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Chapter 3 examples - EEL4657 Dr Haniph Latchman Chapter 3...

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EEL4657 - Dr. Haniph Latchman Chapter 3: Example Problems 1. Given the signal flow diagram in Figure 1, use Mason’s rule to find the trans- fer function T ( s ) = Y ( s ) X ( s ) Figure 1: Signal flow diagram for example problem 1, where X ( s ) is the input and Y ( s ) is the output Solution We realize that there is only one forward path from X ( s ) to Y ( s ) in the flow diagram. This path is shown below in Figure 2: Figure 2: The existing path in the system, P 1 1
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The total path gain, P 1 , is given by: P 1 = (1)( k 1 )( k 2 )( G 1 )( G 2 ) 1 s 2 = k 1 k 2 G 1 G 2 s 2 There are a total of four loops in the signal flow diagram, L 1 , L 2 , L 3 and L 4 . Each individual loop is shown in Figure 3. Figure 3: Shown above are each of loops in the system a) L 1 , b) L 2 , c) L 3 , d) L 4 By examinining each of the individual loops, we can derive the following loop gains: 2
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Loop 1 : L 1 = ( k 1 )( k 2 )( G 1 )( G 2 ) 1 s 2 ( - H 1 ) = - k 1 k 2 G 1 G 2 H 1 s 2 Loop 2 : L 2 = ( k 2 )( G 1 )( G 2 )( - H 2 ) = - k 2 G 1 G 2 H 2 Loop 3 : L 3 = ( G 1 )( - H 3 ) = - G 1 H 3 Loop 4 : L 4 = ( G 1 )( G 2 )( - H 4 ) = - G 1 G 2 H 4
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