Editing Electrical reactance (section)

== Inductive reactance ==
{{main|Inductance}}

Inductive reactance is a property exhibited by an inductor, and inductive reactance exists based on the fact that an electric current produces a magnetic field around it. In the context of an AC circuit (although this concept applies any time current is changing), this magnetic field is constantly changing as a result of current that oscillates back and forth. It is this change in magnetic field that induces another electric current to flow in the same wire (counter-EMF), in a direction such as to oppose the flow of the current originally responsible for producing the magnetic field (known as [[Lenz's law]]). Hence, ''inductive reactance'' is an opposition to the change of current through an element.

For an ideal inductor in an AC circuit, the inhibitive effect on change in current flow results in a delay, or a phase shift, of the alternating current with respect to alternating voltage. Specifically, an ideal inductor (with no resistance) will cause the current to lag the voltage by a quarter cycle, or 90°.

In electric power systems, inductive reactance (and capacitive reactance, however inductive reactance is more common) can limit the power capacity of an AC transmission line, because power is not completely transferred when voltage and current are out-of-phase (detailed above). That is, current will flow for an out-of-phase system, however real power at certain times will not be transferred, because there will be points during which instantaneous current is positive while instantaneous voltage is negative, or vice versa, implying negative power transfer. Hence, real work is not performed when power transfer is "negative". However, current still flows even when a system is out-of-phase, which causes transmission lines to heat up due to current flow. Consequently, transmission lines can only heat up so much (or else they would physically sag too much, due to the heat expanding the metal transmission lines), so transmission line operators have a "ceiling" on the amount of current that can flow through a given line, and excessive inductive reactance can limit the power capacity of a line. Power providers utilize capacitors to shift the phase and minimize the losses, based on usage patterns.

Inductive reactance <math>X_L</math> is [[Proportionality (mathematics)|proportional]] to the sinusoidal signal [[frequency]] <math>f</math> and the [[inductance]] <math>L</math>, which depends on the physical shape of the inductor:

<math>X_L = \omega L = 2\pi f L</math>.

The average current flowing through an [[inductance]] <math>L</math> in series with a [[sinusoidal]] AC voltage source of RMS [[amplitude]] <math>A</math> and frequency <math>f</math> is equal to:
:<math>I_L = {A \over \omega L} = {A \over 2\pi f L}.</math>

Because a [[Square wave (waveform)|square wave]] has multiple amplitudes at sinusoidal [[harmonic]]s, the average current flowing through an [[inductance]] <math>L</math> in series with a square wave AC voltage source of RMS [[amplitude]] <math>A</math> and frequency <math>f</math> is equal to:
:<math>I_L = {A \pi^2 \over 8 \omega L} = {A\pi \over 16 f L}</math>
making it appear as if the inductive reactance to a square wave was about 19% smaller <math>X_L = {16 \over \pi} f L</math>  than the reactance to the AC sine wave.

Any conductor of finite dimensions has inductance; the inductance is made larger by the multiple turns in an [[electromagnetic coil]]. [[Faraday's law of induction|Faraday's law]] of electromagnetic induction gives the counter-[[Electromotive force|emf]] <math>\mathcal{E}</math> (voltage opposing current) due to a rate-of-change of [[magnetic flux density]] <math>\scriptstyle{B}</math> through a current loop.

:<math>\mathcal{E} = -{{d\Phi_B} \over dt}</math>

For an inductor consisting of a coil with <math>N</math> loops this gives:

:<math>\mathcal{E} = -N{d\Phi_B \over dt}</math>.

The counter-emf is the source of the opposition to current flow.  A constant [[direct current]] has a zero rate-of-change, and sees an inductor as a [[short-circuit]] (it is typically made from a material with a low [[resistivity]]).  An [[alternating current]] has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

== Impedance ==
{{main|Electrical impedance}}
Both reactance <math>{X}</math> and [[Electrical resistance|resistance]] <math>{R}</math> are components of [[Electrical impedance|impedance]] <math>{\mathbf{Z}}</math>.
:<math>\mathbf{Z} = R + \mathbf{j}X</math>
where:
*<math>\mathbf{Z}</math> is the complex [[Electrical impedance|impedance]], measured in [[ohm]]s;
*<math>R</math> is the [[Electrical resistance|resistance]], measured in ohms. It is the real part of the impedance: <math>{R=\text{Re}{(\mathbf{Z})}}</math>
*<math>X</math> is the reactance, measured in ohms. It is the imaginary part of the impedance: <math>{X=\text{Im}{(\mathbf{Z})}}</math>
*<math>\mathbf{j}</math> is the [[square root of minus one]], usually represented by <math>\mathbf{i}</math> in non-electrical formulas. <math>\mathbf{j}</math> is used so as not to confuse the imaginary unit with current, commonly represented by <math>\mathbf{i}</math>.

When both a capacitor and an inductor are placed in series in a circuit, their contributions to the total circuit impedance are opposite. Capacitive reactance <math>X_C</math> and inductive reactance <math>X_L</math> contribute to the total reactance <math>X</math> as follows:
:<math>{X = X_L + X_C = \omega L -\frac {1} {\omega C}}</math>
where:
*<math>X_L</math> is the [[Inductance|inductive]] reactance, measured in ohms;
*<math>X_C</math> is the [[Capacitance|capacitive]] reactance, measured in ohms;
*<math>\omega</math> is the angular frequency, <math>2\pi</math> times the frequency in [[Hertz|Hz]].

Hence:<ref name="Glisson"/>
*if <math>\scriptstyle X > 0</math>, the total reactance is said to be inductive;
*if <math>\scriptstyle X = 0</math>, then the impedance is purely resistive;
*if <math>\scriptstyle X < 0</math>, the total reactance is said to be capacitive.

Note however that if <math>X_L</math> and <math>X_C</math> are assumed both positive by definition, then the intermediary formula changes to a difference:<ref name="Hughes"/>

:<math>{X = X_L - X_C = \omega L -\frac {1} {\omega C}}</math>

but the ultimate value is the same.

=== Phase relationship ===

The phase of the voltage across a purely reactive device (i.e. with zero [[Parasitic element (electrical networks)|parasitic resistance]]) ''lags'' the current by <math>\tfrac{\pi}{2}</math> radians for a capacitive reactance and ''leads'' the current by <math>\tfrac{\pi}{2}</math> radians for an inductive reactance. Without knowledge of both the resistance and reactance the relationship between voltage and current cannot be determined.

The origin of the different signs for capacitive and inductive reactance is the phase factor <math>e^{\pm \mathbf{j}{\frac{\pi}{2}}}</math> in the impedance.

:<math>\begin{align}
  \mathbf{Z}_C &= {1 \over \omega C}e^{-\mathbf{j}{\pi \over 2}} = \mathbf{j}\left({ -\frac{1}{\omega C}}\right) = \mathbf{j}X_C \\
  \mathbf{Z}_L &= \omega Le^{\mathbf{j}{\pi \over 2}} = \mathbf{j}\omega L = \mathbf{j}X_L\quad
\end{align}</math>

For a reactive component the sinusoidal voltage across the component is in quadrature (a <math>\tfrac{\pi}{2}</math> phase difference) with the sinusoidal current through the component. The component alternately absorbs energy from the circuit and then returns energy to the circuit, thus a pure reactance does not dissipate power.