CONCEPTS OF MEASURING SYSTEMS Units= The result of a measurement of a physical quantity must be defined both in kind and magnitude. The standard measure of each kind of physical quantity is called a unit. Magnitude of a physical quantity = (Numerical ratio) X (Unit) Absolute Units An absolute system of units is defined as a system in which the various units are all expressed in terms of a small number of fundamental units. Fundamental and Derived Units In Science and Technology two kinds of units are used (i) Fundamental units (ii) Derived units The fundamental units in mechanics are measures of length, mass and time. The sizes of fundamental units, whether centimeter or meter or foot, gram, or kilogram or pound, second or hour are quite arbitrary and can be selected to fit a certain set of circumstances. Since length, mass and time are fundamental to most other physical quantities besides those in mechanics; they are called the Primary fundamental units. Measures of certain physical quantities in the thermal, electrical, illumination fields are also represented by fundamental units. These units are used only where these particular disciplines are involved and therefore they are called Auxiliary Fundamental Units All other units which can be expressed in terms fundamental units with the help of physical quantities are called Derived Units. Every derived unit originates from some physical law or equation which defines that unit. The volume V of a room is equal to the product of its length (l), width (b), and height (h) therefore V=lbh If meter is chosen as the the unit of length, then the volume of a room 6m X 4m X 5m is 120 m3. The number of measures (6 X 4 X 5 =120) as well as units (m x m x m = m3) are multiplied. The derived unit for volume is thus m3. Some fundamental units S. No Name Unit Symbol 1 Length Metre M 2 Mass Kilogram kg 3 Time Second sec 4 Electric Current Ampere A 5 Temperature Kelvin K 6 Luminous Intensity Candler Cd Supplementary Units S. No Name Unit Symbol 1 Plane angle radian rad 2 Solid angle steradian sr Derived Units S. No Name Unit 1 Area m2 2 Volume m3 3 Density kg/m3 4 Angular velocity rad/sec 5 Angular acceleration rad/sec2 6 Pressure, Stress kg/m2 7 Energy Joule(Nm) 8 Charge Coulomb 9 Electric Field Strength V/m 10 Capacitance (ASec/V) 11 Frequency Hz 12 Velocity m/sec 13 Acceleration m/sec2 14 Force Kg-m(N) 15 Power Watt (J/sec) 16 EMF Volt (W/A) Dimensions Disregarding the problem of measurement and the concept of magnitude, it is evident that every quantity has a quality which distinguishes it from all other quantities. This unique quality is called Dimension. The dimension is written in a characteristics notation, example [L] for length, [T] for time and so on. A derived unit is always recognized by its Dimensions, which can be defined as the complete algebraic formula for the derived unit. Thus when quantity such as area A of a rectangle is measured in terms of other quantities i.e. length l, and width b, in this case, the relationship is expressed mathematically as; Area A = l x b Since l and b each have the dimensions of a length, [L], the dimensions of area is [A] = [L] x [L] = [L2] If meter (m) is unit of length, then meter square (m2) can be used as unit of area. In mechanics has the three fundamental units are length, mass and time. The dimensional symbols are length [L], Mass [M], time [T] Conversions A Few Standard Conversions 1 ft = 30.48 cm= 12 inches 1 m = 3.28 ft 1 kg = 2.2 pounds 1 hp = 746 W Standards and their Classification A standard is a physical representation of a unit of measurement. The term standard is applied to a piece of equipment having a known measure of physical quantity. They are used for the purpose of obtaining the values of the physical properties of other equipment of by comparison methods The classification of standards is based on the function and the application of the standards. (a) International Standards International standards are defined on the basis of international agreement. They represent the units of measurements which are closest to the possible accuracy attainable with present day technological and scientific methods. International standards are checked and evaluated regularly against absolute measurement in terms of the fundamental units. The international standards are maintained at the International Bureau of Weights and Measures and are not available to the ordinary user of measuring instruments for the purposes of calibration or comparison (b) Primary Standards. Primary standards are absolute standards of such high accuracy that they can be used as the ultimate reference standards. These standards are maintained by national standards laboratories in different part of the world. The primary standards, which represent the fundamental units and some of the derived electrical and mechanical units, are independently calibrated by absolute measurements at each of the national laboratories. (c) Secondary Standards The secondary standards are the basic reference standards used in industrial measurement laboratories. The responsibility of