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The unit step response of a second order system is = 1-e-5t-5te-5t . Consider the following statements:
1. The under damped natural frequency is 5 rad/s.
2. The damping ratio is 1.
3. The impulse response is 25te-5t.
Which of the statements given above are correct?
1. The under damped natural frequency is 5 rad/s.
2. The damping ratio is 1.
3. The impulse response is 25te-5t.
Which of the statements given above are correct?
- a)Only 1 and 2
- b)Only 2 and 3
- c)Only 1 and 3
- d)1,2 and 3
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The unit step response of a second order system is = 1-e-5t-5te-5t. Co...
Answer: d
Explanation: C(s) = 1/s-1/s+5-5/(s+5)^2
C(s) = 25/s(s2+10s+25)
R(s) = 1/s
G(s) = 25/(s2+10s+25 )
w= √25
w = 5 rad/sec
G = 1.
Explanation: C(s) = 1/s-1/s+5-5/(s+5)^2
C(s) = 25/s(s2+10s+25)
R(s) = 1/s
G(s) = 25/(s2+10s+25 )
w= √25
w = 5 rad/sec
G = 1.
Most Upvoted Answer
The unit step response of a second order system is = 1-e-5t-5te-5t. Co...
Explanation:
To analyze the given second-order system, let's first write the transfer function of the system:
Transfer function:
The unit step response can be expressed as the inverse Laplace transform of the transfer function. Let's assume the transfer function to be G(s):
G(s) = 1 / (s^2 + 2ξω_ns + ω_n^2)
where ξ is the damping ratio and ω_n is the natural frequency.
Statement 1: The underdamped natural frequency is 5 rad/s.
From the given unit step response, we can see that the natural frequency (ω_n) of the system is 5 rad/s. Therefore, statement 1 is correct.
Statement 2: The damping ratio is 1.
To find the damping ratio, we need to compare the given unit step response with the standard form of the unit step response for a second-order system:
y(t) = 1 - e^(-ξω_nt) * (cos(ω_d*t) + (ξ/sqrt(1-ξ^2)) * sin(ω_d*t))
Comparing this with the given unit step response: 1 - e^(-5t) - 5te^(-5t), we can conclude that the damping ratio (ξ) is 1. Therefore, statement 2 is correct.
Statement 3: The impulse response is 25te^(-5t).
The impulse response can be determined by taking the derivative of the unit step response. Let's differentiate the given unit step response:
y(t) = 1 - e^(-5t) - 5te^(-5t)
dy(t)/dt = 0 - (-5e^(-5t)) - 5e^(-5t) + (-5t)(-5e^(-5t)) = 10e^(-5t) - 25te^(-5t)
Therefore, the impulse response is 10e^(-5t) - 25te^(-5t), which is not equal to 25te^(-5t). Hence, statement 3 is incorrect.
Therefore, the correct statements are:
1. The underdamped natural frequency is 5 rad/s.
2. The damping ratio is 1.
Hence, the correct option is (d) 1, 2, and 3.
To analyze the given second-order system, let's first write the transfer function of the system:
Transfer function:
The unit step response can be expressed as the inverse Laplace transform of the transfer function. Let's assume the transfer function to be G(s):
G(s) = 1 / (s^2 + 2ξω_ns + ω_n^2)
where ξ is the damping ratio and ω_n is the natural frequency.
Statement 1: The underdamped natural frequency is 5 rad/s.
From the given unit step response, we can see that the natural frequency (ω_n) of the system is 5 rad/s. Therefore, statement 1 is correct.
Statement 2: The damping ratio is 1.
To find the damping ratio, we need to compare the given unit step response with the standard form of the unit step response for a second-order system:
y(t) = 1 - e^(-ξω_nt) * (cos(ω_d*t) + (ξ/sqrt(1-ξ^2)) * sin(ω_d*t))
Comparing this with the given unit step response: 1 - e^(-5t) - 5te^(-5t), we can conclude that the damping ratio (ξ) is 1. Therefore, statement 2 is correct.
Statement 3: The impulse response is 25te^(-5t).
The impulse response can be determined by taking the derivative of the unit step response. Let's differentiate the given unit step response:
y(t) = 1 - e^(-5t) - 5te^(-5t)
dy(t)/dt = 0 - (-5e^(-5t)) - 5e^(-5t) + (-5t)(-5e^(-5t)) = 10e^(-5t) - 25te^(-5t)
Therefore, the impulse response is 10e^(-5t) - 25te^(-5t), which is not equal to 25te^(-5t). Hence, statement 3 is incorrect.
Therefore, the correct statements are:
1. The underdamped natural frequency is 5 rad/s.
2. The damping ratio is 1.
Hence, the correct option is (d) 1, 2, and 3.
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Solutions for The unit step response of a second order system is = 1-e-5t-5te-5t. Consider the following statements:1. The under damped natural frequency is 5 rad/s.2. The damping ratio is 1.3. The impulse response is 25te-5t.Which of the statements given above are correct?a)Only 1 and 2b)Only 2 and 3c)Only 1 and 3d)1,2 and 3Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Electrical Engineering (EE).
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