Consider a negative unity feedback system with the forward path transfer function s2-s-1s3-2s2-2s-k- where K is a positive real number- The value of K for which the system will have some of its poles on the imaginary axis is

CE is 1 + G(s) H(s) = 0 ⇒ 1+s^2+s+1s^3+2s^2+2s+K=0 ⇒ s^3+3s^2+3s+(1+K)=0 R.H. criteria: . s^3 nbsp; s^2 nbsp; s^1| 1 amp;3 3 nbsp; nbsp; am ...

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Standard XII

Physics

Electric Potential at a Point in Space Due to a Point Charge Q

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Question

Consider a negative unity feedback system with the forward path transfer function $\frac{s^{2} + s + 1}{s^{3} + 2 s^{2} + 2 s + k},$ where K is a positive real number. The value of K for which the system will have some of its poles on the imaginary axis is

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Solution

The correct option is A 8
CE is
$1 + G (s) H (s) = 0$

$\Rightarrow 1 + \frac{s^{2} + s + 1}{s^{3} + 2 s^{2} + 2 s + K} = 0$

$\Rightarrow s^{3} + 3 s^{2} + 3 s + (1 + K) = 0$

R.H. criteria:

$\begin{matrix} s^{3} s^{2} s^{1} \end{matrix} ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 3 3 & (1 + K) 9 - (1 + K) & 0 \end{matrix}$

For marginal stability,

$9 - (1 + K) = 0$

$\Rightarrow K = 8$

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Consider a negative unity feedback system with the forward path transfer function s2+s+1s3+2s2+2s+k, where K is a positive real number. The value of K for which the system will have some of its poles on the imaginary axis is

The correct option is A 8CE is 1+G(s)H(s)=0 ⇒1+s2+s+1s3+2s2+2s+K=0 ⇒s3+3s2+3s+(1+K)=0 R.H. criteria: s3s2s1∣∣ ∣ ∣∣133(1+K)9−(1+K)0 For marginal stability, 9−(1+K)=0 ⇒K=8

Consider a negative unity feedback system with the forward path transfer function $\frac{s^{2} + s + 1}{s^{3} + 2 s^{2} + 2 s + k},$ where K is a positive real number. The value of K for which the system will have some of its poles on the imaginary axis is