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Instrument Calibration Fundamentals


Inspiration for this article came from a recent visit to a glass fiber manufacturer who was evaluating the use of an IR instrument to continually measure and potentially control the moisture of glass fiber products through different stages of manufacture. The instrument had been evaluated for a number of weeks and a considerable volume of data recorded. Attempts were made to calibrate the instrument but had not been successful due to a lack of knowledge of the fundamentals of calibration.

The purpose of calibration is to adjust the response of an instrument such that it displays a true value relative to a traceable reference.

With regard to continuous moisture instruments, regardless of technology, they must be calibrated relative to primary laboratory methods such as loss on drying, Karl Fischer titration and toluene distillation. These laboratory methods are classified as primary analyses and are all destructive, time consuming and generally require the skills of a laboratory technician. They also use relatively small samples of test material with a potential sampling error.


There are many powerful statistical analysis tools available today helping to facilitate instrument calibration. Although easy to use, results may be misleading without some knowledge of statistics. Much relies on common sense and does not require formal training.

The objective of this paper is to try to educate users in measurement fundamentals and to offer some ideas in how to replace a spreadsheet of incomprehensible numbers into usable, correlatable data using a little experimental forethought.

The principles described may be applied to almost any other type of measurement instrument.

Primary Laboratory Method

Primary moisture measurements require taking samples of material from the process and measuring them, usually in a laboratory, using a standard method. Since the continuous moisture measuring instrument is being calibrated against a laboratory method, the implied accuracy of that instrument may never be better than the reference against which it is calibrated. For this reason, it is important to understand the errors associated with the reference methods.

The most common method used for moisture determination in virtually all types of solid products is loss on drying (LOD). A sample is weighed, dried for a period of time, then reweighed. Air oven or vacuum oven may be used. The loss of weight, presumed to be that of evaporated water, will provide the moisture content using the equation:

Moisture Content = (Total Product Weight – Dry Product Weight)/Total Product Weight

In a few industries such as wood and textile, a variation known as “Dry Basis Moisture” or “Atro Moisture” uses the equation:

Moisture Content = (Total Product Weight – Dry Product Weight)/Dry Product Weight

In both cases a percentage measurement is obtained by multiplying the fraction by 100.

Errors with Gravimetric Moisture Analysis

Weigh Scale

An inherent error results from the accuracy and resolution of the weigh scale used. Accuracy is usually specified by the manufacturer as a percentage of the reading or as an absolute number such as ±0.01%

Resolution describes the number of digits following the decimal point. For example, a weigh scale with a 1mg resolution would need three decimal places. Accuracy of the weigh scale can never be better than its resolution and, though often claimed to be the same, in the authors experience is usually less. Example: a weigh scale with 2dp resolution and assuming the same accuracy (best case), with a sample weighing 10g before drying and 9.5g dry weight. Theoretical moisture content will be [(10 - 9.5)/10]*100 = 5.0% Given weigh scale accuracy of ±0.01g, however, the moisture reading may range between: [(10.01 – 9.49)/9.99]*100 = 5.2% & [(9.99 – 9.51)/10.01]*100 = 4.8%

So a 2 decimal place scale and a 10g sample offers a best case moisture content accuracy of ±0.2% This accuracy may only be increased by increasing the sample weight with the resultant increase in drying time.

Operator Error

Often a weight display may flicker between two values, say 9.49 – 9.50 and the operator records his best estimate. A good operator will always bias the reading in one direction, that is always using the lower number or always using the higher. A less experienced operator may randomly mix values and even an experienced operator may have difficulty sometimes calling the correct number. This uncertainty applied to the previous example could double the error to ±0.4%

Another operator error occurs when product is “cherry-picked” for testing. This refers to selectively rejecting product that is not considered “Normal”. An example is a powdered product containing lumps. An operator may be trying to weigh 10g of product and he discards a 3g lump. It is likely the lump has a higher moisture content than the finer powder so the operator has just biased the sample result.

