# Name Date Day/Time of Lab Partner(s) Lab TA

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1 Name Date Day/Time of Lab Partner(s) Lab TA Objectives LAB 7: AC CIRCUITS To understand the behavior of resistors, capacitors, and inductors in AC Circuits To understand the physical basis of frequency-dependent impedance To observe and understand the phase differences between current and voltage for resistors, capacitors, and inductors. To observe oscillations in an RCL circuit Introduction In this lab you will explore the behavior of standard circuit elements in cases where the input voltage varies and the time structure of the voltage becomes important. In AC circuits, the behavior of circuit elements like inductors and capacitors is in some ways similar to the behavior of resistors. With a resistor in a DC circuit, the resistance determines how much current will flow through it when a voltage is applied to it, according to Ohm s Law. In AC circuits there is a quantity called impedance associated with each circuit element that acts like the resistance in Ohm s Law. In fact, the peak voltage Vmax is related to the peak current, Imax by the relationship Imax = Vmax/Z, where Z is the impedance. For a resistor, the impedance is just the same as the resistance. As you will see, the capacitor and inductor have more complicated behavior which is related to their ability to store and release electrical energy. In this lab, some of the results are based on ratios of measured voltages. In this experiment, all voltage measurements with the oscilloscope are peak-to-peak values, and can be left in units of cm of vertical deflection on the oscilloscope. Since amplitudes, rms values and peak-to-peak values are all proportional to one another, the same vector addition rules apply to all. However, if you want to compare voltages on two oscilloscope channels in terms of vertical deflection, their vertical deflection scales must be the same. Note that phase difference measurements can also be made by looking at the difference in the times where the two signals cross the axis. The time difference!t = t2-t1 can be related to the phase difference between the two signals (!") by dividing by the period (T) [which gives the difference in terms of a fraction of one oscillation] and multiplying by 2#:!t/T =!"/2#. A schematic diagram of the components used in the lab and the connections between them is shown below. The series circuit consists of the function generator and two or three of the elements R, L, C, labeled in the diagram as A, B, or C. Either channel of the oscilloscope can be used to measure the voltage across C, C + B, or C + B + A (= Vo, where Vo is the output voltage of the function generator). The ground connections of the function generator and the oscilloscope must always be connected together since this connection already exists through the grounding connections in the power cord (round prong of plug) of each unit. In order to measure the voltage across either B or A alone, that element must be placed in the position of element C, so that one of its terminals is connected to ground. These permutations in the positions of the circuit elements are required throughout the experiment in order to make the appropriate measurements and to maintain the ground connections as mentioned above.

2 Overloading of the function generator may cause distortion of the sinusoidal voltage waveform. If this occurs, turn down the amplitude control on the function generator until the proper waveform is restored while increasing the sensitivity of the oscilloscope to obtain a reasonable height on the scope. INVESTIGATION 1: IMPEDANCE OF CIRCUIT ELEMENTS You will need the following materials for this investigation: Oscilloscope (Picoscope) Function Generator Lucite boards with resistors, capacitors, and inductors Two scope probes Leads to build circuits The values of the L, C, and R components, and their uncertainties, are listed on the frames on which the components are mounted (as well, each resistor is color coded with its resistance). The internal resistance, RL, of each inductor is labeled with its value; the nominal internal resistance of the function generator, RG, is 600 ohms. Record all these values. Activity 1-1: Impedance of a Resistor 1 Assemble the simple circuit shown below. In addition to the appropriate set-up for the first activity, it is also a good circuit to build just to get a decent signal on the oscilloscope and to check that everything is working properly.

3 2 Set the frequency generator to create a 50 khz sine wave (5 on the 10 khz scale), and display both the input signal (Vin, measured between a and c ) and the output signal (Vout measured between b and c ) on the scope. Set the oscilloscope trigger controls to cha, Auto, Rising, and + slope. Adjust the time base to 5 µs/div. You should see about three full periods of the oscillation. Make sure both scope channels are set to AC with the same voltage setting (not Auto ). Question 1-1: Is there any observable difference in phase (time offset) between the two signals? Is this what you would have expected? Why or why not? 3 Now, vary the signal frequency over a wide range using the function generator. Observe the ratio of the amplitudes Vin and Vout as you do this. Note any differences as the frequency varies. Question 1-2: Did you see any variation in the amplitude Vout as you changed the frequency? Is there any evidence that the impedance of a resistor is frequency-dependent? Was this what you would have expected? Why? Activity 1-2: Impedance of a Capacitor 4 Assemble the RC circuit shown below.

4 Prediction 1-1: 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: current in the circuit or 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. Prediction 1-2: The actual AC voltage applied to the circuit by the signal generator is shown on the axes that follow. Use your answers from the above questions to sketch with dashed lines your prediction for the current as a function of time. 5 To observe VRC (= Vin) connect the oscilloscope probe at point a (across R and C) and the probe ground connected at point c, where the black ground lead from the function generator is connected. 6 To measure \$, the phase difference between VR (which, remember, is proportional to the current in the circuit) and VRC, let probe #1 remain connected as before across R and C, and connect probe #2 to point b. Why is it important that both traces are centered vertically as shown in the figure below on the left? (They need not have the same amplitude.)

