5
RC Circuit
5.1
Theory
• A capacitor stores electrical energy in an electric field.
• There are many ways to make capacitors, one popular way is made by placing
two conducting plates separated by a dielectric.
• A circuit containing a resistor and capacitor is called an RC circuit.
• The purpose of this experiment is to investigate the different properties of the
capacitance. Specifically to determine the capacitive time constant and the
half-life of an RC circuit.
• As the capacitor is being charged it takes more and more work to accumulate
the charges on the capacitor because the electrons in the capacitor repel the
new electrons.
• When a switch is closed on any circuit, there is always some time required for
the current to reach some steady value.
• The purpose of this experiment is to investigate the capacitance of a capacitor,
capacitive time constant, and half-life of an RC Circuit. All three will describe
how the voltage across a capacitor varies as the capacitor charges.
2
9
6
1
3
4
5
7
8
10
Figure 5.1: The numbers on this RLC Board correspond to the numbers on the circuit
diagrams throughout this experiment.
33
34
CHAPTER 5. RC CIRCUIT
• When an uncharged capacitor is connected across a DC voltage source, the
rate at which it charges up decreases 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 the plates already have like charges
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 = qmax 1 − e−t/τ ,
(5.1)
where qmax is the maximum charge on the plates, t is time, 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.
• Consider the extreme limits: notice that when t = 0, q = 0 that there is no
charge on the plates initially. Also notice that when t goes to infinity, q goes
to qmax , which means it takes an infinite amount of time to completely charge
the capacitor.
• 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:
t1/2 = τ ln 2 = R C ln 2.
(5.2)
• You will derive this equation from the charge equation as part of your lab
experiment.
• In this experiment 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.
(5.3)
• Also, an AC current will be supplied in order to have an “off” time just prior to
sending the signal. The charging portion of the AC current will simulate turning
on a DC current.
– Note: One of the benefits of this laboratory is the fact that the student
sets up each lab. So much can be learned from setting up the equipment.
Yet, so much is also at stake. Be very careful to set up circuits in the
proper way. Always connect a sensor (A Voltage Sensor, for example) so
that the positive end of the sensor matches the positive side of the circuit.
With some experiments the results will not make sense if the circuit is
connected improperly.
35
5.2. EQUIPMENT
A charging capacitor
q = qmax (1− e−t/τ ) .
q[C]
qmax = CVs
0
1
2
3
4
5
t[ s ]
Figure 5.2: Charging Capacitor
5.2
Equipment
Computer
SW750 Interface; Capstone
Voltage sensor cable (8 pin-DIN)
RLC Board (CI-6512)
Patch cords (2)
Shorting cable (1)
Digital Multimeter
5.3
Procedure
Numbers in brackets [1] correspond to the RLC Board picture located on the previous page.
Set-up
• Turn on the Interface box.
• Open Excel templates.
• Open the Capstone Template for RC Circuits named RC.cap
36
CHAPTER 5. RC CIRCUIT
• Connect the Voltage Sensor to Analog Channel A. The voltage sensor is a set
of two wires. One is has banana plugs. The other side combines the wires into
an 8-pin-DIN.
• Set up the circuit board.
to the
– Connect one patch cord from the power output "positive"
jack for the 100 ohm resistor [4].
– Connect the other patch cord from the power output "negative", or "ground"
to the jack for the 330 microF capacitor [10].
– Use your finger to trace the current path from [4] to [10]. Notice that the
current path goes through the coil. We don’t want an RLC (L is inductance
and is from the coil.) We want an RC circuit.
– Use a third banana plug patch cord to “jump” over the inductor coil on
the RLC board, from [2] to [9].
– Attach the voltage sensor from Channel A across the 330 µF capacitor.
Keep in mind the direction of the current! The positive (red) Voltage
Sensor plug should be on the positive side of the circuit [9].
Shorting wire
2
Red
8.2 mH
9
750 Interface
330 µF
100 Ω
A B C
4
10
Black
Red
Black
Figure 5.3: RC circuit lab set-up
• Start recording data with the computer.
• Rename the data run.
– Use “100[Ohm] 330[microF]”, or something similar.
5.4
Analysis
37
5.4. ANALYSIS
After collecting your data you will need to measure the capacitance, and resistance
with the Digital Multimeter. Please see page 40 for instructions.
• Measure Beginning Voltage/time as well as Peak (Max) Voltage.
