# Experiment 8: Undriven & Driven RLC Circuits

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1 Experiment 8: Undriven & Driven RLC Circuits Answer these questions on a separate sheet of paper and turn them in before the lab 1. RLC Circuits Consider the circuit at left, consisting of an AC function generator ( Vt () = V0 sin( ωt), with V 0 = 5 V), an inductor L = 8.5 mh, resistor R = 5 Ω, capacitor C = 47 µf and switch S. The circuit has been running in equilibrium for a long time. We are now going to shut off the function generator (instantaneously replace it with a wire). (c) (d) Assuming that our driving frequency ω is not necessarily on resonance, what is the frequency with which the system will ring down (in other words, that current will oscillate at after turning off the function generator)? Feel free to use an approximation if you wish, just make sure you know you are. At what (numerical) frequency f should we drive to maximize the peak magnetic energy in the inductor? In this case, if we time the shut off to occur when the magnetic energy in the inductor peaks, after how long will the electric energy in the capacitor peak? Approximately how much energy will the resistor have dissipated during that time? 2. Phase in an RLC Circuit Using the same circuit as in problem 1, only this time leaving the function generator on and driving below resonance, which in the following pairs leads (if either): (c) (d) Voltage across the capacitor or voltage across the resistor Voltage across the function generator or voltage measured across the inductor Current or voltage across the resistor Current or voltage across the function generator E07-7

2 IN-LAB ACTIVITIES EXPERIMENTAL SETUP 1. Download the LabView file from the web and save the file to your desktop. Start LabView by double clicking on this file. 2. Set up the circuit pictured in Fig. 5 of the pre-lab reading (no core in the inductor!) 3. Connect a voltage probe to channel A of the 750 and connect it across the capacitor. MEASUREMENTS Part 1: Free Oscillations in an Undriven RLC Circuit In this part we turn on a battery long enough to charge the capacitor and then turn it off and watch the current oscillate and decay away. 1. Press the green Go button above the graph to perform this process. Question 1: What is the period of the oscillations (measure the time between distant zeroes of the current and divide by the number of periods between those zeroes)? What is the frequency? Question 2: Is this frequency the same as, larger than or smaller than what you calculated it should be? If it is not the same, why not? Part 2: Energy Ringdown in an Undriven RLC Circuit 1. Repeat the process of part 1, this time recording the energy stored in the capacitor U CV U = LI, and the sum of the two. Question 3: 2 ( C 2 = ) and inductor ( L 2 ) The circuit is losing energy most rapidly at times when the slope of total energy is steepest. Is the electric (capacitor) or magnetic (inductor) energy a local maximum at those times? Briefly explain why. E07-9

3 Part 3: Driving the RLC Circuit on Resonance Now we will use the function generator to drive the circuit with a sinusoidal voltage. 1. Enter the frequency that you measured in part 1 of the lab as a starting point to find the resonant frequency. 2. Press GO to start recording the function generator current and voltage vs. time, as well as a phase plot of voltage vs. current. 3. Adjust the frequency up and down to find the resonant frequency and observe what happens when driving above and below resonance. Question 4: What is the resonant frequency? What are two ways in which you know? Question 5: What is the impedance of the circuit when driven on resonance (hint: use the phase plot)? Question 6: When driving on resonance, insert the core into the inductor. Are you now driving at, above or below the new resonant frequency of the circuit? How can you tell? Why? Part 4: What s The Frequency? For the remainder of the lab you will make some measurements where you are given incomplete information (for example, you won t be shown the frequency or won t be told what is being plotted). From the results you must determine the missing information. If you find this difficult, play with the circuit using the further questions tab to get a better feeling for how the circuit behaves. 1. Remove the core from the inductor 2. Press GO to record the function generator current and voltage Question 7: At this frequency is the circuit capacitor- or inductor-like? Are we above or below resonance? E07-10

4 Part 5: What s That Trace? 1. Press GO to record the function generator current and voltage as well as the voltage across the capacitor. Note that you are not told which trace corresponds to which value. Question 8: What value is recorded in each of the three traces (I, V FG or V C )? How do you know? Question 9: Are we above, below or on resonance? How do you know? Further Questions (for experiment, thought, future exam questions ) What happens if we insert the coil core in the undriven circuit? For a random frequency can you bring the circuit into resonance by slowly inserting the core into the coil? Are there any conditions on the frequency (e.g. does it need to be above or below the resonant frequency of the circuit with the empty coil)? Could you do part 5 if you were given only two traces instead of three? Would it matter which two you were given? What is the energy doing in the driven case? Is the resistor still dissipating power? If so, where is this power coming from? What happens to the resonant frequency of the circuit if a resistor is placed in series with the capacitor and coil? In parallel? NOTE: You can use the variable resistor, called a potentiometer or pot (just to the left of the coil, connect to the center and right most contacts, allowing you to adjust the extra resistance from 0 Ω to 3.3Ω by simply turning the knob). With the resistor in series with the coil and capacitor, at what frequency is the energy dissipation a maximum? How could you verify this experimentally? E07-11

