ABSTRACT The theory of beats is investigated experimentally with sound and is compared with amplitude modulation using electronic signal generators and modulators. Observations are made by ear, by oscilloscope and, if possible, by dual trace, storage, and digital oscilloscopes. GENERAL BEATS Beats occur when two vibrations of nearly the same frequency are added. The resultant is equivalent to a vibration of the average frequency modulated by an envelope of the difference frequency. Surprisingly this rather abstract sounding effect has many applications ranging from tuning instruments to police radar. MODULATION Modulation the process of changing a carrier signal to send information. A carrier wave is transmitted at a constant frequency, amplitude, polarization, etc. It transmits no information except that the transmitter is turned on. In order to send information the carrier must be changed in some way. This change in the carrier is modulation. There are many modulation schemes. A common scheme is Amplitude Modulation. Suppose we want to amplitude modulate a 1MHz radio signal with speech. As the speech gets loud the radio carrier increases in amplitude, if there is no speech the amplitude of the radio signal is zero. The amplitude of the speech forms an envelope for the much higher frequency of the carrier. Another common modulation scheme is Frequency Modulation. This time only changes in frequency transmit information. Any changes in amplitude are ignored. If a loud signal is to be sent then the frequency is changed a great deal from the carrier frequency. If a zero signal is to be sent then the exact carrier frequency is sent. APPLICATIONS TUNING Tuning an instrument such as a violin or guitar is often done by matching the pitch of one string with the pitch of another. If the strings are close in pitch a beat frequency will be heard, the smaller the difference in pitch the slower the beat. HETRODYNING It is often desirable to change the carrier frequency of signals, such as a radio station. The carrier frequency that the station broadcasts on is usually quite inconvenient for the tuner, the capacitors and inductors required would be awkward values. The signal from the radio station is mixed with a signal generated by the tuner chosen so the beat frequency will be just the frequency desired for further amplification etc. The resulting frequency is called the intermediate frequency since it is intermediate between the original carrier, MHz, and the audio frequencies that carry the information, KHz. The amplitude of the intermediate frequency is still modulated by the audio information. This process is called heterodyning. LB1BBEAT.DOC 1.069, May 13, 00 Ken Cheney [SECTION]-1
THEORY BEATS We have two waves of identical amplitude y 0 but different angular frequencies ω 1 and ω. The amplitude y of the sum is given by: y = y 0 cos(ω 1 t)+y 0 cos(ω t) If ω 1 and ω are nearly the same we might suspect that expressing them as functions of their average and difference would be useful. i.e. ω 1 = ω+ ω where ω= ω 1 +ω ω = ω ω ω = ω 1 ω Using the trig identity cos(a + b) = cos(a) cos(b) sin(a) sin(b) we find: y = y 0 [cos(ωt) cos( ωt) sin(ωt) sin( ωt)] + y 0 [cos(ω) cos( ωt) sin(ωt) sin( ωt)] Since cosine is an even function cos( ωt)=cos( ωt). Sine is on odd function so sin( ωt)= sin( ωt). The nice result is that the sines all cancel out and the cosine are identical giving: Amplitude with beats y = y 0 cos(ωt) cos( ωt) Equation 1 ω= ω 1 +ω ω = ω 1 ω The interesting cases occur when ω 1 ω. Then ω >> ω. In this case it is quite easy to visualize what the resulting wave will be like. The cosines oscillate between plus and minus one. The cos(ωt) oscillates much faster than the cos( ωt) so the later, slow, oscillation forms an envelope for the faster oscillation. If you ignore the fast ωt oscillation you just see y = y 0 cos( ωt). The fast oscillation looks just like the carrier in a modulated wave: Carrier frequency for beats ω= ω 1 +ω Equation Surprisingly the beat frequency is not ω but is twice ω or just ω 1 ω. The reason for this is that while cos( ωt) goes through one cycle the carrier cos(ωt) has gone through many cycles. Consider a half cycle of the envelope, say from y=zero through a positive half cycle back to y=zero. The actual y goes from plus to minus many times due to the high frequency of cos(ωt) so what we actually see or hear is a double sided envelope going both plus and minus, this looks exactly the same as the second half of the cycle. The effect is that each half cycle of cos( ωt) becomes a whole cycle of the resultant wave. [SECTION]-
