Bandpass and Bandwidth
The quality factor sets the band’s width
The graph is the band-pass curve around the resonant frequency, and the shaded region is the passband. Raise the quality factor Q and the sides of the peak steepen while the passband narrows. Set Q so that the bandwidth BW reaches the target 0.25. Here the bandwidth is measured as a fraction of the resonant frequency.
The passband is set at half power
A band-pass filter takes its output across the resistor of a series RLC circuit; at the resonant frequency ω_0 the gain reaches its maximum of 1, and it falls off on either side. You have to decide where the passband ends, and by convention it runs between the two frequencies where the gain is 1/√2 of the maximum, that is, where the power is half. These two edges are the half-power (−3dB) frequencies, and the band between them is the passband.
The bandwidth is BW = ω_0/Q
The spacing between the two half-power frequencies is the bandwidth BW. Working it out, BW comes out neatly as the resonant frequency divided by the quality factor, BW = ω_0/Q. A large Q steepens the sides of the peak and narrows the bandwidth, while a small Q flattens the peak out and widens it. The height of the peak's top stays the same whatever Q is; only the width changes.
One Q designs the selectivity
A radio or communications receiver must pick out just the channel it wants and reject the neighbors. That calls for a narrow bandwidth matched to the channel width, and the required Q is worked backward from BW = ω_0/Q. This plot sets ω_0 to 1 and measures the bandwidth as a fraction of it, so the target 0.25 corresponds to Q = 4. Raise Q too far, though, and the band becomes so narrow that information at the edges of the signal is cut off, so Q is a compromise between the width to pass and the width to block.
Back to the first screen
As you raised the quality factor Q, the sides of the band-pass peak steepened and the shaded passband narrowed, and BW reached 0.25 at Q = 4. The shape of the peak and the width of the band it passes were tied together by BW = ω_0/Q alone. If ω_0 decided where resonance happens, the quality factor Q decides how narrowly the band around it is passed.