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Name: Lab Partners:
Date:
Pre-Lab Assignment
Capacitors, Inductors & Resonance
(Due at the beginning of lab)
Directions: Read over Lab and then answer the following questions about the procedures.
Question 1 What is the effect of an inductor filter on a multi-frequency AC signal?
Question 2 What does resonance mean? Give an example of resonance in a mechanical sys- tem.
Question 3 What do you change when you tune a radio to your favourite radio station?
Question 4 The unit of resistance is the ohm, denoted by the greek letter Ω.
a) What is the unit of capacitance?
b) What is the unit of capacitive reactance?
c) What is the unit of inductance?
d) What is the unit of inductive reactance?
PHYS-204: Physics II Laboratory i
Name: Lab Partners:
Date:
Capacitors, Inductors, and Resonance
Objectives
- To understand the frequency response of capacitors and inductors.
- To understand resonance in circuits driven by AC signals.
Overview
In a previous lab you studied the time-dependent behavior of Alternating Current (AC) signals that exist all around you. Electronic devices in everyday use such as radio receivers and am- plifiers, computers and televisions use AC signals that are manipulated in very precise ways. Resistors, capacitors, and inductors are used in those manipulations and constitute a very important part of electronic circuits. This lab continues the investigation of the role of resistors, capacitors and inductors in circuits powered by AC signals. In Investigation 1, you will study the role of capacitors and inductors as AC filters where voltage of certain frequency ranges of AC signals are filtered out while leaving signals of other frequencies relatively unchanged. In Investigation 2, you will discover the relationship between peak current and peak voltage for a series circuit containing a resistor, an inductor, and a capacitor and examine the phase relationship between the current and the voltage in such circuits. You will also study the situation that maximizes the current in the circuit for a narrow range of frequencies at the “resonance condition”. The phenomenon of resonance is the central concept underlying the tuning of a radio or television to a particular frequency. These labs have been adapted from the Real Time Physics Active Learning Laboratories [1]. The goals, guiding principles and procedures of these labs closely parallel the implementations found in the work of those authors [1, 2, 3].
Investigation 1:
Introduction to AC Filters
In the previous lab, you explored the relationship between impedance (the AC equivalent of resistance) and the frequency for a resistor, capacitor, and inductor. These relationships play a very important in the design of electronic equipment, in particular audio and video equipment. You can predict many of the basic characteristics of simple audio circuits based on what you have learned in previous labs. In this lab you will create circuits that filter out those AC signals of frequencies that lie outside the range of interest. In this lab, a filter is a circuit that suppresses the voltage of certain signal frequencies, leaving other frequency ranges relatively unaffected. You will need the following materials:
- computer-based laboratory system with Logger Pro software.
- voltage probe.
- multimerter.
- 10 ohm resistor.
Explain how you arrived at your graphs. Is the graph of the current qualitatively similar or different from the graph of the voltage? Explain your answer.
Test your predictions.
Step 1: Open the experiment file L9A1-1 (Capacitive Filter).
Comment: In this experiment we use just the voltage probe, even though we want to know about both the current and the voltage. Remember that in the previous experiment you found that for a resistor, I = V /R , with the same value of the resistance, R, at all frequencies. Thus we can use the software to calculate the current by dividing the voltage measurements by R.
Step 2: Measure the actual resistance of the 100 ohm resistor using the multimeter. On the computer select Modify Column under the Data menu and choose current. In the formula that defines the current replace the number 1 in the divisor by your measured value for R.
Step 3: Zero the voltage probe with it disconnected from the circuit.
Step 4: Connect the resistor, capacitor, signal generator and probe as shown in Fig. 1.
Step 5: Set the signal generator to a frequency of 500Hz. Adjust the amplitude control to half of its maximum value (the mark on the amplitude knob should be vertical).
Step 6: Collect data. When you have a good graph, click on stop to capture the graph.
Step 7: Use the statistics command in the analysis menu to determine the peak voltage and peak current, and enter these data in table 1.
Fsignal (Hz) Vmax (V) Imax (A)
Table 1:
Step 8: Reduce the frequency of the signal generator to another value of frequency (pick a value significantly less than 500 Hz). Begin graphing. Again use the statistics command to determine the peak voltage and current and enter in table 1.
Step 9: Continue making measurements of peak voltage and current at different frequencies, as in the previous step. You should have well distributed measurements over the range
of frequencies from 500 Hz down to 100 Hz or lower. If you find a range of frequencies in which small changes in frequency produce large changes in the peak voltage, you should be sure to make measurements at several frequencies within that range.
Step 10: Plot the data from Table 1 on the axes below. Mark scales on the vertical axes.
fsignal (Hz)
V
max
(V)
fsignal (Hz)
Imax
(A)
Question 1.1 If you could continue taking data up to very high frequencies, what would happen to the peak current, Imax, through the resistor and the peak voltage, Vmax, across the resistor?