Beat Frequency Beat Frequency = ω 1 ω Equation3 Beats or Amplitude Modulation The trace above was photographed from an oscilloscope screen showing an amplitude modulated wave. The carrier frequency was 480 Hz and the modulating frequency was 44 Hz. The modulating wave is shown on the top, the modulated wave on the bottom. As show in the example below this is the same wave we would expect to get by beating together waves of 458 Hz and 50 Hz. EXAMPLE: If we combine a wave with a frequency of 458 Hz with a wave of frequency 50 Hz we will hear a beat frequency of 44 Hz and a carrier of 480 Hz. Equations and 3 apply just as well in cycles per second, ν, as in radians per second, ω. Beat Frequency = ν 1 ν =50 Hz 458 Hz = 44 Hz ν= ν 1 +ν = 458 Hz + 50 Hz = 480 Hz EQUIPMENT SIGNAL GENERATORS The beat portion of this experiment can be done with two signal generators but to compare the results with amplitude modulation it is necessary to have some modulator. We will use a device built just for this type of experiment, the Pasco Dual Function Generator. [SECTION]-3
This function generator has two separate signal generators and can accept external signals or a microphone. One of the signal generators can be used to modulate the other. The modulation can be turned off, Amplitude Modulation, Frequency Modulation, Double Side Band, or Gate. The signal generators can produce sine, triangular, square, pulse, or ramp wave forms. The amplitude and frequency can be controlled and the phase can be inverted. A trigger output is also provided to provide stable triggering of an oscilloscope in synchronization with the output of the signal generator. This trigger output is a simple pulse which is very easy for the oscilloscope to use. The wave forms produced by the modulated waves may be very difficult to trigger on reliably. The outputs of the individual function generators as well as the modulated output can be monitored on an oscilloscope. There are also amplified outputs designed to drive speakers so you can hear what the wave form sounds like. All this versatility comes at the price of a good deal of complexity. There are lines showing the signal path but it is still easy to get lost. Some warnings: WARNINGS ABOUT THE PASCO MODEL 9301 DUAL FUNCTION GENERATOR ON-OFF: There is no pilot light so you cannot be sure the function generator is turned on. Check before panicking FREQUENCY: The frequency step knobs and continuous knobs work together in just the opposite way expected. If the step knob is on 30 you read the frequency from the scale on the continuous knob that ends in 10. Conversely if the stepped knob is on 1 K you read the frequency from the scale that ends a little past 3. You will want to use a frequency counter to get the exact frequency, don t be too surprised at this strange arrangement on the function generator. AMPLITUDE: The amplitude of the signal generators can be varied with the amplitude knobs, just as expected. How much effect the modulating signal has on the carrier depends on how much amplitude the modulating signal has. If the modulating amplitude is zero there will be no modulation. MODULATOR AND CARRIER: If you are using the built in function generators the lower generator () is the carrier and the upper generator (1) is the modulator. Be sure to use a lower frequency on the modulator than on the carrier. Reversing the frequencies leads to very strange wave forms. SUMMING AMPLIFIERS: It would seem logical to sum the two signal generators using the small slide switches on the right under the Summing Amplifier label. For some reason doing this yields no output. The outputs can be easily summed however by simply plugging wires from the outputs near each generator into the same oscilloscope input. TRIGGERING: Triggering on the modulated wave form will probably not be at all satisfactory since the voltage changes in a very complex way. It is best to use the external trigger on the oscilloscope and trigger either on the carrier or on the modulator. Generally if you trigger on the modulator the envelope will stand still but the carrier will drift past. If you fiddle with the frequencies you may be able to get both patterns stable at once for a short time. If you trigger on the carrier then it will stand still while the envelope drifts past. [SECTION]-4