Use the scroll on the mouse to zoom in on the beginning voltage value (just
before the signal starts to leave the x-axis.) Use the "Highlight"
tool to
hover over the data in question. Now find the mean voltage value by using
the "Statistics"
voltage.
tool. Record this in the spreadsheet as the minimum
Record the time where the data leave the x-axis as t0 using the "Add MultiCoordinates Tool"
.
Use the "Highlight" tool again to select the region for the top of Capacitor
Voltage graph. Use Stats to find the mean value. Record this in the spreadsheet
data table as the max voltage.
• Use the spreadsheet to calculate Voltage Values for where you will measure the
time at V1/2 and V3/4 . Go back to Capstone and measure the time values on the
graph at those Voltage values using the "Add Multi-Coordinates Tool". Zoom
Way In using the scroll on the mouse.
• Find the values for your calculations and uncertainties.
Resistor (Use the digital multimeter and determine uncertainty.)
Capacitor (Use the digital multimeter, See page 40 for instructions.)
Voltage (uncertainty is the std-dev from a constant voltage section.)
Time (uncertainty is 1/(sample rate).)
• Follow the Spreadsheet Calculations from this point forward.
• You need to print out the graph. Annotate the location of t1/2 and t3/4 with
those titles and the values for Voltage at those places.
38
CHAPTER 5. RC CIRCUIT
100 Ω
+
330 µF
V
-
Voltmeter
Channel A
-
+
Wave form = "positive-only" Square Wave
Frequency = 0.4 Hz
Amplitude = 4.00 V
Derivation of t1/2 :
1
qm = qm 1 − e−t/τ
→
2
Taking natural log of both sides
1
→
ln e−t/τ = ln
2
1
= 1−e−t/τ
2
we get
−
t
= |{z}
ln 1 − ln 2
τ
0
Solving for t we get
t1/2 = τ ln 2.
Example for 4t1/2 :
q = qm 1 − e−(4τ ln 2)/τ
= qm 1 − e−4 ln 2
= qm 1 − eln 1/16
1
= qm 1 −
16
15
=
qm
16
= 0.9375 qm .
e−t/τ = 1−
→
→
−
3
1
=
4
2
t
= − ln 2.
τ
Dashed lines mean to use MSExcel to
calculate the value using cell references!
Uncertainty
Minimum Voltage=
Maximum Voltage=
1/2 Maximum Voltage Value=
Minimum + ((Maximum - Minimum) * 1/2)
3/4 Maximum Voltage Value=
Minimum + ((Maximum - Minimum) * 3/4)
t0 =
t1 =
t2
=
Time VALUE at beginning of charge cycle
Time VALUE at 1/2 max voltage
Time VALUE at 3/4 max charge
t1/2 =
Time to reach half maximum voltage (t1-to)
t3/4 =
Time to reach 3/4 maximum charge: (t2-to)
Resistance of Circuit =
(via Multimeter: watch movie, if necessary)
As calculated from t1/2 measured and R
Capacitance =
measured. Report this in micro Farads.
(t1/2=RCLn2)
Capacitance (from digital multimeter) =
Report this in micro Farads.
% Difference =
Using t1/2 from your data, how long should
b/w two capacitance values
Show your calculation below:!
it have taken to reach 75% of max charge?
How long did it take?
% Difference =
Theoretically (not according your data),
after four t1/2's, to what percentage is the
According to t3/4 from Capstone
b/w two t3/4 values.
Show your calculation below:!
capacitor charged?
What is the maximum charge in Coulombs
for the capacitor in this experiment (use
your data)?
Don’t forget to derive t1/2 on the next page.
Use Equation Editor. There is a movie for how to use it.
Show your calculation below:!
41
5.5. WRITE-UP GRADING
5.5
Write-up Grading
Content
Cover Page
Points
5
Explanation
Properly filled out cover page
Data Table
65
Vmin and Vmax with uncertainty (5)
1/2 Max. voltage and 3/4 Max. voltage. (5)
t0 , t1 , t2 (5)
t1/2 and t3/4 (5)
Resistance measured (5)
Capacitance & % difference (10)
t3/4 & % difference with work shown (10)
4 t1/2 charge time with work shown (10)
Charge and no. of electrons with work shown (10)
t1/2 derivation
10
Correct derivation with enough details
Sample plot
15
Correct graph image into Excel (8)
points annotated (7)