They are compared with the frequency comp onen of sine of the same frequency B Tw sin usoidal es with sligh tly di57355eren frequencies are added and the frequency comp onen ts observ ed and compared with those obtained passing the signal through di ID: 26035 Download Pdf

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They are compared with the frequency comp onen of sine of the same frequency B Tw sin usoidal es with sligh tly di57355eren frequencies are added and the frequency comp onen ts observ ed and compared with those obtained passing the signal through di

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Ph ysics 263 Exp erimen requency Comp onen ts of Non-Sin usoidal es Purp ose (A) The frequency comp onen ts of square are obtained capturing the eform and using the FFT algorithm. They are compared with the frequency comp onen of sine of the same frequency (B) Tw sin usoidal es with sligh tly dieren frequencies are added, and the frequency comp onen ts observ ed, and compared with those obtained passing the signal through dio de. In tro duction It is ossible to add up sine es of dieren amplitude, frequency and phase to obtain almost an or pulse shap e. This pro

cedure is called ourier syn thesis "; the rev erse, extracting the frequency comp onen ts whic mak up an arbitrary shap e, is ourier analysis ". What will do in this exp erimen is to capture some eforms with the digital oscilloscop e, and use the \F ourier Analysis" feature of the spreadsheet Excel to study the frequency comp onen ts. Pure Sine Connect the eform generator directly to the oscilloscop e: Set the frequency to ab out Scope 200 Hz Figure 1: Setup for single eform. 200 and set the eform to sin usoidal. The amplitude should ab out olt. Set the time scale so that the scop sho ws at

least cycles. Read this eform out with precise, when the form, is erio dic, it ma written as an (innite) ourier series. When the form is ap erio dic, i.e. single \pulse", is expressed as an in tegral, the ourier transform, instead of series. And when is only discrete set of \samples", as ould get from an oscilloscop capture, it is expressed as nite series called the \discrete ourier transform" or DFT. sp ecial algorithm to erform the DFT rapidly is the \fast ourier transform" or FFT.

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the eform Data Capture program. Cop the data to the clipb oard and paste it in

to Excel. Square No simply switc the eform generator eform to square e, hanging nothing else. Rep eat the ab capture pro cedure. This time lea at least columns free et een the rst data set and where ou paste in this one. Finding the requency Comp onen ts Select Data, Data Analysis, ourier Analysis The instructions in this and the follo wing section assume that the times are put in column A, and the signal oltages in column B. As Input Range, en ter B1:B2048, if our oltage data is in column B. (W can't use all 2500 oin ts from the scop ecause the algorithm requires that the um er of oin

ts er of 2.) Select Output Range and en ter C1:C2048. What ou will see next is set of um ers for eac cell. This is ecause the frequency decomp osition algorithm (the FFT) pro duces them. They can put in to the form of magnitude and phase. Excel's algorithm pro duces comp onen ts, the real and imaginary parts of complex um er. In this exp erimen t, are only in terested in the magnitude, so in the adjacen column, (D), en ter =IMABS(C1) and cop this to the rest of the column. requencies The FFT algorithm pro duces frequency comp onen ts er nite set of frequencies. The lo est one is 0,

corresp onding to constan lev el. The highest one is: max 2 where is the time in terv al et een samples. In addition, the second half of the output column is asso ciated with negativ frequencies. So in this data set, there are only 1024 ositiv frequencies. are only in terested in the lo er part of the frequency sp ectrum, so in the next column, compute the frequency for eac comp onen for the rst 200 um ers in the magnitude column. do this, en ter in cell E1. In cell E2 en ter =E1 1./((A3 2)*2048.) Then cop cell E2 to cells E3 to E200. This will calculate the correct frequency for the

corresp onding comp onen in column D. Con tin uing Next, rep eat the ab pro cedure for the square data. ou don't ha to construct the frequency column, ecause it is the same as for the sine e. Mak graph of the FFT magnitude vs. frequency for the rst 200 oin ts. If the sine-w frequency is

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do ou see eaks in the square-w sp ectrum at ev ery in teger ultiple of If not, for what \harmonics" do the eaks ccur? Making Beats No tak eform generators, and add their signals as sho wn in Figure elo w. Mak the generator amplitudes eac ab out olts. Figure 2: Setup for addition of sin

usoidal signals. On the scop e, ou can see the \in terference" et een the signals. Set the time scale so that ou can see at least eats. An example of this is sho wn in Figure 3. No rep eat the capture and FFT analysis ou did ab e. Is there eak in the frequency sp ectrum corresp onding to the \b eat" frequency? Eect of non-linear elemen non-linear elemen (one for whic the output curren is not prop ortional to the oltage drop) has in teresting eects on the frequency resp onse. The simplest non-linear circuit elemen is dio de. Insert dio de in our circuit as sho wn in Figure 4.

dio de conducts in only one direction, and in the forw ard direction, the resp onse is non-linear. Rep eat the capture and FFT analysis. Mak plot comparing the frequency sp ectrum with and without the dio de. What additional frequencies do es the dio de pro duce?

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-5 -4 -3 -2 -1 -0.0015 -0.001 -0.0005 0.0005 0.001 0.0015 Sum signal, volts Time, sec Sum of two Sine Waves Figure 3: Sum signal vs. time Figure 4: Signal addition with dio de included.

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10 requency Comp onen ts of Rectied Beat Signal Wh should rectied sine con tain harmonics? ma write

it as: sin sin This is sho wn in Figure 5. sqrt(sin(omega*t)^2) sin(omega*t) Figure 5: Left: rectied signal; Righ t: sin sin or sine es, (sin sin sin sin Expanding the square: (sin sin sin sin 2(sin sin rigometric iden tities can used to sho that sin (1 cos and sin (1 cos and that 2(sin sin cos cos So the rectied sine con tains at least and The in the equation generates additional harmonics.

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