Filters AC & Resonance
July 21
167
Name ___________________________ Date ____________ Partners__________________________________
INTRODUCTION TO AC
FILTERS AND RESONANCE
OBJECTIVES
To understand the design of capacitive and inductive filters
To understand resonance in circuits driven by AC signals
OVERVIEW
In a previous lab, you explored the relationship between impedance
(the AC equivalent of resistance) and frequency for a resistor,
capacitor, and inductor. These relationships are very important to
people designing electronic equipment, particularly audio equipment.
You can predict many of the basic characteristics of simple audio
circuits based on what you have learned in previous labs.
Recall that if there is a current of the form
I (t ) = I max sin (t )
flowing through a circuit containing resistors, capacitors and/or
inductors, then the voltage across the circuit will be of the form
V ( t ) = I max Z sin (t + ) .
Z is called the impedance and is called the phase shift. The
maximum voltage will be given by
Vmax = I max Z .
When is zero, the voltage and current will be in phase. When is
less than zero, the voltage will reach its peak before the current and we
say that the current lags the voltage. When is greater than zero, the
voltage will reach its peak after the current and we say that the current
leads the voltage.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
168
AC Filters & Resonance
July 21
For a series combination of a resistor, a capacitor and an inductor,
Z = R2 + ( X L X C )
2
and
tan( ) =
X L XC
R
where
Xc
1
C
and
X L L .
XC is called the capacitive reactance and XL is called the inductive
reactance. If there is only a capacitor or only an inductor, the
impedance is simply the corresponding reactance.
In this lab you will continue your investigation of the behavior of
resistors, capacitors and inductors in the presence of AC signals. In
Investigation 1, you will see how capacitors and inductors can act as
filters. More precisely, you will see how these elements can be used
to suppress the voltage of certain frequency ranges of AC signals,
while leaving other signals relatively unchanged.
In Investigation 2, you will explore the relationship between peak
current and peak voltage for a series circuit composed of a resistor,
inductor, and capacitor. You will also explore the phase difference
between the current and the voltage. This circuit is an example of a
resonant circuit. The phenomenon of resonance is a central concept
underlying the tuning of a radio or television to a particular frequency.
IMPORTANT NOTE: In the experiments today, we want to compare
the current through the circuit with the voltage across it. Normally we
would use a current probe, but the probes 1 Ohm internal resistance is not
negligible with respect to the circuit resistances.
All of the circuits today are series circuits and so the same current will
flow through each element. Hence, we can (and will) simply measure the
voltage drop across a resistor in the circuit and calculate the current from
Ohms law.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
AC Filters & Resonance
July 21
169
INVESTIGATION 1: INTRODUCTION TO AC FILTERS
The purpose of this lab is for you to create circuits that filter out AC
signals with frequencies outside the range of interest. In the context of
these activities, a filter is a circuit that attenuates the voltage of some
range of signal frequencies, while leaving other frequency ranges
relatively unaffected.
You will need the following materials:
Voltage probe
Multimeter
RLC Circuit Board
Alligator clip leads
Activity 1-1: Capacitors as Filters
In this activity, you will investigate how a circuit containing a resistor,
capacitor, and signal generator responds to signals at various
frequencies.
Consider the circuit in Figure 1-1 with a resistor, capacitor, signal
generator and voltage probe.
+
Vsignal
R
C
-
VPA
R = 33
C = 1.23 F
Vsignal = 5 V
fsignal = 200 Hz
Figure 1-1: Capacitive filter circuit
Prediction 1-1: On the axes that follow, use dashed lines to sketch your
qualitative prediction for the peak current through the circuit, Imax , as the
frequency of the signal from the signal generator is increased from zero.
[Remember that = 2f.]
Imax
fsignal
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
170
AC Filters & Resonance
July 21
Test your predictions.
1. Open the experiment file AC Filter.
2. We will use the internal signal generator of the computer interface.
Note that the signal generator parameters will appear on the
computer screen.
3. The signal generator should already be set to a frequency of
200 Hz and amplitude of 5 V (+5 V maximum and -5 V
minimum).
4. Before setting up the circuit, use the multimeter to measure the
value of the resistor, R, and the capacitor, C.
R __________
C __________ F
5. Connect the resistor, capacitor, signal generator and probe as
shown in Figure 1-1.
6. Press Start to turn on the scope display.
7. You should see two displays on the scope display. One will be the
voltage produced by the signal generator. This is the input
(source) voltage for the circuit. It should be 5 V at its peak. The
other voltage, sensed by VPA, will be the voltage across the
resistor R and is proportional to the current through the circuit.
8.Remember, we are explicitly using the voltage across R to
measure the current through the circuit.
