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1 10. A ircuits* Objective: To learn how to analyze current and voltage relationships in alternating current (a.c.) circuits. You will use the method of phasors, or the vector addition of rotating vectors that represent the voltages or currents. This method does not require calculus. Reading assignment: All students should read the Appendix about using oscilloscopes. The learning objectives are the following: 1. To learn how alternating currents or voltages can be represented as the y-components of rotating vectors, or phasors. 2. To understand the amplitude and phase relationships amongst the different components (resistors, capacitors, inductors) of a.c. circuits. 3. To learn effective use of a typical oscilloscope. 4. To understand the phenomenon of resonance in series RL circuits. (This material is presented in your textbook iunder alternating current circuits.) Reading assignment Before you come to the lab study the sections on alternating current, capacitor circuits, inductors and inductor circuits, Read the following sections. (Section numbers may be slightly different depending on the edition of your textbook: heck the section titles.) Knight, Jones and Field : 26.1 Alternating urrent, capacitor circuits 26.4, inductors and inductor circuits 26.5, oscillation circuits 26.6 Serway and uille (212): 21.1 Resistors in an A ircuit, 21.2: apacitors in an A ircuit, 21.3 Inductors in an A ircuit, 21.4 The RL Series ircuit, 21.5: Power in an A ircuit, 21.6 Resonance in a Series RL ircuit Serway and Jewett (252): 33.1 A Sources, 33.2 Resistors in an A ircuit, 33.3: Inductors in an A ircuit, 33.4 apacitors in an A ircuit, 33.5 The RL Series ircuit, 33.6 Power in an A ircuit, 33.7 Resonance in a Series RL ircuit Pre-lab assignment: 1. The vector shown is rotating counter-clockwise around the circle shown with an angular frequency of 2 f. At time t 0sec, the angle is. Write a formula for the x and y components of the vector (a phasor ) at general time t. The length of the vector is I o. I x = I y = * William A Schwalm

2 The idea behind a current phasor is that either I x or I y can be used to represent an alternating current as the phasor rotates, since these components vary 2. (Phys 252 only) At the right is a schematic circuit di sinusoidally. 3. Is t 2. (Phys 252 only) At the right is a schematic circuit diagram showing a resistor, capacitor, inductor and alternating voltage source. (a) Label these with R,, L and a.c. (b) At a given instant t in time, suppose is the current flowing out of the source, as shown. Let, and be the currents flowing through R, L and at this instant. Write the equations that hold between each pair currents. Explain why. (c) Suppose the capacitor starts out with zero charge. As the current flows in the direction shown, what happens to the right and to the left capacitor plate? (Two separate answers.) (d) As we have learned, the voltage drop across the coil, that is to say across the inductor L, is proportional to the rate of change of the current.. What is the relation between the current through the resistor and the voltage drop across the resistor? Give an equation. (e) The voltage across the capacitor is related to the charge Q on either of its plates by the definition of capacitance. Thus give a formula for the voltage drop across the capacitor. 10-2

3 (f) Write an equation (one equation) relating the voltage to the three voltage drops across the three other circuit components. s t of the alternating voltage source 3. (Phys 212 and 252) In the following figure you see the source current as a function of time. The other curves represent voltages across the resistor, capacitor and inductor, in some order. Based on your answers above, identify which curve represents which voltage drop. Then explain briefly each choice. (Don t just say that current leads voltage across the grob, (whatever) because you also have to explain why that is, based on how grob works.) source current time voltage A B time A = B = = 4. Below left is a current phasor, a vector rotating counter-clockwise with angular frequency such that its vertical component (y component) gives the source current. At the right are three voltage phasors marked A, B, and. These relate to voltages. Based on your responses above, identify which of the three phasors corresponds to the voltage drop across each of R, L and? (The letters A, B, are not necessarily going to correspond to the curves in the previous question.) Explain your choices. For instance, which voltage phasor should line up with the current phasor because the voltage drop is proportional to current? I o A B A? B?? Phasor for source current Phasors for voltage drops 10-3

4 Exploring equipment: Read the Appendix on the use of oscilloscopes The hold introduces extra time delay between sweeps by missing some sweeps. Pulling the hold knob you can chop the input between H1 and H2. That means the input (not triggering) is sampled by rapidly alternating back and forth between the two inputs. This is handy for very low sweep rates and is used in place of alternate sweeps in this case. Equipment: The a.c. voltage source or signal generator (a) The same current flows at each instant through each circuit element. This means the current through the capacitor I c is the same at each instant as the current I R through the resistor. (b) The sum at each instant of the voltage drops around the circuit is zero. Thus if c is the voltage drop across the capacitor, R EMF provided at the same instant by the signal generator, then at each instant in time Resistor Resisto r apacitor apacitor apacito r Inductor The photo on the left shows the signal generator attached to a resistor and capacitor. The signal generator is in the rear and there is a resistor, capacitor and an inductor (not part of the circuit) each mounted on a Plexiglas box. The wires complete the circuit, so that during half the cycle current flows out one terminal of the signal generator, through the resistor, into the capacitor (charging the capacitor) and back into the generator. Then on the other half of the cycle it flows the other way. Therefore notice from the way the circuit is set up that. - - R = 0. (Yes) But and here s the tricky part R0 0. (No! ) (c) The maximum voltages across each circuit element do not add up. This is because these maximum voltages occur at different times during the cycle. 10-4

