basicfundamental
basic fundamental
Electric power
Electric power is defined as the amount of work done by an electric current in a unit time.
Explanation
Electrical power is distributed via cables and electricity pylons like these in Brisbane, Australia.
When a current flows in a circuit with resistance, it does work. Devices can be made that convert this work into heat (electric heaters), light (light bulbs and neon lamps), or motion (electric motors). In our modern society, electrical power or electricity is a term not just for physical work but also for a public utility of modern infrastructure.
When a loss of electrical power occurs over a large region of a country, whether by the actions of people or nature, it is a disaster. Geomagnetic storms (solar storms) have previously caused power outages of this scale, and similar events can be anticipated in the future. More recent power outages have been primarily caused by human action and inaction . A few locations for power production and transmission facilities and hardware are classified in the United States, to prevent tampering or sabotage, because of the necessity of electrical power.
Mathematics of electric power
In circuits
Electric power, like mechanical power, is represented by the letter P in electrical equations, and is measured in units called watts (symbol W), named after Scottish engineer James Watt. The term wattage is used colloquially to mean 'electric power in watts'.
In resistive circuits, instantaneous electrical power is calculated using Joule's Law, which is named after the British physicist James Joule, who first showed that electrical and mechanical energy were interchangeable.
where
P = power in watts
I = current in amperes
V = potential difference in volts
For example: 2 amperes × 12 volts = 24 watts.
Joule's law can be combined with Ohm's law to produce two more equations:
and
where
R = resistance in ohms.
For example:
(2 amperes)2 × 6 ohms = 24 watts
and
(12 volts)2 / 6 ohms = 24 watts.
In reactive 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.
The unit for reactive power is given the special name VAR, which stands for volt-amperes-reactive. In reactive circuits, the watt unit (symbol W) is generally reserved for the real power component. The vector sum of the real power and the reactive power is called the apparent power. Apparent power is conventionally expressed in volt-amperes (V·A) since it is the simple multiple of rms voltage and current.
Power triangle
The relationship between real power, reactive power and apparent power can be expressed by representing the quantities as vectors. Real power is represented as a horizontal vector and reactive power is represented as a vertical vector. The apparent power vector is the hypotenuse of a right triangle formed by connecting the real and reactive power vectors. This representation is often called the power triangle. Using the Pythagorean Theorem, the relationship among real, reactive and apparent power is shown to be:
In space
Electrical power flows wherever electric and magnetic fields exist in the same place. The simplest example of this is in electrical circuits, as the preceding section showed. In the general case, however, the simple equation P = IV must be replaced by a more complex calculation, the integral of the vector cross-product of the electrical and magnetic fields over a specified area, thus:
The result of this integral is a vector, since power has both magnitude and direction. For more information on this vector.
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 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).
In power transmission and distribution, significant effort is made to control the reactive power flow. This is typically done automatically by switching inductors (also commonly called reactors) 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 (especially larger customers).
Kilowatt-hour
When paired with a unit of time the term watt is used for expressing energy consumption. For example, a kilowatt hour, is the amount of energy expended by a one kilowatt device over the course of one hour; it equals 3.6 megajoules. A megawatt day (MWd or MW·d) is equal to 86.4 GJ. These units are often used in the context of power plants and home energy bills.
Voltage
.
International danger high voltage symbol.
Voltage is the difference of electrical potential between two points of an electrical network, expressed in volts. It is a measure of the capacity of an electric field to cause an electric current in an electrical conductor.
Explanation
Between two points in an electric field, such as exists in an electrical circuit, the difference in their electrical potentials is known as the electrical potential difference. This difference is proportional to the electrostatic force that tends to push electrons or other charge-carriers from one point to the other. Potential difference, electrical potential and electromotive force are measured in volts, leading to the commonly used term voltage. Voltage is usually represented in equations by the symbols V or E. (E is often preferred in academic writing, because it avoids the confusion between V and the SI symbol for the volt, which is also V).
Electrical potential difference can be thought of as the ability to move electrical charge through a resistance. At a time in physics when the word force was used loosely, the potential difference was named the electromotive force or emf - a term which is still used in certain contexts.