maintenance and calibration of these standards lies with the particular industry involved. These standards are checked locally against reference standards available in the area. Secondary standards are normally sent periodically to the national standards laboratories for calibration and comparison against primary standards. (d) Working Standards The working standards are the major tools of a measurement laboratory. These standards are used to check and calibrate general laboratory instruments for their accuracy and performance. For example, a manufacturer of precision resistances may use a Standard Resistance (which may be working standard) in the quality control department for checking the values of resistors that are being manufactured. This way, he verifies that his measurement set up perform within the limits of accuracy that are specified. Measurements and Measuring Systems Measurements The measurement of a given quantity is essentially an act or the result of comparison between the quantity and a predefined standard. Since two quantities are compared the result is expressed in numerical values Methods of Measurement The methods of measurement may be broadly classified into two categories. (i) Direct methods (ii) Indirect methods (i) Direct Method In these methods, the unknown quantity (also called the measurand) is directly compared against a standard. The result is expressed as a numerical number and a unit. Direct methods are quite common for the measurement of physical quantities like length, mass and time (ii) Indirect methods Measurements by direct methods are not always possible, feasible and practicable. These methods in most of the cases, are inaccurate because they involve human factors. They are also less sensitive. Hence direct methods are not preferred and are rarely used. In engineering applications Measurement systems are used. measurement systems use indirect methods for measurement purposes. These A measurement system consists of a transducing element which converts the quantity to be measured in an analogous form. The analogous system is then processed by some intermediate means and is then fed to the end devices which present the result of the measurement. Instrument An instrument may be defined as a device for determining the value or magnitude of a quantity or variable. Basic Types of Measuring instruments are (i) Mechanical measuring instruments (ii) Electrical measuring instruments (iii) Electronic measuring instruments Classification of Instruments There are many way in which instruments can be classified. Broadly, instruments are classified into two categories. (i) Absolute Instruments (ii) Secondary Instruments (i) Absolute Instruments These instruments give the magnitude of the quantity under measurements in terms of physical constants of the instrument. The examples of this class of instruments are Tangent Galvanometer and Rayleigh’s current balance (ii) Secondary Instruments These instruments are so constructed that the quantity being measured can only be measured by observing the output indicated by the instrument. These instruments are calibrated by comparison with an absolute instrument or another secondary instrument which has already been calibrated against an absolute instrument. A voltmeter, a glass thermometer, and a pressure gauge are typical examples of secondary instruments. Functions of Instruments and Measurement Systems Instruments may be classified based on their function. Three main functions are (i) Indicating Function (ii) Recording Function (iii) Controlling Function (i) Indicating Function These instruments provide information regarding the variable quantity under measurement and most of the time this information are provided by the deflection of the pointer. This kind of function is known as the indicating function of the instruments. (ii) Recording Function These instruments usually use the paper in order to record the output. This type of function is known as the recording function of the instruments. (iii) Controlling Function This is function is widely used in industrial world. In this these instruments controls the processes. Characteristics of Instruments and Measurement Systems The performance characteristics of an instrument are mainly divided into two categories: 1. Static characteristics 2. Dynamic characteristics Set of criteria defined for the measurements, which are used to measure the quantities, which are slowly varying with time or almost constant, i.e., do not vary with time, are called Static Characteristics. While when the quantity under measurement changes rapidly with time, the relation existing between input and output are generally expressed with the help of differential equations and are called “Dynamic Characteristics”. Static Characteristics Desirable Undesirable Accuracy/ Precision Drift/ Hysteresis Resolution/ Threshold Dead Zone Sensitivity Static Error Repeatability Dynamic Characteristics Desirable Undesirable Speed of Response Lag Fidelity Dynamic Error Calibration The various performance characteristics are obtained in one form or