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The response of a first-order measurement system to a unit step input is 1 – e– 0.5t, where t is in seconds. A ramp of 0.1 units per second is given as the input to this system. The error in the measured value after transients have died down is
- a)0.02 units
- b)0.1 units
- c)0.2 units
- d)1 unit
Correct answer is option 'D'. Can you explain this answer?
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The response of a first-order measurement system to a unit step input...
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The response of a first-order measurement system to a unit step input...
Given:
- First-order measurement system
- Response to a unit step input: 1 - e^(-0.5t)
- Input: Ramp of 0.1 units per second
To Find:
- Error in the measured value after transients have died down
Solution:
To find the error in the measured value, we need to determine the steady-state value of the output when the input is a ramp.
Steady-State Value:
The steady-state value of the output can be found by taking the limit of the response as t approaches infinity.
lim (t→∞) (1 - e^(-0.5t)) = 1
Therefore, the steady-state value of the output is 1.
Error:
The error is defined as the difference between the desired value and the measured value. In this case, the desired value is the input ramp rate of 0.1 units per second.
The measured value can be found by evaluating the response at t = ∞:
Measured value = lim (t→∞) (1 - e^(-0.5t)) = 1
Error = Desired value - Measured value
= 0.1 - 1
= -0.9
Since the error is negative, it means that the measured value is lower than the desired value.
To find the absolute error, we take the absolute value of the error:
Absolute error = | -0.9 |
= 0.9
Conclusion:
The error in the measured value after transients have died down is 0.9 units. Therefore, none of the given options (a, b, c, d) match the correct answer.
- First-order measurement system
- Response to a unit step input: 1 - e^(-0.5t)
- Input: Ramp of 0.1 units per second
To Find:
- Error in the measured value after transients have died down
Solution:
To find the error in the measured value, we need to determine the steady-state value of the output when the input is a ramp.
Steady-State Value:
The steady-state value of the output can be found by taking the limit of the response as t approaches infinity.
lim (t→∞) (1 - e^(-0.5t)) = 1
Therefore, the steady-state value of the output is 1.
Error:
The error is defined as the difference between the desired value and the measured value. In this case, the desired value is the input ramp rate of 0.1 units per second.
The measured value can be found by evaluating the response at t = ∞:
Measured value = lim (t→∞) (1 - e^(-0.5t)) = 1
Error = Desired value - Measured value
= 0.1 - 1
= -0.9
Since the error is negative, it means that the measured value is lower than the desired value.
To find the absolute error, we take the absolute value of the error:
Absolute error = | -0.9 |
= 0.9
Conclusion:
The error in the measured value after transients have died down is 0.9 units. Therefore, none of the given options (a, b, c, d) match the correct answer.
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The response of a first-order measurement system to a unit step input is 1 – e– 0.5t, where t is in seconds. A ramp of 0.1 units per second is given as the input to this system. The error in the measured value after transients have died down isa)0.02 unitsb)0.1 unitsc)0.2 unitsd)1 unitCorrect answer is option 'D'. Can you explain this answer?
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