AC power

This article deals with power in AC systems. See Mains electricity for information on utility-supplied AC power.

Usually hidden from the unaided eye, the blinking of (non-incandescent) lighting powered by AC mains is revealed in this motion-blurred long exposure of city lights. Light is emitted twice each cycle.

Power is defined as the rate of flow of energy past a given point. It is measured in watts. 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. On the other hand, the portion of power flow due to stored energy, which returns to the source in each cycle, is known as reactive power.

Energy is power multiplied by time. All real work, including lighting, heating or mechanical movement, is performed by energy. Mechanically, energy is the product of Force and Distance. So, one Newton-Meter is a Joule, which is one watt-second. Energy is often measured mechanically, thermally, or electrically. These forms are not practically equivalent, because no machine is perfectly efficient at changing one form of energy to another form.

Real, reactive, and apparent power

The apparent power is the vector sum of real and reactive power

Consider a simple alternating current (AC) circuit consisting of a source and a load, where both the current and voltage are sinusoidal.

A normal multimeter can measure volts, electrical pressure, and amps, electrical current. The product of these two measurements is apparent power, because it is the apparent power transfer. Apparent power is measured in volt-amps, because it is the product of volts (root-mean-squared) and amps (also root-mean-squared). The problem with a multimeter is that it averages these quantities over time.

If the waves of voltage and current coincide, then the multiplication of voltage and current physically occurs, and the real power is the same as the apparent power. Real power is measured in "watts." As the waves of current and voltage coincide less, less real power is transferred. When the voltage and current are a quarter out of phase (90 degrees, or pi/2), then the voltage peak occurs where the current shifts from positive to negative (or vice-versa). At this offset, no real power transfers at all; as much power is returned as is drawn.

Note that even though no net power is being transferred, the circuit carries current back and forth during the waveform cycle. This current heats wires. So, electrical equipment usually must have wires large enough to carry the required current. So this out-of-phase current costs money for equipment, but has no benefit because it doesn't move real power. Naturally power engineers want to measure this waste in order to reduce it.

As a mathematical convenience for engineering design, the nontransfer of power is described as the product of a physically nonexistent (imaginary) current wave that is a quarter out of phase with the actual voltage wave. This nonexistent current is called reactive current because it is normally caused by loads that react to voltage by storing current. When multiplied by voltage, this quantity is called reactive power. It is measured by a quantity called volt-amps-reactive, or VARs.

VARs are mathematically convenient because the apparent power is the vector sum of VARs and Watts. Since VARs and watts are orthogonal, the pythagorean theorem can be used to perform the vector sums or differences. So, as a practical matter, mechanical meters for VARs often calculate the VARs as the square root of VA squared minus Wh squared.

The nature of the electrical load determines whether the voltage and current coincide. If the load is purely resistive, the two quantities become negative and positive at the same time, the direction of energy flow does not reverse, and only real power flows. If the load is purely reactive, then the voltage and current are 90 degrees out of phase and there is no net power flow over an entire cycle. A practical load will have resistive, inductive, and capacitive parts, and so both positive and negative real and reactive power will flow to the load.

If a capacitor and an inductor are placed in parallel, then the currents flowing through the inductor and the capacitor tend to cancel out rather than adding. Conventionally, capacitors are considered to generate reactive power and inductors to consume it. This is the fundamental mechanism for controlling the power factor in electric power transmission; capacitors (or inductors) are inserted in a circuit to partially cancel reactive power of the load.

Apparent power is handy for sizing of equipment or wiring. However, adding the apparent power for two loads will not accurately give the total apparent power unless they have the same displacement between current and voltage (the same power factor).

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

• Real power (P) - unit: watt (W)
• Reactive power (Q) - unit: volt-amperes reactive (var)
• Complex power (S) - unit: volt-ampere (VA)
• Apparent Power (|S|) , that is, the absolute value of complex power S - unit: volt-ampere (VA)

Since apparent power, watts and VARs are vectors, a vector diagram in the complex plane is very helpful to understand them. In the diagram, P is the real 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 transfer energy, so it is represented as the imaginary basis. Real power moves energy, so it is the real basis.

