Chpter. Spline Method of Interpoltion More Exmples Electricl Engineering Exmple Thermistors re used to mesure the temperture of bodies. Thermistors re bsed on mterils chnge in resistnce with temperture. To mesure temperture, mnufcturers provide you with temperture vs. resistnce clibrtion curve. If you mesure resistnce, you cn find the temperture. A mnufcturer of thermistors mkes severl observtions with thermistor, which re given in Tble. Tble Temperture s function of resistnce. R ohm T C. 9. 66. 4... 4..8..
.. Chpter. Figure Resistnce vs. temperture. Determine the temperture corresponding to 74.8 ohms using liner splines. Solution Since we wnt to find the temperture t R 74. 8 nd we re using liner splines, we need to choose the two dt points tht re closest to R 74. 8 tht lso brcket R 74. 8 to evlute it. The two points re R 9. nd R 66.. Then R 9., ( ). T R R 66., ( ) 4. T R given T ( R ) T ( R ) T ( R) T ( R ) ( R R ) R R 4... ( R 9.), 66. R 9. 66. 9. Hence T ( R)..684( R 9.) At R 74.8, T ( 74.8)..684(74.8 9.).89 C Liner spline interpoltion is no different from liner polynomil interpoltion. Liner splines still use dt only from the two consecutive dt points. Also t the interior points of the dt,
Spline Method of Interpoltion-More Exmples: Electricl Engineering.. the slope chnges bruptly. This mens tht the first derivtive is not continuous t these points. So how do we improve on this? We cn do so by using qudrtic splines. Exmple Thermistors re used to mesure the temperture of bodies. Thermistors re bsed on mterils chnge in resistnce with temperture. To mesure temperture, mnufcturers provide you with temperture vs. resistnce clibrtion curve. If you mesure resistnce, you cn find the temperture. A mnufcturer of thermistors mkes severl observtions with thermistor, which re given in Tble. Tble Temperture s function of resistnce. R ohm T C. 9. 66. 4... 4..8 ) Determine the temperture corresponding to 74.8 ohms using qudrtic splines. Find the bsolute reltive pproximte error for the qudrtic pproximtion. b) The ctul clibrtion curve used by industry is given by ln R ln R T substituting y, nd x ln R, T the clibrtion curve is given by y x x Tble Mnipultion for the given dt. R ohm T C x ln R y T. 9. 66. 4... 4..8 7.4 6.849 6.4 6.7.98.88.49.9949 Find the clibrtion curve using qudrtic splines, nd find the temperture corresponding to 74.8 ohms. Wht is the difference between the results from prt ()? Is the difference lrger using the results from prt () or prt (b), if the ctul mesured vlue t 74.8 ohms is.8 C?
..4 Chpter. Solution ) Since there re four dt points, three qudrtic splines pss through them. T R) R b R,. R 9. ( c R br c R br c, 9. R 66., 66. R 4. The equtions re found s follows.. Ech qudrtic spline psses through two consecutive dt points. R b R psses through R. nd R 9.. c (.) b (.) c ( 9.) b (9.) c R br c ( 9.) b (9.) c.. psses through R 9. nd R 66.. ( 66.) b (66.) c R br c ( 66.) b (66.) c. 4. psses through R 66. nd R 4.. ( 4.) b (4.) c 4..8 () () () (4) () (6). Qudrtic splines hve continuous derivtives t the interior dt points. At R 9. (9.) b (9.) b (7) At R 66. (66.) b (66.) b (8). Assuming the first spline R b R c is liner, (9)
Spline Method of Interpoltion-More Exmples: Electricl Engineering... 8.47 8.6 6. 9. 8.47 4.4 8.6 7 9. 66. 4.4.49 7 66. 4. b c b c b c... 4. 4..8 Solving the bove 9 equtions gives the 9 unknowns s i i bi ci.64 4.7.7.94 8.89 4.. 86.974 Therefore, the splines re given by T ( R).64R 4.7, 9. R..7 R.94R 8.89, 66. R 9. 4. R.R 86.974, 4. R 66. At R 74. 8 T (74.8).7 (74.8).94(74.8) 8.89.4 C The bsolute reltive pproximte error obtined between the results from the liner nd qudrtic splines is.4.89.4.889% b) Since there re four dt points, three qudrtic splines pss through them. y x) x b x, 7.4 x 6. 849 ( c x b x c, x b x c 6.849 x 6. 4, 6.4 x 6. 7 The equtions re found s follows.. Ech qudrtic spline psses through two consecutive dt points. x b x psses through x 7. 4 nd x 6. 849. c ( 7.4) b (7.4) c.98 ( 6.849) b (6.849) c.88 () ()
..6 Chpter. x b x c 6. ( 6.849) b (6.849) c psses through x 849 nd x 6. 4..88 ( 6.4) b (6.4) c.49 x b x c 6. ( 6.4) b (6.4) c psses through x 4 nd x 6. 7..49 ( 6.7) b (6.7) c.9949. Qudrtic splines hve continuous derivtives t the interior dt points. At x 6. 849 (6.849) b (6.849) b At x 6. 4 (6.4) b (6.4) b () (4) () (6) (7) (8). Assuming the first spline x b x c is liner, 49.6 46.44.6 7.4 6.849 46.44 4.67.6.9 6.849 6.4 4.67 7..9 6.4 6.7 (9).98 b.88 c.88.49 b.49 c.9949 b c Solving the bove 9 equtions gives the 9 unknowns s i i.69.8.67.46.9..4646.4889 Therefore, the splines re given by y ( x).69x.8, 7.4 x 6. 849.67x.46x.9, 6.849 x 6. 4 bi ci
Spline Method of Interpoltion-More Exmples: Electricl Engineering..7.x.4646x.4889, 6.4 x 6. 7 At x ln(74.8) y (ln(74.8)).67(ln(74.8)).46(ln(74.8)).9.777 Since T, y T.78 6.4C 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.8.94% nd for prt (b) is.8 6.4 t.8.8% Therefore, the spline method of interpoltion using qudrtic splines, tht is, T R R R obtined more ccurte results thn the clibrtion curve of ln R ln R T