Scheme & Solutions of 14EI 505 (OCT-2018)

Transcription

1 Scheme & Solutions of 14EI 505 (OCT-2018) Prepared By: P. Vinodh Babu Associate Professor Department of EIE Bapatla Engineering College Bapatla Phone numbers: ,

2 Hall Ticket Number: 14EI505 November, 2018 Fifth Semester Time: Three Hours Answer Question No.1 compulsorily. III/IV B.Tech (Regular/Supplementary) DEGREE EXAMINATION Electronics and Instrumentation Engineering Transducers Maximum : 60 Marks (1X12 = 12 Marks) Answer ONE question from each unit. (4X12=48 Marks) 1. Answer all questions (1X12=12 Marks) a) Distinguish between static and dynamic characteristics of an instrument. b) What is the difference between transducer and sensor? c) How sensitivity of an instrument is related to the linearity of the system? d) Define magnetostrictive effect. e) List the applications of LVDT. f) Give two examples of active transducer. g) What is the difference between PTC and NTC type thermistors? h) What is piezo electric effect? i) What is the need of cold junction compensation in thermocouple? j) What is the principle of Electro dynamic acceleration transducer? k) What is Piezo resistive effect? l) Define permeability. UNIT I 2. a) Draw the block diagram of generalized instrument system and explain it in detail. b) Briefly discuss various static characteristics of instrumentation system. (OR) 3. a) Three resistors have the following values: R %, R % and R3 50 5%. Determine the magnitude of the resultant resistances and limiting errors, if they are connected in series and parallel. b) Write short notes on the following errors (i) Gross errors (ii) Systematic errors (iii) Environmental errors UNIT II 4. a) Describe the constructional details of a resistive potential divider and derive the expression for its output voltage when connected across a meter of finite impedance. b) Explain NTC and PTC type thermister with a neat sketch. How the characteristics can be linearized. (OR) 5. a) Explain reluctance type inductive transducer with a neat sketch. b) Classify the transducers. Briefly discuss each of them with examples. UNIT III 6. a) Discuss the working of electrostatic pressure transducer. b) Explain piezoelectric transducer with example. (OR)

3 14EI a) Explain the principle of electro dynamic acceleration transducer. b) Piezoelectric crystal of 1cm 2 area, 0.1 cm thick has been subjected to a force. Two metal electrodes measure the changes in the crystal. Young s Modulus of the material = 9 x Pa. Charge sensitivity 2p C/N, relative permittivity is 5; the applied force is 0.01N. Find the voltage across the electrodes and change in crystal thickness. UNIT IV 8. a) With a neat sketch explain the basic principle of operation of bimetallic thermometer. b) Explain the basic principle of operation of liquid in glass thermometer. (OR) 9. a) Explain the working principle of total radiation pyrometer 5M b) What is the basic principle of selective wavelength optical pyrometer? Explain about the selective radiation pyrometer with a neat sketch. 7M

4 Scheme of 14EI 505 (OCT-2018) Q.No:1 (a) to (l) carries 12 marks, one mark for each question Q.No:2 (a) Block Diagram - 2M, Explanation - 4M (b) Any six static characteristics of the instrument - Q.No:3 (a) Limiting error calculation for series combination-3m, Limiting error calculation for parallel combination-3m (b) Gross errors-2m, Systematic errors-2m, Environmental errors-2m Q.No:4 (a) Construction of Resistive potentiometer-2m, Derivation of output voltage-4m (b) NTC thermistor-2m, PTC thermistor-2m, Characteristics Linearization -2M Q.No:5 (a) Reluctance type inductive transducer diagram-2m, Explanation-4M (b) Classification transducers with examples- Q.No:6 (a) Electro static pressure transducer-2m, Explanation -4M (b) Diagram-2M, Working of Piezo-electric transducer-3m Q.No:7 (a) Electro dynamic acceleration transducer-2m, Explanation -4M (b) Voltage calculation-3m, Change in thickness calculation-3m Q.No:8 (a) Diagram of Bi-metallic sensors-2m, Explanation-4M (b) Diagram of Liquid-in-glass thermometer -2M, Explanation-4M Q.No:9 (a) Diagram of total radiation pyrometer-2m, working-4m (b) Principle-1M, Diagram-2M, working-4m

