8/1/18 Cours nstructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) 747 6958 E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 4 mpdanc Transformation and Paramtr Rlations mpdanc Ths Transformation nots may conta & Paramtrs copyrightd matrial obtad undr fair us ruls. Distribution of ths matrials is strictly prohibitd Slid 1 ctur Outl nput mpdanc, Paramtr Rlations Spcial Cass of Tratd Transmission s Shortd l ( = ) Opn circuit l ( = ) Matchd l ( = ) mpdanc Transformation & Paramtrs Slid 1
8/1/18 nput mpdanc, mpdanc Transformation & Paramtrs Slid 3 Problm Stup Gnrator Transmission oad g g, Gnrator g nput mpdanc Th put impdanc is th impdanc obsrvd by th gnrator. g Th put impdanc is NOT ncssarily th l s charactristic impdanc or th load impdanc. mpdanc Transformation & Paramtrs Slid 4
8/1/18 Animation of mpdanc Transformation m 4 nput impdanc vrts 15 j15 m nput impdanc rpats 1 j1 mpdanc Transformation & Paramtrs Slid 5 Drivation of nput mpdanc, (1 of ) Th rflction coffict at any pot from th load is Backward Wav Forward Wav This mans that from th prspctiv of th gnrator, th rflction gog to th transmission l will chang dpndg on th lngth of th transmission l. This can only happn of th put impdanc to th transmission l is changg. mpdanc Transformation & Paramtrs Slid 6 3
8/1/18 Drivation of nput mpdanc, ( of ) W df th impdanc of th l at position to b W prviously wrot () and () as Substitutg our xprssions for () and () givs t maks sns that th impdanc is not a function of voltag a lar systm. mpdanc Transformation & Paramtrs Slid 7 Sanity Chck: nput mpdanc at oad Th put impdanc at th load can b dtrd by sttg = our prvious quation. 1 1 1 1 1 Cancl from th quation. Multiply numrator and dnoator by. Cancl numrator and dnoator. W got th answr w wr xpctg! mpdanc Transformation & Paramtrs Slid 8 4
8/1/18 nput mpdanc at Th put impdanc at location is A Not About Sign: Backg away from th load, bcoms ngativ. Howvr, w dfd so stays positiv this quation and for quations that follow. mpdanc Transformation & Paramtrs Slid 9 mpdanc Transformation Formula (1 of ) Rcall that W can liat from th put impdanc quation by substitutg our xprssion for. cosh sh sh cosh cosh sh mpdanc Transformation & Paramtrs Slid 1 5
8/1/18 mpdanc Transformation Formula ( of ) Rcall that tanh sh cosh This lts us writ th put impdanc xprssion as sh tanh tanh sh cosh cosh sh cosh sh cosh mpdanc Transformation & Paramtrs Slid 11 nput Transformation for osslss Th losslss l has j Puttg ths valus to our impdanc transformation formula givs tanh j tanh j Rcognig that tanh(j) = jtan(), th xprssion for losslss ls bcoms j tan j tan mpdanc Transformation & Paramtrs Slid 1 6
8/1/18 nput mpdanc Rpats for osslss s For losslss ls, th tan function th impdanc transformation quation tlls us that th function is priodic and rpats. Th function rpats vry tgr multipl of. m m,, 3,, 1,,1,, 3,, Rcognig that = /, th abov xprssion lads to m Not: is th wavlngth th transmission l, not th fr spac wavlngth. This mans th put impdanc rpats for vry half wavlngth long th transmission l is. W will rvisit this whn w covr Smith charts, which will giv you a way to visuali th impdanc transformation phnomnon. mpdanc Transformation & Paramtrs Slid 13 Exampl: mpdanc Transformation (1 of 3) A transmission l with 5 charactristic impdanc is connctd to a 1 nf capacitor as th load. f th phas constant of th transmission l is = 6 m -1, what is th put impdanc of a 1 ch sction of l opratg at 4. GH? What quivalnt circuit would th sourc s? Transmission oad 5 1 nf 1 ch mpdanc Transformation & Paramtrs Slid 14 7
