96.6. Comparison of Poles in Different Cases

In the next slide we are going to compare the location of poles. So we like to compare location of poles for these two cases. So let me try to see here this A(s), the forward amplifier circuit it is having 2 poles p and p . So it is let me consider this case first.

In this initial segment, we set the stage for comparing pole locations in feedback systems. We start by discussing the two poles of the forward amplifier circuit, denoted as p1 and p2. This sets up the context for analyzing how these poles behave in different scenarios within the system.

Imagine a seesaw with two children seated at different distances from the pivot point. The relative positions of the children (representing the poles) determine how easily the seesaw tips in one direction or the other. Similarly, the placement of poles significantly influences system behavior.

Now let us consider case II-A. So, we do have situation for case II-A and here again, this is the real part of the s and this is the imaginary part of s and in this case, what we said is p it is say let me again use the blue colour for A(s). So we do have p here, but then p it is quite close.

In Case II-A, we analyze a situation where the poles are far apart in the complex plane, which typically represents a stable system behavior. The blue representation visually indicates the poles of the forward amplifier. This distinction is important because if the poles are separated adequately, it tends to lead to a predictable and reliable frequency response of the system.

Consider two cars traveling on parallel lanes but at different speeds. If one car is significantly faster than the other, it’s easier to predict the distance between them at any given time. Similarly, widely spaced poles make the system’s behavior easier to predict and manage.

In this case, what we are expecting is that this p′ , it may be in the near vicinity. So this is p′ and since it is very close to this p, it will interfere with that and p will also be changing.

In contrast, Case II-B outlines a scenario where the poles are close together, which significantly affects the system response and stability. As they get closer, their interaction leads to potential complex conjugate pole formation, indicating a more complex behavior that could introduce instability into the system.

Think of two musicians trying to play a duet. If they are too close in pitch (the poles in this analogy), they might start to sound dissonant or out of harmony, resulting in a less pleasant experience. This reflects how closely placed poles can disrupt system stability and performance.

So we can say that this is maybe p′, it is having ‒ sign here. So, I am considering this is p′ and this is p′. So, if we take this p and p closer and closer, then bifurcation of this complex conjugate pole pair, it will be more.

This section emphasizes the consequences of pole proximity. When poles approach each other closely enough, they can create complex conjugate pairs, leading to oscillatory behavior in the system’s response. The extraction and factorization of these poles become critical for understanding their influence on feedback stability.

Imagine a crowd at a concert. If two friends near the stage start pushing together against each other, they can create a ripple effect through the crowd, changing the atmosphere of the concert. Similar to this, closely positioned poles influence the overall stability and response of the system.

One important point, you we have simply missed it if you see this corner point and the frequency at which the loop gain it is becoming crossing 0 dB level.

In summarizing the discussions on pole locations, we highlight critical points of interest, such as where loop gain crosses the 0 dB line, signifying stability limits. This corner point is significant as it delineates stable from unstable system behavior, serving as a reference for analyzing the response to frequency changes.

This can be likened to a company marking a crucial revenue milestone—once they cross that threshold, their strategies must change to maintain growth. Similarly, in feedback systems, crossing the 0 dB threshold necessitates adjustments in design to preserve stability.

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