ELECTROMAGNETIC COMPATIBILITY HANDBOOK 1. Chapter 12: Spectra of Periodic and Aperiodic Signals

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1 ELECTOMAGNETIC COMPATIBILITY HANDBOOK Chapr : Spcra of Priodic ad Apriodic Sigals. Drmi whhr ach of h followig fucios ar priodic. If hy ar priodic, provid hir fudamal frqucy ad priod. a) x 4cos( 5 ) si( 0) b) p 4cos 5 si 0 7 b) k 3cos( π) + 6si( 0π ) 7 cos( 5π + 45 ) π j 4cos 5 si si 30π + 6si 30π 7 cos 35π + 5 cos 0π + 85 c) d) ) m 3cos 3 ( 30π ) + 7 cos( 60π + 05 ) si( 5π 85 ) + f) j0 3cos 0 w 5 + g) 3si ( 4 ) + cos( ) p j0. Drmi h fudamal frqucy i Hz of h wavform rprsd by h followig Fourir sris. α + jπ 6 3π 4 + jπ 4 j 3.3 Drmi h (o- or sigl-sidd) ampliud of h 3 rd harmoic of h wavform rprsd by h followig Fourir sris. ( ) 3A π π π cos si + 4 ( ) T T.4C For h priodic fucio, wih a las o discoiuiy pr priod, giv i h Fourir sris abl i his chapr (h spcific fucio providd by h isrucor), plo N 5, 0, 00, ad 500 rms of h sris. Vrify ha h maximum ovrshoo is abou 9% of h high of ach discoiuiy for ach N..5C For h priodic fucio giv i h Fourir sris abl i his chapr (h spcific fucio providd by h isrucor), plo h prc rgy coaid wihi h firs N rms of h sris vrsus N for N o 500 i rm icrms. Th, drmi h valu of N corrspodig o abou 90% prc rgy (ovr o priod)..6c For h priodic fucio giv i h Fourir sris abl i his chapr (h spcific fucio providd by h isrucor), vrify via h igral rlaioships, all forms of h sris providd i h abl. Do o us ay Copyrigh 00 by Kh L. Kaisr, Vrsio /0/04

2 ELECTOMAGNETIC COMPATIBILITY HANDBOOK symmry argums o rduc h umbr or complxiy of h igraios. Also, if spcific dc, fudamal, ad harmoic rms ar giv, vrify ha hy ar corrc. Fially, if h fucio has ay symmry, vrify ha all of propris (.g., v harmoics ar zro) rlad o h symmry codiio ar saisfid..7 For h priodic fucio giv i h Fourir sris abl i his chapr (h spcific fucio providd by h isrucor), chck h xprssio agais aohr sris giv i h abl by usig h dc shif, liariy, im rvrsal, im shifig, im diffriaio, im igraio, fucio muliplicaio propris or ay combiaio of hs propris. Do o us h dfiiio for h Fourir sris as a chck..8 Usig h covrsio rlaioships, vrify ach of h complx cofficis i Tabl usig h rigoomric cofficis. Th, vrify ach of h rigoomric cofficis usig h complx cofficis. Tabl a b F F j 0.488j j 0.0j j j.9 Usig h covrsio rlaioships, vrify ach of h complx cofficis i Tabl usig h rigoomric cofficis. Th, vrify ach of h rigoomric cofficis usig h complx cofficis. Tabl a b F F j j j j For h priodic fucio giv i h Fourir sris abl i his chapr (h spcific fucio providd by h isrucor), drmi or vrify all forms of h sris (rigoomric, ampliud/phas, ad xpoial) usig h covrsio rlaioships. Do o us h dfiiios for h Fourir sris..c By usig h propris of boh liariy, combi wo priodic fucios (boh providd by h isrucor) giv i h sris abl. Provid a aalyical Copyrigh 00 by Kh L. Kaisr, Vrsio 09/4/05

3 ELECTOMAGNETIC COMPATIBILITY HANDBOOK 3 dscripio of his w priodic fucio ovr o compl priod. Plo his fucio usig 5 ad 0 rms..c For h fracioal rcifid cosi wav whr A, T µs, ad k 0., drmi, sarig from h Fourir sris dfiiios, boh h rigoomric ad xpoial Fourir sris for h dc rm ad h firs 5 ozro cofficis. Th, compar h magiud of h cofficis wih h xprssio providd i h abl. Fially, plo boh h xac wavform ad h rucad rigoomric ad xpoial sris ovr wo priods..3c For h cosi puls rai whr A, T ms, ad τ 0.3 ms, drmi, sarig from h Fourir sris dfiiios, boh h rigoomric ad xpoial Fourir sris for h dc rm ad h firs 5 ozro cofficis. Th, compar h magiud of h cofficis wih h xprssio providd i h abl. Fially, plo boh h xac wavform ad h rucad rigoomric ad xpoial sris ovr wo priods..4c For h cosi-squard puls rai whr A, T ms, ad τ 0.3 ms, drmi, sarig from h Fourir sris dfiiios, boh h rigoomric ad xpoial Fourir sris for h dc rm ad h firs 5 ozro cofficis. Th, compar h magiud of h cofficis wih h xprssio providd i h abl. Fially, plo boh h xac wavform ad h rucad rigoomric ad xpoial sris ovr wo priods..5ec For h filr giv i Figur, plo boh h oupu rspos i h im domai ad h frqucy domai (i.., ampliud ad phas spcrums) for a priodic ipu volag providd by your isighful isrucor. Thr diffr valus of should b usd. Afr slcig a rasoabl valu for C, h hr valus for should b such ha () h lows cuoff frqucy is o-half of h fudamal frqucy, () midway bw h fudamal ad h firs harmoic frqucy, ad (3) ims h firs harmoic frqucy. Th, comm o h im rsposs basd o h spd ad smoohss of h oupu. C Figur.6EC pa Problm.5 for h circui giv i Figur. C C Figur Copyrigh 00 by Kh L. Kaisr, Vrsio 09/4/05

