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

Famous quotes containing the words complex and/or current:

    When distant and unfamiliar and complex things are communicated to great masses of people, the truth suffers a considerable and often a radical distortion. The complex is made over into the simple, the hypothetical into the dogmatic, and the relative into an absolute.
    Walter Lippmann (1889–1974)

    Phlebas the Phoenician, a fortnight dead,
    Forgot the cry of gulls, and the deep sea swell
    And the profit and loss.
    A current under sea
    Picked his bones in whispers. As he rose and fell
    He passed the stages of his age and youth
    Entering the whirlpool.
    —T.S. (Thomas Stearns)