5 Solutions-October-2018 Q.No: 1 Answer all questions Each question carries one mark 1x12=12M (a) Difference between Static and dynamic characteristics of an instrument: Static characteristics play important role when the desired input to the instrument may be constant or varying slowly with respect to time Dynamic characteristics play important role when the desired input to the instrument varying with respect to time (b) Difference between transducer and sensor: The main difference between sensor and transducer is that a transducer is a device that can convert energy from one form to another, whereas a sensor is a device that can detect a physical quantity and convert the data into an electrical signal. Sensors are also a type of transducers. (c) Sensitivity relationship with linearity: The sensitivity is represented by the slope of the input-output curve if the ordinates represented in actual units. With a linear calibration curve, the sensitivity is a constant one. However, if the relationship between the input and output is not linear, the sensitivity varies with the input. (d) Magnetostrictive effect: Magnetostriction is a property of ferromagnetic materials that causes them to change their shape when subjected to a magnetic field. (e) List the applications of LVDT: Used for the measurement of linear displacement primarily Secondarily these are used for the measurement of: Force Pressure Vibration Acceleration etc. (f) Examples of active transducer: Piezo-electric transducer Thermo electric transducer (Thermo couple) (g) Difference between PTC and NTC type thermistors: The NTC thermistors are ceramic semiconductors that have a high Negative Temperature Coefficient of resistance. NTC thermistors decrease in resistance as the temperature increases. PTC thermistors are Positive Temperature Coefficient resistors generally made of polycrystalline ceramic materials that have a high positive temperature coefficient, PTC thermistor increases in resistance as the temperature increases.

6 (h) Piezo electric effect: Certain materials, especially the crystalline ones, produce an e.m.f. when deformed by an application of pressure along the specific axes. This phenomenon is known as piezoelectric effect (i) Need of cold junction compensation in thermocouple: Also called reference junction compensation. When measuring temperature using thermocouples, the reference terminal may not be held at 0 C, but at the surrounding temperature of T 1 C instead. Without any compensation, the thermocouple output will be reduced by T 1 C. (j) Principle of Electro dynamic acceleration transducer: Force balance principle is the basis for electro dynamic acceleration transducer. (k) Piezo resistive effect The piezoresistive effect is a change in the electrical resistivity of a semiconductor or metal when mechanical strain is applied. In contrast to the piezoelectric effect, the piezoresistive effect causes a change only in electrical resistance, not in electric potential. (l) Permeability: Magnetic permeability (μ) is the ability of a magnetic material to support magnetic field development. In other words, magnetic permeability is the constant in the proportionality between magnetic induction and magnetic field intensity.

7 UNIT-I Q.No:2. (a) Draw the block diagram of generalized instrument system and explain it in detail. A generalized measurement system consists of the following: (i) Basic functional elements (ii) Auxiliary elements. Basic functional elements are those that form the integral parts of all instruments. They are (i) Transducer element (ii) Signal conditioning element (iii) Data presentation element. Auxiliary functional elements are those which may be incorporated in a particular system depending on the type of requirement, the nature of measurement technique, etc. They are (i) Calibration element (ii) External power element (iii) Feedback element (iv) Microprocessor element Transducer element: This element senses and converts the desired input to a more convenient and practicable form to be handled by the measurement system Signal Conditioning element: This element is for manipulating/processing the output of the transducer in a suitable form Data presentation element: This element is for giving the information about the measurand or measured variable in the quantitative form Calibration element: This element is to provide a built-in calibration facility.

8 External power element: This element is to facilitate the working of one or more of the elements like the transducer element, the signal conditioning element, the data processing element or the feedback element. Feedback element: This element is to control the variation of the physical quantity that is being measured. Microprocessor element: This element is to facilitate the manipulation of data for the purpose of simplifying or accelerating the data interpretation. (b) Briefly discuss various static characteristics of instrumentation system The static performance characteristics accuracy, precision, resolution, threshold, sensitivity, linearity, hysteresis, dead zone, drift, overload capacity, loading effect...etc are usually good enough to give meaningful quantitative descriptions of the instrument. In these characteristics the desirable characteristics are accuracy, precision, resolution, sensitivity and linearity. Threshold, Hysteresis, dead zone, drift, loading effect are undesirable characteristics. Accuracy: Accuracy of measuring system is defined as the closeness of instrument output to the true value of the measured quantity. Accuracy of the instrument mainly depends on the inherent limitations of the instrument as well as on the short comings in the measurement process. Precision: Precision is defined as the ability of the instrument to reproduce a certain set of readings within a given accuracy. The degree of repeatability of the instrument is termed as precision. Precision is used in measurements to describe the consistency or the reproducibility of results. A precise measurement may not necessarily be accurate and vice versa. An indication of the precision of the measurement is obtained from the number of significant digits. Significant digits convey actual information regarding the magnitude and the measurement precision of a quantity. Resolution: It is defined as the smallest increment in the measured value that can be detected with certainty by the instrument. A high resolution instrument is one that can detect smallest possible variation in the input. Threshold: It is defined as the minimum value of input below which no output can be detected. It is a particular case of resolution. Both threshold and resolution are not zero because of various factors such as (i) Friction between moving parts (ii) Inertia of the moving parts.

