A lamp switched on is not the lamp measured

Three essays on this site are about the fact that the eye takes time to settle. An appearance model has no clock; a gain has a time constant; the slowest of them is chemical. All three assume a fixed stimulus and a moving observer.

A luminaire switched on is not a fixed stimulus. It warms from ambient to its working point over minutes, and everything about its output moves while it does.

The claim #

The lamp’s clock is slower than the eye’s, and every appearance judgement on this site assumes both have stopped.

• The transient is minutes, not seconds. A thermal time constant of about a hundred and fifty seconds puts nine tenths of the colour change at four hundred and five seconds — nearly seven minutes.
• It is not small. The output falls 25.4 per cent and the white moves ΔE00 2.6, mostly in Duv: from +0.0068 to +0.0039, crossing toward the locus while the correlated colour temperature barely moves at all.
• And it overlaps every clock in the eye. The fast neural constant is one second, the slow is sixty, the pigment’s is a hundred and twenty. The lamp’s is a hundred and fifty, and it is the largest of the four.

What is warming and what that does #

Three quoted slopes act at once, and they pull in different directions, which is why the net result is not obvious.

The die’s peak moves to longer wavelengths as the junction heats, at about 0.035 nanometres per kelvin. Over a fifty-five kelvin rise that is nearly two nanometres — small in absolute terms and not small relative to a narrow blue line whose position decides how much of it the converter absorbs.

The die loses radiant efficiency, at about 0.3 per cent per kelvin, which is sixteen per cent over the rise. This is the largest single contribution to the output loss and it is the reason thermal management dominates LED luminaire design.

And the converter loses quantum yield, at about 0.2 per cent per kelvin, while its own emission band moves slightly. So the blue is falling and the converted light is falling faster, which shifts the mixture back toward the blue — in the opposite direction to what the die’s red-shifting peak does.

The two effects nearly cancel in colour temperature, which is why the CCT moves only 27 K, and they do not cancel in Duv, which moves 0.0029 — a shift that would be a specification failure if it were a shelf-to-shelf variation and is unremarked because it happens to one lamp over seven minutes.

What was computed, and how #

The junction is first-order. Temperature rises from ambient toward a stated working point with a single exponential, at a time constant quoted between sixty and three hundred seconds for a luminaire of this size. That is a caricature — a real fixture has at least two thermal masses, the die and the heat sink, with different constants — and the caricature’s shape is right and its detail is not.

The three slopes are data-sheet quantities and every one is a slope rather than a curve, which is what makes them safe to use: each result here is a difference between two evaluations of the same slope, so a result scales with the coefficient rather than depending on it in some hidden way.

The lamp is the same device as the other two essays in this group. A pump at 452 nm, a broad converter at 565 with red nitride at 630, a warm-neutral 4,044 K on the axis at full power. Nothing was retuned for this measurement.

And the settling time is measured against the colour rather than against the temperature, which is the useful definition: the time at which the correlated colour temperature has covered nine tenths of its total excursion. It is longer than the temperature’s own ninety-per-cent point because the colour follows a combination of three slopes rather than one.

atJunction refuses a temperature that would take a slope past zero. At six hundred degrees the quoted efficiency slope would predict negative output, which is not a physical result but a statement that the slopes are local — and the rejection is asserted, because a model that silently extrapolates a linear coefficient to extinction is a model that will do so quietly on the day somebody asks it about a hot day.

How large the transient is against the things it is compared with #

Two-point-six ΔE00 is a number without a scale until it is put beside something, and this site has several things to put it beside.

It is larger than any tolerance a supplier is held to. A graphic-arts specification asks for a unit or two on a print; the light the print is judged under moves by more than that between switch-on and settled.

It is a quarter of a white bin. Two lamps at opposite corners of one 4000 K bin are ΔE00 11.3 apart on a shelf, and a bin is meant to be the granularity at which a purchaser stops caring. One lamp moves a quarter of that distance on its own, over seven minutes, every time it is switched on.

It is the same size as the observer’s own spread. Two people picked at random disagree about an ordinary pair by rather less than this and about a display white by rather more, so the lamp’s transient sits squarely inside the range of effects this phase has been measuring — which is the argument for it belonging in the same phase rather than in a lighting-engineering footnote.

And it is comparable with the beam’s angular variation divided by a few. The same device is ΔE00 16 from its axis to seventy-five degrees. So a lamp has at least three colours — the axis, the edge and the cold start — and the specification names one.

What a measurement made during the overlap is about #

This is the part with a consequence.

A proofing booth’s warm-up is specified and the reason usually given is the wrong one. Graphic arts standards require a booth to be on for a stated period before use — typically a few minutes for fluorescent tubes, whose warm-up is dominated by mercury pressure and is dramatic. The reason given is that the lamp needs to stabilise, and that is exactly right. What the standards do not say is that the observer needs to stabilise too, on constants of the same order, and that the two settle independently.

