Sensors: Loggers: Voltage Any EASYSENSE Capacitor investigations Logging time: EasyLog (20 s) Teacher s notes 01 Time constant for a capacitor - resistor circuit Theory The charging and discharging of a capacitor are exponential changes. The following equations represent the way in which the voltage across the capacitor changes during the charge and discharge cycle. Discharging V = V max e -t/cr Charging V = V max (1-e -t/cr ) The time constant "T" is derived from these equations: T = C x R Considering the charging cycle, when the time t = T = C x R, then the equation becomes: V = V max e -1 V V = max e Hint: If t = CR, then substitute t in the charge and discharge cycles to get cr/cr = 1, i.e. anything divided by itself = 1. e -1 is shorthand for e per, so V = V max e -1 is translated as V is equal to V max per e or, V max divided by e, in which case the superscript -1 is no longer required to describe its function and is removed. V max e = 2.718 V = V = V max x 0.368 2.718 The time constant T, for charging, is the time taken for the voltage across the capacitor to fall to 0.368 of its maximum value V max. Conversely, for discharging, the time constant T is the time taken for the voltage to reach 0.632 of the maximum value V max. When the capacitor is used in timing circuits it will be arranged so that the switching occurs on the steeply rising or steeply falling part of the curve, i.e. somewhere around the time constant. The use of the Lascells Capacitor investigation module significantly reduces set up time and creates an environment where the results are standard and there is a high degree of control. The guarantee for results allows you, the teacher, to devote time to explanation of the experiment (and mathematics) and not on getting it to work; all students have the same results for analysis allowing group discussion and peer assistance. Set up of the software Record data in EasyLog, this gives the students the visual feed back of the electrical event being studied, while the data is being logged. Recording method EasyLog Time About 20 seconds Teacher s notes for 01 Time constant for a capacitor - resistor circuit 1
Apparatus 1. An EASYSENSE logger. 2. A Smart Q Voltage sensor ±12 or 20 V. 3. Lascells Capacitor investigation module*. 4. A DC power supply or dry cells in good condition work well. Refer to notes below for fuller advice. 5. Patch connecting cables with 4 mm plugs. 6. Optional: Lascells Resistance selector module (to investigate change of resistance on the constant). * If a Lascells Resistance selector module isn t available the value of R can be altered by connecting a resistor in parallel to the sockets either side of the internal resistor on the Capacitor investigation module. A capacitor takes 4 to 8 times the time constant to approach full charge. When doing repeats you can speed up recharge time by switching to the lowest resistance on the selector. Remember to switch back to the correct resistance for the discharge cycle. 30 seconds gives a reasonable time to get two full charge discharge cycles for a time constant of about 1 second. Notes A capacitor takes the 4 to 8 times the time constant to approach full charge. Use EasyLog to collect data. Time between samples is: 30 ms for 0-30 second recording. 60 ms for 0 60 second recording (after 30 seconds have passed). 120 ms for 0 120 second recording (after 60 seconds have passed). Alternatively choose a 20 ms intersample time and select an appropriate time using Graph mode. All of the suggested times above allow the student to see the curve developing in real time on the logger. Default TC of the apparatus with the 470 ohm resistance only is 470 x 0.01 = 4.7 seconds. Sample of expected time constants with selected values from Lascells resistance selector: Resistance selector Module resistance Resistance (ohms) Capacitance μf Time constant (s) value (ohms) value (ohms) 100 470 82 10,000 0.82 330 470 194 10,000 1.94 470 470 235 10,000 2.35 680 470 278 10,000 2.78 1000 470 320 10,000 3.20 3,300 470 411 10,000 4.11 4,700 470 427 10,000 4.27 The use of the additional resistance selector allows the students to change the time constant by variance of the resistor value, the capacitor is hard wired into the apparatus. Students will need to be reminded of how to calculate parallel resistance values. R1 R2 Resistances calculated as R = as there are only 2 resistors in parallel in the circuit. R1 + R2 Power can be provided by dry cells in good condition, the voltages and currents used are low. GCSE circuit kits often provide holders for cells that can be used as power packs. If a low power supply unit is used you need to be aware of the effect of ripple voltages. Even good quality units can produce large ripple voltages. Some units described as DC are unsmoothed half wave rectified power supplies; they will not be suitable for this exercise. Teacher s notes for 01 Time constant for a capacitor - resistor circuit 2