8. You may need to adjust the time and voltage scales on the scope
display so that both the waveforms are visible. You may also need
to adjust the trigger level on the left part of the screen to see the
waveforms. Play with the trigger level a bit to see how it operates.
9. Use the Smart Tool to determine the peak (maximum) voltage,
Vmax, across the resistor (not the signal voltage, which should
remain at 5 V), write it in Table 1-1 with f = 200 Hz. Then
calculate the maximum current from the maximum voltage using
the value of the resistor you measured in step 4.
10. Increase the frequency of the signal generator to 1,200 Hz. Be
sure that the peak signal amplitude is still 5 V. Repeat step 9.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
AC Filters & Resonance
July 21
171
11. Repeat step 9 for 2,200 Hz, 4,200 Hz and 8,200 Hz.
Table 1-1
fsignal (Hz)
Vmax (V)
Imax (A)
12. Sketch the data from Table 1-1 on the axes below. Mark scales on
the vertical axes.
Imax
(m A )
1
2
3
4
5
6
7
8
9
10
fsignal (kHz)
Question 1-1: If you could continue taking data up to very high
frequencies, what would happen to the peak current, Imax through the
circuit?
Question 1-2: At very high frequencies, does the capacitor act more
like an open circuit (a break in the circuits wiring) or more like a short
circuit (a connection with very little resistance)? Justify your answer.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
172
AC Filters & Resonance
July 21
13. Now note the phase difference (in the next question) between the
peaks of the signal generator voltage and the voltage across the
resistor (~circuit current) at the frequency 8,200 Hz that you should
still have (note that they should be close to being in phase). Then
go back to a frequency of 200 Hz and observe the phase difference.
Question 1-3: What phase difference do you observe between the
peaks of the signal voltage and circuit current for low and high
frequency?
Question 1-4: What would the current be through the circuit if we
applied only a DC voltage? Explain.
Question 1-5: At very low frequencies, does the capacitor act more
like an open circuit (a break in the circuits wiring) or more like a short
circuit (a connection with very little resistance)? Justify your answer.
Comment: In the circuit in Figure 1-1, since the peak signal voltage
from the signal generator remains unchanged, the peak current in the
circuit must increase as the total impedance decreases. Therefore, the
peak voltage across the resistor increases as the frequency of the signal
increases. This type of circuit is an example of a high-pass circuit or
filter.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
AC Filters & Resonance
July 21
173
Activity 1-2: Inductors as Filters
This activity is very similar to the previous one except that you will
replace the capacitor with an inductor and determine the filtering
properties of this new circuit.
Consider the circuit containing a resistor, inductor, signal generator
and probes shown in Figure 1-2 below.
+
R
Vsignal
-
VPA
L
L = 8.2 mH
R = 33
Vsignal = 5 V
fsignal = 20 Hz
Figure 1-2: Inductive Filter Circuit
Prediction 1-2: On the axes that follow, use dashed lines to sketch
your qualitative prediction for the peak current through the circuit,
Imax, as the frequency of the signal from the signal generator is
increased from zero.
Imax
fsignal
Test your predictions.
1. You can continue to use the experiment file AC Filter.
2. Set the signal generator to a frequency of 20 Hz and amplitude of
5 V.
3. Before setting up the circuit, use the multimeter to measure the
inductance L and resistance RL of the inductor
L __________ mH
University of Virginia Physics Department
PHYS 636, Summer 2006
RL _______________
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
174
AC Filters & Resonance
July 21
4. Connect the resistor, inductor, signal generator and probe as shown
in Figure 1-2. Simply replace the capacitor in the previous setup
with the inductor.
5. Press Start to turn on the scope display.
6. Adjust the time and voltage scales on the scope so that both
waveforms are visible. Remember the trigger level.
7. Use the Smart Tool to determine the peak voltage and peak
current, and enter in Table 1-2. Then calculate the maximum
current from the maximum voltage using the value of the resistor.
Table 1-2
fsignal (Hz)
Vmax (V)
Imax (mA)
8. Increase the frequency of the signal generator to 200 Hz. Make
sure that the amplitude is still 5 V.
9. Repeat step 6 with 1,200 Hz, 2,200 Hz, 4,200 Hz and 8,200 Hz.
10. Sketch the data from Table 1-2 on the axes below.
Imax
(mA)
1
University of Virginia Physics Department
PHYS 636, Summer 2006
2
3
4
5
6
fsignal (kHz)
7
8
9
10
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
AC Filters & Resonance
July 21
175
Question 1-6: If you could continue taking data up to very high
frequencies, what would happen to the peak current, Imax, through the
resistor?
