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The relation between the resonant frequency f0 and the half-power frequencies f1 and f2 is
- A\({f_0} = \sqrt {{f_1}{f_2}}\)
- B\({f_0} = \frac{{{f_1} + {f_2}}}{2}\)
- C\({f_0} = {f_1}{f_2}\)
- D\(f_0^2 = f_1^2f_2^2\)
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CCommunity
We know that,
$({\omega _1} = - \frac{R}{{2L}} \pm \sqrt {{{\left( {\frac{R}{{2L}}} \right)}^2} + \frac{1}{{LC}}})$
$({\omega _2} = \frac{R}{{2L}} \pm \sqrt {{{\left( {\frac{R}{{2L}}} \right)}^2} + \frac{1}{{LC}}})$
$({\omega _1}{\omega _2} = {\left( {\frac{R}{{2L}}} \right)^2} + \frac{1}{{LC}} - {\left( {\frac{R}{{2L}}} \right)^2} = \frac{1}{{LC}} = \omega _0^2)$
$(\Rightarrow \omega _0^2 = {\omega _1}{\omega _2})$
$(\Rightarrow {\left( {2\pi {f_0}} \right)^2} = 2\pi {f_1} \times 2\pi {f_2})$
$(\Rightarrow {f_0} = \sqrt {{f_1}{f_2}})$
$({\omega _1} = - \frac{R}{{2L}} \pm \sqrt {{{\left( {\frac{R}{{2L}}} \right)}^2} + \frac{1}{{LC}}})$
$({\omega _2} = \frac{R}{{2L}} \pm \sqrt {{{\left( {\frac{R}{{2L}}} \right)}^2} + \frac{1}{{LC}}})$
$({\omega _1}{\omega _2} = {\left( {\frac{R}{{2L}}} \right)^2} + \frac{1}{{LC}} - {\left( {\frac{R}{{2L}}} \right)^2} = \frac{1}{{LC}} = \omega _0^2)$
$(\Rightarrow \omega _0^2 = {\omega _1}{\omega _2})$
$(\Rightarrow {\left( {2\pi {f_0}} \right)^2} = 2\pi {f_1} \times 2\pi {f_2})$
$(\Rightarrow {f_0} = \sqrt {{f_1}{f_2}})$