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 A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in mat ...

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. It is related to a more general concept called

analytic representation In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hilber ...

,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 networks 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 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. The originator of the phasor transform was Charles Proteus Steinmetz working at General Electric 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, which additionally can be used to (simultaneously) derive the transient response 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 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

(\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 are magnitude

A

and angle

\theta

is written

A \angle \theta.

The angle may be stated in degrees with an implied conversion from degrees to radians. 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, 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 ''

'' 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;
* \arctan\left(\frac\right), if A_1 \cos\theta_1 + A_2 \cos\theta_2 > 0;
* \pi + \arctan\left(\frac\right), if A_1 \cos\theta_1 + A_2 \cos\theta_2 < 0 .

or, via the law of cosines 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, 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, 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. 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 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 or integral 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 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 in an RC circuit:

\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: 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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

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 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 with phasors to analyze single frequency linear 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 (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. 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 this case, the phase angle is the phase difference between the voltage applied to the impedance and the current driven through it.

Power engineering

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 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 mathemati ...

s, and the magnitude in rms value rather than the peak amplitude of the sinusoid. The technique of

synchrophasor A phasor measurement unit (PMU) is a device used to estimate the magnitude and phase angle of an electrical phasor quantity (such as voltage or current) in the electricity grid using a common time source for synchronization. Time synchronization i ...

s 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.

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

{{Sister project, project = wikiversity, text = Wikiversity has a lesson on ''{{#if:{{{1, {{{1}, Phasor algebra''
Phasor Phactory

Electrical circuits AC power Interference Trigonometry