95.3. Case II: Two Poles

In this case, we consider A is having two poles namely p1 and p2. We can see what is the corresponding location of the pole of the feedback system. Here again, we consider it is –ve feedback system in DC condition β also remaining independent of frequency and the forward amplifier it is having two poles p1 and p2.

In this section, we examine a scenario involving an amplifier that has two distinct poles, denoted as p1 and p2. The analysis assumes that the feedback system operates under negative feedback conditions in a direct current (DC) state. We also indicate that the feedback factor (β) remains constant with respect to frequency. Understanding the effects of these two poles will provide insights into how feedback alters the amplifier's performance in various frequency domains.

Think of a two-pole feedback system like a seesaw with two weights on board. Each weight represents a pole; the placement and mass of each weight will affect how the seesaw moves (i.e., the overall system response). When feedback is applied, it's similar to someone pushing down on one side of the seesaw, changing how the seesaw balances and swings.

The location of the p2 is not only much higher than p1 magnitude-wise, but let us consider p1(1 + βA) is also lower than p2; which means, we are expecting this p1 will be getting shifted by this factor and probably this will be the shifted version of p1.

Here, the analysis elaborates on the relationship between the two poles p1 and p2 in the system. It states that pole p2 is at a higher frequency than p1, and when we apply feedback, we expect the first pole (p1) will shift due to the factor (1 + βA). Essentially, this indicates that feedback influences the position of the poles, leading to changes in system behavior at different frequencies. This shifting is crucial when analyzing the stability and the performance of the amplifier.

Imagine driving on two parallel roads, where the lower road represents p1 and the upper road represents p2. If the lower road (p1) begins to incline due to added feedback (like pressure from a heavy load on a vehicle), you'd expect it to rise slightly but remain lower than the upper road (p2). This change illustrates how the feedback can alter the characteristics of the system's response in a comparable way.

To start with, again let us consider A and the expression of A is given here. The numerator is A and in the denominator what we get is (1 + βA)i. This part if I put it here, then from this denominator this factor and this factor are getting cancelled here.

This chunk introduces the transfer function associated with the feedback system having two poles. It highlights that the transfer function can be simplified by examining the relationship between the numerator and the denominator. When we derive this function, specific terms in the denominator are cancelled out, which ultimately helps in formulating the feedback behavior more straightforwardly. Recognizing how to manipulate the transfer function is essential for determining the amplifier’s frequency response and improving performance.

Consider making a smoothie where the ingredients (the numerator) blend together smoothly with the liquid base (the denominator). If the blender's power (feedback) is too high, it might cause the base to swirl effectively, allowing the ingredients to mix better than if it were moving too slowly. This analogy illustrates how feedback can enhance or modify the output of a system.

Now, this updated pole and this is also updated pole and interestingly this is we call say p′ which is the case similar to a single pole situation. And then p2 may say that now this is a pole second pole of the A which is again approximately equal to p2.

After applying feedback to the transfer function, the analysis indicates that the poles of the feedback system exhibit behaviors akin to a single pole system. The pole p1 is substantially impacted and adjusted to a new pole, designated as p′, while the second pole p2 remains approximately equal in position. This outcome highlights the feedback’s significant role in reshaping the amplifier's dynamic characteristics, influencing its stability and frequency response profile.

Imagine tuning a musical instrument; as you twist a knob (feedback), one string immediately resonates correctly (shifts to p′), while another string stays roughly the same (remains as p2). The adjustment illustrates how feedback can fine-tune a system's responses while leaving other aspects unchanged.

In summary we can say that A shows the approximation we do have and this helps me for the factorization, and as I said that p′ is approximately equal to p1. This approximation indicates that one of these two poles remains the same, while the other shifts and potentially affects system dynamics.

In conclusion, the section reviews the results of the feedback analysis, particularly focusing on the close approximation of one of the poles (p') remaining largely unchanged compared to its original position (p1). This approximation allows us to maintain a clearer understanding of the system's dynamics and facilitates the factorization of the transfer function for easier analysis. The transformations of these poles are further illustrated through Bode plots, showcasing how the visual representation aligns with our theoretical predictions.

Returning to our earlier analogy of the seesaw, we can visualize the transformed system by recognizing that while one weight remains in place, another has adjusted. The seesaw now balances differently, representing how feedback effectively reshapes system dynamics while maintaining certain stabilizing features.

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