Feedback and Control System Signal Flow Graphs Page 1 of 15 Topic: Signal Flow Graphs Welcome Notes: WELCOME ENGINEERING STUDENTS. I. INTRODUCTION: Block diagram reduction is the excellent method for determining the transfer function of the control system. However, in a complicated system, it is very difficult and time-consuming process that is why an alternate method, i.e., SFG was developed by S.J Mason which relates the input and output system variables graphically. In the signal flow graph, the transfer function is referred to as transmittance. Characteristics of SFG: SFG is a graphical representation of the relationship between the variables of a set of linear algebraic equations. It doesn't require any reduction technique or process. o It represents a network in which nodes are used for the representation of system variable which is connected by direct branches. o SFG is a diagram which represents a set of equations. It consists of nodes and branches such that each branch of SFG having an arrow which represents the flow of the signal. o It is only applicable to the linear system. II. OBJECTIVES: At the end of this module, you should be able to: 1. Define and enumerate the basic elements of a Signal Flow Graph (SFG). 2. Identify the different common terms used in a signal flow graph 3. Illustrate and solve the algebraic equation base on the Signal Flow Graph (SFG).. Feedback and Control System Signal Flow Graphs Page 2 of 15 III. PRELIMINARY ACTIVITIES: Before you proceed to the main lesson, test yourself in this activity. Give the Algebraic Expression Study the figure carefully and write the common function in the space below. _________________________________ ANSWER GREAT!!! You may now proceed to the main lesson. Feedback and Control System Signal Flow Graphs Page 3 of 15 IV. LESSON PROPER Based on the preliminary activities, what did you notice about it? ________________________________________________________ CONGRATULATIONS! You may now proceed to the lesson. Signal flow graph is a graphical representation of algebraic equations. In this module, let us discuss the basic concepts related signal flow graph and also learn how to draw signal flow graphs. Basic Elements of Signal Flow Graph Nodes and branches are the basic elements of signal flow graph. Node Node is a point which represents either a variable or a signal. There are three types of nodes — input node, output node and mixed node. Input Node − It is a node, which has only outgoing branches. Output Node − It is a node, which has only incoming branches. Mixed Node − It is a node, which has both incoming and outgoing branches. Example Let us consider the following signal flow graph to identify these nodes. The nodes present in this signal flow graph are y1, y2, y3 and y4. y1 and y4 are the input node and output node respectively. y2 and y3 are mixed nodes. Feedback and Control System Signal Flow Graphs Page 4 of 15 Branch Branch is a line segment which joins two nodes. It has both gain and direction. For example, there are four branches in the above signal flow graph. These branches have gains of a, b, c and -d. Construction of Signal Flow Graph Let us construct a signal flow graph by considering the following algebraic equations − There will be six nodes (y1, y2, y3, y4, y5 and y6) and eight branches in this signal flow graph. The gains of the branches are a12, a23, a34, a45, a56, a42, a53 and a35. To get the overall signal flow graph, draw the signal flow graph for each equation, then combine all these signal flow graphs and then follow the steps given below − Step 1 − Signal flow graph for y2 = a13y1 + a42y4 is shown in the following figure. Step 2 − Signal flow graph for y3 = a23y2 + a53y5 is shown in the following figure. Step 3 − Signal flow graph for y4 = a34y3 is shown in the following figure. Feedback and Control System Signal Flow Graphs Page 5 of 15 Step 4 − Signal flow graph for y5 = a45y4 + a35y3 is shown in the following figure. Step 5 − Signal flow graph for y6 = a65y5 is shown in the following figure. Step 6 − Signal flow graph of overall system is shown in the following figure. Conversion of Block Diagrams into Signal Flow Graphs Follow these steps for converting a block diagram into its equivalent signal flow graph. 1. Represent all the signals, variables, summing points and take-off points of block diagram as nodes in signal flow graph. 2. Represent the blocks of block diagram as branches in signal flow graph. Feedback and Control System Signal Flow Graphs Page 6 of 15 3. Represent the transfer functions inside the blocks of block diagram as gains of the branches in signal flow graph. 4. Connect the nodes as per the block diagram. If there is connection between two nodes (but there is no block in between), then represent the gain of the branch as one. For example, between summing points, between summing point and takeoff point, between input and summing point, between take-off point and output. Sample Problem: Let us convert the following block diagram into its equivalent signal flow graph. Represent the input signal R(s) and output signal C(s) of block diagram as input node R(s) and output node C(s) of signal flow graph. Just for reference, the remaining nodes (y1 to y9 ) are labelled in the block diagram. There are nine nodes other than input and output nodes. That is four nodes for four summing points, four nodes for four take-off points and one node for the variable between blocks G1 and G2. The following figure shows the equivalent signal flow graph. With the help of Mason’s gain formula (discussed in the next chapter), you can calculate the transfer function of this signal flow graph. This is the advantage of signal flow graphs. Here, we no need to simplify (reduce) the signal flow graphs for calculating the transfer function. Feedback and Control System Signal Flow Graphs Page 7 of 15 Terms used in Signal Flow Graph The common terms used in a signal flow graph are as follows: 1. Node: The small points or circles in the signal flow graph are called nodes and these are used to show the variables of the system. Each node corresponds to an individual variable. In the figure given below x1 to x8 are nodes. 