Capacitors, Inductors & Resonance: A Physics II Laboratory Experiment, Lab Reports of Physics

Download Capacitors, Inductors & Resonance: A Physics II Laboratory Experiment and more Lab Reports Physics in PDF only on Docsity! 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. PHYS-204: Physics II Laboratory 1 Capacitors, Inductors & Resonance v 0.1 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. 100 200 300 400 500 fsignal (Hz) V m a x (V ) 100 200 300 400 500 fsignal (Hz) I m a x (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? PHYS-204: Physics II Laboratory 4 Capacitors, Inductors & Resonance v 0.1 Test your prediction by acquiring a DC data point (by using batteries). Step 11: Replace the signal generator with two D-cells in series. Step 12: Begin graphing. Step 13: Determine the peak voltage and current. Step 14: Enter this data point in Table 1 and plot it on your graph. Question 1.4 At very low 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. Comment: In a previous lab you learned that a capacitor’s impedance, its capacitive reac- tance, decreases as the frequency of the AC signal increases and that the impedance of a resistor is independent of the signal frequency. Note that even though these elements are in series, their impedances do not simply add together since the current and voltage are not in phase. The actual relation is: Z = √ R2 + X2C . Nevertheless, as you can see from this expression for Z, the impedance of the series circuit decreases as the reactance of the capacitor decreases. Since XC = 1 2πfC , the impedance of the circuit decreases as the signal frequency increases. The peak voltage applied to the circuit by the signal generator and peak current are related to each other by: Vapplied = ImaxZ. The peak voltage applied by the signal generator remains unchanged in the circuit shown in Fig. 1, and therefore the peak current in the circuit must increase as the total impedance decreases. As a result, 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” filter. Activity 1.2: Inductors as Filters The activity that follows is similar to the previous one except that you will replace the capacitor with an inductor and determine the filtering properties of the inductor circuit. Consider the circuit containing a resistor, inductor, signal generator and probes shown in Fig. 2 below. Vsignal(t) ∼ AC input L R VP1 + − Figure 2: Inductive filter circuit.L = 8mH, R = 10 Ω PHYS-204: Physics II Laboratory 5 Capacitors, Inductors & Resonance v 0.1 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 = 2πfL.] 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. 0 100 200 300 400 500 fsignal (Hz) V m a x (V ) 0 100 200 300 400 500 fsignal (Hz) I m a x (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). PHYS-204: Physics II Laboratory 6 Capacitors, Inductors & Resonance v 0.1 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 VP1 + − 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. PHYS-204: Physics II Laboratory 9 Capacitors, Inductors & Resonance v 0.1 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 (Ω ) 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 = √ R2 + (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. PHYS-204: Physics II Laboratory 10 Capacitors, Inductors & Resonance v 0.1 Prediction 2.6 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 your reasoning. The point you identified for Predictions 2.5 and 2.6 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 minimum. This occurs at a frequency where the values of XL and XC are equal. Prediction 2.7 On the axes above draw a curve that represents XL − XC vs. frequency. Be sure to label it. Prediction 2.8 Use your results from Predictions 2.5 and 2.7 to determine the general ex- pression for the resonant frequency, f0, as a function of L and C. (Hint: you will need the expressions for XL and XC given below.) XL = 2πfL and XC = 1/(2πfC) Test your predictions: Activity 2.1: The Resonant Frequency of a Series RLC circuit Step 1: Open the experiment file L9A2-1 (RLC Resonance). Step 2: 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 the value you measured for the resistance of your 10 Ω resistor. Step 3: Zero the voltage probe while disconnected from the circuit. Step 4: Connect the circuit with resistor, capacitor, inductor, signal generator and probes shown in Fig. 3. 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. PHYS-204: Physics II Laboratory 11 Capacitors, Inductors & Resonance v 0.1 Step 1: Remove the connections from the coil in your circuit. Use the multimeter set to measure ohms to measure the resistance of the coil. Add it to the value you measured for the resistance to get the total resistance of the series circuit. Rtot = ? Question 2.5 If the peak value of the signal voltage is V0, calculate the peak current at the resonant frequency, using your equation from question 2.4 and the total resistance you determined in step 1. Calculated current at resonance, I0 = A. Step 2: Reconnect the coil to your resonant circuit. Step 3: Move the clips of the voltage probe to measure the voltage provided by the signal generator, as shown in Fig. 4. Vsignal(t) ∼ AC input L C RVP1 + − Figure 4: RLC resonant circuit with voltage probe connected to measure the applied signal. Step 4: Set the signal generator to the resonant frequency that you measured in the previous activity. Step 5: Collect data. Adjust the amplitude of the signal generator to obtain a peak value of 1 volt for the applied signal. You can be more accurate if you change the scale of the voltage graph. Stop collecting data after you have the amplitude set. Warning: Once you have set the amplitude be careful not to move the amplitude control during the remaining steps in this activity. Step 6: Move the voltage probe clips to measure the voltage across the resistor, as in Fig. 3. Step 7: Collect data. When you have a good graph, click on stop to capture the graph. Step 8: Use the statistics command in the analysis menu to determine the peak current measured current at resonance: I0 = A Question 2.6 Does the measured value of the current at the resonant frequency agree with the value you calculated in question 2.5? PHYS-204: Physics II Laboratory 14 Capacitors, Inductors & Resonance v 0.1 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 VP1 − + 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. PHYS-204: Physics II Laboratory 15 Capacitors, Inductors & Resonance v 0.1 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. PHYS-204: Physics II Laboratory 16