Parallel Resistance across Inductance

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

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Parallel Resistance across Inductance, Rp

We have derived the equation for $R_p$ in the topic, which is given as

$$ R_p = { R_l^2 + X^2_L \over R_l} \tag{1}$$

$$ \begin{split} R_p &= { R_l^2 + X^2_L \over R_l}= R_l + {X^2_L \over R_l }({R_l \over R_l})\\ &= R_l + R_l{X^2_L \over R_l^2 } = R_l + Q^2_l R_l = (1 - Q_l^2)R_l \end{split}$$

For $Q_l \geq 10$, $1 + Q^2_l \approx Q^2_l $ and

$$ \bbox[10px,border:1px solid grey]{R_p = Q^2_l R_l} \tag{2}$$

Substituting $Q_l = { X_L \over R_l}$ into Eq. (2),

$$\begin{split} R_p &= Q^2_l R_l = ({ X_L \over R_l})^2 R_l\\ &={ X_L^2 \over R_l} = { X_L X_C \over R_l} \\ &= { 2 \pi fL \over R_l( 2 \pi f C)} \end{split}$$

and

$$ \bbox[10px,border:1px solid grey]{R_p = { L \over R_l C}} \tag{3}$$

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