# Power Calculator in Series RLC

Table of Contents

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A calculator to calculate the average power delivered to a resistor, a capacitor and an inductor in series, as shown below, is presented.

The calculator gives the impedance of the series circuit as a complex numbers in standard form , its modulus and argument, the power factor and the average power.

## Formula for The Average Power Delivered to a series RLC Circuit

The formula of the average power delivered to an impedance \( Z \) as shown in the circuit below is given by

\[ \displaystyle \quad \quad P_a = \dfrac{V_0^2}{2 |Z|} \cos \theta \]

where \( V_0 \) is the peak voltage of the source voltage \( v_ i\). \( |Z| \) is the modulus of \( Z \) and \( \theta \) its argument.

The term \( \cos \theta \) is called the power factor.

and define the following parameters used in the calculations

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

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

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

The formula of the impedance \( Z \) of the series RLC circuit shown above and write it in standard complex form as follows

\( Z = R + (X_L - X_C) j \)

and in polar form as follows

\[ Z = |Z| e^{j \theta} \]

The formulae for the modulus \( r \) and argument \( \theta \) are given by (see proof at the bottom of the page)

Modulus: \( |Z| = r = \sqrt {R^2 + (X_L - X_C)^2 } \) in ohms \( (\Omega) \)

Argument: \( \theta = \arctan \left(\dfrac{X_L - X_C}{R} \right) \) in radians or degrees

## Use of the calculator

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

## Results

## More References and links

Power in AC Circuits

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