Experiment 2: Waveform Analysis

Transcription

1 Specrum aalyzer Experime : Waveform Aalysis 8/3/6 his experime is o familiarize you wih he aalysis of sigals ad eworks usig he HP 358A Specrum Aalyzer. he aalysis of sigals is achieved i he ope loop mode while he aalysis of eworks (Bode plo) uses he closed loop mode. I eiher case he daa ca be obaied i liear form measured i rms vols or i log form measured i db (decibel) where db = log ( V /V ). () he decibel scale is commoly used for describig he gai or loss of amplifiers ad oher circuis. I is also used for describig he ampliudes of sigals. Is advaages over a liear scale are: Whe a sigal is amplified or aeuaed by a gai G i deci-bells (db), he oupu sigal level is he sum of he ipu sigal level ad he gai. For example, if he ipu sigal is 3 db ad he gai is + db, he oupu sigal is +7 db. For example, if a relaive deermiaio were required, he a volage ha is oe-half of is origial value would be db = log ( ½ ) = -6 db. () Ope Loop for Sigal Aalysis Ipu Sigal 5 o 5 khz VCO khz o 5 khz Mixer or Muliplier f LO + f i f LO - f i KHz Badpass Filer ad deecor CR Figure Ope loop block diagram of specrum aalyzer Figure shows a block diagram of he specrum aalyzer i ope loop mode. he volage-ued local oscillaor (VCO), which varies i frequecy from f LO equals khz o 5 khz i a sweep period,, is applied o a muliplier. A ipu sigal f i ragig from 5 Hz o 5 khz is also applied o he muliplier where i is mixed wih he VCO sigal accordig o he rigoomeric ideiy

2 cos (πf LO ) cos (π f i ) = cos[π (f LO + f i ) ] + cos[π (f LO - f i ) ], (3) o produce sum ad differece frequecies. hese frequecies are he fed io a bad pass iermediae frequecy (IF) filer wih a passbad ceered a khz wih a 6dB badwidh as idicaed o he specrum aalyzer. he oupu of he IF filer is he preseed o he cahode ray ube (CR) a he ime give by ( / f i ) = (/5 khz) (4) where is he sweep period. hese resuls are summarized i Figure assumig a siusoidal ipu frequecy less ha 5 khz. If he ipu sigal is o siusoidal, he each harmoic of he ipu sigal mixes wih he VCO producig a sigal a he CR a he appropriae ime i he sweep,. Ipu Mixer Oupu f LO + f i 5 khz khz 35 khz khz f LO - f i 5 khz 35 khz f i 65 khz f LO Filer Oupu Figure Frequecy respose of specrum aalyzer compoes

3 = a je ω )(x j e ω d Closed Loop for Nework Aalysis (Bode Plo) I he closed loop cofiguraio of Figure 3, a khz local oscillaor ieral o he specrum aalyzer is mixed wih he VCO oupu o produce a sum frequecy ha varies from khz o 5 khz ad a differece frequecy ha varies from o 5 khz. A low pass filer removes he sum frequecy ad leaves he differece frequecy. his differece frequecy is available a he back of he specrum aalyzer ogeher wih a ampliude corol. Whe his sweep frequecy ragig from o 5 khz is applied o he ipu of a es circui ad he oupu of he es circui is se o he ipu of he specrum aalyzer, he specrum aalyzer displays he respose of he circui over his frequecy rage. his is referred o as a Bode ampliude plo. Crysal Oscillaor khz VCO khz o 5 khz Mixer Or Muliplier f LO + khz f LO - khz IF Lowpass Filer 5 khz o 5kHz Circui Uder es o Specrum Aalyzer Ipu Figure 3 Block diagram obaiig a Bode Plo wih he Specrum Fourier Aalysis I sigal aalysis i is well kow ha a arbirary periodic sigal ca be represeed as a sum of frequecy compoes = jω x () = a e (5) where ω = πf ad f = / is he fudameal frequecy, ad is he period. he complex coefficie, a, is obaied from a = x()e jω d (6) 3

4 his compac formulaio of Fourier aalysis implies egaive frequecies ad complex volages. he specrum aalyzer measures he magiude of a, ad gives o iformaio abou he phase. I should be oed ha he magiudes of he Fourier compoes as give by (6) are peak values while he specrum aalyzer measures rms values. o relae he complex coefficies, a, o measurable values, le a = a e j θ he a * = a = a e jθ (7) where a is he magiude of he complex coefficie ad θ is he phase agle of ha coefficie. Usig equaio (7), equaio (5) becomes x( ) = a x( ) = a j( ω+ θ ) j( ω + θ ) + a[ e + e ] = + a cos( + θ ). = Comparig equaio (9) o a periodic sigal volage + V cos(ω + θ ) = ω (9) v () = V () where V is he average or dc compoe ad V is he rms value (ha value measured by he specrum aalyzer) of he -h harmoic (8) jω V = v() d () V = v()e d. () Sie Wave As a simple example of he compariso of calculaed Fourier coefficies o experimeal rms values from he specrum aalyzer, le v( ) = cos( ω) Equaio () simply represes he area uder a cosie curve over he period, so he dc compoe of V is jus zero as expeced. he rms values from equaio () are jω V = cos( ω)e d (4) V = [cos( ω) cos(ω) jcos( ω) si( ω)] d (5) 4 (3)

