Experiment 10 RC and RL circuits: Measuring the time constant.
Object: The object of this lab is to measure the time constant of an RC circuit and a
LR circuit. In addition, one can observe the characteristics of these two circuits and
compare them.
Pre Lab Assignment: Refer to the theory below.
1. If the time to achieve ½ the maximum voltage (refer to step 11 of the procedure
of the RC circuit) across the capacitor in an RC circuit is 0.045 s, what is the
capacitance in the circuit if the resistance is 150 Ω?
2. For an RL circuit, the voltage across the inductor drops from 4.0 V to a
constant 1.5 V as a current is established. In addition the voltage across a 4.7
Ω resistor builds to a steady 2.5 V. What is the resistance of this less than
perfect inductor? (Refer to step 13 of the procedure of the RL circuit for help
with this problem.)
3. If it takes 1.2 ms for the voltage across the inductor to drop from 4.0 V to 2.75
V, what is the time constant for the RL circuit in question 2? Hint: What is the
fraction of the total change in voltage across the inductor represented by a drop
from 4.0 V to 2.75 V?
4. What is the inductance of the RL circuit in questions 2 and 3?
Theory:
RC Circuit
The RC circuit was discussed earlier in the semester. To review:
A voltage source will charge a capacitor by moving charge through a resistor until the
capacitor is fully charged. The potential difference across the resistor is proportional to
the current through the resistor while the potential difference across the capacitor is
proportional to the charge on the capacitor. The equations for current through the
resistor, and charge on the charging capacitor are
I=
t
− 

V0 −τt
e and Q = CV0 1 − e τ 
R


where t is time, I is the current in the circuit, Q is the charge on the capacitor, V0 is
the source voltage, R is the resistance, C is the capacitance and τ is the time constant
and equal to RC.
If the voltage source is now disconnected, and the capacitor discharged through the
resistor, the potential difference and current through the resistor is now reversed. The
current in the resistor and the charge on the capacitor decrease according to
I=
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t
−
Vmax −τt
e and Q = CVmax e τ
R
1
LR Circuit
The LR circuit, shown schematically in Figure 1
behaves similarly. If we close the switch so the
voltage source provides current to the circuit
(position 1), Kirchhoff’s loop rule yields
Vs − IR − EL = 0
emf
1
R
2
L
where V0 is the source voltage, I is the current in the
circuit, R is the resistance and EL is the induced
emf of the inductor. Substituting for the emf of the
inductor gives us
V0 − IR − L
dI
=0
dt
We can solve this differential equation readily if we
divide by R and then let a new variable x be defined
as
V
V0
dx
dI
− I . Since 0 is a constant,
= − and the
R
dt
dt
R
differential equation becomes
x+
Figure 1. An RL circuit. When the
switch is closed toward position 1, a
current will begin to flow through the
resistor and inductor. If, after a
current is established in the circuit,
the switch is quickly thrown to
position 2, the magnetic field in the
inductor will continue to push a
current through the circuit until the
energy in that field is exhausted.
L dx
=0
R dt
Separating variables and integrating with x = xo when t = 0, yields
ln
First we define τ =
to give us
x
R
=− t
xo
L
L
, then we rewrite the above equation as an exponential equation
R
t
−
x
=e τ
xo
We note that I = 0 when t = 0, so xo =
V0
. We then substitute the values for x and xo
R
from above and rearrange the equation to give us the final result that
t
− 
L
V0 
τ
I = 1 − e  where τ = = the time constant of the circuit.
R
R

dI
The emf across the inductor is determined by EL = − L
dt
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2
EL = − L
V0  1 −τt
− e
R τ

