Let vi as input node, the output node v, can be found as follows 1R1 1/R2 1/R 1/E2 1/R1 1/2 I/R Ra 1∥R2 2022-2-3 26
2022-2-3 26 Let as input node, the output node can be found as follows: 1 v 3 v
2.5 General input-output gain transfer function o Let ir denote the ration between the input and the output. For the signal flow diagram representation, it becomes T=xI Definition 10: Non-touching two loops, paths, or loop and path are said to be non-touching if they have no nodes In common 2022-2-3 27
2022-2-3 27 2.5 General input-output gain transfer function ¨ Let denote the ration between the input and the output. For the signal flow diagram representation, it becomes ¨ Definition 10: Non-touching two loops, paths, or loop and path are said to be non-touching if they have no nodes in common. R C T l n X X T
Definition 11: Signal Flow Graph Determinant(or characteristic function); A is defined as follows A=1-(1)∑∑Pk the ith forward path gain Pk= jth possible product of k non-touching loop gains ∑P+∑P2-∑ P,+ 2022-2-3 28
2022-2-3 28 P i P jk j jk k k P 1 1 (1) j j j j Pj1 Pj 2 P 3 1 = the ith forward path gain = jth possible product of k non-touching loop gains Definition 11: Signal Flow Graph Determinant (or characteristic function); is defined as follows:
1-(sum of all loop gains)+( sum of all gain-products of 2 non-touching loops)-(sum of all gain-products of 3 non- touching loops )+ A =A evaluated with all loops touching Pi eliminated The general formula for any signal flow graph is ∑ P△ R △ 2022-2-3 29
2022-2-3 29 1-(sum of all loop gains) + (sum of all gain-products of 2 non-touching loops) -(sum of all gain-products of 3 nontouching loops) +… i i Pi i R C T The general formula for any signal flow graph is = evaluated with all loops touching Pi eliminated
Example: 1 =H There is only one forward path; hence P=G 0 There is only one(feedback)loop. Hence 11= IGH k=0,j≠1,k≠1 2022-2-3 30
2022-2-3 30 Example: 2 3 0 1 P P P G 0, 1, 1, 11 P j k P GH jk There is only one forward path; hence There is only one (feedback) loop. Hence