ECE 300 Measurement of Fourier Coefficients Lab 4 ECE 300 Signals and Systems In this laboratory you will observe the frequency spectrum of periodic signals comparing theoretical results to measured data You will learn to use the spectrum analyzer as a tool to investigate signal spectral content Objectives This laboratory project has two objectives 1 To become acquainted with the Tektronix 7L5 Spectrum Analyzer 2 To measure the Fourier coefficients of several waveforms and compare the measured values with theoretical values Background This week in class we are learning how to calculate the spectrum of a periodic signal by using the Fourier series representation We have in the lab spectrum analyzers that can display the spectrum of a signal in real time The Tektronix 7L5 Spectrum analyzer can be used to view the power spectrum of any signal having a bandwidth of less than 5 MHz The 7L5 displays a one sided spectrum in decibels versus frequency In this lab we will observe the magnitude spectra of sinusoidal square and triangle waves as well as pulse trains but first we must learn how to convert the Fourier series coefficients that we calculate to the dB values displayed by the spectrum analyzer Refer to the document Guide to dBs included in Lab 2 to refresh your memory on the use of dBs Recall that any periodic signal x t can be written as x t C n n e j 2 nf 0t where f0 1 T0 the period of x t The Cn are the Fourier coefficients Writing out a few terms gives x t K C 2 e j 2 2 f 0t C 1e j 2 f 0t C 0 C1e j 2 f 0t C 2 e j 2 2 f 0t K K C 2 e j C2 e j 2 2 f 0t C1 e j C1 e j 2 f 0t C 0 C1 e j C1 e j 2 f 0t C 2 e j C2 e j 2 2 f 0t K where we have used the fact the C n Cn for a real valued waveform x t Note that aside from C0 the terms come in pairs A power spectrum for x t based on this Fourier series representation is shown in Figure 1 This is shown as a two sided or double sided spectrum in which the power associated with the complex exponential at frequency nf0 is seen to be Cn 2 C0 is the coefficient for DC C1 for the fundamental frequency f0 C2 for the second harmonic and so forth BAF 1 of 5 6 January 2002 ECE 300 Sx f C2 2 DS C2 2 C0 2 C3 2 C3 2 C1 2 C1 2 3f0 f0 2f0 f0 0 2f0 3f0 f Hz Figure 1 Double sided Power Spectrum We can combine each positive frequency term with its matching negative frequency term to obtain x t C 0 2 C1 cos 2 f 0 t C1 2 C 2 cos 2 2 f 0 t C 2 K The corresponding one sided or single sided spectrum is shown in Figure 2 To make the one sided spectrum the powers associated with complex exponentials at frequencies nf0 and nf0 are added The result representing the average power in the sinusoid 2 Cn cos 2 fnt Cn is displayed at the frequency nf0 Sx f SS 2 C2 2 C0 2 2 C3 2 2 C1 2 0 f0 2f0 3f0 f Hz Figure 2 Single sided Power Spectrum The Tektronix 7L5 spectrum analyzer displays a one sided spectrum as shown in Figure 2 but instead of showing the value of 2 Cn 2 at each frequency the spectrum analyzer shows the average power in decibels with respect to a one Volt reference dBV For the sinusoid at frequency nf0 the average power in decibels is given by BAF 2 of 5 6 January 2002 ECE 300 Pn dB 10 log10 Pn PREF where the power Pn represents the power spectrum coefficient 2 Cn 2 and the power Pref is the average power delivered to a one Ohm resistor by a one Volt RMS sinusoid We have Pn dB 10 log10 2 Cn 1 2 dBV The units dBV indicate that the reference for the decibels is a one volt sinusoid Note The Tektronix 7L5 spectrum analyzer will not display the DC power term C0 2 even when one is present in the waveform This is because the spectrum analyzer input is AC coupled Also because it is displaying a power spectrum the spectrum analyzer does not measure or display the phase angles Cn Prelab Exercises Perform each of the following two calculations for each of the three signals given below You may use Maple or MATLAB to perform the calculations 1 Calculate the Fourier series coefficients Cn for n 0 1 2 9 2 Calculate the decibel values that you expect will be displayed by the spectrum analyzer at each of the harmonic frequencies nf0 The signals are a x a t 0 1 cos 2 10 x10 3 t V b xb t is a square wave of period 100 s and peak to peak amplitude 0 2 V c xc t is a triangle wave of period 100 s and peak to peak amplitude 0 2 V d For each signal create a table in your lab notebook containing a column of values of Cn and a column of values of predicted decibels Leave two additional blank columns one for measured decibels and one for dB difference as a measure of accuracy Read the first two sections of The Spectrum Analyzer Operating Principles and Instructions Pay particular attention to the Cautions on the first page BAF 3 of 5 6 January 2002 ECE 300 Equipment HP 8116A Function Generator Tektronix 7L5 Spectrum Analyzer Oscilloscope BNC T connector Procedures This laboratory project will be performed in the Circuits Laboratory Getting Ready Before connecting any input to the spectrum analyzer be sure that the TerminZ switch is set to provide an input impedance of 1 M The spectrum analyzer is a little more robust when its input impedance is high and it will not be subject to damage from small DC voltages Follow the calibration steps in Section 2 of The Spectrum Analyzer Operating Principles and Instructions Be sure to calibrate the spectrum analyzer every time you turn it on Measuring the Power Spectrum coefficients 1 Use the HP 8116A Function generator to generate a sinusoid of frequency 10 kHz and open circuit amplitude 0 1 V which is waveform a Make sure the duty cycle on the FG is set to the correct setting Use the oscilloscope to verify the amplitude Now observe the sinusoid on the spectrum analyzer Measure the amplitude and frequency of the coefficients and compare with your pre lab calculations Why are there extra peaks in the spectrum Informally vary the frequency and amplitude of the sinusoid and observe how the spectrum analyzer display changes Do not move on until you understand these variations 2 Use the function generator to generate a square wave of period 100 s and peak to …
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