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Instantaneous power in an electric circuit is the rate of flow of energy past a given point of the circuit. In
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which ...
circuits, energy storage elements such as
inductor An inductor, also called a coil, choke, or reactor, is a incremental passivity, passive two-terminal electronic component, electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically c ...

inductor
s and
capacitor A capacitor is a device that stores electrical energy Electrical energy is energy derived from electric potential energy or kinetic energy. When used loosely, ''electrical energy'' refers to energy that has been converted ''from'' electric p ...

capacitor
s may result in periodic reversals of the direction of energy flow. The portion of power that, averaged over a complete cycle of the
AC waveform When describing a periodic function A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the stud ...
, results in net transfer of energy in one direction is known as active power (more commonly called real power to avoid ambiguity especially in discussions of loads with non-sinusoidal currents). The portion of power due to stored energy, which returns to the source in each cycle, is known as ''instantaneous reactive power'', and its amplitude is the absolute value of reactive power.


Active, reactive, and apparent power

In a simple alternating current (AC) circuit consisting of a source and a linear load, both the current and voltage are
sinusoidal A sine wave or sinusoid is a curve, 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 of a function, graph. It occurs often in both p ...

sinusoidal
. If the load is purely
resistive In electronics and electromagnetism, the electrical resistance of an object is a measure of its opposition to the flow of electric current. The Multiplicative inverse, reciprocal quantity is , and is the ease with which an electric current passe ...
, 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''
''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 In electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identi ...
). 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. 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 The watt (symbol: W) is a unit of power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal ...

watt
(W); * Reactive power, ''Q'':
volt-ampere reactive A volt-ampere ( SI symbol: V⋅A or V A; also VA) is the unit used for the apparent power in an electrical circuit An electrical network is an interconnection of electronic component, electrical components (e.g., battery (electricity ...
(var); * Complex power, ''S'':
volt-ampere A volt-ampere ( SI symbol: V⋅A or V A; also VA) is the unit used for the apparent power Instantaneous power in an electric circuit is the rate of flow of energy past a given point of the circuit. In alternating current Alternating ...
(VA); * Apparent power, , ''S'', : the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
of complex power ''S'': volt-ampere (VA); * Phase of voltage relative to current, ''φ'': the angle of difference (in degrees) between current and voltage; \varphi=\arg(V)-\arg(I). Current lagging voltage (
quadrant Quadrant may refer to: Companies * Quadrant Cycle Company, 1899 manufacturers in Britain of the Quadrant motorcar * Quadrant (motorcycles), one of the earliest British motorcycle manufacturers, established in Birmingham in 1901 * Quadrant Private ...
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 The watt (symbol: W) is a unit of power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal ...

watt
(symbol: W). Apparent power is often expressed in
volt-ampere A volt-ampere ( SI symbol: V⋅A or V A; also VA) is the unit used for the apparent power Instantaneous power in an electric circuit is the rate of flow of energy past a given point of the circuit. In alternating current Alternating ...
s (VA) since it is the product of rms
voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

voltage
and rms
current Currents or The Current may refer to: Science and technology * Current (fluid) A current in a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. ...
. The unit for reactive power is var, which stands for
volt-ampere reactive A volt-ampere ( SI symbol: V⋅A or V A; also VA) is the unit used for the apparent power in an electrical circuit An electrical network is an interconnection of electronic component, electrical components (e.g., battery (electricity ...
. 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 grid An electrical grid is an interconnected network for from producers to consumers. Electrical grids vary in size and can cover whole countries or continents. It consists of:Kaplan, S. M. (2009). Smart Grid. Electrical Power Transmission: Backgr ...
s and its lack has been cited as a significant factor in the
Northeast Blackout of 2003 The Northeast blackout of 2003 was a widespread power outage throughout parts of the northeastern United States, Northeastern and Midwestern United States, and the Canadian province of Ontario on August 14, 2003, beginning just after 4:10 p ...
. 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 The imaginary unit or unit imaginary number () is a solution to the quadratic equation . Although there is no real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be ...
).


Calculations and equations

The formula for complex power (units: VA) in
phasor In and , a phasor (a of phase vector), is a representing a whose (''A''), (''ω''), and (''θ'') are . It is related to a more general concept called ,Bracewell, Ron. ''The Fourier Transform and Its Applications''. McGraw-Hill, 1965. p2 ...

phasor
form is: :S=VI^*=, S, \angle\varphi, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of ''I'' is used, which is denoted I^* (or \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). :S=, I, ^2 Z= \frac. Consequentially, with reference to the power triangle, real power (units in watts, W) is derived as: : P=, S, \cos=, I, ^2 R=\frac \times. For a purely resistive load, real power can be simplified to: : P = \frac. ''R'' denotes resistance (units in ohms, Ω) of the load. Reactive power (units in volts-amps-reactive, var) is derived as: :Q=, S, \sin=, I, ^2 X=\frac \times. For a purely reactive load, reactive power can be simplified to: : Q = \frac, where ''X'' denotes
reactance
reactance
(units in ohms, Ω) of the load. Combining, the complex power (units in volt-amps, VA) is back-derived as :S=P+jQ, and the apparent power (units in volt-amps, VA) as :, S, =\sqrt. These are simplified diagrammatically by the power triangle.


