Harmonic Signal Analysis

Fall 2006

 

OBJECTIVES

 

1.      Learn to perform Fourier Analysis on acquired signals.

2.      Learn to take a measured signal that has been modified by your measurement system and reconstruct the actual signal using harmonic analysis.

3.      Prepare an objective of your own.

 

 

BACKGROUND

 

The basic concept that a measurement system has a complex harmonic response, Ĝ(w), leads to the necessity to represent arbitrary signals in terms of their frequency components.  Fourier analysis is the tool that allows us to do this, and the process is shown conceptually in Figure 1.

Figure 1.  Reconstruction process for a measurement system.

 

 

PREPARATORY EXERCISES

 

1.      You are to read the following handout on Mathematical Treatment of Data: Periodic Function Analysis to prepare you to do the other exercises:

http://www.engr.uidaho.edu/thompson/courses/ME330/lecture/FourierIntro.html

2.      Study the Excel spreadsheet for converting data into a set of Fourier coefficients and phase shifts at specific multiples of the fundamental frequency.

http://www.engr.uidaho.edu/thompson/courses/ME330/lecture/HarmonicResponseExample.xls  

3.      Go through the circuit model to get the formula for the complex gain () of the filter circuit assuming R1 = R2 = R.  Design the filter to have a corner frequency of as close to 1 Hz as possible by selecting the proper capacitance value, but using capacitance values of even orders of magnitude (0.001, 0.01, 0.1, 1, 10 mfarad).  Use 300kΩ as the resistance value for R1 and R2

4.      Derive the formulae for |G| and qG and plot these versus frequency for the range 0.1Hz to 1000Hz using the capacitance value you selected.

 

LABORATORY INSTRUCTIONS

 

In the laboratory, you will be given a pre-assembled filter circuit as shown in Figure 2.  The capacitor is removable and you will be able to insert the value you selected.  The pre-lab calculations are intended to prepare you for calculating the gain and corner frequencies for the actual filter values used in the lab.

Figure 2. RC Filter Daughterboad.

 

The overall experimental setup is shown schematically in Figure 3, with the Mystery Signal going to the oscilloscope, the PICaxe08M, and the filter.  The output also goes to the PICaxe and the oscilloscope so that both the input and output can be measured.

Figure 3. Experimental setup for Mystery Signal measurement.

(1)      Assemble and test the filter circuit that you designed in the preparatory exercise using the prototyping breadboard on the PICaxe board.

a.       Assemble your setup, paying special attention to grounding method.

b.      Record the necessary information for your setup.

Text Box: NOTE: Circuit connections are important. Leads and wires should be short and tight to the breadboard. Use the wire cutters and extra wire if necessary for a good layout.

(2)      Take harmonic response calibration data using the oscilloscope.  Ascertain the corner frequency, wc, and the DC Gain, G0.

 

 

POST LAB CALCULATIONS

 

(1)      Using the serial data recorded from the terminal, use Excel to build a table of voltage-time values for the Mystery Signal and the filter output signal.

(2)      Using this table, perform a Fourier analysis on the Mystery Signal and the filter output.

(3)      Using the filter’s DC gain, G0, and corner frequency, w, reconstruct the input waveform from the Fourier characterization of the output waveform.

(4)      Compare the voltage-time of your reconstructed input to the Mystery Signal.

(5)      Compare the Fourier coefficients, Cn, for the Mystery Signal and your reconstructed signal.

 

 

DISCUSSION QUESTIONS

 

(1)      Based on your results, were the resistors of equal value?

(2)      Discuss the accuracy of your DC gain and corner frequency measurements.  Indicate any possible additional sources for error.

(3)      Did your Fourier analysis yield a quality signal compared to what you observed on the oscilloscope?  Did the Fourier coefficients of the reconstructed signal agree well with those of the Mystery Signal?

(4)      How were the objectives met?  What happened if they weren’t met?

 

 

Laboratory Grading Form

 

Specific Item

Possible

Ntbk

Table of Contents

2

 

Experiment Titled

3

 

Preparatory Exercises

20

 

Objectives

3

 

Date, Time, Place, Ambient Conditions

2

 

Apparatus/Equipment Sketches.  Make the sketch such that you could set up this experiment from you notebook.

5

 

Equipment Description.  Write a sentence for each piece of equipment – to describe what it was used for.

5

 

Lab Procedure that you did.  Record the steps that you made, data that you took, and any comments that would help you conduct this type of work in the future.

15

 

Calculations

20

 

Discussion of Results

10

 

Discussion Questions

10

 

Organization and Readability

5

 

Total

100