AC CURRENTS, VOLTAGES, FILTERS, and RESONANCE

Transcription

1 July 22, 2008 AC Currents, Voltages, Filters, Resonance 1 Name Date Partners AC CURRENTS, VOLTAGES, FILTERS, and RESONANCE V(volts) t(s) OBJECTIVES To understand the meanings of amplitude, frequency, phase, reactance, and impedance in AC circuits. To observe the behavior of resistors in AC circuits. To observe the behaviors of capacitors and inductors in AC circuits. To examine the resonant behavior of RLC series circuits. OVERVIEW Until now, you have investigated electric circuits in which a battery provided an input voltage that was effectively constant in time. This is called a DC or Direct Current signal. A steady voltage applied to a circuit eventually results in a steady current. Steady voltages are usually called DC voltages as shown in Figure 1. voltage DC Signal Figure 1 time

2 2 July 22, 2008 AC Currents, Voltages, Filters, Resonance Signals that change over time (see Figure 2) exist all around you, and many of these signals change in a regular manner. For example, the electrical signals produced by your beating heart change continuously in time. Examples of Time-Varying Signals voltage time voltage time Figure 2 There is a special class of time-varying signals. These signals can be used to drive current in one direction in a circuit, then in the other direction, then back in the original direction, and so on. They are referred to as AC or Alternating Current signals as seen in Figure 3. Examples of AC Signals voltage time voltage time Figure 3 It can be shown that any periodic signal can be represented as a sum of weighted sines and cosines (known as a Fourier series). It can also be shown that the response of a circuit containing resistors, capacitors, and inductors (an RLC circuit) to such a signal is simply the sum of the responses of the circuit to each sine and cosine term with the same weights. We further note that a cosine is just a sine that is shifted back in time by ¼ cycle (a phase shift of -90 or -π/2 radians). So, to analyze an RLC circuit we need only find the response of the circuit to an input sine wave of arbitrary frequency.

3 July 22, 2008 AC Currents, Voltages, Filters, Resonance 3 Let us suppose that we have found a way to generate an input current of the form: I ( t) = I sin ωt (1) max ( ) Note: Here we use the angular frequency, ω, which has units of radians per second. Most instruments report f, which has units of cycles per second or Hertz (Hz). Clearly, ω = 2πf. We have already seen that the voltage across a resistor is then given by: R( ) max sin ( ω ) V t = RI t (2) Without proof we will state that the voltage across a capacitor is given by: Imax Imax π VC ( t) = cos( ωt) = sin ωt (3) ωc ωc 2 and the voltage across an inductor is given by: π VL ( t) = ωlimax cos( ωt) = ωlimax sin ωt + 2 These are can all be written in the form (a generalized Ohm's Law): ( ) sin ( ω ϕ ) max (4) V t = I Z t + (5) Arbitrary combinations of resistors, capacitors and inductors will have voltage responses of this form. Z is called the impedance and φ is called the phase shift. The maximum voltage will be given by V = I Z (6) max Consider a series circuit with a resistor, capacitor, and inductor as shown in Figure 4. max V R L C Figure 4. Series circuit of AC voltage and R, L, and C.

4 4 July 22, 2008 AC Currents, Voltages, Filters, Resonance The impedance for a RLC series circuit is given by and the phase shift φ series by where and ( ) 2 Z = R + X X (7) 2 series L C X L X C tan( ϕseries ) = (8) R X c X L 1 (9) ωc ωl (10) X C is called the capacitive reactance and X L is called the inductive reactance. If there is only a capacitor or only an inductor, the impedance is simply the corresponding reactance. If we rearrange Equation (6) and solve for the current I max, we have I V Z max max = (11) We obtain the maximum current with the impedance Z is a minimum. If we examine Equation (7) we see that this occurs when X L = X C or ωl 1 1 ω ωc LC 2 = or =. (12) The condition for resonance in an RLC series circuit is then 1 1 ω = and f = (13) LC 2π LC In Investigation 1, you will explore how a time-varying signal affects a circuit with a resistor. In Investigations 2 and 3, you will explore how capacitors and inductors influence the current and voltage in various parts in an AC circuit. In Investigation 4, you will look at the resonance in an RLC series circuit.

