Note that the argument inside the second square root is always positive since R L > Z 0. The series reactance can be found as

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1 Ipedace Matchig Ipedace Matchig Itroductio Ipedace atchig is the process to atch the load to a trasissio lie by a atchig etwork, as depicted i Fig Recall that the reflectios are eliiated uder the atched Fig : Ipedace atchig coditio Ipedace atchig is iportat for the followig reasos: To achieve axiu power trasfer ad iiize power loss To iprove sigal-to-oise ratio To reduce aplitude ad phase errors for power distributio etworks, eg, atea arrays There are ay choices regardig atchig etwork desig, but the followig factors ust be cosidered i the selectio of the etwork: Coplexity Badwidth Ipleetatio Adjustibility Matchig with uped Eleets ( etworks) The -sectio is cosidered the siplest type of atchig etwork There are two possible cofiguratios, as depicted i Fig (a) is the etwork for Re[ ] >, while (b) is the etwork for Re[ ] < ote that i both cofiguratios, two copoets (jx, jb) are required i order to have degree of freedo, sice the load ipedace is geerally coplex Cosider Fig (a) et = R +jx, the the ipedace see lookig ito the atchig etwork followed by the load ipedace ust be equal to, ie, = jx + jb+ /( R + jx ) Rearragig ad separatig ito real ad iagiary parts yield B( XR X ) R = ; X ( BX ) = BR X Solvig the above equatios yields B X ± R / + X = R + X R R (a) (b) Fig : -sectio atchig etworks ote that the arguet iside the secod square root is always positive sice R > The series reactace ca be foud as X X = + B R BR ote also that two solutios are geerally possible Oe ust cosider the above factors i decidig which etwork to use ikewise, for the etwork i Fig (b), the atched coditio is give by = jb+ R + j( X + X ) Rearragig ad separatig ito real ad iagiary parts yield

2 Ipedace Matchig B ( X + X ) = R ; X + X = BR Solvig for X ad B gives ± ( R ) / R X = ± R ( R ) X ; B= ote that R < i this case, so the arguet of the square root is always positive Exaple Desig a -sectio atchig etwork to atch a series RC load with a ipedace = j Ω, to a Ω lie, at a frequecy of 5 MHz 9 8 sectio atchig : reflectio coefficiet plot solutio solutio Frequecy [Hz] x 8 Sigle-Stub Tuig The ipedace atchig usig -sectios discussed previously requires luped eleets that ight ot be available, thus it is ot practical i soe cases The sigle-stub tuig is the atchig techique that uses a sigle ope-circuited or short-circuited legth of trasissio-lie (a stub ), coected either i parallel or i series with the trasissio feed lie at a certai distace fro the load ote that there are two desig paraeters, aely the legth of the stub ad the distace fro the load, which cotribute degree of freedo, as i the atchig with -sectios The choice of ope-circuited stub or short-circuited stub depeds o the type of trasissio lie edia For icrostrip lies, ope stubs are preferred due to ease of fabricatio, while for coaxial lies

Ipedace Matchig or waveguides, short stubs are ore desirable sice such ope-circuited stubs ted to radiate, resultig i reactace chages Shut Stubs The sigle-stub shut tuig circuit cofiguratio is show i

susceptace of Y ust cacel out the susceptace of Y stub, resultig i Y i becoes Y Usig the Sith chart akes the desig process easier The first step is to fid the distace such that the oralized adittace

