Capacitors

(last edited May 28, 2013)

Dr. Larry Bortner

 

 Learning Objectives

 

 

Find the capacitance of an element by analyzing the voltage decay in an RC circuit.

 

 

Explain equivalent capacitance of capacitors in series and in parallel.

 

 

Translate schematics to real-world connections: resistors, capacitors, series and parallel circuits.

 

 

 Purpose

 

 

To measure capacitance. To investigate the exponential time dependence of charge accumulation and discharge on a capacitor plate. To find the effective capacitance of the series and parallel combinations of two different capacitors.

 

 

 Background

 

 

Please read the section on Circuit Diagram Basics in the Background of the Potential Differences lab.

 

 

 

 

 

The electric force between two charges is very strong. The net charge of the universe is zero and it is the natural state of a single charge to be as close as possible to one of the opposite polarity. Of course, there can be instances when the net charge in a particular area is nonzero.

 

 What is a Capacitor?

A capacitor presents a case where the charge is nonzero in a localized area. This physical entity is an electrical energy storage device that is best visualized by two parallel metallic plates separated by a given distance. (Other capacitor configurations are used but the concepts laid out in the following paragraphs for the two plates hold for different geometries.) A battery or some other generator of electric potential difference is used to separate tightly bound positive and negative charges, pushing charges of one sign to one plate and charges of the other sign to the second plate. The amount of charge that eventually accumulates on either of the plates is proportional to the charging voltage V0. There is a nonzero charge on either plate but the net charge of the capacitor (both plates) is still zero.

 Capacitance

In general, capacitors have different capacities to store charge, depending on the geometry and the material between the plates. This charge storage capacity, or capacitance, is the constant of proportionality C between the magnitude Q of the charge on one of the plates and the potential V across the plates. In equation form, this relationship is

 

(1)                     

 

The SI unit for capacitance is the farad, which is a coulomb per volt (1 F = 1 C/1 V). The unit we will use in this experiment is the microfarad (μF).

 Charging and Discharging
Exponential Decay and the Time Constant


Simple enough measure the charge on a capacitor for different charging voltages, plot the line and the slope is then the capacitance, right? The experimental problem comes in measuring Q. Although charge-measuring instruments exist, they are not useful for much else and are expensive and imprecise.

 

How then do we measure the capacitance? First look at how the charge collects on a plate as a function of time.

Figure 1.jpg

Figure 1 RC circuit for charging and discharging a capacitor.

 

We set up a circuit as in Fig. 1. Connect a resistor R and a capacitor C in series to a voltage source through a two-pole switch. This is called an RC series circuit. Connecting the switch to terminal 1 charges the capacitor, with the rate of the charging decreasing as time passes. At first, the capacitor is easy to charge because there is very little charge on the plates. But as charge accumulates on the plates, the voltage source must do more work to move additional charges onto the plates because like charges repel each other. Thus the capacitor charges in an exponential manner, quickly at the beginning and more slowly as the capacitor gets closer to being fully charged. If this connection is made at t = 0, the charge Q on one of the plates after a time t is

 

(2)                     

 

Q0 is the magnitude of the maximum charge that accumulates on a plate.

 

Since the voltage across the plates and the charge on one plate are proportional, we can write

 

(3)                     

             

Resistance is an electrical property that impedes the flow of charges through a circuit. It is measured in ohms (signified by the Greek letter Ω). A series connection of two things means that whatever goes through one has to go through the other.

 

When the capacitor is fully charged and the switch is flipped to 2 at t = 0, there is an exponential decay of the charge:

 

(4)                     

 

with the time-dependent voltage being

 

(5)                     

 

The constant RC in the exponent of all these equations is called the time constant. It is a measure of how fast the circuit reacts. It is also a number that characterizes the exponential decay. Note that an ohm times a farad is a second. (1 Ω × 1 F = 1 s)

 

We use either Eq. 3 or Eq. 5 to find the capacitance, by fitting voltage and time data to these equations. Experimentally, we charge up a capacitor with a potential of V0  while monitoring the voltage across the capacitor as a function of time. An exponential fit of V0 − V(t) gives the time constant RC. If R is known, we can then calculate C. The discharge of a capacitor is easier to analyze, since an exponential fit of V(t) should give the same time constant.

 

We use a signal generator to electronically flip the switch to 1 then 2 in Fig. 1 then back to 1 and again to 2. A signal generator, clearly, is a voltage source that supplies a time-varying voltage. (It's no longer DC, it's AC.) The symbol for this in a schematic is a squiggle inside a circle. A voltmeter (something that measures voltage) is a capital V inside a circle. The schematic of our test circuit for measuring capacitance becomes Fig. 2.

