Last weekend BINSIC drove me mad in a desperate rush to finish. This weekend I have been obsessed by this problem, but have finally worked it out.

The Q factor measures the width of a signal at half the resonant power (it is a dimensionless quantity and measures lots of things, but we’ll stick with this for now).

For an AC LRC series circuit, at half power the impedance equals and so the reactances equal R or minus R, i.e.,

at the high frequency end and (1)

at the low frequency end. (2)

Looking at the angular frequencies, this gives for (1)

and for (2)

Subtract (2) from (1) and we get

Now is the bandwidth and we will label this for ease of exposition.

And we can see that can be expanded to:

And (here’s one of the parts I stumbled on), is the square of the resonant frequency (as this is a geometric and not arithmetic mean).

So now we have:

and so and

Now, , so…

Now, (from the resonant frequency definition, see the previous blog)

And so we get , the definition you’ll see in a textbook.

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