Capacitors and RC Circuits: Pre-Lab Guide and Questions

Updated by: Claude Cournoyer-Cloutier 1 February 9, 2024 LAB 3: CAPACITORS AND THE RC CIRCUIT PRE-LAB QUESTIONS Please complete the following questions before coming to the lab, and bring your answers with you to the lab: 1. What is the main purpose and function of a capacitor? Provide a practical example of how capacitors are used within circuits. 2. When we double the separation between capacitor plates, what happens to the capacitance, C? 3. What is the time constant of a series RC circuit, and what unit does it have? What about for a parallel RC circuit? Explain. 4. When building an RC circuit, does the polarity of the capacitor with respect to the voltage source need to be considered? You can find the answer to this within the lab manual along with specifics regarding our capacitors. 5. Within Microsoft Excel, if you wanted to take a column of over a thousand rows and apply a natural logarithm function to each row value, how would you go about doing that in an efficient manner? Write out the explicit command for the natural logarithm function in Excel along with the process for doing this quickly. a. Note: Don’t worry about the version, the specific command will be given in the manual for the version you will work with.

2 Set yourself the following objectives for this lab: ● Learn the basic properties of a capacitor experimentally and theoretically ● Analyze circuits which have current/voltage changing in time ● Understand the meaning of a time constant and how to find it for half an initial voltage ● Discharge a capacitor in parallel to a resistor and find the cumulated charge through integration ● Charge a capacitor through a resistor in a series circuit and experimentally analyze the results The following shows the value of all the questions in this lab: Laboratory 3, Question Grading Scheme Totals Part 1 Question 1 Question 2 Question 3 Question 4 Points /2 /1 /2 /1 /6 Part 2 Question 5 Question 6 Question 7 Question 8 Points /1 /1 /2 /1 /5 Total number of points in this lab /11 The following shows the value of all the tables and graphs in this lab: Laboratory 3, Results Grading Scheme Totals Part 1 Results/Table Graph Points /3 /2 /5 Part 2 Results/Table Graph Points /1 /3 /4 Total number of points in this lab /9

3 THEORY ELECTROLYTIC CAPACITOR A capacitor is a device which stores charge. Its capacitance (C) is the charge it will hold per volt of applied potential, so that Q = CV, (1) where Q is the charge, and V is the potential difference. Capacitance has SI units in the form of Farads (F) which are equivalent to Coulombs divided by Volts. The simplest form of a capacitor is a pair of parallel conducting plates with surface area (A) separated by a distance (d). In this case, the capacitance is ࠵? = ! ! " # , (2) for plates in air or vacuum, ࠵? $ = 8.85 × 10 %&’ ࠵? %( ࠵?࠵? %& ࠵? ) ࠵? ’ and it is called the permittivity of free space. It is a fundamental constant in electricity and magnetism as it describes the capability of an electric field to permeate a vacuum. Large values of C are obtained by increasing A and minimizing the separation d. Rolling thin foil plates into a cylinder with a plastic layer in between gives a large area in a small package. But for capacitances much greater than 1 μF, a thinner insulator is needed. The electrolytic capacitor replaces one of the plates with a conducting solution of aluminum hydroxide. An extremely thin layer of oxide formed on the other (aluminum foil) plate insulates it from the electrolyte. By reducing the separation d in this manner, capacitances of thousands of μF are readily obtained. However, some compromises are made. First, if the applied voltage makes the electrolyte positive with respect to the foil, the aluminum oxide will be broken down (chemically reduced), and the capacitor will short-circuit internally and possibly explode. All electrolytic capacitors have the polarity of the leads indicated on the package. Those used in this lab have a white band with minus signs indicating the negative side. Always connect the circuit so that the voltage across the capacitor is negative at this end, positive at the other. It is a good idea to check with a voltmeter before the second capacitor lead is connected. Second, there is inevitably some internal leakage through the imperfect insulation layer. This effect is insignificant in this lab, as the capacitors will hold their charge for a few days. There is a third effect which you may notice in your measurements. Because of chemical reactions between the electrolyte and the oxide layer, the capacitor does not strictly follow equation (1). In some ways it acts as a cross between a rechargeable battery and an ideal capacitor. This behavior is worse if the device has been charged up for several minutes. When no measurements

