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The characteristic equation of a closed loop control system is given by:
s2 + 2s +10 + K(s2 + 6s + 10) = 0
The angle of asymptotes for the root loci for K > 0 are given by
s2 + 2s +10 + K(s2 + 6s + 10) = 0
The angle of asymptotes for the root loci for K > 0 are given by
- a)180°, 360°
- b)90°, 270°
- c)90°, 180°
- d)none of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The characteristic equation of a closed loop control system is given b...
Given, s2 + 2s + 10 + K(s2 + 6s + 10) = 0
or,
or, 1 + G(s)H(s) = 0
Thus, G(s)H(s) =
Here, number of open loop poles, P = 2. Number of open loop zero, Z = 2
∴ P - Z = 0.
Angle of asymptotes are given by:
Sines P - Z= 0, therefore there are no angle of asymptotes for the root locus of the given system.
or,
or, 1 + G(s)H(s) = 0
Thus, G(s)H(s) =
Here, number of open loop poles, P = 2. Number of open loop zero, Z = 2
∴ P - Z = 0.
Angle of asymptotes are given by:
Sines P - Z= 0, therefore there are no angle of asymptotes for the root locus of the given system.
Most Upvoted Answer
The characteristic equation of a closed loop control system is given b...
To find the angle of asymptotes for the root loci, we need to determine the number of poles and zeros that approach infinity.
The given characteristic equation is:
s^2 + 2s + 10 + K(s^2 - 6s + 10) = 0
The characteristic equation can be rewritten as:
(s^2 + 2s + 10) + K(s^2 - 6s + 10) = 0
s^2 + 2s + 10 + Ks^2 - 6Ks + 10K = 0
(s^2 + Ks^2) + (2 - 6K)s + (10 + 10K) = 0
Comparing coefficients, we have:
s^2 + Ks^2 = s^2(1 + K)
(2 - 6K)s = s(2 - 6K)
10 + 10K = 0
For there to be asymptotes, the number of poles that approach infinity should be equal to the number of zeros that approach infinity. In other words, the number of poles at infinity should be equal to the number of zeros at infinity.
The number of poles at infinity is equal to the difference between the number of poles and the number of zeros. Let's determine the number of poles and zeros.
The characteristic equation can be rewritten as:
(s^2(1 + K) + (2 - 6K)s + (10 + 10K)) = 0
The coefficient of s^2 is (1 + K), which is equal to 1 + K = 0, therefore K = -1.
Substituting K = -1 into the characteristic equation, we have:
(s^2(1 - 1) + (2 - 6(-1))s + (10 + 10(-1)) = 0
s^2 + 8s = 0
The characteristic equation has two poles at the origin (s = 0) and no zeros at infinity. Therefore, the number of poles at infinity is 2 and the number of zeros at infinity is 0.
The difference between the number of poles and zeros is 2 - 0 = 2.
Since the number of poles at infinity is equal to the number of zeros at infinity, there are no asymptotes for the root loci.
Therefore, the angle of asymptotes for the root loci for K is undefined or not applicable in this case.
The given characteristic equation is:
s^2 + 2s + 10 + K(s^2 - 6s + 10) = 0
The characteristic equation can be rewritten as:
(s^2 + 2s + 10) + K(s^2 - 6s + 10) = 0
s^2 + 2s + 10 + Ks^2 - 6Ks + 10K = 0
(s^2 + Ks^2) + (2 - 6K)s + (10 + 10K) = 0
Comparing coefficients, we have:
s^2 + Ks^2 = s^2(1 + K)
(2 - 6K)s = s(2 - 6K)
10 + 10K = 0
For there to be asymptotes, the number of poles that approach infinity should be equal to the number of zeros that approach infinity. In other words, the number of poles at infinity should be equal to the number of zeros at infinity.
The number of poles at infinity is equal to the difference between the number of poles and the number of zeros. Let's determine the number of poles and zeros.
The characteristic equation can be rewritten as:
(s^2(1 + K) + (2 - 6K)s + (10 + 10K)) = 0
The coefficient of s^2 is (1 + K), which is equal to 1 + K = 0, therefore K = -1.
Substituting K = -1 into the characteristic equation, we have:
(s^2(1 - 1) + (2 - 6(-1))s + (10 + 10(-1)) = 0
s^2 + 8s = 0
The characteristic equation has two poles at the origin (s = 0) and no zeros at infinity. Therefore, the number of poles at infinity is 2 and the number of zeros at infinity is 0.
The difference between the number of poles and zeros is 2 - 0 = 2.
Since the number of poles at infinity is equal to the number of zeros at infinity, there are no asymptotes for the root loci.
Therefore, the angle of asymptotes for the root loci for K is undefined or not applicable in this case.
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The characteristic equation of a closed loop control system is given by:s2+ 2s +10 + K(s2+ 6s + 10) =0The angle of asymptotes for the root loci for K > 0 are given bya)180°, 360° b)90°, 270°c)90°, 180° d)none of theseCorrect answer is option 'D'. Can you explain this answer?
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