A system is given by G(s) = \(\rm \frac{K}{s(s+2)(s+4)}\) and H(s) = 1. What should be the value of K for the system to have stability? 

Explanation:

To determine the value of \( K \) for the system to be stable, we first need to analyze the open-loop transfer function of the system. The given transfer function of the system is:

\[ G(s) = \frac{K}{s(s+2)(s+4)} \] and the feedback transfer function is \( H(s) = 1 \).

The closed-loop transfer function for a unity feedback system is given by:

\[ T(s) = \frac{G(s)}{1 + G(s)H(s)} = \frac{\frac{K}{s(s+2)(s+4)}}{1 + \frac{K}{s(s+2)(s+4)}} \]

To simplify, multiply the numerator and the denominator by the denominator of \( G(s) \):

\[ T(s) = \frac{K}{s(s+2)(s+4) + K} \]

For the system to be stable, all the poles of the closed-loop transfer function must lie in the left half of the s-plane (i.e., all the poles must have negative real parts). The poles of the closed-loop transfer function are the roots of the characteristic equation:

\[ 1 + G(s)H(s) = 0 \rightarrow 1 + \frac{K}{s(s+2)(s+4)} = 0 \]

Multiplying through by \( s(s+2)(s+4) \):

\[ s(s+2)(s+4) + K = 0 \]

Expanding the polynomial:

\[ s^3 + 6s^2 + 8s + K = 0 \]

To determine the range of \( K \) for stability, we can use the Routh-Hurwitz criterion. The Routh-Hurwitz criterion provides a systematic way to determine the number of roots of the characteristic equation that lie in the right half-plane, left half-plane, and on the imaginary axis. The characteristic equation for the system is:

\[ s^3 + 6s^2 + 8s + K = 0 \]

The Routh array for this characteristic equation is constructed as follows:

For the system to be stable, all the elements in the first column of the Routh array must be positive:

1. The coefficient of \( s^3 \): \( 1 \) (which is positive)
2. The coefficient of \( s^2 \): \( 6 \) (which is positive)
3. The coefficient of \( s^1 \): \(\frac{48 - K}{6} > 0 \rightarrow 48 - K > 0 \rightarrow K < 48\)
4. The coefficient of \( s^0 \): \( K > 0 \)

So, for the system to be stable:

\[ 0 < K < 48 \]

The correct option is therefore option 3: \( K < 48 \).

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \( K > 48 \)

This option is incorrect because if \( K > 48 \), the coefficient of \( s^1 \) would become negative (\(\frac{48 - K}{6} < 0\)), indicating that the system would be unstable.

Option 2: \( K < 24 \)

While \( K < 24 \) is within the stability range, it is not the complete condition for stability. \( K \) must be less than 48 for the system to be stable, so this option is incomplete and not the best representation of the stability condition.

Option 4: \( K > 24 \)

This option is also incorrect because \( K > 24 \) does not guarantee stability. The critical upper limit for stability is \( K < 48 \).

Conclusion:

The stability of a control system is crucial for its correct operation. By applying the Routh-Hurwitz criterion to the characteristic equation, we determined that the system is stable for \( 0 < K < 48 \). This comprehensive analysis helps ensure that the system operates correctly within the specified range of \( K \).