5 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2006 OBJECTIVES Experiment 8: Microwaves 1. To observe the polarization and angular dependence of radiation from a microwave generator 2. To measure the wavelength of the microwave radiation by analyzing an interference pattern similar to a standing wave INTRODUCTION PRE-LAB READING Heinrich Hertz first generated electromagnetic waves in 1888, and we replicate Hertz s original experiment here. The method he used was to charge and discharge a capacitor connected to a spark gap and a quarter-wave antenna. When the spark jumps across the gap the antenna is excited by this discharge current, and charges oscillate back and forth in the antenna at the antenna s natural resonance frequency. For a brief period around the breakdown ( spark ), the antenna radiates electromagnetic waves at this high frequency. We will detect and measure the wavelength λ of these bursts of radiation. Using the 10 relation f λ= c= 3 10 cm/s, we will then deduce the natural resonance frequency of the antenna, and show that this frequency is what we expect on the basis of the very simple considerations given below. Figure 1 Spark-gap transmitter. The 33 is a 33 pf capacitor. It is responsible for storing energy to be rapidly discharged across a spark gap, formed by two tungsten cylinders pictured directly above it (one with a vertical axis, one horizontal). Two MΩ resistors limit current off of the capacitor and back out the leads, protecting the user from shocks from the 800 V to which the capacitor will be charged. They also limit radiation at incorrect frequencies. E08-1

9 Experiment 8: Microwaves Answer these questions on a separate sheet of paper and turn them in before the lab 1. Spark Gap Distance and Timing The time to charge the transmitter capacitor until it discharges depends on the resistance in the charging circuit (R = 4.5 MΩ), the capacitance (C = 33 pf) and the voltage required to initiate breakdown. Assume that the power supply supplies 800 V but that breakdown typically occurs at a voltage of about 500 V on the capacitor. (c) Thinking of the tungsten electrodes as parallel plates, how far apart must they be in order generate a spark at 500 V? In reality, the electrodes aren t parallel plates, but rather cylinders with a fairly small radius of curvature. Given this, will the distance needed between the electrodes to generate sparking be smaller or larger than you calculated in? Why? About how much time will it take for the power supply to charge the capacitor from empty to discharge? 2. Wavelength and Frequency of the Radiation The spark-gap antenna is a quarter-wavelength antenna, radiating as described above. Using l = 31 mm for the length of one of the arms of the antenna, what is the wavelength of the emitted radiation? the frequency of the emitted radiation? 3. Reflections Now place the transmitter some distance in front of a perfectly conducting sheet, oriented so that the propagation direction of the waves hitting the reflector is perpendicular to the plane of the reflector (so that they ll reflect straight back out towards the transmitter). For example, place the transmitter at z = -D with the antenna parallel to the x-axis, and have the reflector fill the z=0 (xy-) plane. (c) (d) (e) (f) (g) Write an equation for the electric field component of the radiation from the transmitter (the incident wave). Treat the field as plane wave, with a constant amplitude E 0 and angular frequency ω 0. What condition must the total electric field satisfy at the surface of the conductor (z=0)? What is the direction of propagation of the reflected wave? Write an equation for the time dependent amplitude & direction of the reflected wave, making the same assumptions as above. Write an equation for the total amplitude of the electric field as a function of position, by adding (c) and (d). Nodes are locations (in this case planes) where the electric field is zero at all times. What is the distance between nodes along the z-axis? What is the numerical distance you thus expect for our transmitter (i.e. use 2a)? E08-5

11 Part 2: Angular Dependence of the Emitted Radiation 1. Now measure the angular dependence of the radiation intensity by moving the receiver along the two paths indicated in the below figures. Angular dependence - Horizontal Angular dependence - Vertical Question 4: Which kind of motion, horizontal or vertical, shows a larger change in radiation intensity over the range of motion? Part 3: Wavelength of the Emitted Radiation Now we will use the reflector to create an interference pattern that we can use to measure the wavelength of the radiation. We will do this in two ways. 1. Position the reflector so that it is about 30-40cm from the spark gap transmitter (note that you can insert the tube into notch on the bottom of the reflector in order to make a stable base for it to stand on). You may need to change this distance slightly to get good results. Experiment! 2. Orient the transmitter so that the propagation direction of the waves hitting the reflector is perpendicular to the plane of the reflector (so that they ll reflect straight back out towards the transmitter). 3. Place the receiver between the transmitter and the reflector and then on the other side of the reflector to see that the waves are reflected (not at all transmitted). 4. Place the receiver, in the best orientation, between the reflector and the transmitter very near the reflector. E08-8

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