Signal flow graph Mason’s Gain Formula

Signal flow graph
Mason’s Gain Formula

1. Topic Signal Flow Graph

2. What is Signal Flow Graph? SFG is a diagram which represents a set of simultaneous equations. This method was developed by S.J.Mason . This method does n’t require any reduction technique. It consists of nodes and these nodes are connected by a directed line called branches. Every branch has an arrow which represents the flow of signal. For complicated systems, when Block Diagram (BD) reduction method becomes tedious and time consuming then SFG is a good choice.

3. Comparison of BD and SFG block diagram: signal flow graph: In this case at each step block diagram is to be redrawn. That’s why it is tedious method. So wastage of time and space. Only one time SFG is to be drawn and then Mason’s gain formula is to be evaluated. So time and space is saved.

4. SFG

5. Node : It is a point representing a variable. x 2 = t 12 x 1 +t 32 x 3 X 2 X 1 X 2 X 3 t 12 t 32 X 1 Branch : A line joining two nodes. Input Node : Node which has only outgoing branches. X 1 is input node. In this SFG there are 3 nodes. Definition of terms required in SFG

6. Output node/ sink node : Only incoming branches. Mixed nodes : Has both incoming and outgoing branches. Transmittance : It is the gain between two nodes. It is generally written on the branch near the arrow. t 12 X 1 t 23 X 3 X 4 X 2 t 34 t 43

7. Path : It is the traversal of connected branches in the direction of branch arrows, such that no node is traversed more than once. Forward path : A path which originates from the input node and terminates at the output node and along which no node is traversed more than once. Forward Path gain : It is the product of branch transmittances of a forward path. P 1 = G 1 G 2 G 3 G 4, P 2 = G 5 G 6 G 7 G 8

8. Loop : Path that originates and terminates at the same node and along which no other node is traversed more than once. Self loop : Path that originates and terminates at the same node. Loop gain : it is the product of branch transmittances of a loop. Non-touching loops : Loops that don’t have any common node or branch. L 1 = G 2 H 2 L 2 = H 3 L 3 = G 7 H 7 Non-touching loops are L1 & L2, L1 & L3, L2 &L3

9. SFG terms representation input node (source) mixed node mixed node forward path path loop branch node transmittance input node (source)

10. Mason’s Gain Formula A technique to reduce a signal-flow graph to a single transfer function requires the application of one formula. The transfer function, C(s)/R(s), of a system represented by a signal-flow graph is k = number of forward path P k = the kth forward path gain ∆ = 1 – (Σ loop gains) + (Σ non-touching loop gains taken two at a time) – (Σ non-touching loop gains taken three at a time)+ so on . ∆ k = 1 – (loop-gain which does not touch the forward path)

11. Ex: SFG from BD

12. Construction of SFG from simultaneous equations

13. t 21 t 23 t 31 t 32 t 33

14. After joining all SFG

15. SFG from Differential equations Consider the differential equation Step 2: Consider the left hand terms (highest derivative) as dependant variable and all other terms on right hand side as independent variables. Construct the branches of signal flow graph as shown below:- 1 -5 -2 -3 (a) Step 1: Solve the above eqn for highest order

16. 1 -2 -5 -3 1/s 1/s 1/s Step 3: Connect the nodes of highest order derivatives to the lowest order der.node and so on. The flow of signal will be from higher node to lower node and transmittance will be 1/s as shown in fig (b) (b) Step 4: Reverse the sign of a branch connecting y’’’ to y’’, with condition no change in T/F fn.

17. Step5: Redraw the SFG as shown.

18. Problem: to find out loops from the given SFG

19. Ex: Signal-Flow Graph Models

20. P 1 = P 2 =

21. Individual loops L 1 = G 2 H 2 L 4 = G 7 H 7 L 3 = G 6 H 6 L 2 = G 3 H 3 Pair of Non-touching loops L 1 L 3 L 1 L 4 L 2 L 3 L 2 L 4

22. Block Diagram Reduction Example _ + _ + + +

23. R R

24. R

25. _ +

26. Solution for same problem by using SFG

27. Forward Path P 1 = G 1 G 2 G 3

28. Loops L 1 = G 1 G 2 H 1 L 2 = - G 2 G 3 H 2

29. L 3 = - G 1 G 2 G 3 P 1 = G 1 G 2 G 3 L 1 = G 1 G 2 H 1 L 2 = - G 2 G 3 H 2 L 3 = - G 1 G 2 G 3 ∆ 1 = 1 ∆ = 1- (L 1 + L 2 +L 3 ) T.F= (G 1 G 2 G 3 )/ [1 -G 1 G 2 H 1 + G 1 G 2 G 3 + G 2 G 3 H 2 ]

30. SFG from given T/F

31. Thanks ……….