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Transcript
LabPro Measurement of RC Time Constant and Energy Stored in Capacitor by Jim Sizemore Contents Introduction ......................................................................................................................................1 Objectives ........................................................................................................................................1 Limitations .......................................................................................................................................2 Equipment ........................................................................................................................................2 RC Time Constant and Capacitance Measurement .........................................................................2 Creating a Calculated Column .................................................................................................. 2 Collect Data .............................................................................................................................. 3 Add Graph and Change Scale ................................................................................................... 4 Fit Data to Curve ....................................................................................................................... 6 Repeat for New Resistance Values and Charging .................................................................... 8 Results ............................................................................................................................................13 Part II – Energy Stored in a Capacitor ...........................................................................................14 Integration ............................................................................................................................... 14 Conclusion .....................................................................................................................................17 List of Figures ................................................................................................................................17 List of Links ...................................................................................................................................17 Introduction Using the LabPro system (LabPro device and LoggerPro software) we will examine the charging and discharging of a capacitor through a resistor. The objective is to measure, plot and observe the form of the curve. We will transform voltage vs. time curve in LoggerPro using the natural logarithm function. We will discover a linear relationship between time and logarithm of voltage. The time constant, and therefore capacitance, is found from the slope. We compare the measured capacitance to the value marked on the capacitor and attempt to explain discrepancies. Using LoggerPro we will transform and integrate to find stored energy and compare this to standard formulas for energy stored in a capacitor. Objectives Setting up LabPro to measure charging and discharging characteristics of RC circuits Creating a calculated column Adding graphs and changing scales Fitting data to models (curve fitting) Understanding differences between charging and discharging circuits Measuring time constant and capacitance Limitations The current probe was only used for the energy stored in a capacitor and not for the RC time constant. For the RC time constant, current is found from the voltage drop across the resistor. As will be seen, the current probe is very noisy at low currents and was therefore omitted when measuring RC time constant. However to measure power, and since power is IV, we must measure current. The noise in the current probe became apparent when measuring power. Two things are critically important: 1. The current probe cannot exceed 5 V from ground. 2. Ground of the voltage probes MUST be at the same potential or damage to the LabPro instrument may result. Equipment LabPro system, 2 LabPro voltage probes, electrical breadboard, electrical leads as required, RC time constant PC board (2.7 k & 47 k resistors, 10 F capacitor), electrical power supply, SPST switch, DMM. RC Time Constant and Capacitance Measurement 1. Connect a voltage probe to the LabPro (refer to Vernier LabPro System Basic Setup and Operation document for instructions how to plug in, turn on and calibrate LabPro system). 2. Before making electrical connections, measure the actual value of the resistors using the DMM. “2.7 k” = _______________________________ k (example 2.69 k) “47 k” = _______________________________ k (example 47.1 k) 3. Construct an electrical circuit according to the schematic shown in Figure 1 as follows. LabPro’s voltage probes are rated at ±10 V, therefore, do not exceed 10V on the power supply. 10 F <10 V A 2.7 k or 47 k V The voltmeter and ammeter are the probes connected to the LabPro system. Ammeter is not always present. A wire replaces the ammeter, if absent. Figure 1 – Schematic for discharging capacitor. 4. Set up Experiment > Data Collection at 15 seconds, 250 samples per second as described in Vernier LabPro System Basic Setup and Operation. Creating a Calculated Column 5. Go to Data > New Calculated Column as shown in Figure 2 as follows. RC Time Constant Experiment Using LabPro page 2 of 17 Figure 2 – Menu navigation to Data > New Calculated Column 6. Create a new calculated column named Log V as shown in Figure 3 as follows. Note that the natural logarithm function was selected to make data analysis later a little easier. Figure 3 – New calculated column dialog box Collect Data 7. Collection may begin with the switch open or closed, however you must transition from closed to open to observe the RC decay. Note from the schematic in Figure 1 that the RC Time Constant Experiment Using LabPro page 3 of 17 capacitor charges nearly instantaneously when the switch is closed, but will discharge slowly through the resistor when the switch is opened. 