V0  R −τt 
=
−
L

− e 
R L


EL = V0 e
−
t
τ
If the switch in Figure 1 is quickly changed to remove the voltage source and connect
the inductor directly to the resistor (Position 2), the magnetic field that was
established in the inductor will collapse, driving a current through the circuit. The
loop equation for the voltages now becomes
IR − EL = 0 or IR + L
We separate variables and integrate to yield
I = I 0e
−
dI
=0
dt
t
τ
where τ =
L
.
R
Note: The inductor used in our experiment isn’t perfect. There is a resistance in the
inductor also. All resistances in the above theory are total resistances for the circuit.
Overview of the experiment: We will use Data Studio to generate graphs of the time
dependent voltages across the resistor and capacitor for an RC circuit and the time
dependent voltages across the resistor and inductor for an RL circuit. These graphs
created by Data Studio will be printed and used to answer several questions about the
time dependent circuits. In addition, we will use some of the software functions to
measure the time for the voltage to change by one half its total change.
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Experimental Setup and Procedure:
The RC circuit
Figure 2 shows the set up for the RC circuit.
750 Interface Box
Ch A
+ -
Ch B
+ -
gnd
.
pos. sq. wave output
jumper wire
Figure 2a. A picture of the
circuit board wired for the RC
time constant measurements. A
jumper wire short circuits the
inductor while the voltage
probes are connected across
the resistor and capacitor.
Figure 2b.
1. Turn on the computer and start Data Studio. Click on Create Experiment.
2. Select the voltage sensor and drag the icon to channel A. Repeat for channel B.
Double click on one of the sensor icons to get the sensor properties window. Set
the sample rate to 1000 Hz, Low (1X).
3. Click on the output (right side of 750 interface box image) and set up the
following in the signal generator window: Positive Square Wave Function with
an Amplitude of 4.0 V. Set the frequency (of the square wave) to 0.4 Hz. Close
the window.
4. Select Sampling Options from the Experiment Setup window and set Automatic
Stop to 4 seconds. Click on OK.
5. Wire the output of the Science Workshop interface to the circuit board as
shown in Figure 2. Note that the interface has an output terminal and a ground
terminal. The output terminal will be positive in this experiment. The 100 Ω
resistor and 330 µF capacitor are used in this part of the experiment.
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6. Connect the wires of the voltage sensor from one channel to across the resistor
and the wires of the voltage sensor from the second channel across the
capacitor. Be sure to observe the polarity so the red wire is connected to the
positive side of each element.
7. Set up to graph the data of both inputs from the voltage sensor on the same
graph.
8. Click start to record the data.
9. Click on the Data button (right most button at the top of the graph) and
deselect data grouping. Lock the origins of the horizontal axes. Enlarge the data
portion of the graph appropriately and print the result. Label your graph as
indicated in the analysis section. Maximize the graph window.
10. Turn off the Data Point Gravity by double clicking on the graph display,
select Tools and set Data Point Gravity to 0.
11. In the Data window (upper left) double click on Voltage, ChA V). In the Data
Properties window that opens select the Numeric tab and choose Scientific
Notation. Set the number of digits to 3. Repeat the step for Channel B.
12. Click on the smart tool button located at the top of the graph. It looks like
coordinate axes. Click and drag it to locate the coordinate for the time at the
start of the charge cycle for the capacitor. (Hint: Enlarge portions of the graph
as necessary so you can read the position accurately.) Record this time.
13. Move the cursor until you get a hand with a triangle near the index finger. Click
and drag that icon to determine the change in time needed to obtain ½ the
maximum voltage displayed. Use the fact that t1/2 = τ ln 2 to calculate the time
constant. If it is difficult to get an accurate position for the time, you may want
to enlarge your graph at this time.
14. Use the same click and drag process to determine t1/2 for the discharge portion
of your graph.
15. Repeat steps 10 through 12 to measure the t1/2 times for the voltage across the
resistor for both parts of the square wave cycle.
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The RL circuit.
Figure 3 shows the set up for the RC circuit.
750 Interface Box
Ch A
+ -
Ch B
+ -
gnd
.
pos. sq. wave output
Figure 3a. A picture of the
circuit board wired for the RL
time constant measurements.
Figure 3b.
