hw2s - MEEN 651 Control System Design Homework 2: Modeling...

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Unformatted text preview: MEEN 651 Control System Design Homework 2: Modeling of Electro-Mechanical Systems and Linearization Assigned: Tuesday, 11 Sept. 2007 Due: Tuesday, 18 Sept. 2007, 5:00 pm Problem 1) For the electrical circuit shown below, a. Derive the governing equations (Hint: define I 1 and I 2 as the currents through inductors L 1 and L 2 , respectively). b. Determine a state space representation. Assume the outputs of interest are V R and V C2 . Figure 1: Electrical circuit for Problem 1 Solution: Problem 2) A graphic equalizer is an audio amplifier (filter) which allows you to select different levels of amplification within different frequency regions. The following OP amp circuit is an example of a circuit element in a graphic equalizer. Determine the transfer function from the input voltage, to the output voltage. v i OP amp 1 OP amp 2 OP amp 3 v 1 v 2 Figure 2: Electrical circuit for Problem 2 Solution: The resistors are numbered from left to right: R 1 =20k ; R 2 =20k ; R 3 =1k ; R 4 =1k ; R 5 =1k ; R 6 =1k ; R 7 =3k ; C=0.5 F Op-Amp 1: ) ( 1 ) ( 2 1 2 1 s V Cs R R R s V i + = Op-Amp 2: ) ( 1 ) ( 3 4 2 s V Cs R Cs R s V i + = Op-Amp 3: )) ( ) ( ( 3 ) ( 2 1 s V s V s V + = Combining ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + + + + = + + + + + = + + + = 1 0005 . 1 01 . 1 01 . 0015 . 1 0005 . 3 ) ( ) ( 1 1 1 3 1 3 ) ( ) ( ) ( 1 1 3 ) ( 3 2 2 4 3 1 2 3 4 2 1 2 s s s s s s V s V Cs R Cs R Cs R Cs R Cs R R R s V s V s V Cs R Cs R Cs R R R s V i i i Problem 3) A DC motor-driven lifting system is shown below. A DC motor drives, through a rigid shaft with equivalent viscous damping coefficient, b , a large wheel of radius r and inertia J , which rotates with an angular velocity...
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hw2s - MEEN 651 Control System Design Homework 2: Modeling...

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