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Electrical reactance

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Opposition to current by inductance or capacitance

"Reactance (physics)" redirects here. For other uses, seeReactance.

In electrical circuits,reactance is the opposition presented toalternating current byinductance andcapacitance.^[1] It is measured inohms.^[1] Along with resistance, it is one of two elements ofimpedance; however, while both elements involve transfer of electrical energy, nodissipation 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 appliedvoltage.

Reactance is used to computeamplitude andphase changes ofsinusoidal alternating current going through a circuit element. Like resistance, reactance is measured inohms, with positive values indicatinginductive reactance and negative indicatingcapacitive reactance. It is denoted by the symbol $X {\displaystyle X}$ . An idealresistor has zero reactance, whereas ideal reactors have no shunt conductance and no series resistance. Asfrequency increases, inductive reactance increases and capacitive reactance decreases.

Comparison to resistance

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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, butcomplex numbers are typically needed. This is treated below in the section onimpedance.

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 termreactance was first suggested by French engineer Édouard Hospitalier inL'Industrie Electrique on 10 May 1893. It was officially adopted by theAmerican Institute of Electrical Engineers in May 1894.^[2]

Capacitive reactance

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Main article:Capacitance

A capacitor consists of twoconductors separated by aninsulator, also known as adielectric.

Capacitive reactance is an opposition to the change of voltage across an element. Capacitive reactance $X_{C}$ isinversely proportional to the signalfrequency $f {\displaystyle f}$ (orangular frequency $\omega$ ) and thecapacitance $C {\displaystyle C}$ .^[3]

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,^[3]^[4]^[5]

X_{C}=-{\frac {1}{\omega C}}=-{\frac {1}{2\pi fC}}

.

Another choice is to define capacitive reactance as a positive number,^[6]^[7]^[8]

X_{C}={\frac {1}{\omega C}}={\frac {1}{2\pi fC}}

.

In this case however one needs to remember to add a negative sign for the impedance of a capacitor, i.e. $Z_{c}=-jX_{c}$ .

At $f=0$ , the magnitude of the capacitor's reactance is infinite, behaving like anopen circuit (preventing anycurrent from flowing through the dielectric). As frequency increases, the magnitude of reactance decreases, allowing more current to flow. As $f {\displaystyle f}$ approaches $\infty$ , the capacitor's reactance approaches $0 {\displaystyle 0}$ , behaving like ashort circuit.

The application of aDC voltage across a capacitor causes positivecharge to accumulate on one side and negativecharge to accumulate on the other side; theelectric field due to the accumulated charge is the source of the opposition to the current. When thepotential 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

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Main article: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 asLenz'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 $X_{L}$ isproportional to the sinusoidal signalfrequency $f {\displaystyle f}$ and theinductance $L {\displaystyle L}$ , which depends on the physical shape of the inductor:

$X_{L}=\omega L=2\pi fL$ .

The average current flowing through aninductance $L {\displaystyle L}$ in series with asinusoidal AC voltage source of RMSamplitude $A {\displaystyle A}$ and frequency $f {\displaystyle f}$ is equal to:

I_{L}={A \over \omega L}={A \over 2\pi fL}.

Because asquare wave has multiple amplitudes at sinusoidalharmonics, the average current flowing through aninductance $L {\displaystyle L}$ in series with a square wave AC voltage source of RMSamplitude $A {\displaystyle A}$ and frequency $f {\displaystyle f}$ is equal to:

I_{L}={A\pi ^{2} \over 8\omega L}={A\pi  \over 16fL}

making it appear as if the inductive reactance to a square wave was about 19% smaller $X_{L}={16 \over \pi }fL$ 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 anelectromagnetic coil.Faraday's law of electromagnetic induction gives the counter-emf ${\mathcal {E}}$ (voltage opposing current) due to a rate-of-change ofmagnetic flux density $\scriptstyle {B}$ through a current loop.

{\mathcal {E}}=-{{d\Phi _{B}} \over dt}

For an inductor consisting of a coil with $N {\displaystyle N}$ loops this gives:

{\mathcal {E}}=-N{d\Phi _{B} \over dt}

.

