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:

“All propaganda or popularization involves a putting of the complex into the simple, but such a move is instantly deconstructive. For if the complex can be put into the simple, then it cannot be as complex as it seemed in the first place; and if the simple can be an adequate medium of such complexity, then it cannot after all be as simple as all that.”
—Terry Eagleton (b. 1943)

“The current flows fast and furious. It issues in a spate of words from the loudspeakers and the politicians. Every day they tell us that we are a free people fighting to defend freedom. That is the current that has whirled the young airman up into the sky and keeps him circulating there among the clouds. Down here, with a roof to cover us and a gasmask handy, it is our business to puncture gasbags and discover the seeds of truth.”
—Virginia Woolf (1882–1941)