Nuclear reaction

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Nuclear_reaction

In this symbolic representing of a nuclear reaction, lithium-6 (⁶
₃Li
) and deuterium (²
₁H
) react to form the highly excited intermediate nucleus ⁸
₄Be
which then decays immediately into two alpha particles of helium-4 (⁴
₂He
). Protons are symbolically represented by red spheres, and neutrons by blue spheres.

In nuclear physics and nuclear chemistry, a nuclear reaction is a process in which two nuclei, or a nucleus and an external subatomic particle, collide to produce one or more new nuclides. Thus, a nuclear reaction must cause a transformation of at least one nuclide to another. If a nucleus interacts with another nucleus or particle and they then separate without changing the nature of any nuclide, the process is simply referred to as a type of nuclear scattering, rather than a nuclear reaction.

In principle, a reaction can involve more than two particles colliding, but because the probability of three or more nuclei to meet at the same time at the same place is much less than for two nuclei, such an event is exceptionally rare (see triple alpha process for an example very close to a three-body nuclear reaction). The term "nuclear reaction" may refer either to a change in a nuclide induced by collision with another particle or to a spontaneous change of a nuclide without collision.

Natural nuclear reactions occur in the interaction between cosmic rays and matter, and nuclear reactions can be employed artificially to obtain nuclear energy, at an adjustable rate, on-demand. Nuclear chain reactions in fissionable materials produce induced nuclear fission. Various nuclear fusion reactions of light elements power the energy production of the Sun and stars.

History

In 1919, Ernest Rutherford was able to accomplish transmutation of nitrogen into oxygen at the University of Manchester, using alpha particles directed at nitrogen ¹⁴N + α → ¹⁷O + p.  This was the first observation of an induced nuclear reaction, that is, a reaction in which particles from one decay are used to transform another atomic nucleus. Eventually, in 1932 at Cambridge University, a fully artificial nuclear reaction and nuclear transmutation was achieved by Rutherford's colleagues John Cockcroft and Ernest Walton, who used artificially accelerated protons against lithium-7, to split the nucleus into two alpha particles. The feat was popularly known as "splitting the atom", although it was not the modern nuclear fission reaction later (in 1938) discovered in heavy elements by the German scientists Otto Hahn, Lise Meitner, and Fritz Strassmann.

Nuclear reaction equations

Nuclear reactions may be shown in a form similar to chemical equations, for which invariant mass must balance for each side of the equation, and in which transformations of particles must follow certain conservation laws, such as conservation of charge and baryon number (total atomic mass number). An example of this notation follows:

6 3Li

+

2 1H

→

4 2He

+

?.

To balance the equation above for mass, charge and mass number, the second nucleus to the right must have atomic number 2 and mass number 4; it is therefore also helium-4. The complete equation therefore reads:

6 3Li

+

2 1H

→

4 2He

+

4 2He .

or more simply:

6 3Li

+

2 1H

→

2 4 2He .

Instead of using the full equations in the style above, in many situations a compact notation is used to describe nuclear reactions. This style of the form A(b,c)D is equivalent to A + b producing c + D. Common light particles are often abbreviated in this shorthand, typically p for proton, n for neutron, d for deuteron, α representing an alpha particle or helium-4, β for beta particle or electron, γ for gamma photon, etc. The reaction above would be written as ⁶Li(d,α)α.

Energy conservation

Kinetic energy may be released during the course of a reaction (exothermic reaction) or kinetic energy may have to be supplied for the reaction to take place (endothermic reaction). This can be calculated by reference to a table of very accurate particle rest masses, as follows: according to the reference tables, the ⁶
₃Li
nucleus has a standard atomic weight of 6.015 atomic mass units (abbreviated u), the deuterium has 2.014 u, and the helium-4 nucleus has 4.0026 u. Thus:

• the sum of the rest mass of the individual nuclei = 6.015 + 2.014 = 8.029 u;
• the total rest mass on the two helium-nuclei = 2 × 4.0026 = 8.0052 u;
• missing rest mass = 8.029 – 8.0052 = 0.0238 atomic mass units.

In a nuclear reaction, the total (relativistic) energy is conserved. The "missing" rest mass must therefore reappear as kinetic energy released in the reaction; its source is the nuclear binding energy. Using Einstein's mass-energy equivalence formula E = mc², the amount of energy released can be determined. We first need the energy equivalent of one atomic mass unit:

1 u c² = (1.66054 × 10⁻²⁷ kg) × (2.99792 × 10⁸ m/s)² 

= 1.49242 × 10⁻¹⁰ kg (m/s)² = 1.49242 × 10⁻¹⁰ J (joule) × (1 MeV / 1.60218 × 10⁻¹³ J)

= 931.49 MeV,

so 1 u c² = 931.49 MeV.

Hence, the energy released is 0.0238 × 931 MeV = 22.2 MeV.

Expressed differently: the mass is reduced by 0.3%, corresponding to 0.3% of 90 PJ/kg is 270 TJ/kg.