another by a process called “Calibration”. It is the process of making an adjustment or marking a scale so that the readings of an instrument agree with the accepted and the certified standard. Some important definitions are Static Error: It is the difference between the measured value and true value of the quantity Mathematically δA = Am − At ----------- eq (1.1) δA: Absolute error or Static error Where, Am: Measured value of the quantity At : True value of the quantity . Static Correction: It is the difference b/w the true value & measured value of the quantity mathematically δC=(−δA)=(At−Am) Limiting error or Relative error: (εr) = δA/At εr=(Am − At)/At Percentage relative error: % εr = (δA/At) × 100 From relative percentage error, accuracy is expressed as A = 1 − |εr| Where A: relative accuracy And a = A × 100% Where a = Percentage accuracy Error can also be expressed as percentage of Full Scale Deflection (FSD) as, Am At X 100 F .S .D Example: The expected value of voltage to be measured is 150 V. However, the measurement gives a value of 149 V. Calculate (i) Absolute error (ii) Percentage error, (iii) Relative accuracy (iv) Percentage accuracy (v) Error expressed as percentage of full scale reading if scale range is 0 – 200 V. Solution: Expected value implies true value At= 150 V Am = 149 V (i) Absolute error = Am − At = −1 V (ii) % εr = (Am − At )/At ×100= 1/150 ×100=− 0.66% (iii) A = 1 − |εr| = 1 − |−1/150|= 0.9933 (iv) % a = A × 100 = 99.33% (v) F.S.D= [(Am − At )/F.S.D] × 100 = −1/200 ×100 = − 0.5 % Example: A Voltage has a true value of 1.50 V. An analog indicating instrument with a scale range of 0 – 2.50 V shows as a voltage of 1.46 V. What are the values of absolute error and correction. Express the error as a fraction of the true value and the full scale deflection (f.s.d.). : Absolute error δA = Am − At = 1.46 – 1.50 = − 0.04 V Absolute correction δC = − δA = + 0.04 V Relative error,εr= δA/At= (− 0.04 /1.50) ×100= −2.67 % Relative error (expressed as a percentage of F.S.D.) = (− 0.04 / 2.5) ×100 =−1.60 % Where F.S.D. is the Full Scale Deflection. Example: A meter reads 127.50 V and the true value of the voltage is 127.43 V Determine (a) The static error, (b) The static correction for this instrument Solution: From Eqn. 1.1, the error is δA = Am − At= 127.50 – 127.43 = + 0.07 V Static Correction δC = − δA = − 0.07 V Example: A thermometer reads 95.45°C and the static correction given in the correction curve is –0.08°C. Determine the true value of the temperature. Solution: True value of the temperature At= Am + δC = 95.45 – 0.08 = 95.37°C 3. Accuracy: It is the degree of closeness with which the instrument reading approaches the true value of the quantity. Accuracy is expressed in the following ways: Accuracy as “Percentage of Full Scale Reading. In case of instruments having uniform scale, the accuracy can be expressed as percentage of full scale reading. Example: The accuracy of an instrument having full scale reading of 50 units is expressed as ±0.1% of full scale reading. Note: This form of notation indicates the accuracy is expressed in terms of limits of error. So for the accuracy limits specified above, there will be ±0.05 units of error in any measurement. So for a reading of 50 units, there will be a error of ±0.05 units i.e., ±0.1% while for a reading of 25 units, there will be a error of ±0.05 units and i.e., ±0.2%. Thus as reading decreases, error in measurement is ±0.05 units but net percentage error is more. Hence specification of accuracy in this manner is highly misleading. Accuracy as “Percentage of True Value”. This is the best method of specifying the accuracy. Here it is specified in terms of true value of quantity being measured. Example: Accuracy can be specified as ±0.1% of true value. This indicates that as readings gets smaller, error also gets reduced. Accuracy as “Percentage of Scale Span”: For an instrument with a max , amin representing full scale and lowest reading on scale, then (amax−amin) is called span of the instrument (or) scale span. Accuracy of an instrument can be specified as percent of such scale span. Example: For an instrument having scale span from 25 to 225 units, then accuracy can be specified as ±0.2% of scale span i.e., ± [(225−25)× 0.2/100] which is ±0.4 units of error in every measurement. Point Accuracy: Here accuracy is specified at only one particular point of scale. It does not give any information about accuracy at any other point on scale. Example: A wattmeter having a range 1000 W has an error of ± 1% of full scale deflection. If the true power is 100 W, what would be the range of readings? Suppose the error is specified as percentage of true value, what would be the range of the readings? Solution: When the error is specified as a percentage of full scale deflection, the magnitude of limiting error at full scale = ± 1/100 ×1000= ± 10 W Thus the Wattmeter reading when the true reading is 100 W may be 100 ± 10 W i.e., between 90 to 110 W Relative error = ± 10/100 ×100= ± 10% Now suppose the error is specified as percentage of true value. The magnitude of error = ± 1/100 ×100= ± 1 W Therefore the meter may read 100 ± 1 W or between 99 to 101 W Accuracy can also be defined in terms of static error. 