A colour measurement taken during warm-up is not a measurement of the lamp. An instrument reading a booth two minutes after switch-on is reading a light that is still moving, and averaging over the reading period does not help — it produces the average of a transient, which is a number describing no state the lamp will ever be in.

And a comparison between two rooms is a comparison between two thermal histories. Two identical fixtures, one running all day and one just switched on, are ΔE00 2.6 apart and a quarter of a stop different in output. That is larger than the tolerance any colour-critical viewing standard allows for the illuminant itself.

The direction nobody predicts #

The interesting part of the arithmetic is that the colour temperature barely moves and the Duv moves a lot, which is the opposite of what an intuition about “warming up” produces.

The intuition is that a lamp getting hotter should get warmer in colour, because hot things are warm. It does not, because a phosphor-converted lamp’s colour is a ratio between two emitters and both of them are losing. The die loses radiant efficiency; the converter loses quantum yield; the die’s peak moves toward the converter’s absorption band, which raises the fraction converted and pulls the other way.

Three slopes, two of them subtractive and one of them additive, and their net effect on the ratio is nearly nothing while their net effect on the shape of the spectrum is substantial. A shape change with no ratio change is precisely a Duv shift: the chromaticity moves perpendicular to the locus while its projection onto the locus stays where it was.

That has a practical consequence for anybody trying to catch this. A specification written in colour temperature reports the warm-up as 27 K and dismisses it. A specification written as a chromaticity, or in colour temperature and Duv, reports it as a shift five times the width of a binning tolerance. The two documents describe the same seven minutes.

Seven minutes is measured on the quantity that barely moves #

The settling time is defined as the time at which the correlated colour temperature has covered nine tenths of its total excursion, and that excursion is 27 kelvin — the number this essay elsewhere says a specification would dismiss.

The arithmetic makes the choice visible. A first-order rise with a 150-second constant reaches nine tenths of the way in 345 seconds; the quoted colour settling time is 405, at which point the junction is 93.3 per cent of the way. So the colour’s last tenth arrives during the temperature’s last seven per cent, and the colour excursion goes roughly as the 1.5 power of the temperature rise rather than in step with it.

That superlinearity is a property of the residual rather than of any slope. All three data-sheet coefficients are linear in temperature; what is not linear is a near-cancellation between them, and the correlated colour temperature is exactly that cancellation — 27 kelvin left over from two effects pulling in opposite directions. A quantity that is the small difference of two large ones does not inherit their shape, and its ninety-per-cent point is a much less robust number than the junction’s own.

So the headline seven minutes is the least stable figure in the essay. Nudge any of the three slopes by a few per cent and the 27 kelvin moves by a large fraction of itself and the 405 seconds moves with it; the junction’s 345 seconds moves by nothing, because it depends only on the thermal constant.

The Duv would have been the better clock. It moves 0.0029, which is the essay’s own argument for specifying in it, and it is a residual of one effect rather than of two cancelling ones — so its trajectory should track the temperature’s closely and its ninety-per-cent point should sit near 345 seconds. Defining the settling time on the quantity a specification should use, rather than on the one it should not, would make the headline number both smaller and firmer.

Which slope costs the quarter #

The output loss decomposes cleanly and it is worth having, because it says where thermal management buys anything.

The die loses 0.3 per cent per kelvin, which over a 55-kelvin rise is 16.5 per cent. The converter loses 0.2 per cent per kelvin, which is 11.0 per cent. Compounding the two gives a loss of 25.7 per cent against the reported 25.4 — a third of a point apart, which pins how the model apportions them: both slopes act on essentially the whole beam rather than the converter’s acting only on the converted fraction.

On that accounting the die is 61 per cent of the loss and the converter 39, and the ranking is the one the essay asserts without a number when it says thermal management dominates LED luminaire design. It also says how much a better heat sink is worth: halving the temperature rise recovers about an eighth of the output, and it recovers it in the same 61-to-39 proportion, so nothing about the split changes with the fixture.

The peak shift is the third slope and it contributes nothing to this total. Nearly two nanometres — 1.93 at 0.035 per kelvin — moves the colour by changing how much of the blue the converter catches, and moves the total radiant output by nothing at all. Two of the three slopes are about the level and one is about the shape, which is the structural reason the output falls by a quarter while the colour temperature moves by twenty-seven kelvin.

Where the two clocks actually overlap #

The lamp’s is a hundred and fifty, and it is the largest of the four is right and understates the overlap, because a time constant is not when something finishes.

At ninety per cent: the fast neural clock is done in 2 seconds, the slow one in 138, the pigment in 276, and the junction in 345. So the eye is finished at 276 seconds, and at that moment the lamp is 84 per cent of the way through its own transient.