Voltage is a property of an electric field, not individual electrons. An electron moving across a voltage difference experiences a net change in energy, often measured in electron-volts. This effect is analogous to a mass falling through a given height difference in a gravitational field.
When using the term 'potential difference' or voltage, one must be clear about the two points between which the voltage is specified or measured. There are two ways in which the term is used. This can lead to some confusion.
Voltage with respect to a common point
The first way the term voltage is used is when specifying the voltage of a point in a circuit. When this is done, it is understood that the voltage is usually being specified or measured with respect to a stable and unchanging point in the circuit that is known as ground or common. We say that a point in a circuit has a particular voltage relative to ground when we take the time to say all the clarifying words. This voltage is really a voltage difference, one of the two points being the reference point, that is, ground.
Voltage between two stated points
The second way the term voltage is used is when specifying how many volts are dropped across an electrical device (such as a resistor). In this case, the voltage (loosely stated) or the voltage drop across the device (better, but not always stated for brevity) is really the first voltage taken (relative to ground) on one terminal of the device minus (hence a voltage difference) a second voltage taken (relative to ground) on the other terminal of the device. In practice, the voltage drop across a device can be measured directly and safely using a voltmeter (such as a battery-powered meter) that is isolated from ground, provided that the maximum voltage capability of the voltmeter is not exceeded.
Addition of voltages
Voltage is additive in the following sense: the voltage between A and C is the sum of the voltage between A and B and the voltage between B and C. The various voltages in a circuit can be computed using Kirchhoff's circuit laws.
Two points in an electric circuit which are connected by an ideal conductor, without resistance and without the presence of a changing magnetic field, have a potential difference of zero. But other pairs of points may also have a potential difference of zero. If two such points are connected with a conductor, no current will flow through the connection.
Hydraulic analogy
If one imagines water circulating in a network of pipes, driven by pumps in the absence of gravity, as an analogy of an electrical circuit, then the potential difference corresponds to the fluid pressure difference between two points. If there is a pressure difference between two points, then water flowing from the first point to the second will be able to do work, such as driving a turbine.
This hydraulic analogy is a useful method of teaching a range of electrical concepts. In a hydraulic system, the work done to move water is equal to the pressure multiplied by the volume of water moved. Similarly, in an electrical circuit, the work done to move electrons or other charge-carriers is equal to 'electrical pressure' (an old term for voltage) multiplied by the quantity of electrical charge moved. Voltage is a convenient way of quantifying the ability to do work. In relation to electric current, the larger the gradient (voltage or hydraulic) the greater the current (assuming resistance is constant).
Mathematical definition
The electrical potential difference is defined as the amount of work needed to move a unit electric charge from the second point to the first, or equivalently, the amount of work that a unit charge flowing from the first point to the second can perform. The potential difference between two points a and b is the line integral of the electric field E:
Useful formulas
DC circuits
Where V=voltage, I=current, R=resistance, P=power
AC circuits
Where V=voltage, I=current, R=resistance, P=true power, Z=impedance, θ=phasor angle between I and V
AC conversions
Where Vpk=peak voltage, Vppk=peak-to-peak voltage, Vavg=average voltage over a half-cycle, Vrms=effective (root mean square) voltage
Total voltage
Voltage sources and drops in series:
Voltage sources and drops in parallel:
Voltage drops
Across a resistor (Resistor R):
Across a capacitor (Capacitor C):
Across an inductor (Inductor L):
Where V=voltage, I=current, R=resistance, X=reactance.
Measuring instruments
A multimeter set to measure voltage.
Instruments for measuring potential differences include the voltmeter, the potentiometer (measurement device), and the oscilloscope. The voltmeter works by measuring the current through a fixed resistor, which, according to Ohm's Law, is proportional to the potential difference across it. The potentiometer works by balancing the unknown voltage against a known voltage in a bridge circuit. The cathode-ray oscilloscope works by amplifying the potential difference and using it to deflect an electron beam from a straight path, so that the deflection of the beam is proportional to the potential difference.
Current definition
The magnitude of an electric current is defined as the time derivative of electric charge:
Formally this is written as:
or inversely as
The amount of charge Q flowing per unit of time t is I, standing for the intensity of the current.
