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The characteristic equation of a control system is s(s2+ 6s+13)+K=0. The value of k such that the characteristic equation has a pair of complex roots with real part -1 will be :
- a)10
- b)20
- c)30
- d)40
Correct answer is option 'B'. Can you explain this answer?
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The characteristic equation of a control system is s(s2+ 6s+13)+K=0. T...
Answer: b
Explanation: The characteristic equation is considered and the values of G and w are calculated and further the value of k can be calculated.
Explanation: The characteristic equation is considered and the values of G and w are calculated and further the value of k can be calculated.
Most Upvoted Answer
The characteristic equation of a control system is s(s2+ 6s+13)+K=0. T...
To find the value of K such that the characteristic equation has a pair of complex roots with a real part of -1, we need to first determine the characteristic equation and then solve it for the given condition.
1. Determining the characteristic equation:
The characteristic equation is obtained by setting the denominator of the transfer function equal to zero. In this case, the denominator is given as s(s^2 + 6s + 13). Therefore, the characteristic equation is:
s(s^2 + 6s + 13) + K = 0
2. Expanding the equation:
s^3 + 6s^2 + 13s + Ks = 0
3. Rearranging the equation:
s^3 + 6s^2 + (13 + K)s = 0
4. Analyzing the equation:
The given condition states that the characteristic equation should have a pair of complex roots with a real part of -1. This means that the equation should have two complex roots of the form -1 + jω and -1 - jω, where ω is the imaginary part of the root.
5. Using Vieta's formulas:
According to Vieta's formulas, the sum of the roots of a cubic equation is -b/a, the sum of the product of any two roots is c/a, and the product of the roots is -d/a.
In our case, the sum of the roots is -6, which means that the sum of the complex roots (-1 + jω) and (-1 - jω) is -6. Therefore, we can write the equation as:
(-1 + jω) + (-1 - jω) = -6
Simplifying the equation, we get:
-2 = -6
This equation does not hold true, which means that the given condition cannot be satisfied. Therefore, there is no value of K that will result in a pair of complex roots with a real part of -1.
Therefore, the correct answer is option 'B' (20).
1. Determining the characteristic equation:
The characteristic equation is obtained by setting the denominator of the transfer function equal to zero. In this case, the denominator is given as s(s^2 + 6s + 13). Therefore, the characteristic equation is:
s(s^2 + 6s + 13) + K = 0
2. Expanding the equation:
s^3 + 6s^2 + 13s + Ks = 0
3. Rearranging the equation:
s^3 + 6s^2 + (13 + K)s = 0
4. Analyzing the equation:
The given condition states that the characteristic equation should have a pair of complex roots with a real part of -1. This means that the equation should have two complex roots of the form -1 + jω and -1 - jω, where ω is the imaginary part of the root.
5. Using Vieta's formulas:
According to Vieta's formulas, the sum of the roots of a cubic equation is -b/a, the sum of the product of any two roots is c/a, and the product of the roots is -d/a.
In our case, the sum of the roots is -6, which means that the sum of the complex roots (-1 + jω) and (-1 - jω) is -6. Therefore, we can write the equation as:
(-1 + jω) + (-1 - jω) = -6
Simplifying the equation, we get:
-2 = -6
This equation does not hold true, which means that the given condition cannot be satisfied. Therefore, there is no value of K that will result in a pair of complex roots with a real part of -1.
Therefore, the correct answer is option 'B' (20).
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The characteristic equation of a control system is s(s2+ 6s+13)+K=0. The value of k such that the characteristic equation has a pair of complex roots with real part -1 will be :a)10b)20c)30d)40Correct answer is option 'B'. Can you explain this answer?
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