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Consider the following block diagram of an internally controlled rotational system. This system has a tunable parameter ka 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 Ta(s) is a disturbance. The transfer functions G₁(s) and G₂(s) are: G₁(s) 5000 s+ 1000 1 G₂(s) s(s +20) Td(s) R(s) Ka G₁(s) Y(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) = 5000Ka R(s) s31020s2 + 20000s + 5000K 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 Ka. 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 Ka. 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 Y(s) response can be safely assumed to be the following second-order system: R(s) Y(s) R(s) 5Ka s220s+5Ka 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 Y(s) frequencies) in the response of R(s)' 3.5. Determine the value of Ka for which the second-order response Y(s) R(s) of will be critically damped.

Consider the following block diagram of an internally controlled rotational system. This system has a tunable parameter ka 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 Ta(s) is a disturbance. The transfer functions G₁(s) and G₂(s) are: G₁(s) 5000 s+ 1000 1 G₂(s) s(s +20) Td(s) R(s) Ka G₁(s) Y(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) = 5000Ka R(s) s31020s2 + 20000s + 5000K 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 Ka. 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 Ka. 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 Y(s) response can be safely assumed to be the following second-order system: R(s) Y(s) R(s) 5Ka s220s+5Ka 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 Y(s) frequencies) in the response of R(s)' 3.5. Determine the value of Ka for which the second-order response Y(s) R(s) of will be critically damped.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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Consider the following block diagram of an internally controlled rotational system. This system has a tunable parameter ka 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 Ta(s) is a disturbance. The transfer functions G₁(s) and G₂(s) are: G₁(s) 5000 s+ 1000 1 G₂(s) s(s +20) Td(s) R(s) Ka G₁(s) Y(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) = 5000Ka R(s) s31020s2 + 20000s + 5000K 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 Ka. 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 Ka. 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 Y(s) response can be safely assumed to be the following second-order system: R(s) Y(s) R(s) 5Ka s220s+5Ka 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 Y(s) frequencies) in the response of R(s)' 3.5. Determine the value of Ka for which the second-order response Y(s) R(s) of will be critically damped.
Transcribed Image Text:Consider the following block diagram of an internally controlled rotational system. This system has a tunable parameter ka 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 Ta(s) is a disturbance. The transfer functions G₁(s) and G₂(s) are: G₁(s) 5000 s+ 1000 1 G₂(s) s(s +20) Td(s) R(s) Ka G₁(s) Y(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) = 5000Ka R(s) s31020s2 + 20000s + 5000K 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 Ka. 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 Ka. 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 Y(s) response can be safely assumed to be the following second-order system: R(s) Y(s) R(s) 5Ka s220s+5Ka 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 Y(s) frequencies) in the response of R(s)' 3.5. Determine the value of Ka for which the second-order response Y(s) R(s) of will be critically damped.
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