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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

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 Correct Answer the origin of the G(s)-plane once in the clockwise direction

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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