In physics and engineering, a phasor (a portmanteau of phase vector^{[1]}^{[2]}), is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are timeinvariant. It is related to a more general concept called analytic representation,^{[3]} which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude,^{[4]}^{[5]} and (in older texts) sinor^{[6]} or even complexor.^{[6]}
A common situation in electrical networks is the existence of multiple sinusoids all with the same frequency, but different amplitudes and phases. The only difference in their analytic representations is the complex amplitude (phasor). A linear combination of such functions can be factored into the product of a linear combination of phasors (known as phasor arithmetic) and the time/frequency dependent factor that they all have in common.
The origin of the term phasor rightfully suggests that a (diagrammatic) calculus somewhat similar to that possible for vectors is possible for phasors as well.^{[6]} An important additional feature of the phasor transform is that differentiation and integration of sinusoidal signals (having constant amplitude, period and phase) corresponds to simple algebraic operations on the phasors; the phasor transform thus allows the analysis (calculation) of the AC steady state of RLC circuits by solving simple algebraic equations (albeit with complex coefficients) in the phasor domain instead of solving differential equations (with real coefficients) in the time domain.^{[7]}^{[8]} The originator of the phasor transform was Charles Proteus Steinmetz working at General Electric in the late 19th century.^{[9]}^{[10]}
Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the Laplace transform, which additionally can be used to (simultaneously) derive the transient response of an RLC circuit.^{[8]}^{[10]} However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required.^{[10]}
Euler's formula indicates that sinusoids can be represented mathematically as the sum of two complexvalued functions:
or as the real part of one of the functions:
The function is called the analytic representation of . Figure 2 depicts it as a rotating vector in a complex plane. It is sometimes convenient to refer to the entire function as a phasor,^{[11]} as we do in the next section. But the term phasor usually implies just the static vector . An even more compact representation of a phasor is the angle notation: . See also vector notation.
Multiplication of the phasor by a complex constant, , produces another phasor. That means its only effect is to change the amplitude and phase of the underlying sinusoid:
In electronics, would represent an impedance, which is independent of time. In particular it is not the shorthand notation for another phasor. Multiplying a phasor current by an impedance produces a phasor voltage. But the product of two phasors (or squaring a phasor) would represent the product of two sinusoids, which is a nonlinear operation that produces new frequency components. Phasor notation can only represent systems with one frequency, such as a linear system stimulated by a sinusoid.
The time derivative or integral of a phasor produces another phasor.^{[b]} For example:
Therefore, in phasor representation, the time derivative of a sinusoid becomes just multiplication by the constant .
Similarly, integrating a phasor corresponds to multiplication by . The timedependent factor, , is unaffected.
When we solve a linear differential equation with phasor arithmetic, we are merely factoring out of all terms of the equation, and reinserting it into the answer. For example, consider the following differential equation for the voltage across the capacitor in an RC circuit:
When the voltage source in this circuit is sinusoidal:
we may substitute
where phasor , and phasor is the unknown quantity to be determined.
In the phasor shorthand notation, the differential equation reduces to
Solving for the phasor capacitor voltage gives
As we have seen, the factor multiplying represents differences of the amplitude and phase of relative to and .
In polar coordinate form, it is
Therefore
The sum of multiple phasors produces another phasor. That is because the sum of sinusoids with the same frequency is also a sinusoid with that frequency:
where
or, via the law of cosines on the complex plane (or the trigonometric identity for angle differences):
where .
