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A linear time invariant system initially at rest, when subjected to unit step input gives a response y(t) = te-t, t > 0 The corresponding transfer function is
- a)1/(S+1)2
- b)1/S(S+1)2
- c)S/(S+1)2
- d)1/S(S+1)
Correct answer is option 'C'. Can you explain this answer?
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A linear time invariant systeminitially at rest, when subjected tounit...
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Explanation:
To find the transfer function of the given linear time-invariant system, we need to analyze the response of the system to a unit step input.
The given response is y(t) = te^(-t), t >= 0
Step 1:
We know that the Laplace transform of the unit step function u(t) is given by U(s) = 1/s.
Step 2:
Let's find the Laplace transform of the given response y(t).
Taking the Laplace transform of y(t), we get:
Y(s) = L{te^(-t)}
Using the time-shifting property of the Laplace transform, we have:
Y(s) = L{t} * L{e^(-t)}
= (1/s^2) * (1/(s+1))
Simplifying the expression, we have:
Y(s) = 1/(s^2 * (s+1))
Step 3:
Now, let's find the transfer function H(s) of the system.
The transfer function H(s) is defined as the ratio of the output Y(s) to the input U(s). In this case, the input is the Laplace transform of the unit step function, which is 1/s.
Therefore, we have:
H(s) = Y(s) / U(s)
= (1/(s^2 * (s+1))) / (1/s)
= (1/(s^2 * (s+1))) * s
= s/(s^2 * (s+1))
= S/(S+1)^2
Hence, the correct transfer function corresponding to the given response is option 'C': S/(S+1)^2.
To find the transfer function of the given linear time-invariant system, we need to analyze the response of the system to a unit step input.
The given response is y(t) = te^(-t), t >= 0
Step 1:
We know that the Laplace transform of the unit step function u(t) is given by U(s) = 1/s.
Step 2:
Let's find the Laplace transform of the given response y(t).
Taking the Laplace transform of y(t), we get:
Y(s) = L{te^(-t)}
Using the time-shifting property of the Laplace transform, we have:
Y(s) = L{t} * L{e^(-t)}
= (1/s^2) * (1/(s+1))
Simplifying the expression, we have:
Y(s) = 1/(s^2 * (s+1))
Step 3:
Now, let's find the transfer function H(s) of the system.
The transfer function H(s) is defined as the ratio of the output Y(s) to the input U(s). In this case, the input is the Laplace transform of the unit step function, which is 1/s.
Therefore, we have:
H(s) = Y(s) / U(s)
= (1/(s^2 * (s+1))) / (1/s)
= (1/(s^2 * (s+1))) * s
= s/(s^2 * (s+1))
= S/(S+1)^2
Hence, the correct transfer function corresponding to the given response is option 'C': S/(S+1)^2.
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