During lecture we examined a case study in which we apply spline interpolation to temperature measurements as a function of time, however, no mention was made to how the temperature was recorded. In this Mini Project we will assume that the temperature was measured using a thermistor, which is a device with a resistance that decreases as temperature increases. In the thermistor, the relationship between the temperature and resistance is given by an empirical equation known as the Steinhart-Hart equation ABInR+C (InR)3. (1) where T is the temperatures in degrees Kelvin, R is the resistance in ohms and A, B and C are coefficients that must be determined for the thermistor being used. In most cases the thermistor resistance is not measured directly. Instead, the thermistor is connected in series to a known resistor in a circuit known asa voltage divider. In Figure 1 we can see a resistor arrangement in which a resistor R1 is followed by the thermistor R(T) A/D Converter R(T) Figure 1. Schematic diagram of circuit used to measure voltage as a function of temperature Since the two resistors share the same current delivered by the voltage source, V, they are connected in series and the voltage V will be distributed between R1 and R(T). The voltage measured at R(T) has been labeled Vo and is given by the voltage divider equation R(T) R1+R(T) In the Figure the voltage Vo is connected to and analog to digital (A/D) converter which is device that samples a voltage and converts it to a digital variable that is loaded to a computer and then processed using Matlab, or other programs. In particular, for the ENG6 Final Project, the voltage supply and A/D conversion is provided by an Arduino microprocessor In order to be able to use the Seinhart-Hart, and hence get useful data form the thermistor, it is necessary to calculate the calibration coefficients A, B and C Since we have three unknowns we can measure the resistance of the thermistor at three different known temperatures and obtain the system of equations T: where R1, R2 and R3 are the temperatures measured at T1, T2 and T3, respectively. However, keep in mind that we have voltage measurements that can be converted to resistance using equation (2) Tasks 1. For the Vishay 10K NTC thermistor, using a calibrated thermometer, a thermoelectric cooler and a ohmeter we have obtained the following table of calibration measurements Table 1. Calibration measurements for Vishay 10K NTC thermistor T (oK) R (k2) 262.9 55.519 293.7 12.105 352.9 1.256 Calculate the calibration coefficients A, B and C of the Steinhart-Hart equation and calculate the residuals between the predicted and experimental measurements. Use fprintf to display in separate lines the coefficients and the residuals 2. Using the Arduino commands readVoltage ( 'com#', 'A#', now to obtain a time stamp and a while loop we obtained the table of voltage measurements vs. time shown below Table 2: Voltage measurements as function of time for the Vishay 10K NTC thermistor in series with a 10K2 resistor and biased with 5V Time (s) Voltage (V) 2.623 2.458 2.211 2.042 |1.569 1.548 1.600 1.578 1.571 1.585 1.578 2 8 10 Based on the table, do a spline fit to interpolate voltage measurements every 100 ms 3. Convert the interpolated voltages to resistance assuming a 10K2 series 4. Convert the interpolated resistances to temperature, in degrees Celsius, 5. Identify the maximum overshoot temperature and steady-state resistor and 5V applied to the series arrangement. and plot the temperature vs. time. Remember to label the axes temperature and use fprintf to display the values During lecture we examined a case study in which we apply spline interpolation to temperature measurements as a function of time, however, no mention was made to how the temperature was recorded. In this Mini Project we will assume that the temperature was measured using a thermistor, which is a device with a resistance that decreases as temperature increases. In the thermistor, the relationship between the temperature and resistance is given by an empirical equation known as the Steinhart-Hart equation ABInR+C (InR)3. (1) where T is the temperatures in degrees Kelvin, R is the resistance in ohms and A, B and C are coefficients that must be determined for the thermistor being used. In most cases the thermistor resistance is not measured directly. Instead, the thermistor is connected in series to a known resistor in a circuit known asa voltage divider. In Figure 1 we can see a resistor arrangement in which a resistor R1 is followed by the thermistor R(T) A/D Converter R(T) Figure 1. Schematic diagram of circuit used to measure voltage as a function of temperature Since the two resistors share the same current delivered by the voltage source, V, they are connected in series and the voltage V will be distributed between R1 and R(T). The voltage measured at R(T) has been labeled Vo and is given by the voltage divider equation R(T) R1+R(T) In the Figure the voltage Vo is connected to and analog to digital (A/D) converter which is device that samples a voltage and converts it to a digital variable that is loaded to a computer and then processed using Matlab, or other programs. In particular, for the ENG6 Final Project, the voltage supply and A/D conversion is provided by an Arduino microprocessor In order to be able to use the Seinhart-Hart, and hence get useful data form the thermistor, it is necessary to calculate the calibration coefficients A, B and C Since we have three unknowns we can measure the resistance of the thermistor at three different known temperatures and obtain the system of equations T: where R1, R2 and R3 are the temperatures measured at T1, T2 and T3, respectively. However, keep in mind that we have voltage measurements that can be converted to resistance using equation (2) Tasks 1. For the Vishay 10K NTC thermistor, using a calibrated thermometer, a thermoelectric cooler and a ohmeter we have obtained the following table of calibration measurements Table 1. Calibration measurements for Vishay 10K NTC thermistor T (oK) R (k2) 262.9 55.519 293.7 12.105 352.9 1.256 Calculate the calibration coefficients A, B and C of the Steinhart-Hart equation and calculate the residuals between the predicted and experimental measurements. Use fprintf to display in separate lines the coefficients and the residuals 2. Using the Arduino commands readVoltage ( 'com#', 'A#', now to obtain a time stamp and a while loop we obtained the table of voltage measurements vs. time shown below Table 2: Voltage measurements as function of time for the Vishay 10K NTC thermistor in series with a 10K2 resistor and biased with 5V Time (s) Voltage (V) 2.623 2.458 2.211 2.042 |1.569 1.548 1.600 1.578 1.571 1.585 1.578 2 8 10 Based on the table, do a spline fit to interpolate voltage measurements every 100 ms 3. Convert the interpolated voltages to resistance assuming a 10K2 series 4. Convert the interpolated resistances to temperature, in degrees Celsius, 5. Identify the maximum overshoot temperature and steady-state resistor and 5V applied to the series arrangement. and plot the temperature vs. time. Remember to label the axes temperature and use fprintf to display the values