Bandpass and Bandwidth
The quality factor sets the band’s width
Slide the quality factor Q and watch the bandpass curve change. The larger Q is, the narrower the passband (shaded) becomes. Set Q so the half-power bandwidth BW reaches the target of 0.25.
The passband is set at half power
A bandpass takes the output across the resistor of a series RLC: the gain is at its maximum of one at the resonant frequency ω_0 and falls off to either side. You must decide where the passband ends, and the convention is the two frequencies where the gain is 1/√2 of the peak — half the power. These two edges are the half-power (−3 dB) frequencies, and between them is the passband.
The bandwidth is BW = ω_0/Q
The gap between the two half-power frequencies is the bandwidth BW. Working it out, BW comes out cleanly as the resonant frequency divided by the quality factor, BW = ω_0/Q. Since Q measures the sharpness of the resonance, a large Q gives a tall, narrow peak and a small bandwidth, while a small Q gives a low, broad peak and a wide bandwidth. So Q and BW are inversely proportional.
One Q designs the selectivity
A radio or communications receiver must pick out one desired channel and reject the neighbors. That calls for a narrow bandwidth matched to the channel’s width, and from BW = ω_0/Q you work backward to the Q you need. But push Q too high and the band gets so narrow that even the edges of the signal are cut off, so Q is a compromise between the width you want to pass and the width you must block.
Back to the first screen
As you raised the quality factor Q, the bandpass peak grew tall and steep, the shaded passband kept narrowing, and BW reached 0.25 at Q = 4. The shape of the peak and the width of the band it passes were bound together by the single relation BW = ω_0/Q. Where the resonance happens was ω_0; how narrowly it passes the band around it was held by the quality factor Q.