PID Tuning Calculator Formula

Understand the math behind the pid tuning calculator. Each variable explained with a worked example.

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Formulas Used

Proportional Gain (Kp)

kp = 0.6 * ku

Integral Gain (Ki)

ki = 1.2 * ku / tu

Derivative Gain (Kd)

kd = 0.075 * ku * tu

Integral Time (Ti)

ti = tu / 2

Derivative Time (Td)

td = tu / 8

Variables

Variable	Description	Default
`ku`	Ultimate Gain (Ku)	10
`tu`	Ultimate Period (Tu)(s)	2

How It Works

Ziegler-Nichols PID Tuning

The Ziegler-Nichols closed-loop method determines PID gains from two measurements: the ultimate gain Ku (gain at which the system oscillates) and the ultimate period Tu (period of those oscillations).

Formulas

Kp = 0.6 × Ku

Ti = Tu / 2, so Ki = Kp / Ti = 1.2 × Ku / Tu

Td = Tu / 8, so Kd = Kp × Td = 0.075 × Ku × Tu

These gains provide a starting point; fine-tuning is usually needed to balance response speed, overshoot, and stability.

Worked Example

A system oscillates at Ku = 10 with period Tu = 2 seconds.

ku = 10tu = 2

01Kp = 0.6 × 10 = 6.0
02Ti = 2 / 2 = 1.0 s, Ki = 1.2 × 10 / 2 = 6.0
03Td = 2 / 8 = 0.25 s, Kd = 0.075 × 10 × 2 = 1.5

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