6 In-class group response: Go back to your circuit diagram and figure out which point you would like to connect the grounded sides of the H1 and H2 inputs (same point). Indicate this point on the diagram. Then explain why this makes H2 measure the capacitor voltage upside down! (Dang.) Thus it seems you will have to see t on the screen. This is not you t and R assignment. Explain here and on the white board. A diagram showing both the R circuit and the oscilloscope connections will be appropriate. Ok, here is a possible solution to the problem: If you look carefully on the control panel you may find a control (often a push-pull thing) that will reverse the sign of the display for H2. Thus it will flip the H2 trace on the screen vertically. This should allow you to show what your supervisor needs to see. Try to find such a control. In-class response: Draw a picture of how you expect the two traces on the oscilloscope screen to look. Indicate clearly which trace is which. In-class response: Draw a phasor diagram showing voltage drops across the resistor and across the capacitor as phasors of length R0 and 0 respectively. Draw them at a time when the current phasor points along the x axis. Measurement plan: Describe very briefly how you will set up the circuit and create the display the supervisor wants. Recall that an R circuit has a time constant, which is the product R of resistance and capacitance. Plan to observe the voltage signals in three cases, (a) when the angular frequency 2 f applied by the source is less than 1/R, (b) when it equals 1/R and (c) when it is greater than 1/R. Record your plan here. 10-6

7 Implementation: arry out your plan. Record the results as graphs of what you actually observe in each case. Include the actual numbers on the scales taking into account the time base and voltage per division. Graph at least two cycles for each. Be sure your supervisor sees the display for at least one of the three cases. Analysis: Write a summary of what you learned. In particular, describe the phase relation between the two voltage drops. In each of three experiments, did the capacitor or the resistor voltage reach a maximum first, and was this what you expected? an you say how much (what fraction of a cycle) one voltage leads the other by? Why or why not? Also, how do the voltage maxima (the amplitudes) compare in each of the three cases? an you explain this? 10-7

8 Problem 2 Now your group is asked to perform a similar study and to provide a similar demonstration for a series RL. You are to demonstrate to your supervisor using the oscilloscope how the voltages across the R, the inductor L and the capacitor vary in a series RL circuit as a function of the applied angular frequency. (Recall the pre-lab exercises.) Further, you should compare to predictions that you make using phasor pictures. Work assignment: You need first to be able to show on the screen: (a) both the resistor voltage drop (You have to rewire between a) and b)) (b) both the inductor voltage drop (Rewire again. This time it s a little different.) t in H1 and the capacitor voltage drop R L t in H1 and the capacitor voltage drop Measure the phase angle from the current to the source voltage in the cases (a) when 1/ L, (b) when.2/ L, (c) when 5./ L. t in H2. t in H2. Finally you need to produce a nice, professional-looking graph showing the ratio of the current amplitude I o divided by the source voltage amplitude o as a function of, where 2 f. That is, make a graph of I o o versus. The supervisor wants this written up, of course, so you will hand it in as part of your report, as follows. You should be aware from your reading that the length of the voltage phasors representing voltage drops across each of the circuit elements are Of course the phasor, which are rotating vectors, do not all point in the same direction. In-class response: (a) The voltage phasor for which one of the circuit elements (R, L or ) is parallel to the current phasor at every instant in time? (b) Why? (c) Also, how are the directions of the other two voltage phasors related to the direction of the current phasor? Draw a picture of the three voltage phasors in this situation, representing phasors as arrows with proper labels, at an instant when the current phasor points directly along the positive x axis. R0 0 L0 0 0 I R, I L, I

9 In-class response: The source voltage at any instant is the y component (in some textbooks, the x components are used) of the vector sum of the voltage phasors. Referring to your drawing and to the formulas for the phasor lengths, what are the components of the total voltage phasor at the instant shown in your figure? ox = oy = In-class response: Thus what is the magnitude of the source voltage phasor as a function of frequency? (This gives the amplitude of the source voltage.) In-class response: What is the phase angle between the source voltage and the source current? (Remember, the current phasor is along the x axis at the moment, so this asks the angle between the axis and the total voltage phasor. It s easy to get its tangent.) Measurement plan: Your group should agree on a measurement plan for performing the assigned work. You will turn in the graph as part of your report. Include a brief outline of the measurement plan here, devoting at least some space to each point. Be sure in particular to tell us how you will wire the circuits in each case. Implementation: arry out your plan. Record the necessary data and ancillary observations here and on additional sheets. Before leaving lab, you should also have a rough sketch of the curve representing data from your resonance experiment. Analysis: You have worked out a prediction of what to expect in each case using phasors. Thus, for each of the points above, the analysis may consist of comparing what you would expect to see (which of course you must state, basing it on the phasors) with what you actually did see. As usual, you should comment especially on things that didn t go as expected. In particular, your analysis should include some discussion of the form of the graph you have constructed, and how this relates to the phenomenon of resonance as described in the text. What is the resonance frequency? What happens to the current there? etc. 10-9

10 onclusion: Usually the last five minutes are devoted to discussing the lab activity and how the activities related to the learning objectives. Give here a brief discussion of how the objectives were addressed, noting anything of particular interest. Include the main points of the class discussion

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