This is a large amount of energy for a nuclear reaction; the amount is so high because the binding energy per nucleon of the helium-4 nucleus is unusually high because the He-4 nucleus is "doubly magic". (The He-4 nucleus is unusually stable and tightly bound for the same reason that the helium atom is inert: each pair of protons and neutrons in He-4 occupies a filled 1s nuclear orbital in the same way that the pair of electrons in the helium atom occupy a filled 1s electron orbital). Consequently, alpha particles appear frequently on the right-hand side of nuclear reactions.

The energy released in a nuclear reaction can appear mainly in one of three ways:

• kinetic energy of the product particles (fraction of the kinetic energy of the charged nuclear reaction products can be directly converted into electrostatic energy);
• emission of very high energy photons, called gamma rays;
• some energy may remain in the nucleus, as a metastable energy level.

When the product nucleus is metastable, this is indicated by placing an asterisk ("*") next to its atomic number. This energy is eventually released through nuclear decay.

A small amount of energy may also emerge in the form of X-rays. Generally, the product nucleus has a different atomic number, and thus the configuration of its electron shells is wrong. As the electrons rearrange themselves and drop to lower energy levels, internal transition X-rays (X-rays with precisely defined emission lines) may be emitted.

Q-value and energy balance

In writing down the reaction equation, in a way analogous to a chemical equation, one may, in addition, give the reaction energy on the right side:

Target nucleus + projectile → Final nucleus + ejectile + Q.

For the particular case discussed above, the reaction energy has already been calculated as Q = 22.2 MeV. Hence:

6 3Li

+

2 1H

→

2 4 2He

+

22.2 MeV.

The reaction energy (the "Q-value") is positive for exothermal reactions and negative for endothermal reactions, opposite to the similar expression in chemistry. On the one hand, it is the difference between the sums of kinetic energies on the final side and on the initial side. But on the other hand, it is also the difference between the nuclear rest masses on the initial side and on the final side (in this way, we have calculated the Q-value above).

Reaction rates

If the reaction equation is balanced, that does not mean that the reaction really occurs. The rate at which reactions occur depends on the energy and the flux of the incident particles, and the reaction cross section. An example of a large repository of reaction rates is the REACLIB database, as maintained by the Joint Institute for Nuclear Astrophysics.

Charged vs. uncharged particles

In the initial collision which begins the reaction, the particles must approach closely enough so that the short-range strong force can affect them. As most common nuclear particles are positively charged, this means they must overcome considerable electrostatic repulsion before the reaction can begin. Even if the target nucleus is part of a neutral atom, the other particle must penetrate well beyond the electron cloud and closely approach the nucleus, which is positively charged. Thus, such particles must be first accelerated to high energy, for example by:

• particle accelerators;
• nuclear decay (alpha particles are the main type of interest here since beta and gamma rays are rarely involved in nuclear reactions);
• very high temperatures, on the order of millions of degrees, producing thermonuclear reactions;
• cosmic rays.

Also, since the force of repulsion is proportional to the product of the two charges, reactions between heavy nuclei are rarer, and require higher initiating energy, than those between a heavy and light nucleus; while reactions between two light nuclei are the most common ones.

Neutrons, on the other hand, have no electric charge to cause repulsion, and are able to initiate a nuclear reaction at very low energies. In fact, at extremely low particle energies (corresponding, say, to thermal equilibrium at room temperature), the neutron's de Broglie wavelength is greatly increased, possibly greatly increasing its capture cross-section, at energies close to resonances of the nuclei involved. Thus low-energy neutrons may be even more reactive than high-energy neutrons.

Notable types

While the number of possible nuclear reactions is immense, there are several types that are more common, or otherwise notable. Some examples include:

• Fusion reactions — two light nuclei join to form a heavier one, with additional particles (usually protons or neutrons) emitted subsequently.
• Spallation — a nucleus is hit by a particle with sufficient energy and momentum to knock out several small fragments or smash it into many fragments.
• Induced gamma emission belongs to a class in which only photons were involved in creating and destroying states of nuclear excitation.
• Alpha decay — Though driven by the same underlying forces as spontaneous fission, α decay is usually considered to be separate from the latter. The often-quoted idea that "nuclear reactions" are confined to induced processes is incorrect. "Radioactive decays" are a subgroup of "nuclear reactions" that are spontaneous rather than induced. For example, so-called "hot alpha particles" with unusually high energies may actually be produced in induced ternary fission, which is an induced nuclear reaction (contrasting with spontaneous fission). Such alphas occur from spontaneous ternary fission as well.
• Fission reactions — a very heavy nucleus, after absorbing additional light particles (usually neutrons), splits into two or sometimes three pieces. This is an induced nuclear reaction. Spontaneous fission, which occurs without assistance of a neutron, is usually not considered a nuclear reaction. At most, it is not an induced nuclear reaction.

Direct reactions

An intermediate energy projectile transfers energy or picks up or loses nucleons to the nucleus in a single quick (10⁻²¹ second) event. Energy and momentum transfer are relatively small. These are particularly useful in experimental nuclear physics, because the reaction mechanisms are often simple enough to calculate with sufficient accuracy to probe the structure of the target nucleus.

Inelastic scattering

Main article: Inelastic scattering

Only energy and momentum are transferred.