5 July 22, 2008 AC Currents, Voltages, Filters, Resonance 5 INVESTIGATION 1: AC SIGNALS AND RESISTANCE In this investigation, you will consider the behavior of resistors in a circuit driven by AC signals of various frequencies. You will need the following materials: current probe and voltage probe 100 Ω resistor multimeter alligator clip leads internal Data Studio signal generator ACTIVITY 1-1: RESISTORS AND TIME-VARYING (AC) SIGNALS. Consider the circuit in Figure 5 with a signal generator and resistor. CP A V signal R _ V VP B Figure 5 Question 1-1: What is the relationship between the input signal, V signal, and the voltage measured by the voltage probe, V? (Hint: remember that CP A has a very small resistance compared to R.) Prediction 1-1: On the axes that follow, sketch, with dotted lines, your qualitative prediction for the current I through the resistor (100 Ω) and the voltage across the resistor V R vs. time. [Hint: consider Ohm s Law]. Assume V signal has frequency of 20 Hz and amplitude (peak voltage) of 5 V. Draw two complete periods and don t forget to label your axes. Do this before coming to lab. Your TA will check all predictions during the first few minutes of lab.

6 6 July 22, 2008 AC Currents, Voltages, Filters, Resonance Voltage (V) Current (A) 0 0 Test your predictions. Time (ms) 1. Open the experiment file L07.A1-1 Resistor with AC. 2. Measure the resistance of the nominal 100 Ω resistor: R: 3. Connect the circuit in Figure 5.. Check SETUP. We are using the internal signal generator of the PASCO interface. The controls should appear on the computer screen. 4. Set the signal generator to 20 Hz and 5 volts amplitude (+5 V maximum and -5 V minimum). We call this 10 V peak-to-peak. 5. Begin graphing. When you have a good graph of the signal, stop graphing. Expand the graph to look at the same time range as above. 6. Print one set of graphs for your group report. Do not erase data. 7. On the printed graph of voltage vs. time, identify and label a time or two when the current through the resistor is maximum. Depending on the way you hooked up the voltage probe across the resistor, you may have current and voltage in or out of phase. If out of phase, you may want to switch the voltage probe and repeat. 8. On your graph of current vs. time, identify and label a time or two when the voltage across the resistor is maximum.

7 July 22, 2008 AC Currents, Voltages, Filters, Resonance 7 Question 1-2: Does a voltage maximum occur at the same time as a current maximum, or does one maximum (current or voltage) occur before the other? Explain. 9. Use the Smart Tool to find the period (time from one peak to the next), T, of the voltage. T: 10. Use your graph to complete Column I in Table 1-1. To obtain information from the graph, you can use the Smart Tool or you can select several cycles by highlighting them, and then use the statistics feature to find the maximum values for the voltage and current. 11. Now set the frequency of the signal generator to 30 Hz. Check that the amplitude is still 5 V. Graph I and V as before. Use the analysis feature to complete Column II in Table 1-1. Do not print out this graph. 12. Set the frequency of the signal generator to 40 Hz. Check that the amplitude is still 5 V. Graph I and V as before, and complete Column III in Table 1-1. Do not print graph. Table 1-1 Column I Column II Column III 20Hz 30Hz 40Hz At maximum voltage, current is (circle one): maximum, minimum, zero and increasing, zero and decreasing, nonzero and increasing, nonzero and decreasing, other At maximum voltage, current is (circle one): maximum, minimum, zero and increasing, zero and decreasing, nonzero and increasing, nonzero and decreasing, other At maximum voltage, current is (circle one): maximum, minimum, zero and increasing, zero and decreasing, nonzero and increasing, nonzero and decreasing, other max. voltage (V max ) = max. voltage (V max ) = max. voltage (V max ) = max. current (I max ) = max. current (I max ) = max. current (I max ) = R = V max /I max = R = V max /I max = R = V max /I max =

