Half power points, bandwidth and effect of Q

Half power points are the two frequencies f1 and f2 (one below and one above the resonant frequency f0) at which the power in the circuit is half of the maximum power at resonance, i.e. P = I0²R/2. The current is then I0/√2 = 0.707 I0, and the voltage-to-current ratio |Z| = √2 R.

Bandwidth is the width of the frequency band between them: BW = f2 − f1.

Derivation

At a half power point |X_L − X_C| = R.

ωL − 1/ωC = ±R   →   LCω² ∓ RCω − 1 = 0

ω2 = R/2L + √[(R/2L)² + 1/LC]      (upper)
ω1 = −R/2L + √[(R/2L)² + 1/LC]     (lower)

BW = ω2 − ω1 = R/L rad/s  =  R/(2πL) Hz
ω1 ω2 = 1/LC = ω0²

Effect of quality factor

Q = ω0L/R, so R/L = ω0/Q and

BW = ω0/Q    (or f0/Q)

Bandwidth is inversely proportional to Q.

Example: f0 = 1 kHz; Q = 10 gives BW = 100 Hz; Q = 100 gives BW = 10 Hz.

So increasing Q (by decreasing R for fixed L, C) narrows the bandwidth and increases the selectivity.

Answer: BW = f2 − f1 = R/(2πL) = f0/Q, so bandwidth decreases as Q increases.