Transmission Lines. Introduction to Transmission Lines (T.L.) Exercise Common Transmission Lines. Transmission Lines (TL) Don t worry about the

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3 A 40m g T ha g 15 rm, 30j60 Ω, ad 5 j48 rm.

I dtai, i I tach yu g 30j60 γα j β D t wrry

11 63 Tramii i I. T paramtr (R,,C, G II.

1 Tramii i Itrducti t Tramii i (T.. Hi frqucy r hi pwr rquir T.. TEM wav prpagat thru T.. W wi dvp T.. thry t hw wav prpagat thru thm Dr. Sadra CruzP ECE Dpt. UPRM Sm t b i para, but th ar t qua! Exrci 11.3 A 40m g T ha g 15 rm, 30j60 Ω, ad 5 j48 rm. If th i i matchd t th ad ad th gratr, fid: th iput impdac i, th digd currt ad tag, th prpagati ctat γ. I dtai, i I tach yu g 30j60 γα j β D t wrry abut th abut vig thi typ g f prbm prtty. Awr: 40 m i 30 j60ω, A, rm, γ j m γ Tramii i I. T paramtr (R,,C, G II. T Equati (fr ad I III. 3 Ccpt: I. Iput Impdac, II. Rfcti Cfficit ad III. Charactritic Impdac I. SWR, Pwr. Smith Chart I. I. Appicati Quartrwav trafrmr, Sttd i, Sig tub Micrtrip Tramii i (T fr iuati utr cductr T hav cductr i para with a dictric paratig thm Thy tramit TEM wav iid th i Cmm Tramii i Twwir (ribb twiad Micrtrip Caxia Stripi (Tripat

Othr T (highr rdr [Chaptr 1] Fid iid th T prprtia t E, I prprtia t H I H E d d Ditributd paramtr Th paramtr

R hm/mtr Hri/ m C Farad/m G mh/m At high frquci w r daig with wavgth cmparab t th iz f th circuit.

th matria prprti. R,, G, ad C ca b cacuatd uig fudamta EMAG tchiqu.

2 Othr T (highr rdr [Chaptr 1] Fid iid th T prprtia t E, I prprtia t H I H E d d Ditributd paramtr Th paramtr that charactriz th T ar giv i trm f pr gth. R hm/mtr Hri/ m C Farad/m G mh/m At high frquci w r daig with wavgth cmparab t th iz f th circuit. λ λ 60Hz GHz c / 60 5,000km c / 000,000,000 15cm Cmm Tramii i R,, G, ad C dpd th particuar tramii i tructur ad th matria prprti. R,, G, ad C ca b cacuatd uig fudamta EMAG tchiqu. Paramtr TwWir i Caxia i ParaPat i R G C 1 πaσ cd δ µ π ach D a πσ di ( ach D/ ( a πε ach D/ ( a ( 1 1 πσ cd δ a 1 b µ π b a πσ di b /a ( πε b /a ( wσ cd δ Uit Ω/m µ d w H /m σ di w d S /m ε w F /m d T rprtati Ditributd i paramtr Uig K:

3 Ditributd paramtr Takig th imit a Δz td t 0 ad t ( z, t I ( z, t I RI RI ( z, t z z t t Simiary, appyig KC t th mai d giv I ( z, t ( z, t I G G ( z, t C C z z t t Wav quati Uig phar ( z, t R[ ( z I( z, t R[ I ( z Th tw xpri rduc t z ( R jω I I z z ( G jωc R jω t jωt ] ] γ 0 γ ( jω( G jωc Wav Equati fr vtag Tramii i I. T paramtr (R,,C, G II. T Equati (fr ad I III. 3 Ccpt: I. Iput Impdac, II. Rfcti Cfficit ad III. Charactritic Impdac I. SWR, Pwr. Smith Chart I. I. Appicati Quartrwav trafrmr, Sttd i, Sig tub Micrtrip T Equati Nt that th ar th wav q. fr vtag ad currt iid th i. d dz γ d I 0 dz γ I 0 Th prpagati ctat i γ ad th wavgth ad vcity ar γ α β j (R jω(g jωc λ π β u ω β f λ Wav mv thrugh i Th gra uti i γz γz I tim dmai i (z,t R[ (z jω t ] αz c(ωt βz αz c(ωt βz z Wav mv thrugh i Fr Currt I I γz I γz Simiary fr timdmai, I I(z,t I αz c(ωt βz I αz c(ωt βz z