1. Double click on one of the voltage sensor icons to get the sensor properties
window. Change the sampling rate to 4000 Hz, Low (1X).
2. Double click on the output window, and set up the following in the signal
generator window: Positive Square Wave with an Output of 4.0 V. Change the
frequency (of the square wave) to 50 Hz.
3. Select Options from the menu and set auto stop to 0.05 second.
4. Wire the output of the Science Workshop interface to the circuit board as
shown. Note that the interface has an output terminal and a ground terminal.
The output terminal will be positive in this experiment. The 10 Ω resistor and
8.2 mH inductor are used in this part of the experiment. The inductor will
have a resistance as well.
5. Connect the wires of the voltage sensor from one channel to across the resistor
and the wires of the voltage sensor from the second channel across the
inductor. Be sure to observe the polarity so the red wire is connected to the
positive side of each element.
6. Set up to graph the data of both inputs from the voltage sensor on the same
graph. Click on the Data button (right most button at the top of the graph) and
deselect data grouping. Lock the origins of the horizontal axes. Enlarge the data
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portion of the graph appropriately and print the result. Label your graph as
indicated in the analysis section. Maximize the graph window.
7. Turn off the Data Point Gravity. Set the data display to scientific notation
with at least 3 digits.
8. Click start to record the data.
9. Maximize the graph window. Enlarge the data portion of the graph
appropriately and print the result. Label your graph as indicated in the analysis
section.
10. Click on the smart tool button located at the top of the graph. Locate the where
the lines cross which show the coordinate values on your graph, click it and
drag it to locate the coordinate for the time at the start of the positive voltage
cycle for the resistor. Record this time.
11. Repeat the click and drag procedure to determine the maximum voltage across
the resistor.
12. Repeat the click and drag procedure to determine the time when the voltage
across the resistor was ½ of the maximum. Use the fact that t1/2 = τ ln 2 to
calculate the time constant. If it is difficult to get an accurate position for the
time, you may want to enlarge your graph at this time.
13. Use the same click and drag process to determine t1/2 for the decreasing resistor
voltage portion of your graph.
14. Repeat the steps 9 through 11 for the inductor voltages.
Note that the inductor has a resistance and will not have a zero voltage when the
current reaches a steady positive value. You must determine the minimum voltage
across the inductor and locate the time taken to determine ½ the change of
inductor voltage for increasing current in the circuit. When the current is
decreasing the voltage across the inductor will, of course, drop to zero. Again
you want to determine the time for the inductor voltage to drop to ½ its
maximum value.
15. To determine the resistance of the inductor, first determine the steady constant
voltage across resistor R. Apply Ohm’s Law to calculate the current through
the 10 Ω resistor. Nest divide this same current into the steady constant voltage
across the inductor (Ohm’s Law again) to determine the inductor’s resistance.
Analysis and Report:
For the RC circuit:
1. Determine the average time constant from your measurements. How does this
average compare to the time constant calculated using the values of R and C for
the circuit elements you used?
2. On the printed graph of voltage vs. time, label the curve showing the voltage
across the resistor, and the curve showing the voltage across the capacitor.
Label the period of time corresponding to charging the capacitor, and the period
of time corresponding to discharging the capacitor.
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3. What are the similarities and differences regarding the voltage across the
resistor during each of these time periods?
For the RL circuit:
1. Determine the average time constant from your measurements. How does this
average compare to the time constant calculated using the values of R and L for
the circuit elements you used?
2. On the printed graph of voltage vs. time, label the curve showing the voltage
across the resistor, and the curve showing the voltage across the inductor.
Label the period of time corresponding to establishing a magnetic field in the
inductor, and the period of time corresponding to discharging the magnetic field
in the inductor.
3. What are the similarities and differences regarding the voltage across the
resistor during each of these time periods?
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