MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Fall 2003
Experiment 7: Time Constant of an RC Circuit
OBJECTIVE
To show that the theoretical curves we derive in class for the charging and discharging of
a capacitor actually apply to the real world! (or, how does Nature know the value of e?)
To measure the time constants associated with a discharging and charging RC (resistivecapacitive, or resistor-capacitor) circuit.
To use auxiliary features of DataStudio for enhanced data analysis of exponential decays.
INTRODUCTION
A. Capacitor
(See the 8.02T Study Guide, Section 5.1, for a more extensive discussion of capacitors
and capacitance.)
Capacitors are circuit elements that store electric charge Q , and hence energy, according
to the expression
Q = C V,
(7.1)
where V is the voltage across the capacitor and C is the constant of proportionality
called the capacitance. The unit of capacitance is the farad, [F] = [C]/[V];
1 farad = (1 coulomb)/(1 volt).
Capacitors come in many shapes and sizes, but the basic idea is that a capacitor consists
of two conductors separated by a spacing, which may be filled with an insulating material
(dielectric). One conductor has charge + Q and the other conductor has charge −Q . The
conductor with positive charge is at a higher voltage then the conductor with negative
charge. Most capacitors have capacitances in the range between picofarads
(1pF = 10−12 F) and millifarads (1mF = 10−3 F = 1000 µ F) . Note that we’ve also used
the notation for a microfarad, 1µF=10-6 F =10-3 mF .
E7-1
B. Charging a Capacitor
Consider the circuit shown in Figure 7.1. The capacitor is connected to a DC voltage
source of emf E . At t = 0 , the switch S is closed. The capacitor initially is uncharged
with q ( t = 0 ) = 0 . (In the following discussion, we’ll use the convention that represents
a time-varying charge as “q” instead of “Q”.)
Figure 7.1 (a) RC circuit (b) Circuit diagram for t < 0 (c) Circuit diagram for t > 0
The expressions for the charge on, and hence voltage across a charging capacitor, and the
current through the resistor, are derived in the 8.02T Study Guide, Section 7.6.1. These
instructions will use the notation τ = RC for the time constant of either a charging or
discharging RC circuit.
Figure 7.2 Voltage across capacitor as a function of time
The capacitor voltage as a function of time is given by
VC ( t ) =
q(t )
C
(
)
= E 1 − e − t /τ ;
(7.2)
a graph of this function is given in Figure 7.2.
E7-2
The current that flows in the circuit is equal to the derivative with respect to time of the
charge,
I=
dq  E  −t τ
=
e = I 0 e −t τ
dt  R 
(7.3)
where I 0 is the initial current that flows in the circuit when the switch was closed at
t = 0 . The graph of current vs. time is shown in Figure 7.3:
Figure 7.3 Current as a function of time
After one time constant τ has elapsed, the voltage has increased by a factor
(1 − e −1 ) = 0.632 ;
(
)
VC ( τ ) = E 1 − e − 1 = 0.632 E
(7.4)
and the current has increased by a factor of e −1 = 0.368 ,
I ( τ ) = 0.362 I 0 .
(7.5)
.
D. Discharging a Capacitor
Suppose we initially charge a capacitor to a charge Q0 through some charging circuit.
Suppose at time t = 0 the switch is closed (Figure 7.4). The capacitor will begin to
discharge.
E7-3
The expressions for the charge on, and hence voltage across a discharging capacitor, and
the current through the resistor, are derived in the 8.02T Study Guide, Section 7.6.1.
Figure 7.4 RC circuit with discharging capacitor
The voltage across the capacitor in a discharging RC circuit is given by
VC ( t ) =
q(t )
Q
= 0
C
 C
 −t τ
e .

(7.6)
A graph of voltage across the capacitor vs. time for the discharging capacitor is shown in
Figure 7.5:
Figure 7.5 Voltage vs. time for discharging capacitor
The current also exponentially decays in the circuit as can be seen by differentiating the
charge on the capacitor;
I (t ) = −
dq  Q0
=
dt  RC
 −t τ
e .

(7.7)
A graph of the current flowing in the circuit as a function of time also has the same form
as the voltage graph depicted in Figure 7.6.
E7-4
Figure 7.6 Current vs. time for discharging capacitor
EXPERIMENT OVERVIEW
In this experiment, you will assemble circuits with resistors and a capacitor, and apply a
voltage in a manner that alternately charges the capacitor and allows the capacitor to
discharge. DataStudio will be used to determine the time constant of the circuits, both
graphically and analytically.
Three resistor combinations will be used:
1. A single 100-Ω resistor in series with a 330-µF capacitor
2. Two 100-Ω resistors in parallel, with the parallel combination in series with a
330-µF capacitor.
3. Two 100-Ω resistors in series with a 330-µF capacitor.
APPARATUS/DataStudio SET-UP
A. AC/DC Electronics Lab Circuit Board
1. Connect the banana plug patch cords from the ‘OUTPUT’ ports of the 750
Interface to the banana jacks on the AC/DC Electronics Lab circuit board.
