Question

In a series LCR circuit connected to an ac source of variable frequency and voltage ν = v_m sin ωt, draw a plot showing the variation of current (I) with angular frequency (ω) for two different values of resistance R₁ and R₂ (R₁ > R₂). Write the condition under which the phenomenon of resonance occurs. For which value of the resistance out of the two curves, a sharper resonance is produced? Define Q-factor of the circuit and give its significance.

Answer 1

Figure shows the variation of i_m with ω in a LCR series circuit for two values of Resistance R₁ and R₂ (R₁ > R₂),

The condition for resonance in the LCR circuit is, `ω_0 = 1/(sqrtLC)`

We see that the current amplitude is maximum at the resonant frequency ω. Since i_m = v_m / R at resonance, the current amplitude for case R₂ is sharper to that for case R₁.

Quality factor or simply the Q-factor of a resonant LCR circuit is defined as the ratio of voltage drop across the capacitor (or inductor) to that of applied voltage.

It is given by `Q = 1/RsqrtL/C`

The Q factor determines the sharpness of the resonance curve and if the resonance is less sharp, not only is the maximum current less, the circuit is close to the resonance for a larger range Δω of frequencies and the tuning of the circuit will not be good. So, less sharp the resonance, less is the selectivity of the circuit while higher is the Q, sharper is the resonance curve and lesser will be the loss in energy of the circuit.