8B: ACOUSTIC AND ELECTRIC RESONANCES A. INTRODUCTION: Often a system has natural frequencies of vibration. In the case of the column of air in the PVC tube we are studying, these natural frequencies are called the harmonic frequencies, f 1, f 2,. A vibrating source, such as the speaker, is said to drive the column of air. As the frequency, f, of the driver (speaker) is slowly varied, the amplitude of the driven system (air column) gets larger and larger until it reaches a peak at one of its own resonant frequencies, f 1, f 2,. Therefore, as we sweep the speaker frequency past one of the harmonic frequencies of the air column, we hear the increase in amplitude of the vibrating air column as an increase in loudness of the radiated sound from the tube. At resonance the driving system passes energy very efficiently on to the driven system and the amplitude of the driven system reaches a maximum called A max. (See Rossing, The Science of Sound, Ch. 4.) A graph of the amplitude as a function of frequency is shown here. File Name : resfig.eps Title : Clipboard Creator : Mathematica CreationDate : Sometime Sometime Pages : 0 0 Note that the resonant amplitude peaks at frequencyf 1 and that the peak has a width, f called the linewidth. The linewidth is usually measured at an amplitude of 71% of A max. For a system which loses energy rapidly through damping or friction, the maximum amplitude, A max, is small and the linewidth large, and the resonance is said to be broad. Similarly, for a resonating system which loses energy very slowly, the maximum amplitude is very large and the linewidth is very small. The resonance is said to be sharp. 8B: Acoustic and Electric Resonances - 1
The quality, Q, characterizes the sharpness of one of the resonances. For example, for the fundamental frequency the Q value is: Q = f 1 f. (1.0) A high Q circuit is one with a sharp resonance. Once set oscillating it loses energy very slowly. A tuning fork is an example of an object with a very high Q. To drive an object with high Q, the driving frequency must be very close to the resonant frequency, f 1. B. MEASUREMENT OF THE Q OF A RESONANCE: 1. High Q System: It is convenient to measure the Q of one of the harmonics of your long PVC tube with one end closed. Tape over one of your tube and glue a glass plate to it. a. Set up the scope with channel 1 looking at the signal from the signal generator (2002) and with channel 2 looking at the signal from the microphone. Set the scope to trigger on channel 1. With a small sine wave signal (about 1 volt) from the signal generator, attach the small loudspeaker to the generator. Also, attach the frequency counter to the generator. b. Find a convenient harmonic of your open tube with a frequency above 350 Hz. (The small speakers we are using have a resonant frequency of their own at about 260 Hz. It s nice to avoid this frequency region for this reason.) Choose a sensitivity (VOLTS/DIV) for the microphone input into CH2 so that the full signal can be seen on the scope face at resonance. 8B: Acoustic and Electric Resonances - 2
c. Now, starting below the resonance frequency and scanning over the resonance in 10 or 12 steps, record the amplitude of the microphone signal in millivolts. As a first try you might start about 50 Hz below resonance and step over the resonance in 10 Hz steps. You may want to fill in a few extra points where the amplitude changes rapidly. Driving Frequency Amplitude (mv) Comments d. On a separate sheet of graph paper make a plot of amplitude (on the verticle scale) against frequency (on the horizontal scale). Using the example plots above, connect the points of your plot, calculate the width of your response curve at 71% of A max, f, and the Q for this resonance. Notice that A max is usually measured as the amplitude above the background level. (See the diagram on the first page.) i. f: ii. f: iii. Q = f f : iv. Your comments: 8B: Acoustic and Electric Resonances - 3