8/1/18 Exampl: mpdanc Transformation ( of 3) oss was not spcifid so w assum a losslss transmission l. Our impdanc transformation quation is thrfor j tan j tan Th variabls this quation ar 5 1.54 cm 1 m 6 m 1 ch 1.54 1 ch 1 cm 1 1 1 9 1 9 jc j fc j 4.1 s 11 F j.4 mpdanc Transformation & Paramtrs Slid 15 Exampl: mpdanc Transformation (3 of 3) Substitutg ths valus to th impdanc transformation quation givs j.4 j 5 tan 6.54 5 5 jj.4 tan 6.54 3 j1.71 Th put impdanc is purly imagary and positiv. Thus, th put impdanc looks lik an ductor to th gnrator. j q q j j f j1.71 j 4.1 s 3 9 1 8 4.41 H 4.4 nh mpdanc Transformation & Paramtrs Slid 16 8
8/1/18 Paramtr Rlations mpdanc Transformation & Paramtrs Slid 17,, & Trms of SWR and 1 1 SWR SWR1 SWR1 and 1 1 SWR SWR1 SWR1 mpdanc Transformation & Paramtrs Slid 18 9
8/1/18 Trms of SWR Th charactristic impdanc can b calculatd from and or and. Th put impdanc rpats as you back away from th load. W can calculat th imum and imum impdanc as SWR SWR mpdanc Transformation & Paramtrs Slid 19 Exampl (1 of 3) A 5 impdanc transmission l is connctd to an antnna with a 7 put impdanc. A sourc provids an put signal of 4 pak to pak. What is th rflction coffict at th antnna? 7 5.183 n this cas, th antnna is th load. 7 5 What fraction of th put powr is dlivrd to th antnna? R.183.35 Dspit th mismatch, almost all powr is still dlivrd to th antnna. This still T 1R1.35.9675 96.7% dos not man th antnna will radiat! What is th SWR on th l fdg th antnna? 1 1.183 SWR 1.44 1 1.183 SWR log SWR log 1.44 3.17 db db 1 1 mpdanc Transformation & Paramtrs Slid 1
8/1/18 Exampl ( of 3) What is th imum and imum voltag on th l? First, w nd to convrt voltag pak to pak p-p to voltag magnitud. p-p 4 1 Now w ar a position to calculat and. Whn w ar utilig high voltags, w want to 1 1 1.183 9.84 b sur will not caus arcg or any othr 1 1 1.183 14.16 brakdown problms. What is th imum and imum currnt on th l? 9.84.1967 A 5 At high powr, w want 14.16 to b sur will not.833 A caus hatg problms. 5 mpdanc Transformation & Paramtrs Slid 1 Exampl (3 of 3) What is th total rang of put impdancs a sourc could s? 9.84 34.7.833 A 14.16 7.1967 A 34.7 7 mpdanc Transformation & Paramtrs Slid 11
8/1/18 Spcial Cass of Tratd Transmission s mpdanc Transformation & Paramtrs Slid 3 Shortd, = Rflction from oad 1 and oltag Standg Wav Ratio SWR Thr xists som whr () =. and nput mpdanc j tanh lossy tan losslss Not 1: for th losslss l is purly imagary. This mans it is purly ractiv and no dissipation occurs th l. Th put impdanc altrnats btwn bg capacitiv and ductiv as you back away from th load. [ ] and [ ] short circuit opn circuit Not : Th shortd l bhavs much th sam way as th opn circuit l. W also obsrv that,short,opn mpdanc Transformation & Paramtrs Slid 4 1
8/1/18 Opn Circuit, = Rflction from oad 1 and oltag Standg Wav Ratio SWR Thr xists som whr () =. and nput mpdanc coth lossy j cot losslss Not 1: for th losslss l is purly imagary. This mans it is purly ractiv and no dissipation occurs th l. Th put impdanc altrnats btwn bg capacitiv and ductiv as you back away from th load. [ ] and [ ] short circuit opn circuit Not : Th opn circuit l bhavs much th sam way as th shortd l. W also obsrv that,short,opn mpdanc Transformation & Paramtrs Slid 5 Matchd, = Rflction from oad oltag Standg Wav Ratio SWR 1 bcaus = and and nput mpdanc [ ] and [ ] Not: For th matchd l, thr ar no rflctions and all of th powr is dlivrd to th load. mpdanc Transformation & Paramtrs Slid 6 13