4 Tramii i I. T paramtr (R,,C, G II. T Equati (fr ad I III. 3 Ccpt: I. Iput Impdac, II. Rfcti Cfficit ad III. Charactritic Impdac I. SWR, Pwr. Smith Chart I. I. Appicati Quartrwav trafrmr, Sttd i, Sig tub Micrtrip W wi dfi 3 ccpt: Charactritic impdac, Rfcti Cfficit, Γ Iput Impdac, i W wi dfi 3 ccpt: Charactritic impdac, Rfcti Cfficit, Γ Iput Impdac, i Charactritic Impdac f a i, I th rati f pitivy travig vtag wav t currt wav at ay pit th i d(z (R jωi(z dz ubtitutig I (z γz I(z I γz ( γ γz (R jωi γz R jω R jω γ G jωc R jx I z Charactritic Impdac, I I R jω G jωc Diffrt ca f T Fid th charactritic impdac ad prpagati ctat fr ach: Tramii i I I γz I γz I γz γz Ditrti y Tramii i Tramii i

Charactritic Impdac R jω G j ω C R C R C R G P.E. 11. A tph i ha R30 Ω/km, 100 mh/km, G0, ad C 0µF/km. At 1kHz, FIND: th charactritic impdac f th i, th prpagati ctat, th pha vcity. I it ditrti?

5 i (R0G Hav prfct cductr ad a prfct dictric mdium btw thm. Prpagati: α 0, γ, β ω C Ditrti i (R/ G/C I i which th attuati i idpdt frqucy. γ α Prpagati: α RG β ω C Wavgth & city: u ω β 1 C f λ, λ π β city: u ω β 1 C f λ Impdac R C X 0 Impdac X 0 R C R G Summary Gra (y Ditrti RC G γ α Prpagati Ctat γ ( R jω( G jωc γ 0 jω C γ RG jω C Charactritic Impdac R jω G j ω C R C R C R G P.E. 11. A tph i ha R30 Ω/km, 100 mh/km, G0, ad C 0µF/km. At 1kHz, FIND: th charactritic impdac f th i, th prpagati ctat, th pha vcity. I it ditrti? Suti: R jω G jωc γ ( R jω ( G jωc ( 30 jπ (100( 0 jπ j8.88 / km N 30 jπ (1k(100m 0 jπ (1k Ω u ω β 707 km / W wi dfi 3 ccpt: Charactritic impdac, Rfcti Cfficit, Γ Iput Impdac, i W wi dfi 3 ccpt: Charactritic impdac, Rfcti Cfficit, Γ Iput Impdac, i

impdac ad th charactritic impdac. Rca fr th ca, Th z z ( z ( Γ I ( z Γ z z ( Γ γ 0 jω C Dfiiti: Matchd i Ma that Γ Thrfr thr ar rfcti!

6 Rfcti cfficit at th ad, Γ ad i uuay tak at z0 ad gratr at z (z γz γz ( z Γ γ z γz ( Γ z z0 Fr a T trmiatd with a ad Th, Simiary, ( z I ( z γ z γz ( Γ γ z γz ( Γ Th impdac aywhr ag th i i giv by γz γz ( Γ γz γz ( Γ ( z ( z I ( z Th impdac at th ad d,, i giv by ( 1 Γ ( 0 ( 1 Γ Trmiatd, T Svig fr Γ Ccui: Th rfcti cfficit i a fucti f th ad impdac ad th charactritic impdac. Rca fr th ca, Th z z ( z ( Γ I ( z Γ z z ( Γ γ 0 jω C Dfiiti: Matchd i Ma that Γ Thrfr thr ar rfcti! Γ 0 What happ wh yu cct th wrg T t a pakr? Trmiatd, T Uig cvti th crdiat ytm, z, at iput. z ( d I( d z d d d ( Γ d d d ( Γ Rwritig th xpri fr vtag ad currt, w hav ( I( ( Γ ( j β Γ Rarragig, ( I( ( 1 Γ ( j β 1 Γ

