Densitometer: scale error

Lifted from photodiodes, ADCs and measuring light and the troubleshooting table on the densitometer optical head.

What you see

Every density low — or high — by about the same fraction. Unlike high-density compression, which leaves the clear end alone, this fault touches the whole scale. Unlike a drifting zero, it does not change from hour to hour.

Read against a known wedge, the steps are still equally spaced — they are just the wrong size. Part XV’s troubleshooting table puts it in one line: densities read low across the whole scale by a similar fraction is a scale error, not a range error, and the answer is the calibration experiment rather than the black paint.

Likely causes

1. A reference that is not what it claims. The commonest single cause, and the reason the next section spends its length on wedges.
2. A gain or full-scale value misread, so every count is converted by the wrong constant.
3. A wedge whose actual increment differs from its nominal one.
4. A logarithm taken against the wrong zero, which bends the scale as well as scaling it and is therefore the one variant of this fault that is not a simple multiplication.

What is happening: the chemistry and physics

A densitometer does not measure density. It measures a ratio of two electrical readings and takes a logarithm, and every term in that chain is a place a constant can go astray.

The wedge is the usual one. Stouffer publish nominal increments — the T2115 is 21 steps of 0.15 from a nominal 0.05 — and separately record which of their wedges are calibrated, against NIST Standard Reference Material 38120C, by a densitometer conforming to ANSI PH2.19-1986. An uncalibrated wedge is an excellent relative instrument and a poor absolute one. Calibrate against its nominal steps and every density you ever report inherits the difference between nominal and actual, as a scale factor you cannot see.

The electronics contribute the other half. Part XV shows why a published linearity figure does not transfer: the OPT101’s ±0.01 per cent of full scale is specified at a full-scale output of 24 V, and this course runs the part at 3.3 V, where the output cannot exceed about 2.0 V. The course states plainly that it has found no source specifying photodiode-plus-amplifier linearity over the three decades a densitometer works in at the supply a Pico provides, so the linearity of your instrument over the range that matters is unknown and it is yours to measure. A scale error is often that unknown, discovered.

!The confirming test: read a wedge of equal steps and look at the shape

Four instrument faults, four different-shaped errors, and one wedge separates them.

What the equal steps do	The fault
Stay equal; the whole set moved	Drifting zero
Stay equal; all shrunk by one factor	Scale error — this entry
Correct low down, shrinking high up	Stray light
Correct high up, shrinking low down	An amplifier running out of range
Curved throughout	A logarithm against the wrong zero

Then cross-check against a second reference. Two instruments that disagree systematically may both be sound: Part XV’s own worked question puts two heads with different stray-light floors side by side and shows the disagreement growing with density without either being faulty. What each of you must do is write the floor on the certificate.

Diagnostic questions

• Is the wedge calibrated, and by whom? Nominal is not measured, and Stouffer say which of their products is which.
• What is the air reading being divided by, and when was it taken? A zero taken on a different day is a different scale.
• Was the gain changed since calibration? A range switch is a scale factor with a knob on it.
• Does the disagreement grow with density, or hold constant? Growing means stray light. Constant means scale.
• What geometry is each instrument using? ISO 5-2 defines the geometric conditions for transmission density, and two instruments measuring different quantities will disagree honestly and permanently — see density geometry and spectral response.

Corrective action

Recalibrate against a reference you can defend, and record what it was, when, and with what serial number. That is the whole of the fix and it is bookkeeping rather than engineering.

Do not reach for the black paint, which is Part XV’s own instruction: blackening addresses a fault with a different signature.

Re-read anything measured since the last defensible calibration. A constant scale factor is at least correctable after the fact, which is the one mercy this fault offers — provided you know when it started.

Prevention

Own one calibrated reference and treat it as a standard. Keep it clean, keep it flat, keep it in the dark, and do not use it as a working wedge.

Put the reference, the date and the geometry on the instrument certificate, so that a reading can always be traced to the scale it was made on.

Measure your own linearity rather than inheriting a datasheet figure taken at a supply voltage you are not using. Part XV’s calibration experiment is the method: read a wedge whose steps are equal and see whether your readings step equally.