Editing Inductance (section)

== Inductive reactance ==
[[File:Waveforms - inductive reactance.svg|thumb|The voltage ''(<math>v</math>, blue)'' and current ''(<math>i</math>, red)'' waveforms in an ideal inductor to which an alternating current has been applied. The current lags the voltage by 90°]]

When a [[sinusoidal]] [[alternating current]] (AC) is passing through a linear inductance, the induced [[back-EMF|back-{{abbr|EMF|electromotive force}}]] is also sinusoidal.  If the current through the inductance is <math>i(t) = I_\text{peak} \sin\left(\omega t\right)</math>, from (1) above the voltage across it is
<math display=block>\begin{align}
v(t) &= L \frac{\text{d}i}{\text{d}t} = L\,\frac{\text{d}}{\text{d}t}\left[I_\text{peak} \sin\left(\omega t\right)\right]\\
&= \omega L\,I_\text{peak}\,\cos\left(\omega t\right) = \omega L\,I_\text{peak}\,\sin\left(\omega t + {\pi \over 2}\right)
\end{align}</math>

where <math>I_\text{peak}</math> is the [[amplitude]] (peak value) of the sinusoidal current in amperes, <math>\omega = 2\pi f</math> is the [[angular frequency]] of the alternating current, with <math>f</math> being its [[frequency]] in [[hertz (unit)|hertz]], and <math>L</math> is the inductance.

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

<math display=block>V_p = \omega L\,I_p= 2\pi f\,L\,I_p</math>

Inductive [[reactance (electronics)|reactance]] is the opposition of an inductor to an alternating current.<ref name="Gates">{{cite book
 | last1  = Gates
 | first1 = Earl D. 
 | title  = Introduction to Electronics
 | publisher = Cengage Learning
 | date   = 2001
 | pages  = 153
 | url    = https://books.google.com/books?id=IwC5GIA0cREC&pg=PA153
 | isbn   = 0766816982
 }}</ref>  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

<math display=block>X_L = \frac{V_p }{ I_p} = 2\pi f\,L</math>

Reactance has units of [[ohm (unit)|ohm]]s. It can be seen that [[inductive reactance]] of an inductor increases proportionally with frequency {{nowrap|<math>f</math>,}} 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 <math>\phi =\tfrac{1}{2} \pi</math> [[radian]]s or 90&nbsp;degrees, showing that in an ideal inductor ''the current lags the voltage by 90°''.