 

RC test circuit.jpg

Figure 2 Schematic of RC test circuit used in the lab.

 Equivalent Capacitance of Connected Capacitors

We can connect two capacitors C1 and C2 in a simple circuit one of two ways: either in series or in parallel. In either case, the two capacitors act in the circuit like a single equivalent capacitance Ceq. What is the value of this capacitance?

 

Figure 3 Two capacitors connected in parallel.


 

In the parallel case (Fig. 3), there are two possible paths for charges to take coming from a terminal of the power supply. The potential difference across each capacitor is the same. Different amounts of charge collect on the two positive plates. The net charge deposited on the combination is the sum of the two charges. Thus the equivalent capacitance is the total charge divided by the potential difference, or the sum of the individual capacitances.

 

(6)                     

 

Figure 4 Two capacitors in series.


 

In the series case (Fig. 4), there is only one possible path for charges to follow and the charge accumulation has to be the same on both capacitors. The potential difference across both resistors is the sum of the voltages across each separately. The reciprocal of the effective (equivalent) capacitance is the sum of the reciprocals of the two separate values.

 

(7)                     

 

 Pertinent Questions

There are several objectives or questions to answer in this experiment. You must address these points in the Experimental Results and Conclusions section of your report. They are as follows:

 

1.    Does the voltage across a discharging capacitor (and hence the charge on a plate) decay exponentially?

2.    If it is exponential, calculate the capacitance and compare it to the value measured with a meter based on a different technique.

3.    Does the voltage decay in an exponential manner according to Eq. 3 when the capacitor charges? Is the capacitance calculated from this the same as when it discharges?

4.    Measure a second capacitor of a different nominal value, using the technique of analyzing the voltage decay on discharge. Does it give the same result as the capacitance meter?

5.    Connect these two capacitors in parallel and measure their combined capacitance using the voltage decay technique. Is it the same as the theoretical prediction of Eq. 6?

6.    Connect the two capacitors in series and use the decay technique to find the equivalent capacitance. Is it the same as that predicted by Eq. 7?

 

 Procedure

You need the following equipment:


 

*        wood-based circuit board with 20 junctions

*        Wavetek 27XT digital multimeter (DMM) for measuring capacitance (The displayed value has an uncertainty of 5% of the reading +10 μF on the 2000 μF scale, as specified by the manufacturer. An equivalent way of saying this is that the accuracy is 5% +10.))

*        ScienceWorkshop 750 computer interface

*        voltage sensor (analog input to ScienceWorkshop 5-pin DIN plug with two banana plugs)

*        two capacitors, nominally 330 μF and 100 μF

*        330 W resistor (actual value posted on Electronics Laboratory)

*        banana plug leads (4)

 

 Capacitance from Meter

1.     Turn on the Wavetek DMM and turn the dial to the 2000 μF setting to read capacitance.

a.     Use the banana plug leads to connect the two sides of the 100 mF capacitor to the mA Cx Lx and COM inputs of the meter.

b.     Give the meter 20 or 30 seconds to settle down before recording the reading as C1.

c.     Do the same thing for the 330 μF capacitor and record the value as C2.

d.     Record the value of R ± u{R} from the value on the board.

 

meter measurement of capacitor 4.jpg

Figure 5 Measuring the capacitance with the DMM.

 Computer Data Acquistion

2.     You need to construct the test circuit of Fig. 2. A picture of what it should look like when you're done is shown in Fig. 6.

 

series RC circuit phoito 3.jpg

Figure 6 This is a photo of the wiring and connections for the test circuit of Fig. 2. (The top capacitor is not in the circuit.)

 

a.     Connect the DIN plug of the Voltage Sensor to Channel B of the ScienceWorkshop interface.

b.      Connect a banana plug lead from the ~ jack of the OUTPUT on the interface to the banana plug receptacle on the circuit board shown in Fig. 6. (One end of the capacitor is connected to this.)

c.     Connect a banana plug lead from the other jack (the ground) to the bottom left plug on the board. (One side of the resistor.)

d.     Connect the red banana plug of the Voltage Sensor to the ~ OUTPUT on the board, through the banana plug already there.

e.     Connect the black lead of the Voltage Sensor to the other side of the capacitor.

f.      Connect a banana plug lead from the other side of the capacitor to the other side of the resistor.

Checkpoint! Have the TA check your circuit before you proceed further.

