MEASUREMENT AND ERROR CHARACTERISTICS OF MEASURING INSTRUMENTS ANALYSIS OF MEASURED DATA UNCERTAINTY ANALYSIS

CHARACTERISTICS OF MEASURING Express the need for measurement and analysis of measured data Define technical terms related to a measurement such as accuracy, precision, resolution, error, tolerance, etc. Describe the input/output relationship for a measuring equipment (static calibration) Analyze the accuracy and precision of a measurement. Compare and contrast the accuracy and precision for a measurement. Use significant figures to express the precision of a measurement. Classify the measurement errors and list ways of reducing them.

Measurement The process or act of determining the – Range – Dimension – Extent – Volume, or – Capacity Of anything Process of associating a number with quantity by comparing it to a standard

Performance Attributes of Measurement Response Resolution Connection Range Sensitivity Linearity Precision Lag and Settling Time Lag and Settling Time Sampling Rate

Ideal Instrument Instrument Device Under Test Operator Sensing or Stimulus Display and Control Physical World Information World Reference : Electronic Instrument Handbook (Figure 4.1)

Simplified measuring system Physical quantity Transmission path Transducer Signal conditioning Signal processing User Measuring Instrument: A device for determining the value or magnitude of a quantity or variable

Elements of a Measuring Instrument

Signal Flow of Electronic Instruments DUTDUT Information interface Digital Information Processing & Calibration Analog to Digital Conversion Analog Signal Processing & Reference Sensor or Actuator Physical Sensing or Stimulus Electrical Signal Analog Data Digital Data Measurement Information Display and Controls Operator Instrument Reference : Electronic Instrument Handbook (Figure 4.4)

Measurement System DUTDUT Sensing or Stimulus SensorSensor Electrical Signal Instrument Computer to InstrumentInterface Information Display Computer Reference : Electronic Instrument Handbook (Figure 4.8) Device Under Test Device Reference : Electronic Instrument Handbook (Figure 4.9) Sensor or Actuator Analog Signal Processing & Reference Analog to Digital Conversion Digital Calibration & Information Processing Digital Information Processing Information Interface Operator Electrical Signal Sensing or Stimulus Analog Data Digital Data Computer to Instrument Interface Measurement Information Instrument Computer

Instrument Block Diagram User Interface User Interface Analog Section Analog Section ADC or DAC ADC or DAC Power Supply Power Supply Power Source Mechanical Case / Package InputOrOutputSignal ProcessorProcessor RAMRAMROMROMI/OI/O Digital Section Analog Signal Information Digital Data Power ComputerInterface Control Reference : Electronic Instrument Handbook (Figure 4.7)

Information Processing As the information processing needs increase… Real Time: Some measurement systems add a second computer to handle special real-time requirement … Not real time: The completion of a measurement or calculation can take as long as necessary… “Soft” real time: The task must complete within a deadline if the result is to be useful… “Hard” real time: The result of a task is incorrect if the task is not performed at a specific time …

Multiple Computers in a Measurement System Reference : Electronic Instrument Handbook (Figure 4.11) ComputerInterfaceAdaptor Instrument#1 Digital Section Digital Section ADC or DAC ADC or DAC Digital Section Digital Section Input or Output Signal Instrument#2 Digital Section Digital Section ADC or DAC ADC or DAC Digital Section Digital Section Input or Output Signal Instrument#N Digital Section Digital Section ADC or DAC ADC or DAC Digital Section Digital Section Input or Output Signal PowerSupplyPowerSupplyCoolingCooling Digital Data Control Power Power Source ComputerInterface The block diagram for a cardcage instrument system General-Purpose Computer Embedded Real-Time Computer

Measurement Systems Reference : Electronic Instrument Handbook (Figure 4.11) ComputerInterfaceAdaptor Instrument#1 Digital Section Digital Section ADC or DAC ADC or DAC Digital Section Digital Section Input or Output Signal Instrument#2 Digital Section Digital Section ADC or DAC ADC or DAC Digital Section Digital Section Input or Output Signal Instrument#N Digital Section Digital Section ADC or DAC ADC or DAC Digital Section Digital Section Input or Output Signal PowerSupplyPowerSupplyCoolingCooling Digital Data Control Power Power Source ComputerInterface The block diagram for a cardcage instrument system DUTDUT Sensor Instrument#1Instrument#1 ComputerComputer Operator Instrument#2Instrument#2 Instrument#NInstrument#N A measurement system with multiple instruments Reference : Electronic Instrument Handbook (Figure 4.12)

