Proportional Response in Plain Terms

Why this matters

Proportional band is the one control setting almost every tech has changed and almost none can predict the result of. It is not a strength dial and it is not a sensitivity dial. It is a translation ratio, and once you can do the two lines of arithmetic that ratio implies, you can state in advance exactly how far from setpoint a loop will sit at any load, which turns a vague comfort complaint into a number you can quote before you touch anything. Techs who cannot do it end up narrowing the band until the loop goes unstable, then widening it past where they started, and the customer ends up worse off than before the visit.

Before you change a band

• Changing the band changes how close the loop runs to its protective devices. Read the limit trip points and the current operating value first, and know the margin you are working inside. If a limit has been opening, establish why before anything else; replacing a correctly-operating limit reaches the same end state as jumpering it, one step slower and with a part on the invoice.
• Where a setting must be reached inside an energized control enclosure, 29 CFR 1910.333(a)(1) permits energized work only where de-energizing would introduce additional or increased hazards or is infeasible due to equipment design or operational limitations; use a meter and leads rated CAT III at or above the circuit voltage and work to the boundaries and protective equipment NFPA 70E-2021 assigns. Where the panel can be dead, open the disconnecting means, lock and tag under 29 CFR 1910.333(b)(2) in general industry or 29 CFR 1926.417 in construction, and prove dead per NFPA 70E-2021, 120.5.
• On combustion equipment, put a personal carbon monoxide monitor on your body before it fires. A band change alters firing rate and cycle length, which changes how much of the run is spent in startup.
• Keep clear of an actuator's travel path while you test. It moves under power, and a spring-return unit stores energy; to touch a linkage, isolate the actuator, release or restrain the spring, and lock or tag under 29 CFR 1910.147.

What proportional actually computes

One line: output equals bias plus error divided by band, expressed as a percentage of full output.

• Error is the difference between setpoint and measurement, in the units of the controlled variable.
• Band, also called throttling range, is the amount of error that drives the output across its entire range. It is in the same units as the error, which is why a band is 10 F or 4 psi and never a bare number.
• Bias is the output the controller produces when error is zero. On many controllers it ships at 0% and nobody ever looks at it.

Everything proportional does follows from that line. The output exists only because the error exists. A proportional controller cannot produce 40% output while sitting exactly on setpoint unless the bias is set to 40%, because with a bias of zero, zero error computes to zero output.

That is the whole reason for standing offset, and it is not a fault. It is the loop generating the output the load requires, and the only currency it has to generate output with is error.

Two ways the same setting is written

Some controllers ask for a band in engineering units; others ask for a dimensionless gain against the input span. Convert before you compare anything.

Gain equals 100 divided by the band, in percent of output per unit of the measured variable. A 10 F band is a gain of 10% output per degree; a 5 F band is 20% per degree, twice the gain.

Where the controller expresses gain against its own input span, you cannot convert without knowing the span. A controller with an input span of 0 to 200 F showing a gain of 5 is producing 5% of output per 1% of span; 1% of that span is 2 F, so full output arrives at 40 F of error, and the band is 40 F. Read the span off the input configuration, not off the sensor's own printed range, because the two are frequently different and the controller works from the one it is configured with.

Getting this conversion wrong in the wrong direction is how a tech intending a small softening ends up multiplying the gain by four.

The case: chasing offset with the band

A modulating heating loop, setpoint 70.0 F, band 10 F, bias 0%. The complaint is that the space runs cool, and the previous ticket noted the controller reading 66.0 F with the output steady at 40%.

Where the 4.0 F came from. Forty percent of output, from a bias of zero, requires 0.40 times the 10 F band, or 4.0 F of error. The loop was not failing. It was holding exactly the error its own arithmetic requires to make 40% output.

First change: band 10 F to 5 F. Gain doubles, so the same 40% output now needs only 0.40 times 5, or 2.0 F of error. The space came up two degrees and everyone was pleased for a week.

Second change: band 5 F to 2.5 F. Same logic, projected offset 1.0 F. Instead the loop began oscillating, 3.5 F peak to peak, which is a worse outcome than the offset ever was: a 4.0 F steady error puts the space consistently a little cool, while a 3.5 F oscillation puts it above and below the target in turn and the occupants feel both. The band had been narrowed past what the loop's round-trip delay could tolerate, which is a timing limit and is covered on its own in the hunting article.