4. Precision:It is the measure of degree of agreement within a group of measurements. High degree of precision does not guarantee accuracy. Precision is composed of two characteristics (i). Conformity (ii). Number of significant figures (i). Conformity Consider a resistor having value of 2385692 which is being measured by ohmmeter as 2.4M Ω consistently, due to non-availability of proper scale. The error created due to limitation of scale is called precision error. (ii). Significant Figures Precision of the measurement is obtained from the number of significant figures, in which the reading is expressed. Significant figures convey the actual information about the magnitude and measurement precision of the quantity. Example: A resistance of 110 Ω, is specified may be closer to 109 Ω, and 111 Ω. Thus, there is 3 significant figures while if it is specified as 110. 0 Ω, then it may be closer to 110.1 Ω or 109.9 Ω. Thus, there are now 4 significant figures. Thus, more the significant figures the greater is the precision of measurement. Normally, large numbers with zeros are expressed in terms of powers of ten. Example: Approximate population of a city is reported as 4,90,000 which actually is to be read as the population lies between 4,80,000 to 5,00,000 but due to misconception it can also be implied as population lies between 489,999 to 490,001. So it is expressed as 49× 104 or 4.9 × 105, which is 2 digit significant figure. 5. Sensitivity: The sensitivity denotes the smallest change in the measured variable to which the instrument responds. It is defined as the ratio of the changes in the output of an instrument to a change in the value of the quantity to be measured. Mathematically it is expressed as, Sensitivity Infinitest imal Change in output Infinitest imal Change in input Sensitivity q o qi ∆qo Output qo ∆qi Input qi 6. Hysteresis Many times, for the increasing values of input an instrument may indicate one set of output values, and for the decreasing values of input, the same instrument may indicate it different set of output values. When these output values are plotted against the input the following types of graphs are achieved. For the increasing and the decreasing inputs, the output shows a maximum variations at half of full scale, for this reason hysteresis error is specified at 50% of the full scale. 7. Threshold The smallest change in the input that gives a perceivable change on the output of an instrument is called the resolution. In most of the instruments, when the input is increased from zero value there is a small dead band or dead zone for which no perceivable output is indicated by the instrument. Thus the smallest input that gives some perceivable output is the threshold of the instrument. Then we can say the resolution is the smallest change in input that can be measured and threshold is the smallest input that can be measured. Then needless to say that the resolution has a meaning only after the threshold input has been passed. 8. Repeatability Repeatability is a measure of closeness with which a given input may be measured over and over again. Measurement of Errors The measurement error is defined as the difference between the true or actual value and measured value The true value is the average of the finite number of measurement Measured value is the precise value Types of Errors in Measurement The error may arise from the different source and are usually classified into the following types. 1. Gross Error 2. Systematic Error 3. Random Error 1. Gross Errors Gross errors may occur because of the human mistakes. For example consider the person using the instruments takes the wrong reading, or they can record the incorrect data. Such type of error comes under the gross error. The gross error can only be avoided by taking the reading carefully Two methods can remove the gross error. These methods are The reading should be taken carefully. Two or more reading should be taken of the measurement quantity. The readings are taken by the different experimenter and at a different point removing the error. These type of errors include the loading effect and the misuse of the instruments. 2. Systematic Error The systematic errors are mainly classified into three categories (i) Instrumental Errors (ii) Environmental Errors (iii) Observational Errors (i)Instrumental Errors These errors may be due to wrong construction, wrong calibration of measuring instruments. These types of error may arise due to friction or may be due to hysteresis. (ii) Environmental Errors These errors are due to the external condition of the measuring devices. External condition includes temperature, pressure, humidity or it may include magnetic field (iii) Observational Errors Such types of errors are due to the wrong observation of the reading. 3. Random Errors The error which is caused by the sudden change in the atmospheric condition, such as type of error is called random error.