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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
12.7F
11.7F
12.2F
11.8F
12.6F
12.7F
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.
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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.
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Figure 16 – Integration of power leading to stored energy for 2.7 k resistor
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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
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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
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