Name___________________ ID number_________________________
Date____________________ Lab partner_________________________
Lab CRN________________ Lab instructor_______________________
Physics 2306
Experiment 11: Time-dependent circuits, Part 3
Objectives
•
•
To study the response of R-L and R-C circuits to AC signals of varying
frequencies
To study the resonant response of a series R-L-C circuit
Required background reading
Young and Freedman, sections 31.1, 31.2, 31.3, 31.5
Introduction
In this lab you will study the response of circuits where the driving voltage is a
sinusoidally varying voltage, referred to as alternating current or AC. As discussed in the
text, these circuits can be analyzed by using the concept of impedance to describe the
circuit elements.
Consider first the circuit shown in the figure.
C
R V
It consists of an AC voltage source with time dependent voltage v(t) = V cos(ωt). The
current in the circuit will be given by i(t) = I cos(ωt+φ). As described in your text, the
ratio of the amplitudes of the voltage and current (V and I) is given by the impedance. So
we can write I = V/Z, where Z is the impedance. In the case of the RC circuit shown, the
1
2
⎛ 1 ⎞
⎟⎟ , where we have used the
impedance is given by Z = R + X = R + ⎜⎜
⎝ 2π f C ⎠
relation between angular frequency and natural frequency (ω=2πf). The voltage across
the resistor (which we will measure with the oscilloscope) has an amplitude VR = IR. So
putting all of the above together, we can solve for the amplitude of the voltage across the
resistor VR :
2
2
C
VR =
2
V
⎛
⎞
1
⎟⎟
1 + ⎜⎜
2
π
f
RC
⎝
⎠
2
(1)
From the form of equation 1, we can see the following. At very low frequencies the term
in the denominator becomes large so VR → 0 as the driving frequency f → 0. As the
driving frequency becomes large, the term in the denominator goes to 1, so VR → V as
the driving frequency f → ∞. This circuit is referred to as a “high-pass” filter, since the
full voltage is dropped across the resistor only at high frequencies.
A “low-pass” filter can be made by simply replacing the capacitor in the figure with an
inductor. A similar analysis can be used to show that the voltage across the resistor in
that case will be:
VR =
V
⎛ 2π f L ⎞
1+ ⎜
⎟
⎝ R ⎠
2
(2)
In this case, you can convince yourself that VR → V as the driving frequency f → 0, and
VR → 0 as the driving frequency f → ∞. So this circuit is referred to as a low-pass filter
since the full voltage is dropped across the resistor only at low frequencies.
In summary, the RC circuit will behave as a “high-pass” filter because significant current
will only flow in the circuit at high frequencies. The RL circuit behaves as a “low-pass”
filter because significant current will only flow in the circuit at low frequencies. As
discussed in your textbook, these circuits have an important practical application. They
are used to direct signals in loudspeaker systems, as shown in the figure below. In a
loudspeaker system, low-frequency sounds are produced by a woofer, which is a speaker
with large diameter; while, the tweeter, a speaker of smaller diameter, produces high
frequency sounds. A high-pass filter (RC) is used to pass the high frequency components
2
of the sound to the tweeter, while a low-pass filter (RL) is used to pass the low frequency
components of the sound to the woofer.
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Name___________________ ID number_________________________
Date____________________ Lab partner_________________________
Lab CRN________________ Lab instructor_______________________
Physics 2306
Experiment 11: Time-dependent circuits, Part 3
Pre-lab assignment (complete and turn in at the beginning of your lab session)
1. Consider the R-C high pass filter circuit described in the introduction. As the
frequency is increased the amplitude of the voltage across the resistor gradually changes
from 0 to its maximum value of V. At what frequency f will the amplitude of the voltage
across the resistor be 1 / 2 of its maximum value V? Evaluate this frequency
numerically for the case of R = 1kΩ and C=0.27 μF. (Write your answer (formula for f
in terms of R and C) by Prediction 1-5 in the write-up for reference during the lab.)
2. Consider the R-L low pass filter circuit described in the introduction. As the frequency
is increased the amplitude of the voltage across the resistor gradually changes from its
maximum V to zero. At what frequency f will the amplitude of the voltage across the
resistor be 1 / 2 of its maximum value V? Evaluate this frequency numerically for the
case of R = 1kΩ and L=65 mH. (Write your answer (formula for f in terms of R and L)
by Prediction 2-5 in the write-up for reference during the lab.)
3. Consider an R-L-C series circuit as described in section 31.5 of your textbook. At
what frequency f will the circuit achieve resonance if L= 65 mH and C =0.27 μF? (Write
your answer (formula for f in terms of L and C) by Prediction 3-5 in the write-up for
reference during the lab.)
