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 defined as: Transfer function = L { output variable } L { input variable } initial conditions are zero For example, find 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 flow 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 flow 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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