3.     Start the computer program by clicking on Data Studio Experiments> Second Quarter> Capacitors. DataStudio starts up with a Workbook sheet that gives instructions and other information on the experiment.

a.     Double click on the Voltage Graph in the Displays menu of DataStudio to view the data as it is acquired and press Start to begin data acquisition. The "switch" is turned on for about a second then off for the same time and repeated. Voltage is measured across the capacitor and plotted. Your screen should look like Fig. 7. If nothing shows up or you have all negative voltages, you have a connection problem. Consult the TA. Record the Run # and which capacitor you're using.

 

V(t) for C1.jpg

Figure 7 Voltage behavior across a capacitor on discharging and charging.

 

b.     Replace C1 with C2 and repeat the voltage measurement. Don't forget to record the Run # and the capacitor. The plot will be superimposed on the first graph.

c.     Connect C1 and C2 in parallel as drawn in Fig. 3. (Your assembled product should look something like Fig. 8.) Take voltage data for this combination and record the Run # and what you're measuring.

 

capacitors in parallel 3.jpg

Figure 8 Fig. 3 realized.

 

d.     One more experimental run! Connect C1 and C2 in series as in Fig. 4. (Your assembled product should look something like Fig. 9.) Take voltage data for this combination. Record the Run # and the capacitor combination.

 

capacitors in series 2.jpg

Figure 9 Fig. 4 realized.

Checkpoint! Have the TA check the DataStudio graph of all four data runs before you proceed further.

 

4.     We have to be able to get this data into Excel.

a.     In DataStudio, click on File> Export Data...> the Run # for C1 in the Voltage, ChB list> OK.

b.     Save to the Desktop as a text file with a suitably descriptive file name.

c.     Go to the Desktop and right click on this file.

d.     Click on Open With  Excel. An open file cannot be deleted by the clearing routine run every half hour.

5.     Repeat this data transfer for the C2 data.

6.     Transfer data from DataStudio to the Desktop for the parallel combination data.

7.     Repeat for the series combination data. You should have four open Excel files.

 

 Analysis


 

New units to remember:

1.     capacitance: 1 F=1 C/V

2.     capacitance: 1 μF=10-6 F

3.     resistance: 1 Ω=1 s/F

 Transferring Data to Excel

1.     Open the Capacitors template, click in the blue box, enter in your name(s), and save the file following the standard naming conventions.

2.     Go to the Excel file containing the C1 data.

3.     This file contains 2000 points, most of which you don't need. First you need to find V0, the maximum voltage across the capacitor. (Yes, we are applying 4 volts across it, but there's no uncertainty associated with that, so we'll check what we measured.)

a.     Go to the end of the data (hit Ctrl End). This takes you to the last voltage value, for t=4 s.

b.     Go one cell further down (B2004) and click on the fx by the formula bar to bring up the list of Excel functions. Find the MAX function and choose it. A box will appear of Function Arguments, with all the numbers in the column above already chosen.

c.     Click OK. The maximum value appears. Write this down on your data sheet.

4.     The next task is to filter out the data for C1 discharging.

a.     Scroll to where the time is 1.13 seconds. This is about where the switch is first turned off and the voltage starts to decay.

b.     Find the first point where the voltage is 40 or 50 mV less that the maximum.

c.     Select all points (times and voltages), including this one, down to where the voltage is just less than 0.2 V. (There should be 70-80 points.)

d.     Copy this selection.

e.     Return to the ca file started from the template.

f.      Go to cell A22.

g.     Paste the selected points.

5.     Return to the C1 data file to retrieve the charging data. The positive square wave put out by the Signal Generator has a frequency of 0.4 Hz, meaning it has a period of 2.5 s; it is on for half the period and off for the other half. So 1.25 s after the charging signal goes to zero at t=1.13 s, it goes back to 4 V at t=1.13 s + 1.25 s = 2.38 s.

a.     Scroll down until the time is 2.38. Voltages just before this time should be zero, ± a few millivolts. Voltages just after this should increase precipitously.

b.     Find the first point after t=2.38 s where the voltage is at least 0.020 V.

c.     Select this point and all points after it where the voltage is less than 3.8 V.

d.     Copy this selection.

e.     Return to the ca file started from the template.

f.      Go to cell F22.

g.     Paste the selected points.

6.     Go to the C2 Excel data file. We want the data where it is discharging.

a.     Scroll to where the time is 1.13 seconds.

b.     Find the first point where the voltage is 40 or 50 mV less that the maximum.

c.     Select all points down to where the voltage is just less than 0.2 V.

d.     Copy this selection.

e.     Return to the ca file started from the template.

f.      Go to cell K22.

g.     Paste the selected points.

7.     Go to the parallel combination Excel file to retrieve its discharging data. Repeat the procedure in Step 6, pasting the data into P22.

