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.
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:
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.
- Tuning an LRC circuit (cartesianproduct.wordpress.com)
- Accelerometers (leancrew.com)
- Atom and Photon Interactions – The Semi-Classical Model (panpalitta.wordpress.com)