8. If necessary, review Vernier LabPro System Basic Setup and Operation to recall how to collect data using the LabPro system. Begin collection and execute experiment as described above. Add Graph and Change Scale 9. After data is collected, go to Insert > Graph to display Log V column as shown in Figure 4 as follows. Figure 4 – Menu navigation to Insert > Graph 10. Arrange graphs on screen to view both simultaneously. 11. Using mouse, select falling edge (closed switch is being opened) of data and click on magnifying glass icon as shown in Figure 5 as follows. RC Time Constant Experiment Using LabPro page 4 of 17 Figure 5 – Zooming in on range of data 12. Right click on top graph (Potential) as shown in Figure 6 as follows. Figure 6 – Right click on graph to bring up menu and select Graph Options RC Time Constant Experiment Using LabPro page 5 of 17 13. Change the x-axis scale appropriately as shown in Figure 7 as follows. For example, round to the nearest tenth second. Figure 7 – Manually setting x-axis scale 14. Change the bottom (Log V) graph to the same horizontal scale so that there is a one-to-one correspondence between data in the top graph and bottom graph Fit Data to Curve 15. The equation for the falling edge is: V = Vo e-t/RC. Taking the natural logarithm of both sides, one obtains: Ln(V) = (-1/RC)t + Ln(Vo). The slope of the bottom graph is, therefore, -1/RC. To find this slope, select the linear portion of the Log V data. Then go to Analyze > Linear fit as shown in Figure 8 as follows. RC Time Constant Experiment Using LabPro page 6 of 17 Figure 8 – Fitting a line to the data 16. The results of the analysis are shown in Figure 9 following. In the example, the slope was – 29.37 for the “2.7 k” resistor. The RC time constant is, therefore, 34.0 ms and, using the actual resistance of 2.69 k the calculated capacitance is 12.7 mF. RC Time Constant Experiment Using LabPro page 7 of 17 Figure 9 – Results of line fit analysis Repeat for New Resistance Values and Charging 17. Close linear fit dialog box and autoscale (right click on graph for menu or type Ctrl+u) as shown in Figure 10 as follows. RC Time Constant Experiment Using LabPro page 8 of 17 Figure 10 – Menu navigation to autoscale 18. Repeat procedure (Steps 3 to 16) using the “47 k resistor. The results are shown in Figure 11 as follows. The slope is –1.813, RC time constant is 0.552 seconds and, using the actual resistor value of 47.1 k, C is 11.7 F. RC Time Constant Experiment Using LabPro page 9 of 17 Figure 11 – Results of capacitor discharge using 47 k resistor. 19. Note in both Figures 9 and 11 that there is some curvature in the data – it is not exactly a straight line. This is a good opportunity to discuss why with students. Are there parasitic capacitances or resistances? Intentionally adding capacitances or resistances in the circuit where parasitic resistance or capacitance is suspected tests this hypothesis. In theory, voltage probes should not interfere, however, in reality, do they? Probing voltage with a DMM or VOM tests while performing the experiment tests this hypothesis. 20. Connect the circuit shown in the schematic shown in Figure 12 as follows. Note, due to the LabPro requirement that the grounds of voltage probes be at the same potential, Probe 1 reading V1 (the voltage across the capacitor) will read negative values. Damage to LabPro may result if the grounds of the voltage probes are not connected to the same potential. RC Time Constant Experiment Using LabPro page 10 of 17 V2 Switch 1 <10 V 2.7 k or 47 k 10 F Switch 2 V1 These voltmeters are the voltage probes connected to the LabPro system. LabPro requires the ground of the two probes be at the same potential. The heavy black lines show these ground wires, while the heavy red lines are the hot wires. Figure 12 – Schematic of capacitor charging circuit 21. Start with Switch 1 (Figure 12) open. Close, then open, Switch 2 to discharge the capacitor. 22. Repeat the procedure (Steps 4 to 16) with the following exception at Steps 7 and 8. Replace Steps 7 and 8 with the following: Begin data collection and after a few seconds close Switch 1. 23. Unlike the previous, discharging, experiments, Log V1 was not linear. It is necessary to create a new formula. When charging the model is: V = Vo ( 1 – e-t/RC ). Recall that V and Vo are negative due to physical requirements of the LabPro sensors. Rearranging, Ln(| V - Vo |) = Ln(| Vo |) - t/RC. The slope is, therefore, the negative reciprocal of RC. Find Vo from the fully charged (far right) portion of the graph averaging using descriptive statistics as described in Vernier LabPro System Basic Setup and Operation. Change the formula for Log V to Ln(| V - Vo |) as described in Step 6 previously. Both the graph and data are immediately updated. Formulas may be added or edited after data is collected. 