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On studying RLC circuits, I have come across two definitions of Q factor. One is obviously the measure of selectivity and goes as this:

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The other is this:

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Is there any link between these 2 definitions? Can we start from the first one and arrive at the second one or vice-versa?

While I do understand that the first one comes from the half-power points computation and is a measure of selectivity, I don't get the origin of the second definition besides the link between them.

Some insight would be appreciated.

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Yes - there is a link and we can start from the first definition and arrive at the second.

Example: Lossy capacitor (C||R)

(1) Max. Energy stored: W=0.5 * Vmax² * C

(2) Energy dissipated (loss): W_d=Vmax² * Pi / (R * 2Pi * f)

(3) Ratio W/W_d=Qc=R * 2Pi * f * C = R/(1/wC)

Comments:

  • Derivation of (1): Integral (one period) over {v(t) * i(t) * dt}

    with i(t)=ic(t)=Cdv(t)/dt and v(t)=Vmax * sin wt;

  • Derivation of (2): Same integral with i(t)=ir(t)=v(t)/R .

(4) Extension to the parallel resonant circuit:

If we concentrate the inductive and capacitive losses in one single parallel loss resistor R we can calculate the complex impedance Z. The corresponding pole location in the s-plane allows us to find the pole frequency (no surprise) wp=1/LC and the corresponding pole quality factor Qp=R/(1/wpC). This is the same expression as given above under (3) for w=wp.

Finally, it was shown elsewhere that the pole quality factor Qp is identical to the quality factor Q=wr/delta(w) as defined for a second order bandpass (selectivity).

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