2. Input node or source: The one having branches with only outgoing signal is known as the input node or source. For the below shown SFG, x1 is the source. 3. Output node or sink: A node that only contains branches with entering signal is known as a sink or output node. Both input and output nodes act as dependent variables. Here, x8 is the sink. 4. Mixed node: It is also known as chain node and consists of branches having both entering as well as leaving signals. For this SFG, x2 to x7 are mixed nodes. 5. Branch: The lines in the SFG that join one node to the other is known as a branch. It shows the relationship existing between various nodes. 6. Transmittance: The transfer function or gain between two nodes is said to be the transmittance of the SFG. Its value can be either real or complex. These are from ‘a’ to ‘j’ here. 7. Forward path: A path proceeding from input to output is known as forward path. Here one of the paths is x6 – x7 – x8. 8. Feedback loop: A loop which is formed through a path starting from a specific node and ending on the same node after travelling to at least one node of the graph is known as a feedback loop. It is to be noted that in the feedback loop, no node should be traced twice. Here the loop is x2 to x6 then back to x2. Feedback and Control System Signal Flow Graphs Page 8 of 15 9. Self-loop: A self-loop is the one which has only a single node. The paths in such loops are never defined by a forward path or feedback loop as these never trace any other node of the graph. Here it is formed at node x5 having gain f. 10. Path gain: When the signal flows in a forward path then the product of gains of each branch gives the overall path gain. For above-given graph, i*j is path gain for forward path, x6 to x8. 11. Loop gain: The product of individual gains of the branches that are in the form of loop is said to be the overall loop gain. For the loop forming between x3 and x4 the gain is c*d. 12. Non-touching loops: When two or more loops do not share common nodes then such loops are called nontouching loops. For this SFG, the two non-touching loops are x2-x6-x2 and x3-x4-x3. We had just finished the discussion on Signal Flow Graphs. Let’s move on to the next higher level of activity/ies or exercise/s that demonstrates your potential skills/knowledge of what you have learned. Feedback and Control System Signal Flow Graphs Page 9 of 15 V. ANALYSIS, APPLICATION AND EXPLORATION ACTIVITY 1 Name: ________________________________________ Define and illustrate the basic elements of a Signal Flow Graph (SFG). Course & Section: _________________ Feedback and Control System Signal Flow Graphs Page 10 of 15 ACTIVITY 2 Name: ________________________________________ Course & Section: _________________ Enumerate and define the different terminology used in a signal flow graph Feedback and Control System Signal Flow Graphs Page 11 of 15 ACTIVITY 3 Name: ________________________________________ Convert the signal flow graph below to its corresponding block diagram Course & Section: _________________ Feedback and Control System Signal Flow Graphs Page 12 of 15 Finally, let us summarize the lesson of what we had discussed today. VI. GENERALIZATION SUMMARY OF OUR LESSON 1. __________________________________________________________________________________ 2. __________________________________________________________________________________ 3. __________________________________________________________________________________ KUDOS! You have come to an end of Module 9. OOPS! Don’t forget that you have still an assignment to do. Here it is…. Feedback and Control System Signal Flow Graphs Page 13 of 15 VII. ASSIGNMENT Name: ________________________________________ Course & Section: _________________ Direction/Instruction: Use an additional sheet of paper for your answer, (This assignment will be submitted on ________________.) Convert the block diagram into the corresponding signal flow graph. Note: First, represent each and every take-off and summing point of the above-given block diagram as a separate node. Then replace the blocks of the above figure by branches in the signal flow graph. After your long journey of reading and accomplishing the module, let us now challenge your mind by answering the evaluation part of this module. Feedback and Control System Signal Flow Graphs Page 14 of 15 VIII. EVALUATION Name:______________________________ Course & Section: __________________ Direction/Instruction: ENCIRCLE the correct answer ( Any form of erasure are NOT allowed) (This evaluation will be submitted on ________________.) 1. It is a graphical representation of algebraic equations. A. Signal flow graph C. Block B. Block Diagram D. Node 2. It is a point which represents either a variable or a signal. A. Signal flow graph C. Block B. Block Diagram D. Node 3. It is a node, which has only incoming branches. A. Signal flow graph C. Block B. Block Diagram D. Output Node 4. A line segment which joins two nodes. A. Signal flow graph C. Block diagram B. Branch D. Output Node 5. It is a node, which has both incoming and outgoing branches. A. Signal flow graph C. Mixed Node B. Branch node D. Output Node 6. The transfer function or gain between two nodes is said to be the __________ of the SFG A. Node C. Transmittance B. Branch node D. Output Node Feedback and Control System Signal Flow Graphs Page 15 of 15 7. A path proceeding from input to output is known as ___________. A. Node C. Feedback loop B. Forward path: D. Output Node 8. A loop which is formed through a path starting from a specific node and ending on the same node after travelling to at least one node of the graph. A. Feedback loop C. Self-loop B. Forward path D. Output Node 9. Non-touching loops: When two or more loops do not share common nodes then such loops. A. Non-touching loops C. Self-loop B. Forward path D. Output Node 10. It is the one which has only a single node. A. Non-touching loops C. Self-loop B. Forward path D. Output Node CONGRATULATIONS on reaching the end of this module! You may now proceed to the next module. Don’t forget to submit all the exercises, activities and portfolio on ___________________. KEEP UP THE GOOD WORK. Well Done!!!
