Whits, EE 48/58 Lctur 26 Pag f Lctur 26: Quadratur (9º) Hybrid. Back in Lctur 23, w bgan ur discussin f dividrs and cuplrs by cnsidring imprtant gnral prprtis f thrand fur-prt ntwrks. This was fllwd by an analysis f thr typs f thr-prt ntwrks in Lcturs 24 and 25. W will nw mv n t (rciprcal) dirctinal cuplrs, which ar fur-prt ntwrks. As in th txt, w will cnsidr ths spcific typs f dirctinal cuplrs:. Quadratur (9º) Hybrid, 2. 8º Hybrid, 3. Cupld Lin, and 4. Lang Cuplr. W will bgin with th quadratur (9º) hybrid. Fig 7.2 shws this cuplr implmntd with micrstrip as a : pwr dividr: Bcaus f th physical symmtry, w can simplify th analysis f this circuit cnsidrably using vn-dd md analysis. This 26 Kith W. Whits
Whits, EE 48/58 Lctur 26 Pag 2 f prcss is similar t what w did in th last lctur with th Wilkinsn pwr dividr. Evn-Odd Md Analysis f th Quadratur Hybrid Th nrmalizd (wrt Z ) TL circuit is shwn in Fig 7.22, minus th rturn lins: A symmtric (vn md) xcitatin f this circuit is shwn in Fig. 7.23a: B B4 B2 B3 A4 /8 and an anti-symmtric (dd md) xcitatin is shwn in Fig. 7.23b:
Whits, EE 48/58 Lctur 26 Pag 3 f B B4 B2 B3 A4 /8 Obsrv that th circuit and its bundary cnditins rmain th sam in bth th vn and dd md cnfiguratins. It is nly th xcitatin that changs. Bcaus f this and th circuit bing linar, by suprpsitin th ttal slutin is simply th sum f th vn and dd md vltag wav amplitud slutins. Each slutin (vn and dd) is simplr t dtrmin than th cmplt circuit, which is why w mply this tchniqu. Evn md. Bcaus th vltags and currnts must b th sam abv and blw th lin f symmtry (LOS) in Fig 7.23a, thn I at th LOS pn circuit lads at th nds f /8 stubs, as shwn. Rfrring t th dfinitin f B i ( i,,4) in Fig 7.22, w can writ frm Fig 7.23a that fr th vn md xcitatin: B, B2 T (a) B3 B2 T, B 4 B (b) whr 2, and and T ar th rflctin and transmissin cfficints fr th vn md cnfiguratin. (W ll slv fr ths cfficints shrtly.)
Whits, EE 48/58 Lctur 26 Pag 4 f Odd md. Bcaus th vltags and currnts must hav ppsit valus abv and blw th LOS in Fig 7.23b, thn V alng th LOS shrt circuit lads at th nds f /8 stubs, as shwn. Thn, B, B2 T (2a) B3 B2 T, B 4 B (2b) whr 2and and T ar rflctin and transmissin cfficints fr th dd md cnfiguratin. Ttal slutin. Th ttal slutin is th sum f th vltags and vltag wav amplituds in bth circuits. Frm this fact, w can dduc that th ttal B i cfficints will b th sum f () and (2): B B B (7.62a),(3) B2 B2 B2 T T (7.62b),(4) B3 B3 B3 T T (7.62c),(5) B4 B4 B4 (7.62d),(6) Likwis, th incidnt wav cfficints ar A4 A4 A4
Whits, EE 48/58 Lctur 26 Pag 5 f Ths match th assumd xcitatin in th riginal circuit n p. 2. T finish th calculatin f th S paramtrs fr th quadratur hybrid, w nd t dtrmin th rflctin and transmissin cfficints fr th vn- and dd-md cnfiguratins. Ths ar tw-prt ntwrks that ar much asir t slv than th riginal fur-prt Quadratur Hybrid. Yur txt shws that th slutins fr and T ar and T j (7.64),(7),(8) 2 Hr w ll driv slutins fr and T. Frm Fig 7.23b: 2 T 2 W hav thr cascadd lmnts, s w ll us ABCD paramtrs t slv fr th vrall S paramtrs f this circuit. Elmnts and 3. Ths ar shrt circuit stubs f lngth 8, which appar as th shunt impdanc Z jz tan l whr 2 in l 8 4
Whits, EE 48/58 Lctur 26 Pag 6 f Zin Thrfr, j Z, r YN j Frm th insid flap f yur txt: ABCD Y N j (9) Elmnt 2. This is a /4-lngth f TL whr 2 l 4 2 Frm th insid flap f yur txt: Z 2 cs l j sin l j Z ABCD 2 Z j sin l cs l j 2 Z 2 Cascading ths thr ABCD matrics w find th vrall ABCD matrix fr dd md xcitatin: j A B j 2 C D j j 2 j () j 2 Using Tabl 4.2, w can cnvrt ths t S paramtrs (with Z fr th nrmalizd TL): AB Z CZ D 2 j j S () AB Z CZ D 2 j j
Whits, EE 48/58 Lctur 26 Pag 7 f S2 A BZ CZ D 2 j j (2) 2 22j j Sinc th prts ar matchd, thn: S (7.66a),(3) and T S j 2 2 j j j 2 (7.66b),(4) Finally, using (7), (8), (3), and (4) in (3)-(6) w find: B (7.67a),(5) j B2 j j (7.67b),(6) 2 B3 j j (7.67c),(7) 2 B4 (7.67d),(8) Ths B i frm th first clumn f th S matrix fr th Hybrid Quadratur. Whn prprly intrprtd, ths rsults tll us much abut th circuit. In particular, whn prt is xcitd and all thr prts trminatd in matchd lads, thn: B prt is matchd.
