The Binomial Multi- Section Transformer

Transcription

1 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 + e + + e i = j 2ωT j 4ωT j2ωt 2 = e j2ωt T = v p propagatio tie through sectio ote that for a ulti-sectio trasforer, we have degrees of desig freedo, correspodig to the characteristic ipedace values Z. Q: What should the values of (i.e., Z ) be? A: We eed to defie idepedet desig equatios, which we ca the use to solve for the values of characteristic ipedace Z. First, we start with a sigle desig frequecy ω, where we wish to achieve a perfect atch: Ji Stiles The Uiv. of Kasas Dept. of EECS

2 4/4/26 The Bioial Multisectio Matchig Trasforer 2/2 i ( ω = ω ) = That s just oe desig equatio: we eed - ore! These additio equatios ca be selected usig ay criteria oe such criterio is to ake the fuctio ( ω) axially flat at the poit ω = ω. i To accoplish this, we first cosider the Bioial Fuctio: j 2θ ( θ ) A( e ) = + This fuctio has the desirable properties that: ad that: jπ ( θ π 2) A( e ) = = + = ( ) = A d ( θ ) d θ θ = π 2 = for = 23,,,, I other words, this Bioial Fuctio is axially flat at the poit θ π 2 θ = π 2 =. =, where it has a value of ( ) Q: So? What does this have to do with our ulti-sectio atchig etwork? Ji Stiles The Uiv. of Kasas Dept. of EECS

3 4/4/26 The Bioial Multisectio Matchig Trasforer 3/2 A: Let s expad (ultiply out the idetical product ters) of the Bioial Fuctio: j 2θ ( θ ) A( e ) j2θ j 4θ j6θ j2θ = A( C + C e + C2 e + C3 e + + C e ) = + where: C!!! ( ) Copare this to a -sectio trasforer fuctio: ( ω) = + e + e + + e i j 2ωT j 4ωT j2ωt 2 ad it is obvious the two fuctios have idetical fors, provided that: = AC ad ωt = θ Moreover, we fid that this fuctio is very desirable fro the stadpoit of the a atchig etwork. Recall that θ = at θ = π 2--a perfect atch! ( ) Additioally, the fuctio is axially flat at θ = π 2, therefore ( θ ) over a wide rage aroud θ = π 2--a wide badwidth! Q: But how does θ = π 2 relate to frequecy ω? Ji Stiles The Uiv. of Kasas Dept. of EECS

4 4/4/26 The Bioial Multisectio Matchig Trasforer 4/2 A: Reeber that ωt = θ, so the value θ = π 2 correspods to the frequecy: π v p π ω = = T 2 2 This frequecy ( ω ) is therefore our desig frequecy the frequecy where we have a perfect atch. ote that the legth has a iterestig relatioship with this frequecy: v p π π λ π λ = = = = ω 2 β 2 2π 2 4 I other words, a Bioial Multi-sectio atchig etwork will have a perfect atch at the frequecy where the sectio legths are a quarter wavelegth! Thus, we have our first desig rule: Set sectio legths so that they are a quarterwavelegth ( λ 4 ) at the desig frequecy ω. Q: I see! Ad the we select all the values Z such that = AC. But wait! What is the value of A?? Ji Stiles The Uiv. of Kasas Dept. of EECS

5 4/4/26 The Bioial Multisectio Matchig Trasforer 5/2 A: We ca deterie this value by evaluatig a boudary coditio! Specifically, we ca easily deterie the value of ( ω) at ω =. Z Z i Z Z 2 Z R L ote as ω approaches zero, the electrical legth β of each sectio will likewise approach zero. Thus, the iput ipedace Z i will siply be equal to R L as ω. As a result, the iput reflectio coefficiet ( ω ) be: L ( ω ) ( ω ) Zi = Z ( ω = ) = Zi = + Z RL Z = R + Z However, we likewise kow that: ( ) = A + = A 2 j 2 ( ) ( e ) ( ) = A + = ust Ji Stiles The Uiv. of Kasas Dept. of EECS

6 4/4/26 The Bioial Multisectio Matchig Trasforer 6/2 Equatig the two expressios: Ad therefore: R Z ( ) = A 2 = L RL + Z A = 2 L RL R Z + Z (A ca be egative!) We ow have a for for the argial reflectio coefficiets : RL Z! = AC = 2 R + Z!! L ( ) Of course, we also kow that these argial reflectio coefficiets are: Z+ Z = Z + Z + ow, we kow that the values of Z + ad Z are typically very close, such that Z+ Z is sall. It turs out for this case that we ca use a helpful approxiatio for the argial reflectio coefficiet: Z Z Z l = Z+ Z 2 Z (for sall) Ji Stiles The Uiv. of Kasas Dept. of EECS