Fast logging will give more reliable results, but the logging is unseen and the results are not visible until the recording has finished. In reality, EasyLog or a 20 ms intersample recording will give significantly more readings than student will achieve with stop clock and meters. Errors There are a number of sources of error in this experiment. All components have manufacturers tolerance, typically ±1 to 5% for resistors and ±20% for capacitors. Students need to be aware of these tolerances when comparing measured values and calculated values. The chemicals used in electrolytic capacitors will take time to return to their original state and there will be energy losses connected with the chemical changes in each charge-discharge cycle. Errors of 5% are not unexpected. The Voltage sensors have an internal resistance of 1 MΩ. If the load of resistors in the circuit is small compared to this internal resistance then any errors will be very small. It is advised that resistors of less than 10 kω are used to keep errors within 1%. Below is a table showing how the load resistor is linked to errors. Load resistor Ω Error in V% 1k 0.1 10k 1 100k 10 1M 50 Results Typical result using a 10,000 uf capacitor with a 470 ohm and 100 ohm in parallel, using 2 x dry cells (2.7 V) to charge the capacitor. Extension Investigate the effects of (a) resistance on the time constant T. Using faster recording speeds is useful with short time constants. The time span selected needs to be long enough to capture the whole trace. The results will be better defined, but the investigator will be "blind" if the intersample time is less than 20 ms. Good progression would be to use real time and then move onto fast capture when questions about errors arise. A typical FAST set up would be 1. Graph mode. 2. 20 second total recording time, 10 ms interval between samples (therefore 2,000 samples will be collected). Teacher s notes for 01 Time constant for a capacitor - resistor circuit 3
3. Trigger on Level, when Voltage rises above 0.1 volts (this may need to be changed if there is noise). 4. 25% Pre trigger. 5. Overlay mode on. Discharge curves of a 10,000 μf capacitor with different resistances. 2 x dry cells used as the power supply. Shorter time constants from smaller resistance values. Teacher s notes for 01 Time constant for a capacitor - resistor circuit 4
Sensors: Loggers: Voltage Any EASYSENSE Capacitor investigations Logging time: EasyLog (20 s) 01 Time constant for a capacitor - resistor circuit Read Capacitors are devices that can store electric charge; they do not produce the charge. A capacitor was traditionally made from two metal plates separated by an insulating material called a dielectric. They are now made from a wide variety of materials. When a voltage is applied to a capacitor the potential difference (p.d.) across the plates increases as charge flows in. The charge ceases to flow when the p.d. across the plates is equal to the supply voltage, electromotive force (e.m.f). The capacitor in this state is charged. Shorting the terminals of the capacitor will cause it to discharge. Adding a resistance into the charging circuit increases the time taken to reach a full charge. If a resistance is used to discharge the capacitor, the time taken will also increase. In this experiment, the time taken for a capacitor to charge and discharge through resistors will be measured and used to determine the time constant of the capacitor - resistor circuit. RESISTANCE SELECTOR 1k 3.3k 4.7k 680 470 330 100 6.8k 10k 33k 47k 100k The Resistance selector module (optional) Patch cable Power supply +5V charge discharge R A V Voltage sensor 0V What you need 1. An EASYSENSE logger. 2. A Smart Q Voltage sensor ±12 or 20 V. 3. Lascells Capacitor investigation module. 4. DC (5 V max) power supply or dry cells in good condition work well. 5. Patch connecting cables with 4 mm plugs. 6. Optional: Lascells Resistance selector module (to investigate change of resistance on the constant). 01 Time constant for a capacitor - resistor circuit 1
