# Formulas of Impedances in AC Circuits

The formulas for impedances grouped in series and in parallel and the formula for the impedances of basic series and parallel circuits are presented.

## Impedances in Series

The impedance $$Z_{AB}$$ that is equivalent to the impedances $$Z_1$$, $$Z_2$$ .... $$Z_n$$ grouped in series, as shown below, is given by
$$Z_{AB} = Z_1 + Z_2 + ... + Z_n$$

## Impedances in Parallel

The impedance $$Z_{AB}$$ that is equivalent to the impedances $$Z_1$$, $$Z_12$$ .... $$Z_n$$ grouped in parallel, as shown below, is given by
$$\dfrac{1}{Z_{AB}} = \dfrac{1}{Z_1} + \dfrac{1}{Z_2} + ... + \dfrac{1}{Z_n}$$
or
$$Z_{AB} = \dfrac{1}{ \dfrac{1}{Z_1} + \dfrac{1}{Z_2} + ... + \dfrac{1}{Z_n} }$$

## Impedances in Basic Circuits

Circuit Description Impedance Z Magnitude |Z| Phase $$\theta$$
Resistor $$Z = R$$ $$|Z| = R$$ $$\theta = 0$$

Capacitor $$Z = \dfrac{1}{j \omega \; C} = - \dfrac{j}{\omega \; C}$$ $$|Z| = \dfrac{1}{\omega \; C}$$ $$\theta = - 90^{\circ}$$ or $$-\dfrac{\pi}{2}$$

Inductor $$Z = j \omega \; L$$ $$|Z| = \omega \; L$$ $$\theta = 90^{\circ}$$ or $$\dfrac{\pi}{2}$$

$$R$$ and $$L$$ in series $$Z = R + j \omega \; L$$ $$|Z| = \sqrt{R^2 + (\omega \; L)^2}$$ $$\theta = \arctan \left(\dfrac{\omega \; L}{R}\right)$$

$$R$$ and $$L$$ in parallel $$\dfrac{1}{Z} = \dfrac{1}{R} - j \dfrac{1}{\; \omega \; L}$$
$$|Z| = \dfrac{1}{\sqrt{\dfrac{1}{R^2}+\dfrac{1}{(\omega \; L)^2}} }$$ $$\theta = \arctan \left(\dfrac{R}{\omega \; L}\right)$$

$$R$$ and $$C$$ in series $$Z = R - j \dfrac{1}{\omega \; C}$$ $$|Z| = \sqrt{R^2 + \dfrac{1}{(\omega \; C)^2}}$$ $$\theta = \arctan \left(\dfrac{- 1}{R \omega C}\right)$$

$$R$$ and $$C$$ in parallel $$\dfrac{1}{Z} = \dfrac{1}{R} + j \; \omega \; C$$ $$|Z| = \dfrac{1}{\sqrt{\dfrac{1}{R^2}+(\omega \; C)^2} }$$ $$\theta = - \arctan \left( R {\omega \; C}\right)$$

$$L$$ and $$C$$ in series $$Z = j \omega L - j \dfrac{1}{\omega \; C}$$ $$|Z| = \left| \omega L - \dfrac{1}{\omega \; C} \right|$$ $$\theta = \begin{cases} \dfrac{\pi}{2}, & \mbox{if } \omega L - \dfrac{1}{\omega\; C } \gt 0 \\ - \dfrac{\pi}{2} , & \mbox{if } \omega L - \dfrac{1}{\omega \; C} \lt 0 \\ 0 , & \mbox{if } \omega L - \dfrac{1}{\omega \; C} = 0 \end{cases}$$

$$L$$ and $$C$$ in parallel $$\dfrac{1}{Z} = j \; \omega \; C - \dfrac{j}{ \; \omega \; L}$$ $$|Z| = \dfrac{1}{\left| \omega C - \dfrac{1}{\omega \; L} \right|}$$ $$\theta = \begin{cases} -\dfrac{\pi}{2}, & \mbox{if } \omega C - \dfrac{1}{\omega\; L} \gt 0 \\ \dfrac{\pi}{2} , & \mbox{if } \omega C - \dfrac{1}{\omega \; L} \lt 0 \\ 0, & \mbox{if } \omega C - \dfrac{1}{\omega \; L} = 0 \end{cases}$$

$$R$$, $$L$$ and $$C$$ in series $$Z = R + j \omega \; L - j \dfrac{1}{\omega \; C}$$ $$|Z| = \sqrt{R^2 + \left(\omega \; L- \dfrac{1}{\omega \; C} \right)^2}$$ $$\theta = \arctan \left(\dfrac{\omega^2 L C - 1 }{ R \omega C}\right)$$

$$R$$, $$L$$ and $$C$$ in parallel $$\dfrac{1}{Z} = \dfrac{1}{R} + j \omega \; C - j \dfrac{1}{ \omega \; L}$$ $$|Z| = \dfrac{1} { \sqrt{\dfrac{1}{R^2} + \left(\omega \; C- \dfrac{1}{\omega \; L} \right)^2 }}$$ $$\theta = - \arctan \left(\dfrac{R(\omega^2 \; L \; C - 1) }{ \omega \; L}\right)$$