Sixteen per cent of the light’s change therefore happens to a fully settled observer, and the other eighty-four happens to a moving one. That is the sharp version of the essay’s claim: the overlap is not partial, it is nearly total, and the interval in which the existing four-clock model is adequate — a steady observer under a moving light — is the last minute of seven.

It also identifies the awkward interval precisely. Between about 138 and 276 seconds the neural gains have settled and the pigment has not, while the lamp is between 60 and 84 per cent of the way. A judgement made there is made by an observer in one state of partial adaptation, under a light in another, and the two states are independent because nothing couples a photopigment’s recovery to a heat sink.

Where the model stops #

One thermal mass. A real luminaire has a die, a substrate, a heat sink and an enclosure, with time constants spanning perhaps two orders of magnitude, so the true settling curve is a sum of exponentials rather than one. The single constant is fitted to the overall shape and it will be wrong in the first seconds and in the last minutes in opposite directions.

No ambient dependence. The rise at full drive is quoted for a small luminaire in still air at twenty-five degrees. A fixture in a sealed recess reaches a higher junction temperature, so its transient is larger and its settled state is different — which means the settled colour a manufacturer publishes is a property of the test fixture as well as of the lamp.

And nothing here ages. A luminaire’s colour also drifts over thousands of hours, as the converter degrades and the die’s efficiency falls, and that is a third time scale with its own slopes that this model does not have. Every number here is about one switch-on of a new lamp.

The generalisation #

The sentence worth carrying is: every measurement of appearance has two ends, and this site had assumed one of them was fixed.

The four clock essays before this one all take the form the stimulus is steady and the observer is settling. That is the right frame for a display, whose output is stable in milliseconds, and it is the frame every appearance model uses — CIECAM’s viewing condition is a set of constants, and there is no argument in it that could carry a time.

A lamp is not a display. It has a thermal transient longer than any of the observer’s, so for the first minutes of a session the stimulus and the observer are both in motion on comparable time scales, and neither the appearance model nor the observer model has an argument for the other one’s state.

The surprising connection is with the dimming essay next door. Dimming by current changes the junction temperature, so a dimmer is a thermal control as well as an optical one — turning a lamp down starts a warm-up transient in reverse, on the same time constant, and the settled colour at the new level is not reached for minutes. A room dimmed for a photograph is not at its dimmed colour when the shutter opens.

Who found it, and when #

Thermal derating is the central fact of LED luminaire engineering and has been since power LEDs arrived: the light output, the lifetime and the colour all depend on junction temperature, and every data sheet is a set of curves against it. None of that is a discovery here.

Warm-up requirements are in the viewing-condition standards for graphic arts and have been since fluorescent booths, whose transient is much larger — a mercury lamp’s colour changes substantially over its first minutes as the vapour pressure rises, and the standards’ stated warm-up periods date from that technology.

LED booths inherited the requirement, and their transient is smaller and slower: a fluorescent tube is mostly settled in three minutes and an LED fixture’s thermal mass keeps it moving for ten. Whether the inherited period is long enough for the new technology is a question this essay’s numbers bear on and cannot settle, because the answer depends on the fixture.

And nobody has put the lamp’s clock beside the observer’s, as far as this site can find, because they belong to different fields: the thermal transient is a lighting-engineering quantity and the adaptation constants are a vision-science one, and the only place they meet is in a room where somebody is trying to judge a colour.

What the pictures cannot show #

They cannot warm up. A figure is a static drawing of a transient, and the transient’s whole content is that it takes several minutes — which is longer than a reader will look at any figure and much longer than the page takes to render.

And they cannot show the reader’s own light. Every claim in this essay is about the light falling on a scene, and the light falling on this page is whatever room the reader is in, at whatever point of its own thermal history. A reader who switched a lamp on when they sat down is inside the transient this essay describes while reading about it, and there is no way for the page to know.

Where the ladder goes next #

The nearest unfinished piece is the two-mass model. A die and a heat sink with different constants would give a fast transient and a slow one, which is what fixtures actually show, and the fast one would be the part that matters for a photograph while the slow one is the part that matters for a proofing session.

The second is the join proper: an appearance computation with a moving stimulus and a moving observer. Both halves exist on this site — the adaptation-clock machinery has the observer’s gains with their time constants and this file has the lamp’s — and nothing has ever run them together. The prediction would be that a judgement made two minutes after switch-on is displaced from the settled one in a direction neither model alone would give.

And the third is ageing, which is the same physics on a scale of thousands of hours. A luminaire’s colour drift over its life is specified, is a warranty term, and has never been put beside the observer’s own drift over the same period — which is real, is the lens, and is the largest single term in how much two people disagree.