• (p,p') tests differences between nuclear states.
• (α,α') measures nuclear surface shapes and sizes. Since α particles that hit the nucleus react more violently, elastic and shallow inelastic α scattering are sensitive to the shapes and sizes of the targets, like light scattered from a small black object.
• (e,e') is useful for probing the interior structure. Since electrons interact less strongly than do protons and neutrons, they reach to the centers of the targets and their wave functions are less distorted by passing through the nucleus.

Charge-exchange reactions

Energy and charge are transferred between projectile and target. Some examples of this kind of reactions are:

• (p,n)
• (³He,t)

Nucleon transfer reactions

Usually at moderately low energy, one or more nucleons are transferred between the projectile and target. These are useful in studying outer shell structure of nuclei. Transfer reactions can occur, from the projectile to the target; stripping reactions, or from the target to the projectile; pick-up reactions.

• (α,n) and (α,p) reactions. Some of the earliest nuclear reactions studied involved an alpha particle produced by alpha decay, knocking a nucleon from a target nucleus.
• (d,n) and (d,p) reactions. A deuteron beam impinges on a target; the target nuclei absorb either the neutron or proton from the deuteron. The deuteron is so loosely bound that this is almost the same as proton or neutron capture. A compound nucleus may be formed, leading to additional neutrons being emitted more slowly. (d,n) reactions are used to generate energetic neutrons.
• The strangeness exchange reaction (K, π) has been used to study hypernuclei.
• The reaction ¹⁴N(α,p)¹⁷O performed by Rutherford in 1917 (reported 1919), is generally regarded as the first nuclear transmutation experiment.

Reactions with neutrons

	→ T	→ 7Li	→ 14C
(n,α)	6Li + n → T + α	10B + n → 7Li + α	17O + n → 14C + α	21Ne + n → 18O + α	37Ar + n → 34S + α
(n,p)	3He + n → T + p	7Be + n → 7Li + p	14N + n → 14C + p	22Na + n → 22Ne + p
(n,γ)	2H + n → T + γ		13C + n → 14C + γ

Reactions with neutrons are important in nuclear reactors and nuclear weapons. While the best-known neutron reactions are neutron scattering, neutron capture, and nuclear fission, for some light nuclei (especially odd-odd nuclei) the most probable reaction with a thermal neutron is a transfer reaction:

Some reactions are only possible with fast neutrons:

• (n,2n) reactions produce small amounts of protactinium-231 and uranium-232 in the thorium cycle which is otherwise relatively free of highly radioactive actinide products.
• ⁹Be + n → 2α + 2n can contribute some additional neutrons in the beryllium neutron reflector of a nuclear weapon.
• ⁷Li + n → T + α + n unexpectedly contributed additional yield in the Bravo, Romeo and Yankee shots of Operation Castle, the three highest-yield nuclear tests conducted by the U.S.

Compound nuclear reactions

Either a low-energy projectile is absorbed or a higher energy particle transfers energy to the nucleus, leaving it with too much energy to be fully bound together. On a time scale of about 10⁻¹⁹ seconds, particles, usually neutrons, are "boiled" off. That is, it remains together until enough energy happens to be concentrated in one neutron to escape the mutual attraction. The excited quasi-bound nucleus is called a compound nucleus.

• Low energy (e, e' xn), (γ, xn) (the xn indicating one or more neutrons), where the gamma or virtual gamma energy is near the giant dipole resonance. These increase the need for radiation shielding around electron accelerators.

Further information: Spallation § Nuclear spallation

Information warfare

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Information_warfare 

Information warfare (IW) (as different from cyber warfare that attacks computers, software, and command control systems) is a concept involving the battlespace use and management of information and communication technology (ICT) in pursuit of a competitive advantage over an opponent. Information warfare is the manipulation of information trusted by a target without the target's awareness so that the target will make decisions against their interest but in the interest of the one conducting information warfare. As a result, it is not clear when information warfare begins, ends, and how strong or destructive it is. Information warfare may involve the collection of tactical information, assurance(s) that one's information is valid, spreading of propaganda or disinformation to demoralize or manipulate the enemy and the public, undermining the quality of the opposing force's information and denial of information-collection opportunities to opposing forces. Information warfare is closely linked to psychological warfare.

The United States military focus tends to favor technology and hence tends to extend into the realms of electronic warfare, cyberwarfare, information assurance and computer network operations, attack, and defense.

Most of the rest of the world use the much broader term of "Information Operations" which, although making use of technology, focuses on the more human-related aspects of information use, including (amongst many others) social network analysis, decision analysis, and the human aspects of command and control.

Overview

Information war has been described as "the use of information to achieve our national objectives." According to NATO, "Information war is an operation conducted in order to gain an information advantage over the opponent."

Information warfare can take many forms:

• Television, internet and radio transmission(s) can be jammed.
• Television, internet and radio transmission(s) can be hijacked for a disinformation campaign.
• Logistics networks can be disabled.
• Enemy communications networks can be disabled or spoofed, especially online social community in modern days.
• Stock exchange transactions can be sabotaged, either with electronic intervention, by leaking sensitive information or by placing disinformation.
• The use of drones and other surveillance robots or webcams.
• Communication management
• Synthetic media
• The organised use of social media and other online content generation platforms can be used for opinion engineering among masses.

