Question

Which of the following combinations should be selected for better turning of an L.C.R circuit used for communication?
(A) $R = 25\Omega ,L = 1.5H,C = 45\mu F$
(B) $R = 25\Omega ,L = 1.5H,C = 35\mu F$
(C) $R = 25\Omega ,L = 2.5H,C = 45\mu F$
(D) $R = 15\Omega ,L = 3.5H,C = 30\mu F$

Answer 1

Quality factor of resonance for tuning is a parameter do not have dimensions and that defines under-damped nature of oscillator or resonator, and characterizes a resonator bandwidth $BW$ relative to its center value of frequency.
At resonance, ${X_C} = {X_L} \Rightarrow {\omega _0} = \sqrt {\dfrac{1}{{LC}}}$
Quality factor in series $LCR$ circuit $Q = \dfrac{{{\omega _0}L}}{R} = \dfrac{1}{R}\sqrt {\dfrac{L}{C}}$ As quality factor is dimensionless, thus it does not have any SI unit.

Complete step by step answer:
The quality factor shows the maximum or peak energy stored in the circuit (the reactance) to the energy dissipated (the resistance) during each cycle of oscillation per unit time meaning that it is a ratio of resonant frequency upon bandwidth and the higher the circuit calculation of quality factor is dimensionless over the values provided.
From using formula for calculating Quality factor we can find values for all four options provided, and comparing those values we can choose quality factor selectivity of the circuit for the best tuning using formula,
$Q = \dfrac{{{\omega _0}L}}{R} = \dfrac{1}{R}\sqrt {\dfrac{L}{C}}$
As by solving values of quality factor for each option we get,
Substituting value in above equation,
For option (A), Quality factor = $7.3$
For option (B), Quality factor = $8.28$
For option (C), Quality factor = $9.43$
For option (D), Quality factor = $13.67$
As we solve option (D) in detail we get,
$Q = \dfrac{{{\omega _0}L}}{R} = \dfrac{1}{R}\sqrt {\dfrac{L}{C}} = \dfrac{{\sqrt {3.5} }}{{15\sqrt {30 \times {{10}^{ - 6}}} }} = 13.67$

Therefore the correct answer is option D.

Note: The selectivity of an RLC circuit for good tuning is the ability of the circuit component to respond to a particular frequency and had a discrimination against all other of the frequencies. If the bandwidth of frequencies to be selected or rejected is kind of narrow, the quality factor of the resonant circuit must be higher.