Electrical Impedance - Complex Voltage and Current

Complex Voltage and Current

In order to simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as and .

\begin{align} V &= |V|e^{j(\omega t + \phi_V)} \\ I &= |I|e^{j(\omega t + \phi_I)} \end{align}

Impedance is defined as the ratio of these quantities.

Substituting these into Ohm's law we have

\begin{align} |V| e^{j(\omega t + \phi_V)} &= |I| e^{j(\omega t + \phi_I)} |Z| e^{j\theta} \\ &= |I| |Z| e^{j(\omega t + \phi_I + \theta)} \end{align}

Noting that this must hold for all, we may equate the magnitudes and phases to obtain

\begin{align} |V| &= |I| |Z| \\ \phi_V &= \phi_I + \theta \end{align}

The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.

Read more about this topic:  Electrical Impedance

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