ece382-fall2011-block-diagram-signal-flow-graph-slides-4-pages3

Ece382-fall2011-block-diagram-signal-flow-graph-slides-4-pages3

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Transfer Functions Transfer function is deFned as: Transfer function = L { output variable } L { input variable } ° ° ° ° initial conditions are zero ±or example, Fnd the transfer function E o ( s ) E i ( s ) of an RC circuit We shall use the transfer function to determine the stability and response of the system. C. S. George Lee (Purdue Univ.) Block Diagrams & Signal-Flow Graphs ECE382 1 / 31 Block Diagrams A block diagram of a system is a graphical representation of a physical system, illustrating the functional relationship among its components Each functional block is connected by arrows which show the direction of signal ²ow Each block contains information concerning dynamic behavior, but it does not contain any information about the physical structure of the system A block diagram is NOT unique. C. S. George Lee (Purdue Univ.) Block Diagrams & Signal-Flow Graphs ECE382 2 / 31 Components of a Block Diagram Block : C ( s )= G ( s ) E ( s ) ; B ( s H ( s ) C ( s ) Summing point: All signals going into a summing point must have the same unit. E ( s R ( s ) B ( s ) or E ( s R ( s ) C ( s ) . Pick-off point or branch point. Arrows - representing signal ²ow direction (uni-directional). C. S. George Lee (Purdue Univ.) Block Diagrams & Signal-Flow Graphs ECE382 3 / 31 Transfer Function of a Feedback Control System Open-loop transfer function : B ( s ) E ( s ) = G ( s ) H ( s ) Feedforward transfer function : C ( s ) E ( s ) = G ( s ) Closed-loop transfer function : C ( s ) R ( s ) =? C. S. George Lee (Purdue Univ.) Block Diagrams & Signal-Flow Graphs ECE382 4 / 31
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Closed-Loop Transfer Function Determine the closed-loop transfer function : C ( s ) R ( s ) C ( s ) E ( s ) = G ( s ) or C ( s )= G ( s ) E ( s ) (1) and E ( s R ( s ) H ( s ) C ( s ) (2) Substituting Eq. (2) into Eq. (1): C ( s G ( s ) R ( s ) G ( s ) H ( s ) C ( s ) or C ( s )[ 1 + G ( s ) H ( s )] = G ( s ) R ( s ) C ( s ) R ( s ) = G ( s ) 1 + G ( s ) H ( s ) (3) C. S. George Lee (Purdue Univ.) Block Diagrams & Signal-Flow Graphs ECE382 5 / 31 Constructing a Block Diagram from an RC Circuit Find the transfer function E o ( s ) E i ( s ) Equations: e i e o R = i ( t ); e o = 1 C ° i ( t ) dt Taking the Laplace Transform of the above equations, we have E i ( s ) E o ( s ) R = I ( s ) , E o ( s 1 sC I ( s ) C. S. George Lee (Purdue Univ.) Block Diagrams & Signal-Flow Graphs ECE382 6 / 31 Closed-loop System with Disturbance (I) Two inputs : R ( s ) and D ( s ) One output : C ( s ) Two transfer functions : T 1 ( s C 1 ( s ) R ( s ) ± ± ± D ( s )= 0 ; T 2 ( s C 2 ( s ) D ( s ) ± ± ± R ( s )= 0 C 1 ( s ) R ( s ) ± ± ± ± D ( s )= 0 = G 1 ( s ) G 2 ( s ) 1 + G 1 ( s ) G 2 ( s ) H ( s ) C 2 ( s ) D ( s ) ± ± ± ± R ( s )= 0 = G 2 ( s ) 1 + G 1 ( s ) G 2 ( s ) H ( s ) Output, C ( s ) C ( s C 1 ( s )+ C 2 ( s T 1 ( s ) R ( s T 2 ( s ) D ( s ) C. S. George Lee (Purdue Univ.) Block Diagrams & Signal-Flow Graphs ECE382 7 / 31 Closed-loop System with Disturbance (II) C ( s C 1 ( s C 2 ( s T 1 ( s ) R ( s T 2 ( s ) D ( s ) = G 2 ( s ) 1 + G 1 ( s ) G 2 ( s ) H ( s ) [ G 1 ( s ) R ( s D ( s )] If | G 1 ( s ) H ( s ) | ° 1 and | G 1 ( s ) G 2 ( s ) H ( s ) | ° 1, then C 2 ( s ) D ( s ) ± ± ± ± R ( s )= 0 ± G 2 ( s ) G 1 ( s ) G 2 ( s ) H ( s ) ± 1 G 1 ( s ) H ( s ) = 0 This is the advantage of a feedback control system. C 1 ( s ) R ( s ) ± ± ± ± D ( s )= 0 ± G 1 ( s ) G 2 ( s ) G 1 ( s ) G 2 ( s ) H ( s ) ± 1 H ( s ) Hence, it is independent of G 1 ( s ) G 2 ( s ) . If H ( s ) 1, then C 1 ( s R ( s ) ; we have output tracks the input.
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