Ziegler and Nichols do not reject mathematics. They postpone the full derivation and the complete body of laboratory and field evidence so that a usable scheme can reach practising engineers. Beneath the variety of air-, liquid- and electrically operated instruments, they identify at most three principal effects: proportional response, automatic reset and pre-act. The modern reader will recognise the family resemblance to proportional, integral and derivative action, but the original operational units matter.

The authors abstract the measurement scale. A recording pen might represent temperature, pressure, level or flow; one inch might mean one degree or one thousand. The controller turns pen displacement into output pressure and thereby moves a valve, damper or rheostat. By stating each effect as a relation between observable pen behaviour and valve-driving pressure, they create a vocabulary that survives differences in process units and instrument dials.

The paper asks two questions. How can the correct adjustments be found quickly after a controller has been installed? And how can approximate settings be predicted before installation? The first question leads to a closed-loop test at the threshold of sustained oscillation. The second leads to an open-loop step and an S-shaped process-reaction curve. Both compress a complicated plant into two measured numbers.

This compression is the achievement and the boundary. It provides reproducible starting values without claiming a complete dynamic model. The authors explicitly acknowledge that the two reaction-curve descriptors omit shape before and after the inflection point. The rules are approximate and arise from experience across a range of industrial applications. They turn “hit and miss” adjustment into an experiment, a measurement and a formula.

The paper defines sensitivity as output-pressure change per inch of pen travel. Zero means manual control; an arbitrarily large value approaches on–off behaviour. Below a plant-dependent maximum, a disturbance produces a damped oscillation that eventually settles. At one particular setting, called ultimate sensitivity S_u, a small oscillation neither grows nor decays. Above it the waves grow towards a limiting amplitude; below it successive waves shrink.

Ziegler and Nichols measure this behaviour by amplitude ratio: the amplitude of one wave divided by its predecessor. A ratio of one marks the ultimate setting, greater than one growth and less than one decay. Their chart records show a broadly repeatable relation between sensitivity expressed as a fraction of S_u and amplitude ratio. Around one half of S_u, their test application produces a quarter-amplitude ratio.

Why not choose an even lower sensitivity for more damping? A proportional controller can hold only one valve position when the pen is exactly at set point. A load change that requires a different sustained valve position therefore needs a persistent pen displacement. This is offset. For a required 2.8 psi output change and sensitivity 5 psi per inch, the paper’s example yields 0.56 inch offset. Reducing sensitivity further improves damping but enlarges that displacement.

The authors call roughly 25 per cent amplitude ratio a good compromise for many applications, then immediately qualify it. A liquid-level system may value gentle motion downstream more than exact level and should use lower sensitivity. A fast pressure loop may tolerate an 80 or 90 per cent ratio because even several short oscillations die quickly. “Optimum” is thus tied to consequences and timescale, not to one universal curve shape.

The paper defines automatic reset as valve velocity proportional to pen displacement from set point. Its measure is reset rate in inverse minutes: the number of times per minute that reset duplicates the proportional correction created by the current displacement. If proportional action produces a 1 psi output change and reset adds another 1 psi during the next minute, the reset rate is one per minute. If the proportional change is 2 psi and reset adds 2 psi per minute, the rate is still one.

This ratio-based unit matters because the integral contribution scales with proportional response. Modern controller forms may expose integral time instead: minutes per repeat rather than repeats per minute. The values are reciprocals. Some implementations instead expose an independent integral gain. Transferring a Ziegler–Nichols number without identifying the convention can change the loop by orders of magnitude.

More reset returns the variable to set point faster, but it also reduces stability and lengthens the oscillation period. The authors’ chart series begins with proportional action and increases reset rate. Recovery accelerates until additional reset produces excessive oscillation. They use the ultimate period P_u, measured during the proportional-only boundary test, as a timescale for the appropriate rate.

For proportional-plus-reset control, their approximate setting is sensitivity 0.45S_u and reset rate 1.2/P_u per minute. Sensitivity is reduced below the proportional-only half because adding reset would otherwise increase the amplitude ratio. A higher reset rate paired with lower sensitivity can produce a similar ratio, but the paper reports longer periods and larger initial deviation. The parameters interact; tuning them independently misses the point.

If pen motion would make proportional output change at 1 psi per minute, one minute of pre-act contributes an additional 1 psi. Doubling pen velocity doubles the addition. Graphically, the combined output runs ahead of the proportional line by the specified pre-act time. The response is used with proportional action; the paper does not treat it as a stand-alone controller.

