1.1 Transmission line basic concepts: Introduction to narrow-band matching networks
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1 . Transmission lin basic concpts: ntroduction to narrow-band matching ntworks March Francsc Torrs, luís Pradll, Jorg Miranda oltag and currnt in th transmission lin For any losslss transmission lin: whr At = mpdanc: load = Priodicity of and : wavlngth n, n n n Rflction cofficint: load
2 Rflction cofficint in th transmission lin At any point of th transmission lin th impdanc is computd as: Modulus At any point of th transmission lin th rflction cofficint is: Phas inar ncrasing with + towards load Priodicity: half a wavlngth Constant in rad 8º dg n, n n n n Th transmission lin as impdanc transformr i, β Γ = Γ = Eampl: Comput th input impdanc i of a circuit formd by a transmission lin of lngth is λ/8 and loadd with = sc. i i i i 8
3 3 Standing wavs in th transmission lin i At any point whr th trm +Γ is ral, voltag and currnt ar ral an thir magnitud is ithr maimum or imum: rad ma ma ma ma ma rad 4 ma Standing wavs in th transmission lin ii At any point whr th voltag is maimum th impdanc is ral an maimum, if th voltag is imum th impdanc is ral and imum: ma ma ma ma ma ma ma Th voltag standing wav ratio SWR or S is dfind as, ma S S S S Matchd load: =, Γ=, SWR=
4 Standing wavs in th transmission lin iii Maimum and imum impdancs at in th transmission lin: λ/4 λ/4 ma ma = ma ma S ma S SWR is asy to masur and it is widly usd to spcify mismatch Powr in th transmission lin i At any point of th transmission lin th nt powr is computd as: * * * P o P Sinc th modulus of th rflction cofficint is constant in, th nt transmittd powr is constant at any and quals th powr dlivrd to th load: P P P P P P P P * a a a a r, Whr th powr associatd to th positiv incidnt and ngativ rflctd wavs is: P,, P 4
5 Powr in th transmission lin ii Eampl: P + = Watt in transmission lin of impdanc =5Ω 5,,, P P. 5W Rturn loss dfinition: P R P n this cas, R log, P dbw P dbw R db R 3 db, P dbw 3dB 3dBW Th standing wav ratio SWR in th transmission lin S 5.8 f w hav a mismatchd load Matching ntworks P P P A fraction of th incidnt powr P + is not dlivrd to th load A fraction of th incidnt powr rturns to th gnrator P P A matching ntwork must b Simpl passiv osslss, C, Transformr, transmission lin, wavguid, P + P - Γ P + P - = Γ i = i = Matching Ntwork f losslss: P P i P All powr is dlivrd to th load 5
6 Normalid impdancs and admittancs An impdanc can b normalid to a rfrnc impdanc ; Y Y Y P + P - n this cas, th rflction cofficint: Γ Y Y = P + P - = Matching Ntwork Γ i = i = Working with normalid impdancs is quivalnt to work with rfrnc transmission lins of = Eampl: losslss narrowband matching ntwork i =X = P + P - = Y B =B =4- i = n this cas, th rflction cofficint: X By qualling th ral parts. X Y Y =.+. B.. B By qualling th imaginary parts. B.3. B. B.5 X X.. B X 3 B 6
7 Eampl: losslss narrowband matching ntwork ii Solution Solution =- = = Y B =.3 =4- = Y B =-.5 =4- i = i = Somtims a shunt-sris solution dos not ist and a sris-shunt ntwork must b usd: = = i = i = Th Smith Chart mpdanc at any point of th transmission lin R, m, P.H. Smith, in 939, dvlopd a chart to rprsnt any impdanc as a function of its rlatd rflction cofficint ρ. This graphic tool is basd in th fact that ρ which allows to rprsnt all impdancs in a finit ara. Th Smith chart is currntly usd as a univrsal tool to rprsnt impdancs. P.H. SMTH 7
8 Rlation - ρ An impdanc can b normalid in rlation to a rfrnc impdanc o as r r i Mathmatically, this corrspond to a bilinar transformation which translats a circl in th impdanc domain into a circl in th rflction cofficint domain. r i r r Rlation - ρ Now, if ρ=ρ r +ρ i is substitutd in th prssion of th normalid impdanc, th quations that rlat th loci r and constants as a function of th componnts ρ r and ρ i ar obtaind. This is a st of circumfrncs in th compl domain : r r i ; r i r r r Constant rsistanc circl: CENTRE, RADUS r r, Constant ractanc circl: CENTRE RADUS i r=const i =const. r r r r 8
9 Th rflction cofficint in th compl domain =-l = Rational l= 36º l= 8º l= 9º = i 9º Towards load 4 8º l l º r 4 7º Towards gnrator Chart of impdancs =.5 = = = Circls of constant rsistanc r = r =.5 r = r = r = = Circls of constant ractanc = - = - = -.5 9
