Quiz_4_sol

Quiz_4_sol - transfer function G 1 ( s ) without being...

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ECE 483 Quiz # 4 Solution Problem 1. (12 points) For the above sampled-data system, determine if the transfer function C ( z ) R ( z ) exists. If so, ±nd it; otherwise, derive the expression of C ( z ) instead. Solution: The relevant equations are: C = G 1 E * 1 + G 2 E * 2 E 2 = R - G 2 H 2 E * 2 E * 2 = R * - G 2 H 2 * E * 2 E * 2 = R * 1 + G 2 H 2 * E 1 = R - G 2 H 2 E * 2 - G 1 H 1 E * 1 E * 1 = R * - G 2 H 2 * E * 2 - G 1 H 1 * E * 1 E * 1 = R * 1 + G 1 H 1 * - G 2 H 2 * 1 + G 1 H 1 * R * 1 + G 2 H 2 * E * 1 = R * (1 + G 1 H 1 * )(1 + G 2 H 2 * ) As a result, from C = G 1 E * 1 + G 2 E * 2 , we have C * = G * 1 E * 1 + G * 2 E * 2 = G * 1 R * (1 + G 1 H 1 * )(1 + G 2 H 2 * ) + G * 2 R * 1 + G 2 H 2 * C ( z ) R ( z ) = G 1 ( z ) (1 + G 1 H 1 ( z ))(1 + G 2 H 2 ( z )) + G 2 ( z ) 1 + G 2 H 2 ( z ) . Problem 2. (8 points) For the above sampled-data system, determine if the transfer function C ( z ) R ( z ) exists. If so, ±nd it; otherwise, derive the expression of C ( z ) instead. Solution: The transfer function C ( z ) R ( z ) does not exist, as the input is directly fed into the analog
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Unformatted text preview: transfer function G 1 ( s ) without being sampled. To Fnd C ( z ), we denote the input to the top sampler by A ( s ). Then, C = G 2 A * C * = G * 2 A * A = G 1 ( R-HD * C * ) = G 1 R-G 1 HD * C * A * = G 1 R *-G 1 H * D * C * Therefore, by plugging the second equation into the Frst, we have C * = G * 2 A * = G * 2 ( G 1 R *-G 1 H * D * C * ) C * = G * 2 G 1 R * 1 + G * 2 G 1 H * D * C ( z ) = G 2 ( z ) G 1 R ( z ) 1 + G 2 ( z ) G 1 H ( z ) D ( z ) . 2...
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Quiz_4_sol - transfer function G 1 ( s ) without being...

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