2. In order to measure the current that flows in the circuit, you must connect the
Current Sensor in series with the 100-Ω resistor (color code brown, black,
brown) and the 330-µF capacitor; connect the Current Sensor in series with the
other circuit elements forming a closed circuit using the 750 Interface as the
voltage source. Connect the Current Sensor directly into the Analog Channel A
on the 750 Interface.
E7-5
3. In order to measure the voltage across the capacitor, you must connect the
Voltage Sensor in parallel with the capacitor. Connect the Voltage Sensor
directly into the Analog Channel B on the 750 Interface.
B. Computer: Connect the 750 Interface to your computer, turn on the Interface, and
then turn your computer on.
C. Data Studio File: Download the Data Studio file “ exp07.ds ” from the web page and
save it on your desktop. Open the activity by double clicking on the icon on the desktop.
Your file should have a Signal Generator Display, and a single Graph Display which is
already set up to display Current vs. Time and Voltage Across Capacitor vs. Time. If the
graphs are not present, see part A of DATA ANALYSIS, below. When you take data
you should see something like this
V across C
Current
D. Signal Generator:
1. In the Signal Generator dialog (Figure 7.8) we have chosen “Pos(itive) Square
Wave Function.”
E7-6
2. The Amplitude has been adjusted to 4.000 V , the Frequency to 0.400 Hz and the
sampling rate to 1000 Hz . We chose the output data that you will record by
clicking the plus button (+) beside Measurements and Sample Rate on the Signal
Generator dialog and clicking the appropriate Measure Output Voltage and
Measure Output Current buttons.
Figure 7.8 Signal Generator display
E. Current and Voltage Sensor Setup:
1. Click the Setup button. On the Sensor menu, drag the Current Sensor icon and
place it on the Analog Channel A.
2. Click the Setup button. On the Sensor menu, drag the Voltage Sensor icon and
place it on the Analog Channel B.
Figure 7.9 Current and Voltage Sensor Setup
E7-7
DATA RECORDING
A. Graphs: Here’s how to set up the two graphs if you ever need to (it should already be
set up for you here). Drag the Voltage, ChB icon in the Data Window and drag it into
the Graph icon. This will create a Voltage, ChB vs. Time graph. Grab the Current,
ChA icon in the Data Window and drag it into the Graph icon. This will create a
Current, ChA vs. Time graph.
B. Sampling Options: Click on the drop-down menu labeled “Experiment” on the top
tool bar. In the “Experiment” menu, click on “Set Sampling Option” to open the
Sampling Options dialog. Check that the Delay Choice is on “None”. Check that the
Automatic Stop choice is “Time” with 3.5 seconds in the window. If these options are
not set in this manner, set them to these values.
C. Data Recording: Press “Start” to begin taking data. Once the data has been
recorded, scale the plots to fit the graph screens by clicking on the first icon on the left at
the top of the Graph Window (which is the “Scale to Fit” icon).
DATA ANALYSIS
In this experiment, you are asked to measure the time constants for the three RC circuits
described above in EXPERIMENT OVERVIEW.
In setting up the apparatus, you first should record data for a single 100-Ω resistor in
series with the 330-µF capacitor. Answer the questions for this data, and then repeat the
measurement of the time constant for the other two circuits, as described on page E7-5.
The first time you measure the time-constant, we want you to use both methods of
determining the time constant described below. For the next two circuits, you can use the
method you like best, but state on your tear-sheet which method you use.
There are several ways to measure the time constant for the RC circuits. We describe
two methods below:
Method 1: The current in the charging circuit with initial value I0 at t = 0 decreases
exponentially in time, I ( t ) = I 0 e −t RC = I 0 e −t τ , where τ = RC is the time constant, as
described above and in the 8.02T Study Guide, Section 7.6.
You can determine the time constant τ graphically by measuring the current I ( t1 ) at a
fixed time t1 and then finding the time t1 + τ such that the current has the value
I ( t1 + τ ) = I ( t1 ) e − 1 = 0.368 I ( t1 )
(7.8)
E7-8
Figure 7.10 Current as a function of time in a discharging RC circuit.
In the Current, ChA graph, enlarge the Graph window by clicking and dragging
anywhere on the edge of the graph window. Click on the “Zoom Select” (fourth from
the left) icon in the Graph icon bar and form a box around a region where there is
exponential decay for the current. Click on “Smart Tool” (sixth from the left) icon.
Move the crosshairs to any point (at some time t1 ) on the exponentially decaying function
and record the values of the current and the time. Multiply the current value (displayed in
the Smart Tool feature) by e −1 = 0.368 . (If you don’t have a familiar calculator with
you, the laptop should have this feature; go to “Start” at the lower left, and follow the
prompts through “Accessories” and “Calculator”. DataStudio does have a calculator
feature, but its use may seem somewhat cumbersome.)
