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The damping ratio of a series RLC circuit can be expressed as
1. \(\frac{{{{\rm{R}}^2}{\rm{C}}}}{{2{\rm{L}}}}\)
2. \(\frac{{2{\rm{L}}}}{{{{\rm{R}}^2}}}\)
3. \(\frac{{\rm{R}}}{2}\sqrt {\frac{{\rm{C}}}{{\rm{L}}}}\)
4. \(\frac{2}{{\rm{R}}}\sqrt {\frac{{\rm{L}}}{{\rm{C}}}}\)

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Correct Answer - Option 3 : \(\frac{{\rm{R}}}{2}\sqrt {\frac{{\rm{C}}}{{\rm{L}}}}\)

The characteristic equation of a series RLC circuit is given by,

\({\rm{C}}.{\rm{E}} = {{\rm{s}}^2} + \frac{{\rm{R}}}{{\rm{L}}}{\rm{s}} + \frac{1}{{{\rm{LC}}}}\)

We have \({{\rm{\omega }}_{\rm{n}}} = \frac{1}{{\sqrt {{\rm{LC}}} }}\)

And \(2{\rm{\zeta }}{{\rm{\omega }}_{\rm{n}}} = \frac{{\rm{R}}}{{\rm{L}}}\)

\(\begin{array}{l} \Rightarrow {{\rm{\omega }}_{\rm{n}}} = \frac{{\rm{R}}}{{\rm{L}}} \times \frac{1}{2} \times \sqrt {{\rm{LC}}} \\ \Rightarrow {{\rm{\omega }}_{\rm{n}}} = \frac{{\rm{R}}}{2}\sqrt {\frac{{\rm{C}}}{{\rm{L}}}} \end{array}\)

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