Root Locus Analysis: Concepts u0026 Applications Quiz Quiz

Explore essential concepts in root locus analysis with this targeted quiz focusing on system stability, design principles, and interpretation of root locus plots. Perfect for engineering students and professionals aiming to strengthen their understanding of root locus methods in control systems.

1. Identifying Real-Axis Segments on a Root Locus Plot

   Which of the following best describes how the real-axis segments of a root locus are determined for a transfer function with multiple poles and zeros?

   1. They appear on the real axis to the right of all poles and zeros.
   2. They appear on portions of the real axis to the left of an odd number of real poles and zeros.
   3. They can only be found between zeros if there are more zeros than poles.
   4. They exist only between every pair of poles regardless of zero positions.

   Explanation: The real-axis segments of a root locus plot correspond to the parts of the real axis that are to the left of an odd number of real poles and zeros. This is a core rule used to sketch root loci in control system analysis. Simply being between every pair of poles is not always correct because zeros also influence the segments. Stating they appear to the right of all poles and zeros ignores the actual rule and would lead to incorrect plots. The idea that segments can only be found between zeros if there are more zeros than poles is inaccurate; the count of zeros versus poles changes the asymptotic behavior, not the real-axis segment rule.

2. Determining System Stability from Root Locus Plots

   If all branches of the root locus for a given system remain in the left-half s-plane for all positive gain values, what can be said about the system’s stability?

   1. The system is stable only for small gain values.
   2. The open-loop system becomes unstable as gain increases.
   3. The closed-loop system is stable for all positive gain values.
   4. The system will always be critically damped.

   Explanation: If the root locus branches stay entirely within the left-half of the s-plane for all positive gains, it means that all closed-loop poles have negative real parts, ensuring stability. The open-loop system’s instability depends not on the locus but on pole positions of the open-loop transfer function itself. Stability for only small gain values would be incorrect since the locus confirms stability for any positive gain. A system with poles in the left-half is not always critically damped; damping depends on pole locations, not just their real part.

3. Asymptote Angles in Root Locus Plots

   Given a system with three poles and one zero, what are the angles of the root locus asymptotes as gain approaches infinity?

   1. 45°, 135°, and 225°
   2. 0°, 120°, and 240°
   3. 60°, 180°, and 300°
   4. 90° and 270°

   Explanation: For a root locus where the difference between the number of poles and zeros is two, the asymptotes' angles are calculated as (2k+1)180° divided by the number of asymptotes. Here, the number of asymptotes is 2, but the correct formula for three poles and one zero actually gives three asymptotes at 60°, 180°, and 300°. The other options do not match the calculation and either use incorrect increments or the wrong starting angle, not fitting with the formula for root locus asymptotes.

4. Effect of Adding a Zero on Root Locus Behavior

   What is the primary effect of adding a zero close to the origin on the root locus and corresponding step response of a control system?

   1. It pulls the root locus branches away from the origin, reducing stability.
   2. It pulls the root locus branches toward the zero, making the response faster.
   3. It moves all poles to the left, making the system overdamped.
   4. It introduces an unstable pole to the system.

   Explanation: Adding a zero near the origin attracts root locus branches toward itself and typically increases the speed of the system response by shifting closed-loop poles further left in the s-plane. However, this does not necessarily make the system overdamped, which requires poles to be much farther apart or on the real axis. The zero does not push branches away from the origin nor does it add unstable poles; zeros don't create new poles but influence their paths.

5. Breakaway Points in Root Locus Analysis

   Where do breakaway points typically occur in root locus analysis, and how are they found on the real axis?

   1. They always occur at the intersection of the root locus with the imaginary axis.
   2. They are found by setting the magnitude of the open-loop transfer function to one.
   3. They occur at locations where zeros and poles coincide.
   4. They occur between two real poles on the root locus segment and are found by solving for points where dK/ds = 0.

   Explanation: Breakaway points are points on the real-axis segments of the root locus between two poles (or zeros) where branches split away from or converge onto the real axis. These are mathematically determined by solving dK/ds = 0 for those segments. They do not always appear at the imaginary axis intersection nor specifically where poles and zeros overlap. Setting the magnitude of the open-loop transfer function to one finds points on the locus, but not specifically breakaway points.