Question

Hz(s) + R(s) Gi(s) G2(s) G3(s) G4(s) C(s) Hi(s) Consider the system described by the block diagram above. a. Find the transfer function of the system by reducing the diagram. b. Draw a signal-flow diagram for the given system. c. Using Mason's rule find the transfer function of the system. d. Compare your answers to part (a) and part (c). What do you notice? Explain.

Answer 1

Bolution M213 GS 4215 Gu10C G, (3) R (S) 106) H118) (a) Solving problem from by reducing the block diagram to Canonical Hosing the pick off point A betona block first to eliminate Loop 43 Gudia we move. He behind block 94 32/3) Gils) RCS) G2/5) 9319) Gulsh 72015) 7,19) Eliminating the loop 43444, we obrown M2152 G41s) R(S) 413 421) 34Guls) 1+634 Sels) 4,5) CS Scanned with CamScanner

Now eliminating inner loop con containing tulen we oblown Res) X 9113) 1926) G318) G418) it G3674468)1,4+G23944949 Res) G, (3)6215) G3(3) Guls) Itho LsG4 (3) 4,(s) +G!!44(3) ,(1) (s) (5) T(S) R13) Transfer function Tis) G; (3) G2 (3) G3(s) Gy(s) 1 + 43/s) Gy(s) 4,(s) +G2(5) G3(34.() yang CS Scanned with CamScanner

(6) Signal flow graph .. H215) G, (3) 4215) G415) RCS) xces) is given by (c) using Mason's Pule Given signal flowe graph with e forward paths from an imprit node you to an netput node (no inpect node) youts then the gain between Yin & Yout M=yout yin where yin = input node variable yout E Mu PK K) output node variable M N - gain between for a yout toral no. yin yout of forward paths between Mk = the gowns of the 4th forward path between Yin & Yout CS Scanned with CamScanner

A= 1 - (sum & gains of all in indiredual coope) + ( sum of products of three non 8 gaine of all possible combination oftwa nontouching loops -(sum of products of gains of all all possible combination touching Loops) + Ax = thes for that part of ste that is nontouching with kth forward path In signal how graph , we have la enpul node 1 output node to ă transmetting sales I forward pates 3 summing nodes N =) MI = G, G24344 A, = 1 2) Individual coops a) LI>-924342 b) L27 - G3 G4 Hi 3) No nontouchung loops ensi- cs Scanne Dz1--42.9393,-439440) +0 CamScanner

A= 1 + 42 43 42 +6344H (4) Gain formula Y/R G1G293 Gy 1+924342 +G3 G4 H) Transfer function = G,(8) 42(8) 43(s) Gys) It 4494,4994,59 +9943)Syls),(s) a & parte rime requiement O d) The transfer function obtained in part 30, reults are solome mly in Mason's rule is less. as same 78) = ,() 42(3) 43 (3) 44(8) 1 +424) 43($) *(s) + 4319) Guls) 8 (8) U same in both cases - CS Scanned with CamScanner