Signal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices:
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1 3/3/009 ignl Flow Grphs / ignl Flow Grphs Consider comple 3-port microwve network, constructed of 5 simpler microwve devices: where n is the scttering mtri of ech device, nd is the overll scttering mtri of the entire 3-port network. Q: Is there n w to determine this overll network scttering mtri from the individul device scttering mtrices? n A: Definitel! Note the wve eiting one port of device is wve entering (i.e., incident on) nother (nd vice vers). This is oundr condition t the port connection etween devices. Jim tiles The Univ. of Knss Dept. of EEC
2 3/3/009 ignl Flow Grphs / Add to this the scttering prmeter equtions from ech individul device, nd we hve sufficient mount of mth to determine the reltionship etween the incident nd eiting wves of the remining three ports in other words, the scttering mtri of the 3-port network! Q: Yikes! Wouldn t tht require lot of tedious lger! A: It sure would! We might use computer to ssist us, or we might use tool emploed since the erl ds of microwve engineering the signl flow grph. ignl flow grphs re helpful in (count em ) three ws! W - ignl flow grphs provide us with grphicl mens of solving lrge sstems of simultneous equtions. W We ll see the signl flow grph cn provide us with rod mp of the wve propgtion pths throughout microwve device or network. If we re ping ttention, we cn glen gret phsicl insight s to the inner working of the microwve device represented the grph. W 3 - ignl flow grphs provide us with quick nd ccurte method for pproimting network or device. We will find tht we cn often replce rther comple grph with much simpler one tht is lmost equivlent. Jim tiles The Univ. of Knss Dept. of EEC
3 3/3/009 ignl Flow Grphs 3/ We find this to e ver helpful when designing microwve components. From the nlsis of these pproimte grphs, we cn often determine design rules or equtions tht re trctle, nd llow us to design components with (ner) optiml performnce. Q: But wht is signl flow grph? A: First, some definitions! Ever signl flow grph consists of set of nodes. These nodes re connected rnches, which re simpl contours with specified direction. Ech rnch likewise hs n ssocited comple vlue. 0.7 j -j Q: Wht could this possil hve to do with microwve engineering? A: Ech port of microwve device is represented two nodes the node nd the node. The node simpl represents the vlue of the normlized mplitude of the wve incident on tht port, evluted t the plne of tht port: Jim tiles The Univ. of Knss Dept. of EEC
4 3/3/009 ignl Flow Grphs 4/ ( = ) V z z n + n n np 0n ikewise, the node simpl represents the normlized mplitude of the wve eiting tht port, evluted t the plne of tht port: Vn ( zn = znp ) n 0n Note then tht the totl voltge t port is simpl: ( = ) = ( + ) 0 V z z n n np n n n The vlue of the rnch connecting two nodes is simpl the vlue of the scttering prmeter relting these two voltge vlues: n + Vn ( zn = znp ) mn V ( z = z ) 0n The signl flow grph ove is simpl grphicl representtion of the eqution: m = mn n m m m mp 0m Moreover, if multiple rnches enter node, then the voltge represented tht node is the sum of the vlues from ech rnch. For emple, the signl flow grph: Jim tiles The Univ. of Knss Dept. of EEC
5 3/3/009 ignl Flow Grphs 5/ 3 3 is grphicl representtion of the eqution: = Now, consider two-port device with scttering mtri : o tht: = = + = + We cn thus grphicll represent two-port device s: Jim tiles The Univ. of Knss Dept. of EEC
6 3/3/009 ignl Flow Grphs 6/ Now, consider cse where the second port is terminted some lod : We now hve et nother eqution: ( = ) = ( = ) V z z V z z + P P = Therefore, the signl flow grph of this terminted network is: Now let s cscde two different two-port networks Here, the output port of the first device is directl connected to the input port of the second device. We descrie this mthemticll s: Jim tiles The Univ. of Knss Dept. of EEC
