Discussion
Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : For a system to be stable, all the coefficients of characteristic equation should be positive with no missing term.
Reason (R) : If some of the coefficients are zero or negative, the system is NOT stable.
In light of the above statements, choose the most appropriate answer from the options given below.
Answer: D
Both (A) and (R) are true and (R) is the correct explanation of (A).
The Assertion is true: by the Hurwitz condition, a necessary condition for stability is that every coefficient of the characteristic equation is present and of the same sign, taken as positive.
The Reason is true and explains it: if a coefficient is zero or negative while the others are positive, the equation must have at least one root on the imaginary axis or in the right half of the s-plane, so the system is not stable. That is exactly why all coefficients must be positive with none missing. The condition is necessary but not sufficient; for third and higher order a Routh test is still needed.
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