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

### More References and links

Calculate Equivalent Impedance in AC Circuits
Series and Parallel Impedances Computations
Engineering Mathematics with Examples and Solutions