Q factor final Math

The Q, quality factor, of a resonant circuit is a measure of the “goodness” or quality of a resonant circuit.

        Q = P_stored/P_dissipated = I²X/I²R

        Q = X/R

        where:         X = Capacitive or Inductive reactance at resonance

                       R = Series resistance.

A high Q resonant circuit has a narrow bandwidth as compared to a low Q

   BW = fc/Q

        Where fc = resonant frequency 

        Q = quality factor 

Bandwidth, Δf is measured between the 70.7% amplitude points of series resonant circuit.

        BW = Δf = fh-fl = fc/Q

        Where: fh = high band edge fl = low band edge

        fl = fc - Δf/2

        fh = fc + Δf/2

        Where fc = center frequency (resonant frequency) 

In Figure above, the 100% current point is 50 mA. The 70.7% level is .707(50 mA)=35.4 mA. The upper and lower band edges read from the curve are 291 Hz for f_l and 355 Hz for f_h. The bandwidth is 64 Hz, and the half power points are ± 32 Hz of the center resonant frequency:

        BW = Δf = fh-fl = 355-291 = 64

        fl = fc - Δf/2 = 323-32 = 291

        fh = fc + Δf/2 = 323+32 = 355

Since BW = f_c/Q:

        Q = fc/BW = (323 Hz)/(64 Hz) = 5