Question 1-7: At very high frequencies, does the inductor act more
like an open circuit (a break in the circuits wiring) or more like a short
circuit connection (a with very little resistance)? Justify your answer.
11. Now note the phase difference between the peaks of the signal
voltage and the voltage across the resistor (~circuit current) at the
frequency 8,200 Hz that you should still have and then go back to
a frequency of 20 Hz and observe the phase difference.
Question 1-8: What phase difference do you note between the peaks of
the signal voltage and circuit current for low and high frequency?
Note: We did this in last weeks experiment as well.
Question 1-9: What would the current through the circuit be if we
applied only a DC voltage?
Question 1-10: At very low frequencies, does the inductor act more
like an open circuit (a break in the circuits wiring) or more like a short
circuit (a connection with very little resistance)? Justify your answer.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
176
AC Filters & Resonance
July 21
Comment: In the circuit in Figure 1-2, since the peak voltage from the
signal generator remains unchanged, the peak current in the circuit
must decrease as the total impedance increases. Therefore, the peak
voltage across the resistor decreases as the frequency of the signal
increases. This type of circuit is an example of a low-pass circuit or
filter.
INVESTIGATION 2: THE SERIES RLC RESONANT (TUNER) CIRCUIT
In this investigation, you will use your knowledge of the behavior of
resistors, capacitors and inductors in circuits driven by various AC
signal frequencies to predict and then observe the behavior of a circuit
with a resistor, capacitor, and inductor connected in series.
The RLC series circuit you will study in this investigation exhibits a
resonance behavior that is useful for many familiar applications.
One of the most familiar uses of such a circuit is as a tuner in a radio
receiver.
You will need the following materials:
voltage probe
RLC Circuit Board
Consider the series RLC circuit shown in Figure 2-1 (below).
+
R
Vsignal
-
VPA
L
L = 8.2 mH
C = 1.23 F
R = 33
Vsignal = 5 V
C
Figure 2-1: RCL Series Circuit
Prediction 2-1: At very low signal frequencies (near 0 Hz), will the
maximum values of I through the resistor and V across the resistor be
relatively large, intermediate or small? Explain your reasoning.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
AC Filters & Resonance
July 21
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Prediction 2-2: At very high signal frequencies (well above
3,000 Hz), will the maximum values of I and V be relatively large,
intermediate or small? Explain your reasoning.
Prediction 2-3: Based on your Predictions 2-1 and 2-2, is there some
intermediate frequency where I and V will reach maximum or
minimum values? Do you think they will be maximum or minimum?
1. On the axes below, draw qualitative graphs of XC vs. frequency
and XL vs. frequency. Clearly label each curve.
XC
and
XL
F re q u e n c y
Question 2-1 For what relative values of XL and XC will the total
impedance of the circuit, Z, be a minimum? Explain your reasoning.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
178
AC Filters & Resonance
July 21
2. On the axes above, mark and label the frequency where Z is a
minimum.
Question 2-2 At the frequency you labeled, will the value of the peak
current, Imax, in the circuit be a maximum or minimum? What about
the value of the peak voltage, Vmax, across the resistor? Explain.
Note: The point you identified in step 2 is the resonant frequency.
Label it with the symbol f0. The resonant frequency is the frequency at
which the impedance of the series combination of a resistor, capacitor
and inductor is minimal. This occurs at a frequency where the values
of XL and XC are equal.
3. On the axes above (after step 1) draw a curve that qualitatively
represents XL - XC vs. frequency. Be sure to label it.
4. Use your results from above to determine the general mathematical
expression for the resonant frequency, f0, as a function of L and C.
(Hint: you will need the expressions for XC and XL given to you in
step 1)
Equation for f0 : ___________________________
You will now test your predictions.
Activity 2-1: The Resonant Frequency of a Series RLC Circuit.
1. Open the experiment file RLC Resonance.
2. Adjust the scope display to 1 V/div and 1 ms/div.
3. Connect the circuit with resistor, capacitor, inductor, signal
generator and probe shown in Figure 2-1.
4. Set the signal generator to a frequency of 200 Hz and amplitude of
5 V.
5. Press Start to begin taking data
6. Use the Smart Tool to determine the peak voltage, Vmax.
7. Enter the data in the first row of Table 2-1.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
AC Filters & Resonance
July 21
179
Table 2-1
fsignal (Hz)
Vmax (V)
200
700
1,200
1,700
2,200
2,700
3,200
8. Repeat steps 5 through 7 for the other frequencies in Table 2-1.
Be sure that the amplitude of the signal generator is always 5 V.