7 tag aywhr th i Rca, ( ( Γ ( d z I( d z d d d ( Γ d d d ( Γ λ / 4 W wi dfi 3 ccpt: Charactritic impdac, Rfcti Cfficit, Γ Iput Impdac, i ( (0 ( ( λ / 4 jπ / jπ / (0 Iput vtag dig d, ad tag rcivig d tag quatrwav frm matchdad Impdac ( i Th impdac aywhr ag th i i giv by ( ( 1 Γ ( I ( j β ( 1 Γ Th rfcti cfficit at ay pit ag th i: Γ( Γ Γ jθγ Th, th impdac ca b writt a. ( 1 Γ( ( ( 1 Γ( Aftr m agbra, a atrativ xpri fr th impdac i giv by ( j ta β ( i ( j ta β Ccui: Th ad impdac i trafrmd a w mv away frm th ad. Impdac (y i Th impdac aywhr ag th i i giv by γ ( ( 1 Γ ( γ I( ( 1 Γ Th rfcti cfficit ca b mdifid a fw Γ( Γ Γ ( γ α Th, th impdac ca b writt a ( 1 Γ( ( ( 1 Γ( Aftr m agbra, a atrativ xpri fr th impdac i giv by ( tah γ ( i ( tah γ Ccui: i y i, w d up with th hyprbic tagt. Examp: Matchd Ca A T ha g 10 rm, 50Ω If th i i matchd t th ad ad th gratr, fid: th iput impdac i, th digd tag g g γj β Awr: g 0 Γ 0 tagdividr i 50Ω 5 0 Γ 0 ( i j ta β ( j ta β Examp: Matchd Ca A λ/8 g T ha 5 < 30, 50Ω If th i i matchd t th ad, fid: th iput impdac i, th digd tag, th prpagati ctat γ. g g γj β Awr: 0 i 50Ω Γ ( 0 (0 0, Sv fr : 5 30 ( ( Γ Γ ( i j ta β ( j ta β ( λ / jπ /4 jπ /4 ( ( jπ /4 ( 5 75

8 Exrci 1: Nt Matchd A cm T ha 10 j30, g 60 Ω, 50 Ω ad 100Ω, λ10cm. Fid: th iput impdac i, th digd tag, Exrci : uig frmua A cm T ha g 10 rm, g 60 Ω, 100j80 Ω ad 40Ω, λ10cm. fid: th iput impdac i, th digd tag, g g γj β ( Γ U thi quati at ad ad at iput, fid Fid Fid i (at iput ( Γ ( i j ta β ( j ta β g g γj β π (100 j80 j40 ta 5 i (cm 40 5 tag Dividr: ( 40 j(100 j80 ta π i gi i rad i g 1. j1. 17 Ω Examp 3: Nt matchd t ad A gratr with 10 rm ad R g 50, i cctd t a 75Ω ad thru a 0.8λ, 50Ω i. Fid Γ i j ta β j ta β Γ 0. PwrDividr rad ( 0 g g 50Ω 75,.8λ ( ( ( ( Γ γj β i 35. j8.75ω Exrci 11.3: Matchd Ca A 40m g T ha g 15 rm, 30j60 Ω, ad 5 j48 rm. If th i i matchd t th ad ad th gratr, fid: th iput impdac i, th digd currt ad tag, th prpagati ctat γ. g g Awr: i i 30j60 γα j β 30 j60ω, 7.5 0, rm γ 40 I i 40 m A, γ j m Tramii i I. T paramtr (R,,C, G II. T Equati (fr ad I III. 3 Ccpt: I. Iput Impdac, II. Rfcti Cfficit ad III. Charactritic Impdac I. SWR, Pwr. Smith Chart I. I. Appicati Quartrwav trafrmr, Sttd i, Sig tub Micrtrip Pwr Th avrag iput pwr at a ditac frm th ad i giv by P 1 R[ ( ] * ( av I which ca b rducd t Pav ( 1 Γ Th firt trm i th icidt pwr ad th cd i th rfctd pwr. Maximum pwr i divrd t ad if Γ0

