Capacitors
LBS 272L
Before you start this lab:
• Read the LBS272 lecture material in ‘Time-varying currents”, up to “RLC-AC Circuits”
• Read this manual
• Finish lab quiz 6.
• Note: This lab contains 2 possibilities to score a half a bonus point (so one bonus point in
total)
• Note 2: You will have a break from error analysis this week!
PURPOSE
The purpose of this laboratory activity is to investigate how the voltage across a capacitor
varies as it charges and to find the capacitive time constant. The first measurement will be
described in great detail, after which you have to construct the second part by yourself.
THEORY
When a DC voltage source is connected across an uncharged capacitor, the rate at which
the capacitor charges up decreases as time passes. At first, the capacitor is easy to charge
because there is very little voltage 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 the plates
already have voltage of the same sign on them. As a result, the capacitor charges exponentially,
quickly at the beginning and more slowly as the capacitor becomes fully charged. The charge on
the plates at any time is given by:
q = q 0 (1 − e − t / τ )
Equation (1)
where qo is the maximum charge on the plates and τ is the capacitive time constant ( τ = RC,
where R is resistance and C is capacitance). NOTE: The stated value of a capacitor may vary by
as much as ±20% from the actual value. Taking the extreme limits, notice that when t = 0, q = 0
which means there is not any charge on the plates initially. Note that the capacitor reaches the
applied voltage asymptotically.
The time it takes to charge the capacitor to half full is called the half-life and is related to
the time constant in the following way:
t 1 2 = τ ln2
Equation (2)
BONUS QUESTION (half point): SHOW THAT EQUATION 2 IS VALID.
RC Write-up
Page 1
In this activity the charge on the capacitor will be measured indirectly by measuring the
voltage across the capacitor since these two values are proportional to each other:
q = CV Equation (3)
PROCEDURE
In this activity, the Power Amplifier produces a low frequency square wave (0 to 2 V).
This waveform charges and discharges the capacitor, by imitating the action connecting and then
disconnecting a DC voltage source. The Voltage Sensor measures the voltage across the
capacitor as it charges and discharges.
The Data Studio program records and displays the data. First you will measure the time
for the capacitor to charge to the “half-maximum” voltage with the cursor feature of Science
Workshop. Using the half-life time and the known value of the resistor, you can calculate the
capacitance of the capacitor. Next you will next fit the voltage versus time with a function of the
proper form. The procedure will then be repeated with an increased value for the capacitance.
PART I: Setup
Equipment Needed:
Pasco Interface with voltage probe
Pasco Signal Generator with connection cable
Pasco LRC Circuit Board
Cables
1.
Make sure that the Data Studio interface is connected to the computer, and that it is on.
2.
The Voltage Sensor should be connected to Analog Channel A and the Power Amplifier to
Analog Channel B.
3.
Open the Data Studio document “Capacitors.” The document opens with a Graph display
of Voltage (V) versus Time (sec), and the Signal Generator window which controls the
Power Amplifier.
4.
The Sampling Options for this activity are: Periodic Samples = Fast at 500 Hz and Stop
Condition = Time at 4.00 seconds.
5.
The Signal Generator is set to output 2.00 V, “positive only” square AC Waveform, at 0.40
Hz. It is set to Auto so the Signal Generator will start automatically when you click
START and stop automatically when you click STOP or PAUSE.
RC Write-up
Page 2
6.
The experimental circuit should be set up as shown in figure 1. The resistor (100 Ohm)
should be put in series with the capacitor (330 µF). The voltage is measured over the
capacitor only.
Figure 1.
In practice, the setup is built in the following manner (refer to figure 2) The Voltage
probe measures the voltage across the 330 µF capacitor (it is connected to both ends of
the capacitor, i.e. connecter C and E). The Power Amplifier is applied to the capacitor
and resistor in series. Connect the black outlet of the amplifier to the top plug of the
100Ω resistor and red output of the amplifier to the top plug (C) of the capacitor (you can
connect the banana plugs on top of each other). An additional cable connects the bottom
plug (B) of the resistor to the bottom plug of the capacitor (E). You may notice that the
inductor coil (the large spool of wire at the top of the board) is in parallel with this
circuit. Don't worry about it, it won't affect the results of the experiment.
RC Write-up
Page 3
Figure 2.
Part II: Data Recording
1.
Click the REC button to start collecting data. The Signal Generator output will
automatically start when data recording begins.
2.
Data recording will continue for four seconds and then stop automatically. You should see
a graph something like shown in figure 3. If the signals are negative, exchange the plugs on
the amplifier.
•
Run #1 will appear in the Data list in the Experiment Setup window.
Part III ANALYZING THE DATA
1.
Click the Autoscale (top most-left) button in the Graph to rescale the Graph to fit the data.
2.
Click the Magnifier button (top, fourth from the left). Use the cursor to click-and-draw a
rectangle over a region of the plot of Voltage versus Time that shows the voltage rising
from zero volts to the maximum volts. (The graph shown in figure 1 is for 4V, but the
shape should be the same.)
