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What will be the natural frequency (wn) for the above controller? 

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  1. \(\rm \omega_n=\sqrt{\frac{1}{Jk_p}}\)
  2. \(\rm \omega_n=\sqrt{\frac{k_p}{J}}\)
  3. \(\rm \omega_n=\sqrt{Jk_p}\)
  4. \(\rm \omega_n=\sqrt{\frac{J}{k_p}}\)

Answer (Detailed Solution Below)

Option 2 : \(\rm \omega_n=\sqrt{\frac{k_p}{J}}\)
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Detailed Solution

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Concept:

In control systems, the natural frequency \(\omega_n\) of a second-order system can be found by comparing the characteristic equation to the standard second-order form:

\( G(s) = \frac{K_p(1+T_d s)}{Js^2} \)

The open-loop transfer function becomes:

\( G(s)H(s) = \frac{K_p(1 + T_d s)}{J s^2} \)

The characteristic equation is:

\( 1 + \frac{K_p(1 + T_d s)}{J s^2} = 0 \Rightarrow J s^2 + K_p T_d s + K_p = 0 \)

Comparing this to the standard form:

\( s^2 + 2\zeta \omega_n s + \omega_n^2 = 0 \)

We get: \( \omega_n^2 = \frac{K_p}{J} \Rightarrow \omega_n = \sqrt{\frac{K_p}{J}} \)

 

 

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