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The characteristic equation of a control system is given as 1+ K(s+4)/s(s+7)(s2+2s+2)=0. The real axis intercept for root locus asymptote is:
- a)-2.25
- b)-1
- c)-1.67
- d)0
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The characteristic equation of a control system is given as 1+ K(s+4)/...
Answer: c
Explanation: Real axis intercept =centroid
Zero =-4 and Pole = -7, -1, -1, 0
Centroid =Σreal part of open loop pole-Σreal part of open loop zeroes/P-Z
Centroid = -7-1-1+4/3 = -1.67.
Explanation: Real axis intercept =centroid
Zero =-4 and Pole = -7, -1, -1, 0
Centroid =Σreal part of open loop pole-Σreal part of open loop zeroes/P-Z
Centroid = -7-1-1+4/3 = -1.67.
Most Upvoted Answer
The characteristic equation of a control system is given as 1+ K(s+4)/...
Real Axis Intercept Calculation:
To find the real axis intercept for the root locus asymptote, we need to first determine the number of branches that will approach the real axis. This can be done by looking at the poles and zeros of the system.
Poles and Zeros Analysis:
- Poles: s(s+7)(s^2+2s+2) = 0
- Zeros: -4/K
Number of branches:
- Number of branches = Number of poles - Number of zeros
- Number of branches = 4 - 1 = 3
Real Axis Intercept Formula:
- Real axis intercept = (Sum of poles - Sum of zeros) / Number of branches
Calculation:
- Sum of poles = 0 + 7 + (-1 + j) + (-1 - j)
- Sum of poles = 5
- Sum of zeros = -4/K
- Real axis intercept = (5 - (-4/K)) / 3
Solving for K:
- 1 + K(s+4)/s(s+7)(s^2+2s+2) = 0
- K = -1 / (s+4) * s(s+7)(s^2+2s+2)
- Substitute K back into the real axis intercept equation and solve for the intercept.
Final Answer:
- The real axis intercept for the root locus asymptote is approximately -1.67.
To find the real axis intercept for the root locus asymptote, we need to first determine the number of branches that will approach the real axis. This can be done by looking at the poles and zeros of the system.
Poles and Zeros Analysis:
- Poles: s(s+7)(s^2+2s+2) = 0
- Zeros: -4/K
Number of branches:
- Number of branches = Number of poles - Number of zeros
- Number of branches = 4 - 1 = 3
Real Axis Intercept Formula:
- Real axis intercept = (Sum of poles - Sum of zeros) / Number of branches
Calculation:
- Sum of poles = 0 + 7 + (-1 + j) + (-1 - j)
- Sum of poles = 5
- Sum of zeros = -4/K
- Real axis intercept = (5 - (-4/K)) / 3
Solving for K:
- 1 + K(s+4)/s(s+7)(s^2+2s+2) = 0
- K = -1 / (s+4) * s(s+7)(s^2+2s+2)
- Substitute K back into the real axis intercept equation and solve for the intercept.
Final Answer:
- The real axis intercept for the root locus asymptote is approximately -1.67.
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