physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

and

engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

, a phasor (a

portmanteau A portmanteau word, or portmanteau (, ) is a blend of words) is a

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

representing a sinusoidal function whose

amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...

(''A''),

angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...

(''ω''), and initial phase (''θ'') are

time-invariant In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...

. It is related to a more general concept called analytic representation,Bracewell, Ron. ''The Fourier Transform and Its Applications''. McGraw-Hill, 1965. p269 which decomposes a sinusoid into the product of a complex constant and a factor depending on time and frequency. The complex constant, which depends on amplitude and phase, is known as a phasor, or complex amplitude, and (in older texts) sinor or even complexor. A common situation in

electrical network An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...

s powered by time varying current 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 represented as a linear combination of phasors (known as phasor arithmetic or phasor algebra) 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. An important additional feature of the phasor transform is that differentiation and

integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...

of sinusoidal signals (having constant amplitude, period and phase) corresponds to simple

algebraic operation Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...

s on the phasors; the phasor transform thus allows the

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

(calculation) of the AC

steady state In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...

RLC circuit An RLC circuit is an electrical circuit consisting of a electrical resistance, resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the ...

s by solving simple

algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...

s (albeit with complex coefficients) in the phasor domain instead of solving

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s (with

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

coefficients) in the time domain. The originator of the phasor transform was

Charles Proteus Steinmetz Charles Proteus Steinmetz (born Karl August Rudolph Steinmetz, April 9, 1865 – October 26, 1923) was a German-born American mathematician and electrical engineer and professor at Union College. He fostered the development of alternati ...

working at

General Electric General Electric Company (GE) is an American multinational conglomerate founded in 1892, and incorporated in New York state and headquartered in Boston. The company operated in sectors including healthcare, aviation, power, renewable energ ...

in the late 19th century. Glossing over some mathematical details, the phasor transform can also be seen as a particular case of the

Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integra ...

, which additionally can be used to (simultaneously) derive the

transient response In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. The transient response is not necessarily tied to abrupt events but to any event that affec ...

of an RLC circuit. However, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required. unfasor

Notation

Phasor notation (also known as angle notation) is a

mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematic ...

used in

electronics engineering Electronics engineering is a sub-discipline of electrical engineering which emerged in the early 20th century and is distinguished by the additional use of active components such as semiconductor devices to amplify and control electric current f ...

and

electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

1 \angle \theta

can represent either the

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

(\cos \theta,\, \sin \theta)

or the

\cos \theta + i \sin \theta = e^

, with

i^2 = -1

, both of which have magnitudes of 1. A vector whose

polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...

are magnitude

A

and

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

\theta

is written

A \angle \theta.

The angle may be stated in degrees with an implied conversion from degrees to

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

s. For example

1 \angle 90

would be assumed to be

1 \angle 90^\circ,

which is the vector

(0,\, 1)

or the number

e^ = i.

Definition

A real-valued sinusoid with constant amplitude, frequency, and phase has the form: :

A\cos(\omega t + \theta),

where only parameter

t

is time-variant. The inclusion of an imaginary component: :

i \cdot A\sin(\omega t + \theta)

gives it, in accordance with

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...

, the factoring property described in the lede paragraph: :

A\cos(\omega t + \theta) + i\cdot A\sin(\omega t + \theta) = A  e^ = A e^ \cdot e^,

whose real part is the original sinusoid. The benefit of the complex representation is that linear operations with other complex representations produces a complex result whose real part reflects the same linear operations with the real parts of the other complex sinusoids. Furthermore, all the mathematics can be done with just the phasors

A e^,

and the common factor

e^

is reinserted prior to the real part of the result. The function

Ae^

is called the '' analytic representation'' of

A\cos(\omega t + \theta).

Figure 2 depicts it as a rotating vector in the complex plane. It is sometimes convenient to refer to the entire function as a ''phasor'', as we do in the next section. But the term ''phasor'' usually implies just the static complex number

A e^.

Arithmetic

Multiplication by a constant (scalar)

Multiplication of the phasor

A e^ e^

by a complex constant,

B e^

, produces another phasor. That means its only effect is to change the amplitude and phase of the underlying sinusoid:

\begin
      &\operatorname\left( \left(A e^ \cdot B e^\right) \cdot e^ \right) \\
  = &\operatorname\left( \left(AB e^\right) \cdot e^ \right) \\
  = &AB \cos(\omega t + (\theta + \phi)).
\end

In electronics,

B e^

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 non-linear operation that produces new frequency components. Phasor notation can only represent systems with one frequency, such as a linear system stimulated by a sinusoid.