8.     From the series capacitor combination, we want the discharging data pasted into U22.

9.     In the ca file, enter in the data for V0, R, u{R}, C1, and C2.

Checkpoint! Have the TA check all the data you have entered into Excel before you proceed further.

 Calculations

10.  Calculate the values of t − t0 for all times for each data set. The first value should be zero for each.

11.  Calculate the normalized voltages (V/V0) for each of the four discharging data sets.

12.  Calculate 1-V/V0 for the charging data of C1.

13.   For C1 Discharging, do an exponential least-squares fit using LOGEST of the normalized voltage (as the y-value) vs. the time (as the x-value). (Details on this function can be found in Appendix 3.)

a.     Highlight the two cells just to the right of the e-1/RC and u{1/RC} (1/s) labels.

b.     Perform the LOGEST function evaluation of V/V0 vs. t − t0. Set the two logical flags to true, or one. Hit Shift/Control/Enter. This gives you the two numbers indicated by the labels, for this data set.

c.     Calculate RC = −1 / ln(e−1/RC) .

d.     Calculate the relative error in RC. This is the second number from LOGEST times RC.

 

(8)                     

 

14.  Calculate the capacitance in microfarads from the values of R and RC.

15.  Find the uncertainty (the absolute error) of this calculation of C, given by

 

 (9)        

 

      Note that you have already calculated the value of u{RC}/RC and that u{R} and R are given.

Checkpoint! Have the TA check your calculations before you proceed further.

16.  Hold off on any comparisons for the moment. Repeat the calculations in steps 13-15 for the other four data sets. (Use 1−V/V0 as the y-value for C1 Charging.) To save keystrokes, use the power of Excel...

a.     Highlight all the numbers from e−1/RC through u{C} (μF).

b.     Copy this selection and paste into the corresponding cells for the other four data sets. You are actually copying the formulas, which use relative cell references, so they are good for each data set, except for the LOGEST ranges.

c.     Correct the cell ranges for LOGEST for the other four data sets.

 Comparisons

17.  Now for the theoretical values (with errors) and the comparisons.

a.     The theoretical value for the first capacitor, discharging, as well as the second capacitor, discharging, is the capacitance meter value. Enter the meter values into the appropriate cells, calculate the uncertainties, and compare the experimental and meter values.

b.     For the first capacitor, charging, use the value of the first capacitor, discharging as the theoretical value. Remember that we want to show that the exponential decay is characterized the same way, whether the capacitor is charging or discharging.

c.     For the two combinations of the two capacitors, the theoretical value is calculated from Eq. 7 or 8. Use the experimental values of C1 and C2 in these calculations, not the meter values. You are responsible for propagating the uncertainties for both quantities.

 Plots and Verifying Exponential Dependence

18.  Do a semilog plot of the five data sets (the ones you did the least squares fits of) on the same graph.

a.     Select the two columns for C1 Discharging.

b.     Insert> Charts> Scatter> Scatter with only Markers.

c.     Chart Tools> Design> Move Chart> New Sheet> OK.

d.     Chart Layouts> Layout 9.

e.     Change Chart and Axes Titles appropriately.

f.      Layout 9 automatically includes a Linear Trendline. You want to make this Exponential. Right click on the line and choose Format Trendline…> Exponential> uncheck Display equation on chart and Display R-squared value on chart> Close.

g.     To make this a semilog plot, choose Layout> Axes> Axes> Primary Vertical Axis> Show Axis with Log Scale.

h.     You still want the other four data sets on this plot.

i.      First choose Design> Select Data> Edit Series 1 and go back to the Analysis sheet to choose cell A5 (or you can type in the name)> OK.

ii.     Still in the Select Data Source window, choose Add and from the Edit Series window, choose the name, X-values, and Y-values for C1 Charging.

iii.    Add an Exponential trendline for this data set. (Layout> Trendline> Exponential Trendline.)

iv.    Repeat Steps ii and iii for the remaining data sets.

i.      The point of a graph is to tell at a glance if the data fit the general theory. In this case we expect a straight line on the semilog plot. If it is not a semilog plot, you will get a curve; it is hard to tell if this curve is exponential or hyperbolic or some other strange dependence.

j.      Do the points for each plotted set fall pretty much on a straight line when the y-axis is scaled logarithmically and the x-axis scaled the good old-fashioned linear way?

k.     Delete the references to the trendlines in the legend.

 Questions

1.     Derive Eq. 9 (propagate the error for C).

2.     Propagate the uncertainty of the equivalent capacitance of two capacitors in series.