Phasor

Last updated
An example of series RLC circuit and respective phasor diagram for a specific o. The arrows in the upper diagram are phasors, drawn in a phasor diagram (complex plane without axis shown), which must not be confused with the arrows in the lower diagram, which are the reference polarity for the voltages and the reference direction for the current Wykres wektorowy by Zureks.svg
An example of series RLC circuit and respective phasor diagram for a specific ω. The arrows in the upper diagram are phasors, drawn in a phasor diagram (complex plane without axis shown), which must not be confused with the arrows in the lower diagram, which are the reference polarity for the voltages and the reference direction for the current

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 time-invariant. 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 depending on time and frequency. The complex constant, which depends on amplitude and phase, is known as a phasor, or complex amplitude, [4] [5] and (in older texts) sinor [6] or even complexor. [6]

Contents

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]

Fig 2. When function
A
[?]
e
i
(
o
t
+
th
)
{\displaystyle \scriptstyle A\cdot e^{i(\omega t+\theta )}}
is depicted in the complex plane, the vector formed by its imaginary and real parts rotates around the origin. Its magnitude is A, and it completes one cycle every 2p/o seconds. th is the angle it forms with the real axis at t = n*2p/o, for integer values of n. Unfasor.gif
Fig 2. When function is depicted in the complex plane, the vector formed by its imaginary and real parts rotates around the origin. Its magnitude is A, and it completes one cycle every 2π/ω seconds. θ is the angle it forms with the real axis at t = n•2π/ω, for integer values of n.

Notation

Phasor notation (also known as angle notation) is a mathematical notation used in electronics engineering and electrical engineering. 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 [11]

The angle may be stated in degrees with an implied conversion from degrees to radians. For example would be assumed to be which is the vector or the number

Definition

Euler's formula indicates that sinusoids can be represented mathematically as the sum of two complex-valued functions:

   [lower-alpha 1]

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, [12] as we do in the next section. But the term phasor usually implies just the static complex number .

Arithmetic

Multiplication by a constant (scalar)

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 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 phasors as addition of rotating vectors Sumafasores.gif
The sum of phasors as addition of rotating vectors

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

and, if we take , then:
  • if , then , with the signum function;
  • if , then ;
  • if , then .

or, via the law of cosines on the complex plane (or the trigonometric identity for angle differences):

where .

A key point is that A3 and θ3 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

Another way to view addition is that two vectors with coordinates [ A1 cos(ωt + θ1), A1 sin(ωt + θ1) ] and [ A2 cos(ωt + θ2), A2 sin(ωt + θ2) ] are added vectorially to produce a resultant vector with coordinates [ A3 cos(ωt + θ3), A3 sin(ωt + θ3) ]. (see animation)

Phasor diagram of three waves in perfect destructive interference Destructive interference.png
Phasor diagram of three waves in perfect destructive interference

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° (2π3 radians), or one third of a wavelength λ3. 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

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 . 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 anti-clockwise 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, (+Amax) at 90° or π2 and the negative peak value, (−Amax) at 270° or 3π2. 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.

Differentiation and integration

The time derivative or integral of a phasor produces another phasor. [lower-alpha 2] 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 time-dependent 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

  [lower-alpha 3]

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

Applications

Circuit laws

With phasors, the techniques for solving DC circuits can be applied to solve linear 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 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) 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. [13]

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. 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 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.

Telecommunications: analog modulations

The rotating frame picture using phasor can be a powerful tool to understand analog modulations such as amplitude modulation (and its variants [14] ) and frequency modulation.

, where the term in brackets is viewed as a rotating vector in the complex plane.

The phasor has length , rotates anti-clockwise 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.

See also

Footnotes

    • is the Imaginary unit ().
    • In electrical engineering texts, the imaginary unit is often symbolized by to avoid confusion with the symbol for electrical current, .
    • The frequency of the wave, in Hz, is given by .
  1. This results from , which means that the complex exponential is the eigenfunction of the derivative operation.
  2. Proof

     

     

     

     

    (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
    Q.E.D.

