Bandwidth from Q and Centre Frequency

Bandwidth and quality factor are two ways of saying how sharp a resonance is: $BW = f_0/Q$, measured between the half-power points either side of centre. A 1 MHz tuned circuit with Q of 125 passes an 8 kHz-wide slice, which is roughly what an AM broadcast channel needs. Raise Q and the passband narrows in exact proportion.

The reason a broadcast receiver's intermediate-frequency strip exists is buried in that ratio. Getting a 10 kHz window at 1 MHz needs Q = 100; getting the same window at 100 MHz would need Q = 10,000, which no ordinary LC circuit reaches. So a superheterodyne mixes everything down to a fixed 455 kHz, where Q of only 45.5 does the job, and the selectivity stops depending on where you tuned. That is Edwin Armstrong's 1918 insight and it is in every radio since.

Two cautions. Q is not a property of the coil alone — connect a load across a tank circuit and you have added loss, so the loaded Q is lower and the bandwidth wider than the components suggest. And selectivity is not free: a filter narrower than the signal chews the sidebands off, so an over-sharp IF makes speech muffled and, in a data link, smears symbols into each other. Bandwidth and rise time are the same constraint viewed from two ends.