This Hindi-language video tutorial explains the concept of root locus in control systems. The instructor focuses on a problem without complex poles, guiding viewers through the design steps and calculations involved in constructing a root locus diagram. The video emphasizes the graphical method of representing the roots of a characteristic equation as a system parameter varies.
The video details the following steps for constructing a root locus. Note that these steps are based on a specific example within the video and might need adjustments depending on the complexity of the system's transfer function.
Determine the open-loop transfer function G(s)H(s): This function describes the system's behavior without feedback. The video provides a specific G(s)H(s).
Plot the poles and zeros of G(s)H(s): Locate the poles (represented by 'x') and zeros (represented by 'o') of the open-loop transfer function on the s-plane (real and imaginary axes). The video uses the transfer function G(s) = k/[s(s+4)(s+5)] which has poles at s=0, s=-4, and s=-5, and no zeros.
Identify the root locus on the real axis: Determine the segments of the real axis that belong to the root locus. A section of the real axis is part of the root locus if the total number of poles and zeros to the right of that section is odd. The video demonstrates this by checking the number of poles to the right of different segments on the real axis between the poles.
Determine the number of asymptotes: The number of asymptotes is given by n = p - z, where 'p' is the number of poles and 'z' is the number of zeros of G(s)H(s). In the video's example, n = 3 (3 poles, 0 zeros), meaning there are three asymptotes.
Calculate the centroid of the asymptotes: The centroid is the point where the asymptotes intersect. It's calculated using the formula: Centroid = (Sum of poles - Sum of zeros) / (p - z). The video applies this formula to determine the centroid and uses it as a point of reference for drawing the asymptotes.
Find the angles of the asymptotes: The angles of the asymptotes are given by: φ = (2k + 1) / (p - z) * 180°, where k = 0, 1, 2,... (p-z)-1. The video calculates these angles for the three asymptotes.
Calculate the breakaway points: Breakaway points are points on the root locus where multiple branches meet. They occur where the derivative of the characteristic equation (1 + G(s)H(s) = 0) with respect to s is zero. The video solves the derivative equation to find breakaway points. The video also illustrates how to determine which of these points are valid (located on valid sections of the root locus from step 3).
Determine the points of intersection with the imaginary axis (if applicable): Use the Routh-Hurwitz criterion or other methods to find where the root locus crosses the imaginary axis. This reveals the range of 'k' for which the system is stable. The video uses the Routh-Hurwitz criterion to find this point.
Sketch the root locus: Draw the root locus based on all the information obtained from the previous steps. Begin with the poles and zeros, connect them using the valid segments from step 3, include the asymptotes, plot the breakaway points, and show the intersection with the imaginary axis (if calculated). The video shows the complete construction of the root locus using this information.
The video emphasizes using graph paper and a protractor ('d') for accurate sketching, especially when drawing the asymptotes and their angles. Also, it stresses that only valid sections of the root locus determined in step 3 should be considered during the entire process.
