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Consider the following block diagram of an internally controlled rotational system. This system has a tunable parameter K that assigns the torque of a motor to the rotating body. Once Ka is chosen, it is a constant. Signal R(s) is a reference state, and T¿(s) is a disturbance. The transfer functions G₁(s) and G₂(s) are: G₁(s) 5000 s + 1000 G₂(s): 1 s(s +20) Ta(s) R(s) Y(s) + Ka G₁(s) G₂(s) + 3.1. Using block diagram algebra and the two definitions of G₁(s) and G₂(s), show that the overall transfer functions relating Y(s) to R(s) and T₁(s) are: Y(s) 5000K R(s) s³ + 1020s² + 20000s + 5000Ka Y(s) Ta(s) s+ 1000 s3+1020s² + 20000s + 5000Ka 3.2. For a unit step change in r(t), show that y(t) converges to 1, regardless of the selected value of K. Assume the response is stable for any Ka. This is a good thing! 3.3. For a unit step change in ta(t), show that y(t) DOES NOT converge to 0, unless we choose an infinitely large value of K. That is, the disturbance signal will permanently affect y(t). This is a bad thing! 3.4. The third-order dynamics for these transfer functions are very fast. An analysis of the system shows that the response can be safely assumed to be the following second-order system: Y(s) R(s) Y(s) 5Ka R(s) s² + 20s + 5Ka Y(s) R(s) Using this approximation, argue that increasing Ka, which we have shown is necessary to minimize the impact of Ta(s) on Y(s), will lead to increasingly oscillatory behaviour (that is, oscillations with higher frequencies) in the response of 3.5. Determine the value of Ka for which the second-order response of Y(s) R(s) will be critically damped.

Consider the following block diagram of an internally controlled rotational system. This system has a tunable parameter K that assigns the torque of a motor to the rotating body. Once Ka is chosen, it is a constant. Signal R(s) is a reference state, and T¿(s) is a disturbance. The transfer functions G₁(s) and G₂(s) are: G₁(s) 5000 s + 1000 G₂(s): 1 s(s +20) Ta(s) R(s) Y(s) + Ka G₁(s) G₂(s) + 3.1. Using block diagram algebra and the two definitions of G₁(s) and G₂(s), show that the overall transfer functions relating Y(s) to R(s) and T₁(s) are: Y(s) 5000K R(s) s³ + 1020s² + 20000s + 5000Ka Y(s) Ta(s) s+ 1000 s3+1020s² + 20000s + 5000Ka 3.2. For a unit step change in r(t), show that y(t) converges to 1, regardless of the selected value of K. Assume the response is stable for any Ka. This is a good thing! 3.3. For a unit step change in ta(t), show that y(t) DOES NOT converge to 0, unless we choose an infinitely large value of K. That is, the disturbance signal will permanently affect y(t). This is a bad thing! 3.4. The third-order dynamics for these transfer functions are very fast. An analysis of the system shows that the response can be safely assumed to be the following second-order system: Y(s) R(s) Y(s) 5Ka R(s) s² + 20s + 5Ka Y(s) R(s) Using this approximation, argue that increasing Ka, which we have shown is necessary to minimize the impact of Ta(s) on Y(s), will lead to increasingly oscillatory behaviour (that is, oscillations with higher frequencies) in the response of 3.5. Determine the value of Ka for which the second-order response of Y(s) R(s) will be critically damped.

Electricity for Refrigeration, Heating, and Air Conditioning (MindTap Course List)
Electricity for Refrigeration, Heating, and Air Conditioning (MindTap Course List)
10th Edition
ISBN:9781337399128
Author:Russell E. Smith
Publisher:Russell E. Smith
Chapter8: Basic Electric Motors
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Problem 25RQ: List the five capacitor replacement rules.
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Consider the following block diagram of an internally controlled rotational system. This system has a tunable parameter K that assigns the torque of a motor to the rotating body. Once Ka is chosen, it is a constant. Signal R(s) is a reference state, and T¿(s) is a disturbance. The transfer functions G₁(s) and G₂(s) are: G₁(s) 5000 s + 1000 G₂(s): 1 s(s +20) Ta(s) R(s) Y(s) + Ka G₁(s) G₂(s) + 3.1. Using block diagram algebra and the two definitions of G₁(s) and G₂(s), show that the overall transfer functions relating Y(s) to R(s) and T₁(s) are: Y(s) 5000K R(s) s³ + 1020s² + 20000s + 5000Ka Y(s) Ta(s) s+ 1000 s3+1020s² + 20000s + 5000Ka 3.2. For a unit step change in r(t), show that y(t) converges to 1, regardless of the selected value of K. Assume the response is stable for any Ka. This is a good thing! 3.3. For a unit step change in ta(t), show that y(t) DOES NOT converge to 0, unless we choose an infinitely large value of K. That is, the disturbance signal will permanently affect y(t). This is a bad thing! 3.4. The third-order dynamics for these transfer functions are very fast. An analysis of the system shows that the response can be safely assumed to be the following second-order system: Y(s) R(s) Y(s) 5Ka R(s) s² + 20s + 5Ka Y(s) R(s) Using this approximation, argue that increasing Ka, which we have shown is necessary to minimize the impact of Ta(s) on Y(s), will lead to increasingly oscillatory behaviour (that is, oscillations with higher frequencies) in the response of 3.5. Determine the value of Ka for which the second-order response of Y(s) R(s) will be critically damped.
Transcribed Image Text:Consider the following block diagram of an internally controlled rotational system. This system has a tunable parameter K that assigns the torque of a motor to the rotating body. Once Ka is chosen, it is a constant. Signal R(s) is a reference state, and T¿(s) is a disturbance. The transfer functions G₁(s) and G₂(s) are: G₁(s) 5000 s + 1000 G₂(s): 1 s(s +20) Ta(s) R(s) Y(s) + Ka G₁(s) G₂(s) + 3.1. Using block diagram algebra and the two definitions of G₁(s) and G₂(s), show that the overall transfer functions relating Y(s) to R(s) and T₁(s) are: Y(s) 5000K R(s) s³ + 1020s² + 20000s + 5000Ka Y(s) Ta(s) s+ 1000 s3+1020s² + 20000s + 5000Ka 3.2. For a unit step change in r(t), show that y(t) converges to 1, regardless of the selected value of K. Assume the response is stable for any Ka. This is a good thing! 3.3. For a unit step change in ta(t), show that y(t) DOES NOT converge to 0, unless we choose an infinitely large value of K. That is, the disturbance signal will permanently affect y(t). This is a bad thing! 3.4. The third-order dynamics for these transfer functions are very fast. An analysis of the system shows that the response can be safely assumed to be the following second-order system: Y(s) R(s) Y(s) 5Ka R(s) s² + 20s + 5Ka Y(s) R(s) Using this approximation, argue that increasing Ka, which we have shown is necessary to minimize the impact of Ta(s) on Y(s), will lead to increasingly oscillatory behaviour (that is, oscillations with higher frequencies) in the response of 3.5. Determine the value of Ka for which the second-order response of Y(s) R(s) will be critically damped.
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