RC Write-up
Page 4
Figure 3
•
This will give you an expanded view of the Voltage versus Time plot for that region.
3.
Click the Smart Cursor button (top, sixth from the left). The cursor changes to a crosshair
when you move the cursor into the display area of the Graph.
•
The X and Y-coordinates of the cursor/cross-hair is shown next to it.
4.
Drag the cursor to the point on the plot where the voltage begins to rise. Record the
corresponding time.
5.
Move the Smart Cursor to the point where the voltage is approximately 1.00 volt. Record
the corresponding time.
6.
Find the difference between the two times and record it as the time to “half-max”, or t1/2.
7.
Deselect the Smart Cursor and then click the Magnifier button. Use the cursor to clickand-draw a rectangle over a region of the plot of Voltage versus Time that shows the
voltage rising from zero volts to the maximum volts. Make sure none of the data that is
taken with zero voltage is included in the selected region is (Select something like the dark
area in the figure 4)
8.
Select the “Display” in the top bar of DataStudio and then “Export Data” to export the data
in the selected region to a file in the Student Area of the hard drive. Give it a name you can
remember.
9. Open the table data with Kaleidagraph (use the import function). If you have more than one
entry that has zero voltage at the beginning of the table, you can still remove it in
kaleidagraph. Make a nice plot (use “Gallery->Linear->Scatter”) of voltage versus time and
then fit it. Here is how to do the fit correctly:
a. Click the window that displays your graph. Click “Curve Fit->General->Edit
general…” in the top bar. A window will open. Generate a “New Fit” by clicking the
RC Write-up
Page 5
“Add” button. Rename the “New Fit” to something reasonable (RC for example), by
clicking on “New Fit” in the l.h.s. box and typing the new name in the box on the
l.h.s. bottom corner. Click okay.
b. Make sure that “Display Equation” under “Plot” is selected.
c. Click “Curve Fit->General->RC (or whatever name you chose)”. Select the column
you want to fit (you only have one choice). The click “define”
d. A new window opens in which you have to type the function to fit the data with and
reasonable start parameters for the unknowns.
Type: m1*(1-exp(-(m0-A)/m2));m1=2;m2=0.03
Where A should be replaced with the time in seconds corresponding to the first
data entry in your table. m0 is the running parameter (time).
This equation stands for:
[
V (t ) = Vmax 1 − e − ( t −t0 ) / τ
]
Equation (4)
where V(t) is the voltage as a function of time, Vmax (m1) is the maximum voltage, t0
(A) is the start time of the measurement and τ (m2) the capacitive time-constant
mentioned before. Vmax=2 V, τ=0.03 s and t0=A(your start time) are the starting
values for the fit. Note that this equation really is a combination of Equations (1) and
(3) except for the t0 which is just a constant to take into account that the relevant part
of your measurement didn’t start at t=0.
Click “OK”. If you get an error message, you must have made a typing error.
e. You are now back in the “curve fit selections’ window. Click “Okay”. The fit will be
performed, the fitted curve shown in your plot and the result of the fit for each of the
parameters shown in the plot.
10. The fitted value for m2 is τ (=RC). Multiply this value with ln2 (0.693) to get t1/2. Compare it
with the value that you found earlier and also with the value that you would expect from the
given values of R and C on the circuit board.
RC Write-up
Page 6
Voltage across a capacitor
Figure 4.
Part IV: Change the value of the capacitor.
1.
Add the 100 µF in parallel to the 330 µF capacitor. First, make a drawing of the new circuit
(something like figure 1). Include it in your report. Think carefully how to achieve this
setup with the given equipment. Describe your connections using Figure 2 (i.e. use the
A,B,C,D,E to describe from which plugs are connected, like in Part I, item 6). Have the
instructor check your setup
2.
Repeat the above steps and find the new RC constant and half-life, using both the measured
half-point (steps 5,6,7 of Part II) and the fitting procedure (steps 8,9,10 of Part II). What
would be the expected value (show how you found it and you might want to use it for the
start-fit parameter in the fitting procedure. Realize that the t0 most likely will change as
well) and compare it with the measured results.
PART V QUESTIONS AND BONUS
1. The time to half-maximum voltage is how long it takes the capacitor to charge halfway.
Based on your experimental results, how long does it take for the capacitor to charge to 75%
of its maximum (both for the single and parallel-connected capacitors)?
2. Why is it necessary to include a resistor in your setup? Hint: what happens to equation (1) if
R=0?
3.
After four “half-lives” (i.e., time to half-max), to what percentage of the maximum charge
is the capacitor charged (again for both cases)?
4.
What is the maximum charge stored in the capacitor in this experiment in the case of the
single capacitor (in Coulomb)? To how many electrons does this correspond?
5.
How long would it take to get to 99.9% of the applied voltage for both measured cases?
RC Write-up
Page 7
6.
By what factor would the time to half maximum increase/decrease if instead of a resistor of
100 Ohm, a resistor of 33 Ohm is used. Half a bonus point is given to groups that actually
confirm this by doing the measurement, complete with the fitting procedure.
RC Write-up
Page 8