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95.2.2. Case I: One Pole
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Welcome everyone! Today, we're diving into how feedback networks affect the frequency response of amplifiers, particularly focusing on systems with a single pole. Does anyone remember what a pole is in the context of a transfer function?
Is it the point where the gain drops off?
Exactly! A pole is a frequency point where the output of the system starts to roll off. In our discussion about feedback, we'll see how the pole's location changes when feedback is applied. Can someone tell me what negative feedback means?
Negative feedback is when the output is fed back in a way that reduces the input signal.
Correct! Negative feedback stabilizes the system and can improve bandwidth. Let's discuss how we will analyze this in the Laplace domain.
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Now that we understand poles and feedback, let’s look at the transfer function. How can we represent the gain of an amplifier in the Laplace domain?
I think it can be represented as A(s) in the frequency domain?
Absolutely! For a system with one pole, we can define A(s) as having a pole at s = -p. When feedback is introduced, we also consider β(s). What happens to the gain in the presence of feedback?
The gain will change due to the feedback factor influencing the pole location.
Exactly! The new pole will be at p′, calculated as p(1 + βA_o). This mathematical relationship is crucial for our analysis.
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Create a free accountLet’s move to Bode plots. Why do we use Bode plots in analyzing feedback systems?
Because they help visualize how gain changes with frequency!
Exactly! So for our amplifier with one pole, if we compare the Bode plots of the open-loop gain A(s) and the loop gain, what do we notice about their intersection and how the feedback affects them?
The feedback reduces the gain, but it also shifts the pole to a new location.
Correct! And this shifting can be represented graphically, showing how feedback modifies the system response.
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Create a free accountNow that we’ve gone over feedback systems with a single pole, can someone summarize the impact of feedback on the pole's location?
Feedback shifts the pole to a new location, which is a function of both the original pole and the loop gain.
Well put! So, do we see how feedback not only stabilizes the gain but also adjusts the frequency response? That's key to understanding amplifier design.
Overview
Short Summary
This section explores the effect of feedback on the frequency response of a forward amplifier with one pole.
Medium Summary
In this section, we examine how the presence of a feedback network in an amplifier with a single pole influences its frequency response, focusing on gain and pole location shifting due to feedback.
Detailed Summary
Detailed Summary of Case I: One Pole
In this section, we delve into the impact of feedback on the frequency response of an analog amplifier characterized by having a single pole. We review the definitions of key feedback parameters and their mathematical representations, emphasizing the Laplace domain for analyzing frequency response. The backdrop of our analysis centers on a negative feedback system, where the feedback factor (β) is kept constant.
Key discussions include:
- An explanation of how the location of the pole in the feedback system changes the overall frequency response.
- The transfer function of the amplifier under feedback conditions, illustrating how the gain A(s) and feedback factor β(s) can be manipulated within the system.
- The derivation of the new pole location as a function of the original pole and the feedback loop's gain, represented mathematically.
- Graphical representations (Bode plots) demonstrating the shifts in gain and phase due to feedback, along with plots of the transfer function that show how gain bandwidth characteristics remain stable despite changes to pole locations.
Reference YouTube Videos
Audio Book
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Create a free accountTo start with let you consider case I, what we have in this situation it is when you say case I we assume that β it is independent of frequency. So, we can say that β is remaining constant and in the system, it is ‒ve feedback system in DC condition and let you consider that forward amplifier it is having a transfer function which is having only one pole. Which means that A(s) can be written in this form, A is the low frequency gain or you can say almost in the DC condition, what is the gain and then it is having a pole at s = p rather s = ‒ p.
Detailed Explanation
In this case, we focus on a feedback system where the feedback gain (β) is considered to be constant across frequencies, simplifying our analysis. The forward amplifier has a single pole in its transfer function, which means its behavior can be effectively represented by a simplified mathematical model. The low-frequency gain (A) represents how the amplifier behaves at DC or low frequencies, providing a baseline for its performance.
Examples & Analogies
Think of a car on a flat road (low-frequency conditions) vs. a steep hill (high-frequency conditions). On flat roads, the car (the amplifier) performs steadily without much effort (low-frequency gain). But as the road gets steeper (moving towards the pole), the car requires more power (gain shifts), similar to how the amplifier behaves when it approaches its pole.