9 SWR r SWR r Whvr thr i a rfctd wav, a tadig wav wi frm ut f th cmbiati f icidt ad rfctd wav. Th (tag Stadig Wav Rati SWR (r SWR i dfid a SWR max mi 1 Γ 1 Γ I I max mi Summary Iput Impdac Rfcti Cf SWR 1 Γ 1 Γ ( i j ta β j ta β Γ ( Thr (3 cmm Ca f iad cmbiati: Shrtd i ( 0 i 0 j ta β jb Opcircuitd i ( i j ct β Matchd i ( i Γ 1, Γ 0, 1 Γ 1, Stadig Wav Shrt ( 0 j ta β, Γ 1 i, S ubtitutig i (z ( z [ ( 1 ] ( z ( j i β ( z i ( β π ( z i λ tag maxima z λ λ/ λ/4 *tag miima ccur at am pac that impdac ha a miimum th i (z Stadig Wav Op( i j ct β, Γ 1, (z [ (1 ] S ubtitutig i (z (z (cβ (z (z c( β c π λ tag miima z λ λ/ λ/4 (z Stadig Wav Matchd ( S ubtitutig i (z (z [ (0 ] (z i, Γ 0, 1 (z z λ λ/ λ/4 (z (z

Java appt http://www.amagawa.cm/ tramii.htm http://phyic.uak.ca/~hir/p5/ http://www.ducatrcrr.cm/ idx.cgi?content_id483 Tramii i I. T paramtr (R,,C, G II.

Ud i hitch quipmt fr dig ad ttig f micrwav circuit O tur (360 arud th SC t λ/ Ntwrk Aayzr What ca b th cr? Smith Chart U th rfcti cfficit ra ad imagiary part.

10 Java appt tramii.htm idx.cgi?content_id483 Tramii i I. T paramtr (R,,C, G II. T Equati (fr ad I III. 3 Ccpt: I. Iput Impdac, II. Rfcti Cfficit ad III. Charactritic Impdac I. SWR, Pwr. Smith Chart I. I. Appicati Quartrwav trafrmr, Sttd i, Sig tub Micrtrip Th Smith Chart Smith Chart Cmmy ud a graphica rprtati f a T. Ud i hitch quipmt fr dig ad ttig f micrwav circuit O tur (360 arud th SC t λ/ Ntwrk Aayzr What ca b th cr? Smith Chart U th rfcti cfficit ra ad imagiary part. ( Γ Γ θγ Γr jγi ( Γ i ad dfi th rmaizd : Γ Γ r z z 1 Γ z 1 ( 1 Γ ( 1 Γ 1 Γr jγi z r jx 1 Γ jγ r jx r i

11 Nw ratig t z r jx Aftr m agbra, w btai tw q. Γ r r Γ 1 i 1 r 1 r [ Γ r 1 ] Γ 1 i x 1 x Simiar t gra quati f a circ f radiu a, ctr at (h,k (x h (y k a Circ f r Circ f x Examp f circ f r ad x r 1 Ctr,0 Radiu 1 r 1 r Ctr Ctr r 1 1, 1 x r,0 Radiu Radiu Circ f r 1 x 1 1 r 1 1 Ctr 1, Radiu x x Circ f x Examp Fid z rjx giv Γ A Fid Γ Giv Examp f circ f r ad x r 1 Ctr,0 Radiu 1 r 1 r Circ f r Circ f x Fid SWR th SC Numricay r th axi f Γ r i th SC Prf: Γ z z 1 r 1 (wh z r 1 r 1 1 but th Γ Γr j0 1 Mvig th T th SmithC A T i rprtd a a circ f ctat radiu, Γ, r ctat Γ( Γ Γ jθγ Mvig ag th i frm th ad tward th gratr, th pha dcra, thrfr, i th SC qua t mv cckwiy. T gratr