Addition

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:

= &A_3 \cos(\omega t + \theta_3), \end

where:

A_3^2 = (A_1 \cos\theta_1 + A_2 \cos \theta_2)^2 + (A_1 \sin\theta_1 + A_2 \sin\theta_2)^2,

and, if we take

\theta_3 \in \left \frac, \frac\right /math>, then \theta_3 is:
* \sgn(A_1 \sin(\theta_1)  + A_2 \sin(\theta_2)) \cdot \frac, if A_1 \cos\theta_1 + A_2 \cos\theta_2 = 0, with \sgn the

signum function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avo ...

; *

\arctan\left(\frac\right),

A_1 \cos\theta_1 + A_2 \cos\theta_2 > 0

; *

\pi + \arctan\left(\frac\right),

A_1 \cos\theta_1 + A_2 \cos\theta_2 < 0

. or, via the

law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

on the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

(or the trigonometric identity for angle differences):

A_3^2 = A_1^2 + A_2^2 - 2 A_1 A_2 \cos(180^\circ - \Delta\theta)
        = A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\Delta\theta),

where

\Delta\theta = \theta_1 - \theta_2.

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 re-inserted into the outcome as long as the only operations used in between are ones that produce another phasor. In

angle notation In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sine wave, sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and Phase (waves), initial phase (''θ'') are time-inva ...

, the operation shown above is written:

A_1 \angle \theta_1 + A_2 \angle \theta_2 = A_3 \angle \theta_3.

Another way to view addition is that two vectors with coordinates and are added vectorially to produce a resultant vector with coordinates (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 A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

, so the angle between each phasor to the next is 120° ( radians), or one third of a wavelength . So the phase difference between each wave must also be 120°, as is the case in

three-phase power Three-phase electric power (abbreviated 3φ) is a common type of alternating current used in electricity generation, transmission, and distribution. It is a type of polyphase system employing three wires (or four including an optional neutral ...

. In other words, what this shows is that:

\cos(\omega t) + \cos\left(\omega t + \frac\right) + \cos\left(\omega t - \frac\right) = 0.

In the example of three waves, the phase difference between the first and the last wave was 240°, while for two waves destructive interference happens at 180°. 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

\lambda

. This is why in single slit

diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...

, the minima occur when

light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...

from the far edge travels a full wavelength further than the light from the near edge. As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360° or 2 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, . 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, () at 90° or and the negative peak value, () at 270° or . 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 represents a scaled voltage or current value of a rotating vector which is "frozen" at some point in time, () 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 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, of the waveform. Consider the diagram below from the previous Phase Difference tutorial.

Differentiation and integration

The time

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

of a phasor produces another phasor. For example:

\begin
      &\operatorname\left( \frac \mathord\left(A e^ \cdot e^\right) \right) \\
  = &\operatorname\left( A e^ \cdot i\omega e^ \right) \\
  = &\operatorname\left( A e^ \cdot e^ \omega e^ \right) \\
  = &\operatorname\left( \omega A e^ \cdot e^ \right) \\
  = &\omega A \cdot \cos\left(\omega t + \theta + \frac\right).
\end

Therefore, in phasor representation, the time derivative of a sinusoid becomes just multiplication by the constant

i \omega = e^ \cdot \omega

. Similarly, integrating a phasor corresponds to multiplication by

\frac = \frac.

The time-dependent factor,

e^,

is unaffected. When we solve a

linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...

with phasor arithmetic, we are merely factoring

e^

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 A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...

in an

RC circuit A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC c ...

\frac + \fracv_\text(t) = \frac v_\text(t).

When the voltage source in this circuit is sinusoidal:

v_\text(t) = V_\text \cdot \cos(\omega t + \theta),

we may substitute

v_\text(t) = \operatorname\left( V_\text \cdot e^ \right).

v_\text(t) = \operatorname\left(V_\text \cdot e^ \right),

where phasor

V_\text = V_\text e^,

and phasor

V_\text

is the unknown quantity to be determined. In the phasor shorthand notation, the differential equation reduces to:

i \omega V_\text + \frac V_\text = \fracV_\text.

Solving for the phasor capacitor voltage gives:

V_\text = \frac \cdot V_\text = \frac \cdot V_\text e^.

As we have seen, the factor multiplying

V_\text

represents differences of the amplitude and phase of

v_\text(t)

relative to

V_\text

and

\theta.

In polar coordinate form, it is:

\frac\cdot e^,

where

\phi(\omega) = \arctan(\omega RC)

. Therefore:

v_\text(t) = \frac\cdot V_\text \cos(\omega t + \theta - \phi(\omega)).