4 4 ELECTOMAGNETIC COMPATIBILITY HANDBOOK.7 By comparig o spcifid Fourir rasform pair (providd by h isrucor) wih wo ohr rasform pairs providd i h abl i his chapr, provid wo diffr parial chcks of h Fourir rasform pair. O or mor of h rasform propris giv i h Fourir rasform propris abl mus b usd. Ths parial chcks may o cosis of mrly muliplyig h fucio or rasform by a cosa, sig a variabl o zro, o ifiiy, or o aohr cosa, or spliig or combiig h fucio or rasform (.g., liariy). Do o us h dfiiio for h Fourir rasform or h ivrs rasform as a chck..8 For h Laplac rasform pair umbr # giv i h Laplac Trasform abl i his chapr (# providd by h isrucor), drmi whhr all of h pols of h Laplac rasform hav gaiv ral pars. If so, drmi hrough s jω subsiuio, h corrspodig Fourir rasform..9 For h wo im-domai fucios providd i h Laplac rasform abl i his book giv as # ad # (boh fucios # s providd by h isrucor), drmi whhr all of h pols of boh corrspodig Laplac rasforms hav gaiv ral pars. If boh of h rasforms saisfy his criria, skch h im-domai fucios s() ad k(): # + # ( ) ( ) ( ) ( ) + s f u f u k f u f u # # Th, drmi h Fourir rasforms for boh of hs fucios usig h s jω subsiuio approach (ad h im-rvrsal propry)..0 Workig i h im domai, vrify ha h oal rgy of ach of h giv sigals is qual o a: + 3a < < 0 3a a < < a r x + 0 < < 3a 0 lswhr 3a 0 lswhr 3π 8a 8a + < < < > a cos 0 8a 3 3 z y a 0 lswhr 0. Vrify h Bod magiud plo providd i his chapr for ach of h followig rasforms icludig all slops ad cuoff frqucis. Also, wha is h magiud (i db) of h rasform for frqucis much lss ha h lows cuoff frqucy? Copyrigh 00 by Kh L. Kaisr, Vrsio 09/4/05

5 ELECTOMAGNETIC COMPATIBILITY HANDBOOK 5 a) ( ω) b) X ( ω ) c) Y ( ω ) si a ω 3aω si 3a 3aω 8aω 9π si 3 ω ω 9π 64a ω 4a + 4a ω d) Z ( ω ).EC Compar h ampliud spcrums of a rcagular puls ad a Gaussia puls. Assum ha h rgy ad h maximum ampliud of boh sigals ar abou h sam. Drmi ad plo boh h approxima sraigh-li Bod magiud plos ad h xac plos, ad was do i his chapr, for boh of hs sigals. Th, drmi h prc rgy vrsus ω o h sam s of axs for hs sigals for frqucis ragig from /0 o 0 of h cuoff frqucy of h rcagular puls..3c Th rsuls giv i his chapr for h rgy wihi h scod cuoff frqucy for a rapzoidal wavform ar cosrvaiv. Isad of approximaig h siusoidal fucio as i h book, us a umrical program πτ Hz. r o drmi h fracioal amou of rgy coaid from 0 o Th fracioal amou of rgy should b plod vrsus h ris im for a fixd puls widh. Allow h ris im o vary from o o-hudrdh of h puls widh o is maximum possibl valu. Wha rlaioship bw h puls widh ad h ris im is rquird so ha 90% of h rgy is coaid wihi h scod corr frqucy?.4c Usig a umrical program, plo h magiud spcrum (i db) of a apriodic rapzoidal wavform wh h ris im is boh qual o ad o qual o h fall im. Slc a rasoabl ris im, τ r, fall im, τ f, ad puls widh, τ. Th brak frqucis should b clarly vid. Is hr a simpl rlaioship for h brak frqucis wh h ris im is o qual o h fall im?.5c If I o ka, η 0.7,, τ 0. µs, ad τ 0. µs, drmi h pak curr, maximum curr drivaiv, ad oal charg rasfr if a lighig puls is modld usig h Hidlr fucio: Copyrigh 00 by Kh L. Kaisr, Vrsio 09/4/05