Whits, EE 48/58 Lctur 26 Pag 8 f B2 j 2 9 phas shift frm prt t prt 2, and n half f th tim avrag input pwr is dlivrd t prt 2. B3 2 8 phas shift frm prt t prt 3 (9 phas shift btwn prts 3 and 2), and n half f th input pwr is dlivrd t prt 3. (Hnc, : pwr divisin.) B4 n pwr utput t prt 4. Bcaus f th high dgr f symmtry, w can trat any prt as th input prt. Thn, th islatin is th thr prt n th sam sid as th input and th utputs ar th tw prts n th thr sid f th circuit. Emplying this cncpt and th rsults abv, w can cnstruct th thr thr clumns in th full S matrix fr th quadratur (9º) hybrid by simply transpsing rws f th first clumn: j j S (7.6),(9) 2 j j That is, th first clumn in (9) is th rsults frm (5)-(8) whn th input was assumd at prt. In th scnd clumn, w can dirctly dduc that th utputs ar at prts and 4, th input is at prt 2 and th islatin is at prt 4. Furthr transpsitin f th rws in clumn prducs clumns 3 and 4.
Whits, EE 48/58 Lctur 26 Pag 9 f Exampl N26.. Dsign a branch lin hybrid cuplr using - micrstrip n 32-mil RO43C fr a cntr frquncy f 2.5 GHz. Includ th ffcts f cppr and substrat lsss. Bcaus thr ar tw diffrnt charactristic impdancs ndd fr th 9º hybrid dvic, tw diffrnt widths f micrstrip must b cmputd (bcaus W/d dpnds n Z ) and tw diffrnt /4-lngths must b dtrmind (bcaus r, dpnds n W/d). Z sctins. Using LinCalc, W 8.2 mil and r, 2.424. Th guid wavlngth at this frquncy is thn 8 c 2.998 7.72 cm 9 f 2.5 2.424 Hnc, this branch lin cuplr shuld hav - lins with lngth 4.93 cm. Z 2 7.7 sctins. Using LinCalc, W 39.62 mil and r, 2.545. Th guid wavlngth is thn 8 c 2.998 7.57 cm 9 f 2.5 2.545 Hnc, this branch lin cuplr shuld hav 7.7- lins with lngth 4.88 cm.
Whits, EE 48/58 Lctur 26 Pag f Th fllwing S paramtr rsults wr btaind fr this dsign using ADS..8 m3 m3 frq= 2.5GHz mag(s(2,))=.73 mag(s(4,)) mag(s(3,)) mag(s(2,)) mag(s(,)).6.4.2 m m2 m frq= 2.37GHz mag(s(,))=.2 m2 frq= 2.64GHz mag(s(,))=.2..4.6.8 2. 2..4 2.6 2.8 3. 3.2 3.4 3.6 frq, GHz 2 m5 phas(s(4,)) phas(s(3,)) phas(s(2,)) phas(s(,)) - m4 m5 frq= 2.5GHz phas(s(3,))=78.357 m4 frq= 2.5GHz phas(s(2,))=-9.79-2.4.6.8 2. 2..4 2.6 2.8 3. 3.2 3.4 3.6 frq, GHz
Whits, EE 48/58 Lctur 26 Pag f.8.7.6 mag(s(3,2)) mag(s(4,3)).5.4.3.2...4.6.8 2. 2..4 2.6 2.8 3. 3.2 3.4 3.6 frq, GHz -5 phas(s(3,2)) - -5-2.4.6.8 2. 2..4 2.6 2.8 3. 3.2 3.4 3.6 frq, GHz