Impedance Transformation

Series to Parallel

Fig. 6 Series to Parallel Conversion

Quality Factor

\[\begin{align*} Q_s &= \dfrac{|X_s|}{R_s} \\[1em] Q_P &= \dfrac{R_p}{|X_p|} \\[1em] \end{align*}\]

R & C

\[\begin{align*} Q_s &= \dfrac{1}{\omega C_s R_s}\\[1em] Q_P &= \dfrac{\omega C_p R_p}{1} \\[1em] \end{align*}\]

Equate Impedances

\[\begin{align*} R_s + \dfrac{1}{j\omega C_s} &= R_p \parallel \dfrac{1}{j\omega C_p} \\[1em] \dfrac{j\omega R_s C_s + 1}{j\omega C_s} &= \dfrac{\dfrac{R_p}{j\omega C_p}}{R_p + \dfrac{1}{j\omega C_p}} \\[1em] \dfrac{j\omega R_s C_s + 1}{j\omega C_s} &= \dfrac{\dfrac{R_p}{j\omega C_p}}{\dfrac{j\omega R_p C_p + 1}{j\omega C_p}} \\[1em] \dfrac{j\omega R_s C_s + 1}{j\omega C_s} &= \dfrac{R_p}{j\omega R_p C_p + 1} \\[1em] -\omega^2 R_s C_s R_p C_p & + j \omega R_s C_s + j \omega R_P C_p + 1 = j \omega R_p C_s \\[1em] \text{equate real} & \text{ and imaginary} \\[1em] \omega^2 R_s C_s R_p C_p = 1 \ \ & \text{ & } \ \ R_s C_s + R_P C_p - R_p C_s = 0 \\[1em] \cdots \\[1em] R_p &= (Q_s^2 + 1)R_s \\[1em] C_p &= \dfrac{Q_s^2}{Q_s^2+1}C_s \end{align*}\]

Thus, as long as \(Q_s^2 \gg 1\) then

\[\begin{align*} R_p &\approx Q_s^2 R_s \\[1em] C_p &\approx C_s \end{align*}\]

R & L

\[\begin{align*} Q_s &= \dfrac{\omega L_s}{ R_s}\\[1em] Q_P &= \dfrac{R_p}{\omega L_p} \\[1em] \end{align*}\]

Equate Impedances

\[\begin{align*} R_s + j\omega L_s &= R_p \parallel j\omega L_p \\[1em] R_s + j\omega L_s &= \dfrac{j\omega R_p L_p}{R_p + j\omega L_p} \\[1em] &= \dfrac{j\omega R_p L_p}{R_p + j\omega L_p} \dfrac{R_p - j\omega L_p}{R_p - j\omega L_p}\\[1em] &= \dfrac{(\omega L_p)^2 R_p + j\omega L_p R_p^2}{R_p^2 + \omega^2 L_p^2}\\[1em] \text{equate real} & \text{ and imaginary} \\[1em] \cdots \\[1em] R_p &= (Q_s^2 + 1)R_s \\[1em] L_S &= \dfrac{Q_s^2}{Q_s^2+1} L_P \end{align*}\]

Thus, as long as \(Q_s^2 \gg 1\) then

\[\begin{align*} R_p &\approx Q_s^2 R_s \\[1em] L_S &\approx L_P \end{align*}\]