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In electrical and electronic systems, **reactance** is the opposition of a circuit element to a *change* in current or voltage, due to that element's inductance or capacitance. The notion of reactance is similar to electrical resistance, but it differs in several respects.

An **electric current** is a flow of electric charge. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

**Voltage**, **electric potential difference**, **electric pressure **or **electric tension** is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named *volt*. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for *volt* uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by ∆*V*, but more often simply as *V*, for instance in the context of Ohm's or Kirchhoff's circuit laws.

In electromagnetism and electronics, **inductance** describes the tendency of an electrical conductor, such as coil, to oppose a change in the electric current through it. The change in current induces a reverse electromotive force (voltage). When an electric current flows through a conductor, it creates a magnetic field around that conductor. A changing current, in turn, creates a changing magnetic field, the surface integral of which is known as magnetic flux. From Faraday's law of induction, any change in magnetic flux through a circuit induces an electromotive force (voltage) across that circuit, a phenomenon known as electromagnetic induction. Inductance is specifically defined as the ratio between this induced voltage and the rate of change of the current in the circuit

- Capacitive reactance
- Inductive reactance
- Impedance
- Phase relationship
- See also
- References
- External links

In phasor analysis, reactance is used to compute amplitude and phase changes of sinusoidal alternating current 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. The magnitude of the reactance of an inductor rises in proportion to a rise in frequency, while the magnitude of the reactance of a capacitor decreases in proportion to a rise in frequency. As frequency goes up, inductive reactance also goes up and capacitive reactance goes down.

In physics and engineering, a **phasor**, is a complex number representing a sinusoidal function whose amplitude (*A*), angular frequency (*ω*), and initial phase (*θ*) are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as **phasor**, **complex amplitude**, and **sinor** or even **complexor**.

A **sine wave** or **sinusoid** is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (*t*) is**:**

**Alternating current** (**AC**) is an electric current which periodically reverses direction, in contrast to direct current (**DC**) which flows only in one direction. Alternating current is the form in which electric power is delivered to businesses and residences, and it is the form of electrical energy that consumers typically use when they plug kitchen appliances, televisions, fans and electric lamps into a wall socket. A common source of DC power is a battery cell in a flashlight. The abbreviations *AC* and *DC* are often used to mean simply *alternating* and *direct*, as when they modify *current* or *voltage*.

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

A **dielectric** is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing **dielectric polarization**. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

*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 ω) and the capacitance .^{ [1] }

**Frequency** is the number of occurrences of a repeating event per unit of time. It is also referred to as **temporal frequency**, which emphasizes the contrast to spatial frequency and angular frequency. The

In physics, **angular frequency***ω* is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

**Capacitance** is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. There are two closely related notions of capacitance: *self capacitance* and *mutual capacitance*. Any object that can be electrically charged exhibits *self capacitance*. A material with a large self capacitance holds more electric charge at a given voltage than one with low capacitance. The notion of *mutual capacitance* is particularly important for understanding the operations of the capacitor, one of the three elementary linear electronic components.

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 a negative number:^{ [1] }^{ [2] }^{ [3] }

Another choice is to define capacitive reactance as a positive number,^{ [4] }^{ [5] }^{ [6] }

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

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 usage 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 through 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 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:
- is the square root of minus one, usually represented by in non-electrical formulas ( is used so as not to confuse the imaginary unit with current, commonly represented by ).

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 and inductive reactance contribute to the total reactance 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:^{ [3] }

- if , the total reactance is said to be inductive;
- if , then the impedance is purely resistive;
- 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:^{ [5] }

but the ultimate value is the same.

The phase of the voltage across a purely reactive device (a capacitor with an infinite resistance or an inductor with a resistance of zero) *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.

**Electrical impedance** is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term *complex impedance* may be used interchangeably.

In electrical engineering, **admittance** is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance. The SI unit of admittance is the siemens ; the older, synonymous unit is mho, and its symbol is ℧. Oliver Heaviside coined the term *admittance* in December 1887.