A key point is that A_{3} and θ_{3} do not depend on ω or t, which is what makes phasor notation possible. The time and frequency dependence can be suppressed and reinserted into the outcome as long as the only operations used in between are ones that produce another phasor. In angle notation, the operation shown above is written
Another way to view addition is that two vectors with coordinates [ A_{1} cos(ωt + θ_{1}), A_{1} sin(ωt + θ_{1}) ] and [ A_{2} cos(ωt + θ_{2}), A_{2} sin(ωt + θ_{2}) ] are added vectorially to produce a resultant vector with coordinates [ A_{3} cos(ωt + θ_{3}), A_{3} sin(ωt + θ_{3}) ]. (see animation)
In physics, this sort of addition occurs when sinusoids interfere with each other, constructively or destructively. The static vector concept provides useful insight into questions like this: "What phase difference would be required between three identical sinusoids for perfect cancellation?" In this case, simply imagine taking three vectors of equal length and placing them head to tail such that the last head matches up with the first tail. Clearly, the shape which satisfies these conditions is an equilateral triangle, so the angle between each phasor to the next is 120° (^{3π}⁄_{2} radians), or one third of a wavelength ^{λ}⁄_{3}. So the phase difference between each wave must also be 120°, as is the case in threephase power
In other words, what this shows is that
In the example of three waves, the phase difference between the first and the last wave was 240 degrees, while for two waves destructive interference happens at 180 degrees. In the limit of many waves, the phasors must form a circle for destructive interference, so that the first phasor is nearly parallel with the last. This means that for many sources, destructive interference happens when the first and last wave differ by 360 degrees, a full wavelength . This is why in single slit diffraction, the minima occur when light from the far edge travels a full wavelength further than the light from the near edge.
As the single vector rotates in an anticlockwise direction, its tip at point A will rotate one complete revolution of 360° or 2π radians representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When the vector is horizontal the tip of the vector represents the angles at 0°, 180°, and at 360°.
Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +A_{max} ) at 90° or ^{π}⁄_{2} and the negative peak value, ( −A_{max} ) at 270° or ^{3π}⁄_{2}. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represent a scaled voltage or current value of a rotating vector which is “frozen” at some point in time, ( t ) and in our example above, this is at an angle of 30°.
Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the alternating quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time t = 0 with a corresponding phase angle in either degrees or radians.
But if a second waveform starts to the left or to the right of this zero point, or if we want to represent in phasor notation the relationship between the two waveforms, then we will need to take into account this phase difference, Φ of the waveform. Consider the diagram below from the previous Phase Difference tutorial.
With phasors, the techniques for solving DC circuits can be applied to solve AC circuits. A list of the basic laws is given below.
Given this we can apply the techniques of analysis of resistive circuits with phasors to analyze single frequency AC circuits containing resistors, capacitors, and inductors. Multiple frequency linear AC circuits and AC circuits with different waveforms can be analyzed to find voltages and currents by transforming all waveforms to sine wave components with magnitude and phase then analyzing each frequency separately, as allowed by the superposition theorem.
In analysis of three phase AC power systems, usually a set of phasors is defined as the three complex cube roots of unity, graphically represented as unit magnitudes at angles of 0, 120 and 240 degrees. By treating polyphase AC circuit quantities as phasors, balanced circuits can be simplified and unbalanced circuits can be treated as an algebraic combination of symmetrical components. This approach greatly simplifies the work required in electrical calculations of voltage drop, power flow, and shortcircuit currents. In the context of power systems analysis, the phase angle is often given in degrees, and the magnitude in rms value rather than the peak amplitude of the sinusoid.
The technique of synchrophasors uses digital instruments to measure the phasors representing transmission system voltages at widespread points in a transmission network. Differences among the phasors indicate power flow and system stability.
The rotating frame picture using phasor can be a powerful tool to understand analog modulations such as amplitude modulation (and its variants ^{[12]} ) and frequency modulation.
, where the term in brackets is viewed as a rotating vector in the complex plane.
The phasor has length , rotates anticlockwise at a rate of revolutions per second, and at time makes an angle of with respect to the positive real axis.
The waveform can then be viewed as a projection of this vector onto the real axis.


(Eq.1) 
Since this must hold for all , specifically: , it follows that


(Eq.2) 
It is also readily seen that
Substituting these into Eq.1 and Eq.2, multiplying Eq.2 by and adding both equations gives
Additive white Gaussian noise (AWGN) is a basic noise model used in Information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:
Additive because it is added to any noise that might be intrinsic to the information system.