Question 1.2 At very high frequencies, does the capacitor act pretty much like an open circuit (an incomplete or broken circuit) or rather like a short circuit (a wire with a very small or negligible resistance)? Justify your answer.
Question 1.3 What signal frequency does a DC signal have?
Prediction 1.3 What would be the current through and voltage across the resistor if you replaced the signal produced by the AC signal generator with a DC signal?
Prediction 1.4 Make a qualitative prediction for the behavior of the peak current through the resistor, Imax, as the frequency of the signal is increased from zero and sketch it in the left panel below. [Hint: recall that the inductor’s impedance is related to the frequency of the signal by the expression XL = 2πf L.]
Prediction 1.5 Make a qualitative prediction for the behavior of the peak voltage across the resistor, Vmax, as the frequency of the signal is increased from zero and sketch it in the right panel below.
fsignal (Hz)
V
max
(V)
fsignal (Hz)
Imax
(A)
Explain how you arrived at your graphs. Is the graph of the current qualitatively similar or different from the graph of the voltage? Explain your answer.
Test your predictions:
Step 1: Open the experiment file L9A1-2 (Inductive Filter).
Step 2: Measure the actual resistance of the 10 ohm resistor using the multimeter. On the computer select Modify Column under the Data menu and choose current. In the formula that defines the current replace the number 1 in the formula by your measured value for R.
Step 3: Zero the voltage probe with the probe disconnected from the circuit.
Step 4: Connect the resistor, inductor, signal generator and probe as shown in Fig. 1.
Step 5: Set the signal generator to a frequency of 100Hz. Adjust the amplitude control to half of its maximum value (the mark on the amplitude knob should be near vertical).
Step 6: Collect data. When you have a good graph, click on stop to capture the graph.
Step 7: Use the statistics command in the analysis menu to determine the peak voltage and peak current, and enter the data in table 2.
Fsignal (Hz) Vmax (V) Imax (A)
Table 2:
Step 8: Increase the frequency of the signal generator to a value greater than 100 Hz. Begin graphing. Again use the statistics command to determine the peak voltage and current and enter in table 1.
Step 9: Continuing making measurements of peak voltage and current at different frequencies, as in step 8. You should have measurements over the range of frequencies from 100 Hz up to 500 Hz. If you find a range of frequencies in which small changes in frequency produce large changes in the peak voltage, you should be sure to make measurements at several frequencies within that range.
Step 10: Plot the data from Table 2 on the axes below. Mark scales on the vertical axes.
fsignal (Hz)
V
max
(V)
fsignal (Hz)
Imax
(A)
Question 1.5 If you continued taking data up to very high frequencies, what would happen to the peak voltage Vmax, across the resistor, and peak current, Imax, through the resistor?
Question 1.8 Into which circuit should you wire the tweeter-the capacitive filter circuit or the inductive filter circuit? Briefly explain your reasoning.
Investigation 2:
The Series RLC Resonant Circuit
In this investigation you will use your knowledge of the behavior of resistors, capacitors and inductors in circuits driven by AC signal of various 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 used in a radio receiver to tune to a particular radio station.. You will need the following materials:
- computer-based laboratory system with Logger Pro software.
- voltage probe.
- multimeter.
- 10 ohm resistor.
- 7 μF capacitor.
- 8 mH inductor.
- Low impedance signal generator.
Consider the series RLC circuit shown in Fig. 3.
Vsignal(t) ∼
AC input
L
C
R VP
−
Figure 3: RLC Series Circuit
Prediction 2.1 At very low signal frequencies (near 0 Hz), will the maximum values of I and V across the resistor be relatively large, intermediate or small? Explain your reasoning.
Prediction 2.2 At very high signal frequencies (well above 100Hz), 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?
Prediction 2.4 On the axes below, draw qualitative graphs of XC vs. frequency and XL vs. frequency. Clearly label each curve. (Hint: base your answers on your observations in the previous lab and in investigation 1 of this lab.)
f (Hz)
X
Comment: As we noted earlier in this lab, the relationship between the total impedance, Z, for a series combination of a resistor, capacitor, and inductor is not just the sum of the impedances of the three circuit elements. Instead, because of phase differences, Z is given by the following expression:
Z =
R^2 + (XL − XC )^2
Note that the combination XL − XC appears because the phase current is behind the phase of the voltage in the inductor, while the phase of the current is ahead in the capacitor.
Prediction 2.5 For what values of XL and XC will the total impedance of the circuit, Z, be a minimum? On the axes above, mark and label the frequency where this occurs. Explain your reasoning.
Fsignal (Hz) Vmax (V) Imax (A)
Table 3:
Step 7: Use the statistics command in the analysis menu to determine the peak voltage and peak current, and enter these in Table 3.
