Cohen–Coon PID Tuning Calculator — Kp, Ki, Kd from Process Gain, Time Constant and Dead Time
Calculate Kp, Ki and Kd using process gain, time constant and dead time based on the Cohen–Coon PID tuning formula.
Enter the known values and review the calculated result
Input parameters
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Process parameters
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
Kp, Ki and Kd formula:
R = L / T
Kp = (1 / K) · (T / L) · (4/3 + R/4)
Ti = L · (32 + 6R) / (13 + 8R)
Td = L · 4 / (11 + 2R)
Ki = Kp / Ti
Kd = Kp · Td
where:
- Kp — proportional gain (—)
- Ki — integral gain (1/s)
- Kd — derivative gain (s)
- K — process gain (—)
- T — time constant (s)
- L — dead time (s)
- R — dead time ratio L/T (—)
- Ti — integral time (s)
- Td — derivative time (s)
When to use this calculator
When to use this calculator:
- To calculate Cohen–Coon PID proportional gain from process gain, time constant and dead time.
- To calculate integral gain from proportional gain and integral time.
- To calculate derivative gain from proportional gain and derivative time.
- To classify a process as lag-dominant, balanced or dead-time dominant using the dead time ratio.
- To check whether the calculated PID settings exceed the gain, integral time, derivative time or dead time ratio limits defined in the calculator logic.
How to interpret the result
Kp is defined as the proportional gain calculated from process gain, time constant and dead time. Ki is defined as proportional gain divided by integral time. Kd is defined as proportional gain multiplied by derivative time.
The result depends on process gain, time constant and dead time. Increasing process gain decreases proportional gain. Increasing time constant relative to dead time increases proportional gain. Increasing dead time changes the dead time ratio and directly affects integral time and derivative time.
- Safe — proportional gain, integral gain and derivative gain are not greater than 50, integral time is not below 0.1 × dead time, derivative time is not greater than time constant, and dead time ratio is not greater than 1.
- Limit — dead time ratio is greater than 1 and less than or equal to 1.5, with no unsafe condition.
- Warning — derivative time is greater than time constant, or dead time ratio is greater than 1.5, with no unsafe condition.
- Unsafe — proportional gain, integral gain or derivative gain is greater than 50, or integral time is below 0.1 × dead time.
- Invalid — process gain, time constant or dead time is not finite or is less than or equal to zero, or the dead time ratio is invalid.
The result is used to evaluate calculated PID gain values, integral time, derivative time, dead time ratio and process classification.
Calculation example
Example:
A user wants to calculate Cohen–Coon PID settings for a process with measured gain, time constant and dead time.
- K — Process gain = 2
- T — Time constant = 30 s
- L — Dead time = 5 s
Result: Kp = 4.125, Ki = 1.627 1/s, Kd = 6.964 s.
Assumptions and limitations
- Process gain must be finite and greater than zero.
- Time constant must be finite and greater than zero.
- Dead time must be finite and greater than zero.
- The calculated PID settings are based only on process gain, time constant and dead time.
- Process classification is based only on the dead time ratio: below 0.1, from 0.1 to 1, and above 1.