PID Ziegler-Nichols Quick Table: Ku and Pu to Kc, Ti, Td

Classic Ziegler-Nichols ultimate-gain PID tuning gives (K_c = 0.6K_u), (T_i = P_u/2), and (T_d = P_u/8). For (K_u = 2.0) and (P_u = 1.0) minute, the starting settings are (K_c = 1.2), (T_i = 0.5) minute, and (T_d = 0.125) minute. These are starting values—not confirmed settings for every process. The coefficients follow Ziegler and Nichols’ original ultimate-sensitivity recommendations, expressed here as gain, integral time, and derivative time.

Ziegler-Nichols PID Quick Table

Use this table for the closed-loop ultimate-gain method, not the open-loop reaction-curve method.

The original paper specifies sensitivity as (0.6S_u), reset rate as (2/P_u), and pre-act time as (P_u/8). Integral time is the reciprocal of reset rate, so (T_i = P_u/2). See the “Optimum Pre-Act Time Adjustment” discussion on page 763 of Optimum Settings for Automatic Controllers.

What Ku and Pu Mean

• (K_u), the ultimate gain, is the proportional-only controller gain at which the loop exhibits sustained, approximately constant-amplitude oscillation.
• (P_u), the ultimate period, is the time for one complete oscillation at that gain—for example, from one peak to the next corresponding peak. Some references call it (T_u).
• (K_u) is controller gain, not the process gain obtained from a manual output step. Record its input/output scaling as well as its numerical value.

The ultimate-gain test deliberately approaches the stability boundary. Do not increase gain toward sustained oscillation unless an approved procedure permits that disturbance and qualified personnel have established operating limits and a recovery method. If that condition cannot be reached safely, do not invent (K_u) or substitute an unrelated oscillation period.

Worked Example: Ku = 2.0 and Pu = 1.0 Minute

Assume the controller uses compatible gain-and-time settings, with both error and output expressed on percentage-of-span scales. On that basis, (K_u = 2.0) means 2.0 percentage points of output change per percentage point of error.

Calculate:

[
K_c = 0.6 \times 2.0 = 1.2
]

[
T_i = \frac{1.0}{2} = 0.5\ \text{min} = 30\ \text{s}
]

[
T_d = \frac{1.0}{8} = 0.125\ \text{min} = 7.5\ \text{s}
]

These are equivalent representations of the same calculated settings. Entering 0.5 and 0.125 into fields expecting seconds would make both time settings 60 times shorter than intended.

If the Controller Uses Proportional Band

For the normalized percentage-based gain convention:

[
PB(%) = \frac{100}{K_c}
]

Thus:

[
PB = \frac{100}{1.2} = 83.333\ldots% \approx 83.3%
]

NI’s Ziegler-Nichols documentation gives the reciprocal relationship (K_c = 100/PB). Apply it only when the controller’s proportional-band definition and scaling match this convention; an engineering-unit gain cannot automatically be treated as a normalized gain.

The original “83 percent band” is therefore a whole-percent approximation. Here, proportional band is rounded to one decimal place; the other example values are shown without additional rounding. Final entries must respect the controller’s available resolution.

Match the Controller Equation Before Entering Values

A familiar parameter name does not establish that two controllers use the same PID equation. For an ideal, noninteracting representation:

[
u(t) = u_b + K_c\left[e(t)+\frac{1}{T_i}\int e(t),dt+T_d\frac{de(t)}{dt}\right]
]

Here:

• (u(t)) is controller output, and (u_b) is output bias.
• (e(t)) is control error under the controller’s configured action convention.
• (t), (T_i), and (T_d) use one consistent time unit.
• (K_c) has units of output change per unit of error.

If the controller instead asks for independent parallel gains in

[
u(t)=u_b+K_p e(t)+K_i\int e(t),dt+K_d\frac{de(t)}{dt},
]

matching coefficients gives:

[
K_p=K_c,\qquad K_i=\frac{K_c}{T_i},\qquad K_d=K_cT_d
]

For the example, using seconds:

[
K_p=1.2,\qquad K_i=\frac{1.2}{30}=0.04\ \text{s}^{-1},
\qquad K_d=1.2\times7.5=9.0\ \text{s}
]

Those units assume normalized input and output. With engineering-unit scaling, each gain also carries the corresponding output-to-error unit ratio.

If the integral field specifies reset rate in repeats per minute rather than integral time:

[
R=\frac{1}{T_i}=\frac{1}{0.5}=2.0\ \text{repeats/min}
]

Do not enter 0.5 into that reset-rate field: integral time and reset rate are reciprocals. The original paper explicitly defines reset rate in repeats per minute.

Before transferring any result, check the actual controller manual for:

• Ideal/noninteracting, series/interacting, or independent parallel PID form.
• Gain versus proportional band, including input/output normalization.
• Integral time versus reset rate and the configured time units.
• Direct/reverse action, derivative implementation, filtering, and output limits.

These conversions do not establish equivalence with an interacting algorithm or every vendor-specific PID block.

Check the Inputs and the Actual Loop Response

Use positive, finite (K_u) and (P_u) values for this magnitude-based table, and configure controller action separately. A zero or negative period is not a valid oscillation period; a gain without a known scaling basis is insufficient for transferring settings.

Before using recorded test data, confirm that integral and derivative action were disabled according to the controller manual. Do not assume that setting integral time to zero disables integral action. Also confirm that the observed cycling was a valid ultimate-gain response rather than output clipping or another source of repetitive behavior.

The arithmetic provides useful consistency checks:

[
\frac{K_c}{K_u}=0.6,\qquad
\frac{T_i}{P_u}=0.5,\qquad
\frac{T_d}{P_u}=0.125
]

Changing (P_u) from minutes to seconds must leave (K_c) unchanged and multiply both displayed time settings by 60. These relationships check the calculation and unit handling, not the quality of the process test.

For a related calculation page, see SenseCalc’s PID Tuning (Z-N Ultimate). Its indexed listing identifies it as an ultimate-method tuning calculator; that listing alone does not verify its calculation implementation.

After any approved tuning change, assess the actual process-variable and controller-output trends against site-defined limits. A controlled response test may help assess overshoot, settling, and actuator behavior, but “finish with a bump test” is not a universal acceptance rule. Ziegler and Nichols themselves describe approximate settings and note that acceptable tuning can change with load conditions. Correct arithmetic does not establish acceptable plant performance, instrument calibration, or safety-function validity.