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Quality Factor | Quality Factor Formula | What Is Quality Factor

Quality Factor | Quality Factor Formula | What Is Quality Factor :- The ratio of the resonant frequency (ω_r) to the bandwidth (β) of a circuit is called the Quality Factor (Q).

$\displaystyle Q=\frac{{{\omega }_{r}}}{{{\omega }_{2}}-{{\omega }_{1}}}=\frac{{{\nu }_{r}}}{{{\nu }_{2}}-{{\nu }_{1}}}$ …..(1)

We know that at the half-power points P₁ and P₂, the value of current becomes $\displaystyle \frac{1}{\sqrt{2}}$ times the current at resonance (in a Series Resonance Circuit). Therefore,

$\displaystyle {{I}_{rms}}=\frac{{{\left( {{I}_{rms}} \right)}_{_{max}}}}{\sqrt{2}}$

$\displaystyle \Rightarrow \frac{{{E}_{rms}}}{\sqrt{{{R}^{2}}+{{\left( \omega L-\frac{1}{\omega C} \right)}^{2}}}}=\frac{{{E}_{rms}}}{\sqrt{2}R}$

$\displaystyle \Rightarrow {{R}^{2}}+{{\left( \omega L-\frac{1}{\omega C} \right)}^{2}}=2{{R}^{2}}$

$\displaystyle \Rightarrow {{\left( \omega L-\frac{1}{\omega C} \right)}^{2}}={{R}^{2}}$

$\displaystyle \Rightarrow \omega L-\frac{1}{\omega C}=\pm R$ …..(2)

At point P₁ ,

$\displaystyle {{\omega }_{\text{1}}}L-\frac{1}{{{\omega }_{1}}C}=-R$ …..(3)

At point P₂ ,

$\displaystyle {{\omega }_{\text{2}}}L-\frac{1}{{{\omega }_{2}}C}=+R$ …..(4)

Adding equations (3) and (4),

$\displaystyle \left( {{\omega }_{\text{1}}}+{{\omega }_{\text{2}}} \right)L-\frac{\left( {{\omega }_{\text{1}}}+{{\omega }_{\text{2}}} \right)}{\left( {{\omega }_{\text{1}}}{{\omega }_{\text{2}}} \right)}\frac{1}{C}=0$

$\displaystyle \Rightarrow L=\frac{1}{C\left( {{\omega }_{\text{1}}}{{\omega }_{\text{2}}} \right)}$

$\displaystyle \Rightarrow {{\omega }_{\text{1}}}{{\omega }_{\text{2}}}=\frac{1}{LC}=\omega _{r}^{2}\text{ }\left( \because {{\omega }_{r}}=\frac{1}{\sqrt{LC}} \right)$

$\displaystyle \Rightarrow {{\omega }_{\text{1}}}{{\omega }_{\text{2}}}=\omega _{r}^{2}$ …..(5)

Subtracting equation (3) from equation (4), we get

$\displaystyle \left( {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}} \right)L+\frac{1}{C}\left( \frac{1}{{{\omega }_{\text{1}}}}-\frac{1}{{{\omega }_{\text{2}}}} \right)=2R$

$\displaystyle \Rightarrow \left( {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}} \right)L+\frac{\left( {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}} \right)}{C\left( {{\omega }_{\text{1}}}{{\omega }_{\text{2}}} \right)}=2R$

$\displaystyle \Rightarrow \left( {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}} \right)L+\frac{\left( {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}} \right)}{C\left( \frac{1}{LC} \right)}=2R$

$\displaystyle \Rightarrow \left( {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}} \right)L+\left( {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}} \right)L=2R$

$\displaystyle \Rightarrow 2L\left( {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}} \right)=2R$

$\displaystyle \Rightarrow {{\omega }_{\text{2}}}-{{\omega }_{\text{1}}}=\frac{R}{L}=\beta =Band\text{ }Width$ …..(6)

Using equation (6) in equation (1), we get

$\displaystyle Q=\frac{{{\omega }_{r}}}{{{\omega }_{2}}-{{\omega }_{1}}}=\frac{{{\omega }_{r}}}{{}^{R}\!\!\diagup\!\!{}_{\text{L}}\;}=\frac{{{\omega }_{r}}L}{R}=\frac{{{X}_{L}}}{R}$ …..(7)

Since at resonance X_L = X_C, therefore

$\displaystyle {{\omega }_{r}}L=\frac{1}{{{\omega }_{r}}C}\Rightarrow L=\frac{1}{\omega _{r}^{2}C}$ …..(8)

Substituting equation (7) into equation (8),

$\displaystyle Q=\frac{{{\omega }_{r}}}{R}\times \frac{1}{\omega _{r}^{2}C}=\frac{1}{R{{\omega }_{r}}C}=\frac{\left( {}^{1}\!\!\diagup\!\!{}_{{{\omega }_{r}}C}\; \right)}{R}=\frac{{{X}_{C}}}{R}$ …..(9)

Thus, from equations (7) and (9), in a series resonant circuit, the ratio of the inductive reactance X_L (or capacitive reactance X_C) to the resistance of the circuit at resonance is called the quality factor or Q-factor of the circuit.

Actually, the quality factor (Q) represents the sharpness of the resonance curve. At resonance, the maximum current in the circuit is inversely proportional to the resistance (R) of the circuit. For lower values of R, the current is high, and for higher values of R, the current is low, as shown in the figure below :-

From the above article, it is clear that when the resistance (R) is small, the current in the circuit becomes large, and when the bandwidth (β = Δω) is small, a sharp curve is obtained. For high resistance (R), the current becomes small and the bandwidth (β = Δω’) also increases, which reduces the sharpness of the curve.

Other Definitions of Quality Factor and Quality Factor Formula

Multiplying the numerator and the denominator in equations (7) and (9) by the current value $I$ :

$\displaystyle Q=\frac{I{{X}_{L}}}{IR}=\frac{{{V}_{0L}}}{{{V}_{0R}}}$ …..(10)

Similarly,

$\displaystyle Q=\frac{I{{X}_{C}}}{IR}=\frac{{{V}_{0C}}}{{{V}_{0R}}}$ …..(11)

Hence, from equations (10) and (11), the ratio of the potential difference across the inductor V_0L (or potential difference across the capacitor V_0C ) to the potential difference across the resistor (V_0R) at resonance is called the quality factor.

Substituting the value $\displaystyle {{\omega }_{r}}=\frac{1}{\sqrt{LC}}$ into equation (7),

$\displaystyle Q=\frac{{{\omega }_{r}}L}{R}=\frac{1}{\sqrt{LC}}\times \frac{L}{R}$

$\displaystyle \therefore Q=\frac{1}{R}\sqrt{\frac{L}{C}}$ …..(12)

Similarly, from equation (9),

$\displaystyle Q=\frac{1}{R{{\omega }_{r}}C}=\frac{\text{1}}{RC}\times \frac{1}{\frac{1}{\sqrt{LC}}}$

$\displaystyle \therefore Q=\frac{1}{R}\sqrt{\frac{L}{C}}$

Note :-

Quality factor (Q) is a dimensionless and unitless quantity.
From equation (2), at the half-power frequencies : the total reactance (X) equals the total resistance (R).

Next Topic :- Parallel Resonance Circuit

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Quality Factor | Quality Factor Formula | What Is Quality Factor

Other Definitions of Quality Factor and Quality Factor Formula

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