9 Sensitivity: It is defined as the ratio of magnitude of response to the magnitude of the quantity being measured. In, other words the sensitivity is represented by the slope of the input-output curve if the ordinates represented in actual units. With a linear calibration curve, the sensitivity is a constant one. However, if the relationship between the input and output is not linear, the sensitivity varies with the input. Linearity: It is defined as the ability to reproduce the output symmetrically, and this can be expressed by the straight line equation y mx c. A linear indicating scale is one of the most desirable features of any instrument. Therefore, manufacturers of the instruments always attempt to design their instruments so that output is linear function of the input. However linearity is never completely achieved and the deviations from the ideal are termed as linearity tolerances.

10 Hysteresis: It is defined as the magnitude of error caused in the output for a given value of input, when this value is approached from opposite directions i.e. from ascending order and then descending order. This occurs due to various factors such as backlash, elastic deformations, magnetic characteristics and frictional effects. Hysteresis effects are best eliminated by taking the observations both for ascending and descending values of input and then taking the arithmetic mean. Dead band: It is defined as the largest change of the measurand to which the instrument does not respond. Drift: It is defined as the variation of the output for a given input caused due to change in the sensitivity of the instrument to certain interfering inputs like temperature changes, component instabilities...etc. Loading effect: The incapability of the system to faithfully measure, record, or control the input signal in undistorted form is called loading effect. This occurs due to extraction of energy from the mesurand. (OR) Q.No:3. (a) Three resistors have the following values: R %, R % and R %. Determine the magnitude of the resultant resistances and limiting errors, if they are connected in series and parallel.

11 Solution: The magnitude of resultant resistance when connected in series: R S R R R3 The magnitude of resultant resistance when connected in parallel: R P R1R2 R3 28. R R R R R R Percentage limiting error in R S 10% 5% 5% % 7.86% Percentage limiting error in R P 10% 5% 5% 15% 10% 15% % (b) Write short notes on the following errors (i) Gross errors (ii) Systematic errors (iii) Environmental errors Q.No:4. Gross errors: These are largely human errors. This type of errors occurs due to: Misreading of instruments In correct adjustment Improper application of instruments Computational mistakes These errors can be avoided only by taking care in reading and recording of the measurement data Systematic errors: These errors are due to poor design/construction of the instrument. The other errors are calibration errors. These errors can be avoided by: Selecting a suitable instrument for the particular measurement application Applying correction factors after determining the instrumental error. Calibrating the instrument against a standard. Environmental errors: Environmental errors are due to conditions external to the measuring device, including the conditions in the area surrounding the instrument, such as the effects of: Change in temperature Change in humidity, Barometric pressure Electric and magnetic fields These errors can be avoided by: Air-conditioning Hermetically sealing certain components in the instruments. Using proper magnetic shields. UNIT-II (a) Describe the constructional details of a resistive potential divider and derive the expression for its output voltage when connected across a meter of finite impedance. Generally potentiometers are used for the purpose of voltage division. The resistive potentiometer consists of resistive element with a sliding contact called wiper. The motion of the sliding contact may be translational (or) rotational. But some potentiometers use both translational and rotational motions.