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

4 are being taken, you should disconnect the capacitor from the circuit and short (i.e. connect together) the leads with a wire; this will help to give reproducible results. THE RC CIRCUIT The circuit above is what we call an RC circuit (you might have guessed it, a Resistor and Capacitor circuit). If the capacitor has an initial charge (Q) and voltage (V) associated with it, then the charge will begin to decline once it is connected to the resistor (R). The rate of loss of Q through the resistor is the current I, which in turn depends on V through Ohm’s Law. This gives us a differential equation of (3) where the - sign indicates that Q decreases. Substituting for V using (1) (4) which has the solution (verify by substitution) ࠵?(࠵?) = ࠵?(0)࠵? % " #$ . (5) The potential V therefore declines exponentially as well, ࠵?(࠵?) = ࠵?(0)࠵? % " #$ , (6) Or ࠵?࠵?9࠵?(࠵?): = ࠵?࠵?(࠵? $ ) − * +, , where V 0 =V(0). (6a) The product, RC, has units of seconds (check this for yourself, with R is in Ω and C is in Farads) and is called the time constant of the circuit. It is the time at which the voltage or charge declines by a factor of e ≈ 2.718. For simplicity in the experiment, we are going to find a different time constant. Instead of the time at which V or C declines by a factor of e , our time constant will be when the voltage in dQ dt I V R = - = - dQ dt Q RC = -

5 the capacitor drop to half of its initial value. We will call this time constant ࠵? % & (the Greek letter ࠵? is pronounced “tau”). Analytically, we can find this value by: ࠵?(࠵?) = ࠵? $ ࠵? % * +, , ࠵? = ࠵? & ’ ⇒ ࠵?(࠵?) = ࠵? $ 2 . ∴ ࠵? $ 2 = ࠵? $ ࠵? % - % & +, . ln(2) = - % & +, (7) ⇒ ࠵? = - % & +./(’) (8)

6 SETUP There are only two new circuit elements that we are going to introduce in this lab. One of them is the capacitor, while the other is the switch. Figure 1 shows an image with the capacitor and switch we are using in this lab. We will be using an external capacitor for our circuit. The capacitor has a white band which indicates the negative sign. This switch does not have a polarity associated with it and can be oriented in either direction. Figure 1: Image showing the capacitor and switch that its being used in this experiment. The capacitor is shown on the left, while the switch is shown on the right. Note: the white band on the capacitor indicates the negative terminal for the capacitor.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

7 PART 1: Discharging the Capacitor Part 1 is divided into two sections. The first section of Part 1 will involve determining the time constant ࠵? % & for the discharging RC circuit. The second section is dedicated to finding the capacitance for our capacitor through linear regression of the voltage versus time plot. PROCEDURE Construct Circuit_1 as shown above. Make sure the capacitor’s polarity is correctly indicated as shown in Circuit_1. The white band with minus signs on the capacitor indicates the negative side. Always connect the circuit so that the voltage across the capacitor is negative at this end and positive at the other. The SW1 circuit element is a switch which allows you to open and close the circuit as intended. There is a PASCO circuit building block which acts like a switch. Finding the Time Constant ࠵? ࠵?/࠵? This part of the procedure is dedicated to finding the time it takes for the voltage on the capacitor to reduce to half of its initial value (the time constant ࠵? % & ). 1. Construct Circuit_1 by using the R1=100 ࠵?࠵? resistor and making sure the capacitor’s polarity is correctly oriented.