5 I equaio (5) he cosie-sie erm iegraes o zero for all values of, so here is o imagiary compoe of V ad he magiude sigs are reduda. he cosie-cosie erm iegraes o zero for all values of excep =. For = he cosie-cosie erm iegraes o a value of /. So V = = =.77 (6) Which is he expeced rms value for a sigal wih peak value of uiy. Periodic Expoeial Decay As a more complicaed example cosider a periodic expoeial decay wih period of he form From equaio () v() = exp( ) where <<. V = e d = e = e. (8) Sice by assumpio he period is much greaer ha he characerisic decay ime, he V = /. (9) Usig equaio () + jω jω V = e e d = e d () V = [ e + jω + jω ]. () Sice is much greaer ha, he expoeial erm i equaio () is srogly damped ad V = = () + jω + jω 5

6 V = + ω. (3) hese are he rms values of volage ha we would expec o measure o he specrum aalyzer. he phase agle of each harmoic compoe ca be deermied from he real ad imagiary compoes i equaio (). Experime: Equipme Lis hp 358A Specrum Aalyser Screwdriver Schoky Pulse Circui Miibox (Labeled Schoky) Solderless Wirig Fixure 47 mh Iducor (I drawer labeled Waveform Aalysis Experime) 3 Assored Capaciors Procedure he objecive of his experime is o familiarize you wih he use of a specrum aalyzer o deermie he properies of periodic sigals. For his purpose perform he ampliude calibraio procedure idicaed i paragraph 3- of he HP 358A Operaig Maual (his is also reproduced i Appedix I) usig he Hz BANDWIDH ad he LOG db AMPLIUDE MODE. Whe his is compleed perform he familiarizaio exercise idicaed i paragraph 3-5 wih he khz ieral calibraio sigal. (a) Periodic Waveforms. Keepig he calibraio badwidh he same, examie he ampliudes ad frequecies of he firs harmoics of square, riagular, ad sie waves from he HP 33A fucio geeraor wih oupu se o High Z. Apply a khz, V peak-o-peak, square wave o he specrum aalyzer ad measure he ampliude ad frequecy of harmoics hrough usig boh LIN ad db modes. Your primary ieres here is he relaive ampliude of each higher harmoic ( hrough ) o he fudameal o a liear ad a log scale. Repea hese measuremes for riagular ad sie waves of he same frequecy ad ampliude. 6

7 (b) Soliary Pulse rai. Coec he Schoky Pulse Circui Miibox i he circui show i Figure. Drive i wih a V peak o peak, 5 Hz, square wave from he fucio geeraor ad observe he shape of he square wave from he fucio geeraor ad pulses from he Schoky Pulse Circui o he oscilloscope per Figure. Record he ampliude ad ime decay of he pulses. Make sure he resoluio badwidh of he specrum aalyzer is appropriae, ad he measure he ampliude i db of harmoics hrough for hese pulses. o scope o scope HP33A.47 µf N58.8 kω HP358 o scope A Figure - Schoky Pulse Circui (c) Bode Plo. Followig Figure 3-6B ad paragraph 3-3 of he hp 358A Operaig Maual examie ad record he frequecy respose up o 5 khz of he circui show i Figure above. For C use capaciace values of. µf,. µf, ad pf ad record he resoa frequecy for each value. Use he db AMPLIUDE MODE ad he REPEIIVE SWEEP MODE. HP 358A rackig Oscillaor Oupu 47 mh C HP 358A Ipu Figure - Bode Plo Circui Repor (a) For he square wave daa calculae he relaive ampliudes of harmoics hrough o he fudameal frequecy i boh liear ad log form. Calculae he heoreical values expeced from a Fourier aalysis of he square wave ad compare hese o he experimeal daa i a able. Discuss ad explai ay differeces bewee heory ad experime. Repea his process for he riagular ad sie waves usig a separae able for each wave form. (b) For he rai of expoeial pulses explicily derive a expressio for he expeced specrum showig all seps i he derivaio. Wih his equaio calculae he expeced relaive ampliudes of all harmoics ad compare hese values o he experimeal daa i a able. Discuss ad explai ay differeces bewee heory ad experime. 7

8 (c ) Prese o he same graph dimesioed skeches of he hree Bode plos. Explai qualiaively he sigifica feaures ad differeces amog he plos. Compare he experimeal resoa frequecies o hose expeced heoreically i a able ad discuss ay differeces. Suggesed Readig James W. Nilsso, Elecric Circuis, 6 h Ed (Addiso-Wesley, Readig, 993). Operaig ad Service Maual for HP 358A Specrum Aalyzer. Ala V. Oppeheim ad Ala S. Willsky. Sigals ad Sysems, (Preice Hall, Upper Saddle River, 996) A. Papoulis, Probabiliy, Radom Variables, ad Sochasic Processes, (McGraw-Hill, New York, 965). A. Papoulis, he Fourier Iegral ad Is Applicaios, (McGraw-Hill, New York, 96). L. P. Kadaoff, Roads o Chaos Physics oday, (December 983), pp R. W. Rollis ad E. R. Hu, Exacly solvable Model of a Physical Sysem Exhibiig Uiversal Chaoic Behavior, Phys. Rev. Le. 49, (98). 8