3 Ipedace Matchig or waveguides, short stubs are ore desirable sice such ope-circuited stubs ted to radiate, resultig i reactace chages Shut Stubs The sigle-stub shut tuig circuit cofiguratio is show i Fig Refer to the figure, to atch the ipedace, it is required that Y = Yi = Y + Y stub Sice Y stub is purely susceptace (ie, zero coductace), the real part of Y ust be equal to Furtherore, the susceptace of Y ust cacel out the susceptace of Y stub, resultig i Y i becoes Y Usig the Sith chart akes the desig process easier The first step is to fid the distace such that the oralized adittace is o the +jb circle The fid the Fig : Sigle-stub shut tuig legth such that the stub has susceptace jb Exaple For a load ipedace = 6 j8 Ω, desig two sigle-stub (short circuit) shut tuig etworks to atch this load to a 5 Ω lie Assuig that the load is atched at GHz ad the load cosists of a resistor ad a capacitor i series solutio # solutio # 5 4 Series Stubs The sigle-stub series tuig circuit cofiguratio is show i Fig 4 Refer to the figure, to atch the ipedace, it is required that = i = + stub Sice stub is purely reactace (ie, zero resistace), the real part of ust be equal to Furtherore, the reactace of ust cacel out the reactace of stub, resultig i i becoes As i the shut tuig circuit desig, usig the Sith chart akes the desig process easier The first step is to fid the distace such Frequecy [GHz] Fig 4: Sigle-stub series tuig

4 Ipedace Matchig that the oralized ipedace is o the +jx circle The fid the legth such that the stub has reactace jx Exaple For a load ipedace = + j8 Ω, desig two sigle-stub (ope circuit) series tuig etworks to atch this load to a 5 Ω lie Assuig that the load is atched at GHz ad the load cosists of a resistor ad a iductor i series solutio # solutio # 4 Double-Stub Tuig The sigle-stub tuer requires a variable legth of lie betwee the load ad the stub, thus it is difficult to ake it adjustable The double-stub tuig show i Fig 5 uses adjustable shut stubs i fixed positios However, the double-stub tuer caot atch all load ipedaces d Y Y jb jb Y Y Y jb jb Y d Y Frequecy [GHz] Ope or shorted stub l Ope or shorted stub l Ope or shorted stub l Ope or shorted stub l (a) (b) Fig 5: Double-stub tuig (a) Origial circuit with the load a arbitrary distace fro the first stub (b) Equivalet circuit with the load trasfored to the first stub The Sith chart solutio ca be illustrated i Fig 6 First, locate y ad draw the rotated +jb circle with respect to the stub spacig d The ove the load adittace oto the rotated +jb circle (poits y, y ) usig the susceptace b, b of the stub ext, ove the poits y, y oto the +jb circle (poits y, y ) Fially, add the susceptace b, b to atch the load ipedace ote that there are two possible solutios as i the case of sigle-stub tuig 4

Ipedace Matchig otice that if y is iside the shaded regio i the figure, specified by g +jb circle, it is ipossible to ove this adittace oto the rotated circle, which eas that it caot be atched by a

5 Ipedace Matchig otice that if y is iside the shaded regio i the figure, specified by g +jb circle, it is ipossible to ove this adittace oto the rotated circle, which eas that it caot be atched by a double-stub tuer (ie, there is o solutio) Therefore, this shaded regio fors a forbidde rage of load adittaces that caot be atched by this double-stub tuer Reducig the space d ca lead to the reductio i the size of this forbidde rage, however, d ust be kept sufficietly large for fabricatig two separate stubs I additio, spacigs ear or λ/ load to atchig etworks that are very frequecy sesitive I practice, stub spacigs are usually chose as λ/8 or λ/8 Furtherore, if the legth of lie betwee the load ad the first stub ca be adjusted, the y ca always be oved out of the forbidde regio Fig 6: Sith chart diagra for the operatio of a doublestub tuer Exaple 4 For a load ipedace = 6 - j8 Ω, desig a shut double-stub tuer to atch this load to a 5 Ω lie The stubs are to be ope-circuited ad are spaced λ/8 apart Also, the load is assued to cosist of a 6Ω-resistor ad a 995pF-capacitor l=489, l=498, lp=465, lp=4 6+8i 9 8 ψ ψ Frequecy [Hz] x 9 5 Quarter-Wave Trasforer Recall that, for a quarter-wavelegth trasissio lie (l = λ/4), the iput ipedace becoes Solutio # Solutio # i = / or = i Therefore, a quarter-wavelegth trasissio lie ca be used to covert a resistive load to atch a trasissio lie by choosig the proper characteristic ipedace of the quarter-wavelegth lie This is called a quarter-wave trasforer The geeral cofiguratio of this quarter-wave trasforer is show i Fig 6, where 5