Its reputation in 1942 was unstable in two senses. On some processes it looked almost mythically anticipatory; on others it was worse than useless. Ziegler and Nichols strip away the mythology by defining the effect operationally and tying its setting to a measured plant timescale. The optimum pre-act time is approximately one eighth of the ultimate oscillation period P_u.

With all three effects, the initial rules are sensitivity 0.6S_u, reset rate 2/P_u and pre-act time P_u/8. The sensitivity can sometimes rise towards S_u; the test application tolerated 0.9S_u. If oscillation remains excessive, the paper recommends reducing the settings. Its diagnostic heuristic uses period: excessive proportional sensitivity tends to create a period near P_u, excessive reset lengthens the period, and excessive pre-act shortens it.

Pre-act also illustrates why a beneficial mechanism can be overused. Increasing it initially adds phase lead and permits stronger proportional and reset action. Beyond a useful point it again reduces stability. Modern derivative action is especially vulnerable to noisy measurements because differentiation magnifies rapid variation. The original paper notes application limits: pre-act was used mainly for temperature control, not generally for fast pressure, flow or level loops, and available hardware imposed a minimum practical setting.

The resulting sensitivity S_u describes how much loop gain this plant, valve, sensor and operating condition can sustain. Measure the oscillation period P_u in minutes. Together they supply an action scale and a time scale. The proportional rule uses 0.5S_u. The proportional-plus-reset rule uses 0.45S_u and 1.2/P_u. The three-effect rule uses 0.6S_u, 2/P_u and P_u/8.

These coefficients are not arbitrary independent gains. Adding reset requires a lower proportional setting because it reduces stability. Adding pre-act restores phase lead, allowing a higher proportional setting and faster reset. The formulas encode observed interactions among effects. Their simplicity is a virtue for field work and a warning against false precision.

The procedure requires a small-amplitude limit cycle. Above S_u, oscillations grow; below it, they decay. In a nonlinear plant, saturation may create a sustained large cycle even when the linearised loop is unstable, so amplitude and operating region matter. Stiction, backlash and valve limits can also counterfeit a boundary. The experiment identifies the whole installed loop, not a timeless constant belonging to the process alone.

Modern safety changes the protocol. Do not casually induce sustained oscillation in a reactor, aircraft, fragile actuator or tightly constrained production line. Establish protection limits, supervision and a manual fallback. Use relay feedback or a safe model-based experiment when appropriate. The paper’s conceptual contribution survives: excite the system in a controlled way and infer controller scale from measured feedback behaviour.

Bring the process to the desired chart point under manual valve control. Change controller-output pressure suddenly by ΔF and hold it. The resulting pen trajectory is often S-shaped because several lags accumulate: valve inflation, sensor response and process dynamics. Find the point of inflection where the curve’s slope is greatest. Call that measured slope R.

Extend the tangent backwards until it crosses the initial output level. The time from the input step to that intersection is the apparent lag L. Normalise slope by the size of the valve-pressure step to obtain unit reaction rate R₁=R/ΔF. The product R₁L measures the output change accumulated over one lag per unit input-pressure change.

The curve must be caused by a manipulated valve change, not by a load disturbance. The formula connects the known input step to the observed response. A load-change curve does not provide the same denominator or causal contrast. This distinction is easy to lose when a modern historian redraws only the S-shape.

The authors know that R₁ and L are rough. They say the pair reveals nothing complete about curve shape before and after inflection. Some level processes are not self-controlling: output continues ramping rather than settling. Some plants have a pure dead period before motion begins; others immediately build towards maximum slope. The tangent nevertheless provides a portable classification and approximate controller settings when a full model is unavailable.

Substitute these relations into the closed-loop table. Half of S_u becomes 1/(R₁L), the proportional reaction-curve rule. For proportional plus reset, 0.45S_u becomes 0.9/(R₁L), while 1.2/P_u becomes 0.3/L. For all three effects, 0.6S_u becomes 1.2/(R₁L), 2/P_u becomes 0.5/L, and P_u/8 becomes 0.5L.

This exact arithmetic should not be confused with exact plant theory. The equality of the tables follows after accepting approximate empirical mappings. It is valuable because it makes the paper internally auditable. A software library that produces inconsistent settings from corresponding inputs has a conversion or convention error.