Definition of Terms in Measurement True value – standard or reference of known value or a theoretical value Accuracy: closeness to true value Error: deviation from the true value Precision: a measure of reproducibility or agreement with each other for multiple trials Sensitivity: output(incremental)/input(incremental) Resolution: smallest change responded

Linearity: departure from linear value Tolerance: maximum deviation allowed from the conventional true value. – It is not possible to built a perfect system or make an exact measurement. – All devices deviate from their ideal (design) characteristics and all measurements include uncertainties (doubts). Hence, all devices include tolerances in their specifications. Definition of Terms (Cont.)

IMPORTANT: A measurement isn’t very meaningful without an error estimate! No measurement made is ever exact. The accuracy (correctness) and precision (number of significant figures) of a measurement are always limited by: – Apparatus used – skill of the observer – the basic physics in the experiment and the experimental technique used to access it

Goal of experimenter To obtain the best possible value of some quantity or to validate/falsify a theory. What comprises a deviation from a theory? – Every measurement MUST give the RANGE of possible value

Static Calibration Measuring Device Output (o/p) Input A Input B Output (o/p) = Sensitivity (S) x input (i/p) Input B Linear i/p-o/p relation B = B 1 B = B 2 Output Input A Non-linear i/p-o/p relation A = A 1 A = A 2

Accuracy

Example A voltmeter is used for reading on a standard value of 50 volts, the following readings are obtained: 47, 52, 51, 48

Precision Conformity: – ability of an instrument to produce the same reading, or it is the degree of agreement between individual measurements. So, it is also called repeatability or reproducibility. – (Pr) = max {(V AV – V MIN ), (V MAX – V AV )} # of significant digits

Bias The difference between CTV and average value (V AV ) Ideally, the bias should be zero. For a high quality digital voltmeter, the loading error is negligible yielding bias very close to zero.

Example A known voltage of 100 volts (CTV = 100 V) is read five times by a voltmeter and following readings are obtained: 104, 103, 105, 103, 105 Pr = max {(VAV – VMIN), (VMAX – VAV)} = max {(104 – 103), (105 – 104)} = 1 volt Accuracy = max {(CTV – VMIN), (VMAX - CTV)} = max {(100 – 103), (105 – 100)} = 5 V

Poor accuracy Poor precision Poor accuracy High precision High accuracy High precision Average accuracy Poor precision

Expressing accuracy and precision Mean (average) Percent error Range Deviation Standard deviation Percent coefficient of variation precision accuracy

Graphical methods Scatter plots Most accurate and precise Worst precision Systematic error?

Precision error due to instrument limitation the insulation resistance of a transformer = 2,475,653 .. an ohmmeter consistently & repeatedly indicates 2.5 M.M. M M is as close to the true value as he can read the scale by estimation. Although there are no deviations from the observed value, the error created by the limitation of the scale reading is a precision error. Conformity is a necessary, but not sufficient, condition for precision because of the lack of significant figures obtained. Similarly, precision is a necessary, but not sufficient condition for accuracy.

68  versus 68.0 

Error indication by significant figures Record a measurement with all digits of which we are sure nearest to the true value. A voltage may be read as V. – indicates that the voltage, read by the observer to best estimation, is closer to V than to V or V. – Another way of expressing is that it indicates the range of possible error. The voltage may be expressed as  0.05 V, indicating that the value of the voltage lies between V and V.

Example - Two resistors in series R 1 = 18.7  (3 significant figures) R 2 =  (4 significant figures) R T = R 1 + R 2 =  (5 figures) = 22.3  – no value in retaining the 2 and the 4. – If the (least significant) digit in the first place to be discarded is less than five, it and the following digits are dropped. – to 22.3; and to

sampling preparation analysis Representative sample homogeneous vs. heterogeneous Loss Contamination (unwanted addition) Measurement of Analyte Calibration of Instrument or Standard solutions How about sampling a chocolate chip cookie?

