mws_ele_inp_txt_lagrange_examples

# mws_ele_inp_txt_lagrange_examples - Chapter 05.04 Lagrange...

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05.04.1 Chapter 05.04 Lagrange Method of Interpolation – More Examples Electrical Engineering Example 1 Thermistors are used to measure the temperature of bodies. Thermistors are based on materials’ change in resistance with temperature. To measure temperature, manufacturers provide you with a temperature vs. resistance calibration curve. If you measure resistance, you can find the temperature. A manufacturer of thermistors makes several observations with a thermistor, which are given in Table 1. Table 1 Temperature as a function of resistance. R   ohm T   C 1101.0 911.3 636.0 451.1 25.113 30.131 40.120 50.128 Figure 1 Resistance vs. temperature.

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05.04.2 Chapter 05.04 Determine the temperature corresponding to 754.8 ohms using a first order Lagrange polynomial. Solution For first order Lagrange polynomial interpolation (also called linear interpolation), the temperature is given by 1 0 ) ( ) ( ) ( i i i R T R L R T ) ( ) ( ) ( ) ( 1 1 0 0 R T R L R T R L Figure 2 Linear interpolation. Since we want to find the temperature at 8 . 754 R , we need to choose the two data points that are closest to 8 . 754 R that also bracket 8 . 754 R to evaluate it. The two points are 3 . 911 0 R and 0 . 636 1 R . Then  131 . 30 , 3 . 911 0 0 R T R  120 . 40 , 0 . 636 1 1 R T R gives 1 0 0 0 0 ) ( j j j j R R R R R L 1 0 1 R R R R ( x 0 , y 0 ) ( x 1 , y 1 ) f 1 ( x ) x y
05.04.3 1 1 0 1 1 ) ( j j j j R R R R R L 0 1 0 R R R R Hence ) ( ) ( ) ( 1 0 1 0 0 1 0 1 R T R R R R R T R R R R R T 3 . 911 0 . 636 ), 120 . 40 ( 3 . 911 0 . 636 3 . 911 ) 131 . 30 ( 0 . 636 3 . 911 0 . 636 R R R ) 120 . 40 ( 3 . 911 0 . 636 3 . 911 8 . 754 ) 131 . 30 ( 0 . 636 3 . 911 0 . 636 8 . 754 ) 8 . 754 ( T ) 120 . 40 ( 56847 . 0 ) 131 . 30 ( 43153 . 0 C 809 . 35 You can see that 43153 . 0 ) ( 0 R L and 56847 . 0 ) ( 1 R L are like weightages given to the temperatures at 3 . 911 0 R and 0 . 636 1 R to calculate the temperature at 8 . 754 R . Example 2 Thermistors are used to measure the temperature of bodies. Thermistors are based on materials’ change in resistance with temperature. To measure temperature, manufacturers provide you with a temperature vs. resistance calibration curve. If you measure resistance, you can find the temperature. A manufacturer of thermistors makes several observations with a thermistor, which are given in Table 2. Table 2 Temperature as a function of resistance. R   ohm T   C 1101.0 911.3 636.0 451.1 25.113 30.131 40.120 50.128 Determine the temperature corresponding to 754.8 ohms using a second order Lagrange polynomial. Find the absolute relative approximate error for the second order polynomial approximation. Solution

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mws_ele_inp_txt_lagrange_examples - Chapter 05.04 Lagrange...

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