12 Otur th Smith Chart O tur (360 arud th SC t λ/ bcau i th frmua bw, if yu ubtitut gth fr hafwavgth, th pha chag by π, which i tur. Γ( Γ Fid th pit i th SC whr Γ1,1, j, j, 0, 0.5 What i r ad x fr ach ca? Fu fact : Admittac i th SC Th admittac, yy /Y whr Y 1/, ca b fud by mvig ½ tur (λ/4 th T circ π λ β π λ 4 z( 0 1 Γ z( λ / 4 1 Γ jπ jπ 1 y( 0 Y 1 ( 1 Γ ( 1 Γ 1 Γ( 1 1 Γ 1 Γ( 1 1 Γ 1 Γ 1 Γ ( 1 Γ j0 1 Γ 1 Γ j0 ( 1 Γ 1 Γ 1 Γ 1 Γ 1 Γ j0 j0 1 Γ 1 Γ max ad mi th SmithC Th Γ r axi, whr r > 0 crrpd t max Th Γ r axi, whr r < 0 crrpd t mi mi max (Maximum impdac Exrci: uig S.C. A cm T ha g 10 rm, g 60 Ω, 100j80 Ω ad 40Ω, λ10cm. fid: th iput impdac i, th digd tag, g γj β g cm 100 j80 z.5 j 40 λ Γ λ. Γ( cm ad i S.C. tag Dividr: Mv.λ ad arriv t.4179λ gi Rad z i. 3 j.55 i rad y. 76 j1. 4Ω i g 1 j Ω i i Exrci: ct.uig S.C. z A cm T ha g 10 rm, g 60 Ω, 100j80 Ω ad 40Ω, λ10cm. fid: th iput impdac i, th digd tag, z i 0.λ g g γj β z.5 j λ. λ cm Ditac frm th ad (.179λ t th art miimum & max Mv t hrizta axi tward th gratr ad arriv t.5λ ( max ad t.5λ fr th mi. Ditac t mi λ Ditac t t vtag maximum i.8λ.5λ.48 S drawig Γ

13 Exrci : uig frmua A cm T ha g 10 rm, g 60 Ω, 100j80 Ω ad 40Ω, λ10cm. fid: th iput impdac i, th digd tag, g g γj β cm Γ π (100 j80 j40 ta 5 i (cm 40 ( 40 j(100 j80 ta π Γ(cm Γ 5 tag Dividr: i 1. j1. 17 Ω Γ gi i rad i g Athr xamp: z A 6cm T i cctd t ad 36j44 Ω ad 100Ω, λ10cm. fid: th iput impdac i g γj β g 6cm. 36 j.44.6λ 5(.5λ. λ λ 1 ad i S.C. Ditac t firt max : Mv.1λ ad arriv t.57λ (.07λ Rad z i. 31 j.16 mi 0.5λ.47λ max 0.08λ.5λ. 78 i 31 j16 Ω Γ λ λ Exrci 11.4 A 70 Ω i ha 1.6 ad at th ad θ Γ 300. If th i i 0.6λ g, btai Γ,, i ad th ditac f th firt miimum vtag frm th ad. Awr Γ Γ z 1.15 j.48 1 z 80.5 j33.6ω Th ad i catd at:.3338λ zi 0.68 j.5 Mv t.4338 λ ad draw i frm i 47.6 j17.5 Ω ctr t thi pac, th rad whr it cr yu T circ. Ditac t mi i thi ca, mi.5λ.3338λ ~ λ / 6 Java Appt : Smith Chart

Lectur 22. RF and Microwave Circuit Design Γ-Plane and Smith Chart Analysis. ECE 303 Fall 2005 Farhan Rana Cornell University

Lectur 22. RF and Microwave Circuit Design Γ-Plane and Smith Chart Analysis. ECE 303 Fall 2005 Farhan Rana Cornell University ctur RF ad Micrwav Circuit Dig -Pla ad Smith Chart Aalyi I thi lctur yu will lar: -pla ad Smith Chart Stub tuig Quartr-Wav trafrmr ECE 33 Fall 5 Farha Raa Crll Uivrity V V Impdac Trafrmati i Tramii i ω