Let us look at a linear equation for a one-port network, for example some load with a reflection coefficient s, Figure L6.

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1 ECEN 5004, prng 08 Actve Mcrowve Crcut Zoy Popovc, Unverty of Colordo, Boulder LECURE 5 IGNAL FLOW GRAPH FOR MICROWAVE CIRCUI ANALYI In mny text on mcrowve mplfer (e.g. the clc one by Gonzlez), gnl flow-grph re ued for defnng gn, determnng tblty, etc. h prt of the lecture gve bref revew of gnl flow-grph they pply to mcrowve crcut,.e. -prmeter nd wve vrble. One of the mn reon for ung gnl flow-grph tht, by followng ome mple rule, one cn qucly fnd one of the prmeter of the crcut, o tht there no need to wte tme on olvng for prmeter tht re not of nteret. gnl flow grph were developed few decde go for control theory, nd were not ued n crcut theory becue crcut model re eer to olve. In mcrowve, however, ctterng prmeter nd ncdent nd reflected wve cn be repreented ely wth gnl flow grph. he clc pper on gnl flow grph w wrtten by muel Mon t MI ( Feedbc heory Further Properte of gnl Flow Grph, Proceedng of the Inttute of Rdo Engneer, volume 4, pp , 953). L5.. BRIEF OVERVIEW OF IGNAL FLOW-GRAPH Let u loo t lner equton for one-port networ, for exmple ome lod wth reflecton coeffcent, Fgure L6.: b, where the ncdent wve, nd b the reflected wve. In gnl flow grph, one thn of wve vrble, nd ctterng prmeter re contnt. he vrble become node of the flow grph nd the contnt coeffcent become brnche, Fgure L5.. he rrow on the brnch pont from the vrble on the rght de of the equton to the vrble on the left de. Let u now dd ecton of trnmon lne of electrcl length n front of the one port lod, Fgure L5.. Wht doe the gnl flow grph for th networ loo le? Let u frt wrte the ctterng mtrx for trnmon lne ecton: 0 e e 0. b b Fgure L5.. A one-port networ decrbed wth ctterng prmeter nd t gnl flow grph.

2 Now the vrble need to be lbeled. he forwrd wve for the trnmon lne re the reflected wve for the lod, o the followng cn be wrtten, from Fgure L5.: b e b b e. he gnl flow grph hown n Fgure L5.. It ccde of three brnche. he drecton of the rrow very mportnt, nce t how whch vrble re on whch de of the equton. olvng for b gve: b e e. b Z b 0 b b 0 e - e - Fgure L5.. A ecton of trnmon lne of mpednce Z 0 nd electrcl length connected to lod nd the gnl flow grph. From here we cn ee tht the three brnche form pth from to b. A pth collecton of brnche nd node tht llow one to move from begnnng node to n endng node, followng the rrow from node to node. he gn from to b gven by the product of the coeffcent of the three brnche connectng thee node. You cn ee from here tht the gn of crcut cn be found ely by ut multplyng ccded brnch coeffcent wth rrow n the me drecton. When there loop n the grph, t modfe the gn. Let u now loo t crcut tht h loop n the gnl flow grph, Fgure L5.3. We wll fnd the ctterng prmeter from port of the networ to port I of the networ. he two connected port re on nd m on. Let u cll the combned networ nd fnd t prmeter '. Combnng mller networ nto lrger one n mportnt problem, t ued n number of crcut multor, wll be llutrted t the end of th ecton.