8 8 July 22, 2008 AC Currents, Voltages, Filters, Resonance Question 1-3: Based on the calculations in Table 1-1, what can you say about the resistance of R at different frequencies (does its value appear to increase, decrease, or stay the same as frequency increases)? Explain your answer. Question 1-4: When the input signal is 30 Hz or 40 Hz, does a maximum positive current through R occur before, after, or at the same time as the maximum positive voltage across R? Note: Do not disconnect this circuit as you will be using a very similar one in Investigation 2. Comment: In this Investigation you saw that the resistance of a resistor does not change when the frequency of the AC signal applied to it changes. Ohm s Law, V = I R, holds true at every instant in time. In Investigations 2 and 3, you will examine the behavior of capacitors and inductors with AC signals applied to them. INVESTIGATION 2: AC SIGNALS WITH CAPACITORS You will need the following materials: current probe and voltage probe multimeter 47 µf capacitor seven alligator clip leads internal Data Studio signal generator ACTIVITY 2-1: CAPACITORS AND AC SIGNALS In this investigation we want to see how the impedance of a capacitor changes when the frequency of the applied signal

9 July 22, 2008 AC Currents, Voltages, Filters, Resonance 9 changes. You will investigate this by measuring the behavior of a capacitor when signals of various frequencies are applied to it. Specifically, you will look at the amplitudes and the relative phase of the current through it and the voltage across it. Consider the circuit shown in Figure 6. CP A V signal C _ V VP B Figure 6 Prediction 2-1: Suppose that you replaced the signal generator with a battery and a switch. The capacitor is initially uncharged, and therefore the voltage across the capacitor is zero. If you close the switch, which quantity reaches its maximum value first: a) current in the circuit or b) voltage across the capacitor? As charge builds up on the capacitor and the voltage across the capacitor increases, what happens to the current in the circuit? Explain. Do this before coming to lab. Prediction 2-2: Sketch on the following axes one or two cycles of the current I C through the capacitor and the voltage V C across the capacitor versus time for the circuit in Figure 6. Use your answers to the previous prediction. Assume V signal has frequency of 20 Hz and amplitude of 5 V. Label your axes. Do this before coming to lab.

10 10 July 22, 2008 AC Currents, Voltages, Filters, Resonance Voltage (V) Current (A) Test your predictions. Time (ms) 1. Open the experiment file called L07.A2-1 Capacitor. 2. Measure the capacitance of the capacitor: C: 3. Connect the circuit in Figure Set the signal generator to 20 Hz and amplitude of 5 volts. 5. Begin graphing. When you have a good graph of the signal, stop graphing. Expand the graph to look at the same range as above. 6. Print one set of graphs for your group report. 7. On the graph of voltage vs. time, identify and label a time or two when the current through the capacitor is maximum. 8. On your graph of current vs. time, identify and label a time or two when the voltage across the capacitor is maximum. 9. Clearly mark one period of the AC signals on your graphs. Comment: One way you can determine the phase difference between two sinusoidal graphs with the same period is by finding the time difference between peaks from each graph and dividing that time difference by the time period. This will give you the phase difference as a fraction of a period. For example, if the time difference between two peaks is 0.5 s and the period of the signals is 2.0 s, then the phase difference is 0.25 or ¼ period. Phase differences should be given in degrees or radians by simply multiplying the phase difference in periods by 360 /period or 2π rad/period. In this example, the signals are 90 or π/2 radians out of phase. The signal that reaches its peak first in time is said to lead the other.

11 July 22, 2008 AC Currents, Voltages, Filters, Resonance 11 Question 2-1: Discuss how well your measured voltage graph agrees with your predicted one. Question 2-2: For the capacitor with an input signal of 20 Hz, does a current maximum occur before, after, or at the same time as the maximum voltage? Question 2-3: Calculate the theoretical phase difference between current and voltage for both 20 Hz and 30 Hz. Show your work and put your result in Table Use the various analysis features to help you fill in Column I in Table 2-1. Determine the experimental phase difference. 11. Set the frequency of the signal generator to 30 Hz. Check that the amplitude is still 5 V. Graph current I and voltage V as before. Use the analysis feature to complete Column II in Table 2-1. Do not print graph. Question 2-4: Based on your observations, what can you say about the magnitude of the reactance of the capacitor at 20 Hz compared to the reactance of the capacitor at 30 Hz? Explain.

