CALCULATORS / CONTROL SYSTEMS AND DYNAMIC RESPONSE

PI tuning for a first-order delayed process calculator

PI settings from an identified first-order-plus-delay process; supplied tuning time.

controls-pid-tuning.napkin / editable

SCENARIO COMPARISON

Compare the numbers.

This source describes an analytical model. Its quantities and checks are the view.

Scenario	kp	integralTimes	kis^-1
	0.625	5	0.125
	0.78125	5	0.15625

All calculated quantities are listed with the inputs. Switch to Source to inspect the equations.

SCENARIOS / CALCULATED VALUESSAME SOURCE

Change an input to see the consequences.

positive integral gain0.13 s^-1✓ passes

BASELINE INPUTS

processGain2 timeConstant5 sdelay1 stuningTime3 s

CALCULATED

kp0.625

integralTime5 s

ki0.125 s^-1

Edits change the baseline source.

THE SOURCE

Edits below update the same model.

# PID tuning from a stated model
# Scope: PI settings from an identified first-order-plus-delay process; supplied tuning time.
# Status: illustrative worked example; engineering review pending.
# Inputs are example values. Checks are editable example requirements, not standards.

processGain = 2
timeConstant = 5 s
delay = 1 s
tuningTime = 3 s

kp := timeConstant/(processGain*(tuningTime+delay))
integralTime := timeConstant [s]
ki := kp/integralTime [s^-1]

check positive integral gain: ki > 0 s^-1

scenario alternative: processGain = 1.6

Calculated from your source · not saved

Baseline · Baseline

positive integral gain0.13 s^-1 ✓ passes

alternative · alternative

positive integral gain0.16 s^-1 ✓ passes

Illustrative first-order example. Checks are editable requirements, not design standards. The broader topic may require additional equations, data, and validation.

Formula and units

This example uses the following equations. Input names and units match the editable source above.

kp: kp = timeConstant/(processGain*(tuningTime+delay))
integral time: integralTime = timeConstant [s]
ki: ki = kp/integralTime [s^-1]

Original worked example

These are the baseline values in the downloadable source. Edits above let you test your own values.

Given

process gain: 2
time constant: 5 s
delay: 1 s
tuning time: 3 s

Calculated

kp: 0.625
integral time: 5 s
ki: 0.125 s^-1

Assumptions and limits

PI settings from an identified first-order-plus-delay process; supplied tuning time.

The requirement checks use the limits written in this example. Passing a check does not establish that the model covers every load, material behavior, or operating condition in a real design.

Download this .napkin example ↓

Take it further with your agent

Napkin keeps the inputs, equations, checks and assumptions together in one study. Your agent sets it up; you change the numbers.

Try with Claude ↗

codex 'Use Napkin (https://napkin.engineering/mcp, or the local Napkin plugin) to build a study: PID tuning from a stated model. Inputs: plant parameters; tuning rule; design targets. Outputs: candidate gains; modeled response. Model scope: Declared linear or low-order nonlinear plant models. Read napkin_capabilities and the current snapshot. Ask me for the values that matter, label assumptions with units, author .napkin source, validate it and apply it with napkin_source using the current revision, then open the study. Add checks for the limits I supply. Keep unsupported analyses explicit.'

Claude needs the Napkin connector; Codex needs the Napkin plugin. Set up your agent ↗

The broader study

The topic brief below describes the broader scope. The worked example above states the particular calculation it implements.

Inputs: plant parameters, tuning rule, design targets
Outputs: candidate gains, modeled response
Modes: calculate, compare, sweep
Drawing: Response and stability plots

EXECUTABLE WORKED EXAMPLE · FIRST-ORDER MODEL · CHECK THE SCOPE & ASSUMPTIONS