Usually hidden to the unaided eye, the 60Hz blinking of (non-incandescent) lighting powered by AC mains is revealed in this motion-blurred long exposure of city lights.

Power is defined as the rate of flow of energy past a given point. In alternating current circuits, energy storage elements such as inductance and capacitance may result in periodic reversals of the direction of energy flow. The portion of power flow that, averaged over a complete cycle of the AC waveform, results in net transfer of energy in one direction is known as real power. That portion of power flow due to stored energy, that returns to the source in each cycle, is known as reactive power. Image File history File links Download high resolution version (2048x1536, 707 KB) The city lights of Prince George, British Columbia, motion-blurred by waving the camera during the exposure. ... Image File history File links Download high resolution version (2048x1536, 707 KB) The city lights of Prince George, British Columbia, motion-blurred by waving the camera during the exposure. ... // Mechanical power In physics, power (symbol: P) is the amount of work W done per unit of time t. ... city lights viewed in a motion blurred exposure. ... // Definition Inductance is a measure of the amount of magnetic flux produced for a given electric current. ... Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric potential. ...

Mostly this article will concentrate on single-frequency systems. Unfortunately, unlike voltage and current, power cannot be calculated by superposition of components at different frequencies because its calculation involves the square of voltage or current.

Real, reactive, and apparent power

Engineers use three types of power to describe energy flow in a system: Apparent power S is the vector sum of real power P and reactive power Q This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

• Real power (P)
• Reactive power (Q)
• Apparent power (S; sometimes |S| when regarded as the modulus of complex power)

In the diagram, P is the real power, Q is the reactive power (in this case negative), and the length of S is the apparent power. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

The unit for all forms of power is the watt (symbol: W). In practice, however, this is generally reserved for the real power component. Apparent power is conventionally expressed in volt-amperes (VA) since it is the simple multiple of rms voltage and current. The unit for reactive power is given the special name "VAR", which stands for volt-amperes-reactive in IEC 60027-1. The watt (symbol: W) is the SI derived unit of power. ... The volt (symbol: V) is the SI derived unit of electric potential difference. ... The ampere (symbol: A) is the SI base unit of electrical current equal to one coulomb per second. ... In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ... The volt (symbol: V) is the SI derived unit of electric potential difference. ... The ampere (symbol: A) is the SI base unit of electrical current equal to one coulomb per second. ...

Understanding the relationship between these three quantities lies at the heart of understanding power engineering. The mathematical relationship among them can be represented by vectors and is typically expressed using complex numbers

S = P + jQ (where j is the imaginary unit) In mathematics, the imaginary unit i (sometimes also represented by the Latin j or the Greek iota, but in this article i will be used exclusively) allows the real number system to be extended to the complex number system . ...

This complex value S is often referred to as the complex power.

Consider an ideal alternating current (AC) circuit consisting of a source and a generalized load, where both the current and voltage are sinusoidal. If the load is purely resistive, the two quantities reverse their polarity at the same time; the direction of energy flow does not reverse; and only real power flows. If the load is purely inductive or capacitive, then the voltage and current are 90 degrees out of phase (for a capacitor, current leads voltage; for an inductor, current lags voltage) and there is no net power flow. This energy flowing backwards and forwards is known as reactive power. If a capacitor and an inductor are placed in parallel, then the currents caused by the inductor and the capacitor are in antiphase with each other and therefore partially cancel out rather than adding to each other. Conventionally, capacitors are considered to generate reactive power and inductors to consume it. In reality, the load is likely to have resistive, inductive, and capactive parts; and so both real and reactive power will flow to the load. The apparent power is the result of a naive calculation of power from the voltage and current in which the rms voltage is simply multiplied by the rms current. Apparent power is handy for rough sizing of generators or wiring, especially when the power factor is close to 1. However, adding the apparent power for two loads will not give the total apparent power unless the two loads have the same phase difference between voltage and current. city lights viewed in a motion blurred exposure. ... The term induction has more than one meaning in the English language. ... Various types of capacitors A capacitor is a device that stores energy in the electric field created between a pair of conductors on which equal but opposite electric charges have been placed. ... RMS may mean: root mean square, a concept in statistics and electronics Richard M. Stallman, a computer programmer and founder of the GNU project. ...

Apparent power

The definition of apparent power 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:

that is, the quotient of the sums of the real powers for each phase over the sum of the apparent power for each phase.

that is, the quotient of the sums of the real powers for each phase over the magnitude of the sum of the complex powers for each phase.

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 (Emanuel, 1993). Further resolution of this debate did not come until the late 1990s.

Power factor

The ratio between real power and apparent power in a circuit is called the power factor. Where the waveforms are purely sinusoidal, the power factor is the cosine of the phase angle (φ) between the current and voltage sinusoid waveforms. Equipment data sheets and nameplates often will abbreviate power factor as "cosφ" for this reason. The power factor of an AC electric power system is defined as the ratio of the real power to the apparent power. ...

Power factor equals unity (1) when the voltage and current are in phase, and is zero when the current leads or lags the voltage by 90 degrees. Power factor must be specified as leading or lagging. For two systems transmitting the same amount of real 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 in a practical system may 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 real power transfer.

Capacitive circuits cause reactive power with the current waveform leading the voltage wave by 90 degrees, while inductive circuits cause 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. By convention, capacitors are said to generate reactive power whilst inductors are said to consume it (this probably comes from the fact that most real-life loads are inductive and so reactive power has to be supplied to them from power factor correction capacitors). Power factor correction (PFC) is a technique of counteracting the undesirable effects of electric loads that create a power factor (p. ...

In power transmission and distribution, significant effort is made to control the reactive power flow. This is typically done automatically by switching inductors or capacitor banks in and out, by adjusting generator excitation, and by other means. Electricity retailers may use electricity meters which measure reactive power to financially penalise customers with low power factor loads (especially larger customers). Transmission lines in Lund, Sweden Electric power transmission is one process in the delivery of electricity to consumers. ... Electricity retailing is the final process in the delivery of electricity from generation to the consumer. ... Typical US domestic electricity meter An electricity meter is generally taken to be a device which measures the amount of electrical energy supplied to a customer of an electricity company. ...

Basic calculations using real numbers

A perfect resistor stores no energy, and current and voltage are in phase. Therefore there is no reactive power and P = S. Therefore for a perfect resistor:

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

Where X is the reactance of the capacitor or inductor. In the analysis of an alternating-current electrical circuit (for example a RLC series circuit), reactance is the imaginary part of impedance, and is caused by the presence of inductors or capacitors in the circuit. ...

If X is defined as being positive for an inductor and negative for a capacitor then we can remove the modulus signs from Q and X and get. Mathematical meanings Especially in British/European usage, the modulus of a number is its absolute value. ...

More generally using phasors/complex numbers

(In this section overline will be used to indicate phasor or complex quantities and letters with no annotation will be considered the magnitude of those quantities.) See wikibooks book on Phasors A phasor is a complex number representing a sinusoidal function. ... In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. ...

Say we have a series circuit with some resistance and some reactance. From what has been said before we can make up the expression: Left: Series / Right: Parallel Arrows indicate direction of current flow. ...

which simplifies to:

but the complex impedance is simply: In electrical engineering, Impedance is a measure of opposition to a sinusoidal electric current. ...

so:

However, (multiplying a complex number by its conjugate squares its magnitude and makes its angle 0) and so:

See also

• Audio power