Resonant Frequency (Unity Power Factor)

Electrical Circuit Analysis > Resonance > Effect of Quality Factor greater than or Equal to 10

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Resonant Frequency, $f_p$ (Unity Power Factor)

We have derived the equation for $f_p$, which is given as

$$ f_p = f_s \sqrt{1 - { R_l^2 C \over L}} \tag{1}$$

We can rewrite the factor ${ R_l^2 C \over L}$ of Eq. (1) as

$$\begin{split} { R_l^2 C \over L} &= { 1 \over {L \over R_l^2 C}}= { 1 \over {w L \over w R_l^2 C}}\\ &= { 1 \over {w L \over R_l^2 wC}}={ 1 \over {X_L X_C \over R_l^2}} \end{split}$$

and substitute ($X_L \approx X_C$):

$${ 1 \over {X_L X_C \over R_l^2}} = { 1 \over {X_L^2 \over R_l^2}} = { 1 \over Q_l^2}$$

Equation (1) then becomes

$$ f_p = f_s \sqrt{1 - { 1 \over Q_l^2}} \tag{1}$$

clearly revealing that as $Q_l$ increases, $f_p$ becomes closer and closer to $fs$.

$$ 1 - {1 \over Q_l^2} \approx 1$$

and

$$ f_p \approx f_s$$

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