Experimental Test Plan and Calibration

1. Experimental Test Plan

The design of an experiment requires more than attaching some instruments and turning on the power. Relevant information can only be obtained from a well designed measurement test plan. A test plan will require the following three steps:

I . Parailzeter Test Plan. An identification of the process variables and parameters and a means for their control.

2. System and Tnleraizce Design Plun. This step involves the proper selection of the measurement technique, equipment, and test procedure based on a pre- determined tolerance limits for error.

3 . Data Reductioiz Desigiz Plan. A methodology to analyze, present, and use the anticipated data.

Experimental design includes the development of a measurement test plan. This plan allows the engineer to properly optimize the time, accuracy and costs involved with acquiring the relevant data for analysis.

Variables

The objectives of the measurement program requires the identification of the test variables (variables of interest). In the parameter test plan, all known variables should be listed and evaluated for any possible cause and effect relationships. If a change in one variable does not effect the value of the other variable, i t means the two variables are independent of each other. These variables are known as ir?&peidentvariahles. A variable that changes due to changes in one or more variables is known as a dependeizt variable.

The control of variables is important in any experimental test plan. Variables are defined as coizh-olled whenever they can be held at a constant value for the duration of a measurement. The control may only be for a short duration while a sample is being made.

0.0 0.5 1.0 1.5 2.0

Time lsrcl

Figure 1. Effects of Noise and Interference Superimposed. on the signal y=2+sin2xt

Variables that cannot be controlled during a test but will still affect the magnitude of the measured variable is defined as extraneous variables. The effects of extraneous variables can take the form of signals superimposed over the measured signal in the form of noise or drift. These extraneous variables may impose a false trend onto the behaviour of the variable.

Noise is defined as a random variation of the value of the measured signal as a consequence of the variation of the value of the extraneous variable. Examples would include vibrations and thermal excitations. Interference produces undesirable trends in the data due to extraneous variables. Examples would include variations on environmental conditions (such as changes in temperature or barometric pressure) local AC power noise (50 or 60 Hertz) electromagnetic or radio -frequency interference due to nearby motors, transformers etc. An undesirable situation occurs in cases where the period of interference is longer than the period of the measurement. This situation will result in a false trendin the behaviour of the signal.

In general, as the number of samples is increased, the estimated value of the measured variables improves. Repeated measurements made during any single test are defined as repetitions. Repeating the set of measurements using similar operating conditions is referred to as replication. Repetition and replication are means to reduce and identify sources of noi se and interference.

2. Calibration

In any measurement program, the relationship between the input to the measurement device and the output signal is critical to the success of the experiment. The relationship between input and output is established during a calibration of the measurement system. Calibrationis defined as the act of applying a known value of input to the measurement system for the purpose of observing the system output. The known value used for the calibration is called the staizd'd.

By application of a range of known values for the input and observation of the system output, a calibration curve can be established for the measurement system. In a calibration curve, the controlled independent variable is the input (plotted on the x-axis) and the measured output variable is the dependent variable (plotted on the y-axis). This calibration curve forms the relationship between the variables during the actual experiment. For example, the output from an electronic temperature device would be translated into an output display on the device.

The calibration curve can also be defined by a mathematical relationship, y=.f(x). This f~~nctional relationship is defined as the correlation between input and output. The correlation is generally obtained through normal curve fitting techniques . The correlation is then used later in the experiment to ascertain the unknown input value based on the known output value. Several terms are used to describe the results of a calibration test, including:

Static Calihra~ioiz Test

The most common type of calibration test is known as the static calibration test. In general, static calibration refers to a situation in which all the inputs except one are kept at some constant values. Then the one under study is varied over some range of constant values which causes the output(s) to vary over some range of constant values. This procedure may be repeated by varying each input considered to be of interest and thus developing a family of input-output relationships. The total system behaviour can then be described as the superposition of all the individual static calibration curves. The statement "all other inputs

are held constant" refers to an ideal situation which can only be approached but never > reached in practice"

l n ~ u t value. x lun~tsl

Figure 2, Representative Static Calibration Curve

In many cases, whenever time dependent variables are to be measured, a dynamic calibration is performed in addition to a static calibration. This dynamic calibration determines the relationship between the an input of known dynamic behaviour and the output signal. Usually such calibrations involve either a sinusoidal signal or a step change as the known input signal.

Static Sensitivity

The slope of the static calibration curve is the static sensitivity of the measurement system. The static sensitivity , K, at any particular static input x, is defined as;

Since calibration curves can sometimes be non-linear, the value of K can sometimes vary over the range of input values.

The range is defined the maximum and minimum limits of the static calibration. The input range refers to the range of the independent variable, x, while the output span orfull scale nperatin,p output (FSO) is the range limits of the output variable, y. It is important to avoid exceeding the operating range of the measurement system since the behaviour of the system is unknown beyond the calibration region.

Accuracy

The accuracy of the system can be estimated during a calibration test. If the known input of the calibration is compared with the output from the measurement system then the absolute error E can be estimated by;

E = true value - indicated value

The accuracy is often described as a percentage value by;

A + - 14 \ 100 true value )

Biased and Unbiased (Precision) Errnrs

Repeated observations of a measurement device wiil generate observations that often differ from one another. There are generally two types of error that are identified, hiaxedad unbiased errors. The unbiased errors is a measure of the random variation to be expected during a set of repeated tests. These types of errors may result from incorrect reading of the measurement device (i.e. reading the scale of a micrometer incorrectly). IJnbiased errors . are often referred to as precision or random errors.

