Question

In a series RLC resonant circuit with the resonant frequency f₀, the quality factor is
1. \(\dfrac{1}{2\pi f_0 RC}\)
2. \(\dfrac{L}{2\pi f_0 R}\)
3. \( \dfrac{1}{2\pi f_0 LC}\)
4. \(\dfrac{L}{2\pi f_0 C}\)

Answer 1

Correct Answer - Option 1 : \(\dfrac{1}{2\pi f_0 RC}\)

Concept:

The quality factor is defined as the ratio of the maximum energy stored to maximum energy dissipated in a cycle

\(Q = 2π \frac{{Maximum\;energy\;stored}}{{Total\;energy\;lost\;per\;period}}\)

The quality factor in a series RLC circuit is given by:

\(Q = \frac{1}{R}√ {\frac{L}{C}}\)    ----- (1)

When the RLC circuit is set to resonate (XL = XC), the resonant frequency is expressed as 

 \(f_0 = \frac{1}{{2π }}√ {\frac{1}{{LC}}}\)   ----- (2)

Application:

From (2), we can write

√L = 1/( 2π × f₀ × √C)    ----- (3)

Substitute (3) in (1), we get

Q = [1/( 2π × f0 × √C) ] / (R × √C)

⇒ Q = \(\dfrac{1}{2\pi f_0 RC}\)