Editing Electrical impedance (section)

== Complex voltage and current ==
[[File:Impedance symbol comparison.svg|thumb|right|165px|Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box.]]
To simplify calculations, [[Sine wave|sinusoid]]al voltage and current waves are commonly represented as complex-valued functions of time denoted as <math>V</math> and <math>I</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html#c1 Complex impedance], Hyperphysics</ref><ref name=HH2>{{cite book |last=Horowitz |first=Paul |author2=Hill, Winfield |title=The Art of Electronics |year=1989 |publisher=Cambridge University Press |isbn=978-0-521-37095-0 |pages=[https://archive.org/details/artofelectronics00horo/page/31 31–32] |chapter=1 |chapter-url=https://archive.org/details/artofelectronics00horo |url=https://archive.org/details/artofelectronics00horo/page/31 }}</ref>

:<math>\begin{align}
  V &= |V|e^{j(\omega t + \phi_V)}, \\
  I &= |I|e^{j(\omega t + \phi_I)}.
\end{align}</math>

The impedance of a bipolar circuit is defined as the ratio of these quantities:

:<math> Z = \frac{V}{I} = \frac{|V|}{|I|}e^{j(\phi_V - \phi_I)}.</math>

Hence, denoting <math>\theta = \phi_V - \phi_I</math>, we have

:<math>\begin{align}
     |V| &= |I| |Z|, \\
  \phi_V &= \phi_I + \theta.
\end{align}</math>

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

=== Validity of complex representation ===
This representation using complex exponentials may be justified by noting that (by [[Euler's formula]]):

:<math>\ \cos(\omega t + \phi) = \frac{1}{2} \Big[ e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}\Big]</math>

The real-valued sinusoidal function representing either voltage or current may be broken into two complex-valued functions. By the principle of [[superposition principle|superposition]], we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term. The results are identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that

:<math>\ \cos(\omega t + \phi) = \operatorname{Re} \Big\{ e^{j(\omega t + \phi)} \Big\}</math>

=== Ohm's law ===
[[File:General AC circuit.svg|thumb|right|165px|An AC supply applying a voltage <math>V</math>, across a [[Electrical load|load]] <math>Z</math>, driving a current <math>I</math>]]
{{Main|Ohm's law}}

The meaning of electrical impedance can be understood by substituting it into Ohm's law.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html AC Ohm's law], Hyperphysics</ref><ref name=HH1>{{cite book |last=Horowitz |first=Paul |author2=Hill, Winfield |title=The Art of Electronics |year=1989 |publisher=Cambridge University Press |isbn=978-0-521-37095-0 |pages=[https://archive.org/details/artofelectronics00horo/page/32 32–33] |chapter=1 |chapter-url=https://archive.org/details/artofelectronics00horo |url=https://archive.org/details/artofelectronics00horo/page/32 }}</ref> Assuming a two-terminal circuit element with impedance <math>Z</math> is driven by a sinusoidal voltage or current as above, there holds  

:<math>\ V = I Z = I |Z| e^{j \arg (Z)}</math>

The magnitude of the impedance <math>|Z|</math> acts just like resistance, giving the drop in voltage amplitude across an impedance <math>Z</math> for a given current <math>I</math>. The [[phase factor]] tells us that the current lags the voltage by a phase <math>\theta = \arg(Z)</math> (i.e., in the [[time domain]], the current signal is shifted <math display="inline">\frac{\theta}{2 \pi} T</math> later with respect to the voltage signal).

Just as impedance extends Ohm's law to cover AC circuits, other results from DC circuit analysis, such as [[voltage divider|voltage division]], [[current divider|current division]], [[Thévenin's theorem]] and [[Norton's theorem]], can also be extended to AC circuits by replacing resistance with impedance.

=== Phasors ===
{{Main|Phasor}}
{{Further|Analytic representation}}

A phasor is represented by a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids (such as in AC circuits<ref name=":0" />{{Rp|page=53}}), where they can often reduce a differential equation problem to an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current. This is identical to the definition from [[Electrical impedance#Ohm's law|Ohm's law]] given above, recognising that the factors of <math>e^{j\omega t}</math> cancel.