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Create a free accountNow, if you recall that the feedback system transfer function assuming it is having a minus sign here it is and A(s) it is given here and β is independent of frequency. So, if you write the expression of this A(s) here what we are getting in the numerator it is and in the denominator we do have 1 + β × . Now, this factor we can take it here and so this factor it will be coming in the denominator of denominator. So, that part it is getting cancelled here.
Detailed Explanation
The feedback system's transfer function is analyzed in this context, focusing on how feedback influences the forward amplifier's gain. The expression accounts for the feedback factor (β), which modifies the amplifier's behavior. It simplifies the calculation by reducing the number of elements that affect the overall gain, ultimately yielding a clearer understanding of the system's stability and performance under feedback conditions.
Examples & Analogies
Imagine a director (feedback) managing the stage performance (amplifier). The director's guidance (feedback factor) helps improve the acting (output). If guidance is consistent (β constant), the show becomes smoother and more enjoyable (greater stability and performance), akin to how the feedback stabilizes the amplifier's output.
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Create a free accountSo, what we can get here is that the transfer function of the feedback system, it is having low frequency gain A were, A it is defined here divided by (1 + βA). So, this p′ is the location of the pole of the feedback system which is of course, it is a function of p × (1 + βA). So, this is this is what we do get that the location of the pole of this feedback system starting from primary input to primary output, it is a function of the location of the pole here and also getting multiplied by (1 + βA).
Detailed Explanation
This segment explains how the presence of feedback modifies the pole's location in the feedback system. When the forward amplifier has a pole located at p, under negative feedback, the new pole p' shifts to a location defined by the original pole's position and the feedback factor. This shift affects the system's stability and response, making it crucial for understanding feedback's role in circuit behavior.
Examples & Analogies
Consider a seesaw (the amplifier) where you place weights (the feedback). By moving the weights (changing the feedback), the pivot point (the pole) shifts, altering the seesaw's balance. This represents how feedback modifies circuit behavior by changing where it 'balances' in terms of performance, similar to how the pole moves from p to p′.
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Create a free accountSo, apart from the amplifier gain it is dropping from this level to this level, it is also having a consequence that the pole it is getting shifted from p to p′. And incidentally in this case the reduction of this gain it is happening by this factor (1 + βA) and the increase of this pole location or the pole it is getting shifted by the same factor (1 + βA).
Detailed Explanation
In conclusion, case I illustrates how a single pole within a feedback system not only influences the gain of the amplifier but also shifts the pole's position itself. As feedback is applied, both the gain and pole location change by the same factor, indicating a direct relationship between them. This behavior is essential for determining how the overall system responds to various frequencies and feedback levels.
Examples & Analogies
Think of an athlete adjusting their speed (amplifier gain) according to coach’s feedback (feedback factor). As the athlete speeds up (gain goes down), their stride length (pole location) increases proportionately. The relationship resembles how feedback can boost performance in parallel with modifying inherent traits, mirrored in the circuit's response to feedback.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Feedback Influence: Feedback can modify the location of poles in a system's transfer function.
Gain Shift: The introduction of negative feedback influences the gain and stability of amplifiers.
Bode Plot: A graphical tool that helps to visualize how frequency response changes with feedback.
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
Example 1: Appling feedback to an inverting amplifier shifts its frequency response, requiring designers to recalibrate gain settings.
Example 2: In an operational amplifier feedback system, the original pole at s=-p may shift to p′, impacting overall stability and performance.
Memory Aids
Interactive tools to help you remember key concepts
Stories
Flash Cards
Glossary
Pole
A point in the Laplace domain where the transfer function becomes infinite, affecting the system's frequency response.
Negative Feedback
Feedback that reduces the input to a system based on its output to stabilize and improve performance.
Transfer Function
A mathematical representation in the Laplace domain that describes the relationship between input and output of a system.
Bode Plot
A graphical representation of the frequency response of a system, showing gain and phase shift over a range of frequencies.
Feedback Factor (β)
The ratio of feedback voltage to input voltage in a feedback loop.