9. Calculate the resonant frequency for your circuit. Show your
calculations. Use the formula from step 4 and the actual values of
the capacitance and inductance.)
fresonance = ___________Hz
CALCULATED
10. Measure the resonant frequency of the circuit to within a few Hz.
To do this, press Start to begin taking data and slowly adjust the
frequency of the signal generator until the peak voltage across the
resistor is maximal. It may be helpful to use the scope display for
this. (Use the results from Table 2-1 to help you locate the
resonant frequency.)
fresonance = ___________Hz
EXPERIMENTAL (Amplitude)
Question 2-3: How does this experimental value for the resonant
frequency compare with your calculated one?
Activity 2-2: Phase in an RLC Circuit
In previous labs (and in this one), you investigated the phase
relationship between the current and voltage in an AC circuit
composed of a signal generator connected to one of the following
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
180
AC Filters & Resonance
July 21
circuit elements: a resistor, capacitor, or an inductor. You found that
the current and voltage are in phase when the element connected
to the signal generator is a resistor, the current leads the voltage
with a capacitor, and the current lags the voltage with an inductor.
You also discovered that the reactances of capacitors and inductors
change in predictable ways as the frequency of the signal changes,
while the resistance of a resistor is constant independent of the signal
frequency. When considering relatively high or low signal frequencies
in a simple RLC circuit, the circuit element (either capacitor or
inductor) with the highest reactance is said to dominate" because this
element determines whether the current lags or leads the voltage. At
resonance, the reactances of capacitor and inductor cancel, and do not
contribute to the impedance of the circuit. The resistor then is said to
dominate the circuit.
In this activity, you will explore the phase relationship between the
applied voltage (signal generator voltage) and current in an RLC
circuit.
Consider the RLC circuit shown below.
+
R
Vsignal
-
VPA
L
L = 8.2 mH
C = 1.23 F
R = 33
Vsignal = 5 V
C
Figure 2-2: RLC series circuit
Question 2-4: Which circuit element (the resistor, inductor, or
capacitor) dominates the circuit in Figure 2-2 at frequencies well
below the resonant frequency? Explain.
Question 2-5: Which circuit element (the resistor, inductor, or
capacitor) dominates the circuit in Figure 2-2 at frequencies well
above the resonant frequency? Explain.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
AC Filters & Resonance
July 21
181
Question 2-6a: In the circuit in Figure 2-2, will the current through
the resistor always be in phase with the voltage across the resistor,
regardless of the frequency? Explain your reasoning.
Question 2-6b: If your answer to Question 2-6a was no, then which
will lead for frequencies below the resonant frequency (current or
voltage)? Which will lead for frequencies above the resonant
frequency (current or voltage)?
Question 2-7a: In the circuit in Figure 2-2, will the current through
the resistor always be in phase with applied voltage from the signal
generator? Why or why not?
Question 2-7b: If your answer to Question 2-7a was no, then which
will lead for frequencies below the resonant frequency (current or
voltage)? Which will lead for frequencies above the resonant
frequency (current or voltage)?
Test your predictions.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
182
AC Filters & Resonance
July 21
1. Open the experiment file called RLC Phase.
2. Connect the circuit shown in Figure 2-2.
3. Set the signal generator to a frequency 200 Hz below the resonant
frequency you measured in Activity 2-1, and set the amplitude of
the signal to 5 V.
4. Press Start to begin taking data.
5. Determine whether the current or applied voltage leads.
Question 2-8: Which leads applied voltage, current or neither
when the AC signal frequency is lower than the resonant frequency?
Were your predictions correct? Why or why not? Explain.
6. Set the signal generator to a frequency 200 Hz above the resonant
frequency with the amplitude of signal still 5 V.
7. Determine whether the current or applied voltage leads.
Question 2-9: Which leads applied voltage, current or neither
when the AC signal frequency is higher than the resonant frequency?
Were your predictions correct? Why or why not? Explain.
Prediction 2-5: Which will lead for an applied signal at the resonant
frequency (current or voltage or neither)?
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000
AC Filters & Resonance
July 21
183
8. Set the signal generator to the resonant frequency you measured in
Activity 2-1, and set the amplitude of the signal to 5 V.
9. Determine whether the current or applied voltage leads.
Question 2-10: At resonance, does the current or applied voltage lead
(or neither)?
10. Use this result to find the resonant frequency.
fresonance = ___________Hz
EXPERIMENTAL (Phase)
Question 2-11: How does this experimental value for the resonant
frequency with your calculated one?
Question 2-12: How does this experimental value for the resonant
frequency compare with the one you determined by looking at the
amplitude? Comment on the relative sensitivities of the two
techniques.
University of Virginia Physics Department
PHYS 636, Summer 2006
Modified from P. Laws, D. Sokoloff, R. Thornton
Supported by National Science Foundation
and the U.S. Dept. of Education (FIPSE), 1993-2000