AC power
From Wikipedia, the free encyclopedia
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
cause periodic reversals 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. The portion of power flow due to stored
energy which returns to the source in each cycle is known as reactive power.
Real, reactive,
and apparent power
Engineers
use several terms 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.
The apparent
power is the vector sum of real and reactive 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). However, this unit 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. 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,
(where j is the imaginary
unit).
The complex value S is
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 reactive, then the voltage and current are 90 degrees out
of phase 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 flowing through the inductor and the
capacitor oppose and 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. A practical load will
have resistive, inductive, and capacitive parts, and so both real and reactive
power will flow to the load.
The apparent power is the product
of voltage and current. 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.
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 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 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.
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.
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 penalise 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
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.
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.
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.
More generally
using phasors/complex numbers
(In this section tildes (~) 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:
However, 
(multiplying a complex number by its conjugate
squares its magnitude and makes its angle 0) and
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.
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 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.
References
- "Understanding
Power Factor" ST Application Note
http://www.st.com/stonline/books/pdf/docs/4042.pdf
