# Parallel RL circuit Impedance Calculator

Table of Contents

A calculator to calculate the equivalent impedance of a resistor and an inductor in parallel.
The impedance is given as a complex number in standard form and
polar forms.

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## Formulae for Parallel R L Circuit Impedance Used in Calculator and their Units

Let \( f \) be the frequency, in Hertz.

The angular frequency is given by

\( \omega = 2 \pi f \) , in rad/s

The inductive reactance

\( X_L = \omega L \) , in ohms \( (\Omega) \)

The impedance of the inductor \( L \) is given by

\( Z_L = j \omega L \)

Let \( Z \) be the equivalent impedance to the parallel RL circuit shown above and write it in complex form as follows

\[ \dfrac{1}{Z} = \dfrac{1}{R} + \dfrac{1}{Z_L} \]

\( Z = \dfrac{R Z_L}{R + Z_L} = \dfrac{ j R \omega L }{R+j \omega L } = \dfrac{1}{\dfrac{1}{R} - j \dfrac{1}{\omega L}} \)

The formulae for the modulus \( |Z| \) and argument (or phase) \( \theta \) of \( Z \) are given by

Modulus: \( |Z| = \dfrac{1}{\sqrt{ \dfrac{1}{r^2} + \dfrac{1}{\omega^2 L^2}}} \) in ohms \( (\Omega) \)

Argument (Phase): \( \theta = \arctan \left( \dfrac{ R }{\omega L} \right) \) in radians or degrees

## Use of the calculator

Enter the resistance, the capacitance and the frequency as positive real numbers with the given units then press "calculate".

## Results of Calculations

## More References and links

AC Circuits Calculators and Solvers

Complex Numbers - Basic Operations

Complex Numbers in Exponential Form

Complex Numbers in Polar Form

Convert a Complex Number to Polar and Exponential Forms Calculator

Engineering Mathematics with Examples and Solutions