7 4/4/26 The Bioial Multisectio Matchig Trasforer 7/2 Therefore we ca coclude: Z RL Z 2 Z RL + Z + = l = 2 C Solvig for Z +, we fid: 2 + R Z Z L + = Z exp C RL + Z We ca further siplify this with yet aother approxiatio: R Z 2 L + Z exp l C Z This is our secod desig rule. ote it is a iterative rule we deterie Z fro Z, Z 2 fro Z, ad so forth. The result is a axially flat, Bioial reflectio ω. coefficiet fuctio ( ) Ji Stiles The Uiv. of Kasas Dept. of EECS

8 4/4/26 The Bioial Multisectio Matchig Trasforer 8/2 Figure 5.5 (p. 25) Reflectio coefficiet agitude versus frequecy for ultisectio bioial atchig trasforers of Exaple 5.6 Z L = 5Ω ad Z = Ω. ote that as we icrease the uber of sectios, the atchig badwidth icreases. Q: Ca we deterie the value of this badwidth? A: Sure! But we first ust defie what we ea by badwidth. As we ove fro the desig (perfect atch) frequecy f the value ( f ) will icrease. At soe frequecy (f, say) the agitude of the reflectio coefficiet will icrease to soe Ji Stiles The Uiv. of Kasas Dept. of EECS

9 4/4/26 The Bioial Multisectio Matchig Trasforer 9/2 uacceptably high value (, say). At that poit, we o loger cosider the device to be atched. ( f ) f f f f f 2 ote there are two values of frequecy f oe value less tha desig frequecy f, ad oe value greater tha desig frequecy f. These two values defie the badwidth f of the atchig etwork: ( ) ( ) f = f f = 2 f f = 2 f f 2 2 Q: So what is the uerical value of? A: I do t kow it s up to you to decide! Every egieer ust deterie what they cosider to be a acceptable atch (i.e., decide ). This decisio depeds o the applicatio ivolved, ad the specificatios of the overall icrowave syste beig desiged. However, we typically set to be.2 or less. Ji Stiles The Uiv. of Kasas Dept. of EECS

10 4/4/26 The Bioial Multisectio Matchig Trasforer /2 Q: OK, after we have selected, ca we deterie the two frequecies f? A: Sure! We just have to do a little algebra. We start by rewritig the Bioial fuctio: j 2θ ( θ ) A( e ) jθ + jθ jθ = Ae ( e + e ) jθ + jθ jθ = Ae ( e + e ) = + = Ae jθ ( 2cos θ ) ow, we take the agitude of this fuctio: ( θ ) 2 = = 2 Ae j θ Acosθ cosθ ow, we defie the values θ where ( θ ) = 2 ( θ θ ) = = Acosθ = as θ. I.E., : We ca ow solve for θ (i radias!) i ters of : θ = cos 2 A θ 2 = cos 2 A Ji Stiles The Uiv. of Kasas Dept. of EECS

11 4/4/26 The Bioial Multisectio Matchig Trasforer /2 ote that there are two solutios to the above equatio (oe less that π 2 ad oe greater tha π 2)! ow, we ca covert the values of θ ito specific frequecies. Recall thatωt = θ, therefore: v ω = θ θ T = p But recall also that = λ 4, where λ is the wavelegth at the desig frequecy f (ot f!), ad where λ = vp f. Thus we ca coclude: or: v 4v ω θ θ θ ( 4f ) p p = = = λ ( 4 ) θ ( 2 ) v f f θ f = = = 2π 2π π p θ where θ is expressed i radias. Therefore: f 2f = cos + π 2 A f 2 2f = cos π 2 A Ji Stiles The Uiv. of Kasas Dept. of EECS

12 4/4/26 The Bioial Multisectio Matchig Trasforer 2/2 Thus, the badwidth of the bioial atchig etwork ca be deteried as: ( ) f = 2 f f 4f = 2f cos + π 2 A ote that this equatio ca be used to deterie the badwidth of a bioial atchig etwork, give ad uber of sectios. However, it ca likewise be used to deterie the uber of sectios required to eet a specific badwidth requireet! Fially, we ca list the desig steps for a bioial atchig etwork:. Deterie the value required to eet the badwidth ( f ad ) requireets. 2. Deterie the ipedace of each sectio usig the iterative approxiatio: R Z 2 L + Z exp l C Z 3. Deterie sectio legth = λ 4 for frequecy f. Ji Stiles The Uiv. of Kasas Dept. of EECS