The U.S. Air Force has had Information Warfare Squadrons since the 1980s. In fact, the official mission of the U.S. Air Force is now "To fly, fight and win...in air, space and cyberspace", with the latter referring to its information warfare role.

As the U.S. Air Force often risks aircraft and aircrews to attack strategic enemy communications targets, remotely disabling such targets using software and other means can provide a safer alternative. In addition, disabling such networks electronically (instead of explosively) also allows them to be quickly re-enabled after the enemy territory is occupied. Similarly, counter-information warfare units are employed to deny such capability to the enemy. The first application of these techniques was used against Iraqi communications networks in the Gulf War.

Also during the Gulf War, Dutch hackers allegedly stole information about U.S. troop movements from U.S. Defense Department computers and tried to sell it to the Iraqis, who thought it was a hoax and turned it down. In January 1999, U.S. Air Intelligence computers were hit by a coordinated attack (Moonlight Maze), part of which came from a Russian mainframe. This could not be confirmed as a Russian cyber attack due to non-attribution – the principle that online identity may not serve as proof of real world identity.

New battlefield

The innovation of more advanced and autonomous ICTs has engendered a new revolution in military affairs, which encompasses nations' use of ICTs in both cyberspace and the physical battlefield to wage war against their adversaries. The three most prevalent revolutions in military affairs come in the form of cyberattacks, autonomous robots and communication management.

Within the realm of cyberspace, there are two primary weapons: network-centric warfare and C4ISR, which denotes integrated Command, Control, Communications, Computers, Intelligence, Surveillance and Reconnaissance. Furthermore, cyberspace attacks initiated by one nation against another nation have an underlying goal of gaining information superiority over the attacked party, which includes disrupting or denying the victimized party's ability to gather and distribute information. A real-world occurrence that illustrated the dangerous potential of cyberattacks transpired in 2007, when a strike from Israeli forces demolished an alleged nuclear reactor in Syria that was being constructed via a collaborative effort between Syria and North Korea. Accompanied with the strike was a cyberattack on Syria's air defenses, which left them blind to the attack on the nuclear reactor and, ultimately allowed for the attack to occur (New York Times 2014). An example of a more basic attack on a nation within cyberspace is a distributed denial of service (DDOS) attack, which is utilized to hinder networks or websites until they lose their primary functionality. As implied, cyberattacks do not just affect the military party being attacked, but rather the whole population of the victimized nation. Since more aspects of daily life are being integrated into networks in cyberspace, civilian populations can potentially be negatively affected during wartime. For example, if a nation chose to attack another nation's power grid servers in a specific area to disrupt communications, civilians and businesses in that area would also have to deal with power outages, which could potentially lead to economic disruptions as well.

Moreover, physical ICTs have also been implemented into the latest revolution in military affairs by deploying new, more autonomous robots (i.e. – unmanned drones) into the battlefield to carry out duties such as patrolling borders and attacking ground targets. Humans from remote locations pilot many of the unmanned drones, however, some of the more advanced robots, such as the Northrop Grumman X-47B, are capable of autonomous decisions. Despite piloting the drones from remote locations, a proportion of drone pilots still suffer from stress factors of more traditional warfare. According to NPR, a study performed by the Pentagon in 2011 found that 29% of drone pilots are “burned out” and undergo high levels of stress. Furthermore, approximately 17% of the drone pilots surveyed as the study were labeled "clinically distressed" with some of those pilots also showing signs of post-traumatic stress disorder.

Modern ICTs have also brought advancements to communications management among military forces. Communication is a vital aspect of war for any involved party and, through the implementation of new ICTs such as data-enabled devices, military forces are now able to disseminate information faster than ever before. For example, some militaries are now employing the use of iPhones to upload data and information gathered by drones in the same area.

Notable examples

Russo-Ukrainian War

Main articles: Russian information war against Ukraine and Disinformation in the 2021–2022 Russo-Ukrainian crisis

In 2022 Ukrainian forces have taken advantage of deficiencies in Russian communications by allowing them to piggyback on Ukrainian networks, connect, and communicate. Ukrainian forces then eavesdrop, and cut off Russian communications at a crucial part of the conversation.

In an effort to build support prior to its invasion of Ukraine, Russia perpetuated a narrative that claimed the Ukrainian government was committing violence against its own Russian speaking population. By publishing large amounts of disinformation on the internet, the alternate narrative was picked up in search results, such as Google News.

Russian interference in foreign elections

Russian interference in foreign elections, most notably the Russian interference in the 2016 United States elections, has been described as information warfare.

