Block Diagram Reduction [PDF]

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Topic: Block diagram algebra or Block diagram reduction technique



Block Diagram Models •Block diagrams are used as schematic representations of mathematical models •The various pieces correspond to mathematical entities •Can be rearranged to help simplify the equations used to model the system •We will focus on one type of schematic diagram – feedback control systems



Processes are represented by the blocks in block diagrams:



Process Processes must have at least one input variable and at least one output variable Reclassify processes without input or output:



Input variable



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Output variable



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Basic Elements of Block Diagram The basic elements of a block diagram are a block, the summing point and the take-off point.



Block



The transfer function of a component is represented by a block. Block has single input and single output. The following figure shows a block having input X(s), output Y(s) and the transfer function G(s). Transfer Function, G(s)=Y(s)/X(s), ⇒Y(s)=G(s)X(s)



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Summing Point : The summing point is represented with a circle having cross (X) inside it. It has two or more inputs and single output. It produces the algebraic sum of the inputs.



Take-off Point: The take-off point is a point from which the same input signal can be passed through more than one branch.



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Basic Connections for Blocks



block



Take-off or pickoff point summer



Series Connection



Parallel Connection



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Feedback Connection



Y (s)  G (s) E (s) E ( s )  R ( s )  H ( s )Y ( s )



Y ( s )  G ( s )[ R ( s )  H ( s )Y ( s )]  G ( s ) R ( s )  G ( s ) H ( s )Y ( s ) Y ( s) G( s) T ( s)   R( s) 1  G ( s) H ( s)



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Basic Connections for Blocks



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Block Diagram Reduction Rules Follow these rules for simplifying (reducing) the block diagram, which is having many blocks, summing points and take-off points.      



Rule 1 − Check for the blocks connected in series and simplify. Rule 2 − Check for the blocks connected in parallel and simplify. Rule 3 − Check for the blocks connected in feedback loop and simplify. Rule 4 − If there is difficulty with take-off point while simplifying, shift it towards right. Rule 5 − If there is difficulty with summing point while simplifying, shift it towards left. Rule 6 − Repeat the above steps till you get the simplified form, i.e., single block.



Note − The transfer function present in this single block is the transfer function of the overall block diagram.



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