Quality Factor

Quality Factor

Quality Factor describes the energy storage capability of and Capacitor in RLC circuit. It's also called Q factor.

Series RLC Resonance circuit:

 Q[L] = 2π * {(Energy stored by an Inductor)/(Energy dissipated by R per cycle)}

Q[C] = 2π * {( Energy stored by capacitor)/(Energy dissipated by R per cycle)}

Energy stored by inductor = 1/2 LIo^2

Energy stored by capacitor= 1/2 CVo^2

Where, Vo and Io = peak value)

Energy dissipated by R per cycle = {1/2 (Io^2 * R)}*To

 Now, 

Q[L] = 2π * {(1/2 LIo^2) / (1/2 Io^2*R)*To}

or, Q[L] = 2π /To * (L/R)

or, Q[L] = Wo*(L/R) = XL/R = IXL/IR V=VL/VR

Q[L] = VL /V = WoL/ R = 1/R * √(L/C) ------(1)

And,

Q[C] = 2π * {(1/2 CVo^2) / (1/2 Io^2*R)*To}

or, Q[C] =  2π /To * (CVo^2/Io^2R)

or, Q[C] = Wo * {( C * Io^2 * Xc)/(Io^2*R)

or, Q[C] = 1/(WoRC) = Xc/R = Vc/VR

Q[C] = Vc/V = 1/(WoRC) = 1/ R *√(L/C) -----(2)

From eqn(1) and eqn(2); we can says that,

Q[L] = Q[C] = 1/ R * √(L/C) = Q[S]

Where, Q[S] = Q- factor in series RLC Resonance circuit

Bandwidth (B.W) = R/L

Angular frequency(Wo ) = 1/√(LC) rad/sec.

Q- factor = Wo/ B.W = 1/R * √(L/C)

**Quality Factor can be defined as The Ratio of voltage across inductor or capacitor to the applied voltage.**

Parallel Resonance circuit:

Q - factor in parallel resonance circuit is reciprocal of  Q- factor in series Resonance circuit.

Q[P] = 1 / Q[S] 

or, Q[P] = 1 / {1/R * √(L/C)}

Q[P] = R * √(C/L)

Bandwidth (B.W) = 1/ RC 

Angular frequency (Wo) = 1 / √(LC) rad/sec.

Q- factor = Wo / B.W = R *√(C/L)