AC power

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This article deals with power in AC systems. See Mains electricity for information on utility supplied AC power.
Usually hidden to the unaided eye, the blinking of (non-incandescent) lighting powered by AC mains is revealed in this motion-blurred long exposure of city lights. The blinking is at 120Hz instead of 60Hz because light is emitted during both the positive and negative phases of each cycle.
Usually hidden to the unaided eye, the blinking of (non-incandescent) lighting powered by AC mains is revealed in this motion-blurred long exposure of city lights. The blinking is at 120Hz instead of 60Hz because light is emitted during both the positive and negative phases of each cycle.

Power is defined as the rate of flow of energy past a given point. In alternating current circuits, energy storage elements such as inductors and capacitors may cause periodic reversals in the direction of energy flow. The portion of power flow averaged over a complete cycle of the AC waveform that 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.

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[edit] Real, reactive, and apparent power

Engineers use three types of power to describe energy flow in a system:

  • Real power (P)
  • Reactive power (Q)
  • Complex power (S); |S|, the modulus of complex power, is referred to as apparent 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.

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 product of rms voltage and current. The unit for reactive power is given the special name "VAR", which stands for volt-amperes-reactive.

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)

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 capacitive 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.

[edit] 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.

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 while 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).

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. This is particularly relevant to customers operating highly inductive loads such as motors at water pumping stations.

[edit] 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.

[edit] 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:

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}

[edit] 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.)

Say we have a series circuit with some resistance and some reactance. From what has been said before we can make up the expression:

\overline{S}=I_\mathrm{rms}^2R+jI_\mathrm{rms}^2X\,\!

which simplifies to:

\overline{S}=I_\mathrm{rms}^2(R+jX)\,\!

but the complex impedance \overline Z is simply:

\overline{Z}=R+jX\,\!

so:

\overline{S} = I_\mathrm{rms}^2 \overline{Z}\,\!

However, I^2 = \overline{I} \cdot \overline{I}^* (multiplying a complex number by its conjugate squares its magnitude and makes its angle 0) and \overline{V} = \overline{I} \cdot \overline{Z} so:

\overline{S} = \overline{I}\cdot \overline{I}^* \cdot \overline{Z} = \overline{V} \cdot \overline{I}^* = \frac{\overline{V} \cdot \overline{V}^*} {\overline{Z}^*} = \frac{V_\mathrm{rms}^2} {\overline{Z}^*}\,\!

[edit] Multiple frequency systems

Since a 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_1t+k_1)\cos(\omega_2t+k_2)\,\! =\frac{A}{2}\cos((\omega_1t+k_1) + (\omega_2t+k_2))+\frac{A}{2}\cos((\omega_1t+k_1)-(\omega_2t+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(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 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 single frequency (which it generally is close enough to), this shows that harmonic currents are a bad thing. They will increase the rms current (since that bases on the product of current and current they will create a term with a nonzero average to be added in there) 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.

[edit] See also

[edit] References

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