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If you have read my previous post, you already know what exactly the roots and characteristic equations of CLTF are. So to summarize briefly, the roots associated with each characteristic equation of a system are the poles associated with the system, which are further combined with zeros to determine the answer for that system. The branches related to the root locus tell you how the local area of the poles changes when you increase (increase) the gain (k) of a given system. When plotting on the main trace, let’s plot the roots of this characteristic equation as part of gain rather than frequency. So it’s not a frequency domain system. Now we can see some additional points about the root locus method: 1. If there are more zeros and fewer poles, then the number of zeros is greater, then the number of loci = z. The number of loci ending in ∞ is equal to z – p . All other rules remain the same. 2. Value of Gain Margin: The gain margin for the root locus is Gain Margin = If the root locus does not intersect the j-axis, the gain is ∞. Gain margin proves the system with the highest gain that can be multiplied to be just on the edge (edge) of instability. 3. Phase distance to root locus: For a given value of k, the phase distance can be given as follows: Find by |GH(j)| = 1 represents the design value of k. Find arg GH ( j ). Phase Margin = 180 + arg GH(j). The effect of adding poles and zeros on the root locus: 1. The effect of adding poles: The basic effect of including poles is a tendency to move the entire root locus toward the right half of the S-plane. This reduces stability. Conser a system: G(s)H(s) = , whose root locus looks like this: Now conser the system G(s)H(s) = , whose root locus looks like this: Now we add a pole s = – 1.5, G(s)H(s) = , whose root locus is as follows: Therefore, at the value of k, the system becomes unstable. Without adding -1.5 poles at s = -5, the system would eventually become unstable, but the crossover would occur at higher values of k. The addition of poles moves the closed-loop poles toward the j-axis, making the system more oscillating. Of the poles s = -2 and -1.5, s = -1.5 contributes more to the change because it is near the j-axis and the origin. Hence, it is called the dominant pole. 2. Effect of adding zero: The general effect of adding zero is to tend to pull the root locus to the left half of the S-plane. So adding zeros increases stability. Conser G(s) H(s) = , if we add a zero at s = -5, its root locus is given by Now, if we add an asymptotically dominant pole, the root locus is given by Now, which is H . s = -2 instead of s = -5, we get the root locus as follows: We can see that the system becomes more stable. So adding a zero speeds up the settling time (since it pulls the trajectory to the left) and stabilizes the system. Finally, you should know that this method is used to find the stable “k” value (i.e. system gain) for a given closed loop system. The point (key point) here is that if we add an extra controller to our process (or if we change the system gain), we can tune the system to meet our specific needs. For example, if the desired motor is not moving fast enough, we need to add a controller that simply increases the current or voltage to the motor (so the motor spins faster). That’s what timing analysis is for (I hope you really enjoy it). Starting with my next article, we’ll look at frequency response analysis of a system. report this ad
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