What you need to do 1. Connect the Voltage sensor to the Capacitor investigation module, connecting the red lead to the red socket and the black lead to the black socket. 1. Connect the Voltage sensor to an input of the logger. 2. Put a patch cable across the current measuring sockets (link Red to Blue across the A symbol). Without an ammeter in place the circuit is incomplete and the gap in the circuit needs to be bridged. 3. Connect to the power supply. 4. If you are using the Resistance selector module, write down the value of the resistor selected. 5. From the EasySense software s Home screen select EasyLog. 6. Measure the voltage (emf) of the supply with no current flowing using Test Mode (Tools menu). You can do this by temporarily connecting the Voltage sensor across the supply terminals. Replace the sensor into the circuit after taking this measurement. 7. Put the switch in the discharge position to make sure the capacitor is fully discharged before starting. 8. Click on Start, immediately change the switch to the charge position and watch the recording. When the line of the graph appears to show no further change in value with time, change the switch to the discharge position and click on Stop to finish recording. 9. Select Overlay. Repeat the charge / discharge cycle. 10. Use Save As to save the recording. 11. When you have collected all the data you need, leave the switch in the discharge position. Theory The time constant of a resistor - capacitor circuit is used as a measure of the length of time needed to charge a capacitor to within a percentage of its final maximum value. The time constant = C x R R = the value of the charging resistor in Ω (ohms) C = the capacitance of the capacitor in F (farads). 10,000 μf = 0.01 F The time constant is defined as the time taken to reach 63.2% of its final maximum voltage when charging. When discharging the time constant is the time taken to fall by 63.2% from the maximum value i.e. to fall to 36.8% of the maximum value. For most purposes, the values of 63% and 37% can be used. Analysis of results Finding the time constant for charge mode 1. The maximum voltage is the emf of the supply measured at the beginning of the experiment (this will normally be the supply voltage). 2. Multiply the maximum voltage by 0.63 to find the 63% voltage i.e. calculate 63% of the maximum charge voltage. 3. Use the Interval tool to find the time taken to go from zero volts to the 63% value. (Check in the table that the data selected by the Interval selection has correctly identified the beginning and end of the time period). Finding the time constant for discharge mode 1. Use the 63% voltage value determined above. 2. Use Interval to find the time taken to go from the maximum value to 0.37 of the maximum value, i.e. to reduce the voltage by 63% (Interval will show this value as an increasing negative number). 3. Theoretically, the two values for charge and discharge should be the same; in practice they will be slightly different. Explain why there is a difference (hint: think of energy losses). 01 Time constant for a capacitor - resistor circuit 2
Questions 1. How long did it take the capacitor to reach 95% of its maximum voltage? (Multiply the emf value by 0.95 to calculate the 95% of maximum voltage). 2. How does the value of the time constant relate to the time for reaching 95% of the maximum voltage? 3. If there is unlimited time available, when will the capacitor reach full charge? Explain your reasoning 4. Where / how could you apply the knowledge of time constants? Extension 1. How does increasing the following affect the time constant? a. The value of R* b. The value of C c. The supply voltage * If a Lascells Resistance selector module isn t available you can alter the value of R by connecting a resistor in parallel to the sockets either side of the internal resistor on the Capacitor investigation module. 2. What happens if resistors of different values are used on the charge and discharge sides of the circuit? 01 Time constant for a capacitor - resistor circuit 3
Sensors: Loggers: Voltage, Current Any EASYSENSE Capacitor investigations Logging time: EasyLog (20 s) Teacher s notes 02 Charge stored on a capacitor Apparatus 1. An EASYSENSE logger. 2. A Smart Q Current sensor ±100 ma. 3. A Smart Q Voltage sensor ±12 or 20 V. 4. Lascells Capacitor investigation module. 5. Lascells Resistance selector module. 6. DC (5 V max) power supply or dry cells in good condition work well. Refer to notes below for fuller advice. 7. Patch connecting cables with 4 mm plugs. Set up of the software Record data in EasyLog, this gives the students the visual feed back of the electrical event being studied, while the data is being logged. Recording method EasyLog Time About 20 seconds Component values The use of the Lascells Capacitor investigation module fixes the capacitor and the resistance, check on the module for the values. If a Lascells Resistance selector module isn t available the value of R can be altered by connecting a resistor in parallel to the sockets either side of the internal resistor on the Capacitor investigation module. Students should be reminded of how to calculate resistance with parallel resistors. Resistances calculated as R1 R2 R = as there are only 2 resistors in parallel in the circuit. R1 + R2 Notes A capacitor takes 4 to 8 times the time constant to approach full charge. Use EasyLog to collect data. Time between samples is: 30 ms for 0-30 second recording. 60 ms for 0-60 second recording (after 30 seconds have passed). 120 ms for 0-120 second recording (after 60 seconds have passed). Alternatively choose a time period that lets you have 20 ms intersample time within Graph mode. All of these suggested times allow the student to see the curve developing in real time. Teacher s notes for 02 Charge stored on a capacitor 1