Current in a metal wire
In solid conductive metal, with no external forces applied, there exists a random motion of free electrons created by the thermal energy that the electrons gain from the surrounding medium. When an atom loses a free electron, it acquires a net positive charge. The free electron can move amongst these positive ions, while the positive ions can only oscillate about their mean fixed positions. The free electron is therefore the charge carrier in a typical solid conductor. Given an imaginary plane through which the wire passes, the number of electrons moving from one side to the other in any period of time is exactly equal to the number passing in the opposite direction.
When a wire is connected across the two terminals of a DC voltage source such as a battery, the source places an electric field across the conductor. The moment contact is made, the free electrons of the conductor will drift toward the positive terminal under the influence of this field. For every ampere of current, 1 coulomb of electric charge (which consists of about 6.242 × 1018 electrons) drifts every second at the same velocity through the imaginary plane through which the conductor passes.
The current I in amperes can be calculated with the following equation:
where
is the electric charge in coulombs (ampere seconds)
is the time in seconds
It follows that:
and
Current density
Current density is a measure of the density of electrical current. It is defined as a vector whose magnitude is the electric current per cross-sectional area. In SI unit, the current density is measured in amperes per square meter.
The drift speed of electric charges
The mobile charged particles within a conductor move constantly in random directions. In order for a net flow of charge to exist, the particles must also move together with an average drift rate. Electrons are the charge carriers in metals and they follow an erratic path, bouncing from atom to atom, but generally drifting in the direction of the electric field. The speed at which they drift can be calculated from the equation:
where
is the electric current
is number of charged particles per unit volume
is the cross-sectional area of the conductor
is the drift velocity, and
is the charge on each particle.
Electric currents in solid matter are typically very slow flows. For example, in a copper wire of cross-section 0.5 mm², carrying a current of 5 A, the drift velocity of the electrons is of the order of a millimetre per second. To take a different example, in the near-vacuum inside a cathode ray tube, the electrons travel in near-straight lines ("ballistically") at about a tenth of the speed of light.
However, we know that electrical signals are electromagnetic waves which propagate at very high speed (at the speed of light, as can be deduced from Maxwell's Equations). For example, in AC power lines, the waves of electromagnetic energy propagate rapidly through the space between the wires from a source to a distant load, even though the electrons in the wires only move back and forth over a tiny distance. Although the velocity of the flowing charges is quite low, the associated electromagnetic energy travels at the speed of light. The ratio of the signal velocity through a medium versus the speed of light in a vacuum is called the velocity factor.
Direct current
Direct current is the constant flow of electric charge from high to low potential. This is typically in a conductor such as a wire, but can also be through semiconductors, insulators, or even through a vacuum as in electron or ion beams. In direct current, the electric charges flow in the same direction, distinguishing it from alternating current (AC). A term formerly used for direct current was Galvanic current.
Types of direct current
The first commercial electric power transmission (developed by Thomas Edison in the late nineteenth century) used direct current. Because alternating current is much more convenient than direct current for electric power distribution and transmission, today nearly all electric power transmission uses alternating current.
Various definitions
Within Electrical Engineering, the term DC is a synonym for constant. For example, the voltage across a DC voltage source is constant as is the current through a DC current source. The DC solution of an electric circuit is the solution where all voltages and currents are constant. It can be shown that any voltage or current waveform can be decomposed into a sum of a DC component and a time-varying component. The DC component is defined to be the average value of the voltage or current over all time. The average value of the time-varying component is zero.
Although DC stands for "Direct Current", DC sometimes refers to "constant polarity." With this definition, DC voltages can vary in time, such as the raw output of a rectifier or the fluctuating voice signal on a telephone line.
Some forms of DC (such as that produced by a voltage regulator) have almost no variations in voltage, but may still have variations in output power and current.
Applications
Direct current installations usually have different types of sockets, switches, and fixtures, mostly due to the low voltages used, from those suitable for alternating current. It is usually important with a direct current appliance not to reverse polarity unless the device has a diode bridge to correct for this. (Most battery-powered devices don't.)
High voltage direct current is used for long-distance point-to-point power transmission and for submarine cables, with voltages from a few kilovolts to approximately one megavolt.
