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A closed loop system has the characteristic equation given by s3 + Ks2 + ( K + 2 )s + 3 = 0. For this system to be stable, which one of the following conditions should be satisfied?
- a)0 < K < 0.5
- b)0.5 < K < 1
- c)0 < K < 1
- d)K > 1
Correct answer is option 'D'. Can you explain this answer?
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A closed loop system has the characteristic equation given by s3+ Ks2+...
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A closed loop system has the characteristic equation given by s3+ Ks2+...
Given CE = s3 + ks2 + (k+ 2)s + 3 = 0
For stable
k(k+2) >3
k2 + 2k -3 >0
(k-1)(k+3) > 0
k > -1 ∩ k > -3
k >1
OR
by R-H criteria
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A closed loop system has the characteristic equation given by s3+ Ks2+...
To analyze the stability of a closed-loop system, we need to examine the characteristic equation of the system. The characteristic equation is obtained by setting the denominator of the transfer function to zero.
The given characteristic equation is:
s^3 + Ks^2 + (K + 2)s + 3 = 0
To determine the stability of the system, we can use the Routh-Hurwitz criterion. According to this criterion, for a system to be stable, all the coefficients of the characteristic equation must be positive.
Let's break down the characteristic equation into its coefficients:
Coefficient of s^3: 1
Coefficient of s^2: K
Coefficient of s: K + 2
Coefficient of s^0: 3
Now, we can use the Routh-Hurwitz criterion to check the stability condition.
1. The first row of the Routh-Hurwitz table consists of the coefficients of the s^3, s, s^5, s^7, etc. terms.
Row 1: 1, K
2. The second row is obtained by calculating the coefficients of the s^2, s^4, s^6, etc. terms based on the coefficients of the first row.
Row 2: (K + 2)
3. The remaining rows of the table can be calculated by the following formula:
Row i (i ≥ 3): (Row i-2 coefficient * Row i-1 coefficient - Row i-1 coefficient * Row i-2 coefficient) / Row i-1 coefficient
Since we only have two rows, we can calculate the third row directly:
Row 3: (K + 2)
Now, we can analyze the stability condition based on the coefficients in the Routh-Hurwitz table.
For the system to be stable, all the coefficients in the table must be positive. In this case, we need to ensure that:
1 > 0 (from Row 1)
K + 2 > 0 (from Row 2)
K + 2 > 0 (from Row 3)
From the above conditions, we can conclude that K + 2 > 0, which simplifies to K > -2. Therefore, the stability condition for the given closed-loop system is K > -2.
Among the given options, option D (K ≥ 1) satisfies the stability condition.
The given characteristic equation is:
s^3 + Ks^2 + (K + 2)s + 3 = 0
To determine the stability of the system, we can use the Routh-Hurwitz criterion. According to this criterion, for a system to be stable, all the coefficients of the characteristic equation must be positive.
Let's break down the characteristic equation into its coefficients:
Coefficient of s^3: 1
Coefficient of s^2: K
Coefficient of s: K + 2
Coefficient of s^0: 3
Now, we can use the Routh-Hurwitz criterion to check the stability condition.
1. The first row of the Routh-Hurwitz table consists of the coefficients of the s^3, s, s^5, s^7, etc. terms.
Row 1: 1, K
2. The second row is obtained by calculating the coefficients of the s^2, s^4, s^6, etc. terms based on the coefficients of the first row.
Row 2: (K + 2)
3. The remaining rows of the table can be calculated by the following formula:
Row i (i ≥ 3): (Row i-2 coefficient * Row i-1 coefficient - Row i-1 coefficient * Row i-2 coefficient) / Row i-1 coefficient
Since we only have two rows, we can calculate the third row directly:
Row 3: (K + 2)
Now, we can analyze the stability condition based on the coefficients in the Routh-Hurwitz table.
For the system to be stable, all the coefficients in the table must be positive. In this case, we need to ensure that:
1 > 0 (from Row 1)
K + 2 > 0 (from Row 2)
K + 2 > 0 (from Row 3)
From the above conditions, we can conclude that K + 2 > 0, which simplifies to K > -2. Therefore, the stability condition for the given closed-loop system is K > -2.
Among the given options, option D (K ≥ 1) satisfies the stability condition.
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A closed loop system has the characteristic equation given by s3+ Ks2+ ( K + 2 )s + 3 = 0. For this system to be stable, which one of the following conditions should be satisfied?a)0 < K < 0.5b)0.5 < K < 1c)0 < K < 1d)K > 1Correct answer is option 'D'. Can you explain this answer?
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