6 6 ELECTOMAGNETIC COMPATIBILITY HANDBOOK I τ η + τ o τ I.6S For, drmi or loca h Fourir rasform of h Hidlr fucio: I τ η + τ o τ I Skch h Bod magiud plo of is ampliud spcrum. Compar his spcrum wih h doubl-xpoial s spcrum. [akov].7c Usig h plos giv i his chapr for h doubl-xpoial puls, drmi h valus for α ad β o modl a lighig srok wih a ris im of 00 s ad puls widh of 60 µs. Th, adjus C so ha h maximum ampliud of h srok is 40 ka. Usig hs valus, plo h rsula doubl-xpoial fucio..8 Skch wo diffr impuls im fucios ha hav approximaly h sam 50% dlay ims bu clarly diffr impuls rsposs. Th, skch wo similar impuls rsposs ha hav clarly diffr 50% dlay ims..9 Skch wo diffr impuls im fucios ha hav approximaly h sam croid dlay ims bu clarly diffr impuls rsposs. Th, skch wo similar impuls rsposs ha hav clarly diffr croid dlay ims..30 Skch wo diffr impuls im fucios ha hav approximaly h sam 0-90% ris ims bu clarly diffr impuls rsposs. Th, skch wo similar impuls rsposs ha hav clarly diffr 0-90% ris ims..3 Skch wo diffr impuls im fucios ha hav approximaly h sam maximum slopd-basd ris ims bu clarly diffr impuls rsposs. Th, skch wo similar impuls rsposs ha hav clarly diffr maximum slop-basd ris ims..3 Skch wo diffr impuls im fucios ha hav approximaly h sam sadard dviaio ris ims bu diffr impuls rsposs. Th, skch wo similar impuls rsposs ha hav diffr sadard-dviaio ris ims..33 Vrify ha h avrag im dlay for h rcagular puls wavform is giv by h xprssio x u u( a) Copyrigh 00 by Kh L. Kaisr, Vrsio 09/4/05

7 ELECTOMAGNETIC COMPATIBILITY HANDBOOK 7 ( τ ) ( τ + ) w w a d whr w() is h oupu rspos o h low pass sysm wih a sady-sa valu of o; ha is, h avrag im dlay ca b drmid by xamiig h oupu rspos ad drmiig h im whr h oupu rspos is rpad a scods lar. [Blichikoff].34EC Drmi h 0-90% ris im of a L low-pass filr o a ramp ipu sigal wih a ris im of abou τ r. Compar o h prfc sp ipu rspos..35ec Drmi h 0-90% ris im of a C low-pass filr o a ramp ipu sigal wih a ris im of abou τ r. Compar o h prfc sp ipu rspos..36c For h Laplac rasform pair umbr # giv i h Laplac rasform abl i his book (# providd by h isrucor), drmi h croid dlay im ad h sadard dviaio usig h sris xpasio of h Laplac rasform. Th, rwri h rasform so ha i is i h form H s K d as as as m b s bs bms whr K is a cosa. As a chck o h dlay ad ris im rsuls obaid from h sris xpasio, drmi h dlay im ad h ris im from h xprssios τ d b a, τ r π b a + a b Assumig h giv rasform pair rprss h Laplac rasform of a impuls rspos, is h sp rspos moooic?.37c Sarig from hir dfiiios ad workig wih h im-domai impuls ad sp rspos fucios, vrify ach of h dlay ad ris ims lisd i h abls i his chapr for h a) sigl-pol xpoial puls b) doubl-pol criically dampd puls c) causal Gaussia puls d) ocausal Gaussia puls.38c Sarig from h dfiiios, vrify ach of h badwidhs lisd i h abl i his chapr for h a) sigl-pol xpoial puls, H ( s) ( s + a) b) doubl-pol criically dampd puls, c) ocausal Gaussia puls, ( ω) H s s + a a ω π H a Copyrigh 00 by Kh L. Kaisr, Vrsio 09/4/05

8 8 ELECTOMAGNETIC COMPATIBILITY HANDBOOK d) samplig fucio, H si ( T ).39 Usig h slop-basd ris im ω ω ω τ r h d h ad a modifid vrsio of h quival badwidh o BW q H H ( ω) ( 0) dω drmi whhr h im-badwidh produc is always π. [Solima].40 Vrify ach of h ris im-badwidh producs rms giv i h marix abl i his chapr for h sigl-pol xpoial puls. Which of h producs is h smalls? Which of h producs is h largs? Qualiaivly xplai, wihou prformig ay calculaios or wihou rfrc o his abl, which of hs producs should b h smalls ad largs..4 Vrify ach of h ris im-badwidh producs rms giv i h marix abl i his chapr for h Gaussia puls. Which of h producs is h smalls? Which of h producs is h largs? Qualiaivly xplai, wihou prformig ay calculaios or wihou rfrc o his abl, which of hs producs should b h smalls ad largs. Copyrigh 00 by Kh L. Kaisr, Vrsio 09/4/05