DC is commonly found in many low-voltage applications, especially where these are powered by batteries, which can produce only DC, or solar power systems, since solar cells can produce only DC. Most automotive applications use DC, although the alternator is an AC device which uses a rectifier to produce DC. Most electronic circuits require a DC power supply. Applications using fuel cells (mixing hydrogen and oxygen together with a catalyst to produce electricity and water as byproducts) also produce only DC.
Most telephones connect to a twisted pair of wires, and internally separate the AC component of the voltage between the two wires (the audio signal) from the DC component of the voltage between the two wires (used to power the phone).
Telephone exchange communication equipment, such as DSLAM, uses standard -48V DC power supply. The negative polarity is achieved by grounding the positive terminal of power supply system and the battery bank. This is done to prevent electrolysis depositions.
Alternating current
City lights viewed in a motion blurred exposure. The AC blinking causes the lines to be dotted rather than constant.
An alternating current (AC) is an electrical current whose magnitude and direction vary cyclically, as opposed to direct current, whose direction remains constant. The usual waveform of an AC power circuit is a sine wave, as this results in the most efficient transmission of energy. However in certain applications different waveforms are used, such as triangular or square waves.
Used generically, AC refers to the form in which electricity is delivered to businesses and residences. However, audio and radio signals carried on electrical wire are also examples of alternating current. In these applications, an important goal is often the recovery of information encoded (or modulated) onto the AC signal.
History
William Stanley designed one of the first practical devices to transfer AC power efficiently between isolated circuits. Using pairs of coils wound on a common iron core, his design, called an induction coil, was an early precursor of the modern transformer. The system used today was devised by many contributors including Nikola Tesla, George Westinghouse, Lucien Gaulard, John Gibbs, and Oliver Shallenger from 1881 to 1889. AC systems overcame the limitations of direct current systems, such as that which Thomas Edison first used to distribute electricity commercially.
The first long-distance transmission of alternating current took place in 1891 near Telluride, Colorado, followed a few months later in Germany. Thomas Edison strongly advocated the use of direct current (DC), having many patents in that technology, but eventually alternating current came into general use.
The first modern commercial power plant using three-phase alternating current was at the Mill Creek hydroelectric plant near Redlands, California in 1893. Its designer was Almirian Decker, a brilliant young engineer. Decker's innovative design incorporated 10,000 volt three phase transmission and established the standards for the complete system of generation, transmission and motors used today. And through the use of alternating current, Charles Proteus Steinmetz of General Electric was able to solve many of the problems associated with electricity generation and transmission.
Transmission, distribution, and domestic power supply
AC voltage can be stepped up or down by a transformer to a different voltage. Modern High-voltage, direct current electric power transmission systems contrast with the more common alternating-current systems as a means for the bulk transmission of electrical power over long distances. However, these tend to be more expensive and less efficient than transformers, and did not exist when Edison, Westinghouse and Tesla were designing their power systems.
Use of a higher voltage leads to more efficient transmission of power. The power losses in a conductor are a product of the square of the current and the resistance of the conductor, described by the formula P = I2R. This means that when transmitting a fixed power on a given wire, if the current is doubled, the power loss will be four times greater. Since the power transmitted is equal to the product of the current, the voltage and the cosine of the phase difference φ (P = IVcosφ), the same amount of power can be transmitted with a lower current by increasing the voltage. Therefore it is advantageous when transmitting large amounts of power to distribute the power with high voltages (often hundreds of kilovolts). However, high voltages also have disadvantages, the main ones being the increased insulation required, and generally increased difficulty in their safe handling. In a power plant, power is generated at a convenient voltage for the design of a generator, and then stepped up to a high voltage for transmission. Near the loads, the transmission voltage is stepped down to the voltages used by equipment. Consumer voltages vary depending on the country and size of load, but generally motors and lighting are built to use up to a few hundred volts between phases.
Three-phase electrical generation is very common. Three separate coils in the generator stator are physically offset by an angle of 120° to each other. Three current waveforms are produced that are equal in magnitude and 120° out of phase to each other.
If the load on a three-phase system is balanced equally between the phases, no current flows through the neutral point. Even in the worst-case unbalanced (linear) load, the neutral current will not exceed the highest of the phase currents. For three-phase at low (normal mains) voltages a four-wire system is normally used. When stepping down three-phase, a transformer with a Delta primary and a Star secondary is often used so there is no need for a neutral on the supply side.
