Chpter. Direct Method of Interpoltion More Emples Electricl Engineering Emple hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o mesure temperture, mnufcturers provide ou with temperture vs. resistnce clibrtion curve. If ou mesure resistnce, ou cn find the temperture. A mnufcturer of thermistors mkes severl observtions with thermistor, which re given in ble. ble emperture s function of resistnce. ohm C. 9. 66. 4... 4..8 Figure esistnce vs. temperture...
.. Chpter. Determine the temperture corresponding to 74.8 ohms using the direct method of interpoltion nd first order polnomil. Solution For first order polnomil interpoltion (lso clled liner interpoltion), we choose the temperture given b, f, Figure Liner interpoltion. Since we wnt to find the temperture t 74. 8, nd we re using first order polnomil, we need to choose the two dt points tht re closest to 74. 8 tht lso brcket 74. 8 to evlute it. he two points re 9. nd 66.. hen 9.,. 66., 4. gives 9. 9.. 66. 66. 4. Writing the equtions in mtri form, we hve 9.. 66. 4. Solving the bove two equtions gives 6.97.684 Hence
Direct Method of Interpoltion More Emples: Electricl Engineering.. 6.97.684,66. 9. At 74. 8, 74.8 6.97.68474.8.89C Emple hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o mesure temperture, mnufcturers provide ou with temperture vs. resistnce clibrtion curve. If ou mesure resistnce, ou cn find the temperture. A mnufcturer of thermistors mkes severl observtions with thermistor, which re given in ble. ble emperture s function of resistnce. ohm C. 9. 66. 4... 4..8 Determine the temperture corresponding to 74.8 ohms using the direct method of interpoltion nd second order polnomil. Find the bsolute reltive pproimte error for the second order polnomil pproimtion. Solution For second order polnomil interpoltion (lso clled qudrtic interpoltion), we choose the temperture given b,, f,
..4 Chpter. Figure Qudrtic interpoltion. Since we wnt to find the temperture t 74. 8 nd we re using second order polnomil, we need to choose the three dt points tht re closest to 74. 8 tht lso brcket 74. 8 to evlute it. he three points re 9., 66. nd 4.. hen 9.,. 66., 4. 4.,. 8 gives 9. 9. 9.. 66. 66. 66. 4. 4. 4. 4.. 8 Writing the three equtions in mtri form, we hve 9. 8.47. 66. 4.4 4. 4..49.8 Solving the bove three equtions gives 8.668.967.877 Hence At 74. 8, 74.8.89C he bsolute reltive pproimte error second order polnomil is.89.89.89.4% Emple 8.668.967.877, 4. 9. 8.668.96774.8.877 74. 8 obtined between the results from the first nd hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o mesure temperture, mnufcturers provide ou with temperture vs. resistnce clibrtion curve. If ou mesure resistnce, ou cn find the temperture. A mnufcturer of thermistors mkes severl observtions with thermistor, which re given in ble.
Direct Method of Interpoltion More Emples: Electricl Engineering.. ble emperture s function of resistnce. ohm C. 9. 66. 4... 4..8 ) Determine the temperture corresponding to 74.8 ohms using the direct method of interpoltion nd third order polnomil. Find the bsolute reltive pproimte error for the third order polnomil pproimtion. b) he ctul clibrtion curve used b industr is given b ln ln ln Substituting nd ln the clibrtion curve is given b ble 4 Mnipultion for the given dt. ohm C ln. 9. 66. 4... 4..8 7.4 6.849 6.4 6.7.98.88.49.9949 Find the clibrtion curve nd use it to find the temperture corresponding to 74.8 ohms. Wht is the difference between the results from prt ()? Is the difference lrger using results from prt () or prt (b), if the ctul mesured vlue t 74.8 ohms is.8 C? Solution ) For third order polnomil interpoltion (lso clled cubic interpoltion), we choose the temperture given b
..6 Chpter.,, f,, Figure 4 Cubic interpoltion. Since we wnt to find the temperture t 74. 8, nd we re using third order polnomil, we need to choose the four dt points closest to 74. 8 tht lso brcket 74.8 to evlute it. he four points re., 9., 66. nd 4.. hen.,. 9.,. 66., 4. 4.,. 8 gives..... 9. 9. 9. 9.. 66. 66. 66. 66. 4. 4. 4. 4. 4.. 8 Writing the four equtions in mtri form, we hve 6 9...46 8 9. 8.47 7.68 8 66. 4.4.76 7 4..49 9.79 Solving the bove four equtions gives 9.79.. 4..8
Direct Method of Interpoltion More Emples: Electricl Engineering..7 Hence.9 9.97.74 8 9.79.9 9.97.74 8, 4.. 74.8 9.79.9 74.8 9.97 74.8.74 8 74. 8.4 C he bsolute reltive pproimte error polnomil is.4.89.4.448% b) Finding the cubic interpolnt using the direct method for equires tht we first clculte the new vlues of nd. hen gives ln 7.4 6.849 6.4 6.7 7.4, 6.849, 6.4,.7, 6.98.88.49.9949. 98. 88. 49. 9949 for the results from the second nd third order 7.4 7.4 7.4 7.4. 98 6.849 6.849 6.849 6.849. 88 6.4 6.4 6.4 6.4. 49 6.7 6.7 6.7 6.7. 9949 Writing the four equtions in mtri form, we hve 7.4 49.6 4.8.98 6.849 46.44 6..88 6.4 4.67 68.99.49 6.7 7. 8.9.9949 Solving the bove four equtions gives
..8 Chpter. Hence.964.6. 448.7.964.6.448 However, since nd ln we get.7, 6.7 7.4.964.6(ln ).448(ln ).7(ln ), 4.. or ( ), 4...964.6(ln ).448(ln ).7(ln ) At 74. 8, (74.8).964.6ln74.8.448ln74.8.7ln74. 8.C Since the ctul mesured vlue t 74.8 ohms is.8 C, the bsolute reltive true error between the vlue used for prt () is.8.4 t.8.% nd for prt (b) is.8. t.8.98% herefore, the direct method of cubic polnomil interpoltion, tht is, obtined more ccurte results thn the ctul clibrtion curve of ln ln ln INEPOLAION opic Direct Method of Interpoltion Summr Emples of direct method of interpoltion. Mjor Electricl Engineering Authors Autr Kw Dte November, 9 Web Site http://numericlmethods.eng.usf.edu