GROUNDS: All grounds, black, are common on the Dual Function Generator, if the grounds are also common on your oscilloscope only one ground is necessary, this can cut down considerably on the clutter of wires. STORAGE OSCILLOSCOPE It is very difficult to get the trace to stand still in this experiment. It is very improbable that the two signal generators will maintain exactly the same frequencies for any length of time. As the frequencies of the generators drift the trace on the oscilloscope drifts and you go crazy trying to make any measurements. This is a good spot to use a storage oscilloscope if one is available. A storage scope will store whatever is displayed as long as you like. The beam need make only one pass across the screen to be stored, then you can take as long as you like to measure or photograph the trace. We will be using a Tektronix model 5111 storage oscilloscope. SINGLE SWEEP: If the trace continually goes across the screen the advantages of storage will be lost. The oscilloscope can be set to make only one sweep by turning on the single sweep feature. Nothing will happen until you touch the reset button, then the trace will sweep once when it is next triggered. STORAGE BUTTONS: PROCEDURE BEATS YOU CAN HEAR SET UP The critical part here is to get two frequencies within twenty Hertz or so of each other, and to use frequencies that the speakers reproduce well. Try frequencies around 00 Hz to start. Connect speakers to the two 8Ω outputs on the function generator. Put the speakers as close together as possible. Turn the GAIN knobs next to the 8Ω outputs up as far as possible. Connect the frequency counters to the trigger outputs of the generators. Set the generators to the desired frequencies, say 190 Hz and 10 Hz. Turn the AMPLITUDE knobs all the way up. Turn the MODULATION knobs to OFF. Find the SUMMING AMPLIFIER section on the middle right. Slide the switch from generator to ON and all other switches to OFF. If all has gone well you will now hear a combination sound from both speakers, a rather harsh sound. Gradually make the frequencies closer and closer and the sound will begin to go noticeably load and soft, more and more slowly as the frequencies get closer and closer. DATA TO GATHER Time enough beats to get an accurate period. Compare the frequency of the beat tone with the beat frequency calculated from ν 1 ν. [SECTION]-5
ELECTRONIC BEATS DATA DESIRED Measure the number of cycles of the carrier contained within each cycle of the beat envelope and compare with theory. SET UP To observe the beat envelope on the oscilloscope it is necessary to have ν not too different from ν. If you use frequencies suitable for hearing you can t see both the carrier and the modulation on the screen at the same time. Also frequencies of tens of Hertz make the screen flicker unpleasantly. It is best to have only five to ten cycles of carrier for each cycle of beats. Numbers near those of the example are quite suitable. A carrier of 480 Hz and a beat frequency of 48 Hz gives just 10 cycles of the carrier per cycle of the beat. If you don t like the flicker these frequencies produce try frequencies of a few KHz calculated so there will be about ten carrier cycles per beat period. Triggering on the beat pattern is not at all satisfactory, trigger on either of the original signals. TO MEASURE THE CYCLES CONVENTIONAL OSCILLOSCOPE You must adjust the frequencies until the pattern on the oscilloscope is as stable as possible, then count quickly. The bigger the pattern the better. Fill the screen with one cycle of the beat envelope. You can freely adjust both vertical sensitivity and horizontal speed since you are not measuring either volts or time. STORAGE OSCILLOSCOPE This is much more pleasant, once you get a good trace you can study it as long as desired. Use low frequencies, 500 Hz say, the flicker won t be there on the stored image. At high frequencies the beam is moving too fast to store well. Even at low frequencies some parts of the trace may be dim or absent if the beam is moving too fast in that region. These missing sections are ugly but may not affect your ability to count cycles. First get a good picture using the oscilloscope in the conventional mode, then try storing one pass. CHECKLIST TO STORE A TRACE Trigger on single sweep. UPPER AND LOWER ERASE buttons both in. STORE buttons both in. Push the ERASE button to clear the screen. Touch the RESET button to arm the trigger and produce a single trace. [SECTION]-6
CHECKLIST IF NO TRACE IS VISIBLE BRIGHTNESS (Y-T) knob, turn clockwise. When you want to keep the trace for a time without observing it turn the knob counterclockwise to save wear and tear on the tube. ENHANCE knob (if you have one) turned clockwise, try RESET again to get another trace. Use a slower speed on the time base. DIGITAL OSCILLOSCOPE We will be using a microcomputer based digital oscilloscope. It is not as powerful as one from Hewlett Packard or Tektronix but, on the other hand, we can afford it. The microcomputers are Apple IIs, the Analog to Digital conversion is by Sunset Laboratory s Advanced Interfacing Board, and the software is Vernier s Voltage Plotter III. This combination of hardware and software acquires 40 voltage samples per sweep and easily handles frequencies of hundreds of Hz. Once the voltages are measured they can be displayed on the screen, measured, or printed out. CHECKLIST TO SET UP APPLE DIGITAL OSCILLOSCOPE COMPUTER PROGRAM: Vernier s "Voltage Plotter III". The