For smaller customers (just how small varies by country and age of the installation) only a single phase and the neutral or two phases and the neutral are taken to the property. For larger installations all three phases and the neutral are taken to the main distribution panel. From the three-phase main panel, both single and three-phase circuits may lead off.
Three-wire single phase systems, with a single centre-tapped transformer giving two live conductors, is a common distribution scheme for residential and small commercial buildings in North America. A similar method is used for a different reason on construction sites in the UK. Small power tools and lighting are supposed to be supplied by a local center-tapped transformer with a voltage of 55V between each power conductor and the earth. This significantly reduces the risk of electric shock in the event that one of the live conductors becomes exposed through an equipment fault whilst still allowing a reasonable voltage for running the tools.
A third wire is often connected between non-current carrying metal enclosures and earth ground. This conductor provides protection from electrical shock due to accidental contact of circuit conductors with the case of portable appliances and tools.
AC power supply frequencies
The frequency of the electrical system varies by country; most electric power is generated at either 50 or 60 Hz. See List of countries with mains power plugs, voltages and frequencies. Some countries have a mixture of 50 Hz and 60 Hz supplies, notably Japan.
A low frequency eases the design of low speed electric motors, particularly for hoisting, crushing and rolling applications, and commutator-type traction motors for applications such as railways, but also causes a noticeable flicker in incandescent lighting and objectionable flicker of fluorescent lamps. 16.7 Hz power (approx. ⅓ of the mains frequency) is still used in some European rail systems, such as in Austria, Germany, Norway, Sweden and Switzerland.
Off-shore, textile industry, marine, computer mainframe, aircraft, and spacecraft applications sometimes use 400 Hz, for benefits of reduced weight of apparatus or higher motor speeds.
Effects at high frequencies
A direct, constant, current flows uniformly throughout the cross-section of the (uniform) wire that carries it. With alternating current of any frequency, the current is forced towards the outer surface of the wire, and away from the center. This is due to the fact that an electric charge which accelerates (as is the case of an alternating current) radiates electromagnetic waves, and materials of high conductivity (the metal which makes up the wire) do not allow propagation of electromagnetic waves. This phenomenon is called skin effect.
At very high frequencies the current no longer flows in the wire, but effectively flows on the surface of the wire, within a thickness of a few skin depths. The skin depth is the thickness at which the current density is reduced by 63%. Even at relatively low frequencies used for high power transmission (50-60 Hz), non-uniform distribution of current still occurs in sufficiently thick conductors. For example, the skin depth of a copper conductor is approximately 8.57mm at 60 Hz, so high current conductors are usually hollow to reduce their mass and cost.
Since the current tends to flow in the periphery of conductors, the effective cross-section of the conductor is reduced. This increases the effective AC resistance of the conductor, since resistance is inversely proportional to the cross-sectional area in which the current actually flows. The AC resistance often is many times higher than the DC resistance, causing a much higher energy loss due to ohmic heating (also called I2R loss).
Techniques for reducing AC resistance
For low to medium frequencies, conductors can be divided into stranded wires, each insulated from one other, and the individual strands specially arranged to change their relative position within the conductor bundle. Wire constructed using this technique is called Litz wire. This measure helps to partially mitigate skin effect by forcing more equal current flow throughout the total cross section of the stranded conductors. Litz wire is used for making high Q inductors, reducing losses in flexible conductors carrying very high currents at power frequencies, and in the windings of devices carrying higher radio frequency current (up to hundreds of kilohertz), such as switch-mode power supplies and radio frequency transformers.
Techniques for reducing radiation loss
As written above, an alternating current is made of electric charge under periodic acceleration, which causes radiation of electromagnetic waves. Energy that is radiated represents a loss. Depending on the frequency, different techniques are used to minimize the loss due to radiation.
Twisted pairs
At frequencies up to about 1 GHz, wires are paired together in cabling to form a twisted pair in order to reduce losses due to electromagnetic radiation and inductive coupling. A twisted pair must be used with a balanced signalling system, where the two wires carry equal but opposite currents. The result is that each wire in the twisted pair radiates a signal that is effectively cancelled by the other wire, resulting in almost no electromagnetic radiation.
