In series RLC resonator, if the self inductance and capacitance become double, the new resonant frequency (f₂) and new quality factor (Q₂) will be :

(f₁ = original resonant frequency, Q₁ = original quality factor)

<p>We know,</p> <p>Quality factor (Q factor)</p> <p>${Q_1} = {{{w_1}} \over {\Delta w}}$</p> <p>$= {1 \over {\sqrt {LC} }} \times {L \over R}$</p> <p>$= {1 \over R}\sqrt {{L \over C}}$</p> <p>Now, when $L' = 2L$ and $C' = 2C$ then $${Q_2} = {1 \over R}\sqrt {{{2L} \over {2C}}} = {1 \over R}\sqrt {{L \over C}} = {Q_1}$$</p> <p>$\therefore$ Q₂ remains same as Q₁.</p> <p>Also, as ${w_1} = {1 \over {\sqrt {LC} }}$</p> <p>$\Rightarrow 2\pi {f_1} = {1 \over {\sqrt {LC} }}$</p> <p>$\Rightarrow {f_1} = {1 \over {2\pi \sqrt {LC} }}$</p> <p>$\therefore$ When $L' = 2L$ and $C' = 2C$ then new resonating frequency</p> <p>$${f_2} = {1 \over {2\pi \sqrt {2L \times 2C} }} = {1 \over {2\pi \times 2\sqrt {LC} }} = {1 \over 2} \times {f_1}$$</p>