Ziegler–Nichols PID starting values and control valve Cv sizing

Part 1: what the Ziegler–Nichols calculator does

The classic closed-loop Ziegler–Nichols method starts from two measured quantities: the ultimate gain K_u, the proportional gain at which the loop oscillates steadily, and the ultimate period P_u, the time of one oscillation. The Ziegler–Nichols PID tuning calculator converts these into starting P, PI or PID settings.

• P: K_p = 0.5 K_u
• PI: K_p = 0.45 K_u, T_i = P_u ÷ 1.2
• PID: K_p = 0.6 K_u, T_i = P_u ÷ 2, T_d = P_u ÷ 8
• K_i = K_p ÷ T_i and K_d = K_p × T_d

Worked examples for tuning

Example A. K_u = 4.0 and P_u = 30 s, PID. K_p = 0.6 × 4.0 = 2.4. T_i = 30 ÷ 2 = 15 s. T_d = 30 ÷ 8 = 3.75 s. K_i = 2.4 ÷ 15 = 0.16 per second. K_d = 2.4 × 3.75 = 9 seconds. With the same K_u and P_u as PI: K_p = 1.8, T_i = 25 s, K_i = 0.072; as P: K_p = 2.0.

Example B. K_u = 2.5 and P_u = 48 s, PI. K_p = 0.45 × 2.5 = 1.125. T_i = 48 ÷ 1.2 = 40 s. K_i = 1.125 ÷ 40 = 0.028125, shown by the calculator rounded to 0.0281 per second.

If the controller uses proportional band, band (%) is commonly 100 ÷ gain, so a gain of 2.4 corresponds to about 41.7 %. Confirm this against the controller manual, because conventions differ.

Limits of Ziegler–Nichols tuning

The values are a starting point and the values often give an aggressive response with overshoot, so they usually need to be detuned. Finding K_u and P_u means driving the loop to sustained oscillation, which is not safe or acceptable on every process and should only be attempted under the site procedure with operations agreement. Gain may be expressed as proportional band, time units differ between vendors, and some controllers use series, parallel or other algorithm forms. A few steps are usually needed: record K_u and P_u from a trend, calculate the starting set, apply a reduced gain first (for example by lowering K_p and lengthening T_i) and watch the next setpoint or load change before going further. The method does not account for noise, valve backlash, measurement filtering, deadtime-dominant processes or interacting loops.

Part 2: liquid control valve Cv

The control valve Cv calculator uses the basic equation for turbulent, non-choked liquid flow. Q is flow in m³/h, ΔP is pressure drop across the valve in bar and SG is specific gravity relative to water:

• K_v = Q × √(SG ÷ ΔP)
• C_v = K_v ÷ 0.865, which is about 1.156 × K_v

K_v is in m³/h per √bar and C_v in US gpm per √psi. The calculator does not include viscosity correction, choked or flashing flow, cavitation, pipe reducers or valve recovery factors.

Worked examples for Cv

Example C. Water, Q = 50 m³/h, ΔP = 2 bar, SG = 1. K_v = 50 × √(1 ÷ 2) = 35.3553. C_v = 35.3553 ÷ 0.865 = 40.8732.

Example D. A liquid with SG = 0.85, Q = 30 m³/h, ΔP = 1.5 bar. K_v = 30 × √(0.85 ÷ 1.5) = 22.5832. C_v = 22.5832 ÷ 0.865 = 26.1077.

Example E. Same water service at several flows with ΔP held at 3 bar: 80 m³/h needs K_v = 46.1880 (C_v 53.3966) and 5 m³/h needs K_v = 2.8868 (C_v 3.3373). If a valve had a hypothetical rated C_v of 63, the 40.8732 of example C would use about 65 % of that rating. These values show how widely the required coefficient moves with flow, which is why minimum, normal and maximum conditions all need checking. In practice the available drop usually changes with flow, so using one fixed drop is an approximation.

Common mistakes

• Treating Ziegler–Nichols values as final tuning and not detuning for overshoot.
• Entering gain in the wrong form, such as proportional band as if it were gain.
• Mixing time units between seconds and minutes.
• Forcing a loop to oscillate on a process that cannot tolerate it.
• Entering valve flow in the wrong unit or ΔP in kPa or psi instead of bar.
• Selecting a valve with a rated C_v exactly equal to the result, with no margin.
• Using the equation for gas, steam, flashing or highly viscous liquids.
• Sizing only at the normal flow, so that the valve sits near closed at minimum flow or near fully open at maximum.

Checks before you trust the result

• Confirm the controller algorithm form, time units and gain convention in its documentation.
• Change settings in small steps and observe the response, with the process owner aware.
• For the valve, confirm the pressure drop is across the valve at the stated flow, not a system total.
• Check cavitation, flashing, noise, rangeability and fail position with the manufacturer sizing method; the IEC 60534-2-1 equations are the usual reference for final selection.
• Treat both results as estimates, not as final design data.

Frequently asked questions

Why does Ziegler–Nichols often overshoot?

It was designed for a fast response to disturbances rather than low overshoot, so detuning is normally needed.

What if I cannot oscillate the loop?

Use another tuning approach recommended by the controller or process owner. This calculator needs K_u and P_u.

Is Cv the same as Kv?

No. They use different units. This calculator divides K_v by 0.865 to obtain C_v.

Should the valve be sized exactly to the calculated Cv?

No. Select a valve with a larger rated value and then check the working travel at minimum, normal and maximum flow with the manufacturer method.

Recording data

Record the controller tag, algorithm form, units, K_u, P_u, the settings before and after, and the observed response. For valves record flow, drop, SG, temperature, the calculated K_v and C_v, and the selected valve rating. A dated record of each change to the tuning, with the reason, makes it possible to return to a previous set of values if a change makes the loop worse. Keep the source of each input, such as a datasheet, a measured trend or a process assumption, so that the basis can be reviewed later.