Discussion
The characteristic equation for a feedback control system is Q(s) = s3 + 13s2 + 30s + k = 0.
The range of k for a stable system is :
Answer: A
The system is stable for \(0 \lt k \lt 390\).
Build the Routh array for \(s^{3} + 13s^{2} + 30s + k\):
\[ \begin{aligned} s^{3}&: \quad 1 \quad 30 \\ s^{2}&: \quad 13 \quad k \\ s^{1}&: \quad \frac{13 \times 30 - k}{13} \\ s^{0}&: \quad k \end{aligned} \]
Every entry of the first column must be positive. The \(s^{0}\) row needs \(k \gt 0\), and the \(s^{1}\) row needs \(390 - k \gt 0\), so \(0 \lt k \lt 390\). At \(k = 390\) the \(s^{1}\) row vanishes and the system oscillates.
The other ranges are built from the coefficients themselves (13 and 30) or drop a factor of ten (39); none comes from the stability condition.
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