Sample Representation

The online instrument is measuring product continuously passing beneath it and often has filtering built in such as a moving average often known as damping. If the instrument has a damping value of 20-seconds, the displayed value represents the average measurement of the last 20-seconds of product. The sample taken for lab analysis should similarly represent 20-seconds of product flow. To do this it is necessary to take multiple small grabs over the averaging period and place them into a single bag.

In order to determine probable error due to non-representative sampling, take multiple sequential samples, bag them individually and test individually. If multiple samples show a spread in moisture of 0.5%, then this is the likely error due to not following the correct sampling procedure.

Sample Handling

Most processes involve drying, baking or otherwise cooking the product. Therefore, the product at point of measurement is very likely to be hot. A grab sample should be placed in a sealed container as soon as possible since it may still be losing moisture or, conversely, may be inclined to regain moisture as it cools. A good tool for storing samples is a Ziplock® plastic bag. These are particularly good for storing hot cookies while allowing them to cool. Often a film of condensate will appear on the inside lining. After the product cools, the cookies can be crushed inside the bag and manipulated to re-absorb the moisture condensate prior to weighing and drying. Note: ensure the product temperature is not so high to melt the plastic bag. In these cases, a glass sample jar may be used.

Samples should generally be allowed to come to room temperature prior to weighing so the moisture content will be stable. An example of instability is when a sample is placed on the weigh scale and the value is continually falling. Sample cooling must take place in a sealed container. After oven drying, the sample may be placed into a desiccator in order to stay dry while cooling.

Drying Time and Temperature

There are standard methods approved for almost all materials. The standard generally specifies the size of sample, the drying temperature and the drying time (see ASTMS methods).

The product determines the drying temperature. It is important not to burn the product or damage it in any way. A vacuum oven is often used to dry the product at a lower temperature. If a fixed time is used, then timing should be accurate and a constant temperature used. The best way is to dry to no further loss of weight but this sometimes takes too much time for practical purpose.

Other Volatiles

With loss on drying, it is generally assumed that any lost weight is due to moisture. This may be a bad assumption if the product contains other volatiles. Most organic products contain volatiles, often with lower evaporative temperatures than moisture. A moisture specific gravimetric method involves drying a product in a closed flask while passing a dry gas, such as nitrogen, over the product. The gas is then passed through a hygroscopic medium such as calcium oxide or concentrated sulphuric acid. The desiccant is weighed before and after drying and the gain in desiccant weight may be considered the water loss and used in the moisture equation.

The primary method described here is Loss on Drying and is by far the most common laboratory method used. Other laboratory methods may result in better accuracy but are generally limited to even smaller sample sizes and all of the above error scenarios, particularly sampling errors, apply equally to all laboratory test methods.

Instrument Errors

Every moisture instrument regardless of measurement technology has an inherent accuracy. Accuracy is always application dependent but the best achievable accuracy will be ±1LSD (least significant digit). In reality, few applications will result in such accuracy and will depend largely on laboratory reference procedure, sampling procedure and whether calibrating in-process (online) or offline.

Most other sources of error relate to product presentation and a multitude of application specific aspects.

Pre-Calibrated Instruments

Some manufacturers claim their IR moisture instruments are pre-calibrated while, in reality, this is not possible for anything other than non-processed raw materials or optically translucent materials.

The previous statement is indisputable and is not related to “Quality” or “Design” of gauge. Reflectance infrared spectroscopy is a surface measurement. Some products, such as salt, sugar and transparent films, are more translucent than others and might be pre-calibrated with a degree of accuracy. The majority of products such as sand, wood or baked goods, are not transparent so only surface concentration of moisture or other constituent of interest is possible.

When a product exits a drying or baking process, surface moisture is usually minimal with highest concentration in the depths of the product. There is a moisture gradient through the product. This doesn’t mean we cannot measure the moisture but, for best accuracy, it is important that the moisture gradient is consistent.

Most moisture measurements are made as close as possible to the process in order to best control it. Unfortunately, this is usually the point at which products are most unstable having the maximum moisture gradient. It is impossible to replicate this same moisture distribution off-line in a laboratory or at the instrument manufacturers facility.

It is possible, using experience of previous same applications to install coefficients obtained from those prior experiences and these may be quite close depending on how closely gauges are replicated in manufacture.