The reaction-curve rules predict settings before a controller is installed and can guide equipment design. The ultimate-cycle test characterises the installed loop directly. When both can be performed safely, disagreement is evidence: perhaps the reaction curve is not captured by one lag and slope, the operating point changed, the valve gain is nonlinear, or the observed oscillation was not the intended small boundary cycle.

The units also expose plausible scaling. If unit reaction rate doubles while lag is unchanged, recommended sensitivity halves because the plant responds twice as strongly to valve pressure. If lag doubles while R₁ is unchanged, sensitivity halves and reset rate halves; pre-act time doubles. Slower dynamics call for slower integral action and a longer derivative timescale.

Different valve-pressure steps on a locally linear process should produce reaction curves of similar shape, with slope proportional to ΔF and approximately constant L. Dividing by ΔF creates R₁. This normalisation permits a smaller, safer test and a shared process descriptor. It also creates a diagnostic: if R/ΔF changes with step size, the system is not locally linear over that range.

Valve characteristics are part of the result. A valve whose flow gain varies across travel gives different unit reaction rates at different loads. The authors argue for matching valve flow–lift characteristics to the process so that R₁ remains more nearly constant. That provides a physical reason for characterised valves and explains why one controller setting can fail across the operating range.

Reducing lag permits faster reset and shorter oscillation period. Reducing R₁ without increasing lag permits higher sensitivity. The combined quantity R₁L controls recommended proportional scale. These relations turn controller tuning into a lens on equipment design: move the sensor, speed the valve, reduce transport delay or reshape valve gain, and a better loop may become possible before any coefficient changes.

Some processes do not settle after a step. A tank level may continue changing at its maximum rate until physical limits intervene. The paper calls such a process not self-controlling and treats the usual S-curve as only one case. Modern engineers would distinguish integrating dynamics, dead time, stiction and saturation explicitly. The original two-number classification is a starting map, not permission to force every plant into the same template.

No single integral performance criterion is minimised in the article. The authors do not prove global optimality over all plants, disturbances, nonlinearities, actuator limits and operating goals. They compare chart records, offset, amplitude ratio, recovery speed and oscillation period. Their recommendations are deliberately approximate and admit exceptions based on process consequences.

The paper also states its evidence limit. Apart from an illustrative case, it does not present the complete laboratory and field data or the mathematical derivation; that work was still under way. This candour should survive retelling. Later popularity and thousands of citations do not retroactively turn the coefficients into a theorem stronger than the primary claims.

Quarter-amplitude decay is itself a choice. It produces a lively loop with overshoot. Many modern systems prefer less aggressive tuning to reduce actuator wear, noise amplification, product variability or risk under model uncertainty. A safety-critical system may optimise worst-case constraint margin; a servo may prioritise rise time; a slow production loop may value disturbance rejection per unit movement. The proper comparison uses the operational loss.

The test is also an intervention. The ultimate-cycle route intentionally approaches marginal stability. The reaction-curve route removes automatic feedback and steps the actuator. Neither is observational bookkeeping. Before testing, an engineer must assess whether the plant can tolerate oscillation or open-loop drift, specify abort conditions and preserve independent protection. An empirical method becomes safer when its causal action is named plainly.

A modern implementation begins with convention control. Confirm the sign of error and action, engineering-unit scale, proportional gain versus proportional band, repeats per time versus time per repeat, ideal versus parallel PID form, derivative placement and any built-in filtering. Convert the original sensitivity, reset and pre-act quantities into that exact structure before commanding the plant.

Then preserve experiment provenance. Record operating load, valve position, step magnitude, constraints, sensor range, observed S_u and P_u or R, ΔF and L, plus uncertainty in reading the curve. Store raw traces alongside derived settings. If a later engineer cannot reconstruct the tangent or the sustained cycle, the tuning number has lost its evidence.

Validate under representative disturbances without assuming the identification test covers everything. Check overshoot, settling, offset, actuator movement, noise sensitivity, saturation and constraint margin. Compare alternative settings using the costs that matter. Re-test when equipment or operating conditions change. The paper itself points towards this by relating valve characteristics, lag and unit reaction rate to controller capability.

Finally, keep conceptual scale. A loop can infer useful settings from its own induced response in a modest sense: the experiment informs the parameters. It does not follow that the controller understands the plant, chooses its own ends or experiences the oscillation. Empirical adaptation is a causal and computational achievement. Intelligence, agency and consciousness remain separate questions requiring separate evidence.