Introductory Circuit Analysis (13th Edition)
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ISBN: 9780133923605
Author: Robert L. Boylestad
Publisher: PEARSON
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This is a part of a review I'm studying, NOT a graded assignment, please do not reject. Thank you!
![**Second-order System Performance in Feedback Systems**
---
**Diagram Explanation:**
The diagram displays a block diagram of a feedback system. It consists of the following components:
- **R(s):** Input to the system.
- **K:** Gain block.
- **G(s):** Forward path transfer function.
- **H(s):** Feedback path transfer function.
- **C(s):** Output of the system, with a feedback loop from C(s) to the summing point before K.
The mathematical expression shows the transfer function for the closed-loop system:
\[ C(s) = \left[\frac{KG(s)}{1 + KG(s)H(s)}\right] R(s) \]
The gain \( K \) ranges from \( 0 \leq K < \infty \).
The characteristic equation is given by:
\[ \Delta_{CLTF}(s) = 1 + KG(s)H(s) \]
---
**Example:**
**Objective:** Sketch all the Closed Loop Roots in the S-Plane for \(\Delta_{CLTF}(s)\) for K varying from Zero to Infinite.
---
- **\( G_c(s) = K \):** Controller gain.
- **\( G(s) = \frac{1}{(s+1)(s+2)(s+4)} \):** Transfer function of the plant.
- **\( H(s) = 1 \):** Feedback transfer function.
The closed-loop transfer function is:
\[ T(s) = \frac{KG(s)}{1 + KG(s)H(s)} \]
The characteristic equation remains:
\[ \Delta_{CLTF}(s) = 1 + KG(s)H(s) \]
---
This exercise involves analyzing the roots of the characteristic equation as the gain \( K \) varies from zero to infinity, which is essential for understanding system stability and performance characteristics in control systems.](https://content.bartleby.com/qna-images/question/ecd9d238-92a1-4a47-af9e-c758e887edd1/67b49093-ffd8-48cd-b359-d6c8c8e56df3/xcm98rp_processed.png)
Transcribed Image Text:**Second-order System Performance in Feedback Systems**
---
**Diagram Explanation:**
The diagram displays a block diagram of a feedback system. It consists of the following components:
- **R(s):** Input to the system.
- **K:** Gain block.
- **G(s):** Forward path transfer function.
- **H(s):** Feedback path transfer function.
- **C(s):** Output of the system, with a feedback loop from C(s) to the summing point before K.
The mathematical expression shows the transfer function for the closed-loop system:
\[ C(s) = \left[\frac{KG(s)}{1 + KG(s)H(s)}\right] R(s) \]
The gain \( K \) ranges from \( 0 \leq K < \infty \).
The characteristic equation is given by:
\[ \Delta_{CLTF}(s) = 1 + KG(s)H(s) \]
---
**Example:**
**Objective:** Sketch all the Closed Loop Roots in the S-Plane for \(\Delta_{CLTF}(s)\) for K varying from Zero to Infinite.
---
- **\( G_c(s) = K \):** Controller gain.
- **\( G(s) = \frac{1}{(s+1)(s+2)(s+4)} \):** Transfer function of the plant.
- **\( H(s) = 1 \):** Feedback transfer function.
The closed-loop transfer function is:
\[ T(s) = \frac{KG(s)}{1 + KG(s)H(s)} \]
The characteristic equation remains:
\[ \Delta_{CLTF}(s) = 1 + KG(s)H(s) \]
---
This exercise involves analyzing the roots of the characteristic equation as the gain \( K \) varies from zero to infinity, which is essential for understanding system stability and performance characteristics in control systems.
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