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Equipment
You will use the following equipment:
•
•
•
•
•
•
•
•
•
•
•
•
Pasco EM-8656 AC/DC Electronics Laboratory
Some 10 cm or 25 cm long white wires (in Ziploc bag)
Components: 1 Resistor: 1 kΩ, 1 capacitor: 0.27 μF
Large wire-wound inductor
Two red and two black banana plug to alligator clip
LCR meter (BK Precision 875B)
Manual range digital multimeter (DMM)
One BNC tee (attached to function generator output)
One BNC – BNC cable
Two BNC – banana plug cables
BK Precision 4017A Function Generator
Tektronix TDS 1002 Oscilloscope
Activity 1: A Driven RC circuit
Set up the RC circuit shown in the figure below. Use a resistor with R=1 kΩ and C=0.27
μF. Measure the actual values (using the manual range digital multimeter) of these
components before inserting them in the circuit and record the values below:
R = ______________________
C = ______________________
Use the BNC tee at the function generator output; send the signal both to your circuit and
to the oscilloscope channel 1. Hook up a cable so that you measure the voltage across the
resistor and plug it into oscilloscope channel 2.
red
C
CH1
red
R
black
CH2
black
5
For the oscilloscope, do the following. Press the DEFAULT SETUP button to get rid of
any previous settings.
Press CH1 MENU : Toggle the Probe to 1x
Press CH2 MENU : Toggle the Probe to 1x
For the function generator, set the settings as follows:
Sine wave
- 20 dB switch pushed IN
Press the AUTO SET button to get a stable, triggered signal on the oscilloscope.
Adjust the output level to give about 1 volt from “peak” to “peak” of the sine wave on
Channel 1 on the oscilloscope.
Before taking quantitative data vary the frequency over the range from about 100 – 2000
Hz and see how the voltage across the resistor responds (in channel 2). Recall that the
current in the circuit is simply this voltage signal divided by the constant resistance R.
So this signal shows you the time dependent behavior of the current as well.
Question 1-1: Do you observe that the amplitude of the current in the circuit is greatest
at high or low frequencies?
.
Question 1-2: In your circuit only the capacitor has impedance that varies with frequency
(the impedance of the resistor does not vary with frequency.) Given your observation in
Question 1-1, do you conclude that the capacitor has large or small impedance at high
frequencies? What about low frequencies? Explain.
Question 1-3: Based on your answer to question 1-2, at high frequencies, would you say
that the capacitor behaves more like an open circuit (a break in the circuit’s wiring) or
more like a short circuit (a connection with very little resistance). Justify your answer.
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Question 1-4: At low frequencies, does the capacitor behave more like an open circuit (a
break in the circuit’s wiring) or more like a short circuit (a connection with very little
resistance). Justify your answer.
.
Prediction 1-5: In pre-lab question 1, you determined at what frequency f the amplitude
of the voltage across the resistor will 1 / 2 of its maximum value. Write that answer
here, and evaluate this frequency numerically for the values of the R and C that you have
in your circuit now.
Open the DataStudio file lab11_plots.ds from the ClassNotes folder. This file will help
you to plot data you will take now. You want to take data on the peak-to-peak amplitude
of the voltage across the resistor for a variety of frequency settings. Use the MEASURE
function as you did last week (press MEASURE, select the Source as CH2 and the Type
as Pk-Pk.) The frequencies you choose should cover a large enough range to clearly see
the transition from the low amplitudes of voltage to high amplitudes. Note that the
maximum voltage you observe may not be quite as large as the input voltage. As you
take data you should type your (frequency, amplitude) pairs into the table. They will
automatically plot in the plot window. To make the curve that connects the points
continuous, you need to have the points typed in order from lowest to highest frequency.
If you want to insert a value in between, then just use the insert row command in the table
controls. When you have enough data (about ten) to clearly see the curve, print out a
copy of your plot for you and your partner. Label it at the top as “RC circuit” and include
it after this page.
Question 1-6: Does your plot show the general behavior you expect for a “high-pass”
filter? Why?
Question 1-7: Mark on your plot the point where the voltage appears to reach 1 / 2 of
its maximum value. Read from your plot what the approximate frequency is at that point
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and write it here. How does this observation compare with your prediction in Prediction
1-5?
Activity 2: A Driven RL circuit
Prepare to set up the RL circuit shown in the figure below. Use the same resistor (R=1
kΩ) that you used in the previous activity and use the large wire-wound inductor on your
table (see photo below).
Measure the actual values (using the DMM for the resistor and the LCR meter for the
inductor; use the 200 mH scale on the LCR meter for the inductor) of these components
before inserting them in the circuit and record the values below. (Note: the inductor has a
significant resistance as well, so you should measure it using the DMM and ADD it to the
resistance of your 1 kΩ resistor.)
R = ______________________
L = ______________________
To set up this circuit, you only need to swap in the inductor in place of the capacitor; no
other changes should be necessary from the circuit you used in Activity 1.
red
L
CH1
red
R
black
CH2
black
8
Before taking quantitative data vary the frequency over the range ~ 100 – 20,000 Hz and
see how the voltage across the resistor responds (in channel 2). Recall that the current in
the circuit is simply this voltage signal divided by the constant resistance R. So this
signal shows you the time dependent behavior of the current as well.