6 Ipedace Matchig R = To atch a arbitrary usig the quarterwave trasforer, oe ust soehow odify the load such that it becoes purely resistive This ay be doe by addig certai luped Fig 6 eleets, trasissio lie of certai legth, tuig circuits or stubs Exaple 5 Repeat exaple by usig the quarter-wave trasforer 6

7 Ipedace Matchig 6 The Theory of Sall Reflectios Quarter-wave trasforers provide a siple ea of ipedace atchig, but caot achieve broad badwidth To obtai ore badwidth, ultisectio trasforers ca be used Sigle-sectio Trasforer Cosider the sigle-sectio trasforer show i Fig 7, the partial reflectio ad trasissio coefficiets are give by = ; = ; = ; + + T = + = ; T = + = + + The total reflectio ca the be give i ters of a ifiite su of partial reflectios ad trasissios as follows: j θ j4θ = + T T e + T T e + = + T T j θ e = Usig the geoetric series x =, for x <, x = ca be rewritte as T T e = + jθ jθ e e j θ Usig =-, Τ =+, Τ = yields + e j θ = j θ + e Fig 7 If the discotiuities betwee the ipedaces, ad, are sall, the <<, ad jθ +e Multisectio Trasforer ow cosider the ultisectio trasforer show i Fig 8 This trasforer cosists of equallegth (coesurate) sectios of trasissio lies Partial reflectio coefficiets ca be defied at each juctio as + = ; = ; = Fig 8 We also assue that all icrease or decrease ootoically across the trasforer, ad is real This iplies that will be real ad of the sae sig The the total reflectio coefficiet ca be approxiated as 7

8 Ipedace Matchig j θ j 4θ j θ ( θ ) = + e +e + + e Furtherore, assue that the trasforer ca be ade syetric, so that =, = -, etc (ote that this does ot iply that the s are syetrical) The, jθ jθ j( ) θ j( ) θ { ( e + e ) + ( e + e ) +} jθ ( θ ) = e It follows that for eve, jθ θ ) = e cos θ + cos( ) θ + + cos( ) θ + + ( / ad for odd, θ jθ ) = e { cos θ + cos( ) θ + + cos( ) θ + + cos θ} ( ( ) /, Fro these results, oe ca otice that ay desired reflectio respose (as a fuctio of θ) ca be realized by choosig the proper s ad usig eough sectios Recall the fact that a sooth fuctio ca be approxiated by a Fourier series, if eough ters are used 7 Bioial Multisectio Matchig Trasforers The passbad respose of a bioial trasforer is optiu i the sese that, for a give uber of sectios, the respose is flat as possible ear the desig frequecy Thus, such as respose is also kow as axially flat This type of respose is desiged, for a -sectio trasforer, by settig the first - derivatives of (θ) to zero, at the ceter frequecy f Such a respose ca be obtaied if j θ ( θ ) = A (+ e ) The the agitude (θ) is jθ jθ jθ ( θ ) = A e e + e = A ote that (θ) = for θ=π/ ad that (d (θ) )/dθ = at θ=π/ for =,,, - (θ=π/ correspods to the ceter frequecy f, for which l=λ/4 ad θ = β l = π /) et f, the θ = β l =, ad θ ( = ) = A(+ ) = A =, + sice for f = all sectios are of zero electrical legth Thus, A= + ow expadig (θ) accordig to the bioial expasio yields ( θ ) = A(+ e j θ ) = A = C e jθ,where C! = Sice, ( )!! j θ j θ j 4θ j θ ( θ ) = A C e = +e +e + + e, = = AC If s are assued to be sall, the followig approxiatio ca be applied: + + x = l, sice l x Therefore, + x + l + + AC ( ) C C l = = + To calculate the badwidth, let deote the axiu value of reflectio coefficiet that ca be tolerated over the passbad The, 8

