1 Mid-term exam in FYS-2008, autumn 2016 Department of Physics and Technology University of Tromsø NB! Deadline for handing in: Monday 31st October 23:59 in Fronter This exam paper consists of four pages. Problem 1: Heavy suitcase Setting: You are ready to go to Silicon Valley to make a fortune. However, as you drag your suitcase out to the car, you start thinking about the maximum weight allowed on this flight: 23 kg. As you are broke, an overweight fee will stop your plans. You decide to take the bathroom scales out to the car and measure the weight. These are state-of-the-art bathroom scales with specified maximum error 0.2% of FSD for standard conditions (temperature 20°C, battery voltage 3.0V, range 0-120kg). Ready to measure the suitcase next to the car, the display on the scales says -10°C and the display is flickering due to low battery voltage: 2.5 V (also indicated). With the suitcase on, the scales show 22.8 kg. Note: It is advised to make some plots, e.g. of calibration curves, non-linearity, equations found and deviations from the equations. Table 1 Calibration of bathroom scales. The displayed value Wout is given with certified weights Win on the scales. Standard conditions: 20°C and 3V. To find interfering and modifying inputs, the scales have been calibrated versus temperature and voltage for two input weights: 0 kg and 120 kg, marked as W0 and W120 in the table. For this, a certified temperature chamber and voltage source was used. Std. conditions Temperature Voltage Win (kg) Wout (kg) T (°C) W0 (kg) W120 (kg) V (V) W0 (kg) W120 (kg) 0.00 -0.02 10 -0.25 119.73 2.6 0.02 119.54 15.00 15.08 15 -0.14 119.81 2.8 -0.03 119.78 30.00 30.17 20 0.03 119.96 3.0 -0.04 120.00 45.00 45.17 25 0.15 120.12 3.2 0.05 120.26 60.00 60.22 30 0.29 120.29 75.00 75.22 90.00 90.13 105.00 105.05 120.00 119.982 a) Calibration Being a meticulous and trained measurement specialist, you have previously calibrated the scales, see table 1. Find the ideal equation for the scales for standard conditions (20°C and 3.0V). b) Non-linearity and error What is the maximum non-linearity and error of the scales for standard conditions? Does it comply with the specified maximum error of 0.2% of FSD? Find an expression for the non-linearity. c) Interfering and modifying inputs Are temperature or battery voltage interfering or modifying inputs? Find the relevant coefficients (KI and KM). d) Calibrated expression Put together the answers in a) to c) to give a complete expression for the displayed weight (Wout) as a function of the weight put onto the scales (Win), taking into account any non-linearity, interfering and modifying inputs. e) True value for 22.8 kg displayed Returning to the suitcase, what is the estimated true weight of the suitcase? I.e. for displayed weight 22.8 kg, temperature -10°C and battery voltage 2.5V. Problem 2: Capacitive sensor In this problem we will study two methods for measuring capacitance induced by a capacitive type of sensor. a) Step input Let us first consider the experimental setup shown in Fig.1. Here a signal generator is modeled as a potential source V in series with a resistor RS. The signal generator is then connected to a capacitive sensor yielding a capacitance CS and an output voltage U with respect to ground (GND) is measured by a high impedance oscilloscope. Figure 1 Signal generator and sensor. Problem 1 In this problem we will study some methods for measuring capacitance induced by a capacitive type of sensor. a) Let us first consider the experimental setup shown in Fig.1. Here a signal generator is modelled as a potential source ܸ in series with a resistor ܴ௦. The signal generator is then connected to a capacitive sensor yielding a capacitance ܥ௦ and an output voltage ܷ with respect to ground (GND) is measured by a high impedance oscilloscope. Fig. 1 In order to estimate the sensor capacitance, we program the signal generator to give the following step potential: ܸ = 0 for t < 0 ܸ = ܸ଴ = ܿ ݐݏ݊݋for t >= 03 In order to estimate the sensor capacitance, we program the signal generator to give the following step potential: V = 0, for t < 0 V = V0 = const for t > 0 Obtain the transfer function for the system in Fig. 1 and find the solution for U as a function of time with initial condition U(t = 0) = 0. Illustrate how this solution can be used to estimate Cs if we assume the resistance RS in Fig. 1 to be known. b) Dynamic error Consider the system given in a). What is the dynamic error of the system? Plot the dynamic error for voltage V0 = 5V, resistance RS = 1kΩ and sensor capacitance CS in the range 20-100 nF. Use a few values of the sensor capacitance to illustrate how the error evolves. c) Bridge A second option for measuring CS is shown in Fig. 2. Here the capacitance sensor is inserted in a bridge circuit together with a constant capacitor C0 and two constant resistors that both have a resistivity value R. An AC potential V acts as the potential source as shown in Fig. 2, while the potential difference Ub - Ua is measured by a high impedance voltage probe assumed to put no load on the circuit. Figure 2 Capacitance sensor in bridge Further on, we assume that the sensor gives a linear capacitance CS = C0(1-αΔd) where α is a sensor constant. Write down the potential relation for the suggested bridge and show that the output potential can approximated by Ub - Ua ≈ ¼VαΔd assuming |αΔd|≪1. d) A third option for measuring ܥ௦ is shown in Fig. 3. Here the capacitance sensor is inserted a bridge circuit together with a constant capacitor ଴ܥand two constant resistor that both have a resistivity value ܴ. An AC potential ܸ acts as the potential source as shown in Fig. 3, while the potential difference ܷ௕ െ ܷ௔ is measured by a high impedance voltage probe assumed to put no load on the circuit. Fig. 3 Further on, we assume that the capacitive sensor gives a linear ܥ௦ = (଴ܥ1 െ ߙȟ݀) where ߙis a sensor constant. Write down the potential relation for the suggested bridge and show that the output potential can approximated by ؄ ௔ܷ ܷ௕ െ ܸ 4 ݀ȟߙ assuming |ߙȟ݀| ا1. e) In the last point we will look at the quantization error and other uncertainties involved in the bridge measurement. We will assume that the potential difference ܷ௕ െ ܷ௔ is amplified and then digitized by an AD-converter. The output from the AD-converter is assumed to be within4 Problem 3 Accelerometer According to Wikipedia: An accelerometer is a device that measures proper acceleration; proper acceleration is not the same as coordinate acceleration (rate of change of velocity). For example, an accelerometer at rest on the surface of the Earth will measure an acceleration due to Earth's gravity, straight upwards (by definition) of g ≈ 9.81 m/s2. By contrast, accelerometers in free fall (falling toward the center of the Earth at a rate of about 9.81 m/s2) will measure zero. Find three types of accelerometers and for each of them, describe: - Measurement principle - Typical specifications, e.g.: Range, noise/resolution, frequency range, size, etc. - Typical applications - 'Killer application': One example where this accelerometer is better suited than the other listed. Hints: Types: The three types should be based on different measurement principles. References: Write where you found the information, e.g. link to web-pages. Figures: You can copy figures from internet, books, etc. But remember to tell where you found it => reference. Tables: It is nice to include tables, e.g. with specifications. But do not include all values! Make a critical selection and include e.g. typical, minimum and maximum values.