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530_343lecture12 - ME 530.343 Design and Analysis of...

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ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 12 - Transfer Functions and Block Diagrams Week of March 2, 2009 Today’s Objectives Transfer Functions Transform Block Diagrams Simulink in MATLAB Transfer Functions example: X ( s ) = 1 ms 2 + bs + k F ( s ) let H ( s ) = 1 ms 2 + bs + k X ( s ) = H ( s ) F ( s ) H ( s ) is a transfer function Block Diagrams To represent X ( s ) = H ( s ) F ( s ), use: Say you have: Y ( s ) = X ( s ) + D ( s ) arrows: signals ( F ( s ), X ( s ), D ( s ), Y ( s )) blocks: transfer functions H ( s ) summing junction: add or subtract signals Multiple Transfer Functions X ( s ) F ( s ) = J ( s ), where J ( s ) = H ( s ) G ( s ) 1
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Feedback system X ( s ) = G ( s ) E ( s ) E ( s ) = F ( s ) - U ( s ) U ( s ) = H ( s ) X ( s ) What is the equivalent transfer function for the entire feedback loop? E ( s ) = F ( s ) - H ( s ) X ( s ) X ( s ) = G ( s )[ F ( s ) - H ( s ) X ( s )] X ( s ) = G ( s ) F ( s ) - G ( s ) H ( s ) X ( s ) X ( s )[1 + G ( s ) H ( s )] = G ( s ) F ( s ) X ( s ) F ( s ) = J ( s ) = G ( s ) 1+ G ( s ) H ( s ) Example: Servo motor system L di dt + Ri + e b = e i (electrical circuit) e b = K v d θ dt (back emf) J d 2 θ dt 2 + b d θ dt = τ (rotational mechanical system) τ = k m i (torque constant) Assume zero initial conditions 2
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Take Laplace Transform of each equation L di dt + Ri + e b = e i sLI ( s ) + RI ( s ) + E b ( s ) = E i ( s ) I ( s )( sL + R ) = E i ( s ) - E b ( s ) I ( s ) = 1 sL + R ( E i ( s ) - E b ( s
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