The counter-emf is the source of the opposition to current flow. A constantdirect current has a zero rate-of-change, and sees an inductor as ashort-circuit (it is typically made from a material with a lowresistivity). Analternating current has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

Impedance

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Main article:Electrical impedance

Both reactance ${X}$ andresistance ${R}$ are components ofimpedance ${\mathbf {Z} }$ .

\mathbf {Z} =R+\mathbf {j} X

where:

$\mathbf {Z}$ is the compleximpedance, measured inohms;
$R {\displaystyle R}$ is theresistance, measured in ohms. It is the real part of the impedance: ${R={\text{Re}}{(\mathbf {Z} )}}$
$X {\displaystyle X}$ is the reactance, measured in ohms. It is the imaginary part of the impedance: ${X={\text{Im}}{(\mathbf {Z} )}}$
$\mathbf {j}$ is thesquare root of negative one, usually represented by $\mathbf {i}$ in non-electrical formulas. $\mathbf {j}$ is used so as not to confuse the imaginary unit with current, commonly represented by $\mathbf {i}$ .

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 $X_{C}$ and inductive reactance $X_{L}$ contribute to the total reactance $X {\displaystyle X}$ as follows:

{X=X_{L}+X_{C}=\omega L-{\frac {1}{\omega C}}}

where:

$X_{L}$ is theinductive reactance, measured in ohms;
$X_{C}$ is thecapacitive reactance, measured in ohms;
$\omega$ is the angular frequency, $2\pi$ times the frequency inHz.

Hence:^[5]

if $\scriptstyle X>0$ , the total reactance is said to be inductive;
if $\scriptstyle X=0$ , then the impedance is purely resistive;
if $\scriptstyle X<0$ , the total reactance is said to be capacitive.

Note however that if $X_{L}$ and $X_{C}$ are assumed both positive by definition, then the intermediary formula changes to a difference:^[7]

{X=X_{L}-X_{C}=\omega L-{\frac {1}{\omega C}}}

but the ultimate value is the same.

Phase relationship

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The phase of the voltage across a purely reactive device (i.e. with zeroparasitic resistance)lags the current by ${\tfrac {\pi }{2}}$ radians for a capacitive reactance andleads the current by ${\tfrac {\pi }{2}}$ 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 $e^{\pm \mathbf {j} {\frac {\pi }{2}}}$ in the impedance.

{\begin{aligned}\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{aligned}}

For a reactive component the sinusoidal voltage across the component is in quadrature (a ${\tfrac {\pi }{2}}$ 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.

References

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Shamieh C. and McComb G.,Electronics for Dummies, John Wiley & Sons, 2011.
Meade R.,Foundations of Electronics, Cengage Learning, 2002.
Young, Hugh D.; Roger A. Freedman; A. Lewis Ford (2004) [1949].Sears and Zemansky's University Physics (11 ed.).San Francisco:Addison Wesley.ISBN 0-8053-9179-7.

^^a ^bVeley, Victor F. C. (1987).The Benchtop Electronics Reference Manual (1st ed.). New York: Tab Books. pp. 229, 232.
^Charles Proteus Steinmetz, Frederick Bedell,"Reactance",Transactions of the American Institute of Electrical Engineers, vol. 11, pp. 640–648, January–December 1894.
^^a ^bIrwin, D. (2002).Basic Engineering Circuit Analysis, page 274. New York: John Wiley & Sons, Inc.
^Hayt, W.H., Kimmerly J.E. (2007).Engineering Circuit Analysis, 7th ed., McGraw-Hill, p. 388
^^a ^bGlisson, T.H. (2011).Introduction to Circuit Analysis and Design, Springer, p. 408
^Horowitz P., Hill W. (2015).The Art of Electronics, 3rd ed., p. 42
^^a ^bHughes E., Hiley J., Brown K., Smith I.McK., (2012).Hughes Electrical and Electronic Technology, 11th edition, Pearson, pp. 237-241
^Robbins, A.H., Miller W. (2012).Circuit Analysis: Theory and Practice, 5th ed., Cengage Learning, pp. 554-558