Editing Electrical reactance

{{Short description|Opposition to current by inductance or capacitance}}
{{use dmy dates|date=May 2021}}{{Redirect|Reactance (physics)||Reactance (disambiguation){{!}}Reactance}}

In electrical circuits, '''reactance''' is the opposition presented to [[alternating current]] by [[inductance]] and [[capacitance]].<ref name="veley01">{{cite book |author-first= Victor F. C. |author-last=Veley |title=The Benchtop Electronics Reference Manual |edition=1st |location=New York |publisher=Tab Books |date=1987 |pages=229, 232|url=https://archive.org/details/benchtopelectron00vele }}</ref> It's measured in [[Ohm|Ω (Ohms)]].<ref name="veley01" /> Along with resistance, it is one of two elements of [[Electrical impedance|impedance]]; however, while both elements involve transfer of electrical energy, no [[Joule heating|dissipation of electrical energy as heat]] occurs in reactance; instead, the reactance stores energy until a quarter-cycle later when the energy is returned to the circuit. Greater reactance gives smaller current for the same applied [[voltage]].

Reactance is used to compute [[amplitude]] and [[Phase (waves)|phase]] changes of [[Sine wave|sinusoidal]] alternating current going through a circuit element.  Like resistance, reactance is measured in [[ohm]]s, with positive values indicating ''inductive'' reactance and negative indicating ''capacitive'' reactance.  It is denoted by the symbol <math>X</math>. An ideal [[resistor]] has zero reactance, whereas ideal reactors have no shunt conductance and no series resistance. As [[frequency]] increases, inductive reactance increases and capacitive reactance decreases.

== Comparison to resistance ==
Reactance is similar to resistance in that larger reactance leads to smaller currents for the same applied voltage. Further, a circuit made entirely of elements that have only reactance (and no resistance) can be treated the same way as a circuit made entirely of resistances. These same techniques can also be used to combine elements with reactance with elements with resistance but [[complex number]]s are typically needed. This is treated below in the section on [[electrical impedance|impedance]].

There are several important differences between reactance and resistance, though. First, reactance changes the phase so that the current through the element is shifted by a quarter of a cycle relative to the phase of the voltage applied across the element. Second, power is not dissipated in a purely reactive element but is stored instead. Third, reactances can be negative so that they can 'cancel' each other out. Finally, the main circuit elements that have reactance (capacitors and inductors) have a frequency dependent reactance, unlike resistors which have the same resistance for all frequencies, at least in the ideal case.

The term ''reactance'' was first suggested by French engineer Édouard Hospitalier in ''L'Industrie Electrique'' on 10 May 1893.  It was officially adopted by the [[American Institute of Electrical Engineers]] in May 1894.<ref>[[Charles Proteus Steinmetz]], Frederick Bedell, [https://ieeexplore.ieee.org/document/4763812 "Reactance"], ''Transactions of the American Institute of Electrical Engineers'', vol. 11, pp. 640–648, January–December 1894.</ref>

== Capacitive reactance ==

{{main|Capacitance}}
A capacitor consists of two [[Electric conduction|conductor]]s separated by an [[Electrical insulation|insulator]], also known as a [[dielectric]].

''Capacitive reactance'' is an opposition to the change of voltage across an element. Capacitive reactance <math>X_C</math> is [[inversely proportional]] to the signal [[frequency]] <math>f</math> (or [[angular frequency]] <math>\omega</math>) and the [[capacitance]] <math>C</math>.<ref name="Irwin">Irwin, D. (2002). ''Basic Engineering Circuit Analysis'', page 274. New York: John Wiley & Sons, Inc.</ref>

There are two choices in the literature for defining reactance for a capacitor. One is to use a uniform notion of reactance as the imaginary part of impedance, in which case the reactance of a capacitor is the negative number,<ref name="Irwin"/><ref>Hayt, W.H., Kimmerly J.E. (2007). ''Engineering Circuit Analysis'', 7th ed., McGraw-Hill, p. 388</ref><ref name="Glisson">Glisson, T.H. (2011). ''Introduction to Circuit Analysis and Design'', Springer, p. 408</ref>

:<math>X_C = -\frac {1} {\omega C} = -\frac {1} {2\pi f C}</math>.

Another choice is to define capacitive reactance as a positive number,<ref>Horowitz P., Hill W. (2015). ''[[The Art of Electronics]]'', 3rd ed., p. 42</ref><ref name="Hughes">Hughes E., Hiley J., Brown K., Smith I.McK., (2012). ''Hughes Electrical and Electronic Technology'', 11th edition, Pearson, pp. 237-241</ref><ref>Robbins, A.H., Miller W. (2012). ''Circuit Analysis: Theory and Practice'', 5th ed., Cengage Learning, pp. 554-558</ref>

:<math>X_C = \frac {1} {\omega C} = \frac {1} {2\pi f C}</math>.

In this case however one needs to remember to add a negative sign for the impedance of a capacitor, i.e. <math>Z_c=-jX_c</math>.

At <math>f=0</math>, the magnitude of the capacitor's reactance is infinite, behaving like an [[wikt:open circuit|open circuit]] (preventing any [[Electric current|current]] from flowing through the dielectric). As frequency increases, the magnitude of reactance decreases, allowing more current to flow.  As <math>f</math> approaches <math>\infty</math>, the capacitor's reactance approaches <math>0</math>, behaving like a [[short circuit]].

The application of a [[Direct current|DC]] voltage across a capacitor causes positive [[Electrical charge|charge]] to accumulate on one side and negative [[Electrical charge|charge]] to accumulate on the other side; the [[electric field]] due to the accumulated charge is the source of the opposition to the current. When the [[potential]] associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply (ideal AC current source), a capacitor will only accumulate a limited amount of charge before the potential difference changes polarity and the charge is returned to the source.  The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

== 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 [[Imaginary unit|square root of negative 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.

== See also ==
* [[Magnetic reactance]]
* [[Susceptance]]

== References ==
* Shamieh C. and McComb G., ''Electronics for Dummies,'' John Wiley & Sons, 2011.
* Meade R., ''Foundations of Electronics,'' Cengage Learning, 2002.
* {{cite book | author = Young, Hugh D. |author2=Roger A. Freedman |author3=A. Lewis Ford | title = Sears and Zemansky's University Physics | url = https://archive.org/details/relativity00unse | url-access = registration | orig-year = 1949 | year = 2004 | edition = 11 | publisher = [[Addison Wesley]] | location = [[San Francisco]] | isbn= 0-8053-9179-7}}
{{Reflist}}

== External links ==
*[https://nationalmaglab.org/magnet-academy/watch-play/interactive-tutorials/inductive-reactance/  National magnet lab, inductive reactance ]

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[[Category:Electrical resistance and conductance]]
[[Category:Physical quantities]]