= Electrical impedance
{title2=$Z(\omega)=\widehat V/\widehat I$}
{wiki}
= Complex impedance
{synonym}
<Electrical impedance> is the ratio of terminal <electric potential difference> and <electric current> <phasors> at one specified positive <angular frequency> $\omega$. For an <electrical circuit> modeled as a <linear time-invariant system>, use the sinusoidal <steady state> after any decaying transient has settled. With the <passive sign convention>, positive <electric current> enters the terminal assigned positive <electric potential difference>. Define peak <complex amplitudes> by
$$
v(t)=\operatorname{Re}(\widehat V e^{\mathrm i\omega t}),\qquad
i(t)=\operatorname{Re}(\widehat I e^{\mathrm i\omega t}),\qquad
Z(\omega)=\frac{\widehat V}{\widehat I},\quad \widehat I\ne0.
$$
The physical $v(t)$ and $i(t)$ are real; the <complex numbers> encode their <amplitudes> and <wave phases>. The unit of <electrical impedance> is the <ohm>, $1\,\Omega=1\,\mathrm V/\mathrm A$. Both <phasors> may instead use <root mean square> magnitudes, with the same ratio $Z$. A nonlinear <diode> can generate additional <frequencies>; a single $Z(\omega)$ then does not describe the whole response.
Write the <electrical impedance> in Cartesian and <polar forms> as
$$
Z=R+\mathrm iX=|Z|e^{\mathrm i\phi},\qquad
R=\operatorname{Re}Z,\quad X=\operatorname{Im}Z,\quad
|Z|=\sqrt{R^2+X^2},\quad
\phi=\arg Z=\arg\widehat V-\arg\widehat I.
$$
Here $R$ is the resistive <real part>, $X$ the signed <electrical reactance>, and $\phi$ the voltage-minus-current <phase difference>, defined modulo a full turn. The <Pythagorean theorem> gives the <modulus> from a <right triangle> in the <complex plane>. The <complex argument> gives the rotation from the current <phasor> to the voltage <phasor>. Thus $V_{\rm rms}=|Z|I_{\rm rms}$, while negative $\phi$ means <electric current> leads terminal <electric potential difference>. For $R>0$, $\phi=\arctan(X/R)$ lies between $-\pi/2$ and $\pi/2$; outside that case, use the quadrant of the <complex number> rather than a bare <inverse tangent>. The geometric magnitude and <phase difference> are the two pieces of the AC relation.<University Physics Volume 2 by Samuel J. Ling and others>[§15.3, figures 15.12–15.13 and equations 15.9–15.11: phasor addition, phase angle, impedance magnitude and ohm unit]{ref}
\Image[/impedance-phasor-components-and-timing.png]
{title=<Electrical impedance> as a <complex number> and a voltage-current <phase difference>}
{description=Original ideal <series RC impedance> calculation: $R=100\,\Omega$, $C=1\,\mu\mathrm F$, $f=1000\,\mathrm{Hz}$. Left: $Z=100-159.155\mathrm i\,\Omega$, resolved into its <real part> and signed <electrical reactance>. Right: normalized predicted <sinusoids> with the current <phasor> as the zero-phase reference; their peak separation is $0.160717\,\mathrm{ms}$. The plotted normalized heights do not compare volts with amperes.}
Multiplication by $Z$ acts on <phasors>, not on instantaneous physical signals. For example, when $i(t)=I_0\cos\omega t$,
$$
v(t)=I_0\bigl(R\cos\omega t-X\sin\omega t\bigr).
$$
The second term is in <phase quadrature> with <electric current>; it cannot be represented by pointwise multiplication $v(t)=Zi(t)$ when $X\ne0$. This is the useful compression: one <complex multiplication> replaces separate <amplitude> and <phase difference> calculations, while the <real part> restores the physical signal.
An ideal <resistor>, <inductor> and <capacitor> have $Z_R=R$, $Z_L=\mathrm i\omega L$ and $Z_C=1/(\mathrm i\omega C)$. Their <electrical impedances> add in a <series RLC circuit>, giving $Z=R+\mathrm i(\omega L-1/(\omega C))$. The current <amplitude> is $V_0/|Z|$, so cancellation of the <imaginary parts> explains the <resonant series RLC response>. The component-law derivation and opposite <complex exponential> convention are in <reactance phase convention>.
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