LECTURE THREE SIGNAL FLOW GRAPH
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1 3rd Yearomputer ommunication EngineeringU ontrol Theor LETUE THEE SIGNAL FLOW GAPH 3.1 Introduction The block diagram reduction technique is tedious and time consuming. Signal flow method gives an alternative approach for finding out transfer function of a control sstem. Signal flow graph is a network diagram consisting of nodes, branches and arrows. Nodes represent variables or signals in a sstem. The nodes are connected b branches and arrows marked on branches indicate direction of flow signal. t If =t, then the signal flow graph is; The following rules appl as well for the signal flow graph; t1 t2 t1 t2 t1 t1 t2 t2 t1 t3 t1 / (1t1 t2) t2 1 2 t1 t2 1 2 t1 t3 t2 t3 Dr. Mohammed Saheb Khesbak Page 19
2 3rd Yearomputer ommunication EngineeringU ontrol Theor 1 t1 2 t2 3 1 (t1 t2) / (1t2 t3) 3 t3 Eamples 3.1: Draw signal flow graph for the following equations: 1 d/dt a1 2 d1/dt d/dt d 2 /dt d 2 /d 2 1 d/d 2/3 1 11/2 Another solution ma be introduced b taking the Laplace transform of the equation as follows; ( ) ( ) ( ) ( ) Dr. Mohammed Saheb Khesbak Page 20
3 3rd Yearomputer ommunication EngineeringU ontrol Theor ( ) ( ) ( ) Therefore; the signal flow graph is shown; X(s) 1 1/S 1/S 1 2/3 11/2 Y(s) 3.2 Mason s Gain Formula Mason gave a formula relating the output and input. The formula is: Δ = 1 (sum of all individual loops)(gain product of all possible combinations of two nontouching loops). Δ K = same as for Δ but formed b loops not touching the k th forward path. P K = gain of k th forward paths. Dr. Mohammed Saheb Khesbak Page 21
4 3rd Yearomputer ommunication EngineeringU ontrol Theor Eample 3.2: For the sstem shown, obtain the closed loop transfer function. Solution: Forward paths: P1=G1 G2 G3 and P2= Loops: L1=G2, L2=G1G2, and L3= G2G3H2 Now; Δ=1(L1L2L3) = 1 G2 G1G2 G2G3H2 Δ1= 1 and Δ2= Δ therefore; ( ) Dr. Mohammed Saheb Khesbak Page 22
5 3rd Yearomputer ommunication EngineeringU ontrol Theor Eample 3.3: Find / for the sstem shown below using signal flow graph technique. G2 G1 G3 H2 The signal flow graph for the sstem block diagram above is shown as; G2 1 1 G1 1 G3 H2 Forward paths: P1=G1 G3, Δ1=1 P2=G1 G2, Δ2=1 P3=G1 G3 H2, Δ3=1 P4 =G1 G2 H2, Δ4=1 Dr. Mohammed Saheb Khesbak Page 23
6 3rd Yearomputer ommunication EngineeringU ontrol Theor Loops: L1= G1 G3 H2 and L2= G1 G2 H2 Now; Δ=1 (L1L2) = 1 G1 G3 H2 G1 G2 H2 Therefore; Dr. Mohammed Saheb Khesbak Page 24
7 3rd Yearomputer ommunication EngineeringU ontrol Theor Eample 3.4: Find / for the sstem shown: G2 G6 1 G1 G8 1 G3 G5 G7 H2 Solution: Forward paths: P1=G2 G6, P2=G3 G5 G7, P3= G2 G1 G7, P4=G3 G8 G6 P5=G2 G1 H2 G8 G6, P6=G3 G8 G1 G7 Loops: L1=, L2=G5 H2, and L3= G1 H2 G8 Nontouching Loops: There are one pair of nontouching loops = ( ) (G5 H2) Now; Δ = 1 ( G5 H2 G1 H2 G8 ) G5 H2 Dr. Mohammed Saheb Khesbak Page 25
8 3rd Yearomputer ommunication EngineeringU ontrol Theor Δ = 1 G5 H2 G1 H2 G8 G5 H2 Also; Δ1=1 (G5 H2)= 1 G5 H2, Δ2=1 ( )= 1 and Δ3= Δ4= Δ5= Δ6=1 Therefore; ( ) ( ) Eample 3.5: Using Mason s formula method find the transfer function / of the below block diagram sstem. G1 G2 G3 H2 Dr. Mohammed Saheb Khesbak Page 26
9 3rd Yearomputer ommunication EngineeringU ontrol Theor onverting the block diagram into signal flow graph as shown; 1 G1 G2 G3 H2 1 Forward paths: P1= G1 G2 G3 and P2= G1 G2 Loops: L1= G2 G3 H2, L2= G2 H2, L3= G1 G2, L4= G1 G2 G3 and L5= G1 G2 Now; Δ1= 1 and Δ2= 1 and Δ= 1 (L1L2L3L4L5) Δ=1 G2 G3 H2 G2 H2 G1 G2 G1 G2 G3G1 G2 Dr. Mohammed Saheb Khesbak Page 27
10 3rd Yearomputer ommunication EngineeringU ontrol Theor Eample 3.6: Using Mason s formula method find the transfer function / of the below block diagram sstem. G2 _ G1 H2 _ G5 G6 onverting the block diagram into signal flow graph as shown; G2 ` G1 H2 G5 G6 1 Forward paths: P1= G1 H2 G5 G6 with Δ1=1 Loops: Dr. Mohammed Saheb Khesbak Page 28
11 3rd Yearomputer ommunication EngineeringU ontrol Theor L1= G1 G2, L2=, and L3= G1 H2 G5 therefore; Δ=1G1 G2 G1 H2 G5 G1 G2 then; Eercises: Q1 Prove that the two shown control sstems have different transfer functions. 1 a 1 b 1 c 1 d e f 1 a b c 1 d e f Dr. Mohammed Saheb Khesbak Page 29
12 3rd Yearomputer ommunication EngineeringU ontrol Theor Q2 Find the transfer function for the sstem shown using Mason s formula. G1 G2 G3 H2 Q3 Find the transfer function for the sstem shown using Mason s formula. G1 G2 G3 G6 G7 G5 G8 Q4 Find the transfer function for the sstem shown using Mason s formula. G1 G2 G3 H3 H2 Dr. Mohammed Saheb Khesbak Page 30
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