Sample of expected time constants, peak current with selected values from the Resistance selector and 4.5 V power supply. Resistance selector value (ohms) Module resistance value (ohms) Resistance (ohms) Capacitance μf Time constant (s) Peak current (ma) 100 470 82 10,000 0.8 55 330 470 194 10,000 2 23 470 470 235 10,000 2.3 19 680 470 278 10,000 2.8 16 1,000 470 320 10,000 3.2 14 3,300 470 411 10,000 4.1 11 4,700 470 427 10,000 4.3 10 Power can be provided by dry cells in good condition, the voltages and currents used are low. GCSE circuit kits often provide holders for cells that can be used as power packs. If a low power DC supply unit is used be aware that even good quality units can produce large ripple voltages. Some units described as DC are unsmoothed half wave rectified power supplies; they will not be suitable for this exercise. Fast logging will give more reliable results, but the logging is unseen and the results are not visible until the recording has finished. In reality, EasyLog or a 20 ms intersample recording will give significantly more readings than student will achieve with stop clock and meters. Errors There are a number of sources of error in this experiment. All components have manufacturers tolerance, typically ±1 to 5% for resistors and ±20% for capacitors. Students need to be aware of these tolerances when comparing measured values and calculated values. The chemicals used in electrolytic capacitors will take time to return to their original state and there will be energy losses connected with the chemical changes in each charge-discharge cycle. Errors of 5% are not unexpected. Results When fully charged, the potential difference across the capacitor plates is the same as that across the battery, so the maximum amount of charge that a capacitor can store is proportional to the battery voltage. By measuring the charge flow into and out of the capacitor, the charge stored can be calculated. The Area function of the software enables the charge flow for the time of charge or discharge to be easily calculated. Data as collected with Area tool being used to find the area under the current curve (ma.s). Q = 40.259 ma.s = 40.259 mc by calculation, Q = CV Q = 10000 μf x 4.02 V = 4070 μc = 40.7 mc. The difference can be explained in the tolerances of the devices, capacitors are usually at 20%. The value from the area under the graph is the actual value of the device. Teacher s notes for 02 Charge stored on a capacitor 2
Values obtained for the charge flow in the charge - discharge cycles. Cycle number Charge flow (mc) Discharge flow (mc) 1 41.140 38.697 2 40.259 Not collected Extension The use of fast logging gives a sampling rate of 100 per second compared to the real time sampling of 20 per second. The extra samples will give greater accuracy on the peak values; this is needed if the time constant is low. You will need to put a trigger in place, it is highly unlikely with fast logging that you will be able to start the recording and turn the switch within the time available. A trigger will automate the start recording. Teacher s notes for 02 Charge stored on a capacitor 3
Sensors: Loggers: Voltage, Current Any EASYSENSE Capacitor investigations Logging time: EasyLog (20 s) 02 Charge stored on a capacitor Read Capacitors are devices that can store electric charge; they do not produce the charge. A capacitor was originally made from two metal plates separated by an insulating material called a dielectric. They are now made from a wide range of materials including ceramics, Mylar, polystyrene, tantalum, aluminium and an electrolyte. There are many different types of capacitor available and they come in a variety of shapes and sizes. The unit of capacitance is the Farad and values will range from pico Farads (10-12 ) to Farads. Capacitors are used in: timing circuits, energy storage, frequency filters and for smoothing voltages. In this investigation, you are going to measure the charge that a capacitor can store and then deliver when discharged. RESISTANCE SELECTOR 1k 3.3k 4.7k 680 470 330 100 100k 6.8k 10k 33k 47k Current sensor +5V A R charge Power supply V Voltage sensor discharge 0V What you need 1. An EASYSENSE logger. 2. A Smart Q Current sensor ±100 ma 3. A Smart Q Voltage sensor ±12 or 20 V. 4. Lascells Capacitor investigation module. 5. Lascells Resistance selector module. 6. DC (5 V max) power supply or dry cells in good condition work well. 7. Patch connecting cables with 4 mm plugs. 02 Charge stored on a capacitor 1
