Instrumentation and PID control calculators
4–20 mA signals, DP flow and level, Pt100 sensors, control valves and PID tuning: 18 calculations, each with its formula and practice problems.
Signals and measurement
4–20 mA signal → engineering value
PV is in the same units as LRV and URV. Below 3.8 mA or above 20.5 mA the signal is out of range; below 3.6 mA or above 21 mA it is a failure signal (NAMUR NE 43).
Engineering value → 4–20 mA signal
Handy to check a transmitter or simulate a value with a loop calibrator: inject this current and compare with the control system display.
Flow from differential pressure (square root)
Extract the square root once only: in the transmitter or in the control system, never both. Below 5 to 10 % flow the reading is unstable; a low-flow cutoff is used.
Level from differential pressure
Positive ΔP₀: zero suppression (transmitter below the lower tap); negative: zero elevation (wet leg on the low side). The range runs from ΔP₀ (LRV) to the URV shown. g = 9.80665 m/s².
Pt100 / Pt1000 RTD: resistance ↔ temperature
IEC 60751 coefficients (α = 0.00385), valid from 0 to 850 °C. Below 0 °C a C term is added: use the standard's table. Measure with 3 or 4 wires to cancel the lead resistance.
Instrument accuracy: % of span vs % of reading
A % of span accuracy weighs heavily at the bottom of the range: at 10 % of span, ±0.25 % of span becomes ±2.5 % of reading. Pick a range close to the operating range.
Flow and control valves
Valve flow coefficient Cv / Kv (liquid)
IEC 60534-2-1: liquid in turbulent flow, no flashing or cavitation (not choked), no reducers (Fp = 1). Pick a valve that works between 20 and 80 % open.
Kv ↔ Cv conversion
Kv: water flow in m³/h at 1 bar DP. Cv: water flow in US gallons per minute at 1 psi DP. North American valve data sheets mostly give Cv.
Valve flow at a new pressure drop
Same valve opening. In service the valve DP falls as flow rises (pump curve, line losses): that is why the installed characteristic differs from the inherent one.
Orifice meter: flow at a new DP
Same plate, same discharge coefficient (high enough Reynolds number). Volumetric flow. If the actual density differs from the plate design density, the displayed flow is off by √(ρ₁/ρ₂).
PID tuning
Ziegler–Nichols closed loop (Ku, Pu)
Ku: gain that gives a sustained oscillation in P only; Pu: period of that oscillation. Aggressive tuning (~1/4 decay): start here and lower Kc if the loop oscillates. Kc as gain, not proportional band.
Ziegler–Nichols open loop (reaction curve)
In manual, step the output (5 to 10 %); read Kp = ΔPV % ÷ ΔOP %, dead time θ and time constant τ (63 % of the response). Valid for roughly 0.1 ≤ θ/τ ≤ 1.
Cohen–Coon (PI and PID)
Same test as the reaction curve. Cohen–Coon handles a larger dead time better but stays aggressive: on a real loop, Kc is often cut by 25 to 50 %.
Lambda (IMC) tuning, self-regulating process
Standard (ISA) form PI controller. λ of 1 to 3 × τ gives a smooth, robust loop (keep λ ≥ θ). The larger λ, the slower the loop and the more it tolerates model errors.
Loop settings
Gain ↔ proportional band
PB = error range, in % of span, that moves the output by 100 %. High gain = narrow PB. Check which term the control system uses before entering a setting.
Integral action: min/repeat ↔ repeats/min
Some controllers set integral in min/repeat (Ti): a bigger value slows the integral. Others use repeats/min (1/Ti): a bigger value speeds it up. Never copy the number across without converting.
Steady-state offset with P-only control
Self-regulating process, ΔSP in % of span. Raising Kc shrinks the offset but moves the loop toward instability; integral action removes it. A load disturbance also leaves an offset with P only.
First-order filter: response after t
A filter smooths noise but adds lag to the loop: keep the filter τ well below the process time constant (often under 1/10). Same response for a sensor in a thermowell.
Learning and checking tools. The ranges shown are general guidance: always follow your plant's procedures, permits and set points. Nothing is sent anywhere: the calculations run in your browser.