For what values of K is this system stable
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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VALUES OF K(NEED A NEAT HANDWRITTEN SOLUTION ONLY OTHERWISE DOWNVOTE )
![### Stability Analysis Using Routh-Hurwitz Criterion
**Question:**
For what values of \( K \) is this system stable? (Use Routh-Hurwitz table).
**Block Diagram Explanation:**
The block diagram shown is a closed-loop feedback control system. It contains the following components:
1. **Reference Input** \(R(s)\): The desired output or the setpoint of the system.
2. **Summing Point**: The summing point takes the reference input \(R(s)\) and subtracts the feedback signal \(C(s)\) from it.
3. **Controller**: The controller is represented by a gain \(K\) divided by \(s\).
4. **Plant Transfer Function**: The plant transfer function is given by \(\frac{1}{s^3 + 6s^2 + 11s + 6}\).
5. **Output** \(C(s)\): The actual output of the system.
### Step-by-Step Solution:
1. **Characteristic Equation:**
The characteristic equation of the closed-loop system can be found by analyzing the open-loop transfer function and setting the denominator to zero.
2. **Open-Loop Transfer Function:**
The open-loop transfer function is:
\[
G(s)H(s) = \frac{\frac{K}{s}}{1 \cdot (s^3 + 6s^2 + 11s + 6)}
\]
This simplifies to:
\[
G(s)H(s) = \frac{K}{s(s^3 + 6s^2 + 11s + 6)}
\]
3. **Characteristic Equation:**
\[
1 + G(s)H(s) = 0
\]
\[
1 + \frac{K}{s(s^3 + 6s^2 + 11s + 6)} = 0
\]
By manipulating the equation, the characteristic equation becomes:
\[
s(s^3 + 6s^2 + 11s + 6) + K = 0
\]
\[
s^4 + 6s^3 + 11s^2 + 6s + K = 0
\]
4. **Routh-Hurwitz Table:**
To determine the values of \(K](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa22dd6ba-b194-4859-8383-05d4d2bb5a27%2F1aedff7f-c459-4111-b2e8-bcdb18acca36%2Fuv8hwbn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Stability Analysis Using Routh-Hurwitz Criterion
**Question:**
For what values of \( K \) is this system stable? (Use Routh-Hurwitz table).
**Block Diagram Explanation:**
The block diagram shown is a closed-loop feedback control system. It contains the following components:
1. **Reference Input** \(R(s)\): The desired output or the setpoint of the system.
2. **Summing Point**: The summing point takes the reference input \(R(s)\) and subtracts the feedback signal \(C(s)\) from it.
3. **Controller**: The controller is represented by a gain \(K\) divided by \(s\).
4. **Plant Transfer Function**: The plant transfer function is given by \(\frac{1}{s^3 + 6s^2 + 11s + 6}\).
5. **Output** \(C(s)\): The actual output of the system.
### Step-by-Step Solution:
1. **Characteristic Equation:**
The characteristic equation of the closed-loop system can be found by analyzing the open-loop transfer function and setting the denominator to zero.
2. **Open-Loop Transfer Function:**
The open-loop transfer function is:
\[
G(s)H(s) = \frac{\frac{K}{s}}{1 \cdot (s^3 + 6s^2 + 11s + 6)}
\]
This simplifies to:
\[
G(s)H(s) = \frac{K}{s(s^3 + 6s^2 + 11s + 6)}
\]
3. **Characteristic Equation:**
\[
1 + G(s)H(s) = 0
\]
\[
1 + \frac{K}{s(s^3 + 6s^2 + 11s + 6)} = 0
\]
By manipulating the equation, the characteristic equation becomes:
\[
s(s^3 + 6s^2 + 11s + 6) + K = 0
\]
\[
s^4 + 6s^3 + 11s^2 + 6s + K = 0
\]
4. **Routh-Hurwitz Table:**
To determine the values of \(K
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