Introduction:
In the field of electronics and communication engineering, the negative feedback configuration is widely used to stabilize and improve the performance of systems. The Nyquist plot is a graphical representation of the frequency response of a system, which provides insights into the stability and robustness of the system. In this question, we are given the loop transfer function G(s)H(s) and we need to determine the number of clockwise encirclements of (-1, 0) in the Nyquist plot.

Understanding the Loop Transfer Function:
The loop transfer function is the product of the open-loop transfer function G(s) and the feedback transfer function H(s). In this case, we are given G(s) = s^2 + 0.2s + 100 / s^2 - 0.2s + 100 and H(s) = 0.5.

Constructing the Nyquist Plot:
To construct the Nyquist plot, we substitute s = jω in the loop transfer function and calculate the magnitude and phase of G(s)H(s) for different values of ω. By varying ω from -∞ to +∞, we can obtain the Nyquist plot.

Number of Clockwise Encirclements:
To determine the number of clockwise encirclements of (-1, 0) in the Nyquist plot, we need to analyze the behavior of the plot as ω approaches infinity.

Asymptotes:
The Nyquist plot of G(s)H(s) will have two asymptotes: one passing through the point (-1, 0) and another through the point (0, 0). The angle between these asymptotes is given by θ = (n + p)π, where n is the number of poles of G(s)H(s) in the right-half of the s-plane and p is the number of zeros of G(s)H(s) in the right-half of the s-plane.

Pole-Zero Analysis:
To find the number of poles and zeros in the right-half of the s-plane, we need to find the roots of the denominator and numerator of G(s)H(s) that have positive real parts.

Roots of the Denominator:
The denominator of G(s)H(s) is given by s^2 - 0.2s + 100. Using the quadratic formula, we can find the roots of the denominator as s = 0.1 ± j9.995. Since both roots have negative real parts, there are no poles in the right-half of the s-plane.

Roots of the Numerator:
The numerator of G(s)H(s) is given by s^2 + 0.2s + 100. Using the quadratic formula, we can find the roots of the numerator as s = -0.1 ± j9.995. Since both roots have negative real parts, there are no zeros in the right-half of the s-plane.

Conclusion:
Since there are no poles or zeros in the right-half of the s-plane, the angle θ between the asymptotes is zero. Therefore, there are no clockwise encirclements of (-1, 0) in the Nyquist plot of G(s)H(s). Hence, the correct answer is '0

Given G(s) = s²+0.2s+100/s²−0.2s+100 and H(s) = 0.5