8 ****WARNING**** (Ask the lab coordinator or instructional assistant to check the polarity of your Capacitor in the circuit. YOU MUST, have this cleared by the lab coordinator or instructional assistant before turning on the power supply) 2. Set the power supply within Capstone to the settings: a. Waveform=DC. b. DC Voltage=8V (this will change later on in the procedure) and turn it on. c. At the bottom of the page, set the frequency of measurement for the Wireless Voltage Sensor to 10Hz (this will record a value for voltage every 0.1 seconds). 3. Close the circuit with the switch (closing the circuit means it is connected) and measure the voltage across the resistor. When the capacitor (or resistor, since they are in parallel) is at the voltage of the power supply, open the circuit but continue recording. You can end the recording when the voltage begins to asymptote to zero. a. Note: your voltage should look like a negative exponential function over time which starts at the initial voltage. 4. Double click on any point on the curve and select the “Add Coordinate” option. Here you will find the time it takes for the voltage across the resistor to drop to half of its initial value ( ࠵? % & ) : a. Select the time when the voltage begins to drop from its initial value. b. Select the time when the voltage is half of its initial value. c. Take the difference between the two times found in steps a and b. Record this value in your report. For the R=1000 ࠵? at an initial voltage of 8V, save this run on a separate tab in Capstone. You do not need to save the next set of runs, only this one. We will be using this set of data in the next section.

9 CHECKPOINT (Ask the TA to double check your circuit and the ࠵? ࠵? ࠵? value) 5. Repeat steps 1-3 with the power supply set to 6V and then with the power supply set to 4V. In the report, calculate the average and standard deviation of ࠵? ࠵? ࠵? for that resistor’s value with the different voltage values. Use this standard deviation as the uncertainty for your ࠵? % & value. a. Note: make your life easier and use Excel to calculate these standard deviations for you. You can use the STDEV() function and select the values for ࠵? % & into the functions arguments. You can also calculate the average value using the AVG() function in Excel. 6. Use resistors R1=5000 ࠵? and R1=10k ࠵? and repeat steps 1-4. Calculate the capacitance C for each of these resistors by using equation (8) and your measured ࠵? % & values. Record the value of your capacitance for each resistor in the report. Then calculate the average value for the capacitors and add it to your report. Question 1. Does the ࠵? % & value seem to depend on the initial voltage? Justify your answer. Question 2. Is the average of your calculated capacitance C within 20% of the nominal value, as the manufacturer claims? If the calculated capacitance is not within 20% of the nominal value, provide possible sources for the discrepancy. Linear Regression with Voltage versus Time Using Circuit_1, with R=1k Ω and an initial voltage of 8V, we are going to apply linear regression to a voltage versus time curve. 1. If you did not save your first curve, recreate the voltage versus time graph for R=1k ࠵? at an initial voltage of 8V at 10Hz. You can refer to the previous procedure on how to do this. 2. Take the data and prepare it for linear regression. a. Remove every data point that is at ࠵? ࠵? = ࠵?࠵? besides the last one. This last data point will be your initial time ࠵? ࠵? , so record its value in the report. To remove some data from your graph: i. Select the option “Highlight range of points in active data” (this icon looks