Types of errors (uncertainties) and how to deal with them Gross (human) Systematic (determinate) Random (indeterminate)

Examples Misreading instruments Erroneous calculations Improper choice of instrument Incorrect adjustment, or forgetting to zero Neglect of loading effects Not possible to estimate their value mathematically Human errors (Gross errors)

Human errors (Gross errors) Methods of elimination or reduction Careful attention to detail when making measurements and calculations. Awareness of instrument limitations. Use two or more observers to take critical data. Taking at least three readings or reduce possible occurrences of gross errors. Be properly motivated to the importance of correct results.

Equipment errorsEnvironmental errors Examples Bearing friction Component nonlinearities Calibration errors Damaged equipment Loss during transmission Examples Changes in temperature, humidity, stray electric and magnetic fields. Systematic errors

Equipment errorsEnvironmental errors Systematic errors How to estimate: Compare with more accurate standards Determine if error is constant or a proportional error How to estimate: Careful monitoring of changes in the variables. Calculating expected changes.

Determinate (or Systematic) Errors Sometimes called bias due to error in one direction- high or low Known cause – Result from mis-calibrated device – Experimental technique that always gives a measurement higher (or lower) than the true value – Operator – Calibration of glassware, sensor, or instrument Try a different method for the same measurement When determined can be corrected May be of a constant or proportional nature

constant or proportional error… Constant error influences the intercept. Proportional error influences the slope.

Equipment errorsEnvironmental errors Systematic errors Methods of reduction or elimination: Careful calibration of instruments. Inspection of equipment to ensure proper operation. Applying correction factors after finding instrument errors. Use more than one method of measuring a parameter. Methods of reduction Hermetically seal equipment and components under test. Maintain constant temperature and humidity by air conditioning. Shield components and equipment against stray magnetic fields. Use of equipment that is not greatly effected by the environmental changes.

Examples Unknown events that cause small variations in measurements. Quite random and unexplainable. How to estimate Take many readings and apply statistical analysis to unexplained variations Random errors

Examples Unknown events that cause small variations in measurements. Quite random and unexplainable. Random errors Methods of reduction: Careful design of measurement apparatus to reduce unwanted interference. Use of statistical evaluation to determine best true estimate of measurement readings.

Indeterminate (or Random) Errors Cannot be determined (no control over) Random nature causes both high and low values which will average out Multiple trials help to minimize Deal with those using statistics

Uncertainties in Reading Digital Displays Gate and clock signals are combined in an AND gate, case (b) results 4 pulses while case (a) supplies only 3 pulses. A digital read-out has an uncertainty of  1 digit. The uncertainty in analog displays is accepted as  ½ scale divisions.

Uncertainties in Reading Analog Displays What is the uncertainty of reading voltages using the meter shown?

Example: Uncertainty in analog voltmeter 100 divisions on scale. – voltage read is 6 V and – meter has two ranges: 0 – 10 V and 0 – 100 V Uncertainty =  ½ V FSD / # of divisions – On 10 V range, uncertainty =  ½ (10/100) =  0.05 V yielding V = 6  0.05 volt. – On 100 V range, uncertainty =  ½(100/100) =  0.5 V yielding V = 6  0.5 volt. Relative uncertainty: – on 10 V range, 0.05/6 = 1/120 = ; – on 100 V range, 0.5/6 = 1/12 = Percentage uncertainty: – on 10 V range, (0.05/6)x100 = 0.83%, and – on 100 V range, (0.5/6)x100 = 8.3%

Exercise: Determine the Instrument limit of error and least count For each of the three rulers in figure, determine and record The least count of the scale (smallest division) – scales are all in cm Length of the gray rods Uncertainties in your readings Compare your result with those of the student next to you

How do we assess the total error? One way to assess total error is to treat a reference standard as a sample. The reference standard would be carried through the entire process to see how close the results are to the reference value.

In critical work: Observer make an independent set of measurements, – using different instruments or different measurement techniques, not subject to the same systematic errors. Make sure that instruments function properly and are calibrated against a known standard, Make sure that no outside influence affects accuracy of measurements.