Legal and ethical concerns

While information warfare has yielded many advances in the types of attack that a government can make, it has also raised concerns about the moral and legal ambiguities surrounding this particularly new form of war. Traditionally, wars have been analyzed by moral scholars according to just war theory. However, with Information Warfare, Just War Theory fails because the theory is based on the traditional conception of war. Information Warfare has three main issues surrounding it compared to traditional warfare:

1. The risk for the party or nation initiating the cyberattack is substantially lower than the risk for a party or nation initiating a traditional attack. This makes it easier for governments, as well as potential terrorist or criminal organizations, to make these attacks more frequently than they could with traditional war.
2. Information communication technologies (ICT) are so immersed in the modern world that a very wide range of technologies are at risk of a cyberattack. Specifically, civilian technologies can be targeted for cyberattacks and attacks can even potentially be launched through civilian computers or websites. As such, it is harder to enforce control of civilian infrastructures than a physical space. Attempting to do so would also raise many ethical concerns about the right to privacy, making defending against such attacks even tougher.
3. The mass-integration of ICT into our system of war makes it much harder to assess accountability for situations that may arise when using robotic and/or cyber attacks. For robotic weapons and automated systems, it's becoming increasingly hard to determine who is responsible for any particular event that happens. This issue is exacerbated in the case of cyberattacks, as sometimes it is virtually impossible to trace who initiated the attack in the first place.

Recently, legal concerns have arisen centered on these issues, specifically the issue of the right to privacy in the United States of America. Lt. General Keith B. Alexander, who served as the head of Cyber Command under President Barack Obama, noted that there was a "mismatch between our technical capabilities to conduct operations and the governing laws and policies" when writing to the Senate Armed Services Committee. A key point of concern was the targeting of civilian institutions for cyberattacks, to which the general promised to try to maintain a mindset similar to that of traditional war, in which they will seek to limit the impact on civilians.

AC power

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/AC_power 

 

The blinking of non-incandescent city lights is shown in this motion-blurred long exposure. The AC nature of the mains power is revealed by the dashed appearance of the traces of moving lights.

In an electric circuit, instantaneous power is the time 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. Its SI unit is the watt.

The portion of instantaneous power that, averaged over a complete cycle of the AC waveform, results in net transfer of energy in one direction is known as instantaneous active power, and its time average is known as active power or real power. The portion of instantaneous power that results in no net transfer of energy but instead oscillates between the source and load in each cycle due to stored energy, is known as instantaneous reactive power, and its amplitude is the absolute value of reactive power.

Active, reactive, apparent, and complex power in sinusoidal steady-state

In a simple alternating current (AC) circuit consisting of a source and a linear time-invariant load, both the current and voltage are sinusoidal at the same frequency. If the load is purely resistive, the two quantities reverse their polarity at the same time. At every instant the product of voltage and current is positive or zero, the result being that the direction of energy flow does not reverse. In this case, only active power is transferred.

If the load is purely reactive, then the voltage and current are 90 degrees out of phase. For two quarters of each cycle, the product of voltage and current is positive, but for the other two quarters, the product is negative, indicating that on average, exactly as much energy flows into the load as flows back out. There is no net energy flow over each half cycle. In this case, only reactive power flows: There is no net transfer of energy to the load; however, electrical power does flow along the wires and returns by flowing in reverse along the same wires. The current required for this reactive power flow dissipates energy in the line resistance, even if the ideal load device consumes no energy itself. Practical loads have resistance as well as inductance, or capacitance, so both active and reactive powers will flow to normal loads.

Apparent power is the product of the RMS values of voltage and current. Apparent power is taken into account when designing and operating power systems, because although the current associated with reactive power does no work at the load, it still must be supplied by the power source. Conductors, transformers and generators must be sized to carry the total current, not just the current that does useful work. Failure to provide for the supply of sufficient reactive power in electrical grids can lead to lowered voltage levels and, under certain operating conditions, to the complete collapse of the network or blackout. Another consequence is that adding the apparent power for two loads will not accurately give the total power unless they have the same phase difference between current and voltage (the same power factor).

Conventionally, capacitors are treated as if they generate reactive power, and inductors are treated as if they consume it. If a capacitor and an inductor are placed in parallel, then the currents flowing through the capacitor and the inductor tend to cancel rather than add. This is the fundamental mechanism for controlling the power factor in electric power transmission; capacitors (or inductors) are inserted in a circuit to partially compensate for reactive power 'consumed' ('generated') by the load. Purely capacitive circuits supply reactive power with the current waveform leading the voltage waveform by 90 degrees, while purely inductive circuits absorb reactive power with the current waveform lagging the voltage waveform by 90 degrees. The result of this is that capacitive and inductive circuit elements tend to cancel each other out.

The Power Triangle
The complex power is the vector sum of active and reactive power. The apparent power is the magnitude of the complex power.
  Active power, P
  Reactive power, Q
  Complex power, S
  Apparent power, |S|
  Phase of voltage relative to current, ${\displaystyle \phi }$

Engineers use the following terms to describe energy flow in a system (and assign each of them a different unit to differentiate between them):

• Active power, P, or real power: watt (W);
• Reactive power, Q: volt-ampere reactive (var);
• Complex power, S: volt-ampere (VA);
• Apparent power, |S|: the magnitude of complex power S: volt-ampere (VA);
• Phase of voltage relative to current, φ: the angle of difference (in degrees) between current and voltage; ${\displaystyle \phi =\arg (V)-\arg (I)}$. Current lagging voltage (quadrant I vector), current leading voltage (quadrant IV vector).

These are all denoted in the adjacent diagram (called a Power Triangle).

In the diagram, P is the active power, Q is the reactive power (in this case positive), S is the complex power and the length of S is the apparent power. Reactive power does not do any work, so it is represented as the imaginary axis of the vector diagram. Active power does do work, so it is the real axis.

