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:

    “In the case of all other sciences, arts, skills, and crafts, everyone is convinced that a complex and laborious programme of learning and practice is necessary for competence. Yet when it comes to philosophy, there seems to be a currently prevailing prejudice to the effect that, although not everyone who has eyes and fingers, and is given leather and last, is at once in a position to make shoes, everyone nevertheless immediately understands how to philosophize.”
    —Georg Wilhelm Friedrich Hegel (1770–1831)

    “We set up a certain aim, and put ourselves of our own will into the power of a certain current. Once having done that, we find ourselves committed to usages and customs which we had not before fully known, but from which we cannot depart without giving up the end which we have chosen. But we have no right, therefore, to claim that we are under the yoke of necessity. We might as well say that the man whom we see struggling vainly in the current of Niagara could not have helped jumping in.”
    —Anna C. Brackett (1836–1911)