What you need to do 1. Connect the Voltage sensor to the Capacitor module, connecting the red lead to the red socket and the black lead to the black socket. 2. Connect the Current sensor to the Capacitor module, connecting the red lead to the red socket and the black lead to the blue socket. 3. Connect the Voltage and Current sensor to inputs of the logger. 4. Connect to the power supply. 5. Note the value of the resistor and capacitor. 6. From the EasySense software s Home screen select EasyLog. 7. Measure the voltage (emf) of the supply with no current flowing using Test Mode (Tools menu). You can do this by temporarily connecting the Voltage sensor across the supply terminals. Replace the sensor into the circuit after taking this measurement. 8. Put the switch in the discharge position to make sure the capacitor is fully discharged before starting. 9. Click on Start, immediately change the switch to the charge position and watch the recording. When the line of the graph appears to show no further change in value with time, change the switch to the discharge position (you should find that the capacitor is, in effect, fully charged / discharged within 12 seconds) and click on Stop to finish recording. 10. Select Overlay. Repeat the charge / discharge cycle. 11. Use Save As to save the recording. Use the switch to make sure the capacitor is discharged fully. 12. Connect the Resistor selector module across the two blue sockets to insert a resistance in parallel to that inside the apparatus. Select a suitable value. Calculate and make a note of the resistance created. Note: For the energy calculations and analysis select a resistance to give a good current graph. With higher resistances the current flow will become smaller 13. Repeat the experiment collecting several cycles of charge and discharge. Theory The charge stored on the capacitor is given by the following equation: Q = C x V Q = the charge in coulombs, C C = the capacitance in farads, F V = the voltage in volts, V When a capacitor is charged, current flows, the relationship between charge and a steady current is: Q = I x t I = the current flowing in amps. If the current is variable, as in this experiment, then the charge flow is the area under the Current vs. Time graph. Analysis of results The graphs you obtained are Current vs. Time and Voltage vs. Time. You will calculate the total charge stored from the Current vs. Time graph. Measuring V max Using Values, measure the maximum voltage in the charge cycle of the Voltage vs. Time graph, this is V max. Write down the value. 02 Charge stored on a capacitor 2
Measuring the charge stored The charge stored during the charge cycle, and the charge delivered during the discharge cycle, are found by measuring the area under the Current vs. Time graph. 1. Click in the area to the left of the Y axis so the Current axis is displayed. 2. Select the Area icon from the toolbar. 3. Go to the first charge cycle on the graph, click on the point where the current starts to increase, and drag the cursor across to the point where current has fallen back to zero*. 4. A value will appear in the data value box showing ma.s, i.e. millicoulombs (mc). In the example graph this is 66.589 ma.s. 5. For comparison, the charge on the capacitor can be calculated from V max and the farad rating of the capacitor. Charge = V x C 6. Note the error rating of the capacitor; the farad rating can have a 20% tolerance. The result collected from the graph will be the measured value for the device. The value written on the device will be its design value. * You may need to apply a tare value to get current to read zero. (Use Values to find the current when there is no charge or discharge. From the Tools menu select Post-log Function, General and Tare. Follow the wizard and enter the no charge value as the tare value). Use this new graph for the energy area analysis. Questions 1. Are there any general differences between the charge and discharge values? Explain why this occurs. 2. A flashgun lamp requires a charge of 0.004 coulombs at 6 volts. What sized capacitor (in μf) is needed to store this charge? 02 Charge stored on a capacitor 3
Extension Use fast logging to obtain more reliable results. Select 4000 samples at a 10 ms interval; this will give 40 seconds of logging time chose an additional resistance value to give a time constant enough for two complete charge-discharge cycles. Investigate the relationship between: 1. Charge vs. Voltage; does the charge stored vary with the charging voltage? 2. Charge vs. Capacitance; Measure the charge stored for different values of the capacitance, maintaining the maximum voltage constant. 02 Charge stored on a capacitor 4