24.6
Plotting and Calibrating the Root Locus
Root loci often diverge as system poles shift from the real axis to the complex plane. Key points in this transition are the breakaway and break-in po…
Root loci commonly diverge when system poles transition from the real to the complex plane.
The breakaway and break-in points signal where the locus leaves and rejoins the real axis. The root locus branches form a 180/n degree angle with the real axis.
The gain peaks at the breakaway point between open-loop poles on the real axis, while the minimum gain occurs at the break-in point between two zeros.
Increasing gain can push some system poles into the right half-plane, indicating potential instability. The jω-axis crossings mark the boundary between stable and unstable system operations.
Root locus analysis involves locating specific points and calculating their related gain.
To know the exact coordinates of the root locus as it crosses a certain damping ratio line, several test points along the line are selected, and their angular sum is evaluated.
The root locus exists where the sum of total angles equals an odd multiple of 180 degrees.
The gain at that specific point is calculated by dividing the product of pole lengths by the product of zero lengths to that point.
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Q1: What are breakaway and break-in points in root locus analysis?
Breakaway points are where the root locus leaves the real axis as system poles transition to the complex plane, occurring between open-loop poles where gain peaks. Break-in points are where the locus rejoins the real axis between two zeros, where minimum gain occurs. These critical points signal pole migration and are essential for understanding system behavior.
Q2: How do you determine if a point lies on the root locus?
A point lies on the root locus where the sum of total angles from all poles and zeros to that point equals an odd multiple of 180 degrees. Test points along a damping ratio line are selected and their angular sum is evaluated. Once a point satisfies this angle condition, the gain at that location can be calculated.
Q3: How is gain calculated at a specific point on the root locus?
Gain is calculated by dividing the product of distances from all poles to the point by the product of distances from all zeros to that point. This ratio determines the system gain required for the poles to occupy that specific location. The calculation uses vector representation complex numbers to measure these distances accurately.
Q4: What does it mean when poles cross into the right half-plane?
When increasing gain pushes system poles into the right half-plane, it signals potential instability. The jω-axis serves as the boundary between stable and unstable operations. Crossing this boundary indicates the system may become unstable, requiring careful gain selection during control system design.
Q5: What angle do root locus branches form with the real axis?
Root locus branches form an angle of 180/n degrees with the real axis, where n represents the number of branches at a breakaway or break-in point. This angular relationship is a fundamental property that helps predict how poles diverge as they transition from the real axis to the complex plane.
Q6: How do you find the exact coordinates where a root locus crosses a damping ratio line?
Multiple test points are selected along the damping ratio line, and the angular sum from poles and zeros to each point is evaluated. When the total angle equals an odd multiple of 180 degrees, that point lies on the root locus. The gain at the intersection is then calculated using the pole and zero distance ratio.
Q7: Why is the jω-axis crossing important in root locus analysis?
The jω-axis crossing marks the critical boundary between stable and unstable system operations. This crossing point indicates the gain value at which the system transitions from stability to instability. Identifying this crossing is essential for determining safe operating ranges and ensuring robust control system design.