12 12 July 22, 2008 AC Currents, Voltages, Filters, Resonance Table 2-1 Column I f = 20Hz At maximum voltage, current is (circle one): maximum, minimum, zero and increasing, zero and decreasing, nonzero and increasing, nonzero and decreasing, other max voltage (V max ) = max current (I max ) = Experimental Z = V max /I max = Theoretical Z X C = 1/ωC Theoretical phase diff: Experimental phase diff: Current leads or voltage leads? Column II f = 30Hz At maximum voltage, current is (circle one): maximum, minimum, zero and increasing, zero and decreasing, nonzero and increasing, nonzero and decreasing, other max voltage (V max ) = max current (I max ) = Experimental Z = V max /I max = Theoretical Z X C = 1/ωC Theoretical phase diff: Experimental phase diff: Current leads or voltage leads? Question 2-5: Based on your observations, what can you say about the phase difference between current and voltage for a capacitor at 20 Hz compared to the phase difference at 30 Hz? Explain. INVESTIGATION 3: AC SIGNALS WITH INDUCTORS In addition to the previous material, you will need: current probe and voltage probe multimeter 800 mh inductor seven alligator clip leads internal signal generator

13 July 22, 2008 AC Currents, Voltages, Filters, Resonance 13 ACTIVITY 3-1: INDUCTORS AND AC SIGNALS In this investigation we want to see how the impedance of an inductor changes when the frequency of the applied signal changes. We will follow much the same procedure as for the capacitor. Consider the circuit shown in Figure 7. CP A + - V signal L + _ V VP B L = 800 mh V max = 5 V f = 20 Hz Figure 7 Prediction 3-1: Suppose that you replaced the signal generator with a battery and a switch. The inductor initially has no current through it. If you close the switch, which quantity reaches its maximum value first: current in the circuit or voltage across the inductor? [Hint: recall that when the current through an inductor is changing, the induced voltage across the inductor opposes the change.] As the current builds up in the circuit, what happens to the induced voltage across the inductor? Explain. Do this before coming to lab. Prediction 3-2: Sketch on the following axes one or two cycles of the current I L through the inductor and the voltage V L across the inductor versus time for the circuit in Figure 7. Use your answers to the previous prediction. Assume V signal has frequency of 20 Hz and amplitude of 5 V. Label your axes. Do this before coming to lab. Your TA will check.

14 14 July 22, 2008 AC Currents, Voltages, Filters, Resonance - Voltage (V) Current (A) Time (ms) Test your predictions. 1. Open the experiment file called L07.A3-1 Inductor. 2. Measure the inductance of the inductor: L: NOTE: The internal series resistance of the inductor is not negligible at these low frequencies. Hence we cannot approximate the impedance Z by the reactance X as we did in the case of the capacitor, but we must include the effect of the resistance when considering the impedance. 3. Measure the resistance of the inductor: R: 4. Connect the circuit in Figure Set the signal generator to 20 Hz and amplitude of 5 volts (+5 V maximum and -5 V minimum). 6. Begin graphing. When you have a good graph of the signal, stop graphing. Expand the graph to look at the same range as above. 7. Print one set of graphs for your group report. 8. On your graph of voltage vs. time, identify and label two times when the current through the inductor is maximum. 9. On your graph of current vs. time, identify and label two times when the voltage across the inductor is maximum.