The unit for all forms of power is the watt (symbol: W), but this unit is generally reserved for real power. Apparent power is conventionally expressed in volt-amperes (VA) since it is the product of rms voltage and rms current. The unit for reactive power is expressed as "VAr", which stands for volt-amperes reactive. Since reactive power flow transfers no net energy to the load, it is sometimes called "wattless" power.

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 or expressed using complex numbers,

$S=P+jQ\,\!$ (where j is the imaginary unit).

The complex value S is referred to as the complex power.

Complex power is not the whole answer. In real circuits, the real and reactive power are consumed at a variety of frequencies, as well. Power engineers sometimes analyze these in order to manage particular types of loads. For example, many electronic systems take current only from the peaks of the voltage. Since currents to these loads have a square profile, when analyzed with a fourier transform, they consist of odd-numbered sinusoidal current harmonics. Aircraft, for example, must have controlled current harmonics in order to avoid "cogging" effects that can cause vibrations in the engines that drive the generators.

Power factor

Main article: 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 "$\cos \phi$" for this reason.

Power factor equals 1 when the voltage and current are in phase, and is zero when the current leads or lags the voltage by 90 degrees. Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle, where leading indicates a negative sign. 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 will 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.

Purely capacitive circuits cause reactive power with the current waveform leading the voltage wave by 90 degrees, while purely 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.

Reactive power flow

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 penalize customers with low power factor loads. This is particularly relevant to customers operating highly inductive loads such as motors at water pumping stations.

Unbalanced polyphase systems

While real 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:

• $pf={Pa+Pb+Pc \over Sa+Sb+Sc}$

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

• $pf={Pa+Pb+Pc \over |Pa+Pb+Pc+j(Qa+Qb+Qc)|}$

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.

Basic calculations using real numbers

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

$Q=0\,\!$

$P=S=V_{\mathrm {rms} }I_{\mathrm {rms} }=I_{\mathrm {rms} }^{2}R={\frac {V_{\mathrm {rms} }^{2}}{R}}\,\!$

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:

$P=0\,\!$

$Q=|S|=V_{\mathrm {rms} }I_{\mathrm {rms} }=I_{\mathrm {rms} }^{2}|X|={\frac {V_{\mathrm {rms} }^{2}}{|X|}}\,\!$

Where X is the reactance of the capacitor or inductor.

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.

$Q=I_{\mathrm {rms} }^{2}X={\frac {V_{\mathrm {rms} }^{2}}{X}}$

Multiple frequency systems

Since an RMS value can be calculated for any waveform, apparent power can be calculated from this.

For real power it would at first appear that we would have to calculate loads of product terms and average all of them. However if we look at one of these product terms in more detail we come to a very interesting result.

$A\cos(\omega _{1}t+k_{1})\cos(\omega _{2}t+k_{2})\,\!$	$={\frac {A}{2}}\cos((\omega _{1}t+k_{1})+(\omega _{2}t+k_{2}))+{\frac {A}{2}}\cos((\omega _{1}t+k_{1})-(\omega _{2}t+k_{2}))$
	$={\frac {A}{2}}\cos((\omega _{1}+\omega _{2})t+k_{1}+k_{2})+{\frac {A}{2}}\cos((\omega _{1}-\omega _{2})t+k_{1}-k_{2})$

however the time average of a function of the form $\cos(\omega t+k)$ 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 real (average) power by simply treating each frequency separately and adding up the answers.

Furthermore, if we assume the voltage of the mains supply is 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 real 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. And hence, the different components of current is defined.

References

• "Understanding Power Factor" ST Application Note
• "AC Power Java Applet"

See also

• War of Currents
• Electric power transmission
• Transformer
• Mains electricity