12 Normally a single wire is used as a resistive element. The variation of resistance is step less as the wiper moves over it. Such potentiometers are available but length and diameter restricts their use because of space considerations. To get the higher values of resistance using a small wire, the wire is wounded on to the former (linear or circular). But here the variation of the resistance is in the form of steps as shown in figures below

13 Derivation of expression for output voltage when connected across a meter of finite impedance: R out x i Rp Rm x t x i Rp R x t m krprm kr R p m R out krprm kr R p m Total resistance of the circuit is given by the expression R ( R p krprm krp) kr R p m p 2 p k(1 k) R RmR R kr R m p Current through the potentiometer is I E E ( kr i i p I 2 R k(1 k) Rp R ) m R m R p Output voltage = E o E kr i p IRout 2 k(1 k) Rp R m R p R m (b) Explain NTC and PTC type thermisters with a neat sketch. How the characteristics can be linearized Thermistors: Thermistors are temperature sensitive resistors that have a greater than normal change in resistance value when the temperature changes. The change in resistance is predictable with changes in temperature. The extreme sensitivity to temperature change enables a thermistor to perform many functions and is utilized in an increasing variety of thermal sensing/control applications. The two basic types of thermistors generally available today are the NTC and the PTC types. NTC (Negative Temperature Coefficient) The NTC thermistors are ceramic semiconductors that have a high Negative Temperature Coefficient of resistance. NTC thermistors decrease in resistance as the temperature increases. NTC thermistors are composed of sintered metal oxides such as manganese, nickel, cobalt, iron, copper and aluminum.

14 The exact shape of the finished product such as disc or chip is dictated by the specific applications. The most prevalent configurations are discs and chips both leaded and surface mount varieties. 1 1 The resistance RT of a thermister at temperature T can be written as R T Ro, where T o T To is the reference point and is generally 298K (25 o C) at which temperature the constant β is nearly 4000K PTC (Positive Temperature Coefficient) PTC thermistors are Positive Temperature Coefficient resistors generally made of polycrystalline ceramic materials that have a high positive temperature coefficient, which increases in resistance as the temperature increases. The PTC thermistor is generally a polycrystalline ceramic material composed of oxalate or carbonate with added dopant materials. The PTC thermistor exhibits only a slight change of resistance with temperature until the switching point is reached at which point an increase of several orders of magnitude in resistance occurs. The following figure shows different form of thermisters Resistance temperature Characteristics of NTC and PTC Thermistors:

15 Linearization technique: Two simple techniques exist for linearization: Resistance mode and Voltage mode. In resistance-mode linearization, a normal resistor sits in parallel with the NTC thermistor and linearizes the combined circuit's resistance. If you choose a resistor value that's equal to the thermistor's resistance at room temperature (R 25C), then the region of relatively linear resistance will be symmetrical around room temperature. In voltage-mode linearization, the NTC thermistor connects in series with a normal resistor to form a voltagedivider circuit. A regulated supply or a voltage reference, V REF, biases the divider circuit to produce an output voltage that is linear over temperature. If you choose a resistor value that equals the thermistor's resistance at room temperature (R 25C), then the region of linear voltage will be symmetrical around room temperature

16 (OR) Q.No:5. (a) Explain reluctance type inductive transducer with a neat sketch. The transducer works based on the variation of reluctance. The total reluctance of the circuit is given by the expression R R g Ri where Rg is the reluctance of air gap and R i the reluctance of iron piece. Generally permeability of iron is much greater than air gap permeability, so R i negligible. Therefore the above expression becomes R R g g g R Rg 1 1 L, with the displacement air gap length changes that indicates change in self inductance R l of the coil (L) and hence reluctance. This change gives a measure displacement when calibrated (b) Classify the transducers. Briefly discuss each of them with examples. Transducers can be electrical, mechanical, optical, acoustic, chemical (or) any of their combination. Unless until specified transducer means an electric transducer. These are classified into: Active transducers Passive transducers Active transducers: Active transducers convert input physical quantity into output electrical quantity without external power. Piezo-electric and thermocouple transducers are some of the examples for active transducers. Based on the principle of operation these are classified into: (i) Thermo electric (ii) Piezo-electric (iii) Photovoltaic (iv) Galvanic.etc. Passive transducers: Passive transducers convert the input physical quantity into output electrical quantity with the use of external power. Resistive potentiometers and strain gauges are some of the examples for passive transducers. Based on the principle of operation these are classified into: (i) Resistive (ii) Inductive (iii) Capacitive (iv) Thermo resistive (v) Electro resistive...etc.