White refers to the idea that it has uniform power across the frequency band for the information system. It is an analogy to the color white which has uniform emissions at all frequencies in the visible spectrum.
Gaussian because it has a normal distribution in the time domain with an average time domain value of zero.Wideband noise comes from many natural noise, such as the thermal vibrations of atoms in conductors (referred to as thermal noise or JohnsonNyquist noise), shot noise, black body radiation from the earth and other warm objects, and from celestial sources such as the Sun. The central limit theorem of probability theory indicates that the summation of many random processes will tend to have distribution called Gaussian or Normal.
AWGN is often used as a channel model in which the only impairment to communication is a linear addition of wideband or white noise with a constant spectral density (expressed as watts per hertz of bandwidth) and a Gaussian distribution of amplitude. The model does not account for fading, frequency selectivity, interference, nonlinearity or dispersion. However, it produces simple and tractable mathematical models which are useful for gaining insight into the underlying behavior of a system before these other phenomena are considered.
The AWGN channel is a good model for many satellite and deep space communication links. It is not a good model for most terrestrial links because of multipath, terrain blocking, interference, etc. However, for terrestrial path modeling, AWGN is commonly used to simulate background noise of the channel under study, in addition to multipath, terrain blocking, interference, ground clutter and self interference that modern radio systems encounter in terrestrial operation.
Analytic signalIn mathematics and signal processing, an analytic signal is a complexvalued function that has no negative frequency components. The real and imaginary parts of an analytic signal are realvalued functions related to each other by the Hilbert transform.
The analytic representation of a realvalued function is an analytic signal, comprising the original function and its Hilbert transform. This representation facilitates many mathematical manipulations. The basic idea is that the negative frequency components of the Fourier transform (or spectrum) of a realvalued function are superfluous, due to the Hermitian symmetry of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complexvalued function instead. That makes certain attributes of the function more accessible and facilitates the derivation of modulation and demodulation techniques, such as singlesideband.
As long as the manipulated function has no negative frequency components (that is, it is still analytic), the conversion from complex back to real is just a matter of discarding the imaginary part. The analytic representation is a generalization of the phasor concept: while the phasor is restricted to timeinvariant amplitude, phase, and frequency, the analytic signal allows for timevariable parameters.
Angle notationAngle notation or phasor notation is a notation used in electronics. can represent either the vector or the complex number , with , both of which have magnitudes of 1. A vector whose polar coordinates are magnitude and angle is written To convert between polar and rectangular forms, see Converting between polar and Cartesian coordinates.
In electronics and electrical engineering, there may also be an implied conversion from degrees to radians. For example would be assumed to be which is the vector or the number
Chris OberthChristian H. "Chris" Oberth (May 17, 1953 – July 14, 2012) was a game programmer who created early titles for the Apple II family of personal computers, handheld electronic games for Milton Bradley, and games for coinoperated arcade machines published in the early 1980s. Though not a hit in arcades, Oberth's 1982 Anteater for Stern Electronics was an influential concept, cloned by a number of developers for 8bit home computers, including Sierra OnLine (as Oil's Well). The following year he wrote his own home version as Ardy the Aardvark (Datamost, 1983).
Oberth's first commercial games, Phasor Zap (1978) and 3D Docking Mission (1978) for the Apple II, were published by Programma International, a company which also published games from future arcade game designers Bob Flanagan and Gary Shannon as well as rejecting the first effort from Mark Turmell. His next thirteen Apple II games, in addition to Phasor Zap and 3D Docking Mission, were published by The Elektrik Keyboard, a musical instrument and computer store in Chicago where Oberth was head of the computer department.
Dielectric lossDielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy (e.g. heat). It can be parameterized in terms of either the loss angle δ or the corresponding loss tangent tan δ. Both refer to the phasor in the complex plane whose real and imaginary parts are the resistive (lossy) component of an electromagnetic field and its reactive (lossless) counterpart.
Electrical impedanceElectrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.