2. Lower Q System (optional): A resonating system with low Q is one which loses energy rapidly when not being driven. Friction or resistance can cause such an energy loss. Also, a system can lose energy quickly if it radiates large amounts of its energy as sound. Instrument sounding boards are examples of low Q systems with broad resonances. For a vibrating column of air in a tube, energy can be dissipated through contact with the air outside. One might expect that the Q of a tube would then depend on the ratio of diameter to length of the tube and hence on the areas at each end of the tube compared to the amount of vibrating air in the tube. Similarly, a tube with a horn installed at one end might radiate sound more quickly to the outside air, thus changing the Q value of the tube. This mechanism is often used in the design of brass instruments. The easiest of these system to test is a tube with a larger ratio of end area to air inside, that is a shorter tube where you expect faster dissipation of energy and hence smaller Q value. Pick a short PVC tube with a length of approximately 0.15 m. a. Calculate the expected fundamental frequency, f 1, and record amplitudes around this frequency as you did above. Record your values in a similar table: Driving Frequency Amplitude (mv) Comments b. On a similar sheet of graph paper as above made a plot of these new data. Calculate the width of this response curve at 71% of A max, f, and the Q for this resonance. i. f: ii. f: iii. Q = f f : 8B: Acoustic and Electric Resonances - 4
C. ELECTRIC RESONANCES 1. Introduction In the previous section we saw that acoustic resonances occur when an air column was driven by an external vibrating source with a frequency at or near the natural frequencies of the air column. It is possible to have resonances in electrical circuits as well. An example of an oscillatory circuit is shown in the following diagram. File Name : elecres.eps Title : elecres.draw - /Users/john/Courses/P108/manual Creator : Draw CreationDate : Thu Sep 26 14:41:50 1996 Pages : 0 0 A capacitor C, an inductance L, and a resistor R are in series with the function generator. The resonance frequency of this circuit is f 0 = 1 2π LC As the frequency of the sine wave generator is varied at constant input voltage v in, the output voltage v out shows a maximum at the resonance frequency f 0, which is analog to maximum amplitude we saw before for the resonating air column. The maximum output voltage and the sharpness of the resonance, Q, are determined by the total resistance and the inductance in the circuit. Note that the inductor has some internal resistance. 2. Measurement of the Quality Factor Q a. Use the provided RLC circuit with L = 0.10H and C = 0.01µF. Connect the 2002 Function Generator to the input of the circuit. Also connect the scope to show v in and v out. Observe the resonance without recording any data and determine roughly the range of frequencies required to go completely across the resonance. 8B: Acoustic and Electric Resonances - 5
b. Now decide upon a set of 15 to 20 frequencies over this range for your measurements. Set v in to 0.4 V peak-to-peak and measure v out at each chosen frequency. Be sure to reset v in to the same amplitude each time, if necessary, and verify that it remains sinusoidal. frequency in Hz v in in volts v out in volts v in /v out c. On a similar sheet of graph paper as above made a plot of these new data. Calculate the width of this response curve at 71% of A max, f, and the Q for this resonance. i. f: ii. f: iii. Q = f f : The observed fact that the output voltage can be much greater than the input voltage shows why an LC resonance circuit is so useful in radio communication circuits (transmitters, receivers). In this experimental setup, the resistor was added to make the resonance broader. Otherwise the quality factor Q would have been too high to make the measurements readily. In communication resonance circuits the resistance is kept to a minimum to provide a very sharp resonance and thus a good selectivity of stations on nearby frequencies. 8B: Acoustic and Electric Resonances - 6
D. COMPARISON OF ACOUSTICAL AND ELECTRICAL RESONATING SYSTEMS You have now measured the resonance behavior of an air column and an electrical RLC circuit. In both cases you used a Function Generator to excite the system at various frequencies. You displayed input and output voltages on an oscilloscope and measured thir peak-to-peak amplitudes. The traces on the scope looked the same, except that the air column and the electric circuit resonated at different frequencies. If you now put the air column or the electric circuit into a black box with two sockets, one for the input to drive the system and one for the output to measure the amplitude of the resonating system, you would not be able to distinguish between the two systems. Both systems show the same resonant behavior. However, there are differences between a resonating air column and an electric RLC circuit. What is the major difference? Does a resonating RLC circuit have higher harmonics like the air column? 8B: Acoustic and Electric Resonances - 7