17 Analog Transducers: Both active and passive transducers are classified into analog type based on the output of the transducer. As the name indicates the output is analog in nature for analog transducers. Digital transducers: Both active and passive transducers are classified into digital type based on the output of the transducer. As the name indicates the output is digital in nature for digital transducers. Based on the form of the digital output these are classified into: (i) Direct type (ii) Indirect type (iii) Quasi digital type (iv) Analog transducer with ADC In the direct type the basic parameter is not sensed as displacement or any other analog phenomenon. In straight it gives instantaneous discrete coded output. Example: proximity digital sensor In the indirect type the basic parameters are sensed as displacement or any other analog phenomenon, later the signal is converted into digital format. Example: Shaft encoders. In the Quasi type the output is in the form of pulses. The frequency of pulses is calibrated in terms of measured quantity. Examples: Vibration string transducer, Turbine flow meter. Analog transducer with ADC fabricated on a single chip acts as a digital transducer. Q.No:6. UNIT-III (a) Discuss the working of electrostatic pressure transducer. The functional components of a typical electro static force-balance pressure transducer are shown in figure below. Two plates of area a are fixed at a distance of 2d apart. A third plate, or diaphragm, of negligible thickness is initially positioned in the centre. For simplicity let the diaphragm be displaced in a piston-like manner parallel to itself by a distance ±x in response to a pressure difference ±P = P 1-P 2 across it. According to the figure below, the two outer plates are fixed potentials +V O and V O, respectively, relative to the diaphragm, which is initially at zero potential.

18 The diaphragm displacement is sensed by connecting the two condenser halves to a capacitance bridge, and the bridge output, after amplification and rectification, is fed to the diaphragm. Vo The diaphragm potential v is thus made to interact with the electrostatic field E between the outer plates d in such away as to oppose the diaphragm displacement by electrostatic forces. (b) Explain piezoelectric transducer with example. Piezo electric effect: Certain materials, especially the crystalline ones, produce an emf when deformed by an application of pressure along the specific axes. This phenomenon is known as piezoelectric effect and is widely used for measurement of dynamic pressure. A necessary, but not sufficient, condition of a crystal to be piezoelectric is a charge asymmetry within the crystal. A piezoelectric crystal is represented by a set of three Cartesian coordinates so that the polarization P can be represented in the vector form as P P P P xx yy zz However, P xx, Pyy, Pzz are again related to the stresses, axial and shear,, and in terms of a set of axesdependent coefficients called d-constants of the crystal. With the axial and shear axes as shown in figure below. Piezo electric Load cell (Force Measurement) The piezoelectric effect is direction sensitive which means tensile and compressive forces produce voltages of opposite polarity. The magnitude of the generated surface charge Q is proportional to the applied force F and, therefore Q df, where the proportional constant d is called charge sensitivity constant of the crystal. o A The capacitance of the crystal of thickness t and plate area A can be written as C r Q df dft d Therefore, the resulting voltagevo C r o A r o A r o t voltage sensitivity of the crystal. P is the applied pressure. F t Vo A t gpt. The g is called the

19 (OR) Q.No:7. (a) Explain the principle of electro dynamic acceleration transducer. It works based on the force-balance principle. The following figure shows a typical electro dynamic force balance transducer for the measurement of linear acceleration. The transducer housing contains a seismic mass m which is free to move in the axial direction. Acceleration applied along this line causes a relative displacement of the mass which is sensed by a push-pull variable capacitance detector with a sensitivity K S (v/m). Its output voltage is fed to the servo amplifier of gain K a (A/v), comprising in its simplest form a phase-advancing network for velocity compensation with a transfer function (1+αp), where p is the Laplace operator, and providing feedback damping. The output current i, in turn is fed to the electro dynamic servo actuator with force factor K f (N/A), thus closing the loop by generating the force necessary to balance the input inertia force caused by acceleration. Mathematical Analysis: In an ideal electro dynamic force-balance transducer spring forces can be ignored in comparison with feedback forces, and damping forces are irrelevant at static loading. The remaining restoring force provided by the electro dynamic servo-actuator is given by the expression: F Bil...(1) Where B represents the magnetic flux density in the (annular) air gap of the magnetic assembly and l the active wire length of the moving coil coupled with magnetic field. Equilibrium between the restoring force of equation-(1) and the inertia force derived from the acceleration applied to the seismic mass occurs for, Bil ma and sensitivity becomes From the above equation the output current is linear function of input acceleration, as long as m, B, and l remain constant over the operational displacement range. S i a m Bl m K f (b) Piezoelectric crystal of 1cm 2 area, 0.1 cm thick has been subjected to a force. Two metal electrodes measure the changes in the crystal. Young s Modulus of the material = 9 x Pa. Charge sensitivity 2p C/N, relative permittivity is 5; the applied force is 0.01N. Find the voltage across the electrodes and change in crystal thickness. Solution: Given data: Area (A) =1cm 2 = 10-4 m 2 Thickness (t) = 0.1cm =10-3 m Young s Modulus of material (Y) =9x10 10 Pa