Quantitatively, the impedance of a twoterminal circuit element is the ratio of the complex representation of a sinusoidal voltage between its terminals to the complex representation of the current flowing through it. In general, it depends upon the frequency of the sinusoidal voltage.
Impedance extends the concept of resistance to AC circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude. When a circuit is driven with direct current (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.
The notion of impedance is useful for performing AC analysis of electrical networks, because it allows relating sinusoidal voltages and currents by a simple linear law.
In multiple port networks, the twoterminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are still linearly related by the impedance matrix.Impedance is a complex number, with the same units as resistance, for which the SI unit is the ohm (Ω).
Its symbol is usually Z, and it may be represented by writing its magnitude and phase in the form Z∠θ. However, cartesian complex number representation is often more powerful for circuit analysis purposes.
The reciprocal of impedance is admittance, whose SI unit is the siemens, formerly called mho.
The class of instruments used to measure the electrical impedance is called impedance analyzer.
ImpedanceImpedance is the complexvalued generalization of resistance. It may refer to:
Acoustic impedance, a constant related to the propagation of sound waves in an acoustic medium
Electrical impedance, the ratio of the voltage phasor to the electric current phasor, a measure of the opposition to timevarying electric current in an electric circuit
High impedance, when only a small amount of current is allowed through
Characteristic impedance of a transmission line
Impedance (accelerator physics), a characterization of the self interaction of a charged particle beam
Nominal impedance, approximate designed impedance
Impedance matching, the adjustment of input impedance and output impedance
Mechanical impedance, a measure of opposition to motion of a structure subjected to a force
Wave impedance, a constant related to electromagnetic wave propagation in a medium
Impedance of free space, a universal constant and the simplest case of a wave impedance
KAAYKAAY (1090 kHz) is a commercial AM radio station in Little Rock, Arkansas, owned by Cumulus Media. It airs a religious format of instruction and preaching, with most of the schedule made up of paid brokered programming, featuring local and national religious leaders, including Charles Stanley, Jim Daly, John F. MacArthur and Albert Pendarvis. Overnight, automated Contemporary Christian music is heard. The station's studios are located in West Little Rock, and the transmitter is located off McDonald Road in Wrightsville, Arkansas.KAAY is a 50,000 watt clearchannel Class A radio station. But because 1090 AM is shared with two other Class A stations, WBAL Baltimore and XEPRS RosaritaTijuana, KAAY uses a directional antenna at night, nulling its signal away from the east and west.
MuTronMusitronics, often shortened to Mutron, was a manufacturer of electronic musical effects active in the 1970s. Founded by Mike Beigel and Aaron Newman, the company's product line focused on filtering and processing effects derived from synthesizer components. The company was known for producing highquality products with many useradjustable parameters, but high production costs and a failed product line, the Gizmotron, caused its downfall.
Their bestknown product was the Mutron III envelope filter, "the world's first envelopecontrolled filter", first made in 1972 and quickly becoming an essential effect for many funk musicians. It was taken in production again, in a modified version, in 2014.
OpenPDCThe openPDC is a complete set of applications for processing streaming timeseries data in realtime. The name stands for "open source phasor data concentrator" and was originally designed for the concentration and management of realtime streaming synchrophasors. Due to the system's modular design, the openPDC can be classified as a generic event stream processor.
Phase factorFor any complex number written in polar form (such as re^{iθ}), the phase factor is the complex exponential factor (e^{iθ}). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e., not necessarily part of the circle group). The phase factor is a unit complex number, i.e., of absolute value 1. It is commonly used in quantum mechanics.
The variable θ appearing in such an expression is generally referred to as the phase. Multiplying the equation of a plane wave Ae^{i(k·r−ωt)} by a phase factor shifts the phase of the wave by θ:
In quantum mechanics, a phase factor is a complex coefficient e^{iθ} that multiplies a ket or bra . It does not, in itself, have any physical meaning, since the introduction of a phase factor does not change the expectation values of a Hermitian operator. That is, the values of and are the same. However, differences in phase factors between two interacting quantum states can sometimes be measurable (such as in the Berry phase) and this can have important consequences.
