Signal flow graph

Suthar Chandresh

1. GANDHINAGAR INSTITUTE OF TECHNOLOGY CONTROL ENGINEERING (2151908 ) Active learning assignment On SIGNAL FLOW GRAPH Prepared by:- 1). Sonani Manav 140120119223 2). Suthar Chandresh 140120119229 3). Tade Govind 140120119230 Guided by :-Asst. Prof. Kashyap Ramaiya

2. Outline Introduction to Signal Flow Graphs Definitions Terminologies Signal-Flow Graph Models BD to SFG Example Mason’s Gain Formula Example 6/09/2016 2

3. Introduction 3 Definition:- “ A signal flow graph is a graphical representation of the relationship between variables of a set of linear algebraic equation.” A signal-flow graph consists of a network in which nodes are connected by directed branches. It depicts the flow of signals from one point of a system to another and gives the relationships among the signals. 6/09/2016

4. Fundamentals of Signal Flow Graphs Consider a simple equation below and draw its signal flow graph: The signal flow graph of the equation is shown below; Every variable in a signal flow graph is designed by a Node . Every transmission function in a signal flow graph is designed by a Branch . Branches are always unidirectional . The arrow in the branch denotes the direction of the signal flow. 6/09/2016 4

5. Terminologies An input node or source contain only the outgoing branches. i.e., X 1 An output node or sink contain only the incoming branches. i.e., X 4 A path is a continuous, unidirectional succession of branches along which no node is passed more than ones. i.e., A forward path is a path from the input node to the output node. i.e., X 1 to X 2 to X 3 to X 4 , and X 1 to X 2 to X 4 , are forward paths. A feedback path or feedback loop is a path which originates and terminates on the same node. i.e.; X 2 to X 3 and back to X 2 is a feedback path. X 1 to X 2 to X 3 to X 4 X 2 to X 3 to X 4 X 1 to X 2 to X 4 6/09/2016 5

6. Terminologies A self-loop is a feedback loop consisting of a single branch. i.e.; A 33 is a self loop. The gain of a branch is the transmission function of that branch. The path gain is the product of branch gains encountered in traversing a path. i.e. the gain of forwards path X 1 to X 2 to X 3 to X 4 is A 21 A 32 A 43 The loop gain is the product of the branch gains of the loop. i.e., the loop gain of the feedback loop from X 2 to X 3 and back to X 2 is A 32 A 23 . Two loops, paths, or loop and a path are said to be non-touching if they have no nodes in common. 6/09/2016 6

7. SFG terms representation input node (source) mixed node mixed node forward path path loop branch node transmittance input node (source) 6/09/2016 7

8. Signal-Flow Graph Models b x 4 x 3 x 2 x 1 x h f g e d c a x o is input and x 4 is output

9. Construct the signal flow graph for the following set of simultaneous equations. There are four variables in the equations (i.e., x 1 ,x 2 ,x 3 ,and x 4 ) therefore four nodes are required to construct the signal flow graph. Arrange these four nodes from left to right and connect them with the associated branches. Another way to arrange this graph is shown in the figure.

10. BD to SFG 6/09/2016 10 Block Diagram Signal Flow Graph

11. BD to SFG 6/09/2016 11 Block Diagram Block Diagram Signal Flow Graph

12. Example:- 6/09/2016 12

13. Mason’s Rule (Mason, 1953) The block diagram reduction technique requires successive application of fundamental relationships in order to arrive at the system transfer function. On the other hand, Mason’s rule for reducing a signal-flow graph to a single transfer function requires the application of one formula. The formula was derived by S. J. Mason when he related the signal-flow graph to the simultaneous equations that can be written from the graph. 6/09/2016 13

14. Mason’s Rule: The transfer function T , of a system represented by a signal-flow graph is; Where, n = number of forward paths. P i = the i th forward-path gain. ∆ = Determinant of the system ∆ i = Determinant of the i th forward path ∆ is called the signal flow graph determinant or characteristic function. Since ∆= is the system characteristic equation. 6/09/2016 14

15. Mason’s Rule: ∆ = 1- (sum of all individual loop transmittance) + (sum of the products of loop transmittance of all possible pairs of Non Touching loops) – (sum of the products of loop transmittance of Triple of Non Touching loop) + … ∆ i = Calculate ∆ for i th path =1- All the loops that do not touch the i th forward path 6/09/2016 15

16. Systematic approach Calculate forward path gain P i for each forward path i . Calculate all loop transfer functions. Consider non-touching loops 2 at a time. Consider non-touching loops 3 at a time. etc Calculate Δ from steps 2,3,4 and 5 Calculate Δ i as portion of Δ not touching forward path i 16 6/09/2016

17. 6/09/2016 17 Example 1: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph

18. Continued….. In this system there is only one forward path between the input R(s) and the output C(s). The forward path gain is we see that there are three individual loops. The gains of these loops are Note that since all three loops have a common branch, there are no non-touching loops. Hence, the determinant is given by There is no any non touching loop so we get, Therefore, the overall gain between the input and the output or the closed loop transfer function, is given by   6/09/2016 18  

19. Therefore, There are three feedback loops Example 2 : Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph 6/09/2016 19

20. ∆ = 1- (sum of all individual loop gains) There are no non-touching loops, therefore Example2: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph 6/09/2016 20

21. ∆ 1 = 1- (sum of all individual loop gains)+... Eliminate forward path-1 ∆ 1 = 1 ∆ 2 = 1- (sum of all individual loop gains)+... Eliminate forward path-2 ∆ 2 = 1 Example2 : Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph 6/09/2016 21

22. Example 2: Continue 6/09/2016 22

23. THANK YOU 6/09/2016 23