20 Charge sensitivity of Crystal (d) = 2pC/N Relative permittivity ( r ) = 5 Applied force (F) = 0.01N Absolute permittivity = 8.854x10-12 F/m 12 d F *10 Vo gpt t *10 12 o r A 8.854*10 *5 F Strain = Change in thickness = t = AY Q.No: v * *9 *10 UNIT-IV (a) With a neat sketch explain the basic principle of operation of bimetallic thermometer. It works based on the thermal expansion of solids. i.e. the dimensions of solids change with temperature. Bimetals are formed by firmly bonding two strips of metals A and B having different thermal expansion coefficients α A and α B. If this bimetal is a straight line at temperature T 1, then at an elevated temperature T 2 the strip will form a uniform circular arc of radius of curvature ρ such that 2t T2 T1, where t is the thickness of the strip. 3 A B Different forms-such as cantilever, U-shape, spiral type bimetals are used (b) Explain the basic principle of operation of liquid in glass thermometer. It works based on the thermal expansion of liquids. i.e. the dimensions of liquids change with temperature. The liquid-in-glass thermometer is a well known temperature measuring instrument which is used in a wide range of applications. The fluid used is usually either mercury or coloured alcohol, and this is contained within a bulb and capillary tube, as shown in the below figure. As the temperature rises, the fluid expands along the capillary tube and meniscus level is read against a calibrated scale etched on the tube. The process of estimating the position of the curved meniscus of the fluid against the scale introduces some error into the measurement process and measurement accuracy better than ±1% of full-scale reading is hard to achieve.

21 However, an accuracy of ±0.15% can be obtained in the best industrial instruments. Industrial versions of the liquid-in-glass thermometer are normally used to measure temperature in the range between -200 o C and o C (OR) Q.No:9. (a) Explain the working principle of total radiation pyrometer. 5M It works based on Stefan-Boltzmann law. This figure shows essential elements of Fery s total radiation pyrometer which, incidentally, is the first such instrument ever designed. Radiation from the target falls on the concave mirror, which can be moved back and forth to focus radiation on the surface of the radiation receiver. The hot junction of a thermocouple is attached to the radiation receiver, a shielding element protecting the thermocouple junction from receiving direct radiation. The developed emf is read on a milli voltmeter which may be calibrated to a temperature scale. Because of the fourth power law, such calibration are evidently nonlinear which renders the device unsuitable for measurement below 650 o C because of poor sensitivity. The presence of absorbing media, such as smoke, dust etc, in the intervening space as well as the change in emissivity of the radiation receiver by oxidation or by any other cause may upset the calibration of the instrument. Commercial instruments usable to about 1800 o C are readily available, although theoretically there is no upper limit to the temperature that can be measured in this way.

22 (b) What is the basic principle of selective wavelength optical pyrometer? Explain about the selective radiation pyrometer with a neat sketch. 7M It works based on Plank s formula. This utilizes the photometric principle of comparison of the intensity of incoming radiation at a particular wave band to that of a lamp. Though it is the most accurate of all radiation pyrometers, it cannot be used below 650 o C since it requires a visual brightness match by a human operator. Here an image of the target is superimposed on a heated tungsten filament. The brightness of the tungsten lamp, which is very stable, has been calibrated previously so that when the current through the lamp is known, the temperature corresponding to the generated brightness of the lamp is known. A red filter allows only a narrow band of wavelength around 0.65µm to the eye of the observer who controls the lamp current until the filament disappears in the superimposed target image. Then the brightness of the target and the lamp are equal we can write: 5 5 c1 c1 Where the subscripts t and l correspond to target and lamp c 2 2 exp c 1 exp 1 Tt T l respectively and is the emissivity of the target at wavelength. For T<4000 o C the exponential terms are much greater than 1. Hence neglecting 1 from the denominators of the both sides, we get from above equation c exp and finally T l T t 1 1 ln T T c t l 2 If the target is a blackbody, 0 and hence T t=t l. Otherwise, the temperature can be calculated either from the above equation by knowing, or from the calibration curve.