Question 2-1: Do you observe that the amplitude of the current in the circuit is greatest
at high or low frequencies?
.
Question 2-2: In your circuit, only the inductor has impedance that varies with frequency
(the impedance of the resistor does not vary with frequency.) Given your observation in
Question 2-1, do you conclude that the inductor has large or small impedance at high
frequencies? What about low frequencies? Explain.
Question 2-3: Based on your answer to question 2-2, at high frequencies, would you say
that the inductor behaves more like an open circuit (a break in the circuit’s wiring) or
more like a short circuit (a connection with very little resistance). Justify your answer.
Question 2-4: At low frequencies, does the inductor behave more like an open circuit (a
break in the circuit’s wiring) or more like a short circuit (a connection with very little
resistance). Justify your answer.
9
Prediction 2-5: In pre-lab question 2, you determined at what frequency f the amplitude
of the voltage across the resistor will 1 / 2 of its maximum value. Write that result here,
and evaluate this frequency numerically for the values of the R and L that you have in
your circuit now.
Open the DataStudio file lab11_plots.ds from the ClassNotes folder and take data on the
voltage amplitude versus frequency as you did in the previous exercise. When you have
enough data (about ten) to clearly see the curve, print out a copy of your plot for you and
your partner. Label it at the top as “RL circuit” and include it after this page.
Question 2-6: Does your plot show the general behavior you expect for a “low-pass”
filter? Why?
Question 2-7: Mark on your plot the point where the voltage appears to reach 1 / 2 of
its maximum value. Read from your plot what the approximate frequency is at that point
and write it here. How does this observation compare with your prediction in Prediction
2-5?
10
Activity 3: Resonance Behavior in a Series R-L-C Circuit
Set up the RLC circuit shown in the figure below. Use the exact same components you
have used earlier in the lab. So all you need to do is add the capacitor in series with the
circuit you built for Activity 2.
C
red
L
CH1
red
R
black
CH2
black
Before taking data with this circuit, you will make some predictions based on your
experience with how the capacitor and inductor behaved in the other two circuits you
built in this lab.
Prediction 3-1: At very low signal frequencies (near 0 Hz), do you expect the amplitude
of the voltage across the resistor (and therefore the amplitude of the current in the circuit)
will be relatively large, intermediate, or small? Explain your reasoning.
Prediction 3-2: At very high signal frequencies (above 10 kHz), do you expect the
amplitude of the voltage across the resistor (and therefore the amplitude of the current in
the circuit) will be relatively large, intermediate, or small? Explain your reasoning.
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Prediction 3-3: Based on your predictions in predictions 3-1 and 3-2, is there some
intermediate frequency where the amplitude of the voltage (across the resistor) and the
current will reach maximum or minimum values? Do you think they will be maximum or
minimum?
Before taking quantitative data, vary the frequency over the range ~ 100 – 10,000 Hz
and see how the voltage across (and therefore the current through) the resistor responds
(in channel 2).
Question 3-4: Is the behavior you observe consistent with your predictions in predictions
3-1 through 3-3? Explain.
.
Question 3-5: In pre-lab question 3, you determined at what frequency f the series RLC
circuit will achieve resonance. Write that result here, and evaluate this frequency
numerically for the values of the C and L that you have in your circuit now.
Open the DataStudio file lab11_plots.ds from the ClassNotes folder and take data on the
voltage amplitude versus frequency as you did in the previous exercise. When you have
enough data to clearly see the curve, print out a copy of your plot for you and your
partner. Label it at the top as “RLC series circuit” and include it after this page.
Question 3-6: Does your plot show the “resonant” behavior you expect? Describe what
you see and in what sense the behavior is “resonant”.
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Question 3-7: Mark on your plot the point where the resonant frequency is achieved.
Read from your plot what the approximate frequency is at that point and write it here.
How does this observation compare with your prediction in Prediction 3-5?
Note that this RLC series resonant circuit has many practical uses. One of the most
familiar uses of such a circuit is as a tuner in a radio receiver. In a typical tuner the
inductance of the inductor is varied to “tune” the resonance frequency to the desired
station.
This “electrical” resonance is exactly the same thing as mechanical resonances that you
observed in Ph2305. There you would have a mechanical system like a spring and a
mass, or a person on a swing set. If you drive the system with forced oscillations (like
pushing the person on the swing at a regular frequency), there is a particular frequency
where the system responds with the largest amplitude. That is the resonant frequency.
When you are done, please put you components back in the Ziploc bags they came
in. Please turn off the LCR meter and DMM to save the battery. Turn off your
oscilloscope and function generator. Tidy up your collection of cables and
connectors, so that the work area is neat for the next pair of students.
There are no post-lab questions this week.
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