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Assertion (A): Root locus is a graphical method in which roots of the characteristic equation are plotted in s-plane for the different value of parameter.
Reason (R): The locus of the roots of the characteristic equation when gain is varied from zero to infinity is called root locus.
Reason (R): The locus of the roots of the characteristic equation when gain is varied from zero to infinity is called root locus.
- a)Both A and R are true and R is a correct explanation of A.
- b)Both A and R are true but R is not a correct explanation of A.
- c)A is true but R is false.
- d)A is false but R is true.
Correct answer is option 'A'. Can you explain this answer?
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Assertion (A): Root locus is a graphical method in which roots of the ...
Root Locus
Root locus is a graphical method used in control system analysis and design to determine the behavior of the roots of a characteristic equation as a parameter, typically the gain, is varied. It provides insights into the stability and transient response of a system.
Assertion (A): Root locus is a graphical method in which roots of the characteristic equation are plotted in s-plane for different values of the parameter.
Reason (R): The locus of the roots of the characteristic equation when gain is varied from zero to infinity is called root locus.
Explanation:
Root Locus Method
The root locus method is based on the principle of the characteristic equation. The characteristic equation is obtained by equating the denominator of the transfer function to zero. The roots of this equation determine the poles of the system.
The transfer function of a system is represented as:
G(s)H(s) = K(s + z1)(s + z2)...(s + zn) / (s + p1)(s + p2)...(s + pm)
Where G(s) is the open-loop transfer function, H(s) is the feedback transfer function, and K is the gain.
Plotting the Root Locus
To plot the root locus, the following steps are followed:
1. Determine the characteristic equation by equating the denominator of the transfer function to zero.
2. Identify the open-loop poles and zeros.
3. Draw the real axis and mark the poles and zeros on it.
4. Determine the angles of departure and arrival for each pole and zero.
5. Draw the root locus branches by following the rules:
- The root locus starts from the open-loop poles.
- The root locus moves towards the zeros.
- The root locus asymptotes and intersects the real axis at angles that sum up to 180 degrees.
- The root locus branches do not cross each other.
Explanation of Assertion (A)
The assertion states that the root locus is a graphical method in which the roots of the characteristic equation are plotted in the s-plane for different values of the parameter. This is true because the root locus provides a visual representation of the behavior of the roots as the parameter, typically the gain, is varied.
Explanation of Reason (R)
The reason states that the locus of the roots of the characteristic equation when the gain is varied from zero to infinity is called the root locus. This is true because the root locus represents the path traced by the roots of the characteristic equation as the gain is varied. It shows how the system's poles move in the s-plane as the gain changes.
Conclusion
Both Assertion (A) and Reason (R) are true, and Reason (R) is a correct explanation of Assertion (A). The root locus is indeed a graphical method in which the roots of the characteristic equation are plotted in the s-plane for different values of the parameter, and the locus of the roots when the gain is varied is known as the root locus.
Root locus is a graphical method used in control system analysis and design to determine the behavior of the roots of a characteristic equation as a parameter, typically the gain, is varied. It provides insights into the stability and transient response of a system.
Assertion (A): Root locus is a graphical method in which roots of the characteristic equation are plotted in s-plane for different values of the parameter.
Reason (R): The locus of the roots of the characteristic equation when gain is varied from zero to infinity is called root locus.
Explanation:
Root Locus Method
The root locus method is based on the principle of the characteristic equation. The characteristic equation is obtained by equating the denominator of the transfer function to zero. The roots of this equation determine the poles of the system.
The transfer function of a system is represented as:
G(s)H(s) = K(s + z1)(s + z2)...(s + zn) / (s + p1)(s + p2)...(s + pm)
Where G(s) is the open-loop transfer function, H(s) is the feedback transfer function, and K is the gain.
Plotting the Root Locus
To plot the root locus, the following steps are followed:
1. Determine the characteristic equation by equating the denominator of the transfer function to zero.
2. Identify the open-loop poles and zeros.
3. Draw the real axis and mark the poles and zeros on it.
4. Determine the angles of departure and arrival for each pole and zero.
5. Draw the root locus branches by following the rules:
- The root locus starts from the open-loop poles.
- The root locus moves towards the zeros.
- The root locus asymptotes and intersects the real axis at angles that sum up to 180 degrees.
- The root locus branches do not cross each other.
Explanation of Assertion (A)
The assertion states that the root locus is a graphical method in which the roots of the characteristic equation are plotted in the s-plane for different values of the parameter. This is true because the root locus provides a visual representation of the behavior of the roots as the parameter, typically the gain, is varied.
Explanation of Reason (R)
The reason states that the locus of the roots of the characteristic equation when the gain is varied from zero to infinity is called the root locus. This is true because the root locus represents the path traced by the roots of the characteristic equation as the gain is varied. It shows how the system's poles move in the s-plane as the gain changes.
Conclusion
Both Assertion (A) and Reason (R) are true, and Reason (R) is a correct explanation of Assertion (A). The root locus is indeed a graphical method in which the roots of the characteristic equation are plotted in the s-plane for different values of the parameter, and the locus of the roots when the gain is varied is known as the root locus.
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Assertion (A): Root locus is a graphical method in which roots of the characteristic equation are plotted in s-plane for the different value of parameter.Reason (R): The locus of the roots of the characteristic equation when gain is varied from zero to infinity is called root locus.a)Both A and R are true and R is a correct explanation of A.b)Both A and R are true but R is not a correct explanation of A.c)A is true but R is false.d)A is false but R is true.Correct answer is option 'A'. Can you explain this answer?
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