disk labeled "VP.STARTUP" initially goes in drive 1 (the left drive), if asked by the computer replace it with the disk labeled "VP.PROGRAM" disk. PASCO DUAL FUNCTION GENERATOR connects to a THORNTON AMP POWER SUPPLY, APS. There is no need for amplification but there is some DC in the output of the Dual Function Generator, this can be canceled with the Thornton APS. PASCO DUAL FUNCTION GENERATOR settings. Use any output you desire. If starting from scratch use the 10K outputs at the lower right. Slide the slide switches on the summing amplifier (right center) to "in" for the signal from generator, and to "out" for the other possible inputs. Turn the amplitude knob on generator fully clockwise. To get the simplest signal to start with turn the modulation knob just to the right of generator to "off". Set the frequency to 100 Hz or so. Set the wave form knob on generator to the sine wave. THORNTON AMP POWER SUPPLY SETTINGS Power on. AC-DC slide switch to AC. GAIN to about 7. DC BIAS to zero. ADVANCED INTERFACING BOARD, AIB, (inside the Apple) is connected to the output of the Thornton APS. The AIB will have a wire coming out of the back of the Apple, perhaps even labeled AIB. [SECTION]-7
TURNON, or reboot, the Apple. To do a"warm" reboothold downthe CONTROL and RESET keys at the same time. This is much easier on the computer than turning it on and off. When asked TYPE OF INPUT DEVICE? answer Advanced Interfacing Board or AIB. When asked VOLTAGE RANGE? check on the side of the computer for information, if you don t find anything try -10 to +10 volts. When asked SLOT CONTAINING THE AIB? check on the side of the computer, if you don t find anything try slot 3. From the MAIN MENU select Oscilloscope. When asked about TRIGGERING there are two possibilities. To be sure of getting a trigger you can select the SPACE BAR. Each time you touch the space bar the trace will be triggered. This is annoying to have to do but at least you do get a trace and can replace it with another whenever you want. Or you can trigger on AUTOMATIC AT 0 VOLTS. If the trace passes through zero volts the oscilloscope will continually trigger, just as a conventional oscilloscope. This is quite convenient for adjusting gain, frequencies etc. The only confusing point can be that if the trace never crosses zero the scope will never trigger. It works well to do the sequence suggested below. TRIGGERING on the SPACE BAR. Adjust the Thornton APS to center the trace, use the DC bias knob. Reset the triggering to AUTOMATIC AT 0 CROSSING. You can stop and start the trace with the H key. Set the proper sensitivity etc. SENSITIVITY: the curser keys for up and down. SWEEP SPEED, time/division: the curser keys for left and right. SUGGESTED INITIAL SETTINGS Sensitivity: volts/division Sweep speed: msec/division LEAVE THE OSCILLOSCOPE SCREEN: RETURN or ESC. This gives you a small but useful menu allowing you to PRINT IMAGE, RETURN TO OSCILLOSCOPE, RETURN TO MAIN MENU, OR SAVE DATA IN MEMORY. PRINT IMAGE prints the trace on the printer. The printer is a GRAPPLER and is connected to slot 1. RETURN TO OSCILLOSCOPE takes you back to the oscilloscope but lets you first change the triggering mode. SAVE DATA IN MEMORY takes the data out of temporary storage and saves it so it can be printed out in a table etc. This must be done before returning to the main menu, otherwise the data is lost. [SECTION]-8
MODULATION GENERAL If you are using a dual trace oscilloscope it helps greatly to display the modulating signal on the top of the screen and the modulated signal on the bottom. It works best to trigger on the modulating signal so the envelope holds still on the screen. Expect to twiddle with the frequencies to get the entire trace to hold still. The AMPLITUDE knob on the top generator of the Pasco Dual Function Generator affect how much modulation there will be. Generally leave it fully clockwise. The top generator is the modulator, the bottom generator provides the carrier frequency on the Pasco Dual Function Generator. AM To match the beat frequency pattern set the modulating wave form to the sine shape. Set the carrier and modulating frequencies to match those calculated from the frequencies you used in the beats part of the experiment. The pattern should match the pattern obtained from the beats almost exactly. Try the other wave forms for the modulating wave also. FM Frequency modulation is most easily observed on square waves or a ramp shape for the modulating wave form. It will take a good deal of adjusting of the frequencies to get the pattern to be steady on the screen. GATE A gate is very much like the triggering on an oscilloscope. When the modulating wave form is above a certain level the carrier comes out, when the modulating wave is below that level nothing comes out. [SECTION]-9