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blockdiagram2 - Block Diagrams QUESTION 2 For the block...

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Block Diagrams Q UESTION 2 For the block diagram shown in Figure 2.1, determine the transfer functions ( 29 ( 29 Y s R s , ( 29 ( 29 1 E s R s , ( 29 ( 29 2 E s R s , and ( 29 ( 29 3 E s R s . G 7 (s) G 1 (s) G 8 (s) G 3 (s) G 6 (s) G 4 (s) G 5 (s) G 2 (s) + + - - + + + R(s) Y(s) E 1 (s) E 2 (s) E 3 (s) Figure 2.1 The block diagram with all of the signals labeled is given in Figure 2.2. Writing an equation for the first summing junction 1 2 8 E R E G Y = - - (1) Writing an equation for the second summing junction 2 1 2 1 3 3 E G G E G E = + (2) Writing an equation for the third summing junction ( 29 3 1 4 5 1 1 5 6 7 1 4 6 7 4 6 7 E G G G E G G G G E G Y G G Y G G G Y = + = + = + (3) Writing an equation for the output
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Block Diagrams 1 5 1 Y G G E = (4) Combining equations (2) and (3) ( 29 2 1 2 1 3 4 6 7 E G G E G G G G Y = + + (5) Multiplying equation (1) by G 1 G 5 and combining with equation (5) ( 29 1 5 1 1 5 1 5 1 2 1 3 4 6 7 G G E G G R G G G G E G G G G Y = - + + (6) Combining equations (4) and (6) and rearranging ( 29 1 5 1 2 1 3 5 4 6 7 1 5 8 1 G G Y R G G G G G G G G
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