Lab 0: Orientation. 1 Introduction: Oscilloscope. Refer to Appendix E for photos of the apparatus

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1 Lab 0: Orientation Major Divison 1 Introduction: Oscilloscope Refer to Appendix E for photos of the apparatus Oscilloscopes are used extensively in the laboratory courses Physics 2211 and Physics In these courses, each group consisting of two students will encounter experiments 1-10 in a different prearranged sequence as assigned by the T.A. It is therefore useful to understand how to use an oscilloscope at the very outset. During this session, you will learn how to use an oscilloscope to measure the properties of simple waveforms, which are simply graphic representations of waves. You will also investigate the basic triggering modes of an oscilloscope. In lab 2 (The Oscilloscope - motion of free electrons in an electric field), you will be working with a stripped down version of an oscilloscope. During this experiment you will have the opportunity to understand the principles of how it works. Oscilloscopes are widely used in research and industrial fields. Physicists use it to analyze data from electronic circuits while biologists or medical researchers could use it to measure brain waves. Oscilloscopes are essential in many industrial fields such as the automobile industry, for instance to measure the vibrations in an engine, or in the electronics sector where it can be used by a technician to test or repair a circuit. The oscilloscopes used in these labs are generally analog oscilloscopes. Looking on the display screen, you can readily notice the grid markings on the screen, as shown in figure 1. These markings create a graticule, calibrated in centimeters (cm). The length between the horizontal or vertical lines is called a major division. The tick marks on the central horizontal and vertical graticule lines are called minor divisions. An oscilloscope basically consists of an electron gun enclosed in an evacuated tube. The end of the tube has a wide surface on which the signal is projected, and upon which the grid is superimposed. The elec- Minor Divison Figure 1: An oscilloscope s grid tron beam s behavior and path are controlled by two sets of capacitor plates found inside the tube, a vertical and a horizontal pair. The beam is deflected by a uniform electric field that is produced by applying a voltage between the plates. Visualize how the horizontal pair causes the beam to deflect horizontally, while the vertical plates cause the beam to be scanned in the vertical direction. The voltage on the horizontal plates determines the speed at which the beam sweeps across the tube and the input signal is applied to the vertical plates. First note the location of the on/off switch, intensity and focus adjust knobs on the front panel of the oscilloscope. Then note the locations of the input BNC connectors and their respective labels, Channel 1, Channel 2 and External Input. Locate the gain controls (labeled volts/div) and the time base control (labeled sec/div). These controls manipulate the scale of the grid superimposed on the screen. The horizontal (x) axis and the vertical (y) axis are the time base and voltage axes respectively. Each major division on the screen corresponds to a single unit of time and voltage that is determined using the control knobs. You can change the display of the waveforms by adjusting their scale and position. When you change the scale, the waveform display will increase or decrease in size. When you change the position, using the vertical and horizontal position knobs, the waveform will move up, down, right or left. One can notice the similarity of the grid on the screen to conventional graph paper. Therefore, reading measurements on the screen is done in the same manner as reading data off a graph. The oscilloscope starts acquiring data, which it displays as a waveform, when it is triggered. Locate 0.1

2 Triggered signal Signal Delay Trigger signal Figure 3: A signal and a trigger pulse Untriggered signal Figure 2: Triggering the trigger knob on the oscilloscope. The trigger control lets you stabilize repeating waveforms and capture single-shot waveforms. It makes repeating waveforms appear static on the oscilloscope display. An oscilloscope is triggered when it detects a preset threshold voltage controlled by the trigger level. In other words, the oscilloscope is triggered if the input signal voltage is greater than the value set by the trigger knob. If this voltage is present between the vertical plates, the electron beam sweeps across the screen, returning to its original position and then repeating the sweep. The time base control knob defines the speed at which the beam travels across the screen. Using the trigger knob, you scan for a signal, so that the oscilloscope begins accumulating and processing data. The oscilloscope does not necessarily have to be triggered by the signal being measured. Typically, an experiment involves a starting point that can be used to generate a pulse or waveform. This pulse can be used to trigger the sweep so that a signal occurring after the trigger can be recorded. There are three trigger sources in the oscilloscope, line, external and internal sources. In internal mode the trigger signal is connected to a one of the channels. By varying the trigger knob the oscilloscope searches for an input signal. When it detects the trigger voltage, the oscilloscope continues to acquire data from the source so that the waveform of interest is displayed on the screen. In external mode, on the other hand, the trigger signal is connected to the external input. Hence, the trigger signal is not displayed. It can be used, for instance, when you want to acquire two signals. For example you might want to trigger from an external clock or with a signal from a test circuit and record two synchronous signals on two channels. In order to clarify this further, consider the following scenario. You are conducting an experiment and analyzing data that involves a trigger pulse and a signal, see figure 3. As you can see it is difficult to decide when the signal occurs without the aid of a trigger pulse. In this case, first triggering using internal mode is necessary in order to understand the delay time between the trigger and the signal, after which we can trigger the oscilloscope externally using the trigger pulse. In both cases the oscilloscope begins accumulating data and displaying it on the screen, following the trigger. Line trigger allows the scope to trigger on the relatively stable 60 Hz AC supply used to power the scope. This mode allows you to investigate signals that have a specific time or phase relationship to the AC supply voltage signal. Line trigger can be used to identify the source of the noise signal affecting the data being analyzed. For instance, if a photodiode measuring the light intensity in a laboratory is connected to an oscilloscope, one could deduce if there are any external sources interfering with the diode, by scanning the signal for any frequency higher or lower than 120 Hz (which is the typical frequency of standard indoor lights). In this case, it is possible to identify the source 0.2

3 DC Coupling of a sine wave AC Coupling of signal Figure 4: AC coupling Figure 5: DC coupling of noise by measuring the frequency of the noise signal and comparing it to the frequency of the AC supply voltage. An oscilloscope has two common trigger modes, normal and auto modes. Normal mode will cause the electron beam to sweep when the input signal exceeds a threshold set by the trigger level knob. Auto mode causes the oscilloscope to sweep, even without a trigger. If no signal is present, a timer in the oscilloscope triggers the sweep, causing the signal displayed on the screen to slide across the field of view. You will find both modes useful in different cases; normal mode is more versatile and auto mode requires less adjustment. Note the buttons below the gain control of either channel. They are the coupling modes that can be set to AC, Ground or DC. Coupling involves the method of connecting an electrical signal from one device to another. Ground mode disconnects the input signal from the oscilloscope so that the trace is a horizontal line which represents zero volts. This allows the user to adjust the position of the signal on the screen. DC coupling displays the input signal without any alteration, for instance an AC signal with a DC offset, as shown in figure 4. AC coupling, on the other hand, filters the DC component, leading the AC signal to center about the zero level, as shown in figure 5. Switching between DC and ground is a convenient method of calculating signal voltage levels with respect to ground (zero voltage). Caution: When switching from DC to ground, the function generator needs to be disconnected from the oscilloscope. Grounding causes the oscilloscope s input to have zero resistance. This forces the function generator to send more current in order to maintain the signal s voltage, which could cause the generator to malfunction. When you are investigating the various coupling modes, discuss them further with the TA. Single and dual modes are two terms that are commonly used to describe the number of channels being used. Single, means that only one channel is retrieving data, while dual indicates the use of two channels. In dual mode, the different display options of the scope can be used. You will find the display mode switches above the gain control knobs. One switch allows you to choose which channel signal to display on the screen. For instance, you can view both channels at once if you need to compare the frequencies of the two signals. The other mode switch allows you to choose between add, alternate or chop. Add superimposes or algebraically sums the two signals and displays the resulting wave on the screen. Alternate mode displays each signal alternately, sweeping each channel separately one after the other. This mode is useful when observing high frequency signals. Chop mode, on the other hand, is useful when studying low frequency waves. In this mode, the display consists of short segments of signals from each channel on a single sweep. The time base control has a display mode switch as well. X1 mode allows you to change the time base using the control knob without altering the scale chosen. Switching to MAG gives a magnification scale that can increase the time base by X5, X10 or X50. The middle switch, ALT, displays both the normal 0.3

