PID Parameter Converter

What does this calculator do?

It helps you compare PID controllers written in different conventions. It does not tune the controller for you.

Worked example

Kp = 2, Ki = 1 and Kd = 0.5 are equivalent to Ti = 2 s and Td = 0.25 s.

Practical tips

• Check which convention your PLC or software uses.
• If Ki = 0, Ti is considered infinite.
• The conversion does not replace a stability analysis or a check of actuator saturation.

Common mistake to avoid

Gains do not carry over from one plant to another: tuning must account for the process dynamics, saturation and sampling time.

The forms of a PID controller

The same PID controller can be written in several ways. The parameters change name and value with the form, which is why settings that work in one program can fail when copied into a PLC without conversion:

• Parallel (independent) form: u(t) = K_p · e + K_i · ∫e dt + K_d · de/dt
• Standard or ideal (ISA) form: u(t) = K_p · [e + (1 / T_i) · ∫e dt + T_d · de/dt]
• Parallel to standard: T_i = K_p / K_i and T_d = K_d / K_p
• Standard to parallel: K_i = K_p / T_i and K_d = K_p · T_d
• Proportional band: PB (%) = 100 / K_p (with normalized variables)
• Series (interacting) to standard: K_p = K′_p · (1 + T′_d / T′_i), T_i = T′_i + T′_d, T_d = T′_i · T′_d / (T′_i + T′_d)

This calculator converts the parallel form (K_p, K_i, K_d) into the standard form (T_i, T_d), with times in seconds.

Conversion table

If your controller expects T_i in minutes, divide the seconds by 60. Some controllers use “repeats per minute”, which is 1 / T_i with T_i in minutes.

Worked example: from Ziegler–Nichols to PLC parameters

In a closed-loop test you raise the proportional gain until the process oscillates steadily. You find an ultimate gain K_u = 8 and an oscillation period T_u = 4 s. The classic Ziegler–Nichols PID rules give:

1. K_p = 0.6 · K_u = 4.8
2. T_i = T_u / 2 = 2 s
3. T_d = T_u / 8 = 0.5 s
4. In parallel form: K_i = 4.8 / 2 = 2.4 s⁻¹ and K_d = 4.8 × 0.5 = 2.4 s.

Enter K_p = 4.8, K_i = 2.4 and K_d = 2.4 in the calculator and it returns T_i = 2 s and T_d = 0.5 s. Ziegler–Nichols gives an aggressive tuning with noticeable overshoot; treat it as a starting point and soften it during commissioning.

Common mistakes

• Entering K_i where T_i is expected: they are inverses; a large K_i means a small T_i (stronger integral action).
• Mixing seconds and minutes: many process PLCs work with T_i in minutes.
• Forgetting the sampling time in your own code: in a discrete PID the integral term adds K_i · e · T_s and the derivative uses Δe / T_s.
• Differentiating a noisy error: derivative action amplifies measurement noise; filter the D term or differentiate the measurement instead of the error.
• Not limiting the integral: if the actuator saturates, the integral keeps growing (windup) and causes large overshoot; use anti-windup.