Use the Smart Tool to find the new time t1 + τ such that the current is down by a factor
of e −1 = 0.368.
Determine the time constant and record your value.
Questions First Circuit– Method 1 (answer on the tear-sheet at the end!!!):
1. What is your measured value using Method 1 for the time constant for this first
circuit (a single 100-Ω resistor in series with a 330-µF capacitor).
2. What is the the theoretical value of the time constant for your circuit?
3. How does your measured value compare to the theoretical value for your circuit?
Method 2: A second approach is to take the natural logarithm of the current, using the
facts that ln ( e −t τ ) = − t τ and ln(ab) = ln a + ln b . This leads to
ln ( I ( t ) ) = ln ( I 0 e −t τ ) = ln ( I 0 ) + ln ( e −t τ ) = ln ( I 0 ) − t τ .
(7.9)
E7-9
Thus, the function ln ( I ( t ) ) is a linear function of time.
The y-intercept of this graph is ln ( I 0 ) and its slope is slope = −1 τ . Thus, the time
constant can be found from the slope according to
τ = − 1/ slope
(7.10)
Click the Calculator button on the bar menu. In the Calculator window click New. The
variable x should be highlighted in the Definition window. Click on the Scientific button
and scroll down and click on ln(x) . Then, with the “ x ” highlighted, scroll down the
Scientific options again to select abs ( x ) . Scroll down on the Variables menu and click
on “Data Measurement.” In the “Please Choose a Data Source Window,” scroll
and click on Current, ChA and then click OK. Then, click the “Accept” button in the
calculator window. A calculator icon with your equation should appear in the Data
window. Drag that calculator icon to the Graph icon in the Display window. A fairly
complicated graph (most of which is no use to us, as the current is so small for most of
the run) will appear (see Figure 7.11 below). Use the “Zoom Select” to isolate the small
amount of data where the function is linear. You should see fluctuations in the data due to
approximations associated with the sampling rate. You can use “Zoom Select” to choose
the region where there are the smallest fluctuations. Use the mouse to highlight a region
of data. Once you isolated this region, click on the Fit button, scroll down and click to
“Linear Fit.”
Record the value of the slope. Use your value of the slope to calculate the time
constant.
Useful data
Figure 7.11: The ln(I) versus t plot of all the data. The useful data is indicated.
E7-10
Questions First Circuit – Method 2 (answer on the tear-sheet at the end!!!):
1. What is your measured value using Method 2 for the time constant for this first
circuit (a single 100-Ω resistor in series with a 330-µF capacitor).
2. How does this Method 2 measured value compare to the theoretical value for your
circuit?
Questions Second Circuit– Use Either Method 1 or 2 (answer on the tear-sheet at the
end!!!):
1. What is your measured value using the method of choice for the time constant for
this second circuit (Two 100-Ω resistors in parallel, with the parallel combination
in series with a 330-µF capacitor). State your method of choice.
2. What is the theoretical value of the time constant for your circuit?
3. How does your measured value compare to the theoretical value for your circuit?
Questions Third Circuit– Use Either Method 1 or 2 (answer on the tear-sheet at the
end!!!):
1. What is your measured value using the method of your choice for the time
constant for this third circuit (Two 100-Ω resistors in series with a 330-µF
capacitor). State your method of choice.
2. What is the theoretical value of the time constant for your circuit?
3. How does your measured value compare to the theoretical value for your circuit?
E7-11
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.02
Fall 2003
Experimental Summary 7: Time Constant of an RC Circuit
Group ___________________________________
Names ____________________________________
____________________________________
____________________________________
Questions First Circuit– Method 1 and 2 (answer on the tear-sheet at the end!!!):
1. Enter your measured values and the predicted value for the time constant of the
first circuit (a single 100-Ω resistor in series with a 330-µF capacitor) in the table
below.
Method 1: _____________
Method 2: _______________
Predicted value: _______________
Method 1 value/Predicted value: ________________
Method 2 value/Predicted value: ________________
2. Which method, 1 or 2, do you think gives the best results? Why?
E7-12
Questions Second Circuit– Use Either Method 1 or 2:
1. What is your measured value using your method of choice for the time constant
for this second circuit (Two 100-Ω resistors in parallel, with the parallel
combination in series with a 330-µF capacitor). State your method of choice.
Answer: _______________________ sec
2. What is the theoretical value of the time constant for your circuit?
Answer: _______________________ sec
3. How does your measured value compare to the theoretical value for your circuit?
Measured Value/Predicted Value: __________________________
Questions Third Circuit– Use Either Method 1 or 2:
1. What is your measured value using your method of your choice for the time
constant for this third circuit (Two 100-Ω resistors in series with a 330-µF
capacitor). State your method of choice.
Answer: _______________________ sec
2. What is the theoretical value of the time constant for your circuit?
Answer: _______________________ sec
3. How does your measured value compare to the theoretical value for your circuit?
Measured Value/Predicted Value: __________________________
E7-13