In electronics, a **voltage divider ** is a passive linear circuit that produces an output voltage (*V*_{out}) that is a fraction of its input voltage (*V*_{in}). **Voltage division** is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.

A **gyrator** is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two-(or-more)-port devices which cannot be realized with just the conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op-amps using feedback.

The **Smith chart**, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

The **Butterworth filter** is a type of signal processing filter designed to have a frequency response as flat as possible in the passband. It is also referred to as a **maximally flat magnitude filter**. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".

An **LC circuit**, also called a **resonant circuit**, **tank circuit**, or **tuned circuit**, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.

A **Colpitts oscillator**, invented in 1918 by American engineer Edwin H. Colpitts, is one of a number of designs for LC oscillators, electronic oscillators that use a combination of inductors (L) and capacitors (C) to produce an oscillation at a certain frequency. The distinguishing feature of the Colpitts oscillator is that the feedback for the active device is taken from a voltage divider made of two capacitors in series across the inductor.

**Power** in an electric circuit is the rate of flow of energy past a given point of the circuit. In alternating current circuits, energy storage elements such as inductors and capacitors may result in periodic reversals of the direction of energy flow.

In microwave and radio-frequency engineering, a **stub** or **resonant stub** is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit or short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus function as capacitors, inductors and resonant circuits at radio frequencies.

**Electrical resonance** occurs in an electric circuit at a particular *resonant frequency* when the impedances or admittances of circuit elements cancel each other. In some circuits, this happens when the impedance between the input and output of the circuit is almost zero and the transfer function is close to one.

**Foster's reactance theorem** is an important theorem in the fields of electrical network analysis and synthesis. The theorem states that the reactance of a passive, lossless two-terminal (one-port) network always strictly monotonically increases with frequency. It is easily seen that the reactances of inductors and capacitors individually increase with frequency and from that basis a proof for passive lossless networks generally can be constructed. The proof of the theorem was presented by Ronald Martin Foster in 1924, although the principle had been published earlier by Foster's colleagues at American Telephone & Telegraph.

**Ripple** in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.

A *capacitor* is a passive two-terminal electronic component that stores electrical energy in an electric field. The effect of a capacitor is known as capacitance. While some capacitance exists between any two electrical conductors in proximity in a circuit, a capacitor is a component designed to add capacitance to a circuit. The capacitor was originally known as a

In electronics, a **differentiator** is a circuit that is designed such that the output of the circuit is approximately directly proportional to the rate of change of the input. An active differentiator includes some form of amplifier. A **passive differentiator circuit** is made of only resistors and capacitors.

**Zobel networks** are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.

The **gyrator–capacitor model** is a lumped-element model for magnetic fields, similar to magnetic circuits, but based on using elements analogous to capacitors rather than elements analogous to resistors to represent the magnetic flux path. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

In an electric power transmission system, a **thyristor-controlled reactor** (TCR) is a reactance connected in series with a bidirectional thyristor valve. The thyristor valve is phase-controlled, which allows the value of delivered reactive power to be adjusted to meet varying system conditions. Thyristor-controlled reactors can be used for limiting voltage rises on lightly loaded transmission lines. Another device which used to be used for this purpose is a magnetically controlled reactor (MCR), a type of magnetic amplifier otherwise known as a transductor.

An **RLC circuit** is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

A **frequency-selective surface** (**FSS**) is any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field. In this sense, an FSS is a type of optical filter or metal-mesh optical filters in which the filtering is accomplished by virtue of the regular, periodic pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSS's also have properties which vary with incidence angle and polarization as well - these are unavoidable consequences of the way in which FSS's are constructed. Frequency-selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as the aforementioned microwave oven, antenna radomes and modern metamaterials. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as photonic crystals.

- 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.

- 1 2 Irwin, 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 - 1 2 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 - 1 2 Hughes 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

- Interactive Java Tutorial on Inductive Reactance National High Magnetic Field Laboratory
- Reactance calculator

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