While generally not in favor of offline calibration for an on-line instrument, sometimes in-situ calibration is impossible due to inability to take a representative sample or inability or unwillingness for the operator to make process adjustments in order to produce an adequate moisture range.

As previously stated, a product exiting a drying process will not have uniform moisture distribution. After the sample cools it tends to equilibrate with moisture migrating toward the dryer regions. Since an IR moisture gauge reads primarily surface moisture, a product that has been in storage for some time can never adequately reflect the product with non-uniform moisture distribution exiting the process. Even single-sided RF or Transmissive Microwave measurement are, to a lesser degree, affected by moisture distribution. Sample presentation is also important trying to replicate the online environment and is virtually impossible to achieve for RF sensors due to product temperature differences, sensor/product distance differences in addition to product moisture distribution.

An exception where offline testing may be valid, allowing pre-calibration of the instrument, is in the measurement of raw materials. For example, sand exiting a hopper should have the same characteristics as a sample that is bottled and sent to the instrument manufacturer for offsite calibration.

In most cases, at best, offline calibration may be used as a starting point for calibration which should be further refined in-process over time with collected samples.

Online measurement, if possible, should be used for instrument calibration. Sensortech instruments, both IR and RF feature included calibration software. The software has “start/stop” buttons allowing the operator to start gathering data while taking a grab sample then stopping measurement. The measurements between start and stop are automatically averaged and may be stored in the calibration table. The Sensortech calibration program provides a graphical presentation of the data allowing the user to judge quality of data and delete bad samples. Actual sampling methods have been previously described in “Sample Representation” and “Sample Handling”. If these methods are adhered to, measurement accuracy should be optimized.

If offline measurements are necessary, it is important to establish techniques and to maintain a consistent methodology.

Offline Calibration

Since offline calibration is virtually impossible for RF sensors, we will focus here only on IR. The instrument must be suitably mounted in a table-top configuration and the sample presented in a consistent manner at approximately the same distance as would be experienced in the actual process. The following shows a typical offline instrument set up.

Offline Calibration in Lab This image illustrates a typical offline instrument calibration showing a vacuum oven, weigh scale, an NIR-6000 and PC running the Sensortech Calibration Program.

The product should be contained in an open-topped vessel at least 75mm diameter and preferably 100mm in order to ensure measurement light beam is wholly contained by the product. The sample container should be moved about and rotated but always under the instrument light beam while monitoring displayed moisture. This is best done using the configuration software with its trend plot. If the display varies significantly in relation to the desired accuracy, then multiple instrument readings should be taken for each sample, moving the sample dish through a range of positions. An alternative method is to measure a sample, pour it back into its container and then refill the sample cup. This should be repeated multiple times recording measurement readings each time.

If an accuracy of ±0.1% is required and repeatability of measurement between same sample readings is 0.5%, it is suggested 5 measurements be taken in different sample positions. Number of required measurements = measurement repeatability/required accuracy. Many sample measurements may be needed when a high accuracy is required from a very nonhomogeneous product.

Calibration Math

Whatever sampling measurement methods are used, the result will be two columns of data representing instrument readings and corresponding laboratory reference values. At least 6 data pairs are required for the most basic calibration. The actual reference values should cover the entire moisture range likely to be experienced in use and preferably a little outside that range.

The most common calibration method is the regression analysis. This is a statistical method of developing the curve relating two variables in a scatter plot. Sensortech instruments have the capability of fitting either a linear function of the form: m = b*x + c or a quadratic function: m = a*x2 + b*x + c. The quadratic is a non-linear function and may fit many nonlinear relationships by calculating coefficients a, b & c. The non-linear function should be used very carefully and is generally only meaningful when the moisture range is large (>10%). This will most likely occur when measuring raw materials. By making a = 0, the equation becomes a linear function which is used in most process applications with narrow moisture range.

The following table shows a typical data set from an instrument calibration. Before applying any statistical analysis, the first task is to graph the data in order to judge the quality of data and whether there is the appearance of correlation between the two variables. In this case there is an obvious relationship and it is assumed to be linear as shown in the Calibration Graph.