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

10 like a series of highlighted points) and cover the highlighted box over the desired data to be deleted. You can move and resize the box as needed. Then right click the highlighted box and select “Deletions->Delete Highlighted Data Points”. b. Remove every data point that is after ࠵? ࠵? + ࠵?࠵? ࠵? ࠵? . Use the average value of ࠵? % & you previously calculated for R=1k Ω . 3. Export the data for this graph and import it into Excel. To do this: a. Select “File” tab within Capstone and then “Export Data”. The data file generated should automatically be in a comma separated value (CSV) format. b. Select “Export to file” in bottom right corner. Name the file an appropriate name and save it to your local directory (folder). c. R ight click on the csv file you just saved. Then go to “open with” → “Microsoft Excel”. 4. With the data imported into Excel, now we are going to create new columns for ln (࠵?) and for ࠵? − ࠵? $ . For the ࠵?࠵?(࠵?) column: a. Write a title for ࠵?࠵?(࠵?) at the top of the column (don’t worry about units). b. Write a function in the cell that takes in the adjacent Voltage value cell (probably B2 for your imported data set) and apply the natural logarithm function. You need to do this for the first two cells in the column. Excel’s built in natural logarithm function is given LN() . i. To enter a numeric function into Excel, don’t forget to start with an equal sign, for example, type: “=LN(B2)” to calculate the natural logarithm of the B2 value. ii. Each row entry in the column should look something like: = LN(࠵?2), = LN(࠵?3), …. c. Select two of the cells in the column and autofill the remaining rows by double clicking on the bottom right square. This will automatically fill in each row based on the adjacent row values in the column. 5. For the ࠵? − ࠵? ࠵? column, write every value with respect to ࠵? 6 , since that is where the exponential decay started. To do this within Excel: a. Write a title of ࠵? − ࠵? ࠵? at the top of the column with SI units of seconds. b. Write a function in the cell that takes in the adjacent time value cell (probably A2 for your imported data set) and subtract by the constant value of ࠵? ࠵? . i. Your function should look like: “= ࠵?2 − ࠵? $ ” where t o is your initial time. Each entry in the column should look something like: = A2 − t 7 , = A3 − t 7 , etc.

11 c. Select two of the cells in the column and autofill the remaining rows by double clicking on the bottom right square. This will automatically fill in each row based on the adjacent row values in the column. CHECKPOINT (Ask the TA to check your Excel file and see if you properly calculated the ࠵?࠵?(࠵?) and ࠵? − ࠵? ࠵? columns.) 6. Create a plot of ࠵?࠵?(࠵?) versus time for a domain of ࠵? ∈ [࠵? ࠵? , ࠵? ࠵? + ࠵?࠵? ࠵? ࠵? ] from the loaded data in Excel. There are multiple ways to do this, but the easiest is to: a. Click the “Insert” tab → “Charts” → “Insert Scatter (X,Y) or Bubble Chart” → “Scatter”. b. Right click the empty graph, select “Select Data” → “Add”. c. Left click under “Series X values:” → highlight the current column. Likewise, do this for the “Series Y values:” for the Voltage column. 7. Give your plot a title and appropriate axis labels. To do this: a. Left click anywhere on the graph, select “Chart Design” tab, then select “Add Chart Elements”. 8. Insert a trendline on the graph and display the equation & ࠵? ࠵? value on the graph. To do this: a. Right click the data series in the graph, select “Add Trendline…” → “Display Equation on chart” → “Display R-squared value on chart”. 9. Use the slope of the trendline you just found, along with Equation (6a), to determine the capacitance of your circuit. Add your plot to the report. Make sure your plot has properly labelled axes and titles. ***CAUTION*** Make sure you are using the natural logarithm function in Excel and that your time is properly plotted with respect to ࠵? ࠵? . Equation (6a) is given as ࠵?࠵?9࠵?(࠵?): = ࠵?࠵?(࠵? $ ) − * +, , where ࠵? $ = ࠵?(0) . Question 3. Is this graphically determined value of C consistent with your previous average C

12 value? If not, give reasons why you think this discrepancy might have occurred. Question 4. What is the significance of the y-intercept for your trendline? Recall: time needs to be written with respect to ࠵? $ .