5 The phase difference \$ is then given by the time interval, x, between the x-intercepts of the two traces according to: \$ = ( x / D ) 360 where D is the time for one cycle. (The relative positions in the diagram are for illustration only.) Question 1-3: What phase difference do you measure? Go back to Prediction 1-2 and draw the observed VR with a solid line on your Current graph. Does VR lead VRC, or does VRC reach a maximum before VR? Do these answers match your predictions? Explain any differences you see. 7 Now, switch the positions of the capacitor and resistor in the circuit so that you are measuring the voltage across the capacitor. Draw in and clearly label the voltage across the capacitor VC with respect to the input voltage (VRC) on the axes under Prediction 1-2, above. Measure the phase difference between VC and VRC as you did before with VR and VRC. Question 1-4: What phase difference do you measure? What is the total phase difference between VC and VR (= the current in the circuit)? Is this what you would have expected? Why? 8 While you are measuring the voltage on the capacitor, you can investigate the impedance of the capacitor, ZC. Remember that VC = ICZC. Record VC as a function of frequency for several different frequencies. (Determine the frequency from the period of one cycle).

6 Question 1-5: Does the capacitor s impedance increase or decrease with frequency? Can you explain this using what you know about capacitors? Comment: The impedance of a capacitor, X C, is called the capacitive reactance. Activity 1-3: Impedance of an Inductor 9 Assemble the RL circuit shown below. Prediction 1-3: Suppose 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. Prediction 1-4: At the instant the current reaches its maximum value for this circuit, what do you predict the magnitude of the induced voltage will be---maximum, minimum, or zero? Why?

7 Prediction 1-5: The actual AC voltage applied to the circuit by the signal generator is shown on the axes that follow. Use your answers from the above questions to sketch with dashed lines your prediction for the current as a function of time on the following graph. 10 To observe VRL (= Vin) connect the oscilloscope probe at point "a" (across L and C) and the probe ground connected at point c, where the black ground lead from the function generator is connected. 11 Measure the phase difference between VR (which, remember, is proportional to the current in the circuit) and V RL. (Let probe #1 remain connected as before across R and L, and connect probe #2 to point "b"). Question 1-6: What phase difference do you measure? Go back to Prediction 1-5 and draw the observed VR with a solid line on your Current graph. Does VR lead VRL, or does VRL reach a maximum before VR? Do these answers match your predictions? Explain any differences you see. 12 Now, switch the positions of the inductor and resistor in the circuit so that you are measuring the voltage across the inductor. Draw in and clearly label the voltage across the inductor VC with respect to the input voltage (VRC) on the axes under Prediction 1-2, above. Measure the phase difference between VC and VRC as you did before with VR and VRL. Question 1-7: What phase difference do you measure? What is the total phase difference between VL and VR (= the current in the circuit)? Is this what you would have expected? Why? 13 While you are measuring the voltage on the inductor, you can investigate the impedance of the inductor, ZL. Remember that VL = ILZLL. Determine VL as a function of frequency for several different frequencies, as you did for the capacitor above.

8 Question 1-8: Does the inductor s impedance increase or decrease with frequency? Can you explain this using what you know about inductors? Plot your values of VC and VL versus frequency. Fit the resultant plots to determine the frequency dependence of the capacitive and inductive impedance. INVESTIGATION 2: RESONANCE IN RCL CIRCUITS 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. Activity 2-1: Properties of RCL Circuits 14 Using the 500! resistor, build the RCL circuit shown below: Prediction 2-1: At very low signal frequencies (1kHz), will the maximum values of I through and V across the resistor be relatively large, intermediate, or small? Explain your reasoning. Prediction 2-2: At very high signal frequencies (well above 300 khz), will the maximum values of I and V be relatively large, intermediate, or small? Explain your reasoning.

9 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? 15 On the axes below, draw qualitative graphs for XC and XL as a function of frequency. Clearly label each curve. Remembering that, at what relative values of XC and XL will the total impedance Z be a minimum? Mark these values on the plot. Explain your reasoning here. Question 2-1: 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 max, across the resistor? Explain. 16 The frequency you identified above is the resonant frequency of the circuit, which we will call f0. Using the fact that this occurs when XC = XL and the expressions for XC and XL given in Investigation 1, solve (algebraically) for f0 in terms of L and C and other constants. Then, put in the numerical values of L and C to predict the frequency for your circuit, and determine the error in this value from the uncertainties in L and C. f0 = Hz for our values of L and C.

10 Activity 2-2: Resonance in RCL Circuits 17 Connect the ground leads of both probes to "d" and probe #1 to "c" and observe VR on channel 1 (or A). Connect probe #2 to "a" and observe Vo = Vad = VRLC on channel 2 (B) 18 Start with a function generator at the frequency calculated above, f0. Measure VR and V0 at several frequencies on each side of resonance. Plot VR/V0 as a function of frequency and fit this curve with a Lorentzian using FITYK. (The theoretical shape of a resonance curve is a Lorentizian). 19 Did V0 depend on the frequency? Explain. End-of-lab Checklist: Turn off the scope and the function generator.

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