Formulas of Impedances in AC Circuits

Table of Contents

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 grouped in series

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 grouped in parallel

Impedances in Basic Circuits

Circuit Description Impedance Z Magnitude |Z| Phase \( \theta \)
impedances of resistor Resistor \( Z = R \) \( |Z| = R \) \( \theta = 0\)

impedances of capacitor Capacitor \( Z = \dfrac{1}{j \omega \; C} = - \dfrac{j}{\omega \; C} \) \( |Z| = \dfrac{1}{\omega \; C} \) \( \theta = - 90^{\circ}\) or \( -\dfrac{\pi}{2} \)

impedances of inductor Inductor \( Z = j \omega \; L \) \( |Z| = \omega \; L \) \( \theta = 90^{\circ}\) or \(\dfrac{\pi}{2} \)

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

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

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

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

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

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

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

impedances of RLC in parallel \( R \), \( L \) 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