The unit for power is the watt (symbol: W). Apparent power is often expressed in volt-amperes (VA) since it is the product of RMS voltage and RMS current. The unit for reactive power is var, which stands for volt-ampere reactive. Since reactive power transfers no net energy to the load, it is sometimes called "wattless" power. It does, however, serve an important function in electrical grids and its lack has been cited as a significant factor in the Northeast Blackout of 2003. Understanding the relationship among these three quantities lies at the heart of understanding power engineering. The mathematical relationship among them can be represented by vectors or expressed using complex numbers, S = P + j Q (where j is the imaginary unit).

Instantaneous power in AC systems when the current lags behind the voltage by 50 degrees.

Calculations and equations in sinusoidal steady-state

The formula for complex power (units: VA) in phasor form is:

${\displaystyle S=V{I}^{\ast }=|S|\mathrm{\angle }\phi }$,

where V denotes voltage in phasor form, with the amplitude as RMS, and I denotes current in phasor form, with the amplitude as RMS. Also by convention, the complex conjugate of I is used, which is denoted ${\displaystyle I^{\ast }}$ (or ${\displaystyle \overline{I}}$), rather than I itself. This is done because otherwise using the product V I to define S would result in a quantity that depends on the reference angle chosen for V or I, but defining S as V I* results in a quantity that doesn't depend on the reference angle and allows to relate S to P and Q.

Other forms of complex power (units in volt-amps, VA) are derived from Z, the load impedance (units in ohms, Ω).

${\displaystyle S=|I{|}^{2}Z=\frac{|V{|}^2}{Z^{\ast }}}$.

Consequentially, with reference to the power triangle, real power (units in watts, W) is derived as:

${\displaystyle P=|S|\cos \phi =|I{|}^{2}R=\frac{|V{|}^2}{|Z{|}^2}\times R}$.

For a purely resistive load, real power can be simplified to:

${\displaystyle P=\frac{|V{|}^2}{R}}$.

R denotes resistance (units in ohms, Ω) of the load.

Reactive power (units in volts-amps-reactive, var) is derived as:

${\displaystyle Q=|S|\sin \phi =|I{|}^{2}X=\frac{|V{|}^2}{|Z{|}^2}\times X}$.

For a purely reactive load, reactive power can be simplified to:

${\displaystyle Q=\frac{|V{|}^2}{X}}$,

where X denotes reactance (units in ohms, Ω) of the load.

Combining, the complex power (units in volt-amps, VA) is back-derived as

${\displaystyle S=P+jQ}$,

and the apparent power (units in volt-amps, VA) as

${\displaystyle |S|=\sqrt{P^2+Q^2}}$.

These are simplified diagrammatically by the power triangle.

Power factor

Main article: Power factor

The ratio of active power to apparent power in a circuit is called the power factor. For two systems transmitting the same amount of active power, the system with the lower power factor will have higher circulating currents due to energy that returns to the source from energy storage in the load. These higher currents produce higher losses and reduce overall transmission efficiency. A lower power factor circuit will have a higher apparent power and higher losses for the same amount of active power. The power factor is 1.0 when the voltage and current are in phase. It is zero when the current leads or lags the voltage by 90 degrees. When the voltage and current are 180 degrees out of phase, the power factor is negative one, and the load is feeding energy into the source (an example would be a home with solar cells on the roof that feed power into the power grid when the sun is shining). Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle of current with respect to voltage. Voltage is designated as the base to which current angle is compared, meaning that current is thought of as either "leading" or "lagging" voltage. Where the waveforms are purely sinusoidal, the power factor is the cosine of the phase angle (${\displaystyle \phi }$) between the current and voltage sinusoidal waveforms. Equipment data sheets and nameplates will often abbreviate power factor as "${\displaystyle \cos \varphi }$" for this reason.

Example: The active power is 700 W and the phase angle between voltage and current is 45.6°. The power factor is cos(45.6°) = 0.700. The apparent power is then: 700 W / cos(45.6°) = 1000 VA. The concept of power dissipation in AC circuit is explained and illustrated with the example.

For instance, a power factor of 0.68 means that only 68 percent of the total current supplied (in magnitude) is actually doing work; the remaining current does no work at the load.

Reactive power

In a direct current circuit, the power flowing to the load is proportional to the product of the current through the load and the potential drop across the load. Energy flows in one direction from the source to the load. In AC power, the voltage and current both vary approximately sinusoidally. When there is inductance or capacitance in the circuit, the voltage and current waveforms do not line up perfectly. The power flow has two components – one component flows from source to load and can perform work at the load; the other portion, known as "reactive power", is due to the delay between voltage and current, known as phase angle, and cannot do useful work at the load. It can be thought of as current that is arriving at the wrong time (too late or too early). To distinguish reactive power from active power, it is measured in units of "volt-amperes reactive", or var. These units can simplify to watts but are left as var to denote that they represent no actual work output.