15 July 22, 2008 AC Currents, Voltages, Filters, Resonance Clearly mark one period of the AC signals on your graphs. Question 3-1: Does your measured voltage graph agree with your predicted one? If not, how do they differ? Question 3-2: For the inductor with an input signal of 20 Hz, does a current maximum occur before, after, or at the same time as the maximum voltage? Explain. 11. Use the analysis features to fill in Column I in Table Set the frequency of the signal generator to 30 Hz. Check that the amplitude is still 5 V. Graph I and V as before. Use the analysis features to complete Column II in Table 3-1. Do not print graph. Question 3-3: Calculate the theoretical phase difference between current and voltage for both 20 Hz and 30 Hz. Show your work and put your result in the table. Question 3-4: What can you say about the magnitude of the reactance of the inductor at 20 Hz compared to the reactance of the inductor at 30 Hz? Explain based on your observations.

16 16 July 22, 2008 AC Currents, Voltages, Filters, Resonance Table 3-1 Column I f = 20Hz At maximum voltage, current is (circle one): maximum, minimum, zero and increasing, zero and decreasing, nonzero and increasing, nonzero and decreasing, other max voltage (V max ) = max current (I max ) = Experimental Z = V max /I max = Theoretical X L = ωl = Column II f = 30Hz At maximum voltage, current is (circle one): maximum, minimum, zero and increasing, zero and decreasing, nonzero and increasing, nonzero and decreasing, other max voltage (V max ) = max current (I max ) = Experimental Z = V max /I max = Theoretical X L = ωl = Theoretical 2 2 Z = R + X L = Theoretical 2 2 Z = R + X L = Theoretical phase diff: Experimental phase diff: Current leads or voltage leads? Theoretical phase diff: Experimental phase diff: Current leads or voltage leads? Question 3-5: Based on your observations, what can you say about the phase difference between current and voltage for an inductor at 20 Hz compared to the phase difference at 30 Hz? Explain. Were the phase differences what you expected? Do you think the fact that the inductor you used has a significant resistance plays a role?

17 July 22, 2008 AC Currents, Voltages, Filters, Resonance 17 Question 3-6: Discuss the agreement between your experimental impedances with theoretical impedances [see Tables 2-1 and 3-1]. INVESTIGATION 4: INTRODUCTION TO AC FILTERS So far in this 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 already learned. The purpose of this investigation 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 4-1: 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 4-1 with a resistor, capacitor, signal generator and voltage probe.

18 18 July 22, 2008 AC Currents, Voltages, Filters, Resonance V signal R C - + VP A R = 33 Ω C = 1.23 µf V signal = 5 V f signal = 200 Hz Figure 4-1: Capacitive filter circuit Prediction 4-1: On the axes that follow, use dashed lines to sketch your qualitative prediction for the peak current through the circuit, I max, as the frequency of the signal from the signal generator is increased from zero. [Remember that ω = 2πf.] Do this before coming to lab. I max f signal 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 Ω C µf 5. Connect the resistor, capacitor, signal generator and probe as shown in Figure Press Start to turn on the scope display.

19 July 22, 2008 AC Currents, Voltages, Filters, Resonance 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 VP A, 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, V max, across the resistor (not the signal voltage, which should remain at 5 V), write it in Table 4-1 with f = 200 Hz. Then calculate the maximum current from the maximum voltage using the value of the resistor you measured in step Increase the frequency of the signal generator to 1,200 Hz. Be sure that the peak signal amplitude is still 5 V. Repeat step Repeat step 9 for 2,200 Hz, 4,200 Hz and 8,200 Hz. Table 4-1 f signal (Hz) V max (V) I max (A) 12. Sketch the data from Table 4-1 on the axes below. Mark scales on the vertical axes.

20 20 July 22, 2008 AC Currents, Voltages, Filters, Resonance I max (ma) f signal (khz) Question 4-1: If you could continue taking data up to very high frequencies, what would happen to the peak current, I max through the circuit? Question 4-2: At very high frequencies, does the capacitor act 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. 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 4-3: What phase difference do you observe between the peaks of the signal voltage and circuit current for low and high frequency?

21 July 22, 2008 AC Currents, Voltages, Filters, Resonance 21 Question 4-4: What would the current be through the circuit if we applied only a DC voltage? Explain. Question 4-5: At very low frequencies, does the capacitor act 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. Comment: In the circuit in Figure 4-1, 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. ACTIVITY 4-2: 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 4-2 below.

