Ziegler–Nichols PID Tuning Calculator (Ultimate Gain Method)

How to use the Ziegler–Nichols PID tuning calculator

Enter the ultimate gain Ku and the ultimate period Pu from your closed-loop test. The calculator gives the controller gain for P, PI and PID control and the integral and derivative times in seconds. Settings assume the ideal (non-interacting) PID form with integral time in seconds per repeat; check whether your controller uses gain or proportional band, and repeats per minute or minutes per repeat.

Ziegler–Nichols closed-loop rules

Controller	Kc	Ti	Td
P	0.5 Ku	—	—
PI	0.45 Ku	Pu ÷ 1.2	—
PID	0.6 Ku	Pu ÷ 2	Pu ÷ 8

Example

A loop oscillates steadily at Ku = 2 with a period Pu = 60 s:

Rule	Kc	PB	Ti	Td
P only	1	100%	—	—
PI	0.9	111%	50 s	—
PID (classic)	1.2	83%	30 s	7.5 s
PI (Tyreus–Luyben)	0.625	160%	132 s	—
PID (Tyreus–Luyben)	0.909	110%	132 s	9.52 s

Cautions

• The test drives the loop to the edge of instability — don’t do it on loops where oscillation is unsafe; use a step test and the lambda tuning calculator instead.
• Valve stiction and sensor lag change Ku and Pu; fix mechanical problems first.
• After tuning, test with a small set-point step and reduce gain if overshoot is excessive.

Frequently asked questions

How do I find Ku and Pu?

Put the controller in automatic with integral and derivative off, then raise the gain in small steps until the process variable oscillates with constant amplitude. That gain is Ku and the time between peaks is Pu.

What are the Ziegler–Nichols PID settings?

Kc = 0.6 Ku, Ti = Pu ÷ 2 and Td = Pu ÷ 8. For PI, Kc = 0.45 Ku and Ti = Pu ÷ 1.2.

Why are Ziegler–Nichols settings so aggressive?

They target a quarter-amplitude decay, which overshoots and oscillates. Many engineers halve the gain, or use the gentler Tyreus–Luyben rules shown in the table.

How do I convert gain to proportional band?

PB (%) = 100 ÷ Kc. A gain of 1.2 is a proportional band of about 83%.

Related calculators

• Lambda Tuning Calculator (IMC PI & Cohen–Coon)Calculate PI controller gain and integral time from process gain, time constant and dead time using lambda (IMC) tuning, with Cohen–Coon for comparison.
• FOPDT Step Test Calculator (Gain, Time Constant, Dead Time)Find process gain, time constant and dead time of a first-order-plus-dead-time model from an open-loop step test using the 28.3% and 63.2% times.
• Sensor Response Time Calculator (Time Constant, T63, T90)Calculate T50, T63.2, T90 and T99 response times of a first-order sensor from its time constant, and the reading at any time after a step change.
• 4–20 mA Scaling CalculatorConvert a 4–20 mA loop current to a process value (and back) for any measuring range — with live scaling chart.
• Thermal Expansion CalculatorCalculate how much a pipe, rail or bar expands with temperature for steel, stainless steel, aluminium, copper and PVC.
• 4–20 mA Loop Resistance Calculator (Maximum Loop Load)Check whether a 4–20 mA loop has enough voltage — maximum loop resistance from supply and transmitter lift-off voltage, with cable and barrier resistance.

Embed this calculator