In optics, the phase factor is an important quantity in the treatment of interference.
Phaser PatrolPhaser Patrol, written by Dennis Caswell, is the first numbered release by Arcadia for the Atari 2600 and was the packin game for the Atari 2600 Supercharger accessory in 1982. The company changed its name to Starpath after launch, and the hardware was rebranded the Starpath Supercharger. The game simulates space combat in which the player pilots a ship to destroy the Dracon invaders.
Phasor (disambiguation)Phasor is a phase vector representing a sine wave.
Phasor may also be:
Phasor (sound synthesizer), a stereo music, sound and speech synthesizer for the Apple II computer
Phasor measurement unit, a device that measures phasors on an electricity grid
Phasor (radio broadcasting), a network of inductors and capacitors used to control the relative amplitude and phase of the radio frequency currents driving a directional antenna array.
Phasor (radio broadcasting)A phasor is a network of capacitors and variable inductors used to adjust the relative amplitude and phase of the current being distributed to each tower in a directional array. A typical phasor has separate controls to adjust the phase of the current going to each tower, adjustable power divider controls, and a common point impedance matching network to adjust the system input impedance to 50 ohms with no reactance without disturbing the phase or amplitude of the tower currents.
Phasor (sound synthesizer)Phasor is a stereo music, sound and speech synthesizer created by Applied Engineering for the Apple II family of computers. Consisting of a sound card and a set of related software, the Phasor system was designed to be compatible with most software written for other contemporary Apple II cards, including the Mockingboard, ALF's Apple Music Synthesizer, Echo+ and Super Music Synthesizer.
Phasor ZapPhasor Zap is a game for the Apple II family of computers, created in 1978 by programmer Chris Oberth and published by The Elektrik Keyboard of Chicago, Illinois.
Phasor approach to fluorescence lifetime and spectral imagingPhasor approach refers to a method which is used for vectorial representation of sinusoidal waves like alternative currents and voltages or electromagnetic waves. The amplitude and the phase of the waveform is transformed into a vector where the phase is translated to the angle between the phasor vector and X axis and the amplitude is translated to vector length or magnitude.
In this concept the representation and the analysis becomes very simple and the addition of two wave forms is realized by their vectorial summation.
In Fluorescence lifetime and spectral imaging, phasor can be used to visualize the spectra and decay curves. In this method the Fourier transformation of the spectrum or decay curve is calculated and the resulted complex number is plotted on a 2D plot where the X axis represents the Real component and the Y axis represents the Imaginary component. This facilitate the analysis since each spectrum and decay is transformed into a unique position on the phasor plot which depends on its spectral width or emission maximum or to its average lifetime. The most important feature of this analysis is that it is fast and it provides a graphical representation of the measured curve.
Phasor measurement unitA phasor measurement unit (PMU) is a device used to estimate the magnitude and phase angle of an electrical Phasor quantity like voltage or current in the electricity grid using a common time source for synchronization. Time synchronization is usually provided by GPS and allows synchronized realtime measurements of multiple remote measurement points on the grid. PMUs are capable of capturing samples from a waveform in quick succession and reconstruct the Phasor quantity. The resulting measurement is known as a synchrophasor. These devices can also be used to measure the frequency in the power grid. A typical commercial PMU can report measurements with very high temporal resolution in the order of 3060 measurements per second. This helps engineers in analyzing dynamic events in the grid which is not possible with traditional SCADA measurements that generate one measurement every 2 or 4 seconds. Therefore, PMUs equip utilities with enhanced monitoring and control capabilities and are considered to be one of the most important measuring devices in the future of power systems. A PMU can be a dedicated device, or the PMU function can be incorporated into a protective relay or other device.
RL circuitA resistor–inductor circuit (RL circuit), or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A firstorder RL circuit is composed of one resistor and one inductor and is the simplest type of RL circuit.
A first order RL circuit is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series driven by a voltage source or in parallel driven by a current source.
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