Ratio of phasors

A quantity called complex impedance is the ratio of two phasors, which is not a phasor, because it does not correspond to a sinusoidally varying function.

Applications

Circuit laws

With phasors, the techniques for solving DC circuits can be applied to solve linear AC circuits. ; Ohm's law for resistors: A

resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...

has no time delays and therefore doesn't change the phase of a signal therefore remains valid. ; Ohm's law for resistors, inductors, and capacitors: where is the complex impedance. ;

Kirchhoff's circuit laws Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhof ...

: Work with voltages and current as complex phasors. In an AC circuit we have real power () which is a representation of the average power into the circuit and reactive power (''Q'') which indicates power flowing back and forth. We can also define the complex power and the apparent power which is the magnitude of . The power law for an AC circuit expressed in phasors is then (where is the

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

of , and the magnitudes of the voltage and current phasors and of are the RMS values of the voltage and current, respectively). Given this we can apply the techniques of

analysis of resistive circuits A network, in the context of electrical engineering and electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many ...

with phasors to analyze single frequency linear AC circuits containing resistors, capacitors, and

inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...

s. 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 (using

Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

) with magnitude and phase then analyzing each frequency separately, as allowed by the

superposition theorem The superposition theorem is a derived result of the superposition principle suited to the network analysis of electrical circuits. The superposition theorem states that for a linear system (notably including the subcategory of time-invariant l ...

. This solution method applies only to inputs that are sinusoidal and for solutions that are in steady state, i.e., after all transients have died out. The concept is frequently involved in representing an

electrical impedance In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit. Quantitatively, the impedance of a two-terminal circuit element is the ratio of the comp ...

. In this case, the phase angle is the

phase difference In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...

between the voltage applied to the impedance and the current driven through it.

Power engineering

In analysis of

three phase Three-phase electric power (abbreviated 3φ) is a common type of alternating current used in electricity generation, transmission, and distribution. It is a type of polyphase system employing three wires (or four including an optional neutral ...

AC power systems, usually a set of phasors is defined as the three complex

cube roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...

, 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 In electrical engineering, the method of symmetrical components simplifies analysis of unbalanced three-phase power systems under both normal and abnormal conditions. The basic idea is that an asymmetrical set of ''N'' phasors can be expressed as a ...

. This approach greatly simplifies the work required in electrical calculations of voltage drop, power flow, and short-circuit currents. In the context of power systems analysis, the phase angle is often given in

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

s, 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.

Telecommunications: analog modulations

The rotating frame picture using phasor can be a powerful tool to understand analog modulations such as

amplitude modulation Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting messages with a radio wave. In amplitude modulation, the amplitude (signal strength) of the wave is varied in proportion to ...

(and its variantsde Oliveira, H.M. and Nunes, F.D. ''About the Phasor Pathways in Analogical Amplitude Modulations''. International Journal of Research in Engineering and Science (IJRES) Vol.2, N.1, Jan., pp.11-18, 2014. ISSN 2320-9364 ) and

frequency modulation Frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave. The technology is used in telecommunications, radio broadcasting, signal processing, and Run-length limited#FM: .280. ...

x(t) = \operatorname\left( A e^ \cdot e^ \right),

where the term in brackets is viewed as a rotating vector in the complex plane. The phasor has length

A

, rotates anti-clockwise at a rate of

f_0

revolutions per second, and at time

t = 0

makes an angle of

\theta

with respect to the positive real axis. The waveform

x(t)

can then be viewed as a projection of this vector onto the real axis. A modulated waveform is represented by this phasor (the carrier) and two additional phasors (the modulation phasors). If the modulating signal is a single tone of the form

Am \cos

, where

m

is the modulation depth and

f_m

is the frequency of the modulating signal, then for amplitude modulation the two modulation phasors are given by,

Am e^ \cdot e^

, and

Am e^ \cdot e^

. The two modulation phasors are phased such that their vector sum is always in phase with the carrier phasor. An alternative representation is two phasors counter rotating around the end of the carrier phasor at a rate

f_m

relative to the carrier phasor. That is,

Am e^ \cdot e^

, and

Am e^ \cdot e^

. Frequency modulation is a similar representation except that the modulating phasors are not in phase with the carrier. In this case the vector sum of the modulating phasors is shifted 90° from the carrier phase. Strictly, frequency modulation representation requires additional small modulation phasors at

2f_m, 3f_m

etc, but for most practical purposes these are ignored because their effect is very small.

Footnotes

References

External links

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has a lesson on ''{{#if:{{{1, {{{1}, Phasor algebra''
Phasor Phactory

Electrical circuits AC power Interference Trigonometry