Coax cables
At frequencies above 1 GHz, unshielded wires of practical dimensions lose too much energy to radiation, so coaxial cables are used instead. A coaxial cable has a conductive wire inside a conductive tube. The current flowing on the inner conductor is equal and opposite to the current flowing on the inner surface of the outer tube. This causes the electromagnetic field to be completely contained within the tube, and (ideally) no energy is radiated or coupled outside the tube. Coaxial cables have acceptably small losses for frequencies up to about 20 GHz. For microwave frequencies greater than 20 GHz, the dielectric losses (due mainly to the dissipation factor of the dielectric layer which separates the inner wire from the outer tube) become too large, making waveguides a more efficient medium for transmitting energy.
Waveguides
Waveguides are similar to coax cables, as both consist of tubes, with the biggest difference being that the waveguide has no inner conductor. Waveguides can have any arbitrary cross section, but rectangular cross section are the most common. With waveguides, the energy is no longer carried by an electric current, but by a guided electromagnetic field. Waveguides have dimensions comparable to the wavelength of the alternating current to be transmitted, so are only feasible at microwave frequencies.
Fiber optics
At frequencies greater than 200 GHz, waveguide dimensions become impractically too small, and the ohmic losses in the waveguide walls become large. Instead, fiber optics, which are a form of dielectric waveguides, can be used. For such frequencies, the concepts of voltages and currents are no longer used.
Mathematics of AC voltages
A sine wave, over one cycle (360°). The dashed line represents the root mean square (RMS) value
Alternating currents are accompanied by alternating voltages. An AC voltage v can be described mathematically as a function of time by the following equation:
,
where
Vpeak is the peak voltage (unit: volt),
ω is the angular frequency (unit: radians per second), and
t is the time (unit: second).
Since angular frequency is of more interest to mathematicians than to engineers and technicians, this is commonly rewritten as:
,
where
f is the frequency (unit: hertz).
The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak. Since the maximum value of sin(x) is +1 and the minimum value is −1, an AC voltage swings between +Vpeak and −Vpeak. The peak-to-peak voltage, usually written as Vpp or VP-P, is therefore (+Vpeak) − (−Vpeak) = 2 × Vpeak.
AC voltage is usually expressed as a root mean square (RMS) value, written Vrms. For a sinusoidal voltage:
Vrms is useful in calculating the power consumed by a load. If a DC voltage of VDC delivers a certain power P into a given load, then an AC voltage of Vpeak will deliver the same average power P into the same load if Vrms = VDC. Because of this fact, RMS is the normal means of measuring AC voltage.
Example
To illustrate these concepts, consider a 240 V AC mains supply. It is so called because its RMS value is (at least nominally) 240 V. This means that it has the same heating effect as 240 V DC. To work out its peak voltage (amplitude), we can modify the above equation to:
For our 240 V AC, the peak voltage Vpeak is therefore 240 V × √2, which is about 339 V. The peak-to-peak value VP-P of the 240 V AC mains is even higher: 2 × 240 V × √2, or about 679 V.
Note that non-sinusoidal waveforms have a different relationship between their peak magnitude and effective (RMS) value. This is of practical significance when working with non-linear circuit elements that produce harmonic currents, such as rectifiers.
The European Union (including the UK) has now officially harmonized on a supply of 230 V 50 Hz. However, it made the tolerance bands very wide at ±10%. Some countries actually specify stricter standards than this; for example, the UK specifies 230 V +10% −6%. Most supplies to the old standards therefore conform to the new one and do not need to be changed.
Power factor
The power factor of an AC electric power system is defined as the ratio of the real power to the apparent power.
Explanation
Instantaneous and average power calculated from AC voltage and current with a unity power factor (φ=0, cosφ=1)
Instantaneous and average power calculated from AC voltage and current with a zero power factor (φ=90, cosφ=0)
Instantaneous and average power calculated from AC voltage and current with a lagging power factor (φ=45, cosφ=0.71)
In a purely resistive AC circuit, voltage and current waveforms are in step, changing polarity at the same instant in each cycle. Where reactive loads are present, such as capacitors or inductors, energy storage in the loads result in a time difference between the current and voltage waveforms. Since this stored energy returns to the source and is not available to do work at the load, a circuit with a low power factor will have higher currents to transfer a given quantity of power than a circuit with a high power factor.