9 Ipedace Matchig = A cos θ θ = cos A, where θ < π / is the lower edge of the passbad Thus, /, ad the fractioal badwidth is give by / f ( f f) 4θ 4 = = = cos f f π π A Exaple 6 Desig a three-sectio trasforer to atch a 5 Ω load to a Ω lie, ad calculate the badwidth for = 5 8 Chebyshev Multisectio Matchig Trasforers I cotrast with the bioial atchig trasforer, the Chebyshev trasforer optiizes badwidth at the expese of passbad ripple The Chebyshev trasforer is desiged by equatig (θ) to a Chebyshev polyoial, which has the optiu characteristics eeded for this type of trasforer Chebyshev Polyoial The th order Chebyshev polyoial is a polyoial of degree, ad is deoted by T (x) The first four Chebyshev polyoials are 4 T ( x) = x; T ( x) = x ; T ( x) = 4x x; T ( x) = 8x 8x Higher-order polyoials ca be foud usig the followig recurrece forula: =4 4 T ( x) = xt ( x) T ( x) Soe iportat properties of Chebyshev polyoials are listed here: For - x, T (x) I this rage, the Chebyshev polyoials oscillate betwee ± This is the equal ripple property, ad this - regio will be apped to the passbad of the atchig trasforer For x >, T (x) > This regio will be -4 apped to the frequecy rage outside the passbad For x >, T (x) icreases faster with x as x icreases Fig 8: First four Chebyshev polyoials ow, let x = cos θ for x < The it ca be show that the Chebyshev polyoials ca be expressed as (cos θ ) = cos θ, or ore geerally as T T cos( cos x) ( x) = cosh( cosh x) for x < for x > T (x) = = = 9

10 Ipedace Matchig Sice equal ripple is desirable i the passbad, it is ecessary to ap θ to x = ad π θ to x = -, where θ ad π θ are the lower ad upper edges of the passbad This ca be accoplished by replacig cos θ i the above equatio with cos θ /cos θ : T cos θ = = T (secθ ) cos cos The sec θ cos θ for θ < θ < π θ, so T (sec θ cos θ ) over this sae rage It follows that the first four ters of the Chebyshev polyoials ca be writte as T (secθ ) = secθ ; T (secθ ) = sec θ(cos θ + ) ; T (secθ ) = sec θ ( + ) secθ ; 4 T4 (secθ ) = sec θ(cos 4θ + 4cos θ + ) 4sec θ(cos θ + ) + The above results ca be used to desig atchig trasforers with up to four sectios Desig of Chebyshev Trasforers A Chebyshev equal-ripple passbad ca be sythesized by akig (θ) proportioal to T (secθ ), where deotes the uber of sectios Thus, jθ { cos θ + cos( ) θ + + cos( ) θ + } = Ae T (secθ cos ) jθ ( θ ) = e where the last ter i the series is (/) Ν/ for eve ad (-)/ for odd The costat A ca be foud fro lettig θ = : ( θ = ) = = AT + θ (secθ ), or A = + T (secθ ) ow if the axiu allowable reflectio coefficiet agitude i the passbad is (ie, the ripple), the = A, sice the axiu value of T (secθ ) i the passbad is uity Usig the approxiatio itroduced i the previous sectio yields T (secθ ) = l + It follows that l( / ) secθ = cosh cosh cosh cosh + Oce θ is kow, the fractioal badwidth ca be calculated fro f 4θ = f π Each ca be deteried by expadig T (secθ ) ad equatig siilar ters of the for cos(- )θ The followig approxiatio ca be applied to iprove the accuracy: + + = l + + Exaple 7 Desig a three-sectio Chebyshev trasforer to atch a Ω load to a 5 Ω lie, with = 5

The Binomial Multi- Section Transformer

The Binomial Multi- Section Transformer 4/4/26 The Bioial Multisectio Matchig Trasforer /2 The Bioial Multi- Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where: ( ω ) = + e +