22 22 July 22, 2008 AC Currents, Voltages, Filters, Resonance V signal R + - L VP A L = 8.2 mh R = 33 Ω V signal = 5 V f signal = 20 Hz Figure 4-2: Inductive Filter Circuit Prediction 4-2: On the axes that follow, use dashed lines to sketch your qualitative prediction for the peak current through the circuit, I max, as the frequency of the signal from the signal generator is increased from zero. Do this before coming to lab. I max f signal 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 R L of the inductor L mh R L Ω 4. Connect the resistor, inductor, signal generator and probe as shown in Figure 4-2. 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.

23 July 22, 2008 AC Currents, Voltages, Filters, Resonance Use the Smart Tool to determine the peak voltage and peak current, and enter in Table 4-2. Then calculate the maximum current from the maximum voltage using the value of the resistor. Table 4-2 f signal (Hz) V max (V) I max (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 4-2 on the axes below. I max (ma) f signal (khz) Question 4-6: If you could continue taking data up to very high frequencies, what would happen to the peak current, I max, through the resistor?

24 24 July 22, 2008 AC Currents, Voltages, Filters, Resonance Question 4-7: At very high frequencies, does the inductor act 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. 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 4-8: What would the current through the circuit be if we applied only a DC voltage? Question 4-9: At very low frequencies, does the inductor act 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. Comment: In the circuit in Figure 4-2, 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.

25 July 22, 2008 AC Currents, Voltages, Filters, Resonance 25 INVESTIGATION 5: 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 resonant behavior in a series RLC circuit. 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 or television receiver. Hence, this is sometimes called a tuner circuit. You will need the following materials: voltage probe RLC Circuit Board Consider the series RLC circuit shown in Figure 5-1(below). Nominal values: R = 10 Ω L = 8.2 mh C = 1.2 µf V signal R + - L C VP A Figure 5-1. Series RLC circuit. We measure the current by observing the voltage across a resistor. The AC voltage driving the circuit V signal has the frequency f. NOTE: Do not yet set up the circuit in Figure 5-1. Prediction 5-1: 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 compared to a DC signal? Explain your reasoning. Do this before coming to lab.

26 26 July 22, 2008 AC Currents, Voltages, Filters, Resonance Prediction 5-2: 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. Do this before coming to lab. Prediction 5-3: Based on your Predictions 5-1 and 5-2, is there some intermediate frequency where I and V will reach maximum or minimum values? Do you think they will be maximum or minimum? Do this before coming to lab. Prediction 5-4: On the axes below, draw qualitative graphs of X C vs. frequency and X L vs. frequency f of the AC applied voltage. Clearly label each curve. You may need to go back and look at Equations (9) and (10). Do this before lab. X C an d X L A C fre que nc y f

27 July 22, 2008 AC Currents, Voltages, Filters, Resonance 27 Question 5-1: For what relative values of X L and X C will the total impedance of the circuit, Z, be a minimum? Hint: see Equation (7). Explain your reasoning here. 1. On the axes above, mark and label the frequency where Z is a minimum. Question 5-2 At the frequency you labeled, will the value of the peak current, I max, in the circuit be a maximum or minimum? What about the value of the peak voltage, V R, across the resistor? Explain. ANote: The point you identified in step 1 is the resonant c frequency. Label it with the symbol f 0. The resonant t frequency is the frequency at which the impedance of the i series combination of a resistor, capacitor and inductor is v minimal. This occurs at a frequency where the values of X L i and X C are equal. t 5-1: THE RESONANT FREQUENCY OF A SERIES RLC CIRCUIT. 1. Open the experiment file L07.A5-1 RLC Resonance. 2. Note that the VR scope scale is 0.1 V/div and 1 ms/div and the signal generator scale to 2 V/div. The signal generator will remain at a maximum voltage of 5 V with a frequency of 100 Hz for the entire experiment. 3. You will be finding the maximum resonant current in the circuit shown in Figure 5-1. Remember that the voltage across a resistor is directly proportional to the current through the resistor. So you will measure the voltage V across the 10 Ω resistor to find the resonant frequency.