7 3/3/009 ignl Flow Grphs 7/ Jim tiles The Univ. of Knss Dept. of EEC = nd = Thus, the signl flow grph of this network is: Q: But wht hppens if the networks re connected with trnsmission lines? A: Recll tht length of trnsmission line with chrcteristic impednce 0 is likewise two-port device. Its scttering mtri is: 0 0 j j e e β β = Thus, if the two devices re connected length of trnsmission line: 0
8 3/3/009 ignl Flow Grphs 8/ = e = e j β j β so the signl flow grph is: e j β e j β Note tht there is one (nd onl one) independent vrile in this representtion. This independent vrile is node. This is the onl node of the sfg tht does not hve n incoming rnches. As result, its vlue depends on no other node vlues in the sfg. From the stndpoint of sfg, independent nodes re essentill sources! Of course, this likewise mkes sense phsicll (do ou see wh?). The node vlue represents the comple mplitude of the wve incident on the one-port network. If this vlue is zero, then no power is incident on the network the rest of the nodes (i.e., wve mplitudes) will likewise e zero! Jim tiles The Univ. of Knss Dept. of EEC
9 3/3/009 ignl Flow Grphs 9/ Now, s we wish to determine, for emple:. The reflection coefficient in of the one-port device.. The totl current t port of second network (i.e., network ). 3. The power sored the lod t port of the second () network. In the first cse, we need to determine the vlue of dependent node : in = For the second cse, we must determine the vlue of wve mplitudes nd : I = 0 And for the third nd finl cse, the vlues of nodes nd re required: P s = Q: But just how the heck do we determine the vlues of these wve mplitude nodes? Jim tiles The Univ. of Knss Dept. of EEC
10 3/3/009 ignl Flow Grphs 0/ A: One w, of course, is to solve the simultneous equtions tht descrie this network. From network nd network : = + = + = + = + From the trnsmission line: = e = e j β j β And finll from the lod: = But nother, EVEN BETTER w to determine these vlues is to decompose (reduce) the signl flow grph! Q: Huh? A: ignl flow grph reduction is method for simplifing the comple pths of tht signl flow grph into more direct (ut equivlent!) form. Reduction is rell just grphicl method of decoupling the simultneous equtions tht re descried the sfg. For instnce, in the emple we re considering, the sfg : Jim tiles The Univ. of Knss Dept. of EEC
11 3/3/009 ignl Flow Grphs / e j β e j β might reduce to: j 8 0. e π j From this grph, we cn directl determine the vlue of ech node (i.e., the vlue of ech wve mplitude), in terms of the one independent vrile. j 0. = 0. = 06. = j 0... j π 8 = 005 = 0e = 03. = 0. And of course, we cn then determine vlues like:. 0. = = = 0. in Jim tiles The Univ. of Knss Dept. of EEC
12 3/3/009 ignl Flow Grphs /. I 8 0. e j π 005. = = P s ( 03. ) ( 0. ) = = Q: But how do we reduce the sfg to its simplified stte? Just wht is the procedure? A: ignl flow grphs cn e reduced sequentill ppling one of four simple rules. Q: Cn these rules e pplied in n order? A: No! The rules cn onl e pplied when/where the structure of the sfg llows. You must serch the sfg for structures tht llow rule to e pplied, nd the sfg will then e ( little it) reduced. You then serch for the net vlid structure where rule cn e pplied. Eventull, the sfg will e completel reduced! Q:???? A: It s it like solving puzzle. Ever sfg is different, nd so ech will require different reduction procedure. It requires little thought, ut with little prctice, the reduction procedure is esil mstered. You m even find tht it s kind of fun! Jim tiles The Univ. of Knss Dept. of EEC
4.5 Signal Flow Graphs
3/9/009 4_5 ignl Flow Grphs.doc / 4.5 ignl Flow Grphs Reding Assignment: pp. 89-97 Q: Using individul device scttering prmeters to nlze comple microwve network results in lot of mess mth! Isn t there n
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