Find the range of K for stability. K - 16 (a) A range doesn't exist to stabilize the system. (b) 16 < K (c) K < 64 (d) 16 < K < 64
Find the range of K for stability. K - 16 (a) A range doesn't exist to stabilize the system. (b) 16 < K (c) K < 64 (d) 16 < K < 64
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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![**Problem Statement: Find the Range of K for Stability**
Consider the equation:
\[
\frac{\frac{1}{6}}{\frac{64 - K}{K - 16}}
\]
Identify the range of \( K \) that ensures system stability. Choose the correct option:
(a) A range doesn’t exist to stabilize the system.
(b) \( 16 \leq K \)
(c) \( K < 64 \)
(d) \( 16 \leq K < 64 \)
**Analysis:**
To determine the correct option, analyze the stability of the system by considering the values of \( K \) for which the denominator does not become zero and the fraction is defined.
1. **Denominator Considerations:**
- The expression \( \frac{64 - K}{K - 16} \) necessitates that \( K \neq 16 \).
2. **Range for K:**
- To avoid division by zero, the term \( K \neq 16 \), and since \( 64 - K \) must be positive for overall positivity, we derive:
- \( K < 64 \)
- \( K > 16 \)
Thus, conclude with the correct range that stabilizes the system:
- **Correct Answer: (d) \( 16 \leq K < 64 \)**
This indicates that the system will remain stable when \( K \) is in the range from 16 to less than 64, inclusive of 16 but excluding 64.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ecf1484-f95f-4119-9858-43c8382b92c0%2F7ad50f99-cde6-4622-a1ba-4a01e7b162f0%2F0msbltf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Find the Range of K for Stability**
Consider the equation:
\[
\frac{\frac{1}{6}}{\frac{64 - K}{K - 16}}
\]
Identify the range of \( K \) that ensures system stability. Choose the correct option:
(a) A range doesn’t exist to stabilize the system.
(b) \( 16 \leq K \)
(c) \( K < 64 \)
(d) \( 16 \leq K < 64 \)
**Analysis:**
To determine the correct option, analyze the stability of the system by considering the values of \( K \) for which the denominator does not become zero and the fraction is defined.
1. **Denominator Considerations:**
- The expression \( \frac{64 - K}{K - 16} \) necessitates that \( K \neq 16 \).
2. **Range for K:**
- To avoid division by zero, the term \( K \neq 16 \), and since \( 64 - K \) must be positive for overall positivity, we derive:
- \( K < 64 \)
- \( K > 16 \)
Thus, conclude with the correct range that stabilizes the system:
- **Correct Answer: (d) \( 16 \leq K < 64 \)**
This indicates that the system will remain stable when \( K \) is in the range from 16 to less than 64, inclusive of 16 but excluding 64.
Expert Solution
Step 1
Using given Routh Horwitz criteria find range of K for stability.
According to Routh Horwitz criteria, for stability first coloumn of routh table should not have no sign changes.
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