= Electrical reactance
{title2=$X=\operatorname{Im}Z$}
Electrical reactance $X$ is the signed <imaginary part> of <electrical impedance> $Z=R+\mathrm iX$, measured in <ohms>. It describes the part of the terminal <electric potential difference> in <phase quadrature> with the <electric current> in a sinusoidal <steady state>. In the $e^{\mathrm i\omega t}$ convention, an ideal <inductor> has $X_L=\omega L>0$ and an ideal <capacitor> has $X_C=-1/(\omega C)<0$ for $\omega>0$. A large $|X|$ reduces <electric current> at fixed applied voltage, but an ideal purely reactive component returns the <energy> it receives over each cycle. This is why <electrical reactance> and <electrical resistance> have the same units yet describe different physical behavior. https://www.feynmanlectures.caltech.edu/II_22.html#Ch22-S5[Feynman, Leighton and Sands, The Feynman Lectures on Physics, II.22.5: resistance, reactance and energy loss]{ref}.
\Image[/reactance-frequency-cancellation.png]
{title=Signed <electrical reactances> cancel in a toy <series RLC circuit>}
{description=Calculated ideal curves, not measured data: <inductance> $L=10\,\mathrm{mH}$, <capacitance> $C=1\,\mu\mathrm F$, total <electrical resistance> $R_\Sigma=100+10=110\,\Omega$, and applied <root mean square> voltage $1.00\,\mathrm V$. The $10\,\Omega$ is the sensing <resistor> in <electrical reactance bench measurement>. Top: $X_L$, signed $X_C$, and their sum. Bottom: <electric current> and voltage-minus-current <phase difference>. At $f_0=1.592\,\mathrm{kHz}$ the sum vanishes, the <electric current> peaks, and terminal voltage and <electric current> are in phase.}
In a <series RLC circuit>, <electrical impedances> add because the same <electric current> passes through every component. For ideal positive <inductance> $L$, <capacitance> $C$, and total <electrical resistance> $R_\Sigma>0$,
$$
X(\omega)=\omega L-\frac1{\omega C},\qquad
|Z|=\sqrt{R_\Sigma^2+X^2},\qquad
I_{\rm rms}=\frac{V_{\rm rms}}{|Z|},\qquad
\phi=\arctan\frac{X}{R_\Sigma}.
$$
Here $\phi$ is voltage <wave phase> minus current <wave phase>. Below $\omega_0=1/\sqrt{LC}$, $X<0$ and <electric current> leads terminal voltage; above it, $X>0$ and <electric current> lags. At $\omega_0$, the two signed <electrical reactances> cancel, leaving $Z=R_\Sigma$ and the maximum <electric current> at fixed terminal voltage. This is the <resonant series RLC response>. <Electrical resistance> does not cancel: the toy circuit gives $f_0=\omega_0/(2\pi)=1591.55\,\mathrm{Hz}$ and $I_{\rm rms}=9.091\,\mathrm{mA}$ there. These numbers follow directly from the stipulated component values. https://openstax.org/books/university-physics-volume-2/pages/15-3-rlc-series-circuits-with-ac[Ling, Moebs and Sanny, University Physics 2, §15.3: series impedance and resonance]{ref}.
Cancellation concerns terminal voltage, not the disappearance of stored <energy>. In the toy circuit at <resonance>, the individual <inductor> and <capacitor> voltages are each $0.9091\,\mathrm V$ in <root mean square> magnitude, with opposite <wave phases>, while their sum is zero. Their peak stored <magnetic energy> and <capacitor energy> are each $0.8264\,\mu\mathrm J$, attained at different times. With smaller positive $R_\Sigma$, either component voltage can exceed the applied voltage. This ability to select a <frequency> while sustaining internal <energy> exchange connects <electrical reactance> to tuning and <frequency response>.
For an ideal <inductor> or <capacitor>, the absorbed instantaneous <power> equals the rate of change of stored <energy>:
$$
p_L=v_Li=\frac{d}{dt}\left(\frac12Li^2\right),\qquad
p_C=v_Ci=\frac{d}{dt}\left(\frac12Cv_C^2\right).
$$
Positive <power> stores <energy>; negative <power> returns it. Over a periodic <steady state>, each stored <energy> returns to its initial value, so its cycle-averaged absorbed <power> is zero. An ideal <resistor> instead has $p_R=Ri^2\geq0$, transferring <energy> into <Joule heating>. More generally, the average terminal <power> of this <series RLC circuit> is
$$
\overline p=V_{\rm rms}I_{\rm rms}\cos\phi=I_{\rm rms}^2R_\Sigma.
$$
Zero average <power> in an ideal reactive component does not imply zero instantaneous <power>, zero <electric current>, or zero stored <energy>. Real coils and <capacitors> have losses, so zero ideal reactive loss is a model assumption rather than a claim about every component. https://openstax.org/books/university-physics-volume-2/pages/15-4-power-in-an-ac-circuit[Ling, Moebs and Sanny, University Physics 2, §15.4: instantaneous and average AC power]{ref}.
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