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The pole-zero map of a rational function G(s) is shown below. When the closed contour Γ is mapped into the G(s)-plane, then the mapping encircles
  • a)
    the origin of the G(s)-plane once in the counter-clockwise direction
  • b)
    the origin of the G(s)-plane once in the clockwise direction
  • c)
    the point -1 + j0 of the G(s)-plane once in the counter-clockwise direction
  • d)
    the point -1+ j0 of the G(s)-plane once in the clockwise direction
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
The pole-zero map of a rational function G(s) is shown below. When the...
Concept:
Cauchy principles argument states that the closed contour Γ is mapped into the G(s)-plane will encircle the origin as many times as the difference between the number of poles (P) and zeros (Z) of the open-loop transfer function G(s) that are encircled by the S – plane locus Γ, i.e.
No. of encirclement is given by:
N = P – Z
Calculation:
The closed contour Γ of a pole-zero map of a rational function G(s) contains 2 poles and 3 zeros.
So, the number of encirclement will be:
N = P – Z
N = 2 – 3 = -1
Hence,
It encircles the origin once in the clockwise direction.
Another method to solve:
The closed contour Γ of a pole-zero map of a rational function G(s) is encircling 2 poles and 3 zeros in a clockwise direction, hence the corresponding G(s) plane contour encircles origin 2 times in anti-clockwise direction and 3 times in clockwise direction.
Hence, Effectively it encircles origin once in the clockwise direction.
Special note:
  • If we discuss the stability of the open-loop transfer function then we take encirclement around the origin.
  • If we discuss the stability of closed-loop transfer function then we take encirclement around
 -1 + j0. (∴ Option 3 and 4 are incorrect)
Free Test
Community Answer
The pole-zero map of a rational function G(s) is shown below. When the...
Concept:
Cauchy principles argument states that the closed contour Γ is mapped into the G(s)-plane will encircle the origin as many times as the difference between the number of poles (P) and zeros (Z) of the open-loop transfer function G(s) that are encircled by the S – plane locus Γ, i.e.
No. of encirclement is given by:
N = P – Z
Calculation:
The closed contour Γ of a pole-zero map of a rational function G(s) contains 2 poles and 3 zeros.
So, the number of encirclement will be:
N = P – Z
N = 2 – 3 = -1
Hence,
It encircles the origin once in the clockwise direction.
Another method to solve:
The closed contour Γ of a pole-zero map of a rational function G(s) is encircling 2 poles and 3 zeros in a clockwise direction, hence the corresponding G(s) plane contour encircles origin 2 times in anti-clockwise direction and 3 times in clockwise direction.
Hence, Effectively it encircles origin once in the clockwise direction.
Special note:
  • If we discuss the stability of the open-loop transfer function then we take encirclement around the origin.
  • If we discuss the stability of closed-loop transfer function then we take encirclement around
 -1 + j0. (∴ Option 3 and 4 are incorrect)
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The pole-zero map of a rational function G(s) is shown below. When the closed contour Γ is mapped into the G(s)-plane, then the mapping encirclesa)the origin of the G(s)-plane once in the counter-clockwise directionb)the origin of the G(s)-plane once in the clockwise directionc)the point -1 + j0 of the G(s)-plane once in the counter-clockwise directiond)the point -1+ j0 of the G(s)-plane once in the clockwise directionCorrect answer is option 'B'. Can you explain this answer?
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The pole-zero map of a rational function G(s) is shown below. When the closed contour Γ is mapped into the G(s)-plane, then the mapping encirclesa)the origin of the G(s)-plane once in the counter-clockwise directionb)the origin of the G(s)-plane once in the clockwise directionc)the point -1 + j0 of the G(s)-plane once in the counter-clockwise directiond)the point -1+ j0 of the G(s)-plane once in the clockwise directionCorrect answer is option 'B'. Can you explain this answer? for Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about The pole-zero map of a rational function G(s) is shown below. When the closed contour Γ is mapped into the G(s)-plane, then the mapping encirclesa)the origin of the G(s)-plane once in the counter-clockwise directionb)the origin of the G(s)-plane once in the clockwise directionc)the point -1 + j0 of the G(s)-plane once in the counter-clockwise directiond)the point -1+ j0 of the G(s)-plane once in the clockwise directionCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The pole-zero map of a rational function G(s) is shown below. When the closed contour Γ is mapped into the G(s)-plane, then the mapping encirclesa)the origin of the G(s)-plane once in the counter-clockwise directionb)the origin of the G(s)-plane once in the clockwise directionc)the point -1 + j0 of the G(s)-plane once in the counter-clockwise directiond)the point -1+ j0 of the G(s)-plane once in the clockwise directionCorrect answer is option 'B'. Can you explain this answer?.
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