Carnegie Mellon University

Department of Mechanical Engineering

24-352 Dynamic Systems and Control

Spring 2001

Lab 1 - Dynamic Signals

**
**

**Overview**

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

**Assignment:**

*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;

- On
*SYSTEM*menu, press green*Preset*button, then press*F1*button to*Do Preset*. - Set the analyzer to split screen mode, to do this; on
*DISPLAY*menu, press*Disp Format*button, then press*F2*to select*Upper/Lower*mode. Now, the Fourier transform is in the upper window, and the time record is in the lower window. - Select a frequency range wide enough to capture the
first several harmonics. On
*MEASUREMENT*menu, press*Freq*button, and select*Span*by pressing*F1*, then press 3.2 kHz (for kHz press F1). - Display the spectrum in linear amplitude format; on
*DISPLAY*menu, press*Trace Coord*button, select*Linear Magnitude*by pressing*F1*and also adjust*Y Units*by pressing*F9*, on this new menu, press*F1*to select*Amplitude Peak*and press*F4*to select*Volts*unit. Also, on*DISPLAY*menu, press*Scale*button and then press*F1*to select*Autoscale On*. - Using the cursors, measure the amplitude and frequency of each peak that you see, and record the data in a table. (For a sine wave, you should have only one peak, but for square or triangular waves, there will be more than one peak)
- Then print out a copy of the analyzer's screen to
include in your lab report. To print, on
*SYSTEM*menu, press*Plot/Print*button, then press*F10*for*More Setup*on analyzer screen, now press*F2*to select*Device is prnt*and*F10*to*Return*. Finally press*F1*to*Start Plot/Prnt*.

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.