Biased errors on the other hand can be defined as the difference between the average of the measured values and the true value. Biased errors may be defined as resulting from perturbations in a measurement device that tend to influence a particular quantity equally. An example of a biased error is the effect of temperature on a metal ruler. As the temperature rises the length of the ruler increases which effects all readings to the same degree.

Measured data I = I Apparent measured average

True or known value

1 2 3 4 5 6 7 8 9 1 0 Measured read~ng number

Figure 4. Effects of Biased and Unbiased Errors on Calibration Readings

In the design of an experimental program, the engineer must strive to isolate and t

remove all those perturbations that tend to result in biased errors. The causes of biased errors may be a result of the measurement device itself (i.e. excessive friction in a mechanical gage ) or be induced in the measurement device by external factors (i.e. temperature ). Normally biased errors can be minimized during the calibration phase of the experiment ,

3. Instrument Error

In the design and selection of a measurement instrument, a number of standard errors are listed to assist the engineer in identifying possible errors in hisiher measurement system. These errors are normally listed as part of the manufacturer's specifications derived from calibration tests. An example of such a list is illustrated on Table 1.1

TABLE I. I Manufacturer's Speci tications: Typical Preshure Transducer Oprroriot~

Input range 0 to I000 cm H:O Excitation +I5 V dc Output range O t o 5 V Prty'i)rmuncr

Linearity error +0.5% full scale (FSO) Hysteresis error Less than f 0.15% full scale (FSO) Sensitivity error +0.25% of reading Thermal sensitivity error +0.02% lQC of reading Thermal zero drift 0.02% 1°C full scale (FSO) Temperature range 0 to 50 "C

Operation

The operation parameters refer to the range of performance for the instrument. In this example the pressure transducer must operate with a + 15 volt input voltage to the device. The pressure recording transducer will operate within an input range (primary sensing element) of 0 to 1000 cm of water and will output a signal of 0 to 5 volts corresponding to 0 to 1000 cm of water pressure (variable conversion element). The calibration constant relating the pressure to output signal voltage would be .005 volts/cm water or 5mVIcm water.

Performance

The performance parametersfor the instrument will give some insight into the limitations of the calibration constant calculated from the operation parameters. These performance parameters must meet the tolerance limits for the experiment. A list of the errors follows.

Hysteresis

Hysteresis error refers to the differences between an upscale sequence calibration and a downscale sequence calibration. Hysteresis is usually specified for a measurement system in terms of the maximum hysteresis error as a percentage of full scale output range (FSO).

Hysteresis is normally caused through friction or viscous damping of moving parts or residual charge in electrical components. Some hysteresis is normal for any system and will always effect the precision of the system.

I I

lnput value

( a ) H!.steresis error

Max~mum for

curve devlce

lnput value

(c) Sensiti\fity error

lnput value

( 1 1 ) Linearity error

0 lnput value

((1) Zero shift (null) error

Probable ( 2 2S,)

P data scatter band on successive measurements

lnput value

(0) Repeatability error

Figure 5. Examples of Elements of Instrument Error

Limwity Error

Most instruments are designed to achieve a simple linear relationship between the input and output signals. However, in real systems, truly linear behaviour is only approximately achieved. As a result, measurement device specifications provide a statement as to the expected linearity of the static calibration curve for the device. The linearity error, e,(x) is the difference between the actual calibration line and a best fitting linear line through the data. It is normally expressed as a percentage of full scale output (FSO).

Sensitivity and Zero Errors %

The scatter in the data measured during a calibration affects the precision in the slope of the calibration curve. If we fix the calibration curve to pass through zero and then bound all our calibration data with straight line, we will create an upper and lower bound for the calibration slope . The sensitivity error relates to the error in the calibration slope (i.e. static sensitivity, K) to account for the full range of data collected in the calibration test.

The static sensitivity of most devices is also a function of the temperature of the device. Therefore an estimate of the sensitivity of the device to temperature is published as a thermal sensitivity error. This value relates the change of the calibration slope (static sensitivity, K) as a function of temperature. The change in the slope due to temperature changes are normally referenced to an ambient test temperature (usually 20°C)for the standard tests.

If the zero intercept is not fixed but the sensitivity is constant (i.e. slope K is constant), then drifting of the zero intercept introduces a vertical shift of the calibration curve. This shift of the zero intercept of the calibration curve is known as the zero error, e, or null error. Zero error can usually be reduced by periodically adjusting the output from the measurement system under a zero input condition. However, some random variation in the zero intercept is common, especially with electronic equipment subjected to temperature variations. This condition is normally referred to as thermal zero drift.

Instrumen? Repeala hility

The ability of the instrument to indicate the same value upon repeated but independent applications of the same input is known as instrument repeatability. The instrument repeatability reflects only the error found under controlled calibration conditions and does not include temperature effects or procedural errors.

Instrument Precision

The term "instrument precision" when reported in instrument specifications refers to the results of separate repeatability tests. Manufacturer claims of instrument precision must be based on multiple repeatability tests performed in different labs on different units of the same instrument.

Overall Instrument Error

An estimate of the overall instrument error is made based on all known errors. This error is sometimes referred to as instrumeiztaccurq. The overall error is estimated from the square root of the sum of all the errors. For M known errors, the instrument error, e,, is estimated by: