MASSACHUSETTS INSTITUTE OF TECHNOLOGY 22 071 6 071 Introduction to Electronics Signals and Measurement Spring 2006 Lab2 Introduction to signals Goals for this Lab Further explore the lab hardware The oscilloscope Measure time varying signals Explore time domain and frequency domain representation of signals Explore noise and signal to noise characteristics Practice complex arithmetic Explore the sound of a piano note Listen to the sound look at the time domain representation of it Explore its frequency content This will be a great motivation for the next lesson on Fourier transforms and Fourier series Exercise 1 Before we begin with the experimentation part of the lab let s do a bit of complex algebra on order to recall some of the fundamentals Determine the phase of the complex numbers o c 1 j o d j o a 2 j 1 j Exercise 2 Graph the following function j 6 071 Spring 2006 Chaniotakis Cory e j 4 t e j 4 t 2 1 Exercise 3 The oscilloscope Measure time varying signals We will use the oscilloscope feature of out software Upon initialization the following appears on your screen Courtesy of National Instruments Used with permission Once started the oscilloscope screen looks like the example shown here The oscilloscope is basically a graph displaying apparatus and it represents an electrical signal such as voltage as a function of time Courtesy of National Instruments Used with permission Our oscilloscope has two channels and the ability to perform the basic functions of Triggering and scaling both in the vertical and the horizontal axes In the laboratory environment one usually finds oscilloscopes that perform a variety of specialized functions including various mathematical operations on signals The features of our scope corresponds to the most basic and important characteristics of these instruments For our first experiment with the scope we will measure the various signals from the function generator Start by setting the function generator to manual mode and generate a sinusoidal signal with a frequency of 1 kHz Measure that signal on CHANNEL A of your scope Adjust the TIMEBASE until you see the signal 6 071 Spring 2006 Chaniotakis Cory 2 Your signal at this point may be drifting along the horizontal axis at an annoying rate This is because the specified time base on the scope does not match perfectly with the frequency of signal In order to correct this problem and thus provide a stable trace of the signal we must use the TRIGGER feature of our scope TRIGGER synchronizes the tracing of the signal on the screen with a specified event The event may be provided externally by some other well defined signal or it may be associated with a certain characteristic of the signal under investigation For example the trigger event may be the raising or the falling edge of a pulse or it might be the zero crossing of the signal in the positive or negative direction Naturally the best way to learn how this actually works is to try it Look at the TRIGGER menu and along with the Slope and Type option try to stop the drift of your 1 kHz signal Change the frequency and see what happens What is the highest frequency that you can measure with your scope see specifications Now that you have a steady signal that you can read adjust the amplitude of you sinusoidal signal until you have a signal with an amplitude of 2 Volts Now turn on CHANEL B and use it to measure a DC signal from the Variable Power Supply Try to trigger your scope for a specific voltage value 6 071 Spring 2006 Chaniotakis Cory 3 Exercise 4 As we just learned in class the RMS value of a periodic signal is related to the energy content of the signal For sinusoidal signals the RMS value is 1 2 of the amplitude For other periodic signals X t we calculate the RMS value by performing the integration 1 T 2 X t dt T 0 Consider the sinusoidal signal shown here on the right with an RMS value of 3 2 Construct periodic square wave signals that vary between 0 and 5 and have the following RMS values X RMS 1 3 2 30 3 10 2 2 2 6 071 Spring 2006 Chaniotakis Cory 4 Exercise 5 Exploring Sinusoidal Signals For this exercise we will use a simulation based on LabView The program called Adding Sinusoidal Signals may be downloaded from the class web site it is located in the Material section Start the program by double clicking on the icon of the downloaded file To start the simulation click on the right going arrow on the upper left hand corner of the virtual instrument To stop the simulation click on the STOP button on the lower right hand corner of the instrument Courtesy of National Instruments Used with permission The signals x1 t x2 t and x3 t are added together resulting in the signal displayed on the right Notice the resulting signal obtained with the default parameters What does it remind you of We will see starting next time that the three default sinusoidal signals represent the first three harmonics of the Fourier series expansion of a square wave signal Feel free to play around Change the various parameters of the signals and observe the effect Determine the values of the various signal parameters in order to obtain the signal shown here f1 f2 f3 A1 A2 A3 P1 P2 P3 O1 O2 O3 6 071 Spring 2006 Chaniotakis Cory 5 Exercise 6 Here we will explore the frequency domain representation of signals Answer the following questions Determine the value in Hz and the number of frequencies contained in the following signals 1 sin 6 t 2 sin 6 t 2 cos 10 t 3 sin 2 t 6 071 Spring 2006 Chaniotakis Cory 6 Exercise 7 Listen and Look at Sound For this exercise we will use an actual recorded music note from a piano The note is the middle C which has a well known frequency of 261 Hz A plot of the voltage coming out of the microphone that made the recording is shown on the plot to the right It looks interesting but what does it really tell us about the sound Make and record below your own observation of this signal by looking at this plot 1 2 3 What can you say about the frequency content of this signal The musicians in the group know already that the note coming out of a musical instrument contains a number of frequencies besides the main frequency Where do these additional frequencies come from What is their value At this point we do not know how to answer these questions but at least we are asking them which for engineering which is a field of discovery and innovation is the most fundamental step 6 071 Spring 2006 Chaniotakis Cory 7 We will explore the frequency content or