# Parallel LC circuit Impedance Calculator

Table of Contents

A calculator to calculate the equivalent impedance of an inductor and a capacitor in parallel is presented.
Complex numbers in standard form and
polar forms are used in the calculations and the presentation of the results.

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

Let \( f \) be the frequency, in Hertz, of the source voltage supplying the circuit.

and define the following parameters used in the calculations

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

\( X_L = \omega L \) , the inductive reactance in ohms \( (\Omega) \)

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

\( Z_L = j \omega L \)

\( X_C = 1 / (\omega C) \) , the capacitive reactance in ohms \( (\Omega) \)

The impedance of the capacitor \( C \) is given by

\( Z_C = \dfrac{1}{j \omega C} = -\dfrac{j}{\omega C}\)

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

\[ \dfrac{1}{Z} = \dfrac{1}{ZL} + \dfrac{1}{ZC} \]

which gives

\( Z = \dfrac{ZL \; ZC}{ZL + ZC} = \dfrac{(j \omega L)(-\dfrac{j}{\omega C})}{j \omega L-\dfrac{j}{\omega C}} = \dfrac{-j}{\omega C - \dfrac{1}{\omega L}} \)

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

Modulus: \( |Z| = \dfrac{1}{\left| \omega C - \dfrac{1}{\omega L} \right|} \)

Argument (Phase): \( \theta = - \dfrac{\pi}{2} \) or \( - 90^{\circ} \) if \( \omega C \gt \dfrac{1}{\omega L} \)

Argument (Phase): \( \theta = \dfrac{\pi}{2} \) or \( 90^{\circ} \) if \( \omega C \lt \dfrac{1}{\omega L} \)

Argument (Phase): \( \theta = 0 \) if \( \omega C = \dfrac{1}{\omega L} \)

## 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

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