Carnegie Mellon University
Department of Mechanical Engineering
24-352 Dynamic Systems and Control
Lab 1 - Dynamic Signals
The objective of this lab is for you to develop experience in measuring and interpreting signals in terms of both time and frequency. You will work with a function generator (which produces different types of periodic and non-periodic signals), a digital oscilloscope (which is used to measure signals as a function of time) and a spectrum analyzer (which uses Fourier methods to examine a signal's frequency content). These instruments are the workhorses for characterizing and understanding dynamic signals. The basic layout of the experiment is shown in Figure 1.
Figure 1: Test arrangement.
Part I: Theory
The most basic signal is a sinusoid, which represents oscillation of a dynamic system. When two or more sinusoids of different frequencies and amplitudes appear at the same time, and are simply added algebraically, the result is periodic but is not necessarily a sinusoid. Of particular interest is the case in which sinusoids have different frequencies that are integer multiples of each other. Given a sinusoid at some frequency measured in cycles per second (or Hertz), we say that a sinusoid with frequency n-times that amount is the n-th harmonic.
Any periodic function, even ones that are not sinusoidal, can be expressed as a Fourier sine-cosine series. You should review your mathematics notes for the precise definition of a Fourier series, and for the method for obtaining the Fourier coefficients. In short, a function f(t) (units: volts) that has period T (units: seconds) can be expressed as
here the coefficients are obtained through integration by
Please submit your answers to the following questions with your lab report as a part of analysis section.
1. The square wave signal is shown in Figure 2. By hand calculation, derive the expression for its Fourier series as being
As was the case for the triangle wave, the frequencies of the various sinusoidal components are multiples of each other, namely
but in this case note that the amplitudes of the Fourier components decay in proportion to n. Thus, the second harmonic has one-half the amplitude of the first; the third harmonic has one-third the amplitude; and so on. The fact that the amplitudes decrease at a slower rate than for the triangle wave is attributed to the fact that the signal is discontinuous. As a general rule, the more discontinuous the signal, the more slowly its Fourier coefficients will decrease in amplitude.
Figure 2: Square waveform.
2. For the square wave Fourier series, write a computer program to add up the series, and plot out several cycles. The Fourier representation is exact in the limit of the infinite sum, but when the sum is truncated to a finite number of terms, there are errors in representing the discontinuities. Generate plots for partial sums at 5 and 20 terms.
Part II: Measurement
1. Turn on the function generator and oscilloscope. By using a BNC cable, connect Output of function generator to Channel 1 of oscilloscope.
2. Generate a 2.5kHz sine wave on the function generator, with peak-to-peak amplitude of 2V (Please Note: there is a calibration problem with function generator, for this reason to have 2V peak-to-peak voltage, you need to set the function generator to 1V peak-to-peak), and display it on the digital oscilloscope. After adjusting the trigger, time base, and voltage level, measure the peak-to-peak amplitude, period, and frequency of the signal on the oscilloscope using the cursor keys. Experiment with the trigger, time base, and voltage level to become familiar with displaying waveforms. Print the oscilloscope screen to include in your lab report.
3. Repeat step 2 a) for a square wave with
frequency 200Hz and peak-to-peak amplitude of 1V. b) for a triangle wave with
frequency 432Hz and peak-to-peak amplitude of 1.5V.
4. Using a BNC-tee connector, feed the signal into the spectrum analyzer's Channel 1 as well as the scope. Turn on the spectrum analyzer.
5. Generate a 200Hz sine wave with the peak-to-peak amplitude of 1V by using function generator and confirm these values on the oscilloscope screen. Then, set spectrum analyzer;
6. Repeat step 5, a) instead of sine wave, this time generate a square waveform with same amplitude and frequency as in step 5. b) this time generate a triangular waveform with same amplitude and frequency as in step 5.
7. Repeat step 5 for a signal of your choice chosen from those available on the function generator
Part III: Laboratory Report
Prepare a typewritten laboratory report following the outline given in class. Also, answer the following questions in Results and Discussions part:
1. Give a written discussion explaining the difference between displaying a signal in the time domain and in the frequency domain. Define the term "frequency spectrum," and explain why the two representations of the signal are equivalent.
2. How do the measured results, namely the frequencies and amplitudes of the Fourier coefficients of a square wave, compare with what you expect based on your calculations of the Fourier series? Present a table showing the expected amplitudes and frequencies with the measured amplitudes and frequencies.
3. Is there a difference between how the square wave signal appears on the oscilloscope, and when displayed in the time domain on the spectrum analyzer? Explain and discuss why.
4. Compare the number of terms required to express a sine wave, a square wave and a triangular wave. To answer this question, you should compare the number of peaks and their relative magnitudes shown on the frequency spectrum. Which one does require more terms and explain why.
5. Discuss the spectrum of the waveform that you chose in step 7 above.