Energy stored in capacitive or inductive elements of the network gives rise to reactive power flow. Reactive power flow strongly influences the voltage levels across the network. Voltage levels and reactive power flow must be carefully controlled to allow a power system to be operated within acceptable limits. A technique known as reactive compensation is used to reduce apparent power flow to a load by reducing reactive power supplied from transmission lines and providing it locally. For example, to compensate an inductive load, a shunt capacitor is installed close to the load itself. This allows all reactive power needed by the load to be supplied by the capacitor and not have to be transferred over the transmission lines. This practice saves energy because it reduces the amount of energy that is required to be produced by the utility to do the same amount of work. Additionally, it allows for more efficient transmission line designs using smaller conductors or fewer bundled conductors and optimizing the design of transmission towers.

Capacitive vs. inductive loads

Stored energy in the magnetic or electric field of a load device, such as a motor or capacitor, causes an offset between the current and the voltage waveforms. A capacitor is a device that stores energy in the form of an electric field. As current is driven through the capacitor, charge build-up causes an opposing voltage to develop across the capacitor. This voltage increases until some maximum dictated by the capacitor structure. In an AC network, the voltage across a capacitor is constantly changing. The capacitor opposes this change, causing the current to lead the voltage in phase. Capacitors are said to "source" reactive power, and thus to cause a leading power factor.

Induction machines are some of the most common types of loads in the electric power system today. These machines use inductors, or large coils of wire to store energy in the form of a magnetic field. When a voltage is initially placed across the coil, the inductor strongly resists this change in a current and magnetic field, which causes a time delay for the current to reach its maximum value. This causes the current to lag behind the voltage in phase. Inductors are said to "sink" reactive power, and thus to cause a lagging power factor. Induction generators can source or sink reactive power, and provide a measure of control to system operators over reactive power flow and thus voltage. Because these devices have opposite effects on the phase angle between voltage and current, they can be used to "cancel out" each other's effects. This usually takes the form of capacitor banks being used to counteract the lagging power factor caused by induction motors.

Reactive power control

Main article: Voltage control and reactive power management

Transmission connected generators are generally required to support reactive power flow. For example, on the United Kingdom transmission system, generators are required by the Grid Code Requirements to supply their rated power between the limits of 0.85 power factor lagging and 0.90 power factor leading at the designated terminals. The system operator will perform switching actions to maintain a secure and economical voltage profile while maintaining a reactive power balance equation:

${\displaystyle \mathrm{G}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{M}\mathrm{V}\mathrm{A}\mathrm{R}\mathrm{s}+\mathrm{S}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{g}\mathrm{a}\mathrm{i}\mathrm{n}+\mathrm{S}\mathrm{h}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}=\mathrm{M}\mathrm{V}\mathrm{A}\mathrm{R}\mathrm{D}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d}+\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{s}+\mathrm{S}\mathrm{h}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}$

The "system gain" is an important source of reactive power in the above power balance equation, which is generated by the capacitative nature of the transmission network itself. By making decisive switching actions in the early morning before the demand increases, the system gain can be maximized early on, helping to secure the system for the whole day. To balance the equation some pre-fault reactive generator use will be required. Other sources of reactive power that will also be used include shunt capacitors, shunt reactors, static VAR compensators and voltage control circuits.

Unbalanced sinusoidal polyphase systems

While active power and reactive power are well defined in any system, the definition of apparent power for unbalanced polyphase systems is considered to be one of the most controversial topics in power engineering. Originally, apparent power arose merely as a figure of merit. Major delineations of the concept are attributed to Stanley's Phenomena of Retardation in the Induction Coil (1888) and Steinmetz's Theoretical Elements of Engineering (1915). However, with the development of three phase power distribution, it became clear that the definition of apparent power and the power factor could not be applied to unbalanced polyphase systems. In 1920, a "Special Joint Committee of the AIEE and the National Electric Light Association" met to resolve the issue. They considered two definitions.

${\displaystyle S_A=|S_{\mathrm{a}}|+|S_{\mathrm{b}}|+|S_{\mathrm{c}}|}$

${\displaystyle {\mathrm{p}\mathrm{f}}_A=\frac{P_{\mathrm{a}}+P_{\mathrm{b}}+P_{\mathrm{c}}}{S_A}}$,

that is, the arithmetic sum of the phase apparent powers; and

${\displaystyle S_V=|P_{\mathrm{a}}+P_{\mathrm{b}}+P_{\mathrm{c}}+j(Q_{\mathrm{a}}+Q_{\mathrm{b}}+Q_{\mathrm{c}})|}$

${\displaystyle {\mathrm{p}\mathrm{f}}_V=\frac{P_{\mathrm{a}}+P_{\mathrm{b}}+P_{\mathrm{c}}}{S_V}}$,

that is, the magnitude of total three-phase complex power.

The 1920 committee found no consensus and the topic continued to dominate discussions. In 1930, another committee formed and once again failed to resolve the question. The transcripts of their discussions are the lengthiest and most controversial ever published by the AIEE. Further resolution of this debate did not come until the late 1990s.