Instrument Value 2.9 3.6 1.5 2.0 4.8 3.3
Reference Value 6.2 8.2 3.5 5.1 11.6 7.6

Axis Designation in Calibration Graph

Normal practice dictates that in a graph of two related variables, the x-axis denotes the independent variable and y-axis, the dependent variable. Since the instrument is dependent on the measured variable (moisture content), convention would dictate displaying moisture content on the x-axis and instrument on the y-axis.

We depart from this convention for instrument calibration in order to simplify calculation and to ensure slope (b) and intercept (c) coefficients are independent of each other.

With axis designations shown in the following graph:

Desired Slope Coefficient:
New b = Old b * regression slope

Desired Offset Coefficient:
New c = Old c + regression intercept

Moisture Calibration Graph

Sensortech calibration software facilitates easy data collection, graphing and data selection allowing deletion of bad data points.

Calibration Software

Sensortech calibration software facilitates easy data collection, graphing and data selection allowing deletion of bad data points.

A meaningful regression analysis requires reasonably uniform distribution through range of moisture. Five low samples and one high sample are not suitable. The concept of taking multiple samples is to average experimental errors and to highlight non-representative samples which may be deleted. This is most effective if all data points are equally weighted.

Another important consideration for a meaningful regression is an adequate moisture range. In the opinion of this author, the moisture range must exceed total measurement errors by a ratio 10:1. Thus, if total measurement error is ±0.25% then data range should cover ±2.5% or 5% total. As moisture range reduces, the quality of fit or correlation coefficient (R2) will reduce. A perfect correlation coefficient is R2=1. A 2-point regression will always result in a perfect correlation since 2 points define a line but if one is a bad sample then the entire calibration is in error.

The following graph illustrates how an insufficient moisture range impairs the ability to accurately fit a line. Note the correlation coefficient is barely >0.5 which implies poor or no correlation. It is obvious, therefore, that the meaningfulness of R2 is dependant on data range.

Moisture Calibration Short Range

Obtaining an adequate moisture range requires the cooperation of the process supervisor who may make process adjustments to quickly develop a moisture range. Do not ask the plant to exceed the permissable moisture range of a product but in some cases the plant is willing and even eager to do this in order to achieve a good calibration in a timely manner. If process manipulation is not possible, the alternative may require waiting a long time, sometimes weeks, for it to cover the full moisture range. It is strongly recommended to calibrate over the entire moisture range that may be encountered in production.

Looking again at the preceding graph, if the process moisture range is within the range 4% - 5% then any error due to wrong slope will be very minor. However, if the process range is 3% - 8%, the extrapolated calibration at the higher moisture range could result in a significant error. When the plant is willing to cover the entire moisture range and even go beyond that range, the result will be a strong calibration.

Linear vs. Quadratic Fit

Sensortech instruments feature the ability to fit either linear or quadratic response curves. The calibration software provided with the instrument originally allowed selection of either fit to be used at the discretion of the operator. This feature was removed, however, due to constant misuse by untrained operators. Statistical tools offering polynomial regressions are readily available, probably the best known being Excel®.

The following examples illustrate the danger of using the wrong regression technique based on lack of knowledge and too little data.

As previously stated, two data points define a straight line but error in a single point may result in a seriously compromised calibration. Similarly, three data points will define a curve but an error in one could result in a serious calibration error.

The data displayed below was obtained from a Sensortech ST-2200A RF pipeline sensor measuring caramel exiting the cooker. The 4 data points fit a quadratic curve with a perfect regression coefficient R² = 1.

ST-2200A KF
5.04 6.99
5.10 7.05
5.14 7.18
5.23 7.73
Calibration Graph Quadratic

The calibration engineer may have gone home with a good feeling of success. Note, however, the rather narrow moisture range (0.25%). When more data points are added, extending the moisture range to 1.0%, we have an entirely different result. This is shown in the following graph; a linear fit is apparent when the moisture range is extended.