Third change, the panic move: band 2.5 F to 15 F. The oscillation stopped. Offset at 40% output became 0.40 times 15, or 6.0 F, which is worse than the original complaint by two degrees. Three visits later the loop was less accurate than when the first tech arrived.

What should have happened. Put the band back to 10 F, which was already known to be stable, and set the bias to 30%. Offset is then the required output minus the bias, times the band: at 40% required output, that is 10 percentage points times 10 F per 100 points, or 1.0 F. Same stability as the original setting, one quarter of the original error, and no new hardware.

The part the bias does not fix

This is the sharp edge, and it is where the shop in the case would have gone wrong next.

Bias moves the offset; it does not shrink the spread. The spread of offset across a load range is the output range times the band, and the bias does not appear in it at all.

Work it through on that loop. Over a season the required output runs from 20% on mild days to 70% on cold ones, a 50-point range. With a 10 F band, the spread of offset across that range is 50 points times 10 F per 100 points, or 5.0 F, regardless of bias. With the bias at 0%, the loop sits from 2.0 F to 7.0 F below setpoint. With the bias at 30%, it sits from 1.0 F above setpoint on the mildest days to 4.0 F below on the coldest. Same 5.0 F spread, relocated.

So a bias tuned in November produces a space that is too warm in April, and the complaint reverses without anything having changed. Only two things shrink the spread: a narrower band, which the round-trip delay may not allow, and reset action, which removes the offset entirely and brings its own costs. Knowing that in advance is what stops the third visit from making things worse.

Where the linear relation stops holding

Everything above assumes the final control element and the process have roughly constant gain across the operating range. That assumption is often good enough and it is sometimes badly wrong, and the failure is predictable.

A valve or damper with a strongly non-linear characteristic, driving a coil whose output flattens near the top of its range, gives a loop whose effective gain changes with load even though the band setting never moves. Both effects point the same way. A coil that flattens up high is delivering most of its capacity in the first part of the valve's travel, so a percent of controller output buys a large process change down low and a small one near the top, and the loop's gain is highest at low output. The practical result: a band that is comfortably stable at 80% output can hunt at 30% on the same equipment on the same day. That is also why equal-percentage trim is specified on coil valves: the valve's own low gain down low is there to cancel the coil's high gain down low. If a loop is stable at design load and unstable in mild weather, look at the element characteristic before you look at the band, because no single band value can be right for both ends of a process whose gain doubles across the range.

Choosing a band on purpose

Start from what the process can tolerate, not from a default. The band you can use is bounded above by the offset the application will accept and bounded below by the round-trip delay. If those two bounds do not overlap, proportional-only is the wrong control for that loop and the honest answer is reset action or a different control strategy, not a heroic band value.

Proportional-only is genuinely right where the load is fairly constant, where the acceptable deviation is wide compared with the offset the load requires, and where simplicity is worth something because the equipment is unattended. A great many pressure bypass and minimum-position applications sit here happily.

It is the wrong choice where the load swings widely, where a specification is written on the controlled variable itself, or where the loop has to hold across a season. Those need the offset removed rather than positioned.

Confirming a band change did what you expected

Predict the offset before you change anything, then measure it. Note the current output percentage and the current error, compute the band those two imply, and compare it against the setting. If they disagree, the controller is not doing what its configuration says, and the band is not your problem.

Check the offset at two loads, not one. A single-point check cannot distinguish a band effect from a bias effect, because both produce an offset at the point you measured. Two points separate them: the spread between the two readings is the band's doing, and the position of the pair is the bias.

Record the band, the bias and the input span together. A band recorded without the span it was configured against cannot be reproduced on a replacement controller, and a bias recorded without the load it was set at is a number nobody can safely trust.

References

• 29 CFR 1910.333(a)(1) and (b)(2), OSHA general industry work practices for energized and de-energized electrical work
• 29 CFR 1926.417, OSHA construction lockout and tagging of circuits
• 29 CFR 1910.147, OSHA control of hazardous energy, for actuator spring tension and mechanical isolation
• NFPA 70E-2021, 120.5, verifying an electrically safe work condition
• See related: Why a Control Loop Hunts; What Reset and Integral Action Actually Do; Why Actual Never Equals Setpoint