3 b = m b b m b m b m b t mm t m = b m Fgure L5.3. wo two-port networ ont wth one port nd the gnl flow grph. From the defnton of the ctterng prmeter, we cn wrte expreon for the ncdent nd cttered wve from Fgure L5.3 : b bm tmmm b t m m Further, nce we now tht port connected to port m, we hve tht bm nd b m. Now we cn wrte gnl flow grph wth four node nd four brnche, hown n Fgure L5.3. he brnche pont from the vrble on the rght de of the equton to the vrble on the left. Here we hve node wth two brnche enterng nd two brnche levng. he grph h loop, whch pth tht end where t trted. he gn of the loop t mm. A loop d to touch nother loop or pth when t hre node wth t. We cn olve the equton wrtten bove to get t m ', t mm where you cn ee tht the numertor the gn of the pth from mnu the loop gn. to b, nd the denomntor When there re more thn pth between two node, the contrbuton from ll pth re dded ccordng to Mon' rule: G, G where G the gn of pth, the determnnt of the grph, nd the cofctor of pth. he determnnt of the grph gven by 3

4 P P P... n n n3 n n n where cofctor Pnr the gn product of the n-th poble combnton of r non-touchng loop. he n of pth the determnnt of the loop tht do not touch the pth. b b m b m b m b b = m t mm = b m Fgure L5.4. Combnng two networ nd gnl flow grph. he output nd nput port re on the me networ. A n exmple, let u loo t combnng two networ, Fgure L5.4 nd the octed gnl flow grph, Fgure L5.4. Let u olve for '. here drect pth between b nd, nd lo n ndrect pth t mm wth loop tht h gn of t mm, o from Mon' rule we get: t mm ' t mm nce G, tmm, nd. Notce tht we hve lredy derved th n the pt for the nput nd output -prmeter of blterl two-port networ. A number of crcut multor ue method referred to ub-networ growth nd gnl flow grph. Let u brefly dcu th on n exmple of the drectonl coupler, Fgure L5.5 Frt, the crcut nterpreted eght prt: four trnmon lne nd four tee. hen, the prt re oned n pr to me four new three-port hown n the fgure. Next, two four-port re formed, ABCD nd EFGH, whch re then oned to me x-port. Fnlly, two of the x port re oned. In the proce, only two type of connecton re olved, Fg.L5.6: when the port re on dfferent networ, nd when the onng port re on the me networ. We hve lredy olved ce, where the onng port re nd m. For the reultng mtrx, we got: t mm ', tmm t lm l '. t mm 4

5 A B C H D G F E 3 4 AB G H EF C D 3 4 ABCD EFGH 5 3 ABCD EFGH Fgure L5.5. Illutrton of gnl flow-grph pplcton to crcut multon: ubnetwor growth on the exmple of brnch lne coupler. b b m m b bm bl l b b b l m m Fgure L5.6. A ont between port nd m on two dfferent networ nd on the me networ. For prctce, fnd the gnl flow grph nd olve for ' for n nternl connecton, Fgure L5.6. h bt more complcted thn the prevou ce. L5.. IGNAL FLOW-GRAPH OF A OURCE AND RANDUCER GAIN In mny boo, uch Gonzle mplfer boo, gnl flowgrph re ued to derve the gn defnton tht cn lo be derved lgebrclly. For th, t mportnt to fnd the gnl flowgrph of ource (genertor), hown n Fg.L5.7. 5

6 Fg.L5.7. gnl flow-grph of ource connected to lod. In th ce, the wve c provdng power t the expene of ome non-electrcl energy ource (mechncl, chemcl ) o the rrow drwn f t come from nowhere. We ee tht Conder the defnton of trnducer gn n exmple. he power delvered to the lod found P L b b L, he vlble power from the ource n th ce b Pv,. Now the trnducer gn become PL b G L P v, b. he rto of the two reflected wve found ung gnl flow-grph theory referrng to Fg.L5.7b, where there one drect pth nd two frt-order loop, the cofctor of tht pth, nd the determnnt contn the frt two um n Mon formul: b. b L L L h cn be ubttuted nto the prevou expreon for trnducer gn nd rerrnged to gve the mot commonly ued form of the trnducer gn formul: G L. L L Another common pplcton of gnl flow-grph fndng the error prmeter n networ nlyzer meurement, well the relted problem of de-embeddng. 6