What is the correct formula for quality factor?

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Examoogle · Accepted Answer

$Q = \frac{f_r}{BW}$

Explanation: The Quality Factor (Q) is a dimensionless parameter that characterizes a resonator's bandwidth relative to its center frequency. It indicates the sharpness of the resonance peak; higher Q values signify a narrower bandwidth and lower energy loss per cycle.

Working Principle: In a series RLC circuit, energy is stored in both the inductor and capacitor, while it is dissipated solely by the resistor. The Q-factor is defined as $2\pi 	imes \frac{	ext{Maximum energy stored}}{	ext{Energy dissipated per cycle}}$. Since the bandwidth (BW) is defined as the frequency range between the half-power points, the ratio of the resonant frequency ($f_r$) to the bandwidth ($BW$) provides a measure of circuit selectivity.

Key Formulas: $Q = \frac{f_r}{BW}$ — Definition of Q-factor; $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ — For series RLC circuit; $BW = f_2 - f_1$ — Bandwidth definition

Key Points: Q is directly proportional to resonant frequency and inversely proportional to bandwidth.; A high Q-factor implies a very selective circuit (narrow passband).; In practical inductors, Q accounts for the parasitic resistance of the coil.; The factor represents the 'sharpness' of the resonance curve.

Advantages: Higher Q-factor improves frequency selectivity in tuning circuits.; Essential for minimizing signal interference in radio communications.

Disadvantages: Extremely high Q-factors can make tuning highly sensitive and difficult to maintain.; Physical components have unavoidable resistive losses limiting maximum achievable Q.

Applications: Band-pass filters in radio receivers; Oscillators in signal generators; Power factor correction circuits

Additional Info: Bandwidth ($BW$) is often calculated as $f_2 - f_1$, where $f_1$ and $f_2$ are lower and upper half-power frequencies.. Option A represents the product of frequency and bandwidth, which is physically incorrect. Option B is the inverse of Q. Option D is dimensionally inconsistent.

Answer: C is correct — The quality factor $Q$ is defined as the ratio of the resonant frequency $f_r$ to the bandwidth $BW$.

Exam Tip: Remember that $Q = 2\pi f_r L / R$ for series circuits; always verify units to ensure the result is dimensionless.

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