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Q Factor Formula at Resonance

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skotz

Newbie level 1
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

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.

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.

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