A new definition based on symmetrical components theory was proposed in 1993 by Alexander Emanuel for unbalanced linear load supplied with asymmetrical sinusoidal voltages:

${\displaystyle S=\sqrt{\left(|V_{\mathrm{a}}^2|+|V_{\mathrm{b}}^2|+|V_{\mathrm{c}}^2|\right)\left(|I_{\mathrm{a}}^2|+|I_{\mathrm{b}}^2|+|I_{\mathrm{c}}^2|\right)}}$

${\displaystyle \mathrm{p}\mathrm{f}=\frac{P^{+}}{S}}$,

that is, the root of squared sums of line voltages multiplied by the root of squared sums of line currents. ${\displaystyle P^{+}}$ denotes the positive sequence power:

${\displaystyle P^{+}=3|V^{+}||I^{+}|\cos (\arg (V^{+})-\arg (I^{+}))}$

${\displaystyle V^{+}}$ denotes the positive sequence voltage phasor, and ${\displaystyle I^{+}}$ denotes the positive sequence current phasor.

Real number formulas

A perfect resistor stores no energy; so current and voltage are in phase. Therefore, there is no reactive power and ${\displaystyle P=S}$ (using the passive sign convention). Therefore, for a perfect resistor

${\displaystyle P=S=V_{\mathrm{R}\mathrm{M}\mathrm{S}}{I}_{\mathrm{R}\mathrm{M}\mathrm{S}}=I_{\mathrm{R}\mathrm{M}\mathrm{S}}^{2}R=\frac{V_{\mathrm{R}\mathrm{M}\mathrm{S}}^2}{R}}$.

For a perfect capacitor or inductor, there is no net power transfer; so all power is reactive. Therefore, for a perfect capacitor or inductor:

${\displaystyle \begin{array}{rl}P& =0\\ Q& =|S|=V_{\mathrm{R}\mathrm{M}\mathrm{S}}{I}_{\mathrm{R}\mathrm{M}\mathrm{S}}=I_{\mathrm{R}\mathrm{M}\mathrm{S}}^2|X|=\frac{V_{\mathrm{R}\mathrm{M}\mathrm{S}}^2}{|X|}\end{array}}$.

where ${\displaystyle X}$ is the reactance of the capacitor or inductor.

If ${\displaystyle X}$ is defined as being positive for an inductor and negative for a capacitor, then the modulus signs can be removed from S and X and get

${\displaystyle Q=I_{\mathrm{R}\mathrm{M}\mathrm{S}}^{2}X=\frac{V_{\mathrm{R}\mathrm{M}\mathrm{S}}^2}{X}}$.

Instantaneous power is defined as:

${\displaystyle p(t)=v(t)i(t)}$,

where ${\displaystyle v(t)}$ and ${\displaystyle i(t)}$ are the time-varying voltage and current waveforms.

This definition is useful because it applies to all waveforms, whether they are sinusoidal or not. This is particularly useful in power electronics, where non-sinusoidal waveforms are common.

In general, engineers are interested in the active power averaged over a period of time, whether it is a low frequency line cycle or a high frequency power converter switching period. The simplest way to get that result is to take the integral of the instantaneous calculation over the desired period:

${\displaystyle P_{\text{avg}}=\frac{1}{t_2-t_1}{\int }_{t_1}^{t_2}v(t)i(t)\mathrm{d}t}$.

This method of calculating the average power gives the active power regardless of harmonic content of the waveform. In practical applications, this would be done in the digital domain, where the calculation becomes trivial when compared to the use of rms and phase to determine active power:

${\displaystyle P_{\text{avg}}=\frac{1}{n}\sum _{k=1}^{n}V[k]I[k]}$.

Multiple frequency systems

Since an RMS value can be calculated for any waveform, apparent power can be calculated from this. For active power it would at first appear that it would be necessary to calculate many product terms and average all of them. However, looking at one of these product terms in more detail produces a very interesting result.

${\displaystyle \begin{array}{rl}& A\cos ({\omega }_{1}t+k_1)\cos ({\omega }_{2}t+k_2)\\ =& \frac{A}{2}\cos \left[\left({\omega }_{1}t+k_1\right)+\left({\omega }_{2}t+k_2\right)\right]+\frac{A}{2}\cos \left[\left({\omega }_{1}t+k_1\right)-\left({\omega }_{2}t+k_2\right)\right]\\ =& \frac{A}{2}\cos \left[\left({\omega }_1+{\omega }_2\right)t+k_1+k_2\right]+\frac{A}{2}\cos \left[\left({\omega }_1-{\omega }_2\right)t+k_1-k_2\right]\end{array}}$

However, the time average of a function of the form cos(ωt + k) is zero provided that ω is nonzero. Therefore, the only product terms that have a nonzero average are those where the frequency of voltage and current match. In other words, it is possible to calculate active (average) power by simply treating each frequency separately and adding up the answers. Furthermore, if voltage of the mains supply is assumed to be a single frequency (which it usually is), this shows that harmonic currents are a bad thing. They will increase the RMS current (since there will be non-zero terms added) and therefore apparent power, but they will have no effect on the active power transferred. Hence, harmonic currents will reduce the power factor. Harmonic currents can be reduced by a filter placed at the input of the device. Typically this will consist of either just a capacitor (relying on parasitic resistance and inductance in the supply) or a capacitor-inductor network. An active power factor correction circuit at the input would generally reduce the harmonic currents further and maintain the power factor closer to unity.