Resonant Frequency, fp (Unity Power Factor)

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 $${ R_l^2 C \over L} = { 1 \over {L \over R_l^2 C}}= { 1 \over {w L \over w R_l^2 C}}\\ = { 1 \over {w L \over R_l^2 wC}}={ 1 \over {X_L X_C \over R_l^2}}$$ and substitute ($X_L \appro X_C$): $${ 1 \over {X_L X_C \over R_l^2}} = { 1 \over {X_L^2 \over R_l^2}} = { 1 \over Q_l^2}$$ 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 $fs$. $$ 1 - {1 \over Q_l^2} \appro 1$$ and $$ f_p \appro f_s$$