ECE3050 Mason’s Flo w G raph FormulaAmplifier analysis using superposition is often facilitated by th e use of signal flow graphs.Asignalflo w graph, or flow graph for short, is a grap hical represen tation of a set of linearequations which can be used to write by inspection the solution to the set of equations. Forexamp le, consider the set of equationsx2= Ax1+ Bx2+ Cx5(1)x3= Dx1+ Ex2(2)x4= Fx3+ Gx5(3)x5= Hx4(4)x6= Ix3(5)where x1through x6are variables and A through I are constants. These equations can berepresented graphically as sho w n in Fig. 1. The graph has a node for eac h variable withbranches co nnecting the nodes labeled with the consta nts A through I. The node labeled x1is called a source node becau se it has only outgoing br anches. The node labeled x6is calleda sink node because it has only incoming branches. The p ath from x1to x2to x3to x6iscalled a forward path because it originates at a source node and terminates at a non-sourcenode and along which no node is encountered t wice. The path gain for this forward path isAEI.Thepathfromx2to x3to x4to x5and back to x2is called a feedback path becauseit originates and termina tes on the same node and along which no node is encou ntered morethan once. The loop gain for this feedback path is EFHC.Figure 1: Example signal flo w graph.Mason ’s formula can be used to calculate th e transmission gain from a source node to anynon-source node in a flow graph. The form u la isT =1∆XkPk∆k(6)1where Pkis the gain of the kth forw ar d path, ∆ is the graph determinan t, and ∆kis thedeterm ina nt with the kth forw a rd path erased. The determinant is giv en by∆ =1− (sum of all loop gain s)+µsum of the gain products of all possiblecombinations of two non-touching loop s¶−µsum of the gain products of all possiblecombinations of three non-touc hing loops¶+µsum of the gain products of all possiblecom binations of four non-touc hing loops¶− ··· (7)For the flow graph in Fig. 1, the objectiv e is to solve for the gain from node x1to node x6.There are tw o forward paths from x1to x6and three loops. Two of the loops do not touc heach other. Thus the product of these two loop gains appears in the expression for ∆.Thepath gains and the determinant are given byP1= AEI (8)P2= DI (9)∆ =1− (B + CEFH + GH )+B × GH (10)Path P1touches t wo loops while path P2touc hes one loop. The determinants with each patherased are given by∆1=1− GH (11)∆2=1− (B + GH)+B × GH (12)Thus the o verall gain from x1to x6is given byx6x1=AEI × (1 − GH )+DI × [1 − (B + GH)+B × GH]1 − (B + CEFH + GH)+B ×
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