# Parallel Resistance across Inductance, Rp

### Parallel Resistance across Inductance, Rp

We have derived the equation for $R_p$ in the topic, which is given as $$R_p = { R_l^2 + X^2_L \over R_l} \tag{1}$$ $$R_p = { R_l^2 + X^2_L \over R_l}= R_l + {X^2_L \over R_l }({R_l \over R_l})\\ = R_l + R_l{X^2_L \over R_l^2 } = R_l + Q^2_l R_l = (1 - Q_l^2)R_l$$ For $Q_l \geq 10$, $1 + Q^2_l \appro Q^2_l $$and$$ \bbox[10px,border:1px solid grey]{R_p = Q^2_l R_l} \tag{2}$$Substituting Q_l = { X_L \over R_l} into Eq. (2),$$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)}$$and$$ \bbox[10px,border:1px solid grey]{R_p = { L \over R_l C}} \tag{3}$\$