Q Factor Formula at Resonance

[skotz]

I am looking for the derivation for the Quality Factor as found as equation 4.55 in Peter Vizmuller's book, "RF Design Guide".

Q=(−X+ωdX/dω)/2R

There is no source listed for it in his book and I would like to know more about it.

Thanks,


SPK

 

[D.A.(Tony)Stewart]

For what circuit?

A Voltage source with R ohm impedance and series resonance into R Ohm load might be Q=|X(f)|/2R or simply the ratio of reactive impedance to real impedance.
The other derivative suggests a Q of a filter that is off resonance such that dX/dω is not zero and thus reduces the Q. Considering the sign of -X, it may be referring to a positive reactance and implies the impedance is reduced by slope of the filter. But that doesn't make sense to me.

 

[vfone]

XL and XC are BOTH positive by convention, although the capacitive reactance makes a negative contribution to the total reactance.
So, your -X should be positive, which actually is well explained in P.Vizmuller book.