PID Parameter Translator Loops
One PID tuning, many spellings — gain or proportional band, repeats per minute or integral time, and three algorithm forms that make the same three numbers mean different things.
Inputs are seeded with an example — edit them to your numbers.
Input
0 = integral off.
0 = derivative off.
State the time base — a per-minute Ki and a per-second Ki for the same loop differ by a factor of 60, and Kd the same the other way. 0 = that action off.
Output
Entries are read as the standard (ISA) form — gain multiplies all three terms. Each output group is labeled with the form it belongs to; the same three numbers behave differently on a controller that implements a different form, so match the form to your controller’s documentation before copying anything.
Across algorithm forms
| Parameter | Standard (ISA) | Series (interacting) | Parallel (independent) |
|---|---|---|---|
| P | — | — | — |
| I | — | — | — |
| D | — | — | — |
State the time base whenever you write an independent gain down: a per-minute Ki and a per-second Ki for the same loop differ by 60×, and Kd by the same factor the other way. And if your platform takes a proportional band in engineering units instead of percent, PB in EU = PB % × span ÷ 100.
The three forms
Standard (ISA): u = Kc · (e + (1/Ti) ∫e dt + Td · de/dt) — one gain multiplies all three terms, so retuning Kc rescales P, I, and D together.
Series (interacting): u = Kc′ · (1 + 1/(Ti′·s)) · (1 + Td′·s), a product of stages — the derivative stage feeds the P·I stage, the way pneumatic and early electronic controllers were built, so Ti′ and Td′ interact. Multiply the product out and you get the standard form with slightly different numbers; that arithmetic is exactly what this page does.
Parallel (independent): u = Kp·e + Ki ∫e dt + Kd · de/dt — three separate gains with no shared multiplier, so changing Kp leaves the integral and derivative action untouched.
Real controllers also put a filter on the derivative term (a limited derivative gain) so it doesn’t amplify sensor noise; this page translates the ideal, unfiltered forms, which is what the parameter arithmetic is defined on.
Worked example
A controller hands you an interacting-form tuning: gain 2.0, Ti 4.0 min, Td 1.0 min. What does the same loop need on a standard-form controller — and as independent gains?
- Interaction factor:
F = 1 + Td′/Ti′ = 1 + 1.0/4.0 = 1.25. - Gain:
Kc = 2.0 × 1.25 = 2.5→ PB = 100 ÷ 2.5 = 40 %. - Integral:
Ti = 4.0 × 1.25 = 5.0 min→ 0.2 rep/min → 300 s. - Derivative:
Td = 1.0 ÷ 1.25 = 0.8 min→ 48 s. - Independent gains:
Kp = 2.5·Ki = 2.5 ÷ 5.0 = 0.5 min⁻¹(0.0083 s⁻¹) ·Kd = 2.5 × 0.8 = 2.0 min.
Same loop — and the knob labeled gain legitimately reads 2.0 or 2.5 depending on the form, while the integral knob reads 4.0, 5.0, 0.2, 300, or 0.5 depending on form and units. That spread is the whole reason this page exists.
Verify the translation at the bench, not on paper. The arithmetic on this page is exact, but which algorithm form a controller actually runs — and whether its times read in minutes or seconds — is a documentation question no calculator can settle. A tuning carried across on the wrong form or time base can turn a stable loop into one that hard enough to wear an actuator or trip the equipment on the swings. This page is the conversion math and a second opinion; the controller’s own documentation governs the form, and the trend after the first setpoint change is the proof. Watch the loop settle before you walk away.