Real power is the capacity of the circuit for performing work in a particular time. Due to reactive elements of the load, the apparent power, which is the product of the voltage and current in the circuit, will be equal to or greater than the real power. The reactive power is a measure of the stored energy that is reflected to the source during each alternating current cycle.
AC power flow has the three components: real power (P), measured in watts (W); apparent power (S), measured in volt-amperes (VA); and reactive power (Q), measured in reactive volt-amperes (VAr).
The power factor can be expressed as:
.
In the case of a perfectly sinusoidal waveform, P, Q and S can be expressed as vectors that form a vector triangle such that:
If φ is the phase angle between the current and voltage, then the power factor is equal to , and:
By definition, the power factor is a dimensionless number between 0 and 1. When power factor is equal to 0, the energy flow is entirely reactive, and stored energy in the load returns to the source on each cycle. When the power factor is 1, all the energy supplied by the source is consumed by the load. Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle.
The power factor is determined by the type of loads connected to the power system. These can be
Resistive
Inductive
Capacitive
If a purely resistive load is connected to a power supply, current and voltage will change polarity in phase, the power factor will be unity (1), and the electrical energy flows in a single direction across the network in each cycle. Inductive loads such as transformers and motors (any type of wound coil) generate reactive power with current waveform lagging the voltage. Capacitive loads such as capacitor banks or buried cable generate reactive power with current phase leading the voltage. Both types of loads will absorb energy during part of the AC cycle, only to send this energy back to the source during the rest of the cycle.
For example, to get 1 kW of real power if the power factor is unity, 1 kVA of apparent power needs to be transferred (1 kW ÷ 1 = 1 kVA). At low values of power factor, more apparent power needs to be transferred to get the same real power. To get 1 kW of real power at 0.2 power factor 5 kVA of apparent power needs to be transferred (1 kW ÷ 0.2 = 5 kVA).
It is often possible to adjust the power factor of a system to very near unity. This practice is known as power factor correction and is achieved by switching in or out banks of inductors or capacitors. For example the inductive effect of motor loads may be offset by locally connected capacitors.
Energy losses in transmission lines increase with increasing current. Where a load has a power factor lower than 1, more current is required to deliver the same amount of useful energy. Power companies therefore require that customers, especially those with large loads, maintain the power factors of their respective loads within specified limits or be subject to additional charges. Engineers are often interested in the power factor of a load as one of the factors that affect the efficiency of power transmission.
Non-sinusoidal components
In circuits having only sinusoidal currents and voltages, the power factor effect arises only from the difference in phase between the current and voltage. This is narrowly known as "displacement power factor". The concept can be generalized to a total, distortion, or true power factor where the apparent power includes all harmonic components. This is of importance in practical power systems which contain non-linear loads such as rectifiers, some forms of electric lighting, electric arc furnaces, welding equipment, switched-mode power supplies and other devices.
A particularly important example is the millions of personal computers that typically incorporate switched-mode power supplies (SMPS) with rated output power ranging from 150W to 500W. Historically, these very low cost power supplies incorporated a simple full wave rectifier that conducted only when the mains instantaneous voltage exceeded the voltage on the input capacitors. This leads to very high ratios of peak to average input current, which also lead to a low distortion power factor and potentially serious phase and neutral loading concerns.
Regulatory agencies such as the EC have set harmonic limits as a method of improving power factor. Declining component cost has hastened acceptance and implementation of two different methods. Normally, this is done by either adding a series inductor (so-called passive PFC) or the addition of a boost converter that forces a sinusoidal input (so-called active PFC). For example, SMPS with passive PFC can achieve power factor of about 0.7...0.75, SMPS with active PFC -- up to 0.99, while SMPS without any power factor correction has power factor of about 0.55...0.65 only.
To comply with current EU standard EN61000-3-2 all switched-mode power supplies with output power more than 75W must include at least passive PFC.
A typical multimeter will give incorrect results when attempting to measure the AC current drawn by a non-sinusoidal load and then calculate the power factor. A true RMS multimeter must be used to measure the actual RMS currents and voltages (and therefore apparent power). To measure the real power or reactive power, a wattmeter designed to properly work with non-sinusoidal currents must be used.