From the given system determine the number of loci, starting points, ending points and number of asymptotes.

Explanation:

Number of Loci, Starting Points, Ending Points, and Asymptotes

Understanding the Parameters:

The question involves analyzing a system based on certain parameters: the number of loci, starting points, ending points, and asymptotes. Let us define each of these terms:

• Number of Loci: This refers to the number of branches or paths traced by the roots of the characteristic equation in the complex plane as a specific parameter (such as gain) is varied.
• Starting Points: These are the initial positions of the roots of the characteristic equation (typically the poles of the transfer function) in the complex plane when the gain is zero.
• Ending Points: These are the final positions of the roots as the gain approaches infinity. These points can be either finite (zeros of the transfer function) or infinite, depending on the system configuration.
• Number of Asymptotes: Asymptotes describe the directions in which the loci tend toward infinity in the complex plane. The number of asymptotes is determined by the difference between the number of poles and zeros of the system.

Correct Option Analysis:

The correct option is:

Option 2: 2 loci, 2 starting points (-2, -4), ending points (2 + j4, 2 - j4), and 0 asymptotes.

Let us analyze this in detail:

Step 1: Determining the Number of Loci

The number of loci is equal to the total number of poles in the system, as each pole corresponds to a branch in the root locus. From the given data, there are 2 poles at (-2, -4). Hence, the number of loci is 2.

Step 2: Identifying Starting Points

The starting points are the locations of the poles of the system when the gain is zero. From the given system, the poles are located at (-2, -4). Therefore, the starting points are -2 and -4.

Step 3: Identifying Ending Points

As the gain increases to infinity, the loci of the poles move toward either the zeros of the system or infinity. From the given data, the ending points are specified as (2 + j4, 2 - j4), which are complex conjugate zeros. Hence, the ending points are (2 + j4, 2 - j4).

Step 4: Determining the Number of Asymptotes

The number of asymptotes is given by the formula:

Number of Asymptotes = Number of Poles - Number of Zeros

From the given data:

• Number of poles = 2 (located at -2 and -4).
• Number of zeros = 2 (located at 2 + j4 and 2 - j4).

Substituting these values:

Number of Asymptotes = 2 - 2 = 0

Therefore, there are 0 asymptotes.

Conclusion for Correct Option:

Based on the analysis, the system has:

• Number of Loci: 2
• Starting Points: (-2, -4)
• Ending Points: (2 + j4, 2 - j4)
• Number of Asymptotes: 0

This matches the description provided in Option 2, making it the correct choice.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 1 locus, starting points (2, 4), ending points (2 + j4, 2 - j4), and 0 asymptotes.

This option is incorrect because the number of loci is mismatched. The system has 2 poles, which means it should have 2 loci, not 1 as stated in this option. Additionally, the starting points are incorrectly given as (2, 4) instead of (-2, -4).

Option 3: 1 locus, starting points (-2, -4), ending points (2 + j4, 2 - j4), and 0 asymptotes.

This option is incorrect because, similar to Option 1, it states only 1 locus instead of 2. While the starting points and ending points are correct, the discrepancy in the number of loci makes this option invalid.

Option 4: 2 loci, starting points (2, 4), ending points (2 + j4, 2 - j4), and 0 asymptotes.

This option is incorrect because, although the number of loci and asymptotes are accurate, the starting points are incorrectly given as (2, 4) instead of (-2, -4). This error invalidates the option.

Conclusion:

Option 2 is the correct choice as it accurately describes the system's parameters: 2 loci, starting points at (-2, -4), ending points at (2 + j4, 2 - j4), and 0 asymptotes. The analysis of other options highlights the importance of correctly identifying the number of loci, starting points, ending points, and asymptotes to evaluate the system's behavior.