4 BNC Socket Pinhole BNC Cable End Pin Shield Figure 6: The structure of a BNC socket and cable and magnified waveform using the alternate display mode described above. The oscilloscope has an interesting mode, known as the XY mode, which helps in measuring phase shifts between two signals. It can be chosen using the base time control knob. This mode is called the XY setup since both the x and y-axes trace voltages. When two channels are receiving signals in this mode, each channel is projected on to one of the axes. This creates a waveform commonly known as a Lissajous pattern. From the shape of the pattern, you can tell the phase difference between the two signals, and their frequency ratio. To connect the oscilloscope to other electronic circuits or devices, BNC cables are used. A BNC (Bayonet Neill-Concelman) cable is a type of RF (radio frequency) connector used for terminating a coaxial cable. The coaxial cable is an electrical cable consisting of an insulated conducting wire surrounded by a conducting sheath which in turn is covered by a final insulating layer. Take a look at a BNC cable, illustrated in figure 6. Notice the central pin protected by a shield. The BNC cables have a twist-lock attachment, for secure connections between the devices in use, as illustrated in figure 6. Caution: When using BNC cables to connect a circuit, you should ensure that the circuit is connected properly. The central pin of the cable needs to be firmly connected to the central pinhole of the BNC output/input socket on the device being used. If this is done properly, the shield of the cable should be firmly connected to the outer shield of the device in use. This is necessary to ensure that the grounds of the two devices are at the same potential. This is crucial to protect the devices at hand from damage. Next you will use the oscilloscope to display a waveform on the screen in order to investigate the different features and modes on the oscilloscope. To generate a waveform, you will use a function generator. It is an instrument that produces a variety of waveforms whose amplitudes and frequencies can be varied. Typically, a function generator produces a sine, square and triangular waveforms. The frequencies range from 1 Hz to over 1 MHz. The amplitude can usually be varied from close to zero to about 20 volts, peak to peak. The common features of a function generator are the amplitude and offset knobs. The offset knob needs to be pulled out and then rotated to vary the offset. A signal with a DC offset leads to a waveform that is not centered on zero volts. You can adjust the offset by observing the waveform on the screen, making sure that the coupling mode is on DC input. 2 Apparatus Refer to Appendix E for photos of the apparatus Oscilloscope Function Generator Tee Connector BNC Cables 3 Experiment I: Using an oscilloscope Whenever you connect a circuit using BNC cables, ensure that the central pin of the function generator output is connected to the central pinhole of the BNC input on the oscilloscope. Similarly, make sure that the outer shield of the connector on the function generator is connected to the shield of the BNC cable, which in turn is connected to the outer shield of the BNC connector on the oscilloscope. This will ensure that the ground of the function generator and the ground of the oscilloscope are at the same potential. Turn on the function generator and make sure that it is set to display a sinusoidal waveform with amplitude 1 volt and a frequency 1 KHz. Connect a BNC cable from the 50Ω output of the function generator to either one of the channels. Now vary the trigger level until you observe a steady waveform on the screen, 0.4

5 making sure your display mode and trigger mode are set appropriately. What mode have you triggered the oscilloscope on? After you observe the waveform, keep turning the trigger level knob further. What do you notice on the screen before the waveform disappears? Now unplug the BNC cable from the function generator and plug it into the other output labeled TTL (Transistor Transistor Logic). Again, vary the trigger level until you observe the waveform on the screen. As you can observe, the trigger signal is a square waveform. The TTL output generates square waveforms that are synchronous with the sinusoidal waveform for the 50Ω output. Now, turn the trigger level knob until the waveform disappears. Did the square waveform portray the same behavior as the sinusoidal waveform? Connect the sinusoidal signal to channel 1 and the trigger signal to channel 2, using two BNC cables. Measure the time delay. Unplug the BNC cable from the channel 2 and plug it into the external input. What trigger mode should you set the oscilloscope to? Vary the trigger knob until you observe the waveform on the screen. Why is the square waveform not displayed on the screen? Unplug the two cables from both the generator and the scope. Connect a BNC Tee to the 50Ω output, and connect two cables to the outputs of the Tee. Use one cable to trigger the scope and the other one as the signal. In this case, both channels should display the same signal. In other words, there should be no time delay, since you are using the same waveform, if the two BNC cables have the same length. Use the gain control so that the waveform nearly fills the screen. Vary the base time control until one or two cycles of the signal fills the screen. Record the amplitude and period of the waveform. Now vary the time base control a few times, recording the amplitude and period accordingly. Exercise 1: Make a data table displaying the amplitude and period. Estimate the error in each of these quantities. Calculate the frequency of the waveform from the data. Estimate the percentage error in the frequency. Now investigate what happens if you change the triggering mode to line. Make sure that you understand your observations. Using two function generators you can investigate the XY display mode of the scope. Plug in the output of each function generator to its respective channel. Turn the time base control knob to XY mode. In order to notice a specific phase difference between the two waveforms, they need to have a specific frequency difference. For instance, try setting one generator to 300 Hz and the other to about 900 Hz. What do you observe? Investigate the different patterns displayed by choosing different values for the frequency. Next, use the BNC Tee to connect the same signal to both channels. In what manner does this display differ form the previous one, in which two generators were used? Now change the time base control knob to an appropriate setting where you can fully observe the waveform, keeping both channels connected to the function generator using the BNC Tee. Investigate what the different modes located above the gain control knobs do to the waveforms from the generator. You could also investigate the various time base modes. You are encouraged to discuss the oscilloscope and it s different features with the TA until you understand the functions of all the oscilloscope s control knobs and switches. It is crucial that you understand the oscilloscope clearly since you will be using it in most of the labs in this course and possibly in the years to come. Exercise 2: In your own words, explain the different trigger modes of the oscilloscope in a sentence or two. The next two experiments deal with various numerical methods used for error analysis. These concepts are essential when it comes to estimating errors in a body of data that has been acquired during an experiment. Make sure you understand these basic concepts clearly since they are the backbone of error analysis, a topic that you will discover and study in greater depth in the coming years. 4 Experiment II: Standard Deviation and Poisson Distribution Physics is deeply rooted in experimentation. For an idea or a theory to be accepted it has to withstand and survive detailed and extensive experimentation. No physical law has ever been adopted without experimental verification of its predictions, from New- 0.5

