Reactance (electronics)

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## Comparison to resistance

## Capacitive reactance

## Inductive reactance

## Impedance

### Phase relationship

## See also

## Notes

## References

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Reactance Electronics

In electric and electronic systems, **reactance** is the opposition of a circuit element to the flow of current due to that element's inductance or capacitance. Greater reactance leads to smaller currents for the same voltage applied. Reactance is similar to electric resistance in this respect, but differs in that reactance does not lead to dissipation of electrical energy as heat. Instead, energy is stored in the reactance, and a quarter-cycle later returned to the circuit, whereas a resistance continuously loses energy.

Reactance is used to compute amplitude and phase changes of sinusoidal alternating current (AC) going through a circuit element. It is denoted by the symbol . An ideal resistor has zero reactance, whereas ideal inductors and capacitors have zero resistance - that is, respond to current only by reactance. As frequency increases, inductive reactance also increases and capacitive reactance decreases.

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 elements with no reactance (pure resistance). These same techniques can also be used to combine elements with reactance with elements with resistance but complex numbers are typically needed. This is treated below in the section on 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 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 typically have the same resistance for all frequencies.

The term *reactance* was first suggested by French engineer M. Hospitalier in *L'Industrie Electrique* on 10 May 1893. It was officially adopted by the American Institute of Electrical Engineers in May 1894.^{[1]}

A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

*Capacitive reactance* is an opposition to the change of voltage across an element. Capacitive reactance is inversely proportional to the signal frequency (or "angular frequency" ?) and the capacitance .^{[2]}

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,^{[2]}^{[3]}^{[4]}

Another choice is to define capacitive reactance as a positive number,^{[5]}^{[6]}^{[7]}

In this case however one needs to remember to add a negative sign for the impedance of a capacitor, i.e. ^{[a]}

At low frequencies a capacitor is an open circuit so no current flows in the dielectric.

A DC voltage applied across a capacitor causes positive charge to accumulate on one side and negative 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 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 use patterns.

Inductive reactance is proportional to the sinusoidal signal frequency and the inductance which depends on the physical shape of the inductor.

The average current flowing through an inductance in series with a sinusoidal AC voltage source of RMS amplitude and frequency is equal to:

Because a square wave has multiple amplitudes at sinusoidal harmonics, the average current flowing through an inductance in series with a square wave AC voltage source of RMS amplitude and frequency is equal to:

making it appear as if the inductive reactance to a square wave was about 19% smaller 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 electromagnetic induction gives the counter-EMF (voltage opposing current) due to a rate-of-change of magnetic flux density flowing through the surface of a current loop.

For an inductor consisting of a coil with loops this gives.

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.

Both reactance and resistance are components of impedance

where:

- is the complex impedance, measured in ohms;
- is the resistance, measured in ohms. It is the real part of the impedance:

- is the reactance, measured in ohms. It is the imaginary part of the impedance:
^{[b]}

where is the square root of minus one.^{[c]}

When both a capacitor and an inductor are placed in series in a circuit, their contributions to the total circuit impedance are opposite. Inductive reactance and capacitive reactance contribute to the total reactance (no subscript) as follows.

where:

- is the inductive reactance, measured in ohms;
- is the capacitive reactance, measured in ohms;
- is the "angular frequency": times the frequency in Hz.

Hence:^{[4]}

- if , the total reactance is said to be inductive;
- if , then the impedance is reactance-free, or purely resistive;
^{[d]} - if , the total reactance is said to be capacitive.

Note however that if and are assumed both positive by definition, then the intermediary formula changes to a difference:^{[6]}

but the ultimate value is the same.

The phase of the voltage across a purely reactive device (i.e. with zero parasitic resistance) *lags* the current by radians for a capacitive reactance and *leads* the current by 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 in the impedance.

For a reactive component the sinusoidal voltage across the component is in quadrature (a 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.

**^**Whereas it is a practical necessity in electronics to tolerate others' preferred conventions for whether capacitive reactance, has a minus-sign incorporated into it or not, there is no such tolerance in the case of the impedance, Whichever convention is used, at the point where the capacitive reactance becomes a part of the complex number, the minus sign*absolutely must*be applied, and applied*only once*.**^**Using the formula for reactance only works well when using the convention that is, with the minus sign "built into" the capacitve reactance, which is then always a negative number.**^**The square root of minus one or the "imaginary unit" is usually represented by in non-electrical formulas, but for electrical formulas the symbol is commonly used represent current. To avoid confusion with current, the symbol is used for the imaginary unit in electrical formulas.**^**Zero total reactance in a system is often used to define the word "(electrically) resonant" system.

**^**Steinmetz, Charles P.; Bedell, Frederick (January-December 1894). "Reactance".*Transactions of the American Institute of Electrical Engineers*.**11**: 640-648.- ^
^{a}^{b}Irwin, D. (2002).*Basic Engineering Circuit Analysis*. New York, NY: John Wiley & Sons, Inc. p. 274. **^**Hayt, W.H.; Kimmerly, J.E. (2007).*Engineering Circuit Analysis*(7th ed.). McGraw-Hill. p. 388.- ^
^{a}^{b}Glisson, 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}^{b}Hughes, E.; Hiley, J.; Brown, K.; Smith, I. McK. (2012).*Hughes Electrical and Electronic Technology*(11th ed.). Pearson. pp. 237-241. **^**Robbins, A.H.; Miller, W. (2012).*Circuit Analysis: Theory and Practice*(5th ed.). Cengage Learning. pp. 554-558.

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

- "Interactive Java tutorial on inductive reactance". Magnet Academy. National High Magnetic Field Laboratory.
- "Reactance calculator".
*FXSolver.com*(shareware).

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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