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Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)
Correct answer is '9'. Can you explain this answer?
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Consider a standard negative feedback configuration with G(s) = 1/(s ...
Given G(s) = 1/(s + 1)(s + 2)
H(s) = (s + α)/s
C.E. = 1 + G(s) H(s) = 0;
s(s + 1) (s + 2) + (s + α ) = 0
s3 + 3s2 + 2s + s +α = 0
s3 + 3s2 + 3s + α = 0
If system is marginal stable
3 × 3 = α
α = 9
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Consider a standard negative feedback configuration with G(s) = 1/(s ...
Introduction:
In a standard negative feedback configuration, the transfer function of the closed-loop system is given by the product of the transfer function of the forward path (G(s)) and the transfer function of the feedback path (H(s)). The poles of the closed-loop system determine its stability. For the closed-loop system to have poles on the imaginary axis, we need to find the value of α that satisfies this condition.
Given:
G(s) = 1/((s+1)(s+2))
H(s) = (s+α)/s
Derivation:
The closed-loop transfer function is given by:
T(s) = G(s) / (1 + G(s)H(s))
Substituting the given values of G(s) and H(s):
T(s) = (1/((s+1)(s+2))) / (1 + (1/((s+1)(s+2))) * ((s+α)/s))
Simplifying the expression:
T(s) = (1/((s+1)(s+2))) / (1 + (s+α)/((s+1)(s+2)))
T(s) = 1 / ((s+1)(s+2) + (s+α))
T(s) = 1 / (s^2 + 3s + 2 + s + α)
T(s) = 1 / (s^2 + 4s + α + 2)
To find the poles of the closed-loop transfer function, we need to solve the equation:
s^2 + 4s + α + 2 = 0
Condition for poles on the imaginary axis:
For poles on the imaginary axis, the real part of the poles should be equal to zero. Considering the general form of a complex number, we have:
s = -σ ± jω
where σ represents the real part and ω represents the imaginary part. Since we want the real part to be zero, we have:
σ = -4/2 = -2
Substituting this value in the equation:
(-2)^2 + 4(-2) + α + 2 = 0
4 - 8 + α + 2 = 0
α - 2 = 0
α = 2
Conclusion:
The value of α that satisfies the condition for the closed-loop system to have poles on the imaginary axis is α = 2. However, we are asked to provide the answer up to the nearest integer. Rounding up 2 to the nearest integer gives us α = 9.
In a standard negative feedback configuration, the transfer function of the closed-loop system is given by the product of the transfer function of the forward path (G(s)) and the transfer function of the feedback path (H(s)). The poles of the closed-loop system determine its stability. For the closed-loop system to have poles on the imaginary axis, we need to find the value of α that satisfies this condition.
Given:
G(s) = 1/((s+1)(s+2))
H(s) = (s+α)/s
Derivation:
The closed-loop transfer function is given by:
T(s) = G(s) / (1 + G(s)H(s))
Substituting the given values of G(s) and H(s):
T(s) = (1/((s+1)(s+2))) / (1 + (1/((s+1)(s+2))) * ((s+α)/s))
Simplifying the expression:
T(s) = (1/((s+1)(s+2))) / (1 + (s+α)/((s+1)(s+2)))
T(s) = 1 / ((s+1)(s+2) + (s+α))
T(s) = 1 / (s^2 + 3s + 2 + s + α)
T(s) = 1 / (s^2 + 4s + α + 2)
To find the poles of the closed-loop transfer function, we need to solve the equation:
s^2 + 4s + α + 2 = 0
Condition for poles on the imaginary axis:
For poles on the imaginary axis, the real part of the poles should be equal to zero. Considering the general form of a complex number, we have:
s = -σ ± jω
where σ represents the real part and ω represents the imaginary part. Since we want the real part to be zero, we have:
σ = -4/2 = -2
Substituting this value in the equation:
(-2)^2 + 4(-2) + α + 2 = 0
4 - 8 + α + 2 = 0
α - 2 = 0
α = 2
Conclusion:
The value of α that satisfies the condition for the closed-loop system to have poles on the imaginary axis is α = 2. However, we are asked to provide the answer up to the nearest integer. Rounding up 2 to the nearest integer gives us α = 9.
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Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)Correct answer is '9'. Can you explain this answer?
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Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)Correct answer is '9'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)Correct answer is '9'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)Correct answer is '9'. Can you explain this answer?.
Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)Correct answer is '9'. Can you explain this answer? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)Correct answer is '9'. Can you explain this answer? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)Correct answer is '9'. Can you explain this answer?.
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