Inductance
Inductive reactance

When a sinusoidal alternating current (AC) is passing through a linear inductance, the induced back- EMF is also sinusoidal. If the current through the inductance is i (t) = I peak sin (ω t) {\displaystyle i(t)=I_{\text{peak}}\sin \left(\omega t\right)}, from (1) above the voltage across it is v (t) = L d i d t = L d d t = ω L I peak cos (ω t) = ω L I peak sin (ω t + π 2) {\displaystyle {\begin{aligned}v(t)&=L{\frac {{\text{d}}i}{{\text{d}}t}}=L\,{\frac {\text{d}}{{\text{d}}t}}\left\\&=\omega L\,I_{\text{peak}}\,\cos \left(\omega t\right)=\omega L\,I_{\text{peak}}\,\sin \left(\omega t+{\pi \over 2}\right)\end{aligned}}}

where I peak {\displaystyle I_{\text{peak}}} is the amplitude (peak value) of the sinusoidal current in amperes, ω = 2 π f {\displaystyle \omega =2\pi f} is the angular frequency of the alternating current, with f {\displaystyle f} being its frequency in hertz, and L {\displaystyle L} is the inductance.

Thus the amplitude (peak value) of the voltage across the inductance is

V p = ω L I p = 2 π f L I p {\displaystyle V_{p}=\omega L\,I_{p}=2\pi f\,L\,I_{p}}

Inductive reactance is the opposition of an inductor to an alternating current. It is defined analogously to electrical resistance in a resistor, as the ratio of the amplitude (peak value) of the alternating voltage to current in the component

X L = V p I p = 2 π f L {\displaystyle X_{L}={\frac {V_{p}}{I_{p}}}=2\pi f\,L}

Reactance has units of ohms. It can be seen that inductive reactance of an inductor increases proportionally with frequency f {\displaystyle f}, so an inductor conducts less current for a given applied AC voltage as the frequency increases. Because the induced voltage is greatest when the current is increasing, the voltage and current waveforms are out of phase; the voltage peaks occur earlier in each cycle than the current peaks. The phase difference between the current and the induced voltage is ϕ = 1 2 π {\displaystyle \phi ={\tfrac {1}{2}}\pi } radians or 90 degrees, showing that in an ideal inductor the current lags the voltage by 90°.
combination:repeat_prompt, detectable_content:number_placeholders[8]