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Q2 Amplifier and motor Ua(s) Wind turbine Um(s) 0,(s) E(s) K1 U(s) e(s) 0.1s + 1 s2 + 4s + 40 Desired Actual pitch pitch K2

Q2 Amplifier and motor Ua(s) Wind turbine Um(s) 0,(s) E(s) K1 U(s) e(s) 0.1s + 1 s2 + 4s + 40 Desired Actual pitch pitch K2

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Q2 Amplifier Ua(s) and motor Wind turbine Um(s) O,(s) E(s) U(s) 0(s) 0.1s + 1 s2 + 4s + 40 Desired Actual pitch pitch K2 Sensor Figure Q2 A pitch-angle control system for a wind turbine is shown in Figure Q2. The desired pitch angle is given by 0„(s), while the actual pitch angle is given by 0(s). The error between actual and desired pitch is amplified and used to actuate a motor. The wind turbine is acted upon by the torque provided by the motor Um(s), and also by torque from external wind disturbances Ua(s). The actual pitch angle is measured and fed back by a sensor which has gain K2. Derive the closed-loop transfer function of the system when there are no wind disturbances. (a) (b) For a unit step change in 0,, it is desired that the steady state error in pitch angle should be zero. Show that the condition for this is given by K1 = K, K2 + 8 In addition, it is desired that a steady-state error of exactly 0.1 rad/s is produced when the desired pitch angle is adjusted by a ramp input of 4 rad/s. By substituting the condition derived in part (b) into the closed-loop transfer function, calculate suitable values for the gains K, and K2. (c) Calculate the steady-state pitch angle resulting from a step change in wind disturbance torque Ua of 5Nm when the reference pitch angle 0, is zero. (Find an expression in terms of the gains K, and K2, and then substitute the values from part (c).) (d)
Transcribed Image Text:Q2 Amplifier Ua(s) and motor Wind turbine Um(s) O,(s) E(s) U(s) 0(s) 0.1s + 1 s2 + 4s + 40 Desired Actual pitch pitch K2 Sensor Figure Q2 A pitch-angle control system for a wind turbine is shown in Figure Q2. The desired pitch angle is given by 0„(s), while the actual pitch angle is given by 0(s). The error between actual and desired pitch is amplified and used to actuate a motor. The wind turbine is acted upon by the torque provided by the motor Um(s), and also by torque from external wind disturbances Ua(s). The actual pitch angle is measured and fed back by a sensor which has gain K2. Derive the closed-loop transfer function of the system when there are no wind disturbances. (a) (b) For a unit step change in 0,, it is desired that the steady state error in pitch angle should be zero. Show that the condition for this is given by K1 = K, K2 + 8 In addition, it is desired that a steady-state error of exactly 0.1 rad/s is produced when the desired pitch angle is adjusted by a ramp input of 4 rad/s. By substituting the condition derived in part (b) into the closed-loop transfer function, calculate suitable values for the gains K, and K2. (c) Calculate the steady-state pitch angle resulting from a step change in wind disturbance torque Ua of 5Nm when the reference pitch angle 0, is zero. (Find an expression in terms of the gains K, and K2, and then substitute the values from part (c).) (d)
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