scaling. The equivalent standard-form time parameters are Ti = 1.5 s and Td = 0.375 s.
These values are the direct result of the classic Ziegler–Nichols closed-loop formulas. They do not by themselves verify closed-loop stability, overshoot, robustness or actuator limits, so the resulting controller should be evaluated on the actual system or a representative model before final use.
Assumptions and limitations
- The calculation uses the classic Ziegler–Nichols closed-loop, or ultimate-gain, tuning rules.
- Ultimate gain Ku and ultimate oscillation period Tu are assumed to have been determined before using the calculator. The calculator does not identify these values from measured response data.
- Ku must be a finite positive gain magnitude and Tu must be a finite positive time. The JavaScript rejects values less than or equal to zero.
- Ku is treated as an output-to-error gain magnitude. Its numerical value therefore depends on the scaling and engineering units used for the controller output and error signals.
- The PID calculations use the standard or ideal time parameters Kp, Ti and Td, together with the equivalent continuous parallel-form gains Ki = Kp/Ti and Kd = KpTd.
- The calculator does not convert the continuous gains into discrete-time coefficients and does not account for controller sample time.
- The calculation does not model the plant or predict the resulting transient response, stability margin, overshoot, settling time or disturbance rejection.
- Actuator saturation, integral windup, derivative filtering, sensor noise, nonlinearities, dead zones, hysteresis and operating-point variation are not included.
- The open-loop Ziegler–Nichols reaction-curve method is not implemented by this calculator. Process gain, dead time and time constant are not input variables here.
- The calculated P, PI and PID settings are rule-based starting values. Final controller parameters should be verified against the actual process dynamics, controller structure and acceptable performance limits.
Frequently asked questions
For the classic PID rule, enter the ultimate gain Ku and ultimate oscillation period Tu. The calculator uses Kp = 0.6Ku, Ti = 0.5Tu and Td = 0.125Tu. It then calculates the equivalent parallel gains Ki = Kp/Ti and Kd = KpTd.
Ku is the ultimate proportional gain associated with the ultimate-cycle condition used by the closed-loop Ziegler–Nichols method. This calculator does not find Ku; it uses the positive gain value you have already determined as an input to the P, PI and PID tuning formulas.
Tu is the period of the ultimate oscillation. Some references use Pu for the same quantity. This calculator uses the symbol Tu and treats it as a time input.
Yes. In addition to the primary classic PID results Kp, Ki and Kd, the calculator returns Kp,P = 0.5Ku for a P controller and Kp,PI = 0.45Ku, Ti,PI = Tu/1.2 and Ki,PI = Kp,PI/Ti,PI for a PI controller.
Kp, Ti and Td describe the standard or ideal PID time-constant form. Kp, Ki and Kd describe the equivalent parallel form C(s) = Kp + Ki/s + Kds. The conversion used here is Ki = Kp/Ti and Kd = KpTd.
For the classic PID rule, Ki = 1.2Ku/Tu, so increasing Tu reduces integral gain. Kd = 0.075KuTu, so increasing Tu increases derivative gain. Kp depends on Ku only.
This calculator implements the closed-loop ultimate-gain method using Ku and Tu. It does not implement the open-loop reaction-curve method based on process gain, dead time and time constant.
No. Ku and Tu are required inputs. The calculator only applies the Ziegler–Nichols correlations after those two quantities have been obtained from the control system, test data or another appropriate analysis.
No. A valid result means that positive finite Ku and Tu values were successfully processed by the implemented tuning formulas. The calculator does not simulate the plant or calculate closed-loop stability margins. The resulting controller settings still require verification for the actual system.
No. This calculator performs Ziegler–Nichols PID tuning from an ultimate gain magnitude and oscillation period; it is not a dB-to-linear converter. If a source provides a gain in decibels, convert that value to the appropriate linear gain convention before using it as Ku. A dedicated logarithmic-scale or dB conversion calculator should be used for that conversion.