• nodes : variables• branches : gainse.g. y = a · xe.g. y = 3x + 5z – 0.1ySignal Flow Graphxyaxzy53-0.1e.g.G1ryG2-+ uryuG11-G2e.g. (3.6)Ry++-G1G2G3H1-x+z++NRyzG31-H1xG1G2-1N1Note: One node is introduced after each summationMason’s Rule• A forward path: a path from input to output• Forward path gain Mx: total product of gains along the path• A loop is a closed path in which you can start at any point, follow the arrows, and come back to the same point• A loop gain Li: total product of gains along a loop• Loop i and loop j are non-touching if they do not share any nodes or branches• The determinant ∆:∑∑∑∑−⋅⋅⋅+⋅⋅−⋅+−=∆−loopstnallmkjiloopstnallkjiloopsofpairstouchingnonalljiialliLLLLLLLLLL4..3.....1• ∆x: The determinant of the S.F.G. after removing the k-th forward path• Mason’s Rule:∑∆∆⋅==pathsforwardallxxiMyyFTOI0..e.g. (3.6)Ry++-G1G2G3H1-x+z++NRyzG31-H1xG1G2-1N13.6: Get T.F. from N to y1 f.p.: N yM = 12 loops: L1= -H1G3L2= -G2G3∆ = same∆1: remove N, y, N y∆1= 1321311111GGHGMMNykk++=∆=∆∆=∆∆=∑3.6 (cont.): Get T.F. from R to y2 f.p.: R x z y : M1=G2G3Rz y : M2=G1G32 loops: L1= -G3H1L2= -G2G33213..11 GGHGLLLTNjialli++=⋅+−=∆∑∑HRNyNy+∆=⇒∆=1103.6 (cont.)∆1: remove M1and compute ∆∆1= 1∆2: remove M2and compute ∆∆2= 1321331321 GGHGGGGGMMRyHkkk+++=∆=∆∆==∑∑Figure 3-16 (p. 57)4132124232121:loops FiveGGGGGHGHGGHGG4123211:paths forward TwoGGMGGGM==4132124232121413211R(s)Y(s):gain TotalGGGGGHGHGGHGGGGGGG++++++=41321242321211:tDeterminanGGGGGHGHGGHGG +++++=∆1loops. no 1,path forward removingAfter 1=∆∴1loops. no 2,path forward removingAfter 2=∆∴s1s1xys1b3b2b1-a1-a2-a3x2x1ex3ΣΣx3eyxx2x1b1b3b2-a3-a2-a111s1s1sExample:s1s1xys1b3b2b1-a1-a2-a3x2x1ex3ΣΣ• Forward paths:sbMsbMsbM13222331===• Loops:33322211saLsaLsaL−=−=−=Determinant:∆1: If M1is taken out, all loops are broken.therefore ∆1 = 1∆2: If M2is taken out, all loops are broken.therefore ∆2 = 1∆3: Similarly, ∆3 = 13322111sasasaLialli+++=−=∆∑33221332213211..sasasasbsbsbMMMMFTii+++++=∆=∆∆=∴∑H4H1H2H3H5H6H7Forward path:M1= H1 H2 H3M2= H4 Loops:L1= H1 H5L2= H2 H6L3= H3 H7L4= H4 H7 H6 H5L1and L3are non-touching∆1: If M1is taken out, all loops are broken.therefore ∆1 = 1∆2: If M2is taken out, the loop in the middle (L2) is still there.therefore ∆2 = 1 – L2= 1 – H2H6Total