Normalising PV and output: gains in units and in percent

In short. A controller gain has units of "output per PV". If the PLC block works in percent of span but the tuning was calculated in °C and %, the gain has to be converted through the range widths. The same relation tells what happens to the loop when a transmitter is re-ranged. The gain scaling tool does the conversion.

What normalisation is

A block can work with PV and output:

• in engineering units: PV in °C, output in %, mA or Hz;
• normalised: PV and output are fractions (or percent) of their ranges.

A gain is a ratio of changes, so when the units change it is multiplied by the ratio of range widths. Let a gain relate input X to output Y (ΔX, ΔY are the range widths):

K_norm [%/%] = K_eu [unit Y / unit X] · ΔX / ΔY

The directions are opposite — so the controller gain and the process gain cannot be converted with the same formula.

Example 1: controller

PV: 0–200 °C; output: 4–20 mA; Kc = 0.08 mA/°C.

Kc_norm = 0.08 · 200 / 16 = 1.0 %/%        PB = 100 / 1.0 = 100 %
Kc in % of output per °C = 0.08 · 100 / 16 = 0.5 %/°C

Open the example

Example 2: process and tuning in percent

Section model: K = 1.8 °C/%, τ = 45 s, θ = 12 s; transmitter 0–150 °C, output 0–100 %.

K_norm = 1.8 · 100 / 150 = 1.2 %/%
SIMC in engineering units:  Kc = 45 / (1.8 · 24) = 1.042 %/°C
SIMC on the normalised model: Kc = 45 / (1.2 · 24) = 1.5625 %/%
check: 1.042 %/°C · 150 / 100 = 1.5625 %/%        PB = 64 %

Both describe the same controller. A block that works in %/% gets 1.5625; a block that takes PV in °C and output in % gets 1.042.

Re-ranging a transmitter changes the loop

If the block works in percent and the transmitter is re-ranged from 0–150 °C to 0–300 °C, one percent of PV is now twice as "large". With the same Kc_norm the real gain (% output per °C) halves and the loop becomes slower. To keep the behaviour:

Kc_norm(new) = Kc_norm(old) · ΔPV_new / ΔPV_old = 1.5625 · 300 / 150 = 3.125 %/%

Changing valve travel, drive frequency range or block output limits has the same effect.

Where else scaling hides

• Analog input: raw ADC counts (e.g. 0–27648) are converted to engineering units. If the controller gets PV before scaling, its Kc is in "percent per count". See RAW scaling and analog signals.
• Output: 0–100 % of the block → 4–20 mA → valve travel. A non-linear valve characteristic changes K between operating points.
• Internal normalisation of the block: some blocks convert PV to percent by configured limits. Then Kc is entered in %/% even if PV is displayed in °C.

Typical mistakes

• Converting Kc with the process formula (or the other way round) — an error of (ΔPV/Δu)².
• Taking PB from Kc in engineering units — PB is defined only for %/%.
• Changing a transmitter range or output limits without re-converting the settings.
• Entering a Kc for PV in °C into a block that normalises PV by its own limits.

Limits

The conversion is linear: it assumes a linear relation between units and percent. For non-linear transmitters and valves (square-root extraction, equal-percentage characteristic) the gain depends on the operating point — normalisation does not fix that.