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

13 PART 2: Charging through a Resistor PROCEDURE 1. For this part of the experiment, we need to ensure the capacitor is fully discharged before we construct Circuit_2. To do this, take a cable or wire with small resistance and hold its two ends to the two ends of the capacitor for a few seconds to discharge. This will discharge the capacitor at a relatively fast rate . 2. Now construct Circuit_2 as shown above with the R2=100 ࠵? resistor. Make sure your capacitor is fully discharged before doing this (refer to step 1 if you have not done so). To keep the capacitor discharged, make sure the switch is left open when connecting it to the circuit. 3. On Capstone, create two graphs. One graph will show voltage over time, and the other graph will show current over time. These are measured directly from the wireless voltmeter and the current module. 4. Set the power supply within Capstone to the settings: a. Waveform=DC. b. DC Voltage=1V c. Set the frequency of measurement at the “Common Rate” of 500Hz at the bottom of the page (this will record a value for voltage and current every millisecond simultaneously). 5. When you have Circuit_2 constructed with the power supply connected and turned on, click record. Make sure the voltage still reads ~ 0 V. If it doesn’t, take a cable or wire with small resistance and hold its two ends to the two ends of the capacitor for a few seconds to discharge. Then stop, remove the cable and click record again so that the initial voltage is ~ 0 V. After a few seconds (<5 seconds), close the switch and observe the current and voltage curves. About 10 seconds after closing the circuit, stop recording. a. According to theory, the current should look like a negative exponential - + _ 2

14 ࠵?(࠵?) = ࠵?(0)࠵? % " #$ . (9) The voltage will analogously look like ࠵?(࠵?) = ࠵?2 [1 − ࠵? % " #$ \. (10) where ࠵?2 is the voltage of the power supply. CHECKPOINT (Ask the TA to check your graphs. If they are incorrect, repeat the procedure from step 1. The capacitor will need to be discharged before you can redo these measurements.) ****IMPORTANT **** If you are not getting the desired result, you may need to discharge your capacitor before going through this procedure again. To do this, take a cable or wire with small resistance and hold its two ends to the two ends of the capacitor for a few seconds to discharge. 6. For the current versus time graph, click on “Highlight range of points in active data” at the top of the graph and select the range of points from when the switch was closed to the final time. Then click on “Display area under highlighter data” at the top of the graph (it will look like a parabola with grey shading underneath). When you click this, it should give an area underneath the data for the Current versus time graph for the selected range. 7. Theoretically, the area under the current versus time graph is the total charge accumulated by the capacitor. ࠵? = − ࠵?࠵? ࠵?࠵? ⇒ ࠵? = − ^ ࠵?(࠵?)࠵?࠵? *8 * ! Write down the total charge accumulated from when the switch was closed to the final time. 8. Using the value of ࠵? you found in the previous step, calculate the capacitance and write both of their values in the report. 9. Copy and paste your graphs from Capstone onto your word report. Question 5. What other types of systematic and random uncertainties may exist within this experiment for Circuit_2?

15 Question 6. When calculating the capacitance from the total accumulated charge, does it match the manufacturer’s claim that the capacitance is within 20% of the nominal value? Explain. If the calculated capacitance is not within 20% of the nominal value, provide possible sources for the discrepancy. Notes on the definition of ࠵? relevant to Question 7. The value of ࠵? (this is different from the ࠵? % & value you previously calculated) is what we call the time constant for the RC circuit and it remains the same regardless of if we are charging the capacitor through a resistor or discharging the capacitor through a resistor. We can determine the time constant from Equation (10) and it can be evaluated as ࠵? = ࠵?࠵? with SI units of seconds. For the charging RC circuit (Part 2), ࠵? represents the time it takes the voltage to reach 63% of the power supplies voltage. Question 7. Using the manufacturer’s middle value of ࠵? = 10000࠵?࠵? , what is the value of the time constant ( ࠵? = ࠵?࠵?) for Circuit_2 in SI units? At time ࠵? = ࠵?࠵? on your Voltage versus Time graph, is the value for voltage what you expect? Explain. Note : for voltage of the power supply, use the voltage that your voltage versus time graph asymptotes to. Question 8. Repeat the calculation of the time constant ( ࠵? = ࠵?࠵? ) with your experimentally determined values of ࠵? from Part 1 instead. Do they provide a more accurate result for ࠵? as per the definition of the time constant? Explain.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

lab3_manual_W24

Related Documents