24. Create a new calculated column named Ln V2. V2/R is the charging current and obeys the following formula: I = Io e-t/RC or V2 = Vo e-t/RC. Taking the natural logarithm of both sides yields: Ln( V2 ) = Ln ( Vo ) – t/RC. The slope is the negative reciprocal of RC. 25. Add and arrange the graph for Ln V2. Results are displayed in Figure 13 as follows for the 2.7 k resistor. The slope of Ln(| V - Vo |) is –30.48 yielding RC = 32.8 ms and C = 12.2 F. The slope of Ln( V2 ) is –29.57 yielding RC = 33.8 ms and C = 12.6 F. RC Time Constant Experiment Using LabPro page 11 of 17 Figure 13 – Capacitor charging data for 2.7 k resistor 26. Repeat Steps 20 to 24 with the 47 k resistor. Results are shown in Figure 14 as follows. The slope of Ln(| V - Vo |) is –1.803 yielding RC = 555 ms and C = 11.8 F. The slope of Ln( V2 ) is –1.668 yielding RC = 600 ms (accurate to 3 significant figures) and C = 12.7 mF. RC Time Constant Experiment Using LabPro page 12 of 17 Figure 14 – Capacitor charging data for 47 k resistor Results From charging and discharging the capacitor, multiple values of RC and C were obtained and are tabulated as follows. RC Time Constant (ms) Experiment 2.7 k 47 k Capacitor voltage charging 34.0 ms 552 ms Capacitor voltage discharging 32.8 ms 555 ms Resistor voltage discharge 33.8 ms 600 ms Measured capacitance (F) 2.7 k 47 k 12.7F 11.7F 12.2F 11.8F 12.6F 12.7F As discussed previously (Step 19), the data doesn’t exactly follow a straight line. This provides the opportunity to discuss possible sources of error. The table above also shows variation in the data providing more opportunity to discuss sources of error. RC Time Constant Experiment Using LabPro page 13 of 17 Part II – Energy Stored in a Capacitor This part shows the limitations of the current probe. The LabPro current probe is used as an ammeter and is inserted in the discharging circuit as shown in Figure 1. I believe it became clear in previous discussions that the discharging of a capacitor is an easier experiment. The discharging experiments are repeated yielding “potential” and “current” columns. A created column called “power” is created using the methods discussed previously by multiplying potential and current. A second created column called “power2” is created using the formula V2/R. Integration A new aspect of the LabPro system is introduced at this point – numerical integration. Select the region to be integrated and then navigate to Analyze > Integral as shown in the next Figure. Figure 15 – Navigation to Analyze > Integrate The results for the 2.7 k and 47 k resistors are shown in the following figures. RC Time Constant Experiment Using LabPro page 14 of 17 Figure 16 – Integration of power leading to stored energy for 2.7 k resistor RC Time Constant Experiment Using LabPro page 15 of 17 Figure 17 – Integration of power leading to stored energy for 47 k resistor Figure 17 especially reveals the limitations of the current probe. The data for current and power calculated from current is very noisy. The integration also highly deviates from the integration based on V2/R. The current probe is designed to be a simple current probe and not an autoranging ammeter. I had discussions with Vernier and they indicated the current probe is a differential voltage probe with a shunt resistor. We may, therefore, wish to consider purchasing a differential voltage probe and using our own shunt resistors as suited for each experiment. Tabulation of power results follows: Power (mJ) Experiment (resistor) 2.7 k 47 k IVt (used current probe) 0.6794 2.48 Integrate V2/R 0.5996 0.5111 CV2/2 0.574 0.574 RC Time Constant Experiment Using LabPro page 16 of 17 Although some degree of noise was evident for the 2.7 k resistor, the result using the current probe is still reasonable. The only data point way out-of-line is using the current probe at low currents, that is, with the 47 k resistor. Conclusion This concludes the RC time constant experiment in which we examined charging and discharging of a capacitor through a resistor. We plotted curves and transformed these curves using the natural logarithm function to find slopes and, from the slopes, calculated the RC time constant. A second part of this experiment is to find stored energy through numerical integration and comparing it to the expected value. Although we find the current probe to have some limitations, it still yielded useful results even when our gut instincts might not expect reasonable results. List of Figures Figure 1 – Schematic for discharging capacitor. ............................................................................ 2 Figure 2 – Menu navigation to Data > New Calculated Column ................................................... 3 Figure 3 – New calculated column dialog box ............................................................................... 3 Figure 4 – Menu navigation to Insert > Graph .............................................................................. 4 Figure 5 – Zooming in on range of data ......................................................................................... 5 Figure 6 – Right click on graph to bring up menu and select Graph Options ................................ 5 Figure 7 – Manually setting x-axis scale ........................................................................................ 6 Figure 8 – Fitting a line to the data ................................................................................................. 7 Figure 9 – Results of line fit analysis.............................................................................................. 8 Figure 10 – Menu navigation to autoscale ...................................................................................... 9 Figure 11 – Results of capacitor discharge using 47 k resistor. ................................................ 10 Figure 12 – Schematic of capacitor charging circuit .................................................................... 11 Figure 13 – Capacitor charging data for 2.7 k resistor .............................................................. 12 Figure 14 – Capacitor charging data for 47 k resistor ............................................................... 13 Figure 15 – Navigation to Analyze > Integrate ............................................................................ 14 Figure 16 – Integration of power leading to stored energy for 2.7 k resistor ............................ 15 Figure 17 – Integration of power leading to stored energy for 47 k resistor ............................. 16 List of Links 1. 2. 3. 4. Vernier Web Site LabPro User’s Manual (large, 6.5 Mb file) LabPro Technical Manual (405 kb) Basic LabPro Setup and Operation RC Time Constant Experiment Using LabPro page 17 of 17