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Signal Flow Graph Lecture On Signal Flow Graph Submitted By: Submitted By: Ms. Anupam Mittal A.P., EE Deptt SBSSTC, Ferozepur SBSSTC, Ferozepur

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Flow of PPT What is Signal Flow Graph (SFG)? Definitions of terms used in SFG Rules for drawing of SFG Masons Gain formula SFG from simultaneous eqns SFG from differential eqns Examples Solution of a problem by Block diagram reduction technique and SFG SFG from a given Transfer function Examples

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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 nt 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.

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

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SFG

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Node: It is a point representing a variable. x 2 = t 12 x 1 +t 32 x 3 X2X2 X1X1 X2X2 X3X3 t 12 t 32 X1X1 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

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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 X1X1 t 23 X3X3 X4X4 X2X2 t 34 t 43

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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

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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 dont 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

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

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Rules for drawing of SFG from Block diagram All variables, summing points and take off points are represented by nodes. If a summing point is placed before a take off point in the direction of signal flow, in such a case the summing point and take off point shall be represented by a single node. If a summing point is placed after a take off point in the direction of signal flow, in such a case the summing point and take off point shall be represented by separate nodes connected by a branch having transmittance unity.

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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) Masons Gain Formula

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Ex: SFG from BD

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EX: To find T/F of the given block diagram

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Identification of Forward Paths P 1 = 1.1.G 1.G 2. G 3. 1 = G 1 G 2 G 3 P 2 = 1.1.G 2. G 3. 1 = G 2 G 3

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Individual Loops L 1 = G 1 G 2 H 1 L 2 = - G 2 G 3 H 2 L 3 = - G 4 H 2

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L 4 = - G 1 G 4 L 5 = - G 1 G 2 G 3

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Construction of SFG from simultaneous equations

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t 21 t 23 t 31 t 32 t 33

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After joining all SFG

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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: (a) Step 1: Solve the above eqn for highest order

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/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.

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Step5: Redraw the SFG as shown.

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Problem: to find out loops from the given SFG

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Ex: Signal-Flow Graph Models

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P 1 = P 2 =

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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

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Ex:

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Forward Paths

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L 5 = -G 4 H 4 L 1 = -G 5 G 6 H 1 L 3 = -G 8 H 1 L 2 = -G 2 G 3 G 4 G 5 H 2 L 4 = - G 2 G 7 H 2 Loops

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L 7 = - G 1 G 2 G 7 G 6 H 3 L 6 = - G 1 G 2 G 3 G 4 G 8 H 3 L 8 = - G 1 G 2 G 3 G 4 G 5 G 6 H 3

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Pair of Non-touching loops L 4 L 5 L 3 L 7 L 4 L 5 L 7 L 4 L 5 L 3 L 4

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Non-touching loops for paths 1 = 1 2 = -G 4 H 4 3 = 1

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Signal-Flow Graph Models

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Block Diagram Reduction Example _ + _ + + +

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R

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R

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R

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_ +

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Solution for same problem by using SFG

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Forward Path P 1 = G 1 G 2 G 3

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

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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 ]

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SFG from given T/F

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Ex:

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Example of block diagram Step 1: Shift take off point from position before a block G 4 to position after block G 4

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Step2 : Solve Yellow block. Step3: Solve pink block. Step4: Solve pink block.

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