10 At a point placd in th transmission lin of =5 Ω w masur an impdanc + 5 Ω What is th valu of ρ at this point? ρ =.75 = φ = 6º r =.75 6º f 39º, what is its rlatd normalid impdanc? How dos φ = 9º this impdanc chang if th point is movd along th transmission lin? r =.8 = +.6 As w mov along a transmission lin, th modulus of th rflction cofficint is constant:.33 Th normalid impdanc varis as givn by this circl. ρ =.33
11 transformation: ρ 3 6º SWR = S standing wav ratio RET N OSS, db= = REF. COEFF. P = REF.COEFF, E OR = log ρ φ.75 6º rt S 7.6 db λ nput impdanc = 6 9 = 75 Ω l =.45 λ mín má mín.8 /S=.8 S=3.6 má mín má 3.6 má mín R má R mín 7 S=3.6
12 Admittanc R X Y G B X Bc Bc C C X C Y Y Y Y Rotation: 8º Y.7.7 Y Y YY Y Y.4.34 Y nput admitanc. λ = + 5 Y = 5 Ω l =. λ Y Y Y Y.3.7 Y λ
13 Short-circuitd lin. λ, Y s.c. l =. λ.73 Y.4 Opn circuit lin Short circuit Opn circuit, Y o.c l =.5 λ.5 λ.73 Y.4 Chang of rfrnc.5 λ impdanc 5 Ω Ω 5 Ω. λ.5 λ. λ.5 λ ' ' λ 3
14 MATCHNG. Transmission lin lin plus plus ractanc in in sris sris.4 λ X S l? = =+ X X S = -X..5 =. XS.. C '. X '. S..34 λ MATCHNG. MATCHNG. Transmission lin Transmission plus ractanc lin in plus paralll ractanc in paralll.89 λ BP Y Y Y = l? Y=+ B B P = -B.4.4 Y..65 Y.4 BP.4.4Y Y Y = Y.4 λ Y '.4 B '.4 P C.4Y 4
15 Eampl A n a Smith chart rfrncd to an impdanc =5 Ω, rprsnt th following sts of impdanc loci: a Th impdancs that caus a standing wav ratio SWR= b mpdancs with ral part largr than r 5 ROE= R> c St of impdancs that disipat mw whn a voltag of rms is applid Y + = P = mw - g =.5 r =4 * * * R P R Y G 3 P G.5 4 G g G.5 Y d mpdancs that hav th sam ral and imaginary parts r= 5
16 Eampl B A gnrator at f=3 MH fds an unknown load by mnas of a transmission lin of impdanc =7. n th lin it is masurd ma =5. and =.. Furthrmor whn th load is substitutd by a short circuit th positions of all imum voltags mov 5 cm towards th load. What is th impdanc? ma v p ROE S 4.7,., f 3MH m ROE f S.5 ma =-l.5 3 = ma c.c c.c c.c c.c. =-l = Eampl B
17 Eampl B ma v p ROE S 4.7,., f 3MH m ROE f S.35.5 Th sam rsult is obtaind if th position movs 35 cm towards th gnrator ma =-l = ma c.c c.c c.c c.c. =-l = Eampl B
18 Eampl C An antnna has and input impdanc of 75 at 4 MH. t is fd by mans of a paralll wirs transmission lin with an impdanc =5. a Dsign amatching ntwork compoundd of transmission lin plus a shunt capacitor. Th dilctric constant of th transmission lin is ε r =.. Bc Y Y = l Y=+ B B = -B c B c > l, C? Bc Y l Y=+ B Y = B = -B c B c > 75.5 Y 5 Y Y.7 Y.5 8 c / r 3 /..56 m 6 f 4 Y.7 l=.88 Y.7 Solution with capacitor l=.88 B.7 C l=4.45 cm.7 / C.85 pf f.348 8
19 Comput th lngth of a transmission lin, ndd in short-circuit, that can substitut th capactior. Yc.7 CC Y c =.7 Y Y = l Y=-.7 l=.347 l=.347 =cm b Matching ntwork formd by a quartr wavlngth transformr. l= = n th cas of a quartr wavlngth transformr Y ' ' ' ' 5 ' cm 4 9
20 Matching ntworks utilitis and ampls cturr: Francsc Torrs Thr ar a numbr of on-lin tools for RF dsign and/or ducational purpos
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27 EXERCSES: Microstrip impdanc computation: Glass Substrat thicknss:.59 mm Dilctric prmittivity: 4.5 Strip thicknss:. mm Coppr conductivity: S/m Substrat conductivity:.3-4 S/m a Calculat th substrat width W mm in ordr to hav o=5 Ω at f=.5 GH b n th prvious cas, comput th rturn loss R rfrrd to o=5 Ω at f= GH and f=5 GH: R dgradation du to frquncy dpndnc of c Comput th rturn loss R rfrrd to o=5 Ω at f=.5 Gh if th strip thicknss is nglctd t= Narrowband matching ntworks Slct th narrow band matching structur that provids th bst bandwidth SWR<.5 to adapt an impdanc =+ Ω, rfrrd to o=5 Ω at f= GH, ε r =.4. Quartr wavlngth adaptr Doubl stub adaptr Singl stub short/opn adaptr Givs lin lngth in mm. 7
28 ayout and pictur of a microstrip two-stag amplifir N TRT TRT OUT ntrstag matching ntwork: coupld lins DC block Ouput matching ntwork GND λ/4 λ/4 o.c. GND s.c. λ/4 Bias rsistor ow frquncy s.c. nput matching ntwork Bias ntwork Gat bias voltag Drain bias voltag Othr utilitis: Simpl tool for transmission lin calculations, bias circuits, smith chart,.. 8
29 Erciss Driv th prssion of th input impdanc of a transmission lin of impdanc o, lngth λ/4 and loadd with an impdanc. Dmonstrat that Γ for any load =R+X if R 3 What is th rturn loss of a load =75Ω whn connctd to a transmission lin of =5Ω? What fraction % of th incidnt powr is dlivrd to th load? 4 What is th tolranc ± Ω of a rsistor of noal impdanc R=5Ω whn connctd to a transmission lin of impdanc =5Ω if SWR.? 9
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