Multi-Section Coupled Line Couplers
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1 /0/009 MultiSction Coupld Lin Couplrs /8 Multi-Sction Coupld Lin Couplrs W cn dd multipl coupld lins in sris to incrs couplr ndwidth. Figur 7.5 (p. 6) An N-sction coupld lin l W typiclly dsign th couplr such tht it is symmtric, i..: c = c, c = c, c = c, tc. N N N whr N is odd. Q: Wht is th coupling of this dvic s function of frquncy? A: To nlyz this structur, w mk n pproximtion similr to tht of th thory of smll rflctions. Jim Stils Th Univ. of Knss Dpt. of EECS
2 /0/009 MultiSction Coupld Lin Couplrs /8 First, if c is smll (i.., lss thn 0.), thn w cn mk th pproximtion: Likwis: S ( θ ) = S ( θ ) = c tnθ c + tn c tnθ + tnθ = c sinθ θ c θ cos sin cosθ + sinθ = c θ + θ whr of cours θ = β = ωt, nd T = v p. W cn us ths pproximtions to construct signl flow grph of singl-sction couplr: c sinθ θ c sinθ θ c sinθ θ c sinθ θ Jim Stils Th Univ. of Knss Dpt. of EECS
3 /0/009 MultiSction Coupld Lin Couplrs /8 Now, sy w cscd thr coupld lin pirs, to form thr sction coupld lin couplr. Th signl flow grph would thus : c θ θ c θ θ c θ θ c θ θ c θ θ c θ θ c sinθ θ c θ θ c θ θ c θ θ c sinθ θ c θ θ Not tht this signl flow grph dcoupls into two sprt nd grphs (i.., th lu grph nd th grn grph). c θ θ c θ θ c θ θ c sinθ θ c θ θ c θ θ c θ θ c sinθ θ c θ θ c θ θ c θ θ c θ θ Jim Stils Th Univ. of Knss Dpt. of EECS
4 /0/009 MultiSction Coupld Lin Couplrs /8 Not lso tht ths two grphs r ssntilly idnticl, nd mphsiz th symmtric structur of th coupld-lin couplr. Now, w r intrstd in dscriing th coupld output (i.., ) in trms of th incidnt wv (i.., ). Assuming ports, nd r mtchd (i.., = = = 0 ), w cn rduc th grph to simply: c θ θ c θ θ c θ θ c θ θ c θ θ Now, w could rduc this signl flow grph vn furthr or w could trunct propgtion sris y considring only th dirct pths! W of cours usd this id to nlyz multi-sction mtching ntworks, n pproch dud th thory of smll rflctions. Essntilly w r now pplying thory of smll couplings. In othr words, w considr only th propgtion pths whr on coupling is involvd th signl propgts cross coupld-lin pir only onc! Jim Stils Th Univ. of Knss Dpt. of EECS
5 /0/009 MultiSction Coupld Lin Couplrs 5/8 Not from th signl flow grph tht thr r thr such mchnisms, corrsponding to th coupling cross ch of th thr sprt coupld lin pirs: c θ θ c θ θ c θ θ c θ θ c θ θ θ θ θ θ θ θ θ ( c sin θ c sinθ c sinθ ) θ θ 5θ ( c sin θ c sin θ c sin θ ) + + = + + Not tht ll othr trms of th infinit sris would involv t lst thr couplings (i.., thr crossings), nd thus ths trms would xcding smll (i.., c 0). Thrfor, ccording to this pproximtion: V θ θ S ( θ) = ( θ ) = ( θ) = c sn i θ + c sin θ + c sin θ + V 5θ Morovr, for multi-sction couplr with N sctions, w find: S ( θ) c sinθ c sinθ c sinθ = θ θ 5θ c sinθ N ( N ) θ And for symmtric couplrs with n odd vlu N, w find: Jim Stils Th Univ. of Knss Dpt. of EECS
6 /0/009 MultiSction Coupld Lin Couplrs 6/8 Nθ ( ) = sin cos( ) + cos( ) S θ θ c N θ c N θ + c cos ( N 5) θ + + c M whr M ( N ) = +. Thus, w find th coupling mgnitud s function of frquncy is: c( θ) = S ( θ) ( ) ( ) ( ) = c sinθ cos N θ + c sinθ cos N θ + c sinθ cos N 5 θ + + c sinθ M And thus th coupling in db is: C ( θ ) = c( θ ) 0log 0 Now, our dsign gols r to slct th coupling vlus c, c, cn such tht:. Th coupling vlu C ( θ ) is spcific, dsird vlu t our dsign frquncy.. Th coupling ndwidth is s lrg s possil. Jim Stils Th Univ. of Knss Dpt. of EECS
7 /0/009 MultiSction Coupld Lin Couplrs 7/8 For th first condition, rcll tht th t th dsign frquncy: θ = β = π I.E., th sction lngths r qurtr-wvlngth t our dsign frquncy. Thus, w find our first dsign qution: ( ) = ( ) + ( ) c θ c cos N π c cos N π θ = π ( ) + c cos N 5 π + + cm whr w hv usd th fct tht sin ( π ) =. Not th vlu c ( θ ) θ = π is st to th vlu ncssry to chiv th dsird coupling vlu. This qution thus provids on dsign constrint w hv M- dgrs of dsign frdom lft to ccomplish our scond gol! To mximiz ndwidth, w typiclly impos th mximlly flt condition: d m c ( θ ) d θ m θ = π = 0 m =,, B crful! Rmmr to prform th drivtiv first, nd thn vlut th rsult t θ = π. Jim Stils Th Univ. of Knss Dpt. of EECS
8 /0/009 MultiSction Coupld Lin Couplrs 8/8 You will find for symmtric couplr, th odd-ordrd drivtivs (.g., d c( θ ) dθ, d c( θ ) dθ, d 5 c( θ ) dθ 5 )r uniquly zro. In othr words, thy r zro-vlud t θ = π rgrdlss of th vlus of coupling cofficints c,c,c,! As rsult, ths odd-ordr drivtivs do not impos mximlly flt dsign qution only th vn-ordrd drivtivs do. Kp tking ths drivtivs until your dsign is fully constrind (i.., th numr of dsign qutions quls th numr of unknown cofficints c,c,c, ). On finl not, you my find tht this trig xprssion is hlpful in simplifying your drivtivs: For xmpl, w find tht: sinφ cosψ = sin φ + ψ + sin φ ψ ( ) ( ) ( ) ( ) ( θ) sin ( θ) ( θ) sin ( θ) sinθ cosθ = sin θ + θ + sin θ θ = sin + = sin Jim Stils Th Univ. of Knss Dpt. of EECS
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