6 Table 1: Data from decay experiments f(x) Experiment Average Count(x) Table 2: Data from decay experiments Trial Count(x) Trail Count(x) ton s gravitational laws to Maxwell s electromagnetic laws. There are many types of experiments in physics, one class of which are timing experiments. An important example is the decay measurement of a number of radioactive samples. For instance, a physicist could measure the number of particles emitted by a radioactive sample of uranium in a five-second interval. Usually, a researcher repeats an experiment a few times, in order to acquire an average value of the measurements. The data from decay timing experiments of a number of different radioactive substances is shown in table 1. Each experiment consists of ten separate five-second decay observations, where after each run, the researcher noted the number of particles detected. The second column is the average number of radioactive particles obtained using the ten trials. Table 2 shows the ten trials of experiment 3, which resulted in an average count of 115 particles. The exercises that follow pertain to experiment 3. Calculate the average value x of the number of counts. The mean is not necessarily the correct value, but it can be expected to be more accurate than any single measurement. Exercise 3: For each trial, calculate the deviation x x. What is the sum of x x? Exercise 4: For each trial calculate (x x) 2. Find the sum of (x x) 2. x-s x x+s x Figure 7: Standard deviation The standard deviation S is defined by, (x x) 2 S = N 1 (1) where N is the number of trials (N 1 N, for large number of trials). Exercise 5: Calculate S. S represents the variation of a single data point with respect to the average value. For example, the most accurate value of experiment 3 is 115 ±S. S is a measure of the experimental error. Thus the correct value is within x±s, 68 percent of the time, as illustrated in figure 7. The individual measurements of x are normally distributed about x with width S. For further analysis of standard deviation refer to the textbook listed under suggested reading. Furthermore, if a certain count x is obtained by averaging N runs or trials, the best estimate for statistical variation of the mean value is x±s m. The uncertainty in the average value x is called the standard deviation of the mean and can be calculated by dividing the standard deviation of a single measurement by the root of the number of runs or trials. It is defined as, S m = S N. (2) Exercise 6: Calculate the standard deviation of the mean for experiment 3. There is another interesting approach to error analysis usually called the Square-Root Rule. This applies to a discrete distribution (such as a histogram with an envelope that looks like figure 7) known as the Poisson 0.6

7 Table 3: Data from decay experiments Experiment B(T)(y) I(A)(x) distribution. It describes the results of experiments in which the results measured occur at random, but at a definite average rate, as in the counting experiment mentioned previously. If, for instance, one conducts a decay experiment once and records the number x of emitted particles in a time period T, the best estimate of the error is to take the square root of the value measured. Thus the average number of events is defined as x ± x. On the other hand, if one conducts a decay experiment N times and records a mean value x the best estimate of the error is to take the square root of the mean value measured. In other words, the standard deviation of the mean value is x. Therefore, the uncertainty is defined by x ± x. Based on the above argument, it is evident that if N is large a more accurate estimate of the mean can be obtained. Exercise 7: Assuming that the average counts listed in table 1 obey Poisson statistics, estimate the standard deviation of each experiment. Exercise 10: Eyeball is a common name for the line of best fit. To plot such a line, draw two straight lines one with the maximum slope and the other with the minimum slope such that they both go through the error bars. Determine the slopes and intercepts of these straight lines. The average of the slopes of these lines is the slope of the best fitting line. Draw the best fitting line on the graph. Estimate the error in the slope of this line. Exercise 11: For a straight line of the form y = mx + b, the least squares algorithm gives, m = N xy x y N x 2 ( x) 2. (3) b = x 2 y x xy N x 2 ( x) 2. (4) Find the slope and intercept of the straight line obtained by using the method of least squares (plotting routines usually use this algorithm). Plot this straight line on the graph. Exercise 12: Compare these values with the corresponding values from the best eyeball fit. 6 Suggested Reading J. R. Taylor, An Introduction to Error Analysis (2nd Edition, University Science Books, 1997). Chapters 4, 5 and 11. Your lab report should include: 5 Experiment III: Straight Line Fit This part of the lab will explain how to obtain the best fitting line, when plotting data obtained from an experiment. Table 3 shows the measured variation of a magnetic field as a function of current in an experiment. Assume that the theory predicts a linear relationship for B as a function of I. Answers to exercises 1-12 with relevant data tables, graphs, figures and qualitative comments. Refer to Appendix D for Maple worksheets. Exercise 8: Plot a graph of B versus I. Label the axes. Show units. Exercise 9: Assume that the error in B is ±20%. Assume that the error in I is negligible. Plot error bars on the graph. 0.7