Power factor

The ratio of active power to apparent power in a circuit is called the
power factor In electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identi ...
. 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 Phase or phases may refer to: Science * State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter) In the physical sciences, a phase is a region of space (a thermodynamic system A thermodynamic system is a ...
. 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 (\varphi) between the current and voltage sinusoidal waveforms. Equipment data sheets and nameplates will often abbreviate power factor as "\cos \phi" for this reason. Example: The active power is and the phase angle between voltage and current is 45.6°. The power factor is . The apparent power is then: . 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 32% does no work. Usually, utilities do not charge consumers for the reactive power losses, as they do no real work for the consumer. However, if there are inefficiencies at the customer's load source that cause the power factor to fall below a certain level, utilities may charge customers in order to cover an increase in their power plant fuel use and their worse line and plant capacities.


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 A volt-ampere (International System of Units, SI symbol: V⋅A or V A; also VA) is the unit used for the apparent power in an electrical circuit. The apparent power equals the product of Root mean square AC voltage, root mean square voltag ...
", 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 is 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 an AC 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 An inductor, also called a coil, choke, or reactor, is a incremental passivity, passive two-terminal electronic component, electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically c ...

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 generator Induction may refer to: Philosophy * Inductive reasoning, in logic, inferences from particular cases to the general case Biology and chemistry * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induction period, the t ...
s 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

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: : \mathrm 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 compensator
static VAR compensator
s and voltage control circuits.


Unbalanced 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 Stanley may refer to: People * Stanley (name) Stanley is a toponymic surname dating from the 11/12th century contraction of ''Stan'' (a form of "Stone") and ''wikt:leigh, Leigh'' (meadow), later also being used as a masculine given name. Pers ...
's ''Phenomena of Retardation in the Induction Coil'' (1888) and
Steinmetz
Steinmetz
's ''Theoretical Elements of Engineering'' (1915). However, with the development of
three phase Three-phase electric power (abbreviated 3φ) is a common type of alternating current used in electricity generation, Electric power transmission, transmission, and Electric power distribution, distribution. It is a type of polyphase system emplo ...

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. : S_A = , S_\mathrm, + , S_\mathrm, + , S_\mathrm, : \mathrm_A = , that is, the arithmetic sum of the phase apparent powers; and : S_V = , P_\mathrm + P_\mathrm + P_\mathrm + j(Q_\mathrm + Q_\mathrm + Q_\mathrm), : \mathrm_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 componentsIn electrical engineering, the method of symmetrical components simplifies analysis of unbalanced three-phase power systems under both normal and abnormal conditions. The basic idea is that an asymmetrical set of ''N'' phasors can be expressed as ...
theory was proposed in 1993 by Alexander Emanuel for unbalanced linear load supplied with asymmetrical sinusoidal voltages: : S = \sqrt : \mathrm = , that is, the root of squared sums of line voltages multiplied by the root of squared sums of line currents. P^+ denotes the positive sequence power: :P^+ = 3 , V^+, , I^+, \cos V^+ denotes the positive sequence voltage phasor, and 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 P=S (using the ). Therefore, for a perfect resistor :P = S = V_\mathrm I_\mathrm = I_\mathrm^2 R = \frac \,\!. For a perfect capacitor or inductor, there is no net power transfer; so all power is reactive. Therefore, for a perfect capacitor or inductor: :\begin P &= 0 \\ Q &= , S, = V_\mathrm I_\mathrm = I_\mathrm^2 , X, = \frac \end. where ''X'' is the of the capacitor or inductor. If X is defined as being positive for an inductor and negative for a capacitor, then the signs can be removed from S and X and get :Q = I_\mathrm^2 X = \frac . Instantaneous power is defined as: :P_\text(t) = V(t)\cdot I(t), where V(t) and 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: :P_\text = \frac\int_^ V(t)I(t)\,\operatornamet. 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: : P_\text = \frac \sum_^n V .


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. :\begin &A\cos(\omega_1t+k_1)\cos(\omega_2t + k_2) \\ = &\frac\cos\left left(\omega_1t + k_1\right) + \left(\omega_2t + k_2\right)\right+ \frac\cos\left left(\omega_1t + k_1\right) - \left(\omega_2t + k_2\right)\right\\ = &\frac\cos\left left(\omega_1 + \omega_2\right)t + k_1 + k_2\right+ \frac\cos\left left(\omega_1 - \omega_2\right)t + k_1 - k_2\right\end However, the time average of a function of the form 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.


See also

*
War of the currents The war of the currents was a series of events surrounding the introduction of competing electric power transmission systems in the late 1880s and early 1890s. It grew out of two lighting systems developed in the late 1870s and early 1880s; arc l ...
*
Electric power transmission Electric power transmission is the bulk movement of electrical energy Electrical energy is energy derived as a result of movement of electrically charged particles. When used loosely, ''electrical energy'' refers to energy that has been convert ...

Electric power transmission
*
Transformer A transformer is a passive electrical device that transfers electrical energy from one electrical circuit to another, or multiple Electrical network, circuits. A varying current in any one coil of the transformer produces a varying magnetic flux ...

Transformer
*
Mains electricity Mains electricity (Commonwealth English The use of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become ...
*
Deformed powerDeformed power is a concept in electrical engineering which characterize the distortion to the sinusoidal states in electric network. It was introduced by Constantin Budeanu Constantin Budeanu (28 February 1886 – 27 February 1959) was a Romanian ...


References


External links


"AC Power Java Applet"
{{DEFAULTSORT:Ac Power Electric power