Related Research Articles

In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha-particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specific in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

Eulers formula Expression of the complex exponential in terms of sine and cosine

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 any real number x:

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

Double-sideband suppressed-carrier transmission (DSB-SC) is transmission in which frequencies produced by amplitude modulation (AM) are symmetrically spaced above and below the carrier frequency and the carrier level is reduced to the lowest practical level, ideally being completely suppressed.

In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle θ between the direction of the incident light and the surface normal; I = I0cos(θ). The law is also known as the cosine emission law or Lambert's emission law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

Electrical impedance Opposition of a circuit to a current when a voltage is applied

In electrical engineering, electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied.

Archimedean spiral Spiral named after the 3rd-century BC Greek mathematician Archimedes

The Archimedean spiral is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle from that point.

Synchrotron radiation

Synchrotron radiation is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, the emission is called cyclotron emission. If the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum, which is also called continuum radiation.

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 Hilbert transform.

Linear phase is a property of a filter where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time by the same constant amount, which is referred to as the group delay. Consequently, there is no phase distortion due to the time delay of frequencies relative to one another.

Instantaneous phase and frequency

Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase of a complex-valued function s(t), is the real-valued function:

A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

Axis–angle representation

In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle θ describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector e rooted at the origin because the magnitude of e is constrained. For example, the elevation and azimuth angles of e suffice to locate it in any particular Cartesian coordinate frame.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

The direct-quadrature-zerotransformation or zero-direct-quadraturetransformation is a tensor that rotates the reference frame of a three-element vector or a three-by-three element matrix in an effort to simplify analysis. The DQZ transform is the product of the Clarke transform and the Park transform, first proposed in 1929 by Robert H. Park.

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

References

  1. Huw Fox; William Bolton (2002). Mathematics for Engineers and Technologists . Butterworth-Heinemann. p.  30. ISBN   978-0-08-051119-1.
  2. Clay Rawlins (2000). Basic AC Circuits (2nd ed.). Newnes. p.  124. ISBN   978-0-08-049398-5.
  3. Bracewell, Ron. The Fourier Transform and Its Applications. McGraw-Hill, 1965. p269
  4. K. S. Suresh Kumar (2008). Electric Circuits and Networks. Pearson Education India. p. 272. ISBN   978-81-317-1390-7.
  5. Kequian Zhang; Dejie Li (2007). Electromagnetic Theory for Microwaves and Optoelectronics (2nd ed.). Springer Science & Business Media. p. 13. ISBN   978-3-540-74296-8.
  6. 1 2 3 J. Hindmarsh (1984). Electrical Machines & their Applications (4th ed.). Elsevier. p. 58. ISBN   978-1-4832-9492-6.
  7. William J. Eccles (2011). Pragmatic Electrical Engineering: Fundamentals. Morgan & Claypool Publishers. p. 51. ISBN   978-1-60845-668-0.
  8. 1 2 Richard C. Dorf; James A. Svoboda (2010). Introduction to Electric Circuits (8th ed.). John Wiley & Sons. p.  661. ISBN   978-0-470-52157-1.
  9. Allan H. Robbins; Wilhelm Miller (2012). Circuit Analysis: Theory and Practice (5th ed.). Cengage Learning. p. 536. ISBN   1-285-40192-1.
  10. 1 2 3 Won Y. Yang; Seung C. Lee (2008). Circuit Systems with MATLAB and PSpice. John Wiley & Sons. pp. 256–261. ISBN   978-0-470-82240-1.
  11. Nilsson, James William; Riedel, Susan A. (2008). Electric circuits (8th ed.). Prentice Hall. p. 338. ISBN   0-13-198925-1., Chapter 9, page 338
  12. Singh, Ravish R (2009). "Section 4.5: Phasor Representation of Alternating Quantities". Electrical Networks. Mcgraw Hill Higher Education. p. 4.13. ISBN   0070260966.
  13. Clayton, Paul (2008). Introduction to electromagnetic compatibility. Wiley. p. 861. ISBN   978-81-265-2875-2.
  14. de 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

Further reading