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 points, indicating where the root locus leaves and reenters the real axis. The branches of the root locus form an angle of 180/ndegrees with the real axis, where n is the number of branches at a breakaway or break-in point.

The maximum gain occurs at the breakaway points between open-loop poles on the real axis, while the minimum gain is observed at the break-in points between two zeros. As the gain increases, some system poles may be pushed into the right half-plane, signaling potential instability. The crossings of the jω-axis delineate the boundary between stable and unstable operations.

Root locus analysis involves identifying these critical points and computing the corresponding gain. To determine the exact coordinates of the root locus as it intersects a specific damping ratio line, various test points along this line are selected, and the sum of angles from the system's poles and zeros to these points is evaluated. The root locus exists where the total angles equal an odd multiple of 180 degrees.

Once a point on the root locus is confirmed, the gain at that point can be calculated. This is done by dividing the product of the distances from the poles to the point by the product of the distances from the zeros to the point. This approach helps determine how the system's poles migrate with changing gain and assess the stability and transient response.

In summary, root locus analysis provides a comprehensive method for visualizing and understanding the behavior of a system's poles as gain varies. By examining breakaway and break-in points and calculating gains at specific points, engineers can design and fine-tune control systems to ensure stability and desired performance characteristics. This method is essential for predicting how system poles move and ensuring robust control system design.