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# RLC Parallel Circuit

## Important Formula

## What do the formula symbols mean?

## Impedance

`$$ |Z| $$`Unit

`$$ \mathrm{\Omega} = \frac{ \mathrm{kg} \, \mathrm{m}^2 }{ \mathrm{A}^2 \, \mathrm{s}^3 } $$`

Impedance is the magnitude of the complex resistance \( Z \). This formula gives the total impedance of a driven resistor-coil-capacitor parallel circuit, in short: RLC parallel circuit.

## Inductance

`$$ L $$`Unit

`$$ \mathrm{H} = \frac{ \mathrm{Vs} }{ \mathrm{A} } $$`

The inductance of the coil connected to the RLC parallel circuit.

## Capacitance

`$$ C $$`Unit

`$$ \mathrm{F} = \frac{ \mathrm{C} }{ \mathrm{V} } $$`

Capacitance of the capacitor connected to the RLC parallel circuit.

## Electrical Resistance

`$$ R $$`Unit

`$$ \mathrm{\Omega} = \frac{ \mathrm{kg} \, \mathrm{m}^2 }{ \mathrm{A}^2 \, \mathrm{s}^3 } $$`

Resistance of a resistor connected to the RLC parallel circuit.

## Angular frequency

`$$ \class{green}{\omega} $$`Unit

`$$ \frac{\mathrm{rad}}{\mathrm s} $$`

The angular frequency \( \omega = 2\pi \, f \) indicates the frequency \(f\) at which the LRC parallel circuit oscillates. The frequency \(f\) is specified, for example, by the frequency of the applied AC voltage.

An RLC parallel circuit consists of a resistor \(R\), an inductance \(L\) and a capacitance \( C \), which are connected in parallel. These three electrical components form an electrical circuit in which the current can take different paths through the resistor, the coil and the capacitor.

$$ \begin{align} |Z| ~=~ \frac{ 1 }{ \sqrt{ \left( \frac{1}{R} \right)^2 ~+~ \left( \class{green}{\omega} \, \class{purple}{C} - \frac{1}{\class{green}{\omega} \, \class{brown}{L}} \right)^2 } } \end{align} $$

$$ \begin{align} I_{\text{rms}} ~=~ \sqrt{ \left( \frac{1}{R} \right)^2 ~+~ \left( \frac{1}{\class{green}{\omega} \, \class{brown}{L}} - \class{green}{\omega} \, \class{purple}{C} \right)^2 } \, U_{\text{rms}} \end{align} $$