Electrical reactance

Last updated

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. Larger reactance leads to smaller currents for the same voltage applied. Reactance is similar to electric resistance, but it differs in several respects.

Contents

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. As frequency goes up, inductive reactance also goes up and capacitive reactance goes down.

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 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.

Capacitive reactance

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 ω) and the capacitance . [1]

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

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.

Impedance

Both reactance and resistance are components of impedance .

where:

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:

Hence: [3]

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.

Phase relationship

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.

See also

Related Research Articles

Electrical impedance intensive physical property

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.

Voltage divider linear circuit that produces an output voltage that is a fraction of its input voltage

In electronics, a voltage divider is a passive linear circuit that produces an output voltage (Vout) that is a fraction of its input voltage (Vin). 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.

Gyrator analog circuit

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.

Smith chart graphical aid designed to assist in solving problems with transmission lines and matching circuits in radio engineering

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.

Antenna tuner Telecommunications device

Antenna tuner, matching network, matchbox, transmatch, antenna tuning unit (ATU), antenna coupler, and feedline coupler are all equivalent names for a device connected between a radio transmitter and its antenna, to improve power transfer between them by matching the specified load impedance of the radio to the combined input impedance of the feedline and the antenna.

LC circuit Electrical "resonator" circuit, consisting of inductive and capacitive elements with no resistance

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.

AC power product of the RMS values of voltage and electric current

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.

Stub (electronics) short electrical transmission line

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.

Q meter

A Q meter is a piece of equipment used in the testing of radio frequency circuits. It has been largely replaced in professional laboratories by other types of impedance measuring device, though it is still in use among radio amateurs. It was developed at Boonton Radio Corporation in Boonton, New Jersey in 1934 by William D. Loughlin.

Electrical resonance occurs in an electric circuit at a particular resonant frequency when the imaginary parts of impedances or admittances of circuit elements cancel each other

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.

Capacitor Passive two-terminal electronic component that stores electrical energy in an electric field

A capacitor is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals.

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. A true differentiator cannot be physically realized, because it has infinite gain at infinite frequency. A similar effect can be achieved, however, by limiting the gain above some frequency.
An active differentiator includes some form of amplifier, while a passive differentiator is made only of resistors, capacitors and inductors.

Zobel network type of filter section based on the image-impedance design principle

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 - sometimes also the capacitor-permeance 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.

Primary line constants

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

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.

References

  1. Shamieh C. and McComb G., Electronics for Dummies, John Wiley & Sons, 2011.
  2. Meade R., Foundations of Electronics, Cengage Learning, 2002.
  3. 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. 1 2 Irwin, D. (2002). Basic Engineering Circuit Analysis, page 274. New York: John Wiley & Sons, Inc.
  2. Hayt, W.H., Kimmerly J.E. (2007). Engineering Circuit Analysis, 7th ed., McGraw-Hill, p. 388
  3. 1 2 Glisson, T.H. (2011). Introduction to Circuit Analysis and Design, Springer, p. 408
  4. Horowitz P., Hill W. (2015). The Art of Electronics , 3rd ed., p. 42
  5. 1 2 Hughes E., Hiley J., Brown K., Smith I.McK., (2012). Hughes Electrical and Electronic Technology, 11th edition, Pearson, pp. 237-241
  6. Robbins, A.H., Miller W. (2012). Circuit Analysis: Theory and Practice, 5th ed., Cengage Learning, pp. 554-558