28 28 July 22, 2008 AC Currents, Voltages, Filters, Resonance 4. Before connecting the circuit shown in Figure 5-1, measure with your multimeter the isolated circuit board elements for the nominal values of R, L, C given in Figure 5-1. Write down the measured values here along with their units. R (resistor): L: R inductor : C: 5. Connect the circuit shown in Figure Calculate the expected resonant frequency of your circuit using the measured values of R, L, and C. f calc : 7. Press On for the signal generator and Start to begin taking data. 8. Adjust the VR scope scale to see the VR signal. It is probably quicker to interpolate the vertical scale to obtain the maximum voltage, but you can use the Smart Tool. 9. Enter the data in Table Measure the voltage for the other frequencies in Table 5-1. TABLE 5-1 f signal (Hz) V R (V)

29 July 22, 2008 AC Currents, Voltages, Filters, Resonance Now you should have a good idea of the value of the resonant frequency. If you have time, use steps of 50 Hz on either side of the suspected resonant frequency value and take voltage measurements for perhaps an additional 5 frequencies on each side of the suspected frequency. This should allow you to map out more precisely the resonant frequency. 12. Print out your data at the resonant frequency. 13. What is your experimental resonant frequency? f exp = Question 5-3: How does this experimental value for the resonant frequency compare with your calculated one? What is the percentage difference? Explain any differences greater than 4-5%. Percentage difference:

30 30 July 22, 2008 AC Currents, Voltages, Filters, Resonance 13. Go to Excel and produce a plot of your voltage measurement across the resistor (proportional to current) versus the input signal frequency. 14. Label and print out the plot for your group. The plot you just produced should indicate the resonant behavior of a series RLC circuit. It should be clear to you that by choosing various values of the individual values of R, L, and C we can produce a circuit that passes signals of chosen frequencies. That is, the output voltage V out that we are measuring across the resistor is significant for only a narrow region around the resonant frequency. You can think of the circuit as filtering out unwarranted frequencies. It is called a band-pass filter circuit. Activity 4-2: The Resonant Q Value of a Series RLC Circuit. The plot you just produced should indicate the resonant behavior of a series RLC circuit. It should be clear to you that by choosing various values of the individual values of R, L, and C we can produce a circuit that passes signals of chosen frequencies. That is, the output voltage V out that we are measuring is significant for only a narrow region around the resonant frequency. You can think of the circuit as filtering out unwarranted frequencies. It is called a band-pass filter circuit. The region around the resonant frequency is characterized by the so-called resonant amplification, which we denote as Q. It is called the Q factor. It is defined theoretically in various (practically identical) ways, and we shall use the parameterization L / C Q = (14) R Figure 9 (below) shows the output voltage as a function of frequency for various values of Q.

31 July 22, 2008 AC Currents, Voltages, Filters, Resonance Q =20 15 Voltage 10 Q =10 5 Q =5 Q = f /f 0 Figure 9 1. Calculate Q using R, L, and C and Equation (14). Q calc : 2. There are various ways to determine experimentally the Q factor, because Q is related to the sharpness of the resonant peak. Find V max from your Table 4-1 data and then find the frequencies f low ( f 1 ) and f high ( f 2 ) for the frequencies where the measured voltages were V max. Let f = f2 f1 and then determine Q exp. Q f exp exp = (15) f So f determines the width of the resonance peak, and Q exp represents the sharpness of the peak. Write down your determined values. f 1 : f 2 : f : Q exp :

32 32 July 22, 2008 AC Currents, Voltages, Filters, Resonance Question 4-4: Compare your calculated and experimental values of Q. Our determination of both values has been a little crude, but you should expect agreement to within 25% or so. Determine the percentage difference and discuss your agreement. Explain any differences greater than 50%. Percentage difference: CLEAN UP YOUR LAB AREA BEFORE LEAVING.