ST-2200A KF Calibrated ST-2200A Projected Error
5.04 6.99 6.94 -0.05
5.10 7.05 7.13 0.08
5.14 7.18 7.26 0.08
5.23 7.73 7.55 -0.18
5.63 8.66 8.83 0.17
5.78 9.28 9.31 0.03
5.84 9.67 9.50 -0.17
6.05 10.13 10.17 0.04
Calibration Longer Range

The true fit is a linear function with the calibration equation having a very respectable R² = 0.9903

Number and Weighting of Samples

Theoretically, only two samples are necessary for a linear calibration and three for a curve fit. The minimum number of points do not, however, guarantee a “Good” calibration. Like so many things, the quality of a calibration is proportional to the amount of effort expended on it. If only two points are used, each point has 50% significance or weighting in the calibration. If one or both points are in error, the calibration may be significantly compromised. Also, with so few points, there is no graphical indication of a bad sample.

The following graph illustrates this point. The data point (5.21,9) is obviously an outlier. It is one of 9 coordinate points and, with caution, may be justifiably deleted. If this was one of only 3 or 4 points, then it is too significant to delete until more data points have been obtained.

Calibration with Outliers

As well as number of samples, the data distribution is also very important. The most representative regression analysis will contain data points of equal weighting or equal importance.

The following graph shows a data set containing 8 samples but 7 of those samples are closely grouped with only 1 high sample. The slope of calibration line is effectively determined by the average of the low data group and the single high point. The high point is thus about 50% weighted while the remaining points each have approximately 7% weighting.

Errors in the high point will be just as significant as if it were a 2-point calibration.

Calibration Graph Uneven Weight


Performing a significant regression analysis requires:
  • A minimum of 6 samples
  • Reasonable even moisture sample distribution
  • Moisture range at least 10x total inherent measurement error
  • Moisture range preferably as wide as teh process range

Offset Calibration

If, despite all the forgoing arguments, a satisfactory moisture range is impossible to achieve, then it is pointless running a regression analysis. This situation calls for an offset calibration. In the calibration equation:

y = b*x + c

Slope coefficient b is set to a nominal value, usually factory default, and offset coefficient c is adjusted to minimize overall instrument error.

Calibration Offset

This illustrates how an instrument response with badly calibrated slope can, using only offset adjustment, result in an acceptable accuracy over a narrow process range.

Once an initial calibration has been achieved using full regression analysis, it is usually possible to make future calibration tweaks using only the offset coefficient c.

End Note

Inspiration for this paper came from a recent experience in a glass fiber factory. The following describes that experience.

A substantial amount of data had been collected at considerable expense prior to my visit. The customer was disappointed with R2 values < 0.5 and slopes varying widely from positive to negative coefficients. A quick inspection of the data showed an extremely narrow range of moisture, < 0.08%. Graphing the data showed a scatter gun data pattern. The many possible error sources included insufficient moisture range for a meaningful regression and questionable laboratory accuracy.

The laboratory method used a 50-minute dry time in a microwave oven – acceptable for a product such as glass. A 5dp weigh scale offered adequate resolution but had a very low maximum capacity. Using a very small reference sample, as mentioned earlier contributes to sampling error. A single sample of chopped glass fibers was sub-divided into 5 samples of approximately 100g each for lab testing. The results were returned about 1-hour later varying from 0.01% - 0.07%. A 0.06% spread of results for a single sample when the total calibration data covered a 0.08% range. Clearly the use of regression analysis was not the preferred method. Taking the entire data set of more than 50 samples and averaging both reference and instrument values, it was possible using only an offset adjustment to achieve accuracy of better than 0.03%.

Samples were obtained at a different point in the process with high moisture contents ~8%. Calibration results for this product were also disappointing. Many points had been recorded, at considerable expense. The sample was measured offline using an NIR-6000 moisture gauge. The initial reading showed 6.2% but sample dish rotations resulted in instrument variations of 3%. The variance was due to seriously non-uniform moisture distribution within the sample. In this case multiple sensor readings should have been taken for each sample and the entire sample (~25g) used for lab analysis. The historical data was of no value since each instrument value represented only a single reading with a 3% probable error due to moisture distribution. There was insufficient time to gather sufficient data using the recommended method but it is hoped the customer will follow the recommended procedures in the future.