Step 8: Increase the frequency of the signal generator to a value greater than 100 Hz. Begin graphing. Again use the statistics command to determine the peak voltage and current and enter in table 1.
Step 9: Continuing making measurements of peak voltage and current at different frequencies, as in step 8. You should have measurements over the range of frequencies from 100 Hz up to 1000 Hz. If you find a range of frequencies in which small changes in frequency produce large increases or decreases in the peak voltage, you should be sure to make measurements at several frequencies within that range.
Step 10: Plot the data from Table 3 on the axes below. Mark scales on the vertical axes.
fsignal (Hz)
V
max
(V)
fsignal (Hz)
Imax
(A)
Question 2.1 Does the behavior of the voltage across the resistor and current in the circuit in Fig. 3 agree with your predictions, especially Predictions 2.5 to 2.7? Explain.
Prediction 2.9 Calculate the resonant frequency for your circuit. Show your calcula- tions. (Hint: use the formula from Prediction 2.8 and the actual values of the capacitance and inductance.) f 0 = Hz.
Step 11: Measure the resonant frequency of the circuit to within 1 Hz. To do this, begin graphing and slowly adjust the frequency of the signal generator until the peak voltage across the resistor is maximum. (Use your results from Table 3 to help you locate the resonant frequency.)
f 0 = Hz.
Question 2.2 How does this experimental value for the resonant frequency compare with your Prediction 2.9?
In a radio receiver, the signal generator is replaced by a long antenna, which picks up all of the radio signal frequencies. (This is similar to a microphone, mentioned in the previous activity, detecting all the frequencies, or pitches, of music at once.) By strategically choosing values of C and L you can tune the circuit to the frequency of your favorite radio station, meaning the circuit is at resonance (the amplitude of the voltage across R is a maximum) for that particular station’s frequency. (In many real radio receivers, a variable capacitor is used. When you turn the knob, you change the capacitance and, hence, the resonant frequency.) Of course the signal picked up by the radio receiver is very weak-possibly just a microvolt. An important and surprising property of resonant circuits makes it possible to obtain a much higher voltage. We will explore this property in the next activity.
Activity 2.2: Voltage across the capacitor at resonance
Question 2.3 Suppose that the peak value of the voltage of the signal applied to the series RLC resonant circuit is denoted by V 0 and the total resistance of the circuit is R. You know that Imax = V 0 /Z and Z =
R^2 + (XL − XC )^2. Using the fact that for the resonant frequency XL = XC , write an equation relating the peak current at resonance,I 0 , to the signal voltage, V 0 , and the total resistance of the circuit, R.
Question 2.4 Does your answer to question 2.3 imply that the current at the resonant fre- quency depends in any way on the inductance L or the capacitance C in the circuit? Explain your answer.
Question 2.7 The peak AC current flowing through the capacitor and the peak voltage across the capacitor are related by Ohm’s law, using the impedance of the capacitor: Cal- culate the voltage that should appear across the capacitor for the current you measured in the previous step.
Calculated capacitor voltage at resonance: VC = V.
Step 9: Move the clips of the voltage probe to measure the voltage across the capacitor, as in Fig. 5.
Vsignal(t) ∼
AC input
L
C
R
VP
− +
Figure 5: RLC resonant circuit with voltage probe connected to measure the voltage across the capacitor.
Step 10: Collect data. When you have a good graph, click on stop to capture the graph.
Step 11: Use the statistics command in the analysis menu to determine the peak current
measured capacitor voltage at resonance: VC = V.
Question 2.8 Does the measured value for the voltage across the capacitor agree with the value you calculated for question 2.7?
Question 2.9 Can you explain how it is possible for the voltage across just the capacitor to be larger than the voltage across the entire combination of resistor, inductor and capacitor. In your explanation you will need to make use of the fact that the phase of the voltage is ahead of the current in the inductor, and behind in the capacitor. The total phase difference between the voltage in the inductor and capacitor is almost 180 ◦. It may help to draw sketches of two sine functions out of phase with each other by 180 ◦.
This laboratory exercise has been adapted from the references below.
References
[1] David R. Sokoloff, Priscilla W. Laws, Ronald K. Thornton, and et.al. Real Time Physics, Active Learning Laboratories, Module 3: Electric Circuits. John Wiley & Sons, Inc., New York, NY, 1st edition, 2004.
[2] Priscilla W. Laws. Workshop Physics Activity Guide, Module 4: Electricity and Magnetism. John Wiley & Sons, Inc., New York, NY, 1st edition, 2004.
[3] Lilian C. McDermott, et.al. Physics by Inquiry, Volumes I & II. John Wiley & Sons, Inc., New York, NY, 1st edition, 1996.
